Abstract
We study an optimal mass threshold for normalizability of the Gibbs measures associated with the focusing mass-critical nonlinear Schrödinger equation on the one-dimensional torus. In an influential paper, Lebowitz et al. (J Stat Phys 50(3–4):657–687, 1988) proposed a critical mass threshold given by the mass of the ground state on the real line. We provide a proof for the optimality of this critical mass threshold. The proof also applies to the two-dimensional radial problem posed on the unit disc. In this case, we answer a question posed by Bourgain and Bulut (Ann Inst H Poincaré Anal Non Linéaire 31(6):1267–1288, 2014) on the optimal mass threshold. Furthermore, in the one-dimensional case, we show that the Gibbs measure is indeed normalizable at the optimal mass threshold, thus answering an open question posed by Lebowitz et al. (1988). This normalizability at the optimal mass threshold is rather striking in view of the minimal mass blowup solution for the focusing quintic nonlinear Schrödinger equation on the one-dimensional torus.
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1 Introduction
1.1 Focusing Gibbs measures
In this paper, we continue the study of the focusing Gibbs measures for the nonlinear Schrödinger equations (NLS), initiated in the seminal papers by Lebowitz et al. [51] and Bourgain [9]. A focusing Gibbs measure \(\rho \) is a probability measure on functions / distributions with a formal density:
where Z denotes the partition function and the Hamiltonian functional H(u) is given by
The NLS equation:
generated by the Hamiltonian functional H(u), has been studied extensively as models for describing various physical phenomena ranging from Langmuir waves in plasmas to signal propagation in optical fibers [1, 40, 88]. Furthermore, the study of the Eq. (1.2) from the point of view of the (non-)equilibrium statistical mechanics has received wide attention; see for example [9,10,11,12, 15, 19, 26, 50, 51, 91, 92]. See also [7] for a survey on the subject, more from the dynamical point of view. Our main goal in this paper is to study the construction of the focusing Gibbs measures on the one-dimensional torus \({\mathbb {T}}= {\mathbb {R}}/{\mathbb {Z}}\) and the two-dimensional unit disc \({\mathbb {D}}\subset {\mathbb {R}}^2\) (under the radially symmetric assumption with the Dirichlet boundary condition) and determine optimal mass thresholds of their normalizability in the critical case. In particular, we resolve an issue in the Gibbs measure construction on \({\mathbb {T}}\) [51, Theorem 2.2] and also answer a question posed by Bourgain and Bulut [12, Remark 6.2] on the optimal mass threshold for the focusing Gibbs measure on the unit disc \({\mathbb {D}}\). Furthermore, in the case of the one-dimensional torus, we prove normalizability at the optimal mass threshold in spite of the existence of the minimal mass blowup solution to NLS at this mass, thus answering an open question posed by Lebowitz et al. [51].
We first go over the case of the one-dimensional torus. Consider the mean-zero Brownian loop u on \({\mathbb {T}}\), defined by the Fourier–Wiener series:
where \(\{g_n\}_{n \in {\mathbb {Z}}\setminus \{0\}}\) denotes a sequence of independent standard complex-valuedFootnote 1 Gaussian random variables. Then, the law \(\mu _0\) of the mean-zero Brownian loop u in (1.3) has the formal density given by
The main difficulty in constructing the focusing Gibbs measures comes from the unboundedness-from-below of the Hamiltonian H(u). This makes the problem very different from the defocusing case, which is a well studied subject in constructive Euclidean quantum field theory. In [51], Lebowitz et al. proposed to consider the Gibbs measure with an \(L^2\)-cutoff:Footnote 2
and claimed the following results.
Theorem 1.1
Given \(p > 2\) and \(K > 0\), define the partition function \(Z_{p, K}\) by
where \({\mathbf {E}}_{\mu _0}\) denotes an expectation with respect to the law \(\mu _0\) of the mean-zero Brownian loop (1.3). Then, the following statements hold:
-
(i)
(Subcritical case) If \(2< p<6\), then \(Z_{p,K}<\infty \) for any \(K>0\).
-
(ii)
(Critical case) Let \(p=6\). Then, \(Z_{6,K}<\infty \) if \(K<\Vert Q\Vert _{L^2({\mathbb {R}})}\), and \(Z_{6,K}=\infty \) if \(K>\Vert Q\Vert _{L^2({\mathbb {R}})}\). Here, Q is the (uniqueFootnote 3) optimizer for the Gagliardo–Nirenberg–Sobolev inequality on \({\mathbb {R}}\) such that \(\Vert Q\Vert _{L^6({\mathbb {R}})}^6 = 3\Vert Q'\Vert _{L^2({\mathbb {R}})}^2\).
There remains a question of normalizability at the optimal threshold \(K= \Vert Q\Vert _{L^2({\mathbb {R}})}\) in the critical case (\(p = 6\)). We address this issue in Sect. 1.2.
Lebowitz et al. proved the non-normalizability result for \(K > \Vert Q\Vert _{L^2({\mathbb {R}})}\) in Theorem 1.1 (ii) by using a Cameron–Martin-type theorem and the following sharp Gagliardo–Nirenberg–Sobolev (GNS) inequality on \({\mathbb {R}}^d\):
with \(d = 1\) and \(p = 6\). See Sect. 3 for a further discussion on the sharp GNS inequality.
The threshold value \(p=6\) and the relevance of the GNS inequality can be understood at an intuitive level by formally rewriting (1.4) as a functional integral with respect to the (periodic) Gaussian free field (= the mean-zero Brownian loop in (1.3)):
Applying the GNS inequality (1.5), this quantity is bounded by
Thus, when \(p<6\) or when \(p=6\) and K is sufficiently small, we expect the Gaussian part of the measure to dominate, and hence the partition function to be finite.
Regarding the construction of the focusing Gibbs measure, a pleasing probabilistic proof of Theorem 1.1 based on this idea was given in [51], using the explicit joint density of the times that the Brownian path hits certain levels on a grid. Unfortunately, as pointed out by Carlen et al. [19, p. 315], there is a gap in the proof of Theorem 2.2 in [51]. More precisely, the proof in [51] seems to apply only to the case, where the expectation in the definition of \(Z_{p,K}\) is taken with respect to a standard (“free”) Brownian motion started at 0, rather than the random periodic function (1.3).
Subsequently, a more analytic proof due to Bourgain appeared in [9], establishing normalizability of the focusing Gibbs measure (i.e. \(Z_{p, K} < \infty \)) for (i) \(2< p < 6\) and any \(K > 0\) and for (ii) \(p = 6\) and sufficiently small \(K > 0\). His argument combines basic estimates for Gaussian vectors with the Sobolev embedding to identify the tail behavior of the random variable \(\int _{\mathbb {T}}|u|^p\,\mathrm {d}x\), subject to the condition \(\Vert u\Vert _{L^2({\mathbb {T}})}\le K\). It also applies to the case \(p=6\), but shows only that \(Z_{6,K}<\infty \) for sufficiently small \(K>0\).
As the first main result in this paper, we obtain the optimal threshold when \(p = 6\) claimed in Theorem 1.1 (ii) by proving \(Z_{6,K}<\infty \) for any \(K < \Vert Q\Vert _{L^2({\mathbb {R}})}\). In particular, our argument resolves the issue in [51] mentioned above. Our proof is closer in spirit to Bourgain’s, since it uses the series representation (1.3) of the Brownian loop, as opposed to the path space approach taken in [51]. In Sect. 2, we go over Bourgain’s argument and point out that, in this approach, closing the gap between small K and the optimal threshold seems difficult. We then present our proof of the direct implication of Theorem 1.1 (ii) in Sect. 4.1. As in [51], the idea is to make rigorous the computation suggested by (1.6) by a finite dimensional approximation.
Remark 1.2
Theorem 1.1 also applies when we replace the mean-zero Brownian loop in (1.3) by the Ornstein–Uhlenbeck loop:
where \(\langle n \rangle = (1 + 4\pi ^2 |n|^2)^\frac{1}{2}\) and \(\{g_n\}_{n \in {\mathbb {Z}}}\) is a sequence of independent standard complex-valued Gaussian random variables. See Remark 4.1. The same comment applies to Theorem 1.4 below. The law \(\mu \) of the Ornstein–Uhlenbeck loop has the formal density
As seen in [9], \(\mu \) is a more natural base Gaussian measure to consider for the nonlinear Schrödinger equations, due to the lack of the conservation of the spatial mean under the dynamics.
We also point out that Theorem 1.1 also holds in the real-valued setting. The same comment applies to Theorems 1.3 and 1.4. For example, this is relevant to the study of the generalized KdV equation (gKdV) on \({\mathbb {T}}\):
Our method also applies to the focusing Gibbs measures on the two-dimensional unit disc \({\mathbb {D}}\subset {\mathbb {R}}^2\), under the radially symmetric assumption with the Dirichlet boundary condition. In the subcritical case (\(p < 4\)), Tzvetkov [91] constructed the focusing Gibbs measures, along with the associated invariant dynamics. His analysis was complemented in [12] by a study of the critical case \(p=4\), under a small mass assumption. See also [12, 91, 92] for results in the defocusing case.
Our approach to Theorem 1.1 allows us to establish the optimal mass threshold in the critical case (\(p = 4\)), thus answering the question posed by Bourgain and Bulut in [12, Remark 6.2]. We first introduce some notations. Let \({\mathbb {D}}=\{(x,y)\in {\mathbb {R}}^2: x^2+y^2<1\}\) be the unit disc. Let \(J_0(r)\) be the Bessel function of order zero, defined by
and \(z_n\), \(n\ge 1\), be its successive, positive zeros. Then, it is known [91] that \(\{e_n\}_{n \in {\mathbb {N}}}\) defined by
forms an orthonormal basis of \(L^2_\text {rad}({\mathbb {D}})\), consisting of the radial eigenfunctions of the Dirichlet self-adjoint realization of \(-\Delta \) on \({\mathbb {D}}\). Here, \(L^2_\text {rad}({\mathbb {D}})\) denotes the subspace of \(L^2({\mathbb {D}})\), consisting of radial functions. Now, consider the random series:
where \(\{g_n\}_{n \in {\mathbb {N}}}\) is a sequence of independent standard complex-valued Gaussian random variables.
Theorem 1.3
Given \(p > 2\) and \(K > 0\), define the partition function \(\widetilde{Z}_{p, K}\) by
where \({\mathbf {E}}\) denotes an expectation with respect to the law of the random series (1.10). Then, the following statements hold:
-
(i)
(Subcritical case) If \(p<4\), then \(\widetilde{Z}_{p,K}<\infty \) for any \(K>0\).
-
(ii)
(Critical case) Let \(p=4\). Then, \(\widetilde{Z}_{4,K}<\infty \) if \(K<\Vert Q\Vert _{L^2({\mathbb {R}}^2)}\), and \(\widetilde{Z}_{4,K}= \infty \) if \(K> \Vert Q\Vert _{L^2({\mathbb {R}}^2)}\), where Q is the optimizer for the Gagliardo–Nirenberg–Sobolev inequality (1.5) on \({\mathbb {R}}^2\) such that \(\Vert Q\Vert _{L^4({\mathbb {R}}^2)}^4 = 2\Vert \nabla Q\Vert _{L^2({\mathbb {R}}^2)}^2\).
Part (i) of Theorem 1.3 is due to Tzvetkov [91]. In [12], Bourgain and Bulut considered the critical case (\(p = 4\)) and proved \(\widetilde{Z}_{4,K}<\infty \) if \(K \ll 1\), and \(\widetilde{Z}_{4,K}= \infty \) if \(K\gg 1\), leaving a gap.
Theorem 1.3 (ii) answers the question posed by Bourgain and Bulut in [12]. See Remark 6.2 in [12]. We present the proof of Theorem 1.3 (ii) in Sects. 4.2 and 5. In Sect. 4.2, we prove \(\widetilde{Z}_{4,K}<\infty \) for the optimal range \(K<\Vert Q\Vert _{L^2({\mathbb {R}}^2)}\) by following our argument for Theorem 1.1 on \({\mathbb {T}}\). We point out that some care is needed here due to the growth of the \(L^4\)-norm of the eigenfunction \(e_n\); see (4.15). In Sect. 5, we prove \(\widetilde{Z}_{4,K} = \infty \) for \(K> \Vert Q\Vert _{L^2({\mathbb {R}}^2)}\). Our argument of the non-normalizability follows closely that on \({\mathbb {T}}\) by Lebowitz et al. [51].
1.2 Integrability at the optimal mass threshold
We now consider the normalizability issue of the focusing Gibbs measure on \({\mathbb {T}}\) in the critical case (\(p = 6\)) at the optimal threshold \(K= \Vert Q\Vert _{L^2({\mathbb {R}})}\), at which a phase transition takes place. Before doing so, let us first discuss the situation for the associated dynamical problem, namely, the focusing quintic NLS, (1.2) with \(p = 6\). On the real line, the optimizer Q for the sharp Gagliardo–Nirenberg–Sobolev inequality (1.5) is the ground state for the associated elliptic problem (see (3.3) below). Then, by applying the pseudo-conformal transform to the solitary wave solution \(Q(x)e^{2it}\), we obtain the minimal mass blowup solution to the focusing quintic NLS on \({\mathbb {R}}\). Here, the minimality refers to the fact that any solution to the focusing quintic NLS on \({\mathbb {R}}\) with \(\Vert u\Vert _{L^2({\mathbb {R}})} < \Vert Q\Vert _{L^2({\mathbb {R}})}\) exists globally in time; see [93].
In [70], Ogawa and Tsutsumi constructed an analogous minimal mass blowup solution \(u_*\) with \(\Vert u_*\Vert _{L^2({\mathbb {T}})} = \Vert Q\Vert _{L^2({\mathbb {R}})}\) to the focusing quintic NLS on the one-dimensional torus \({\mathbb {T}}\). It was also shown that, as time approaches a blowup time, the potential energy \(\frac{1}{6}\int |u_*|^6 \mathrm {d}x\) tends to \(\infty \). In view of the structure of the partition function \(Z_{6, K= \Vert Q\Vert _{L^2({\mathbb {R}})}}\) in (1.4), this divergence of the potential energy seems to create a potential obstruction to the construction of the focusing Gibbs measure in the current setting. In [11], Bourgain wrote “One remarkable point concerning the normalizability problem for Gibbs-measures of NLS in the focusing case is its close relation to blowup phenomena in the classical theory. Roughly speaking, this may be understood as follows. After normalization, the measure would be forced to live essentially on “blowup data”, which however is incompatible with the invariance properties under the flow.”
In spite of the existence of the minimal mass blowup solution, we prove that the focusing critical Gibbs measure is normalizable at the optimal threshold \(K=\Vert Q\Vert _{L^2({\mathbb {R}})}\).
Theorem 1.4
Let \(K=\Vert Q\Vert _{L^2({\mathbb {R}})}\). Then, the one-dimensional partition function \(Z_{6,K}\) in (1.4) is finite.
In view of the discussion above, Theorem 1.4 was unexpected and is rather surprising. Theorem 1.4 answers an open question posed by Lebowitz et al. in [51]. See Section 5 in [51]. Moreover, together with Theorem 1.1 (ii), Theorem 1.4 shows that the partition function \(Z_{6, K}\) is not analytic in the cutoff parameter K, thus settling another question posed in [51, Remark 5.2]. Compare this with the subcritical case (\(p< 6\)), where the analyticity result of the partition function on the parameters (including the inverse temperature, which we do not consider here) was proved by Carlen et al. [19] (for slightly different Gibbs measures).
The proof of Theorem 1.4 is presented in Sect. 6 and constitutes the major part of this paper, involving ideas and techniques from various branches of mathematics: probability theory, functional inequalities, elliptic partial differential equations (PDEs), spectral analysis, etc. We break the proof into several steps:
-
(1)
In the first step, we use a profile decomposition and establish a stability result for the GNS inequality (1.5); see Lemma 6.3. When combined with our proof of Theorem 1.1, this stability result shows that if the integration is restricted to the complement of \(U_\varepsilon =\{u \in L^2({\mathbb {T}}) : \Vert u-Q\Vert _{L^2({\mathbb {T}})}< \varepsilon \}\) for suitable \(\varepsilon \ll 1\), then the resulting partition function is finite. (In fact, we must exclude a neighborhood of the orbit of the ground state Q under translations, rescalings, and rotations, but we ignore this technicality here.) Thus, the question is reduced to the evaluation of the functional integral
$$\begin{aligned} \int _{U_\varepsilon } e^{-H(u)}\, {\mathbf {1}}_{\{\Vert u\Vert _{L^2({\mathbb {T}})}\le K \}}\mathrm {d}u \end{aligned}$$(1.12)in the neighborhood \(U_\varepsilon \) of (the orbit of) the ground state Q, where H(u) is as in (1.1) with \(p = 6\).
-
(2)
In the second step, we show that when \(\varepsilon > 0\) is sufficiently small, i.e. when \(U_\varepsilon \) lies in a sufficiently small neighborhood of the (approximate) soliton manifold \({\mathcal {M}}= \big \{ e^{i \theta } Q_{\delta , x_0}^\rho : 0< \delta < \delta ^* , \, x_0\in {\mathbb {T}}, \text { and } \theta \in {\mathbb {R}}\big \}\), where \(Q_{\delta , x_0}^\rho = (\tau _{x_0} \rho ) Q_{\delta , x_0} \) denotes the dilated and translated ground state (see (6.1.5) and (6.1.7)) and \(\rho \) is a suitable cutoff function for working on the torus \({\mathbb {T}}\cong [-\frac{1}{2},\frac{1}{2})\) (see (6.1.6)), we can endow \(U_\varepsilon \) with an orthogonal coordinate system in terms of the (small) dilation parameter \(0< \delta < \delta ^*\), the translation parameter \(x_0 \in {\mathbb {T}}\), the rotation parameter \(\theta \in {\mathbb {R}}/(2\pi {\mathbb {Z}})\), and the component \(v\in L^2({\mathbb {T}})\) orthogonal to the soliton manifold \({\mathcal {M}}\). See Propositions 6.4 and 6.9.
-
(3)
We then introduce a change-of-variable formula and reduce the integral (1.12) to an integral in \(\delta \), \(x_0\), \(\theta \), and v. See Lemma 6.10.
-
(4)
In Sect. 6.6, we reduce the problem to estimating a certain Gaussian integral with the integrand given by
$$\begin{aligned} \exp \left( -(1-\eta ^2) \langle A w, w\rangle _{H^1({\mathbb {T}})}\right) \end{aligned}$$for some small \(\eta > 0\). Here, \(A = A(\delta , \eta )\) denotes the operator on \(H^1({\mathbb {T}})\) defined in (6.7.1):
$$\begin{aligned} Aw= & {} P^{H^1}_{V'}(1-\partial _x^2)^{-1}\\&\times \bigg (\delta ^{-2}P^{H^1}_{V'}w-(1+5\eta )(\rho Q_\delta )^4\Big (2{{\,\mathrm{Re}\,}}(P^{H^1}_{V'}w) +\frac{1}{2}P^{H^1}_{V'}w\Big )\bigg ), \end{aligned}$$where \(V' \subset H^1({\mathbb {T}})\) is as in (6.6.23). See also (6.3.3). We point out that the operator A is closely related to the second variation \(\delta ^ 2 H\) of the Hamiltonian. See Lemma 6.16. In view of the compactness of the operator A, the issue is further reduced to estimating the eigenvalues of \(\frac{1}{2}{{\,\mathrm{id}\,}}+(1-\eta ^2)A\). Sect. 6.7 is devoted to the spectral analysis of the operator A.
For readers’ convenience, we present the summary of the proof of Theorem 1.4 in Sect. 6.8.
Remark 1.5
There exists an extensive literature on the study of soliton-type behavior to dispersive PDEs on \({\mathbb {R}}^d\); see, for example, [66] and the references therein. In particular, there are existing works on \({\mathbb {R}}^d\) which are closely related to Steps (2) and (4) described above. In Remarks 6.5 and 6.17, we provide brief comparison of our work on \({\mathbb {T}}\) with those on \({\mathbb {R}}^d\), pointing out similarities and differences.
We now state a dynamical consequence of Theorems 1.1 and 1.4.
Corollary 1.6
Let \(p = 6\). Consider the Gibbs measure \(\rho \) with the formal density
where \(\mu \) is the law of the Ornstein–Uhlenbeck loop in (1.7). If \(K \le \Vert Q\Vert _{L^2({\mathbb {R}})}\), then the focusing quintic NLS, (1.2) with \(p = 6\), on \({\mathbb {T}}\) is almost surely globally well-posed with respect to the Gibbs measure \(\rho \). Moreover, the Gibbs measure \(\rho \) is invariant under the NLS dynamics.
By imposing that the Ornstein–Uhlenbeck loop in (1.7) is real-valued (i.e. \(g_{-n} = \overline{g_n}\), \(n \in {\mathbb {Z}}\)) or by replacing \(\mu \) in (1.13) with the law \(\mu _0\) of the real-valued mean-zero Brownian loop in (1.3) (with \(g_{-n} = \overline{g_n}\), \(n \in {\mathbb {Z}}\setminus \{0\}\)), a similar result holds for the focusing quintic generalized KdV, (1.9) with \(p = 6\).
When \(K < \Vert Q\Vert _{L^2({\mathbb {R}})}\), Corollary 1.6 follows from the deterministic local well-posedness results for the quintic NLS [8] and the quintic gKdV [20] in the spaces containing the support of the Gibbs measure, combined with Bourgain’s invariant measure argument [9]. When \(K = \Vert Q\Vert _{L^2({\mathbb {R}})}\), the density \(e^{\frac{1}{6}\int _{\mathbb {T}}|u|^6\,\mathrm {d}x} \, {\mathbf {1}}_{\{\Vert u \Vert _{L^2({\mathbb {T}})} \le K\}}\) is only in \(L^1(\mu )\) and thus Bourgain’s invariant measure argument is not directly applicable. In this case, however, the desired claim follows from the corresponding result for \(K = \Vert Q\Vert _{L^2({\mathbb {R}})} -\varepsilon \), \(\varepsilon > 0\), and the dominated convergence theorem by taking \(\varepsilon \rightarrow 0\).
Remark 1.7
(i) A result analogous to Theorem 1.4 presumably holds for the two-dimensional radial Gibbs measure on \({\mathbb {D}}\) studied in Theorem 1.3. Moreover, we expect the analysis on \({\mathbb {D}}\) to be slightly simpler since the problem on \({\mathbb {D}}\) has fewer symmetries (in particular, no translation invariance). In order to limit the length of this paper, however, we do not pursue this issue here.
(ii) In [75], Quastel and the first author constructed the focusing Gibbs measure conditioned at a specified mass, provided that the mass is sufficiently small in the critical case (\(p = 6\)). This answered another question posed in [51]. See also [13, 19]. However, the argument in [75], based on Bourgain’s approach, is not quantitative. Thus, it would be of interest to further investigate this problem to see if the focusing Gibbs measure conditioned at a specified mass in the critical case (\(p = 6\)) can be indeed constructed up to the optimal mass threshold as in Theorem 1.4. We point out that the key ingredients for the proof of Theorem 1.4 (see the steps (1) - (4) right after the statement of Theorem 1.4) hold true even in the case of the focusing critical Gibbs measure (with the critical power \(p = 6\)) restricted to the critical \(L^2\)-norm \(\Vert u\Vert _{L^2({\mathbb {T}})} = \Vert Q\Vert _{L^2({\mathbb {R}})}^2\), and thus we expect that the focusing critical Gibbs measure at a specified mass is normalizable even at the optimal mass threshold. We, however, do not pursue this issue further in this paper.
Remark 1.8
(i) In view of the minimal mass blowup solution to NLS, the normalizability of the focusing critical Gibbs measure at the optimal threshold \(K=\Vert Q\Vert _{L^2({\mathbb {R}})}\) in Theorem 1.4 was somehow unexpected. As an afterthought, we may give some reasoning for this phenomenon, referring to certain properties of the minimal mass blowup solution on the real line (which are not known in the periodic setting).
The first result is the uniqueness / rigidity of the minimal mass blowup solution on \({\mathbb {R}}\) due to Merle [57], which states that if an \(H^1\)-solution u with \(\Vert u\Vert _{L^2({\mathbb {R}})} = \Vert Q\Vert _{L^2({\mathbb {R}})}\) to the focusing quintic NLS on \({\mathbb {R}}\) blows up in a finite time, it must be the minimal mass blowup solution up to the symmetries of the equation.Footnote 4 See also an extension [52] of this result for rougher \(H^s\)-solution, \(s > 0\), which holds only on \({\mathbb {R}}^d\) for \(d \ge 4\) under the radial assumption. While an analogous result is not known in the periodic setting (and in low dimensions), these results may indicate non-existence of rough blowup solutions at the critical threshold \(K=\Vert Q\Vert _{L^2({\mathbb {R}})}\). Theorem 1.4 shows that this non-existence claim holds true probabilistically.
Another point is instability of the minimal mass blowup solution. The minimal mass blowup solution is intrinsically unstable because a mass subcritical perturbation leads to a globally defined solution. See also [63]. Such instability may be related to the fact that we do not see (rough perturbations of) the minimal mass blowup solution probabilistically.
Lastly, we mention the situation for the focusing quintic gKdV on the real line. While finite time blowup solutions to gKdV “near” the ground state are known to exist [54, 58], it is also known that there is no minimal mass blowup solution to the focusing quintic gKdV on the real line [55]. Thus, from the gKdV point of view, the normalizability in Theorem 1.4 is perhaps naturally expected (but we point out that analogues of the results in [54, 55, 58] are not known on \({\mathbb {T}}\)).
(ii) In recent years, there has been a growing interest in studying nonlinear dispersive equations with random initial data; see, for example, [4,5,6, 10, 15, 17, 18, 21, 27, 53, 71, 74, 82]. In particular, there are recent works [14, 34, 71] on stability of finite time blowup solutions under rough and random perturbations. The so-called log-log blowup solutions to the mass-critical focusing NLS on \({\mathbb {R}}^d\) were constructed by Perelman [80] and Merle and Raphaël [59,60,61,62]. When \(d = 1, 2\), these log-log blowup solutions on \({\mathbb {R}}^d\) are known to be stable under \(H^s\)-perturbations for \(s > 0\); see [22, 83]. When \(d = 2\), Fan and Mendelson [34] proved stability of the log-log blowup solutions to the mass-critical focusing NLS under random but structured \(L^2\)-perturbations. (Some of) these results on the log-log blowup solutions (at least the deterministic ones) are expected to hold in the periodic setting due to the local-in-space nature of the blowup profile. See, for example, [81] for the construction of the log-log blowup solutions on a domain in \({\mathbb {R}}^2\). We point out that the log-log blowup solutions mentioned above have mass strictly greater than (but close to) the mass of the ground state, which is complementary to the regime we study in this paper with regard to the construction of the focusing critical Gibbs measure.
Remark 1.9
While the construction of the defocusing Gibbs measures has been extensively studied and well understood due to the strong interest in constructive Euclidean quantum field theory, the (non-)normalizability issue of the focusing Gibbs measures, going back to the work of Lebowitz et al. [51] and Brydges and Slade [16], is not fully explored. In [11], Bourgain wrote “It seems worthwhile to investigate this aspect [the (non-)normalizability issue of the focusing Gibbs measures] more as a continuation of \(\big [\)[51]\(\big ]\) and \(\big [\)[16]\(\big ]\).” See related works [12, 19, 72, 73, 78, 86, 90] on the non-normalizability (and other issues) for focusing Gibbs measures.
In a recent series of works [72, 73], the first and third authors with Okamoto employed the variational approach due to Barashkov and Gubinelli [2] to study the following two critical focusingFootnote 5 models on the three-dimensional torus \({\mathbb {T}}^3\):
-
(i)
the focusing Hartree Gibbs measure with a Hartree-type quartic interaction, formally written as
$$\begin{aligned} \mathrm {d}\rho (u) = Z^{-1} \exp \bigg (\frac{\sigma }{4} \int _{{\mathbb {T}}^3} (V*u^2)u^2 \mathrm {d}x\bigg ) \mathrm {d}\mu _3(u), \end{aligned}$$(1.14)where the coupling constant \(\sigma > 0\) corresponds to the focusing interaction and V is (the kernel of) the Bessel potential of order \(\beta >0\) given by
$$\begin{aligned} V*f = \langle \nabla \rangle ^{-\beta } f = (1-\Delta )^{-\frac{\beta }{2}} f. \end{aligned}$$Hereafter, \(\mu _3\) denotes the massive Gaussian free field on \({\mathbb {T}}^3\). When \(\beta = 2\), the focusing Hartree model (1.14) turns out to be critical.
Recall that the Bessel potential of order \(\beta \) on \({\mathbb {T}}^3\) can be written (for some \(c>0\)) as
$$\begin{aligned} V(x) = c |x|^{\beta -3} + K(x) \end{aligned}$$(1.15)for \(0<\beta <3\) and \(x \in {\mathbb {T}}^3 \setminus \{ 0 \}\), where K is a smooth function on \({\mathbb {T}}^3\). See Lemma 2.2 in [77]. Thus, when \(\beta = 2\), the potential V essentially corresponds to the Coulomb potential \(V(x) = |x|^{-1}\), which is of particular physical relevance.
-
(ii)
the \(\Phi ^3_3\)-measure, formally written as
$$\begin{aligned} \mathrm {d}\rho (u) = Z^{-1} \exp \bigg (\frac{\sigma }{3} \int _{{\mathbb {T}}^3} u^3 \mathrm {d}x\bigg ) \mathrm {d}\mu _3(u), \end{aligned}$$(1.16)where the coupling constant \(\sigma \in {\mathbb {R}}\setminus \{0\}\) measures the strength of the cubic interaction. Since \(u^3\) is not sign definite, the sign of \(\sigma \) does not play any role and, in particular, the problem is not defocusing even if \(\sigma < 0\). We point out that the \(\Phi ^3_3\)-model makes sense only in the real-valued setting.
In the three-dimensional setting, the massive Gaussian free field \(\mu _3\) is supported on \(H^s({\mathbb {T}}^3) \setminus H^{-\frac{1}{2}}({\mathbb {T}}^3)\) for any \(s < -\frac{1}{2}\). Thus, the potentials in (1.14) and (1.16) do not make sense as they are given, and proper renormalizations need to be introduced. Furthermore, due to the focusing nature of the problems, one needs to endow the measures with certain taming. In [72, 73], the first and third authors with Okamoto studied the generalized grand-canonical Gibbs measure formulations of the focusing Hartree Gibbs measure in (1.14) and the \(\Phi ^3_3\)-measure in (1.16). For example, the generalized grand-canonical Gibbs measure formulations of the \(\Phi ^3_3\)-measure in (1.16) is given by
where \(:\! u^k\!:\) denotes the standard Wick renormalization and the term \(- \infty \) denotes another (non-Wick) renormalization. See the work by Carlen et al. [19] for a discussion of the generalized grand-canonical Gibbs measure in the one-dimensional setting. See also Remark 2.1 in [51].
In [72], the first and third authors with Okamoto established a phase transition in the following two respects: (i.a) the focusing Hartree Gibbs measure in (1.14) is constructible for \(\beta > 2\), while it is not for \(\beta < 2\) and (i.b) when \(\beta = 2\), the focusing Hartree Gibbs measure is constructible in the weakly nonlinear regime \(0 < \sigma \ll 1\), while it is not in the strongly nonlinear regime \(\sigma \gg 1\). This shows that the focusing Hartree Gibbs measure is critical when \(\beta = 2\).
In terms of scaling, the \(\Phi ^3_3\)-model corresponds to the focusing Hartree model in the critical case \(\beta = 2\). Indeed, it was shown in [73] that the \(\Phi ^3_3\)-model is also critical, exhibiting the following phase transition; the \(\Phi ^3_3\)-measure is constructible in the weakly nonlinear regime \(0 < |\sigma | \ll 1\), whereas it is not in the strongly nonlinear regime \(|\sigma | \gg 1\). While the focusing Hartree Gibbs measure in (1.14) is absolutely continuous with respect the base massive Gaussian free field \(\mu _3\) even in the critical case (\(\beta = 2\)), it turned out that the \(\Phi ^3_3\)-measure in (1.16) is singular with respect to the base massive Gaussian free field \(\mu _3\). This singularity of the \(\Phi ^3_3\)-measure introduced additional difficulties (as compared to the focusing Hartree Gibbs measures studied in [72]) in both the measure (non-)construction part and the dynamical part in [73]. See [73] for a further discussion.
In view of the aforementioned results in [72, 73], it is of interest to investigate (existence of) a threshold value \(\sigma _* >0\) (depending on the models) such that the construction of the critical focusing Hartree Gibbs measure with \(\beta = 2\) (and the \(\Phi ^3_3\)-measure, respectively) holds for \(0< \sigma < \sigma _*\) (and for \(0< |\sigma | < \sigma _*\), respectively), while the non-normalizability of the critical focusing Hartree Gibbs measure with \(\beta = 2\) (and the \(\Phi ^3_3\)-measure, respectively) holds for \(\sigma > \sigma _*\) (and for \( |\sigma | > \sigma _*\), respectively). If such a threshold value \(\sigma _*\) could be determined, it would also be of interest to study normalizability at the threshold \(\sigma = \sigma _*\) in the focusing Hartree case (and \(|\sigma | = \sigma _*\) in the \(\Phi ^3_3\)-case), analogous to Theorem 1.4 in the one-dimensional case. Such a problem, however, requires optimizing all the estimates in the proofs in [72, 73] and is out of reach at this point.
Several comments are in order. As mentioned above, the \(\Phi ^3_3\)-model can be considered only in the real-valued setting. Furthermore, the critical focusing Hartree model with \(\beta = 2\) and the \(\Phi ^3_3\)-model are mass-subcritical (whereas the critical cases studied in Theorems 1.1 (ii), 1.3 (ii), and 1.4 are all mass-critical). Hence, the critical nature of these models do not seem to have anything to do with finite-time blowup solutions (in particular to NLS) unlike Theorems 1.1, 1.3, and 1.4 studied in this paper. While we mentioned the results on the generalized grand-canonical Gibbs measure formulations (namely with a taming by the Wick-ordered \(L^2\)-norm) of the focusing Hartree measures and the \(\Phi ^3_3\)-measure, analogous results hold even when we consider the (non-)construction of these measures endowed with a Wick-ordered \(L^2\)-cutoff. See, for example, Remark 5.10 in [72]. We point out that even in this latter setting (namely with a Wick-ordered \(L^2\)-cutoff), what matters is the size of the coupling constant \(\sigma \) and the size of the Wick-ordered \(L^2\)-cutoff does not play any role (unlike Theorems 1.1, 1.3, and 1.4). See also [78].
Lastly, let us mention the dynamical aspects of these models. For both of the models (1.14) and (1.16), it is possible to study the standard (parabolic) stochastic quantization [79] (namely the associate stochastic nonlinear heat equation) and the canonical stochastic quantization [87] (namely the associate stochastic damped nonlinear wave equation). In the parabolic case, well-posedness follows easily from the standard first order expansion as in [10, 24, 56]; see [72, Appendix A] and [32]. Due to a weaker smoothing property, the well-posedness issue in the hyperbolic setting becomes more challenging. By adapting the paracontrolled approach, originally introduced in the parabolic setting [37], to the wave setting [38], the first and third authors with Okamoto [72, 73] constructed global-in-time dynamics for the stochastic damped nonlinear wave equations associated with the focusing Hartree model (for \(\beta \ge 2\)) and the \(\Phi ^3_3\)-model. As for the focusing Hartree model endowed with a Wick-ordered \(L^2\)-cutoff, Bourgain [11] studied the associated Hartree NLS on \({\mathbb {T}}^3\) and constructed global-in-time dynamics when \(\beta > 2\). This result was extended to the critical case (\(\beta = 2\)) in [28, 72], where, in [28], Deng, Nahmod, and Yue proved well-posednessFootnote 6 of the associated Hartree NLS on \({\mathbb {T}}^3\) by using the random averaging operators, originally introduced in [26].
1.3 Notations
We write \( A \lesssim B \) to denote an estimate of the form \( A \le CB \) for some \(C> 0\). Similarly, we write \( A \sim B \) to denote \( A \lesssim B \) and \( B \lesssim A \) and use \( A \ll B \) when we have \(A \le c B\) for some small \(c > 0\). We may use subscripts to denote dependence on external parameters; for example, \(A\lesssim _{p} B\) means \(A\le C(p) B\), where the constant C(p) depends on a parameter p.
In the following, we deal with complex-valued functions viewed as elements in real Hilbert and Banach spaces. In particular, with \(M = {\mathbb {T}}\), \({\mathbb {D}}\), or \({\mathbb {R}}\), the inner product on \(H^s(M)\) is given by
Note that with the inner product (1.18), the family \(\{e^{2\pi in x}\}_{n \in {\mathbb {Z}}}\) does not form an orthonormal basis of \(L^2({\mathbb {T}})\). Instead, we need to use \(\{e^{2\pi in x}, i e^{2\pi in x}\}_{n \in {\mathbb {Z}}}\) as an orthonormal basis of \(L^2({\mathbb {T}})\). A similar comment applies to the case of the unit disc \({\mathbb {D}}\). We point out that the series representations such as (1.3) are not affected by whether we use the inner product (1.18) with the real part or that without the real part. For example, in (1.3), we have
Here, the right-hand side is more directly associated with the the inner product (1.18), while the left-hand side is associated with the inner product without the real part.
Given \(N \in {\mathbb {N}}\), we denote by \(\pi _N\) the Dirichlet projection (for functions on \({\mathbb {T}}\)) onto frequencies \(\{|n|\le N\}\):
and we set
We also define \(\pi _{\ne 0}\) to be the orthogonal projection onto the mean-zero part of a function:
and set \(\pi _0 = {{\,\mathrm{id}\,}}- \pi _{\ne 0}\).
Given \(k \in {\mathbb {Z}}_{\ge 0} := {\mathbb {N}}\cup \{0\}\), let \(P_k\) be the Littlewood–Paley projection onto frequencies of order \(2^k\) defined by
Similarly, set
Given measurable sets \(A_1, \dots , A_k\), we use the following notation:
where \({\mathbf {E}}\) denotes an expectation with respect to a probability distribution for u under discussion.
This paper is organized as follows. In Sect. 2, we review Bourgain’s argument from [9], which will be used in our proof of the direct implication of Theorem 1.1 (ii) in Sect. 4.1. In Sect. 3, we go over the Gagliardo–Nirenberg–Sobolev inequality (1.5) on \({\mathbb {R}}^d\) and discuss its variants on \({\mathbb {T}}\) and \({\mathbb {D}}\). In Sect. 4, we then establish the direct implications of Theorem 1.1 (ii) on the one-dimensional torus \({\mathbb {T}}\) (Sect. 4.1) and Theorem 1.3 (ii) on the unit disc \({\mathbb {D}}\) (Sect. 4.2). In Sect. 5, we prove the non-normalizability claim in Theorem 1.3 (ii). Finally, we prove normalizability of the focusing critical Gibbs measure at the optimal mass threshold (Theorem 1.4) in Sect. 6.
2 Review of Bourgain’s argument
In this section, we reproduce Bourgain’s argument in [9] for the proof of Theorem 1.1 (i). Part of the argument presented below will be used in Sect. 4.1. Let \(2 < p \le 6\) and u denote the mean-zero Brownian loop u in (1.3). Rewriting (1.4) as
we see that it suffices to show that there exist \(C>0\) and \(c>\frac{1}{p} \) such that
for all sufficiently large \(\lambda \gg 1\).
Given \(k \in {\mathbb {Z}}_{\ge 0}\), we set
where \(P_k\), \(P_{\le k}\), and \(P_{\ge k}\) are as in (1.22), (1.23), and (1.24). By subadditivity, we have for any k:
where \(\{\lambda _j\}_{j = k}^\infty \) is a sequence of positive numbers such that
Then, by using Sobolev’s inequality in the form of Bernstein’s inequality, we have
Thus, with (1.3), the probability on the right-hand side of (2.2) is bounded by
where \(C_0 =(2\pi )^{-1} C\).
The next lemma follows from a simple calculation involving moment generating functions of Gaussian random variables. See for example [76] for a proof.
Lemma 2.1
Let \(\{X_n\}_{n \in {\mathbb {N}}}\) be independent standard real-valued Gaussian random variables. Then, we have
if \(R\ge 3 M^{\frac{1}{2}}\).
By applying Lemma 2.1, we can bound the probability (2.5) by
provided
By choosing
for \(0<r<\frac{1}{p}\), both conditions (2.7) and (2.3) are satisfied for all large k (and \(j \ge k\)). For such k, the probability in (2.6) is then bounded by
Summing over \(j\ge k\) in (2.2), we find that
By applying Bernstein’s inequality again with the restriction \(\Vert u\Vert _{L^2({\mathbb {T}})}\le K\), we have
Hence, by setting
it follows from (2.9) that
Therefore, from (2.8) and (2.10), we obtain
for all sufficiently large \(\lambda \gg 1\). Note that the exponent \(\frac{4p}{p-2}\) beats the exponent p in (2.1) if (i) \(p<6\) or (ii) \(p=6\) and K is sufficiently small. Determining the optimal threshold for K would presumably require a delicate optimization of \(\lambda _j\) in (2.2), an exact Gaussian tail bound to replace the appraisal (2.6), and an optimal inequality to replace the applications of Bernstein’s inequality in (2.4) and (2.9) to determine the precise tail behavior of \(\Vert u\Vert _{L^p({\mathbb {T}})}\) given \(\Vert u\Vert _{L^2({\mathbb {T}})}\le K\). We did not attempt this calculation. Even if it is possible to carry out, such an approach would likely lead to a less transparent argument than the one we propose in Sect. 4. Moreover, our argument is easily adapted to the case of the two-dimensional unit disc \({\mathbb {D}}\).
3 Sharp Gagliardo–Nirenberg–Sobolev inequality
The optimizers for the Gagliardo–Nirenberg–Sobolev interpolation inequality with the optimal constant:
play an important role in the study of the focusing Gibbs measures. The following result on the optimal constant \( C_{{{\,\mathrm{GNS}\,}}}(d,p)\) and optimizers is due to Nagy [64] for \(d=1\) and Weinstein [93] for \(d\ge 2\). See also Appendix B in [89].
Proposition 3.1
Let \(d \ge 1\) and let (i) \(p > 2\) if \(d = 1, 2\) and (ii) \(2< p < \frac{2d}{d-2}\) if \(d \ge 3\). Consider the functional
on \(H^1({\mathbb {R}}^d)\). Then, the minimum
is attained at a function \(Q\in H^1({\mathbb {R}}^d)\) which is a positive, radial, and exponentially decaying solution to the following semilinear elliptic equation on \({\mathbb {R}}^d\):
with the minimal \(L^2\)-norm (namely, the ground state). Moreover, we have
See [36] for a pleasant exposition, including a proof of the uniqueness of positive solutions to (3.3), following [49].
Remark 3.2
Recall from Theorem 1.1 in [36] that any optimizer u for (3.1) is of the form \(u(x) = cQ(b (x - a))\) for some \(a \in {\mathbb {R}}^d\), \(b > 0\), and \(c \in {\mathbb {C}}\setminus \{0\}\). In particular, u is positive (up to a multiplicative constant).
The scale invariance of the minimization problem implies that these inequalities also hold on the finite domains \({\mathbb {T}}\) and \({\mathbb {D}}\) (essentially) with the same optimal constants.
Lemma 3.3
(i) Let \(p> 2\). Then, given any \(m>0\), there is a constant \(C=C(m) >0\) such that
for any \(u\in H^1({\mathbb {T}})\). We point out that there exists no \(C_0 > 0\) such that the Gagliardo–Nirenberg–Sobolev inequality:
holds for all functions in \(H^1({\mathbb {T}})\).
By restricting our attention to mean-zero functions belonging to
the Gagliardo–Nirenberg–Sobolev inequality (3.1) holds true on \({\mathbb {T}}\). Namely, we have
for any function \(u \in H^1_{0}({\mathbb {T}})\). In fact, (3.7) holds for any u belonging to
(ii) Let \(p> 2\). Then, we have
for any \(u\in H^1({\mathbb {D}})\) vanishing on \(\partial {\mathbb {D}}\).
Proof
(i) For the proof of (3.5), see Lemma 4.1 in [51]. As for the failure of the GNS inequality (3.6) for general \(u \in H^1({\mathbb {T}})\), we first note that (3.6) does not hold for constant functions. Moreover, given a function \(u \in H^1({\mathbb {T}})\), by simply considering \(u + C\) for large \(C\gg 1\) and observing that the left-hand side of (3.6) on \({\mathbb {T}}\) grows faster than the right-hand side as \(C \rightarrow \infty \), we see that the inequality (3.6) on \({\mathbb {T}}\) does not hold for (non-constant) functions in general, unless they have mean zero. Hence, we restrict our attention to mean-zero functions.
For mean-zero functions on \({\mathbb {T}}\), Sobolev’s inequality on \({\mathbb {T}}\) (see [3]) and an interpolation yield the GNS inequality (3.6) with some constant \(C_0 >0\) for any \(u \in H^1_0({\mathbb {T}})\). In fact, the GNS inequality (3.6) on \({\mathbb {T}}\) for mean-zero functions holds with \(C_0 = C_{{{\,\mathrm{GNS}\,}}}(1,p)\) coming from the GNS inequality (3.1) on \({\mathbb {R}}\). Suppose that \(u \in H^1_0({\mathbb {T}})\) is a real-valued mean-zero function on \({\mathbb {T}}\). Then, by the continuity of u, there exists a point \(x_0 \in {\mathbb {T}}\) such that \(u(x_0) = 0\). By setting
we can apply the GNS inequality (3.1) on \({\mathbb {R}}\) to v and conclude that the GNS inequality (3.6) on \({\mathbb {T}}\) holds for u with \(C_0 = C_{{{\,\mathrm{GNS}\,}}}(1,p)\). Now, given complex-valued \(u \in H^1_0({\mathbb {T}})\), write \(u = u_1 + i u_2\), where \(u_1 = {{\,\mathrm{Re}\,}}u\) and \(u_2 = {{\,\mathrm{Im}\,}}u\). Since u has mean zero on \({\mathbb {T}}\), its real and imaginary parts also have mean zero. In particular, there exists \(x_j \in {\mathbb {T}}\) such that \(u_j(x_j)=0\), \(j = 1, 2\). Hence, by the argument above, we see that for \(u_1\) and \(u_2\), the GNS inequality (3.6) holds with \(C_0 = C_{{{\,\mathrm{GNS}\,}}}(1,p)\). We now proceed as in Step 2 of the proof of Theorem A.1 in [36]. By Hölder’s inequality (in j), we have
Then, by the triangle inequality, the GNS inequality (3.6) with \(C_0 = C_{{{\,\mathrm{GNS}\,}}}(1,p)\) for \(u_j\), \(j = 1, 2\), and (3.10), we obtain
This proves the GNS inequality (3.7) for \(u \in H^1_0({\mathbb {T}})\). Note that the argument above shows that the GNS inequality (3.7) on \({\mathbb {T}}\) indeed holds for any u belonging to a larger class \(H^1_{00}({\mathbb {T}})\) defined in (3.8).
(ii) Given a function \(u\in H^1({\mathbb {D}})\) vanishing on \(\partial {\mathbb {D}}\), we can extend u on \({\mathbb {D}}\) to \({\bar{u}}\in H^1({\mathbb {R}}^2)\) by setting \({{\bar{u}}} \equiv 0\) on \({\mathbb {R}}^2\setminus {\mathbb {D}}\). Then, applying (3.1) on \({\mathbb {R}}^2\), we obtain the sharp GNS inequality (3.9) on \({\mathbb {D}}\) \(\square \)
We conclude this section by stating non-existence of optimizers for the Gagliardo–Nirenberg–Sobolev inequality (3.7) on \({\mathbb {T}}\) among mean-zero functions and for the Gagliardo–Nirenberg–Sobolev inequality (3.9) on \({\mathbb {D}}\) among functions in \(H^1({\mathbb {D}})\), vanishing on \(\partial {\mathbb {D}}\).
Lemma 3.4
(i) There exists no optimizer in \(H^1_0({\mathbb {T}})\) for the Gagliardo–Nirenberg–Sobolev inequality (3.7) on \({\mathbb {T}}\).
(ii) There exists no optimizer in \(H^1({\mathbb {D}})\), vanishing on \(\partial {\mathbb {D}}\), for the Gagliardo–Nirenberg–Sobolev inequality (3.9) on \({\mathbb {D}}\).
Proof
(i) Define the functional \(J_{\mathbb {T}}^{1, p}(u)\) by
as in (3.2) and consider the minimization problem over \( H^1_{00}({\mathbb {T}})\):
It follows from (3.11) that this infimum is bounded below by \( C_{{{\,\mathrm{GNS}\,}}}(1,p)^{-1}>0\). Suppose that there exists a mean-zero optimizer \(u_* \in H^1_{0}({\mathbb {T}}) \subset H^1_{00}({\mathbb {T}})\) for (3.12). Then, by adapting the argument in Step 2 of the proof of Theorem A.1 in [36] (see (3.10) and (3.11) above) to the case of the one-dimensional torus \({\mathbb {T}}\), we see that either (i) one of \({{\,\mathrm{Re}\,}}u_*\) or \({{\,\mathrm{Im}\,}}u_*\) is identically equal to 0, or (ii) both \({{\,\mathrm{Re}\,}}u_*\) and \({{\,\mathrm{Im}\,}}u_*\) are optimizers for (3.12) and \(|{{\,\mathrm{Re}\,}}u_*|= \lambda |{{\,\mathrm{Im}\,}}u_*|\) for some \(\lambda > 0\). In either case, one of \({{\,\mathrm{Re}\,}}u_*\) or \({{\,\mathrm{Im}\,}}u_*\) is an optimizer for (3.12). Without loss of generality, suppose that \({{\,\mathrm{Re}\,}}u_*\) is a (non-zero) optimizer for (3.12). Note that \({{\,\mathrm{Re}\,}}u_* \in H^1_{00}({\mathbb {T}})\) and that its positive and negative parts also belong to \(H^1_{00}({\mathbb {T}})\) (but not to \(H^1_{0}({\mathbb {T}})\)). Then, by the argument in Step 2 of the proof of Theorem A.1 in [36] once again, we conclude that \({{\,\mathrm{Re}\,}}u_*\) is either non-negative or non-positive. This, however, is a contradiction to the fact that u (and hence \({{\,\mathrm{Re}\,}}u_*\)) has mean zero on \({\mathbb {T}}\). This argument shows that there is no mean-zero optimizer for the GNS inequality (3.7) on \({\mathbb {T}}\).
(ii) Suppose that there exists an optimizer u in \(H^1({\mathbb {D}})\), vanishing on \(\partial {\mathbb {D}}\). We can extend u on \({\mathbb {D}}\) to a function in \( H^1({\mathbb {R}}^2)\) by setting \( u \equiv 0\) on \({\mathbb {R}}^2\setminus {\mathbb {D}}\), which would be a (non-zero) non-negative optimizer for (3.1) on \({\mathbb {R}}^2\) with compact support, which is a contradiction (see Theorem 1.1 in [36]). \(\square \)
4 Integrability below the threshold
4.1 On the one-dimensional torus
We first present the proof of the direct implication in Theorem 1.1 (ii). Namely, we show that the partition function \(Z_{6, K}\) in (1.4) is finite, provided that \(K < \Vert Q\Vert _{L^2({\mathbb {R}})}\). See [51, Theorem 2.2 (b)] for the converse. All the norms are taken over the one-dimensional torus \({\mathbb {T}}\), unless otherwise stated.
In the following, \({\mathbf {E}}\) denotes an expectation with respect to the mean-zero Brownian loop u in (1.3) (in particular u has mean-zero on \({\mathbb {T}}\)). Given \(\lambda >0\), we use the notation (1.25) and write
Here, we used the fact that \(P_{\ge 0} = \text {Id}\) on mean-zero functions.
For \(k \ge 1\), define \(E_k\) by
Note that the sets \(E_k\)’s are disjoint and that \(\sum _{k=1}^N {\mathbf {1}}_{E_k}\) increases to \( {\mathbf {1}}_{\{\Vert u\Vert _{L^p}>\lambda \}}\) almost surely as \(N \rightarrow \infty \) since u in (1.3) belongs almost surely to \(L^p({\mathbb {T}})\) for any finite p. Hence, by the monotone convergence theorem, we obtain
The first term on the right-hand side of (4.2) is clearly finite for any finite \(\lambda , K > 0\). Hence, in view of (4.1), it suffices to show that
is summable in \(k \in {\mathbb {N}}\).
Given an integer p, we have
Integrating and applying Hölder’s inequality followed by Young’s inequality, we have, for any u satisfying \(\Vert u_{\ge k}\Vert _{L^p}\le \lambda \),
for some small \(\varepsilon > 0\) (to be chosen later), where in the last step we used
Hence, by letting
the quantity in (4.3) is bounded by
Now, let \(\lambda = 1\) and \(p = 6\). By Lemma 3.3 (i) for some small \(m>0\) (to be chosen later) with \(\Vert u\Vert _{L^2}\le K\), there is a constant \(C(m)>0\) such that (4.6) is now bounded by
By Hölder’s inequality,
By (2.8), we have
Recall that
for \(X \sim {\mathcal {N}}_{\mathbb {R}}(0, 1)\) and \(t < \frac{1}{2}\). Then, using (4.9) and (3.4) in Proposition 3.1, the expectation in (4.7) is bounded by
Note that, under \(K < \Vert Q\Vert _{L^2({\mathbb {R}})}\) and (4.5), we can choose \(m, \varepsilon , \eta > 0\) sufficiently small such that
guaranteeing the application of (4.9) in the computation above.
Finally, summing (4.3) over \(k \in {\mathbb {N}}\) with (4.6), (4.7), (4.8), and (4.10), we have
Therefore, we conclude that the partition function \(Z_{6, K}\) is finite for any \(K<\Vert Q\Vert _{L^2({\mathbb {R}})}\). This completes the proof of Theorem 1.1.
Remark 4.1
As mentioned in Remark 1.2, Theorem 1.1 (ii) holds for the Ornstein–Uhlenbeck loop in (1.7). We first note that (2.8) also holds for the Ornstein–Uhlenbeck loop u since (2.5) holds even if we replace \(n^2\) by \((2\pi )^{-2}+ n^2\). Hence, from (1.7) and (4.8), we have
where \({\mathbf {E}}_\mu \) denotes an expectation with respect to the law of the Ornstein–Uhlenbeck loop u and \(\langle \partial _x \rangle = \sqrt{1 - \partial _x^2}\). The rest of the argument follows as above, thus establishing Theorem 1.1 (ii) for the Ornstein–Uhlenbeck loop u in (1.7).
4.2 On the two-dimensional disc
Next, we prove the normalizability of the Gibbs measure on \({\mathbb {D}}\) stated in Theorem 1.3 (ii). Namely, we show that the partition function \(\widetilde{Z}_{4, K}\) in (1.11) is finite, provided that \(K < \Vert Q\Vert _{L^2({\mathbb {R}}^2)}\). The proof is based on a computation analogous to that in Sect. 4.1. As we see below, however, we need to proceed with more care, partially due to the eigenfunction estimate, which makes the computation barely work on \({\mathbb {D}}\); compare (4.11) and (4.23).
We first recall the following simple corollary of Fernique’s theorem [35]. See also Theorem 2.7 in [25].Footnote 7
Lemma 4.2
There exists a constant \(c>0\) such that if X is a mean-zero Gaussian process with values in a separable Banach space B with \({\mathbf {E}}\big [\Vert X\Vert _{B}\big ]<\infty \), then
In particular, we have
for any \(t>1\).
Recall from [91, Lemmas 2.1 and 2.2] the asymptotic formula for the eigenvalue \(z_n\):
and the eigenfunction estimate:
As in Sect. 2, we define the spectral projection of v by
for \(k \in {\mathbb {Z}}_{\ge 0}\), where \(\widehat{v}(n) = \int _{\mathbb {D}}v e_n \mathrm {d}x\). We also define \(v_{\le k}\) and \(v_{\ge k}\) in an analogous manner.
In the following, \({\mathbf {E}}\) denotes an expectation with respect to the random Fourier series v in (1.10) and all the norms are taken over the unit disc \({\mathbb {D}}\subset {\mathbb {R}}^2\) unless otherwise stated. By Minkowski’s integral inequality with (4.14) and (4.15), we have
for any \(j \in {\mathbb {Z}}_{\ge 0}\). Then, applying Lemma 4.2 and (4.16) with suitable \(\varepsilon _j \sim \langle j \rangle ^{-2}\) such that \(\sum _{j\ge k}\varepsilon _j \le 1\), we have
for some constant \(C>0\), uniformly in \(k \in {\mathbb {Z}}_{\ge 0}\) and \(\lambda \ge 1\). Then, it follows from the Borel–Cantelli lemma that
with probability 1.
Define a set \(F_k\) by
By definition, \(F_k\)’s are disjoint and, from (4.18), we have
This implies that \(\sum _{k=1}^N {\mathbf {1}}_{F_k}\) increases to \( {\mathbf {1}}_{\{\Vert v\Vert _{L^4}>1\}}\) almost surely as \(N \rightarrow \infty \).
Starting from \(\widetilde{Z}_{4,K}\) in (1.11), we reproduce the computations in (4.2), (4.4), and (4.6) with \(p=4\) and \(\lambda = 1\) by replacing u in (1.3), the sets \(E_k\), and the integrals over [0, 1] with v in (1.10), the sets \(F_k\), and integrals over \({\mathbb {D}}\), respectively. We find
where \(\delta =\delta (4)=48\varepsilon \) is as in (4.5) with \(p = 4\). As before, it remains to show that the series in (4.19) is convergent.
From Lemma 3.3 (ii), we have
for any \(v\in H^1_0({\mathbb {D}}):=\{u\in H^{1}(\mathbb {D}):u\equiv 0 \text { on } \partial \mathbb {D}\}\) with
where Q is the optimizer for the Gagliardo–Nirenberg–Sobolev inequality (3.1) on \({\mathbb {R}}^2\). Then, by recalling
and applying Hölder’s inequality, the expectation in the summands on the right-hand side of (4.19) is bounded by
The first factor on the right-hand side of (4.20) can be computed exactly as in (4.10), using (4.9). Namely, provided that
it is finite and equals
Given \(K<\Vert Q\Vert _{L^2({\mathbb {R}}^2)}\), we choose \(\delta \) and \(\eta \) sufficiently small such that (4.21) holds. Then, from (4.20) with (4.17) and (4.22), we conclude that \(\widetilde{Z}_{4, K}\) in (4.19) is bounded by
This proves the normalizability of the Gibbs measure on \({\mathbb {D}}\) claimed in Theorem 1.3 (ii).
5 Non-integrability above the threshold on the disc
In this section, we discuss the non-normalizability of the Gibbs measure on \({\mathbb {D}}\) stated in Theorem 1.3 (ii). Namely, when \(K> \Vert Q\Vert _{L^2({\mathbb {R}}^2)}\), we prove
Here, Q is the ground state on \({\mathbb {R}}^2\) as in Proposition 3.1. Since the proof of (5.1) is essentially identical to that for the non-normalizability of the Gibbs measure on the torus \({\mathbb {T}}\) (see [51, Theorem 2.2 (b)]), we keep our presentation brief.
Let \(K_0 = \Vert Q\Vert _{L^2({\mathbb {R}}^2)}\) and fix \(K > K_0\). Choose \(\alpha > 1\) such that
Then, by setting
it follows from Proposition 3.1 (in particular (3.4)) that \(H_{{\mathbb {R}}^2}(Q) = 0\). As a result, we have
Given \(\rho > 0\), define the \(L^2\)-invariant scaling operator \(D_\lambda \) on \({\mathbb {R}}^2\) by setting
Then, we have
where we assume f to be radial for the last identity and \(\partial _r\) denotes the directional derivative in the radial direction.
For each \(\rho \gg 1\), we define \(Q_{\alpha , \rho } \in C^\infty _{\text {rad}}({\mathbb {D}})\) by setting
for \(|x| \le 1 - \frac{1}{\rho }\) and \(Q_{\alpha , \rho } \equiv 0\) on \(\partial {\mathbb {D}}\). Then, thanks to the exponential decay of the ground state Q on \({\mathbb {R}}^2\), it follows from (5.2), (5.3), and (5.4) that
for some \(A_1, A_2, A_3 > 0\) and sufficiently small \(\eta _0 > 0\), uniformly in large \(\rho \gg 1 \).
We need the following simple calculus lemma which follows from a straightforward computation. See Lemma 4.2 in [51].
Lemma 5.1
Given \(\eta > 0\), there exists small \(\varepsilon > 0\) such that for any \(v_1, v_2 \in C_0({\mathbb {D}})\) satisfying \(\Vert v_1 - v_2\Vert _{L^\infty ({\mathbb {D}})} < \varepsilon \), we have
for \(p = 2, 4\). Here, \(C_0({\mathbb {D}})\) denotes the collection of continuous complex-valued functions on \({\mathbb {D}}\), vanishing on the boundary \(\partial {\mathbb {D}}\).
Given \(\varepsilon > 0\), let \(B_\varepsilon (Q_{\alpha , \rho }) \subset C_0({\mathbb {D}})\) be the ball of radius \(\varepsilon \) centered at \(Q_{\alpha , \rho }\):
Let \( \eta \le \frac{\eta _0}{K^2 + 1-\eta _0}\). Then, from Lemma 5.1 and (5.5), there exists small \(\varepsilon > 0\) such that
where \(C_{\text {rad}, 0}({\mathbb {D}}) = \big \{ v\in C_0({\mathbb {D}}): v, \,\text {radial}\big \}\) and \({\mathbf {P}}\) denotes the Gaussian probability measure with respect to the random series (1.10).
Since \(Q_{\alpha , \rho } \in C^\infty _{\text {rad}, 0}({\mathbb {D}}) \subset H^1_{\text {rad, 0}}({\mathbb {D}}) = \big \{ v\in H^1_0({\mathbb {D}}): v,\, \text {radial}\big \}\), we can apply the Cameron–Martin theorem (Theorem 2.8 in [23]) and obtain
where we used (5.5) in the last step.
Hence, by further imposing \(\eta \le \frac{2 A_1}{A_2}\), we conclude from (5.6) and (5.7) with \( {\mathbf {P}}\big (B_\varepsilon (0)\big ) = C_\varepsilon > 0\) and (5.5) that
as \(\rho \rightarrow \infty \). This proves the non-normalizability of the Gibbs measure on \({\mathbb {D}}\subset {\mathbb {R}}^2\), when \(K> \Vert Q\Vert _{L^2({\mathbb {R}}^2)}\).
6 Integrability at the optimal mass threshold
In this section, we present the proof of Theorem 1.4. Namely, we show that \(Z_{6,K}<\infty \) when
We fix \(d = 1\), \(p = 6\), and \(C_{{{\,\mathrm{GNS}\,}}} = C_{{{\,\mathrm{GNS}\,}}}(1, 6)\). In this section, we prove the normalizability when u is the Ornstein–Uhlenbeck loop in (1.7):
where \(\langle n \rangle = (1 + 4\pi ^2 |n|^2)^\frac{1}{2}\). Namely, expectations are taken with respect to the law \(\mu \) of the Ornstein–Uhlenbeck loop in (6.0.1). In this non-homogeneous setting, the problem is not scaling invariant and some extra care is needed. See for example the proofs of Lemma 6.3 and Proposition 6.4. In Remark 6.20, we indicate the necessary modifications for handling the case of the mean-zero Brownian loop in (1.3).
6.1 Rescaled and translated ground state
In the following, we compare a function u on the circle \({\mathbb {T}}\) to translations and rescalings of the ground state Q. For this purpose, we introduce the \(L^2\)-invariant scaling operator \(D_\lambda \) on \({\mathbb {R}}\) by
Then, given \(\delta > 0\), we set
While the ground state Q is defined on \({\mathbb {R}}\), we now interpret it as a function of \(x\in [-\frac{1}{2},\frac{1}{2})\) which we identify with the torus \({\mathbb {T}}\cong [-\frac{1}{2},\frac{1}{2})\). Then, it follows from (3.3) that
for \(x\in (-\frac{1}{2},\frac{1}{2})\). As a consequence, we have
Given \(x_0 \in {\mathbb {T}}\), we also introduce the translation operator \(\tau _{x_0}\) defined by
where \(x- x_0\) is interpreted mod 1, taking values in \([-\frac{1}{2}, \frac{1}{2})\). The rescaled and translated version of the ground state:
plays an important role in our analysis. By this definition, we have \(Q_\delta =Q_{\delta ,0}\). In the following, we use \(\partial _\delta \) and \(\partial _{x_0}\) to denote differentiations with respect to the scaling and translation parameters, respectively. When there is no confusion, we also denote by \(D_\lambda \) and \(\tau _{x_0}\) the scaling and translation operators for functions on the real line.
When restricted to the torus, the rescaled and translated version of the ground state belongs to \(H^1({\mathbb {T}})\), but not to \(H^2({\mathbb {T}})\) (nor to \(H^k({\mathbb {T}})\) for any higher k). This is due to the fact that \(Q'_\delta \) on \([-\frac{1}{2}, \frac{1}{2})\), truncated at \(x = \pm \frac{1}{2}\), does not respect the periodic boundary condition. In order to overcome this problem, we introduce an even cutoff function \(\rho \in C^\infty ({\mathbb {R}}; [0, 1])\) such that
and consider \(\rho Q_\delta \), which is smooth at \(x = \pm \frac{1}{2}\). In the following, we will view
as a function on the torus. By setting
we will also view \( Q^\rho _{\delta ,x_0}\) as a function on the torus.
Note that since Q decays exponentially as \(|x| \rightarrow \infty \) on \({\mathbb {R}}\), we have
as \(\delta \rightarrow 0\).
Remark 6.1
From (6.1.5), we have
which is an even function. Similarly, \( \partial _\delta Q_\delta ^\rho = \partial _\delta (\rho Q_\delta )\) is an even function. On the other hand, we have
which is an odd function. Similarly,
is also an odd function. By writing
we see that \(\partial _\delta Q_{\delta }^\rho \) and \(\partial _{x_0} Q_{\delta }^\rho \) are orthogonal in \(H^{k}({\mathbb {T}})\), \(k \in {\mathbb {Z}}_{\ge 0}\), since \((1 - \partial _x^2) \partial _{x_0} Q_{\delta }^\rho \) is an odd function.
Similarly, for given \(x_0 \in {\mathbb {T}}\), we have
By parity considerations of these functions centered at \(x = x_0\), we also conclude that \(\partial _\delta Q_{\delta , x_0}^\rho \) and \(\partial _{x_0} Q_{\delta , x_0}^\rho \) are orthogonal on \(H^k({\mathbb {T}})\), \(k \in {\mathbb {Z}}_{\ge 0}\).
Given \( \delta > 0\) and \(x_0 \in {\mathbb {R}}\), let \(Q_{\delta , x_0 } = \tau _{x_0} D_\delta Q\) as a function on the real line. Then, a similar consideration shows orthogonality of \(\partial _\delta Q_{\delta , x_0}\) and \(\partial _{x_0} Q_{\delta , x_0}\) in \(\dot{H}^k({\mathbb {R}})\), \(k \in {\mathbb {Z}}_{\ge 0}\).
Remark 6.2
The set
is a smooth manifold of dimension 3, embedded in \(H^1({\mathbb {T}})\). The presence of the cutoff function \(\rho \) is fundamental for this to be true. Indeed, if we instead consider the set
then it turns out that \({\mathcal {M}}'\) is not a Lipschitz submanifold of \(H^1({\mathbb {T}})\) (at least with the parametrization induced by \((\delta , x_0, \theta )\)). Indeed, if \(\delta = 1, 0< x_0 \ll 1\), and \(\theta =0\), recalling that \(Q'\) is odd, we have
and thus \(\Vert Q_{1,0} - Q_{1,x_0} \Vert _{\dot{H}^1({\mathbb {T}})} > rsim |x_0|^\frac{1}{2}\), which shows that the dependence of \(Q_{\delta , x_0}\) in \(x_0\) is not Lipschitz. Similar considerations hold for \(e^{i\theta } Q_{\delta ,x_0}\) for every value of \(\theta , \delta , x_0\). In view of Lemmas 6.10 and 6.11 below, it is crucial that the set \({\mathcal {M}}\) is a \(C^2\)-manifold.
6.2 Stability of the optimizers of the GNS inequality
We begin by establishing stability of the optimizers of the Gagliardo–Nirenberg–Sobolev inequality (3.1); if u is “far” from all rescalings and translations of the ground state Q in the \(L^2\)-sense, then the GNS inequality (3.1) is “far” from being sharp in the sense of (6.2.1) below.
Given \(\gamma > 0\), define the set \(S_\gamma \) by setting
Here, \(P_{\le k} u\) denotes the Dirichlet projector onto the frequencies \(\{|n| \le 2^k \}\) defined in (1.23) and \(\pi _{\ne 0 }\) denotes the projection onto the mean-zero part defined in (1.21).
Fix \(\gamma > 0\). Young’s inequality and (6.2.1) yield
for any \(u \in S_\gamma \), where \(\pi _0 = {{\,\mathrm{id}\,}}- \pi _{\ne 0}\). Then, by repeating the argument in Sect. 4.1 with (6.2.2), we can show that
The main goal of this subsection is to prove the following “stability” result. In view of (6.2.3), this lemma allows us to restrict our attention to a small neighborhood of the orbit of the ground state Q in the subsequent subsections.
Lemma 6.3
Let \(\rho \) be as in (6.1.6). Given any \(\varepsilon > 0\) and \(\delta ^* > 0\), there exists \(\gamma (\varepsilon , \delta ^*) > 0\) such that the following holds; suppose that a function \(u \in L^2({\mathbb {T}})\) with \(\Vert u\Vert _{L^2({\mathbb {T}})} \le \Vert Q\Vert _{L^2({\mathbb {R}})}\) satisfies
for all \(0<\delta < \delta ^*\), \(x_0\in {\mathbb {T}}\), and \(\theta \in {\mathbb {R}}\), where \(Q_{\delta ,x_0}^\rho = \tau _{x_0} (\rho Q_\delta )\) is as in (6.1.7). Then, we have \(u\in S_{\gamma (\varepsilon ,\delta ^*)}\).
Recall that there exists no optimizer for the GNS inequality (3.7) on the torus \({\mathbb {T}}\); see Lemmas 3.3 and 3.4. Lemma 6.3 states that “almost optimizers” of the GNS inequality (3.7) on \({\mathbb {T}}\) exist only in a small \(L^2({\mathbb {T}})\)-neighborhood of \(e^{i \theta } Q_{\delta , x_0}^\rho \) for \(0< \delta < \delta ^*\), where \(\delta ^* > 0\) is any given small number. This is not surprising since (i) the ground state Q (up to symmetries) is the unique optimizer for the GNS inequality (3.1) on the real line \({\mathbb {R}}\) (see Remark 3.2) and (ii) we see from the definition (6.1.5) that, as \(\delta \rightarrow 0\), \(Q_{\delta }^\rho = \rho Q_\delta \) on \( {\mathbb {T}}\cong [-\tfrac{1}{2}, \tfrac{1}{2}) \) becomes a more and more accurate approximation of (a dilated copy of) the ground state Q on \({\mathbb {R}}\) (and thus becomes a more and more accurate almost optimizer for the GNS inequality (3.7) on the torus \({\mathbb {T}}\cong [-\tfrac{1}{2}, \tfrac{1}{2}) \)).
Proof
We first make preliminary computations which allow us to reduce the problem to the mean-zero case. Suppose that \(u \in H^1({\mathbb {T}})\) satisfies
but \(u \notin S_\gamma \) for some \(\gamma > 0\). Then, from the GNS inequality (3.7) on \({\mathbb {T}}\) for mean-zero functions (see Lemma 3.3) and the definition (6.2.1) of \(S_\gamma \), there exists \(k \in {\mathbb {N}}\) such that
Thus, from (6.2.5) and (6.2.6), we obtain
as \(\gamma \rightarrow 0\). Hence, if we have (6.2.4) for some \(\varepsilon > 0\), then there exists \(\gamma _0 = \gamma _0(\varepsilon ) > 0\) such that
for any \(u\not \in S_{\gamma }\) with \(0< \gamma < \gamma _0\).
We prove the lemma by contradiction. Suppose that there is no such \(\gamma (\varepsilon ,\delta ^*)\). Then, there exist \(\varepsilon > 0\), \(\delta ^* > 0\), \(\{u_n\}_{n \in {\mathbb {N}}} \subset L^2 ({\mathbb {T}})\) with
and \(\gamma _n\rightarrow 0\) such that
for any \(0< \delta <\delta ^*, x_{0} \in \mathbb {T}\), and \(\theta \in {\mathbb {R}}\) but \(u_n\notin S_{\gamma _n}\). By the discussion above, in particular from (6.2.7), there exists \(N_0(\varepsilon ) \in {\mathbb {N}}\) such that
for any \(0< \delta <\delta ^*, x_{0} \in \mathbb {T}\), and \(n \ge N_0\) but \(\pi _{\ne 0} u_n\notin S_{\gamma _n}\). Therefore, without loss of generality, we may assume that \(u_n\), satisfying (6.2.8) and (6.2.9), has mean zero (i.e. \(u_n = \pi _{\ne 0} u_n\)) and derive a contradiction.
By definition (6.2.1) of \(S_\gamma \), for each \(n \in {\mathbb {N}}\), there exists \(k_n \in {\mathbb {N}}\) such that
since \(u_n\notin S_{\gamma _n}\). Then, from (3.1) and (6.2.10) with (6.2.8) and \(\gamma _n \rightarrow 0\), we see that
as \(n \rightarrow \infty \). In view of the upper bound (6.2.8), we also have \(\Vert u_n\Vert _{L^2({\mathbb {T}})} \rightarrow \Vert Q\Vert _{L^2({\mathbb {R}})}\). Hence, by the Pythagorean theorem, we obtain
Furthermore, we claim that
Otherwise, we would have \(\Vert P_{\le k_n} u_n \Vert _{\dot{H}^1({\mathbb {T}})} \le C < \infty \) for all \(n \in {\mathbb {N}}\) and thus there exists a subsequence, still denoted by \(\{P_{\le k_n} u_n\}_{n \in {\mathbb {N}}}\), converging weakly to some u in \(\dot{H}^1({\mathbb {T}})\). Then, from the compact embeddingFootnote 8 of \(\dot{H}^1({\mathbb {T}})\) into \(L^2({\mathbb {T}})\cap L^6({\mathbb {T}})\), we see that \(P_{\le k_n} u_n\) converges strongly to u in \(L^2({\mathbb {T}})\cap L^6({\mathbb {T}})\). Hence, from (6.2.8) and then (6.2.10), we obtain
This would imply that u is a mean-zero optimizer of the Gagliardo–Nirenberg–Sobolev inequality (3.1) on the torus \({\mathbb {T}}\), which is a contradiction to Lemma 3.4. Therefore, (6.2.13) must hold.
By continuity, there exists a point \(x_n \in {\mathbb {T}}\) such that
With \(\beta _n = \Vert P_{\le k_n} u_n \Vert _{\dot{H}^1({\mathbb {T}})}^{-1}\rightarrow 0\), define \(v_n:{\mathbb {R}}\rightarrow {\mathbb {C}}\) by
and by linear interpolation for \(\frac{1}{2} < |x| \le \frac{1}{2} + \beta _n\), where the addition here is understood mod 1. Then, from (6.2.14), we have \(|v_n(\pm \frac{1}{2})| \le \Vert P_{\le k_n}u_n \Vert _{L^2({\mathbb {T}})}\). Moreover, from (6.2.15) with (6.2.8) and (6.2.13), we have \(v_n\in H^1({\mathbb {R}})\) with
for any \(n \in {\mathbb {N}}\), and
as \(n \rightarrow \infty \). Hence, from (6.2.10), (6.2.16), and (6.2.17) with (6.2.14) and (6.2.8), we have
where
Since \(\beta _n = \Vert P_{\le k_n} u_n \Vert _{\dot{H}^1({\mathbb {T}})}^{-1}\rightarrow 0\), we have \(\alpha _n \rightarrow 1\) as \(n \rightarrow \infty \). Hence, we conclude from (6.2.18) that there exists \(\widetilde{\gamma }_n \rightarrow 0 \) such that
With the scaling operator \(D_\lambda \) as in (6.1.1), let
Then, from (6.2.11), (6.2.16), and (6.2.19), we have
Since \(\{w_n\}_{n \in {\mathbb {N}}}\) is a bounded sequence in \(H^1({\mathbb {R}})\), we can invoke the profile decomposition [39, Proposition 3.1] for the (subcritical) Sobolev embedding: \(H^1({\mathbb {R}})\hookrightarrow L^6({\mathbb {R}})\). See also Theorem 4.6 in [42]. There exist \(J^* \in {\mathbb {Z}}_{\ge 0} \cup \{\infty \}\), a sequence \(\{\phi ^j \}_{j = 1}^{J^*}\) of non-trivial \(H^1({\mathbb {R}})\)-functions, and a sequence \(\{x^j_n\}_{j = 1}^{J^*}\) for each \(n \in {\mathbb {N}}\) such that up to a subsequence, still denoted by \(\{w_n \}_{n \in {\mathbb {N}}}\), we have
for each finite \(0 \le J \le J^*\), where the remainder term \(r^J_n\) satisfies
Here, \(\lim _{J \rightarrow \infty } f(J) := f(J^*)\) if \(J^* < \infty \). Moreover, for any finite \(0 \le J \le J^*\), we have
as \(n \rightarrow \infty \), and
From (6.2.21) and (6.2.27) and taking \(n \rightarrow \infty \), we obtain
where an equality holds at the first inequality if and only if \(J^* = 0 \) or 1.
From (3.1) and (6.2.23) with \(\widetilde{\gamma }_n \rightarrow 0\), we have
By (6.2.28) followed by (3.1), (6.2.29), and (6.2.26) with \(\ell ^2\subset \ell ^4\),
Here, an equality holds if and only if \(J^* = 0\) or 1. If \(J^* = 0\), then it follows from (6.2.28) that \(w_n\) tends to 0 in \(L^6({\mathbb {R}})\) as \(n \rightarrow \infty \). Then, from (6.2.23), we see that \(w_n\) tends to 0 in \(L^2({\mathbb {R}})\). This is a contradiction to (6.2.22). Hence, we must have \(J^* = 1\). In this case, (6.2.30) (with \(J^* = 1\)) holds with equalities and thus we see that \(\phi ^1\) is an optimizer for the Gagliardo–Nirenberg–Sobolev inequality on \({\mathbb {R}}\).
Hence, we conclude from Remark 3.2 that there exist \(\sigma \ne 0\), \(\lambda >0\), and \(x_0\in {\mathbb {R}}\) such that
From the last step in (6.2.30) (with \(J^* = 1\) and an equality) with (6.2.22), we have
which implies that \(\sigma =e^{i\theta }\) for some \(\theta \in {\mathbb {R}}\). From (6.2.24) and (6.2.25) with \(J = J^* = 1\), we see that \( \tau _{- x_n^1} w_n\) converges weakly to \( \phi ^1\) in \(L^2({\mathbb {R}})\) (which follows from the (weak) convergence of \( \tau _{- x_n^1} w_n\) to \( \phi ^1\) in \(L^6({\mathbb {R}})\)), while (6.2.22) implies convergence of the \(L^2\)-norms. Hence, we obtain strong convergence in \(L^2({\mathbb {R}})\):
Hence, from (6.2.20), we have
where \(y_n = \Vert v_n \Vert _{\dot{H}^1({\mathbb {R}})}^{-1}(x_0 + x_n^1)\) and \(\lambda _n = \Vert v_n \Vert _{\dot{H}^1({\mathbb {R}})}^{-1}\lambda \rightarrow 0\) in view of (6.2.17).
Recall that \(v_n\) is supported on \([-\frac{1}{2} - \beta _n, \frac{1}{2}+ \beta _n]\) with the \(L^2\)-norm on \([-\frac{1}{2}, \frac{1}{2}]^c\) bounded by \(\sqrt{ 2 \beta _n} \, \Vert Q\Vert _{L^2({\mathbb {R}})}\) (which tends to 0 as \(n \rightarrow \infty \)). Thus, it follows from (6.2.31) that
as \(n \rightarrow \infty \). In the following, we denote by \(Q_{\lambda _n, y_n}\) the dilated and translated ground state Q, viewed as a periodic function on \({\mathbb {T}}\), and by \( \tau _{y_n} D_{\lambda _n} Q\) the dilated and translated ground state viewed as a function on the real line. By possibly choosing a subsequence, we assume that \(y_n \ge 0\) without loss of generality. Write
Note that while \(Q_{\lambda _n, y_n}\) and \( \tau _{y_n} D_{\lambda _n} Q\) coincide on \(I_{1, n}\), they do not coincide on \(I_{2, n}\). Thanks to the exponential decay of the ground state Q, we have
On the other hand, on \(I_{2, n}\), we have \(Q_{\lambda _n, y_n}(x) = \lambda _n^{-\frac{1}{2}} Q(\lambda _n^{-\frac{1}{2}}(x+1 - y_n))\). Thus, from a change of variables and (6.2.32), we have
Therefore, from (6.2.31), (6.2.33) and (6.2.34), we obtain
as \(n \rightarrow \infty \). Moreover, since \(\lambda _n \rightarrow 0\), it follows from (6.1.6) that
Finally, by combining (6.2.12), (6.2.15), (6.2.35), and (6.2.36), we obtain a contradiction to (6.2.9). This completes the proof of Lemma 6.3. \(\square \)
6.3 Orthogonal coordinate system in a neighborhood of the soliton manifold
In view of Lemma 6.3 and (6.2.3), in order to prove \(Z_{6,K=\Vert Q\Vert _{L^2({\mathbb {R}})}}<\infty \), it suffices to show that
for some small \(\varepsilon , \delta ^*> 0\), where \( U_{\varepsilon }(\delta ^*)\) is defined by
with \(Q_{\delta ,x_0}^\rho = \tau _{x_0} (\rho Q_\delta )\) as in (6.1.7). Namely, the domain of integration \(U_\varepsilon (\delta ^*) \) is an \(\varepsilon \)-neighborhood of the (approximate) soliton manifold \({\mathcal {M}}= {\mathcal {M}}(\delta ^*)\):
Here, we say “approximate” due to the presence of the cutoff function \(\rho \). See Remark 6.2. Recall that the expectation in (6.3.1) is taken with respect to the Ornstein–Uhlenbeck loop in (6.0.1) whose law is given by the Gaussian measure \(\mu \) in (1.8) with the Cameron–Martin space \(H^1({\mathbb {T}})\). Our main goal in this subsection is to endow \(U_\varepsilon (\delta ^*)\) with an “orthogonal” coordinate system, where the orthogonality is measured in terms of \(H^1({\mathbb {T}})\). This then allows us to introduce a change of variables for the integration in (6.3.1); see Sect. 6.5.
Given \(0 < \delta \ll 1\), \(x_0 \in {\mathbb {T}}\), and \(0\le \theta < 2\pi \), we define \(V_{\delta , x_0, \theta } = V_{\delta , x_0, \theta }({\mathbb {T}})\) by
We point out that, due to the insufficient regularity of \(u \in L^2({\mathbb {T}})\), instead of the \(H^1\)-inner product, we measure the orthogonality in (6.3.3) with respect to the \(L^2\)-inner product with a weight \((1- \partial _x^2)\) so that the inner products in (6.3.3) are well defined for \(u \in L^2({\mathbb {T}})\). Here, we emphasize that the inner product on \(L^2({\mathbb {T}})\) defined in (1.18) is real-valued. It is easy to check that \((\tau _{x_0}\rho )\partial _\delta Q_{ \delta , x_0}\) and \(\partial _{x_0} (\tau _{x_0}\rho )Q_{ \delta , x_0}\) are orthogonal in \(L^2({\mathbb {T}})\) and \(H^1({\mathbb {T}})\) (see Remark 6.1), and similarly that they are orthogonal to \(i(\tau _{x_0}\rho )Q_{ \delta , x_0}\) in \(H^1({\mathbb {T}})\) (viewed as a real vector space with the inner product in (1.18)). Hence, the space \(V_{\delta , x_0,\theta }\) denotes a real subspace of codimension 3 in \(L^2({\mathbb {T}})\), orthogonal (with the weight \((1 - \partial _x^2)\)) to the tangent vectors \(e^{i\theta }\partial _\delta Q_{ \delta , x_0}^\rho \), \(e^{i\theta }\partial _{x_0} Q_{ \delta , x_0}^\rho = e^{i\theta }\partial _{x_0}\big ( \tau _{x_0} (\rho Q_\delta )\big )\), and \(\partial _\theta (e^{i\theta } Q_{\delta , x_0}^\rho ) = ie^{i\theta } Q_{\delta , x_0}^\rho \) of the soliton manifold \({\mathcal {M}}\) (with \(0 < \delta \ll 1\)). The following proposition shows that a small neighborhood of \({\mathcal {M}}\) can be endowed with an orthogonal coordinate system in terms of \(e^{i\theta }\partial _\delta Q_{ \delta , x_0}^\rho \), \(e^{i\theta }\partial _{x_0} Q_{ \delta , x_0}^\rho \), \(ie^{i\theta } Q_{\delta , x_0}^\rho \), and \(V_{\delta , x_0,\theta }\).
Proposition 6.4
Given small \(\varepsilon _1 > 0\), there exist \(\varepsilon = \varepsilon (\varepsilon _1) > 0\) and \( \delta ^*= \delta ^*(\varepsilon _1) > 0\) such that
Remark 6.5
The series of works by Nakanishi and Schlag [65, 67,68,69] and Krieger, Nakanishi, and Schlag [44,45,46,47,48] use a coordinate system similar to the one given by Proposition 6.4,Footnote 9 but centered around a single soliton. See, for example, Section 2.5 in [45]. Thanks to the symmetries of the problem on \({\mathbb {R}}^d\), it is easy to extend the coordinate system to a tubular neighborhood of the soliton manifold in these works. In the setting of Proposition 6.4 on the torus \({\mathbb {T}}\), however, we lack dilation symmetry, which makes it impossible to use such a soft argument to conclude the desired result (namely, endow \(U_\varepsilon (\delta ^*)\) with an orthogonal coordinate system).
Proof
We first show that the claimed result holds true in the case of the real line (without the extra factor \(\tau _{x_0} \rho \)). Given \(\gamma _0 \in {\mathbb {R}}\), \(0 < \delta \ll 1\), \(x_0 \in {\mathbb {T}}\), and \(0\le \theta < 2\pi \), we define \(V^{\gamma _0}_{\delta , x_0, \theta } = V^{\gamma _0}_{\delta , x_0, \theta }({\mathbb {R}})\) by
where the inner product on \(L^2({\mathbb {R}})\) is real-valued as defined in (1.18).
Consider the map \(H: \big ({\mathbb {R}}\times L^2({\mathbb {R}})\big ) \times \big ({\mathbb {R}}_+\times {\mathbb {R}}\times ({\mathbb {R}}/2\pi {\mathbb {Z}})\times V^0_{1,0,0}\big ) \rightarrow L^2({\mathbb {R}})\) defined by
where \(P_{V^{\gamma _0}_{\delta ,x_0,\theta }}\) is the projection onto \(V^{\gamma _0}_{\delta ,x_0,\theta }({\mathbb {R}})\) in \(L^2({\mathbb {R}})\). It is easy to see that H is Fréchet differentiable and \(H(0,Q_{1,0},1,0,0,0) = 0\). Moreover, the Fréchet derivative of H in the \(\big ({\mathbb {R}}_+ \times {\mathbb {R}}\times ({\mathbb {R}}/2\pi {\mathbb {Z}}) \times V^0_{1,0,0}\big )\)-variable at \((0,Q_{1,0},1,0,0,0)\) is
where
for \((\alpha , \beta , \gamma ) \in {\mathbb {R}}\times {\mathbb {R}}\times {\mathbb {R}}\).
The image of \(Z = \mathrm {d}(e^{i\theta }Q_{\delta ,x_0})|_{(\delta , x_0, \theta ) = (1, 0, 0)}\) has dimension 3, while the image of \({{\,\mathrm{id}\,}}_{V^{0}_{1,0,0}}\), namely the subspace \(V^{0}_{1,0,0}\), has codimension 3. Moreover, if u lies in the intersection of the image of Z and the image of \({{\,\mathrm{id}\,}}_{V^{0}_{1,0,0}}\), then the definition (6.3.4) of \(V^{0}_{1,0,0}\) and the orthogonality of \(e^{i\theta }\partial _\delta Q_{ \delta , x_0}\), \(e^{i\theta }\partial _{x_0} Q_{ \delta , x_0}\), and \(ie^{i\theta } Q_{\delta , x_0}\) in \(\dot{H}^1({\mathbb {R}})\) (see Remark 6.1) allow us to conclude that \(u = 0\). Hence, \(\mathrm {d}_2 H\) is invertible at \((0,Q_{1,0},1,0,0,0)\). By the implicit function theorem, ([31, Theorem 26.27]), there exists a neighborhood in \({\mathbb {R}}\times L^2({\mathbb {R}})\) of \((0, Q_{1,0})\) of the form:
and a \(C^1\)-function \(b = (\delta , x_0, \theta , w) : W \rightarrow {\mathbb {R}}_+\times {\mathbb {R}}\times ({\mathbb {R}}/2\pi {\mathbb {Z}})\times V^0_{1,0,0}\) such that
Namely, from (6.3.5) and (6.3.7), we have
for some \((\delta , x_0,\theta , v ) \in {\mathbb {R}}_+\times {\mathbb {R}}\times ({\mathbb {R}}/2\pi {\mathbb {Z}}) \times V_{\delta ,x_0,\theta }^{\gamma _0}\). Moreover, by continuity b with (6.3.7), if we choose \(\gamma _*= \gamma _*(\varepsilon _0)\) and \( \varepsilon =\varepsilon (\varepsilon _0)\) sufficiently small, then we can guarantee
Now, suppose that
for some \(( \delta _0, x_0, \theta _0) \in {\mathbb {R}}_+ \times {\mathbb {R}}\times ( {\mathbb {R}}/2\pi {\mathbb {Z}})\) with \(0< \delta _0 < \gamma _*\). Note that we have
where \(D_{\delta _0}\) is the scaling operator on the real line defined in (6.1.1) and \(\tau _{x_0}\) denotes the translation operator for functions on the real line. By setting
we can rewrite (6.3.10) as
since T is an isometry on \(L^2({\mathbb {R}})\). Then, it follows from (6.3.6) that \((\delta _0, T^{-1} u) \in W\). Hence, from (6.3.8), we have
for some \((\delta , x,\theta ,v) \in {\mathbb {R}}_+\times {\mathbb {R}}\times ({\mathbb {R}}/2\pi {\mathbb {Z}}) \times V^{\delta _0}_{\delta ,x,\theta }\) near (1, 0, 0, 0). Therefore, we obtain
for some \((\delta _1, x_1,\theta _1 ) \in {\mathbb {R}}_+ \times {\mathbb {R}}\times ({\mathbb {R}}/2\pi {\mathbb {Z}})\) such that
From (6.3.12) and (6.3.15) with (6.3.9), we can easily check that \(Tv \in V^1_{\delta _1, x_1,\theta _1}\) and \(\Vert T v\Vert _{L^2({\mathbb {R}})} < \varepsilon _0\) for \(v \in V^{\delta _0}_{\delta ,x,\theta }\) in (6.3.13). Hence, we conclude that u in (6.3.14) has the desired form in this real line case.
We now prove the claim in the case of the torus \({\mathbb {T}}\). In this case, a scaling argument such as (6.3.11) no longer works and we need to proceed with care. By a translation and a rotation, we may assume that \(u\in L^2({\mathbb {T}})\) satisfies
where \(Q_{\delta _0} = Q_{\delta _0,0}\). Extending u by 0 outside \([-\frac{1}{2},\frac{1}{2}]\), we obtain a function in \(L^2({\mathbb {R}})\), which we compare to the rescaled soliton on the real line. From (6.1.6) and (6.3.16), we have
and thus, for sufficiently small \(\delta _0 = \delta _0(\varepsilon ) >0\), we have
Hence, from the discussion above on the real line case, we have
for some \((\delta _1,x_1,\theta _1)\) near \((\delta _0,0,0)\) and \(v\in V^1_{\delta _1,x_1,\theta _1}({\mathbb {R}})\) with \(\Vert v\Vert _{L^2({\mathbb {R}})} < \varepsilon _0 \ll 1\). Since \(u=0\) and \(Q_{\delta _1,x_1}=O_{L^2({\mathbb {R}})}(\exp (-c\delta _1^{-1}))\) outside \([-\frac{1}{2},\frac{1}{2}]\), we have
Moreover, from (6.1.6) and (6.1.7), we have
Define the map \(F = F_{\delta _1, x_1,\theta _1} :{\mathbb {R}}_+\times {\mathbb {R}}\times ({\mathbb {R}}/2\pi {\mathbb {Z}}) \times V_{\delta _1,x_1,\theta _1}({\mathbb {T}})\rightarrow L^2({\mathbb {T}})\) by
where \(Q_{\delta ,x}^\rho = (\tau _x \rho ) Q_{\delta ,x}\) and \(P_{V_{\delta ,x,\theta }}\) is the projection onto \(V_{\delta ,x,\theta }({\mathbb {T}})\) in \(L^2({\mathbb {T}})\). Then, from (6.3.17), we claim that
where \(v_1 = P_{V_{\delta _1,x_1,\theta _1}}(v|_{{\mathbb {T}}})\). Note that we have
Let us verify (6.3.20). For \(v \in V_{\delta _1, x_1,\theta _1}^1({\mathbb {R}})\) in (6.3.17), we have
Recalling from Remark 6.1 the orthogonality of \(e^{i\theta _1}\partial _{\delta } Q_{ \delta _1, x_1}^\rho \), \(e^{i\theta _1}\partial _{x_0} Q_{\delta _1,x_1}^\rho \), and \(ie^{i\theta _1} Q_{ \delta _1, x_1}^\rho \) in \(H^2({\mathbb {T}})\), we have
Then, from the exponential decay of the ground state, (6.3.18), (6.3.19), and (6.3.22), we obtain
for \(0 < \delta _1 \ll 1\). Here, we used the polynomial bounds (in \(\delta _1^{-1}\)) on \(\Vert (1-\partial _x^2)\partial _{\delta } Q_{\delta _1, x_1}^\rho \Vert ^2_{L^2({\mathbb {T}})}\), \(\Vert (1-\partial _x^2)\partial _{x_0}Q_{\delta _1, x_1}^\rho \Vert ^2_{L^2({\mathbb {T}})}\), and \(\Vert (1-\partial _x^2)Q_{\delta _1,x_1}^\rho \Vert _{L^2}^2\). See (6.3.43), (6.3.46), and (6.3.49) below. Hence, from (6.3.17), (6.3.24), and \(F(\delta _1,x_1,\theta _1,v_1) = e^{i\theta _1}Q_{\delta _1, x_1} + v_1\), we obtain (6.3.20).
By another translation and rotation, we may assume that \(x_1 = 0\) and \(\theta _1=0\) in (6.3.20). Namely, we have
Hence, to finish the proof, we show that (6.3.25) guarantees that u lies in the image of \(F = F_{\delta _1, 0, 0}\). For this purpose, we recall the following version of the inverse function theorem; see [31, Theorem 26.29]. See also [31, Lemma 26.28]. In the following, \(\Vert \cdot \Vert \) denotes the operator norm.
Lemma 6.6
Given Banach spaces X and Y, let \(f:U\rightarrow Y\) be a \(C^1\)-map from an open subset \(U\subset X\) to Y. Suppose \(x_0\in U\) such that \(\mathrm {d}f(x_0)\) is invertible. If there exists \(R>0\) such that \(\overline{B^X(x_0,R)}\subset U\) and
then f is invertible (with a \(C^1\)-inverse) on \(B^X(x_0,R)\). Moreover, letting \(y_0=f(x_0)\in Y\), we have
for any \(r<(1-\kappa )R/\Vert \mathrm {d}f(x_0)^{-1}\Vert \). Here, \(B^Z(z_0,R)\) denotes the ball of radius R in \(Z = X\) or Y centered at \(z_0\).
Our goal is thus to estimate the quantity \(\kappa \) in (6.3.26), with f replaced by \(F = F_{\delta _1, 0,0}\), to conclude from (6.3.25) that u is in the image of F. We begin by computing \(\big (\mathrm {d}F(\delta _1,0,0,v_1)\big )^{-1}\) and its norm.
Lemma 6.7
There is a constant \(C>0\) such that
for all sufficiently small \(\delta _1 > 0\) and \(\Vert v_1\Vert _{L^2({\mathbb {T}})} \ll 1\).
We assume Lemma 6.7 for now and proceed with the proof of Proposition 6.4. The proofs of this lemma and Lemma 6.8 below will be presented at the end of this subsection.
Let \((\delta , x_0,\theta _0, w) \in {\mathbb {R}}_+\times {\mathbb {R}}\times ({\mathbb {R}}/2\pi {\mathbb {Z}}) \times V_{\delta _1, 0,0}({\mathbb {T}})\) such that \(|\delta -\delta _1|\ll 1\), \(|x_0|\ll 1\), \(|\theta _0|\ll 1\), and \(\Vert w\Vert _{L^2({\mathbb {T}})}\ll 1\). In the following, we compute
and compare it with the identity operator. Given a vector \(\mathbf{v }=(\alpha , \beta ,\gamma , v)\in {\mathbb {R}}\times {\mathbb {R}}\times {\mathbb {R}}\times V_{\delta _1,0,0}({\mathbb {T}})\), we have
where \(\big (\mathrm {d}P_{V_{\delta , x_0,\theta _0}}(\alpha , \beta ,\gamma )\big )w\) is given by
Suppose that for \(\widetilde{\mathbf{v }} = (\widetilde{\alpha },\widetilde{\beta },\widetilde{\gamma }, \widetilde{v})\), we have
Namely, we have
Then, in view of the hypothesis in Lemma 6.6, our goal is to estimate \(\mathbf{v } - \widetilde{\mathbf{v }}\).
Lemma 6.8
There exist \(0 < \delta _1, \varepsilon _0 \ll 1\) such that given any \(v_1 \in V_{\delta _1, 0, 0}({\mathbb {T}})\) with \(\Vert v_1\Vert _{L^2({\mathbb {T}})} \lesssim \varepsilon _0 \ll 1\) and given any \((\delta , x_0,\theta _0, w) \in {\mathbb {R}}_+\times {\mathbb {R}}\times ({\mathbb {R}}/2\pi {\mathbb {Z}})\times V_{\delta _1,0, 0}({\mathbb {T}})\) with \(|\delta -\delta _1| \ll \delta _1\), \(|x_0|\ll 1\), \(|\theta _0| \ll 1\), and \(\Vert w - v_1\Vert _{L^2({\mathbb {T}})}\ll 1\), we have
for any \((\alpha , \beta , \gamma , v) \in {\mathbb {R}}\times {\mathbb {R}}\times {\mathbb {R}}\times V_{\delta _1,0, 0}({\mathbb {T}})\), where \((\widetilde{\alpha }, \widetilde{\beta },\widetilde{\gamma }, \widetilde{v})\) is defined by (6.3.30) and \( A_{\delta _1, v_1, \delta , x_0, \theta _0, w}( \alpha , \beta , \gamma , v)\) is defined by
Now, given small \(\delta _1 > 0\), let us choose \(|\delta -\delta _1| + |x_0| + |\theta _0|+ \Vert w - v_1\Vert _{L^2({\mathbb {T}})} \lesssim \delta _1^3\). Then, Lemmas 6.7 and 6.8 allow us to apply Lemma 6.6 with \(R \sim \delta _1^3\) and \(\kappa \sim \delta _1\) and conclude that the image of \(F= F_{\delta _1, 0,0}\) contains a ball of radius \(r \sim \delta _1^{3}\) around \(F(\delta _1,0,0,v_1)\). Recalling (6.3.25), we see that u indeed lies in the image of F. Lastly, we need to choose \(\delta _1 = \delta _1 (\varepsilon _1)>0\) sufficiently small such that \(R = c \delta _1^3 \le \varepsilon _1\). This concludes the proof of Proposition 6.4. \(\square \)
We conclude this subsection by presenting the proofs of Lemmas 6.7 and 6.8. In the following, \(\langle \,\cdot , \cdot \, \rangle \) denotes the inner product in \(L^2({\mathbb {T}})\).
Proof of Lemma 6.7
Let \((\alpha , \beta ,\gamma , v) \in {\mathbb {R}}\times {\mathbb {R}}\times {\mathbb {R}}\times V_{\delta _1,0,0}({\mathbb {T}})\). Then, with \(F = F_{\delta _1, 0,0}\), we have
where \(\partial _{x_0} Q_{\delta _1}^\rho = \partial _{x_0} \big ((\tau _{x_0}\rho ) Q_{\delta _1, x_0})\big )|_{x_0 = 0}\) is as in (6.1.11) and the fourth term on the right-hand side is as in (6.3.28).
\(\bullet \) Case 1: We first consider the case when \(v_1 = 0\).
Then, by the orthogonality of \(\partial _{\delta }Q_{\delta _1}^\rho \), \(\partial _{x_0} Q_{\delta _1}^\rho \), and \(i Q_{\delta _1}^\rho \) in \(H^1({\mathbb {T}})\) (see Remark 6.1) and the definition (6.3.3) of \(V_{\delta _1,0, 0}\), we have
From (6.3.36), we obtain
By a direct computation with (6.1.8) and (6.1.9), we have
for \(0 < \delta _1 \ll 1\), where
By a similar computation, we have
for \(0 < \delta _1 \ll 1\). Hence, from (6.3.40), (6.3.41), and (6.3.43) with (6.3.35), we obtain
for \(0 < \delta _1 \ll 1\).
Proceeding analogously, we have
for \(0 < \delta _1 \ll 1\). Hence, from (6.3.37), (6.3.45), and (6.3.46), we obtain
for \(0 < \delta _1 \ll 1\). Similarly, we have
for \(0 < \delta _1 \ll 1\). Hence, from (6.3.38), (6.3.48), and (6.3.49), we obtain
Lastly, from (6.3.39), (6.3.44), (6.3.47), and (6.3.50) with \(\Vert \partial _\delta Q_{\delta _1}^\rho \Vert _{L^2({\mathbb {T}})}\sim \Vert \partial _{x_0} Q_{\delta _1}^\rho \Vert _{L^2({\mathbb {T}})}\sim \delta _1^{-1}\) and \(\Vert Q_{\delta _1}^\rho \Vert _{L^2({\mathbb {T}})} \sim 1\), we obtain
for \(0 < \delta _1 \ll 1\). Therefore, we conclude from (6.3.44), (6.3.47), (6.3.50), and (6.3.51) that the inverse \(\big (\mathrm {d}F(\delta _1,0,0, 0)\big )^{-1}\) of \(\mathrm {d}F\) at \((\delta _1,0,0, 0)\) has a norm bounded by a constant, uniformly in sufficiently small \( \delta _1 >0\).
\(\bullet \) Case 2: Next, we consider the case when \(v_1 \ne 0\) with \(\Vert v_1\Vert _{L^2({\mathbb {T}})} \ll 1\).
In this case, from (6.3.35), we have
where \(V_1 = V_1(\alpha , \beta , \gamma )\) is given by
From (6.3.28) and (6.3.23), we have
The derivatives appearing above are given by
for \(\kappa _1, \kappa _2 \in \{\delta , x_0,\theta _0\}\). By estimating (6.3.58) with (6.3.41), (6.3.43), (6.3.45), (6.3.46), (6.3.48), (6.3.49), and similar computations for \((1-\partial _x^2)\partial ^2_{\kappa _1, \kappa _2} (e^{i\theta _0}Q_{\delta ,x_0}^\rho )\), it is a simple task to show
uniformly in \(x_0\in {\mathbb {T}}\) and \(\theta _0 \in {\mathbb {R}}/2\pi {\mathbb {Z}}\). In particular, with (6.3.56), we have
From (6.3.52), (6.3.41), and (6.3.43), we obtain
Similarly, from (6.3.53), we have
Compare these with (6.3.44) and (6.3.47). From (6.3.61) and (6.3.62) with (6.3.60), we then obtain
provided that \(\Vert v_1\Vert _{L^2({\mathbb {T}})} \ll 1\). By a similar computation with (6.3.54), we have
Then, from (6.3.64), (6.3.60), and (6.3.63), we obtain
Hence, from (6.3.63) and (6.3.65), we obtain
Proceeding as in Case 1 with (6.3.55), (6.3.65), and (6.3.66), we obtain
Then, using (6.3.60), (6.3.65), and (6.3.66), we conclude from (6.3.67) that
This completes the proof of Lemma 6.7. \(\square \)
Next, we present the proof of Lemma 6.8.
Proof of Lemma 6.8
From (6.3.29), we have
where \(V_1 = V_1(\widetilde{\alpha }, \widetilde{\beta }, \widetilde{\gamma }) = \big ( \mathrm {d}P_{V_{\delta _1,0,0}}(\widetilde{\alpha },\widetilde{\beta },\widetilde{\gamma })\big )v_1 \) is as in (6.3.56) with \(\alpha , \beta , \gamma \) replaced by \(\widetilde{\alpha }, \widetilde{\beta }, \widetilde{\gamma }\). Then, using (6.3.27), we can write the first component \(\widetilde{\alpha }\) of \(\widetilde{{\mathbf {v}}}\) as
where \(W = W(\alpha , \beta , \gamma ) = \big (\mathrm {d}P_{V_{\delta , x_0,\theta _0}}(\alpha , \beta ,\gamma )\big )w\).
The main term of the numerator in the last expression is \( \alpha \langle e^{i\theta _0}\partial _\delta Q_{\delta ,x_0}^\rho , (1-\partial _x^2) \partial _\delta Q_{\delta _1}^\rho \rangle \). Since \(\rho \) and Q are real-valued, we see that
for \(|\theta _0|\ll 1\). Recall from (6.1.12) and (6.3.42) that
Without loss of generality, assume that \(0 < x_0 \ll 1\). By the mean value theorem, we have
for \(x \in [- \frac{3}{8}, \frac{3}{8}]\). Then, by (6.3.70), (6.3.71), and repeating a computation analogous to (6.3.41) with (6.1.7) and the mean value theorem applied to \(\tau _{x_0}\rho -\rho \), we have
Hence, from (6.3.41) and (6.3.72), we obtain
A similar computation with the orthogonality of \( \partial _\delta Q_{\delta _1}^\rho \) and \(\partial _{x_0} Q_{\delta _1}^\rho \) in \(H^1({\mathbb {T}})\) (where \(\partial _{x_0} Q_{\delta _1}^\rho \) is as in (6.1.11)) gives
and together with (6.3.41), we obtain
By Cauchy–Schwarz inequality with (6.3.41), (6.3.43), and \(\Vert Q_{\delta ,x_0}^\rho \Vert _{L^2({\mathbb {T}})}^2 \sim 1\), we have
From (6.1.12) and (6.3.42), we have
Then, by (6.3.3) and repeating a similar computation as above with the mean value theorem, we have
We now consider the last term in (6.3.69). We write \(W - V_1\) as
By Cauchy–Schwarz inequality with (6.3.59), (6.3.41), and (6.3.43), we have
Proceeding as above with (6.3.57), (6.3.58), and the mean value theorem, we have
Thus, we obtain
Lastly, we consider \(\text {I}\!\text {I}\!\text {I}\) in (6.3.77). Recalling that \(v_1 \in V_{\delta _1, 0, 0}\), it follows from (6.3.57) with the orthogonality of \( \partial _\delta Q_{\delta _1}^\rho \), \(\partial _{x_0} Q_{\delta _1}^\rho \), and \(i Q_{\delta _1}^\rho \) in \(H^2({\mathbb {T}})\) that
Then, from \(\Vert v_1\Vert _{L^2({\mathbb {T}})} \lesssim \varepsilon _0 \ll 1\), we have
where \(c_j = c_j(\varepsilon _0) = O(\varepsilon _0)\), \(j = 1, 2, 3\).
Combining (6.3.69), (6.3.73) - (6.3.78), (6.3.80), and (6.3.81), we have
By a similar computation, we also obtain
where \(c_j = c_j(\varepsilon _0) = O(\varepsilon _0)\), \(j = 4, 5, 6\).
As for the estimate (6.3.33), it follows from (6.3.68) with (6.3.27) that
Then, proceeding as before with the mean value theorem, (6.3.48), and (6.3.49), the main contribution to (6.3.84) is given by
while the contribution to \(\widetilde{\gamma }\) from the terms involving \(\alpha \) and \(\beta \) can be bounded by
Proceeding as in (6.3.76), we have
By Cauchy–Schwarz inequality with (6.3.77) and (6.3.59), we have
From (6.3.79), we have
Next, we consider \(\text {I}\!\text {I}\!\text {I}\) in (6.3.77). Recalling that \(v_1 \in V_{\delta _1, 0, 0}\), it follows from (6.3.57) with the orthogonality of \( \partial _\delta Q_{\delta _1}^\rho \), \(\partial _{x_0} Q_{\delta _1}^\rho \), and \(i Q_{\delta _1}^\rho \) in \(H^2({\mathbb {T}})\) that
Then, from \(\Vert v_1\Vert _{L^2({\mathbb {T}})} \lesssim \varepsilon _0 \ll 1\), we have
where \(c_j = c_j(\varepsilon _0) = O(\varepsilon _0)\), \(j = 7, 8, 9\).
Hence, from (6.3.84) - (6.3.90), we obtain
By solving the system of linear equations (6.3.82), (6.3.83), and (6.3.91) for \(\widetilde{\alpha }- \alpha \), \( \widetilde{\beta }- \beta \), and \(\widetilde{\gamma }- \gamma \), we obtain (6.3.31), (6.3.32), and (6.3.33).
Finally, we turn to (6.3.34). By (6.3.68) and (6.3.27), \(\widetilde{v}\) is given by
For \(v \in V_{\delta _1, 0, 0}({\mathbb {T}})\), from (6.3.3), (6.3.43), (6.3.46), (6.3.49), and computations similar to (6.3.73), we have
(Note that we could have obtained the same estimate integrating (6.3.59) over the curve \((\alpha _s,\beta _s,\gamma _s) = (s(\delta -\delta _1), sx_0, s\theta _0), 0 \le s \le 1\)).
From (6.1.7), (6.3.70), and (6.3.31), we have
Similarly, from (6.1.11) and (6.3.32), we have
From (6.3.33), we have
Finally, from (6.3.77), (6.3.59), (6.3.79), (6.3.31), (6.3.32), and (6.3.33) with \(\Vert v_1\Vert _{L^2({\mathbb {T}})} \lesssim \varepsilon _0 \ll 1\), we have
Hence, we obtain (6.3.34) from (6.3.92) together with (6.3.93) - (6.3.97). This concludes the proof of Lemma 6.8. \(\square \)
6.4 Orthogonal coordinate system in the finite-dimensional setting
Given \(N \in {\mathbb {N}}\), let \(\pi _N\) and \(E_N\) be as in (1.19) and (1.20), respectively. Given \(\varepsilon \), \(\delta _*\), \(\delta ^* > 0\), we define \(U_\varepsilon (\delta _*, \delta ^*)\) by
By modifying the proof of Proposition 6.4, we obtain the following proposition on an orthogonal coordinate system in the finite-dimensional setting.
Proposition 6.9
Let \(N \in {\mathbb {N}}\). Given small \(\varepsilon _1 > 0\), there exist \(N_0 = N_0 ( \varepsilon _1) \in {\mathbb {N}}\), \(\varepsilon = \varepsilon (\varepsilon _1) > 0\), \( \delta ^*= \delta ^*(\varepsilon _1) > 0\), and \(\delta _* = \delta _* (\varepsilon _1, N)>0\) with
(for each fixed \(\varepsilon _1 >0\)) such that
Before proceeding to the proof of Proposition 6.9, let us first discuss properties of truncated solitons. Note that the frequency truncation operator \(\pi _N\) is parity-preserving; \(\pi _N Q_\delta ^\rho = \pi _N(\rho Q_\delta )\) is an even function for any \(\delta > 0\). It also follows from (6.1.9) and (6.1.10) that \(\pi _N \partial _\delta Q_\delta ^\rho \) is an even function, while \(\pi _N \partial _{x_0} Q_\delta ^\rho \) is an odd function. Hence, they are orthogonal in \(H^k({\mathbb {T}})\), \(k \in {\mathbb {Z}}_{\ge 0}\). Moreover, the operator \(\pi _N\) also commutes with the pointwise conjugation, so \(\pi _N Q_\delta ^\rho \), \(\pi _N \partial _\delta Q_\delta ^\rho \), and \(\pi _N \partial _{x_0} Q_\delta ^\rho \) are all real functions. Therefore, \(\pi _N iQ_\delta ^\rho \) is orthogonalFootnote 10 to both \(\pi _N \partial _\delta Q_\delta ^\rho \) and \(\pi _N \partial _{x_0} Q_\delta ^\rho \) in \(H^k({\mathbb {T}})\), \(k \in {\mathbb {Z}}_{\ge 0}\). By a similar consideration centred at \(x = x_0\), we also conclude that \(\pi _N e^{i\theta } \partial _\delta Q_{\delta , x_0}^\rho \), \(\pi _N e^{i\theta } \partial _{x_0} Q_{\delta , x_0}^\rho \), and \(\pi _N i e^{i\theta } Q_{\delta ,x_0}^\rho \) are pairwise orthogonal in \(H^k({\mathbb {T}})\), \(k \in {\mathbb {Z}}_{\ge 0}\).
In the following, we use \({\mathcal {F}}_{\mathbb {R}}\) to denote the Fourier transform of a function on the real line. By (6.1.6), a change of variables, and the exponential decay of Q, we have
for \(0 < \delta \ll 1\), provided that \(\delta |n| \lesssim 1\). With \(A_1\) as in (6.3.42), an analogous computation yields
for \(0 < \delta \ll 1\), provided that \(\delta |n| \lesssim 1\).
Fix small \(\gamma > 0\) and set \(\delta _* = \delta _*(N) >0 \) such that
Then, by a Riemann sum approximation with (6.4.2), we then have
for \(\delta _* < \delta \ll 1\), where \(\pi ^{\mathbb {R}}_N\) denotes the Dirichlet projection onto frequencies \(\{|\xi |\le N\}\) for functions on the real line. Since the estimate above holds independently of the base point \(x_0 \in {\mathbb {T}}\), we have
uniformly for \(\delta _* < \delta \ll 1\) and \(x_0 \in {\mathbb {T}}\). On the other hand, by integration by parts 2K times together with the exponential decay of the ground state and (6.4.3), we have
Then, with \(\pi _N^\perp = {{\,\mathrm{id}\,}}- \pi _N\), it follows from (6.4.3) and (6.4.5) that
for any \(K \in {\mathbb {N}}\), uniformly in \(\delta _* < \delta \ll 1\) and \(x_0 \in {\mathbb {T}}\). Similarly, we have
for any \(k, K \in {\mathbb {N}}\), uniformly in \(\delta _* < \delta \ll 1\) and \(x_0 \in {\mathbb {T}}\).
Proof of Proposition 6.9
The proof of this proposition is based on a small modification of the proof of Proposition 6.4. We only go over the main steps, indicating required modifications. By a translation and a rotation, we may assume that \(u\in E_N \) satisfies
From the real line case discussed in the proof of Proposition 6.4 (see (6.3.17) above), we have
for some \((\delta _1,x_1,\theta _1)\) near \((\delta _0,0,0)\) and
provided that \(\delta _0 = \delta _0(\varepsilon _0) > 0\) is sufficiently small.
Given \(N \in {\mathbb {N}}\), define the map \(F^N = F^N_{\delta _1, x_1,\theta _1} :{\mathbb {R}}_+\times {\mathbb {R}}\times ({\mathbb {R}}/2\pi {\mathbb {Z}}) \times V_{\delta _1,x_1,\theta _1}({\mathbb {T}})\cap E_N \rightarrow L^2({\mathbb {T}})\) by
where \(P_{V_{\delta ,x,\theta }\cap E_N }\) is the projection onto \(V_{\delta ,x,\theta }\cap E_N \) in \(L^2({\mathbb {T}})\). From (6.4.8) and (6.4.10) with \(v_1 = P_{V_{\delta _1,x_1,\theta _1}\cap E_N}(v|_{{\mathbb {T}}})\), we have
From the orthogonality of \(\pi _N e^{i\theta _1}\partial _{\delta } Q_{ \delta _1, x_1}^\rho \), \(\pi _N e^{i\theta _1}\partial _{x_0} Q_{\delta _1,x_1}^\rho \), and \(\pi _N ie^{i\theta _1} Q_{ \delta _1, x_1}^\rho \) in \(H^2({\mathbb {T}})\), we have
Then, we can proceed as in the proof of Proposition 6.4 with (6.4.4), (6.4.6), (6.4.7), and (6.4.9) to estimate \(\pi _N v - v_1\). For example, the second term on the right-hand side of (6.4.12) can be written as
Thanks to (6.4.4), (6.4.9), and the exponential decay of the ground state, the first term in (6.4.13) is bounded as \(O_{L^2({\mathbb {T}})} \big (\exp (-c\delta _1^{-1})\big )\). On the other hand, from (6.4.4) and (6.4.6), we can bound the second term in (6.4.13) as \(O_{L^2({\mathbb {T}})} (\delta _1^K)\) for any \(K \in {\mathbb {N}}\). By estimating the third and fourth terms on the right-hand side of (6.4.12) in an analogous manner, we obtain
for any \(K \in {\mathbb {N}}\).
From (6.4.8), \(u \in E_N\), and (6.4.7), we have
for any \(K \in {\mathbb {N}}\). Putting (6.4.14) and (6.4.15) together, we obtain
for \(\delta _* < \delta _1 \ll 1\), where \(\delta _* = \delta _*(N)\) satisfies (6.4.3). Finally, we conclude from (6.4.11), (6.3.19), (6.4.7), and (6.4.16) that
for \(\delta _* < \delta _1 \ll 1\), where \(\delta _* = \delta _*(N)\) satisfies (6.4.3).
By our choice of \(\delta _* = \delta _*(N)\) such that (6.4.4) and (6.4.7) hold, we see that Lemmas 6.7 and 6.8 applied to \(F^N = F^N_{\delta _1, 0, 0}\) hold uniformly for \(\delta _* < \delta _1 \ll 1\), since (6.4.4) and (6.4.7) provide lower bounds on the denominators of the various terms appearing in the proofs of these lemmas. This allows us to apply the inverse function theorem (Lemma 6.6) with \(R \sim \delta _1^3\), \(\kappa \sim \delta _1\), and \(r \sim \delta _1^3\). By taking \(K > 3\) in (6.4.17), we conclude that u lies in the image of \(F^N = F^N_{\delta _1, x_1, \theta _1}\). Lastly, we need to choose \(\delta _* = \delta _* (\varepsilon _1, N)>0\) and \(\delta ^* = \delta ^* (\varepsilon _1)>0\) sufficiently small such that \( \delta _*< \delta ^* \lesssim \varepsilon _1^\frac{1}{3}\). This concludes the proof of Proposition 6.9. \(\square \)
6.5 A change-of-variable formula
Our main goal is to prove the bound (6.3.1). In the remaining part of the paper, we fix small \(\delta ^*>0\) and set
where \( U_\varepsilon (\delta ^*)\) is defined in (6.3.2) for given small \(\varepsilon > 0\). Recalling the low regularity of the Ornstein–Uhlenbeck loop in (1.7), we only work with functions \(u \in H^s({\mathbb {T}})\setminus H^\frac{1}{2} ({\mathbb {T}})\), \(s<\frac{1}{2} \), for the \(\mu \)-integration. Thus, with a slight abuse of notations, we redefine \(U_\varepsilon = U_\varepsilon (\delta ^*)\) in (6.3.2) to mean \(U_\varepsilon \setminus H^\frac{1}{2} ({\mathbb {T}})\). Namely, when we write \(U_\varepsilon = U_\varepsilon (\delta ^*)\) in the following, it is understood that we take an intersection with \(\big (H^\frac{1}{2} ({\mathbb {T}})\big )^c\).
Given small \(\delta >0\) and \( x_0 \in {\mathbb {T}}\), let \( V_{\delta ,x_0,0} = V_{\delta ,x_0,0}({\mathbb {T}})\) be as in (6.3.3) with \(\theta = 0\). In view of Proposition 6.4, the convention above: \(U_\varepsilon = U_\varepsilon \cap \big (H^\frac{1}{2} ({\mathbb {T}})\big )^c\), and the fact that \(Q_{\delta , x_0}^\rho \in H^1({\mathbb {T}})\), we redefine \( V_{\delta ,x_0,0}\) to mean \( V_{\delta ,x_0,0} \cap H^1({\mathbb {T}})\). Namely, when we write \( V_{\delta ,x_0,0}\) in the following, it is understood that we take an intersection with \(H^1({\mathbb {T}})\). Now, let \(\mu ^\perp _{\delta , x_0}\) denote the Gaussian measure with \(V_{\delta ,x_0,0} \subset H^1({\mathbb {T}})\) as its Cameron–Martin space. Then, we have the following lemma on a change of variables for the \(\mu \)-integration over \(U_\varepsilon \).
Lemma 6.10
Fix \(K > 0\) and sufficiently small \(\delta ^* > 0\). Let \(F(u)\ge 0 \) be a functional of u on \({\mathbb {T}}\) which is continuous in some topology \(H^s({\mathbb {T}})\), \(s<\frac{1}{2} \), such that \(F\le C\) for some C and \(F(u) = 0\) if \(\Vert u\Vert _{L^2({\mathbb {T}})} > K\). Then, there is a locally finite measure \(\mathrm {d}\sigma (\delta )\) on \( (0,\delta ^*)\) such that
where the domain of the integration on the right-hand side is to be interpreted as
Moreover, \(\sigma \) is absolutely continuous with respect to the Lebesgue measure \(\mathrm {d}\delta \) on \((0, \delta ^*)\), satisfying \(|\frac{\mathrm {d}\sigma }{\mathrm {d}\delta }| \lesssim \delta ^{-20}\).
The proof of Lemma 6.10 is based on a finite dimensional approximation and an application of the following lemma.
Lemma 6.11
Let \(M\subset {\mathbb {R}}^n\) be a closed submanifold of dimension d and \({\mathcal {N}}\) be its normal bundle. Then, there is a neighborhood U of M such that U is diffeomorphic to a subset \({\mathcal {M}}\) of \({\mathcal {N}}\) via the map \(\phi : {\mathcal {N}}\rightarrow {\mathbb {R}}^n\) given by
for \(x\in M\) and \(v\in T_xM^\perp \). Furthermore, the following estimate holds for any non-negative measurable function \(f: {\mathbb {R}}^n \rightarrow {\mathbb {R}}\) and any open set \(V \subset \{(x,v): |v| \le 1\} \subset {\mathcal {N}}\):
Here, the measure \(\mathrm {d}\sigma \) is defined by
where \(\mathrm {d}\omega (x)\) is the surface measure on M and \( \{t_k(x)\}_{k = 1}^d = \{(t_k^1(x), \dots , t_k^n(x))\}_{k = 1}^d \) is an orthonormal basis for the tangent space \(T_x M\) with the expression \(\Vert \nabla t_k(x)\Vert \) defined by
for any coordinate chart \(\varphi \) on a neighborhood of \(x \in M\) such that \(y = (y_1, \dots , y_d) = \varphi (x)\) and \(d (\varphi ^{-1})(y)\) is an isometry with its image. Note that the constant in (6.5.3) is independent of the dimension n of the ambient space \({\mathbb {R}}^n\).
Proof
Given \(x \in M\), let \(w_1(x), \dotsc , w_{n-d}(x)\) be an orthonormal basis of \(T_{x}M^\perp \). Consider the map
where \(\alpha = (\alpha _1, \dots , \alpha _{n-d})\). Recalling that \(\{w_j(x)\}_{j=1}^{n-d}\) is an orthonormal basis of \(T_{x}M^\perp \), it follows from the area formula (see [33, Theorem 2 on p. 99]) with (6.5.2), (6.5.4), and \(V \subset \{(x,v): |v| \le 1\}\) and then applying a change of variables that
where \(J_\psi \) is the determinant of the differential of the map \(\psi \). Hence, the bound (6.5.3) follows once we prove
for every \(\alpha \in {\mathbb {R}}^{n-d}\) with \(|\alpha | \le 1\), where \( \{t_k(x)\}_{k = 1}^d \) is an orthonormal basis for the tangent space \(T_x M\).
Recall that the tangent space of \(M \times {\mathbb {R}}^{n-d}\) at the point \((x,\alpha )\) is isomorphic to \(T_xM \times {\mathbb {R}}^{n-d}\). Then, denoting by \(\{e_j\}_{j = 1}^{n-d}\) the standard basis of \({\mathbb {R}}^{n-d}\), it follows from (6.5.4) thatFootnote 11
Hence, by taking \((t_1,\dots ,t_d, e_1,\dotsc ,e_{n-d})\) (and \((t_1,\dots ,t_d, w_1,\dots ,w_{n-d})\), respectively) as the (orthonormal) basis of the domain \(T_xM \times {\mathbb {R}}^{n-d}\) (and the codomain \({\mathbb {R}}^n\), respectively), the matrix representation of \(\mathrm {d}\psi \) is given by
where \(B = B(\alpha ) = \{B(\alpha )_{h, k}\}_{1\le h, k \le d}\) is given by
Thus, we have
for any \(\alpha \in {\mathbb {R}}^{n-d}\) with \(|\alpha | \le 1\). Therefore, the bound (6.5.5) (and hence (6.5.3)) follows once we prove
uniformly in \(\alpha \in {\mathbb {R}}^{n-d}\) with \(|\alpha | \le 1\).
By differentiating the orthogonality relation \(\langle w_j(x), t_k(x) \rangle _{{\mathbb {R}}^n}= 0\), we obtain
for any \(\tau \in T_xM\). Thus, from (6.5.6), (6.5.8), Cauchy–Schwarz inequality with \(|\alpha | \le 1\), and the orthonormality of \(\{w_j\}_{j= 1}^{n-d}\), we have
where we used Cauchy–Schwarz inequality once again in the last step. This proves (6.5.7) and hence concludes the proof of Lemma 6.11. \(\square \)
We now present the proof of Lemma 6.10.
Proof of Lemma 6.10
Let \(E_N\) and \(U_\varepsilon (\delta _*,\delta ^*)\) as in (1.20) and (6.4.1). By Proposition 6.9, given any \(u\in U_\varepsilon (\delta _*,\delta ^*)\cap E_N\), there exist coordinates \((\delta ,x_0,\theta ) \in (0, \delta ^*) \times {\mathbb {R}}\times ({\mathbb {R}}/2\pi {\mathbb {Z}})\) and \(v\in V_{\delta ,x_0, \theta } \cap E_N\) such that
Given \((\delta , x_0,\theta )\), we let \(v_j=v_j(\delta , x_0,\theta )\), \(j = 1, \dots , 4N-1\), denote an \(H^1({\mathbb {T}})\)-orthonormal basisFootnote 12 of \( V_{\delta , x_0,\theta }\cap E_N\). Then, from (6.0.1) with \(g = \{g_n\}_{ |n|\le N}\), we have
From Lemma 6.11 with \(y = \{y_j\}_{j = 1}^{4N-1} \in {\mathbb {R}}^{4N-1} \),
where \(\mu _{\delta , x_0}^\perp \) denotes the Gaussian measure with \(V_{\delta ,x_0,0} \subset H^1({\mathbb {T}})\) as its Cameron–Martin space.Footnote 13 Here, the measure \(\sigma _N\) is given by
where \(t_k = t_k(N, \delta , x_0, \theta )\), \(k = 1, 2, 3\), are the orthonormal vectors obtained by applying the Gram-Schmidt orthonormalization procedure in \(H^1({\mathbb {T}})\) to \(\big \{\pi _N\partial _{\delta }(e^{i\theta }Q_{\delta , x_0}^\rho )\), \(\pi _N\partial _{x_0}(e^{i\theta }Q_{\delta , x_0}^\rho )\), \(\pi _N\partial _{\theta }(e^{i\theta }Q_{\delta , x_0}^\rho )\big \}\) and the surface measure \(\omega _N\) is given by
with \(\gamma _N(\delta , x_0,\theta )\) given by
Note that \(\{t_k\}_{k = 1}^3\) are chosen so that \(t_k(\delta , x_0, \theta ) = \tau _{x_0}t_k(\delta , 0, \theta )\), where \(\tau _{x_0}\) is the translation defined in (6.1.4). Together with invariance under multiplication by a unitary complex number, we obtain
for any \(x_0 \in {\mathbb {T}}\) and \(\theta \in {\mathbb {R}}/(2\pi {\mathbb {Z}})\). From (6.5.11) and (6.5.10) with (6.3.41), (6.3.45), and (6.3.48), we have
A computation analogous to (6.3.41), (6.3.45), and (6.3.48) together with (6.4.4) and (6.4.7) shows
uniformly in \(N \in {\mathbb {N}}\).
Therefore, by taking the limit \(N\rightarrow \infty \) in (6.5.9), the dominated convergence theorem yields
Finally, in view of Proposition 6.9, by taking \(\delta _*\rightarrow 0\), we obtain (6.5.1). \(\square \)
6.6 Further reductions
The forthcoming calculations aim to prove (6.3.1). We first apply the change of variables (Lemma 6.10) to the integral in (6.3.1). In the remaining part of this section, we set \(K = \Vert Q\Vert _{L^2({\mathbb {R}})}\) and use \(\langle \,\cdot , \cdot \, \rangle \) to denote the inner product in \(L^2({\mathbb {T}})\), unless otherwise specified. Note that the integrand
is neither bounded nor continuous. The lack of continuity is due to the sharp cutoff \({\mathbf {1}}_{\{\Vert u\Vert _{L^2({\mathbb {T}})}\le K\}}\). Since the set of discontinuity has \(\mu \)-measure zero, we may start with a smooth cutoff and then pass to the limit. We can also replace this integrand by a bounded one, as long as the bounds we obtain are uniform, at the cost of a simple approximation argument which we omit.
By applying Lemma 6.10 to the integral in (6.3.1) and using also the translation invariance of the surface measure (namely, the independence of all the quantities from \(x_0\)), we have
where \(\mu _{\delta }^\perp = \,\mu _{\delta , 0}^\perp \) and
A direct computation showsFootnote 14
From (6.6.2) and (6.6.3) with (6.1.3) and (6.1.7), we have
where we used \({{\,\mathrm{Re}\,}}(v^2) = 2({{\,\mathrm{Re}\,}}v)^2 - |v|^2\) to get the seventh term on the right-hand side.
By the sharp Gagliardo–Nirenberg–Sobolev inequality (Proposition 3.1), we have \( \Vert Q_\delta \Vert _{L^6({\mathbb {R}})}^6 = 3 \Vert Q'_\delta \Vert _{L^2({\mathbb {R}})}^2\) and thus we have
uniformly in \(0< \delta \le 1\), thanks to the exponential decay of Q (as in (6.1.8)). Moreover, recalling from (6.1.6) that \(\rho \equiv 1\) on \([-\frac{1}{8},\frac{1}{8}]\) and using the exponential decay of Q again with \(\Vert v\Vert _{L^2({\mathbb {T}})} \le 1\), we obtain
uniformly in \(0 \le \delta \le 1\). By Young’s inequality, the sum of the last four terms in (6.6.4) is bounded by
for any \(0<\eta \ll 1\) and a (large) constant \(C_\eta > 0\). Hence, from (6.6.1), (6.6.4), (6.6.5), (6.6.6), and (6.6.7) together with Proposition 6.4, we have
where G(v) is defined by
Here, \(\varepsilon _1 > 0\) is a small number to be chosen later (see Lemma 6.12 below), which also appears in Proposition 6.4, determining small \(\varepsilon = \varepsilon (\varepsilon _1) > 0\) and \( \delta ^*= \delta ^*(\varepsilon _1) > 0\).
By a slight modification of the argument presented in Sect. 4.1, we have the following integrability result.
Lemma 6.12
Given any \(C'_\eta >0\), there exists small \(\varepsilon _1>0\) such that
uniformly in \(0 < \delta \ll 1\).
Proof
Let W be a finite-dimensional subspace of \(H^1({\mathbb {T}})\) of dimension n with an orthonormal basis \(\{w_1,\dots ,w_n\} \subset H^2({\mathbb {T}})\) (with respect to the \(H^1({\mathbb {T}})\)-inner product). Define the projector \(P_{W^\perp }\) by
where \(\langle \cdot , \cdot \rangle = \langle \cdot , \cdot \rangle _{L^2({\mathbb {T}})}\). On \(H^1({\mathbb {T}})\), this is nothing but the usual \(H^1\)-orthogonal projection onto \(W^\perp \). The definition (6.6.11) allows us to extend \(P_{W^\perp }\) to \(L^2({\mathbb {T}})\). Then, by the definition of \(\mu ^\perp _\delta = \mu _{\delta , 0}^\perp \), it suffices to show that given \(C'_\eta > 0\), there exist \(M \gg 1\) and \(\varepsilon _1>0\), depending only on \(n = \dim W\), such that
Indeed, in view of (6.3.3) and the definition of \(\mu ^\perp _\delta = \mu _{\delta , 0}^\perp \) with the Cameron–Martin space \(V_{\delta , 0, 0}\), by simply setting \(w_1 = \frac{ \partial _\delta Q_\delta ^\rho }{\Vert \partial _\delta Q_\delta ^\rho \Vert _{H^1({\mathbb {T}})}}\), \(w_2 = \frac{\partial _{x_0} Q_\delta ^\rho }{\Vert \partial _{x_0} Q_\delta ^\rho \Vert _{H^1({\mathbb {T}})}}\), and \(w_3 = \frac{i Q_\delta ^\rho }{\Vert Q_\delta ^\rho \Vert _{H^1({\mathbb {T}})}}\) with \(n = 3\), the desired bound (6.6.10) follows from (6.6.12).
The proof of the inequality (6.6.12) follows closely the argument presented in Sect. 4.1. In the following, we point out the modifications required to obtain (6.6.12). First of all, we replace the definition (4.1) of the set \(E_k\) by
Namely, in the definition (4.1) of \(E_k\), we replace u with \(P_{W^\perp }u\). Arguing as in (4.2) with (6.6.13), it suffices to show that
is summable in \(k \in {\mathbb {N}}\). Proceeding in the same way as in (4.4) and (4.5), we obtain the following analogue of (4.6), bounding (6.6.14):
where we used \(\delta _0\) for the constant \(\delta = \delta (6, \varepsilon )\) in (4.5) to avoid confusion with the dilation parameter \(\delta \). Note that from (3.5) in Lemma 3.3 with (6.6.11), we have
where the implicit constant depends only on \(n = \dim W\).
Using the inequality (6.6.16) instead of (3.5), and fixing \(\lambda = 1\), we obtain
for some \(C = C(n) > 0\). Then, by Hölder’s inequality, this implies the following analogue of (4.7):
We note from (6.0.1) that \(\langle u, (1-\partial _x^2) w_j \rangle \) is a mean-zero Gaussian random variable with \({\mathbf {E}}\big [|\langle u,(1-\partial _x^2) w_j \rangle |^2\big ] = \Vert w_j \Vert _{H^1({\mathbb {T}})}^2 = 1\). In particular, if \((n+2)CC'_\eta (1+\delta _0) \varepsilon _1^4 < \frac{1}{2}\), it follows from (4.9) that
Moreover, by Bernstein’s inequality with \(\Vert w_j \Vert _{H^1({\mathbb {T}})}^2 = 1\), there exists \(c = c(n) > 0\) such that
uniformly in \(j = 1, \dots , n\). Hence, from (6.6.11), (2.8), and (6.6.19), we obtain, for some \(c' = c'(n) > 0\),
which is an analogue of (4.8). In addition, if \((n+2)CC'_\eta (1+\delta _0) \varepsilon _1^4 < \frac{1}{2}\), then using (4.9), we can repeat the computations in (4.10) and obtain
Hence, by applying (6.6.21), (6.6.18), and (6.6.20) to (6.6.17) and proceeding as in (4.11), we see that (6.6.14) is summable in \(k \in {\mathbb {N}}\). Therefore, we conclude that
where the implicit constant depends only on the dimension n of W. \(\square \)
By applying Hölder’s inequality and Lemma 6.12 to (6.6.8), we obtain
Our goal in the remaining part of this section is to bound the inner integral on the right-hand side.
Next, we decompose the subspaceFootnote 15\(V_{\delta ,0,0} = V_{\delta ,0,0}({\mathbb {T}}) \subset H^1({\mathbb {T}})\) in (6.3.3) as
where \(\Vert e\Vert _{H^1({\mathbb {T}})}=1\) and e is orthogonal in \(H^1({\mathbb {T}})\) to
Denote by \(P^{H^1}_W\) the \(H^1\)-orthogonal projection onto a given subspace \(W \subset H^1({\mathbb {T}})\). Then, by noting the orthogonality of \( Q_\delta ^\rho \) with \(\partial _{x_0} Q_\delta ^\rho \) and \(i Q_\delta ^\rho \) in \(L^2({\mathbb {T}})\) and by directly computing \(\langle Q_\delta ^\rho , \partial _\delta Q_\delta ^\rho \rangle \) with (6.1.9) and integrating by parts, we have
for \(0 < \delta \ll 1 \). Corresponding to the decomposition (6.6.22) of the Cameron–Martin space for \(\mu _\delta ^\perp \), we have the following decomposition of the measure:
with
Lemma 6.13
Let G(v) be as in (6.6.9). Then, we have
uniformly in \(0 < \delta \ll 1\), where \(\widetilde{H}_\delta (w)\) is given by
Proof
By expanding \(\langle Q_\delta ^\rho +v, Q_\delta ^\rho +v\rangle \) with the decomposition (6.6.25), we have
Together with (6.1.8), we obtain
We also note from (6.6.24) that
uniformly in \(0 < \delta \ll 1\).
We also note that if \(\Vert v\Vert _{L^2({\mathbb {T}})}\le \varepsilon \le \varepsilon _1\), then we have
Indeed, by observing
we have
Then, from (6.6.28) and (6.6.30), we conclude that \(\Vert ge \Vert _{L^2({\mathbb {T}})} \sim |g\langle Q_\delta ^\rho , e \rangle |\lesssim \varepsilon \). By the triangle inequality with \(w = v - ge\), we obtain \(\Vert w\Vert _{L^2({\mathbb {T}})} \lesssim \varepsilon .\) This proves (6.6.29).
Now, from (6.6.9), the decomposition (6.6.25), Cauchy’s inequality, and \(\Vert Q_\delta ^\rho \Vert _{L^\infty ({\mathbb {T}})} \sim \delta ^{-\frac{1}{2}}\), we have
Noting that \((1+\eta ) (1 + 2\eta ) \le (1-\eta ^2) (1+ 5\eta )\) for any \(0 < \eta \ll 1\), in order to obtain (6.6.26), we only need to bound the first term on the right-hand side of (6.6.31).
\(\bullet \) Case 1: \(g \ge 0\).
In this case, from (6.6.27) and (6.6.28) (which implies \(\langle Q_\delta ^\rho , e \rangle >0\)), we have
Then, for sufficiently small \(\delta > 0\), (6.6.32) shows that the hypothesis of (6.6.29) on the size of \(\Vert v\Vert _{L^2({\mathbb {T}})}\) is satisfied with \(\varepsilon \sim \exp (-c\delta ^{-1})\). Thus, from (6.6.29), we have
provided that \(\delta = \delta (\eta , \varepsilon _1) > 0\) is sufficiently small. Hence, from (6.6.32) and (6.6.33) with (6.6.25), we obtain, for some \(C''_\eta > 0\),
Therefore, from (6.6.31) and (6.6.34), the contribution to the left-hand side of (6.6.26) in this case is bounded by
\(\bullet \) Case 2: \( g < 0\).
Under the condition \(\Vert v\Vert _{L^2}\le \varepsilon _1\), it follows from (6.6.27), Cauchy’s inequality, (6.6.28) (which implies \(\Vert ge\Vert _{L^2({\mathbb {T}})} \sim -g \langle Q_\delta ^\rho , e \rangle \)), and (6.6.29) that
By choosing \(\varepsilon _1 = \varepsilon _1(\eta )> 0\) ands \(\delta = \delta (\eta ) > 0\) sufficiently small, we then have
Hence, we obtain
Proceeding as in Case 1, we also obtain (6.6.26) in this case. \(\square \)
6.7 Spectral analysis
Given small \(\delta , \eta > 0\), define an operator \(A = A(\delta , \eta )\) on \(H^1({\mathbb {T}})\) by
where \(V'\) is as in (6.6.23). Then, the integrand of the Gaussian integral on the right-hand side of (6.6.26) can be written as
The main goal of this subsection is to establish the following integrability result.
Proposition 6.14
Given any sufficiently small \(\eta >0\), there exists a constant \(c = c(\eta )>0\) such that
uniformly in \(0 < \delta \ll 1\).
From (6.7.1) with \(\Vert Q_\delta ^\rho \Vert _{L^\infty ({\mathbb {T}})} \sim \delta ^{-\frac{1}{2}}\), we have
Thus, by Rellich’s lemma, we see that A is a compact operator on \(V' \subset H^1({\mathbb {T}})\) and thus the spectrum of A consists of eigenvalues. Recalling that \(V' \subset H^1({\mathbb {T}})\) in (6.6.23) is the Cameron–Martin space for \(\mu _\delta ^{\perp \perp }\), we see that evaluating the integral in (6.7.3) is equivalent to estimating the product of the eigenvalues of \(\frac{1}{2}{{\,\mathrm{id}\,}}+(1-\eta ^2)A\) on \(V'\). Thus, the rest of this subsection is devoted to studying the eigenvalues of A. We present the proof of Proposition 6.14 at the end of this subsection.
Since A preserves the subspace of \({{\,\mathrm{Re}\,}}H^1({\mathbb {T}})\) consisting of real-valued functions and also the subspace \({{\,\mathrm{Im}\,}}H^1({\mathbb {T}})\) consisting of purely-imaginary-valued functions, in studying the spectrum of A, we split the analysis onto these two subspaces. On the real subspace \({{\,\mathrm{Re}\,}}H^1({\mathbb {T}})\), A coincides with the operator
where
On the other hand, identifying \({{\,\mathrm{Im}\,}}H^1({\mathbb {T}})\) with \({{\,\mathrm{Re}\,}}H^1({\mathbb {T}})\) by the multiplication with i, we see that, on the imaginary subspace, A coincides with the operator
where
In the following, we proceed to analyze the eigenvalues of \(A_1\) and \(A_2\). In Proposition 6.15, we provide a lower bound for the eigenvalues of \(\frac{1}{2}{{\,\mathrm{id}\,}}+(1-\eta ^2)A\). Then, by comparing \(A_j\) with simple operators whose spectrum can be determined explicitly (Lemma 6.19), we establish asymptotic bounds on the eigenvalues in Proposition 6.18.
Proposition 6.15
There exists a constant \(\varepsilon _0>0\) such that for any \( \delta , \eta >0\) sufficiently small, the smallest eigenvalue of \(A=A(\delta , \eta )\) on \(V'\), defined in (6.6.23), is greater than \(-\frac{1}{2}+\varepsilon _0\).
Proof
Fix \(0 < \varepsilon _0\ll 1\). Suppose by contradiction that there exists a sequence \(\{ \delta _n\}_{n \in {\mathbb {N}}}\) of positive numbers tending to 0 and \(w_n^j\in V_j(\delta _n)\), \(j = 1, 2\), such that
for \(j = 1, 2\), where \(c_1 = \frac{5}{2}\) and \(c_2=\frac{1}{2}\). Let \(\widetilde{A}^j_n\) denote the Schrödinger operator given by
where \(P_{V_j(\delta _n)}\) denotes the projection onto \(V_j(\delta _n)\) in \({{\,\mathrm{Re}\,}}L^2({\mathbb {T}})\). Then, by the min-max principle (see [85, Theorem XIII.1]), the minimum eigenvalue \(\widetilde{\lambda }_n^j\) is non-positive. We denote by \(\widetilde{w}^j_n \in V_j(\delta _n)\subset L^2({\mathbb {T}})\) an \(L^2\)-normalized eigenvector associated with this minimum eigenvalue \(\widetilde{\lambda }_n^j\):
Recalling \(\Vert Q_\delta ^\rho \Vert _{L^\infty ({\mathbb {T}})} \sim \delta ^{-\frac{1}{2}}\), we have
for any \(w \in V_j(\delta _n)\), which shows that \(\widetilde{A}^j_n\) is semi-bounded on \(V_j(\delta _n)\) with a constant of order \(\delta _n^{-2}\). In particular, we have
When \(j = 1\), the condition \(\langle \widetilde{w}^1_n, Q_{\delta _n}^\rho \rangle =0\) in (6.7.5) together with the positivity of Q and the definition \(Q_{\delta _n}^\rho = \rho Q_{\delta _n}\) implies that \(\widetilde{w}^1_n(x_n)=0\) for some \(x_n\in {\mathbb {T}}\). We define a sequence \(\{v^1_n\}_{n \in {\mathbb {N}}}\) of \(L^2\)-normalized functions on \({\mathbb {R}}\) by
where the addition is understood mod 1. When \(j = 2\), it follows from the continuity of \(\widetilde{w}_n^2\) and \(\Vert \widetilde{w}_n^2\Vert _{L^2({\mathbb {T}})} = 1\) that \(|\widetilde{w}^2(x_n)| \le 1\) for some \(x_n \in {\mathbb {T}}\). Then, we define a sequence \(\{v^2_n\}_{n \in {\mathbb {N}}}\) of functions on \({\mathbb {R}}\) by
and by linear interpolation for \(\frac{1}{2}\delta _n^{-1} < |x| \le \frac{1}{2}\delta _n^{-1} + \delta _n^{\frac{1}{2} } |w(x_n)|\). In both cases, we have \(v_n^j \in H^1({\mathbb {R}})\). In the remaining part of the proof, we drop the superscript j for simplicity of notations, when there is no confusion.
Define \(\widetilde{T}_n^j\) by
where \(\tau _{x_0}\) denotes the translation by \(x_0\) as in (6.1.4). Then, from (6.7.9), (6.7.12), (6.7.13), and (6.7.14), a direct computation shows that
with \(y = \delta _n x + x_n - \frac{1}{2}\), as long as \(|x|\le \frac{1}{2}\delta _n^{-1}\).
Define \(\widetilde{V}_j\subset {{\,\mathrm{Re}\,}}H^1({\mathbb {R}})\), \(j = 1, 2 \), by
Noting that \(\langle v,w \rangle _{\dot{H}^1({\mathbb {R}})} = \langle v,-\partial _x^2 w \rangle _{L^2({\mathbb {R}})}\) and that Q is a smooth function, we can view \(\widetilde{V}_j\) as a subspace of \({{\,\mathrm{Re}\,}}L^2({\mathbb {R}})\). We now define the operator \(\widetilde{T}_j\), \(j = 1, 2\), on \({{\,\mathrm{Re}\,}}L^2({\mathbb {R}})\) by
Here, \(P_{\widetilde{V}_j}^{L^2({\mathbb {R}})}\) denotes the projection onto \(\widetilde{V}_j\) in \({{\,\mathrm{Re}\,}}L^2({\mathbb {R}})\). Then, from (6.7.10), (6.7.14), (6.7.15), and (6.7.18) along with the smoothness and exponential decay of Q and its derivatives, we have
as \(n \rightarrow \infty \), where \(\lambda _n=\delta _n^2\widetilde{\lambda }_n\le 0\). Moreover, (6.7.5), (6.7.7), (6.7.12), (6.7.13), (6.7.16), and (6.7.17) along with (6.1.6) and the exponential decay of Q and its derivatives once again, we have
Note that from (6.7.11), we have
for any \(n \in {\mathbb {N}}\). Thus, passing to a subsequence, we may assume that \(\lambda _n\rightarrow \lambda \) for some \(\lambda \le 0\) and thus from (6.7.19), we obtain
Noting that \(\widetilde{T}_j\) is semi-bounded, let \(\lambda _0\) be the minimum of the spectrum of \(\widetilde{T}_j\) on \(\widetilde{V}_j\). By the min-max principle (see [85, Theorem XIII.1]), we have
for every \(v \in H^1({\mathbb {R}})\). Therefore, from (6.7.21) and (6.7.20), we obtain that \(\lambda _0 \le \lambda \le 0\).
Since Q is a Schwartz function, the essential spectrum of the Schrödinger operator \(S_j\) in (6.7.18) is equal to \([1,\infty )\); see [41, Theorem V-5.7]. Moreover, \(\widetilde{T}_j\) in (6.7.18) differs from \(S_j\) by a finite rank perturbation and thus its essential spectrum is \([1,\infty )\); see [41, Theorem IV-5.35]. In particular, \(\lambda _0 \le 0\) does not lie in the essential spectrum of \(\widetilde{T}_j\). Namely, \(\lambda _0\) belongs to the discrete spectrum of \(\widetilde{T}_j\). Hence, there exists \(v \in H^2({\mathbb {R}})\) such that
In order to derive a contradiction, we invoke the following lemma.
Lemma 6.16
For any sufficiently small \(\varepsilon _0 > 0\) and \(\eta > 0\), the following statements hold. The operator
viewed as an operator on \(L^2({\mathbb {R}})\), is strictly positive on \(\widetilde{V}_1 \) defined in (6.7.16). Similarly, the operator
viewed as an operator on \(L^2({\mathbb {R}})\), is strictly positive on \(\widetilde{V}_2\) defined in (6.7.17).
This lemma shows that (6.7.22) can not hold for \(\lambda \le 0\). Therefore, we arrive at a contradiction to (6.7.8). This concludes the proof of Proposition 6.15 (modulo the proof of Lemma 6.16 which we present below). \(\square \)
We now present the proof of Lemma 6.16.
Proof of Lemma 6.16
In the following, we only prove the strict positivity of \(B_j := B_j(0, 0)\) on \(\widetilde{V}_j\), \(j = 1, 2\), when \(\varepsilon _0 = \eta = 0\). Namely, we show that there exists \(\theta > 0\) such that
for any \(v\in \widetilde{V}_j\). Then, by writing
with \(c_1 = \frac{5}{2}\) and \(c_2=\frac{1}{2}\), the strict positivity of \(B_j(\varepsilon , \eta )\) for sufficiently small \(\varepsilon _0, \eta > 0\) follows from (6.7.23).
Consider the Hamiltonian \(H(u) =H_{\mathbb {R}}(u)\):
By the sharp Gagliardo–Nirenberg–Sobolev inequality (Proposition 3.1), we know that \(H: H^1({\mathbb {R}}) \rightarrow {\mathbb {R}}\) has a global minimum at Q, when restricted to the manifold \(\Vert u\Vert _{L^2({\mathbb {R}})} = \Vert Q\Vert _{L^2({\mathbb {R}})}\). In view of (6.1.2), we see that \(u = Q\) and \(\lambda = -1\) satisfy the following Lagrange multiplier problem:
for any \(v \in H^1({\mathbb {T}})\), where \(G(u) = \Vert u \Vert _{L^2({\mathbb {R}})}^2 - \Vert Q\Vert _{L^2({\mathbb {R}})}^2\). By a direct computation, the second variation of \(H(u) + G(u)\) at Q in the direction v is given by \( 2\langle B_1{{\,\mathrm{Re}\,}}v,{{\,\mathrm{Re}\,}}v \rangle _{L^2({\mathbb {R}})} + 2\langle B_2{{\,\mathrm{Im}\,}}v,{{\,\mathrm{Im}\,}}v \rangle _{L^2({\mathbb {R}})}\), while the constraint \(G(u) = 0\) gives \(\langle v, Q \rangle _{L^2({\mathbb {R}})} = 0\). Then, from the second derivative test for constrained minima, we obtain that
From (3.3), we have \(B_2 Q = 0\). Also by differentiating (6.1.2), we obtain \(B_1 Q' = 0\). Moreover, by the elementary theory of Schrödinger operators,Footnote 16 the kernel of the operator \(B_j\), \(j = 1, 2\), has dimension 1. In particular, when \(j = 2\), \(B_2 Q = 0\) implies (6.7.23) for any \(v \in \widetilde{V}_2\). Hereafter, we use the fact that the essential spectrum of \(B_j\) is \([1,\infty )\) and that the spectrum of \(B_j\) on \((-\infty , 1)\) consists only of isolated eigenvalues of finite multiplicities.Footnote 17
In the following, we focus on \(B_1\). With \(Q^\perp = (\text {span} (Q))^\perp \), let \(P_{Q^\perp }\) denote the \(L^2({\mathbb {R}})\)-projection onto \(Q^\perp \) given by
Then, from (6.7.25), we have
Obviously, the quadratic form (6.7.26) vanishes for \(v=Q\). On \(Q^\perp \), we have \(P_{Q^\perp } B_1 v=0\) if and only if
Recalling that the restriction of \(B_1\) to \((Q')^\perp \) is invertible, we see that the condition (6.7.27) holds at most for a two-dimensional space. Recall from Remark 6.1 that \(Q' \, (= - \partial _{x_0} Q)\) is orthogonal to Q in \(L^2({\mathbb {R}})\). Moreover, by differentiating (6.1.2) in \(\delta \), we have
while \(\partial _\delta Q \perp Q\) in \(L^2({\mathbb {R}})\). Hence, we have \(P_{Q^\perp }B_1P_{Q^\perp }v=0\) on the three-dimensional subspace \(\text {span}\{Q,\partial _{x_0} Q, \partial _{\delta }Q\}\). As a result, it follows from (6.7.16) that there exists \(\theta > 0\) such that
for any \(v\in \widetilde{V}_1\). \(\square \)
Remark 6.17
The spectral analysis of the operators \(B_1\) and \( B_2\) defined in Lemma 6.16 resembles closely the analysis of similar Schrödinger operators that is at the basis of the results in [30, 44,45,46,47,48, 65, 67,68,69]. For example, focusing on the operator \(B_1(0,0)\), we have the following picture:
-
\(B_1(0,0)\) has one negative eigenvalue, whose existence can be inferred from
$$\begin{aligned} \langle B_1(0,0) Q, Q \rangle _{L^2({\mathbb {R}})} = - 2 \int Q^6 < 0. \end{aligned}$$ -
the eigenvalue \(0 \in \sigma (B_1(0,0))\) has multiplicity 2, corresponding to the directions on the tangent space, \(\mathrm {span}\{\partial _\delta Q, \partial _{x_0} Q\}\), to the soliton manifold .
-
On the resulting space of codimension 3, corresponding to the orthogonal complement to the eigenfunctions described above, \(B_1(0,0)\) is strictly positive.
Compare this result with [66, Lemma 3.2] and [30, Lemma 3.7], which depict a very similar picture in their respective cases. We would like to point out that, in the current work, we make use of the orthogonal coordinate system around the soliton manifold to remove the complications coming from the 0 eigenvalues, while the papers cited above deal with the problem differently. In particular, we require the strict positivity (see (6.7.28)) of the operators \(\frac{1}{2}id+A_j\) defined in (6.7.4) and (6.7.6) in order to guarantee that the denominator appearing in (6.7.50) does not vanish. Moreover, in the same expression, we crucially use the asymptotics of the positive eigenvalues provided by (6.7.29), while this finer analysis does not seem to be present in the aforementioned works.
Next, we establish asymptotic bounds on the eigenvalues of \(A_j\), \(j = 1, 2\). We achieve this goal by comparing \(A_j\) to a simpler operator whose spectrum is studied in Lemma 6.19 below.
Proposition 6.18
Let \(j = 1, 2 \). The spectrum of the operators \(A_j\) defined in (6.7.4) or (6.7.6) consists of a countable collection of eigenvalues. Denoting by \(\{-\lambda _n^{j}\}_{n \in {\mathbb {N}}}\) the negative eigenvalues of \(A_j\) and by \(\{\mu _n^{j}\}_{n \in {\mathbb {N}}}\) the positive eigenvalues, we have
Proof
The first bound (6.7.28) is just a restatement of Proposition 6.15. In the following, we establish the asymptotic behavior (6.7.29) of \(\lambda _n^j\) and \(\mu _n^j\).
For \(j = 1, 2\), define an operator \(T_j\) by
where \(c_1=\frac{5}{2}\) and \(c_2=\frac{1}{2}\). Since \(T_j\) is the composition of a bounded multiplication operator and \((1-\partial _x^2)^{-1}\), which is a compact operator on \(H^1({\mathbb {T}})\), it is compact and thus has a countable sequence of eigenvalues accumulating only at zero. We label the negative eigenvalues as \(-{\bar{\lambda }}_1^j\le -{\bar{\lambda }}_2^j \le \cdots \le 0\), while the positive eigenvalues are labeled as \({\bar{\mu }}_1^j\ge {\bar{\mu }}_2^j\ge \cdots \ge 0\). Since \(A_j\) is the composition of \(T_j\) with a projection, \(A_j\) is also compact. Then, by the min-max principle, we have
Hence, it suffices to prove the asymptotic bounds (6.7.29) for the eigenvalues \(- {\bar{\lambda }}_n^j\) and \({\bar{\mu }}_n^j\) of \(T_j\).
We estimate the eigenvalues \({\bar{\lambda }}_n^j\) and \({\bar{\mu }}_n^j\) for large n by comparing \(T_j\) with an operator with piecewise constant coefficients, whose spectrum can be computed explicitly. Fix \(a> 0\) such that \(10Q^4(a)\le 1\). Given small \(\delta > 0\), define the function \(S = S_\delta \) on \([-\frac{1}{2},\frac{1}{2})\) by setting
Recalling the explicit formula for the ground state \( Q(x) = \frac{6^\frac{1}{4}}{\cosh ^\frac{1}{2} (2^\frac{3}{2} x)}\), we have \(\Vert Q\Vert _{L^\infty ({\mathbb {R}})}^4 = Q^4(0) = 6\). Hence, for any sufficiently small \(\eta > 0\), we have \(S\le \delta ^{-2}-\frac{5}{2}(1+\eta )( Q_\delta ^\rho )^4\) pointwise. Therefore, from (6.7.30), we have
for \(j = 1, 2\) in the sense of operators on \(H^1({\mathbb {T}})\). By the min-max principle, it then suffices to establish the asymptotic behavior (6.7.29) for the eigenvalues of \((1-\partial _x^2)^{-1}S\). This is an explicit computation which we carry out in the following lemma.
Lemma 6.19
Given \(0 < \delta \ll 1\), define the operator \(R = R_\delta \) by
on \(H^1({\mathbb {T}})\), where S is as in (6.7.32). Then, R has a countable sequence of eigenvalues \(\{-\widetilde{\lambda }_n\}_{n \in {\mathbb {N}}}\cup \{ \widetilde{\mu }_n\}_{n \in {\mathbb {N}}}\) with \(\widetilde{\lambda }_n\), \(\widetilde{\mu }_n\ge 0\). Moreover, we have the following asymptotics for the eigenvalues:
where the implicit constants are independent of \(0 < \delta \ll 1\).
We first complete the proof of Proposition 6.18, assuming Lemma 6.19. It follows from the asymptotics (6.7.33) and the min-max principle that
for the eigenvalues \(- {\bar{\lambda }}_n^j\) and \({\bar{\mu }}_n^j\) of \(T_j\). Then, the asymptotic bounds (6.7.29) for the eigenvalues of \(- \lambda _n^j\) and \(\mu _n^j\) of \(A_j\) follows from (6.7.31) and (6.7.34). \(\square \)
We now present the proof of Lemma 6.19.
Proof of Lemma 6.19
Noting that S defined in (6.7.32) is a bounded operator, we see that \(R=(1-\partial ^2_{x})^{-1}S\) is a compact operator on \(H^1({\mathbb {T}})\) and thus has a countable sequence of eigenvalues \(\{\lambda _n\}_{n \in {\mathbb {N}}}\) accumulating only at 0 with the associated eigenfunctions \(\{f_n\}_{n \in {\mathbb {N}}}\) forming a complete orthonormal system in \(H^1({\mathbb {T}})\). Moreover, since the operator R has even and odd functions as invariant subspaces, we assume that any such eigenfunction \(f_n\) is even or odd. The eigenvalue equation:
shows that the second derivative of any eigenfunction is piecewise continuous. With \(C_0 = 5\Vert Q\Vert _{L^\infty ({\mathbb {R}})}^4-2 = 5\cdot 6-2 = 28\), define
Then, by dropping the subscript n, we can rewrite the eigenvalue equation (6.7.35) as
Let \(A = \sqrt{|A_0|}\) and \(B = \sqrt{|B_0|}\). In the following, we carry out case-by-case analysis, depending on the signs of \(A_0\) and \(B_0\). It follows from (6.7.36) that \(\lambda > 0\) implies \(A_0 > B_0\), while \(\lambda < 0\) implies \(B_0 < 0\). This in particular implies that the case \(A_0< 0 < B_0\) can not happen for any parameter choice.
\(\bullet \) Case 1: \(A_0,B_0 < 0\) and f is even. Without loss of generality, we may assume that
From solving (6.7.37) on \([-a\delta , a\delta ]\), we have
On the other hand, the general solution to (6.7.37) on \((a\delta ,\frac{1}{2}]\) is given by
Thus, with the notations:
we have the following “transmission conditions” at \(x=a\delta \):
Hence, on \((a\delta ,\frac{1}{2}]\), we have
To enforce the periodicity condition \(f'(\frac{1}{2}) = 0\), we need
By rearranging, we obtain the condition
Our goal is to show that for every \(k \in {\mathbb {Z}}_{\ge 0}\), there exists exactly one \(A_k^1\) such that (6.7.38) is satisfied and
From (6.7.36), we have
Thus, we can rewrite (6.7.38) as \(F_1(A) = 1\), where \(F_1(A)\) is given by
The existence of \(A^1_k\) satisfying \(F_1(A^1_k) = 1\) follows from continuity of \(F_1\), together with the limits
In order to show uniqueness, it suffices to show that for every A with \(F_1(A) = 1\), we have \(F_1'(A) > 0\). We first note that, for \(0\le A \ll \delta ^{-1}\), we have
Thus, in order to have \(F_1(A) = 1\), we must have \(A > rsim \delta ^{-\frac{1}{2}}\). Moreover, for \(A > rsim \delta ^{-\frac{1}{2}}\), we have
from which we obtain \(A > rsim \delta ^{-1}\). Namely, \(F_1(A)= 1\) implies \(A > rsim \delta ^{-1}\).
From (6.7.40), we have
for \(A \gg 1\). Therefore, whenever \(F_1(x) = 1\), we have
for \(A > rsim \delta ^{-1}\). Note that we used the relation (6.7.38) in handling the first, second, and third terms after the first equality in (6.7.42).
\(\bullet \) Case 2: \(A_0,B_0 < 0\) and f is odd. Without loss of generality, we may assume that
By solving the eigenfunction equation (6.7.37) and solving for the transmission conditions as in Case 1, we have
for \( x \in (a \delta , \frac{1}{2}]\). Therefore, the periodicity condition \(f(\tfrac{1}{2})=0\) becomes
where \(B = B(A)\) is as in (6.7.40). As in Case 1, we want to show that for every \(k \in {\mathbb {Z}}_{\ge 0}\), there exists exactly one value \(A^2_k\) that satisfies (6.7.43) with
From (6.7.43), we have \(\cot (Aa\delta ) < 0\), which implies \(A a \delta > \frac{\pi }{2}\). Then, using (6.7.41) and (6.7.44) with \(A > rsim \delta ^{-1}\), we can proceed as in Case 1 and show existence and uniqueness of such \(A^2_k\), \(k \in {\mathbb {Z}}_{\ge 0}\). Note that in this case, we show \(F_2'(A) < 0\) whenever \(F_2(A) = -1\).
\(\bullet \) Case 3: \(B_0< 0 <A_0\) and f is even. By solving the eigenfunction equation (6.7.37), we obtain
for \(x \in ( a\delta , \frac{1}{2}]\). By imposing the periodicity condition \(f'(\frac{1}{2} ) = 0\), we obtain
However, since \(A, B, a > 0\), we see that this condition can never be satisfied for \(0 < \delta \ll 1\).
\(\bullet \) Case 4: \(B_0< 0<A_0\) and f is odd. In this case, we have
for \(x \in ( a\delta , \frac{1}{2}]\). By imposing the periodicity condition \(f(\frac{1}{2}) = 0\), we obtain
Since \(A, B, a > 0\), we once again see that this condition can not be satisfied for \(0 < \delta \ll 1\).
\(\bullet \) Case 5: \(A_0, B_0 > 0\) and f is even. By solving the eigenfunction equation (6.7.37), we have
for \(x \in ( a\delta , \frac{1}{2}]\). Then, by imposing the periodicity condition \(f'(\frac{1}{2} ) = 0\), we obtain
By writing
we can rewrite (6.7.45) as \(G_1(B) = 0\), where
As in Case 1, we need to show that for every \(k \in {\mathbb {Z}}_{\ge 0}\), there exists exactly one value \(B^1_k\) such that \(G(B^1_k) = 0\) with
Existence follows from continuity of \(G_1\), together with the limits
As for uniqueness of \(B^1_k\), it suffices to show \(G_1'(B) > 0\) whenever \(G_1(B) = 0\). Using (6.7.45), we have
for \(0 < \delta \ll 1\).
\(\bullet \) Case 6: \(A_0, B_0 > 0\) and f is odd. By solving the eigenfunction equation (6.7.37), we have that
for \(x \in ( a\delta , \frac{1}{2}]\). Then, by imposing the periodicity condition \(f(\frac{1}{2}) = 0\), we obtain
In view of (6.7.46), define \(G_2(B)\) by
As in the previous cases, we show that for every \(k \in {\mathbb {Z}}_{\ge 0}\), there exists exactly one value \(B^2_k\) such that \(G_2(B_k^2) = 0\) and
Noting \(G_2(0) > 0\), existence follows from the same argument as in the previous cases. As for uniqueness, we show that \(G_2'(B) < 0\) whenever \(G_2(B) = 0\). Using (6.7.48) (which implies \(\cot (B(2^{-1}-a\delta )) < 0\)) and noting that \(\tanh \big (\sqrt{2(B^2+1)C_0 + 1}\,a\delta \big )\) is strictly increasing in B, we have
for \(0 < \delta \ll 1\).
Conclusion: Fix \(0 < \delta \ll 1\). Putting all the cases together, we conclude that every \(\lambda \) such that in (6.7.36), we have \(A_0 = -(A_k^1)^2\), \(A_0 = -(A_k^2)^2\), \(B_0 = (B_k^1)^2\), or \(B_0 = (B_k^2)^2\) for some \(k \in {\mathbb {Z}}_{\ge 0}\) corresponds to an eigenvalue for R. Moreover, this exhausts all the possibilities for the eigenvalues of R, with possible exceptions of \(\lambda \) such that \(A_0 = 0\) or \(B_0 = 0\). By inverting the formulas of \(A_0= -(A_k^j)^2\) and \(B_0= (B_k^j)^2\) in (6.7.36) for \(\lambda \) together with (6.7.39), (6.7.44), (6.7.47), (6.7.49), and \(A_k^j > rsim \delta ^{-1}\), \(j = 1, 2\), we obtain the asymptotics:
This completes the proof of Lemma 6.19. \(\square \)
Finally, we conclude this subsection by presenting the proof of Proposition 6.14.
Proof of Proposition 6.14
Given \(N\in {\mathbb {N}}\), let \(W_N\) be the subspace spanned by the eigenvectors of A corresponding to the eigenvalues \(\lambda _n^j\) and \(\mu _n^j\), \(1\le n\le N\), \(j = 1, 2\). Define \(A_N=AP_N\), where \(P_N = P_{W_N}^{H^1}\) is the projection onto \(W_N\) in \(H^1({\mathbb {T}})\). Then, by Proposition 6.18 with \(\mathrm {d}\alpha \mathrm {d}\beta = \prod _{j = 1}^2 \prod _{n = 1}^N \mathrm {d}\alpha _n^j\mathrm {d}\beta _n^j\) and (4.9), we have
where we used \(\log (1+x) \ge x - \frac{x^2}{2} > rsim x\) for \(|x|\ll 1\) in the penultimate step.
For \(v, w \in L^2({\mathbb {T}})\), we have
Then, by a density argument, we see that \(\langle A_N w, w \rangle \) converges to \(\langle A w, w \rangle \) as \(N \rightarrow \infty \) for each \(w \in L^2({\mathbb {T}})\). Hence, the estimate (6.7.3) follows from (6.7.50) and Fatou’s lemma. \(\square \)
6.8 Proof of Theorem 1.4
We now put all the steps together and present the proof of of Theorem 1.4. Let \(K = \Vert Q\Vert _{L^2({\mathbb {R}})}\). By Lemma 6.3, we have
where \(S_\gamma \) and \(U_\varepsilon (\delta ^*)\) are as in (6.2.1) and (6.3.2). From (6.2.3), we have
for any \(\gamma > 0\). As for \(\text {I}\!\text {I}(\varepsilon , \delta ^*)\), it follows from (6.6.8), Hölder’s inequality, and Lemma 6.12 that
provided that \(\varepsilon _1 = \varepsilon _1(\eta ) > 0\) is sufficiently small. Note that in obtaining (6.6.8), we applied Proposition 6.4 which gives an orthogonal coordinate system in a neighborhood of the soliton manifold. Then, from Lemma 6.13, (6.7.2), and Proposition 6.14, we obtain
provided that \(\eta > 0\) is sufficiently small, where we used Lemma 6.10 in the last step. Therefore, from (6.8.1), (6.8.2), and (6.8.3), we conclude that
For readers’ convenience, we go over how we choose the parameters. We first choose \(\eta > 0 \) in (6.6.7) sufficiently small such that Proposition 6.14 holds. Next, we fix small \(\varepsilon _1 = \varepsilon _1(\eta ) > 0\) such that Lemma 6.12 holds. Then, Proposition 6.4 determines small \(\varepsilon = \varepsilon (\varepsilon _1)>0 \) and \(\delta ^* = \delta ^*(\varepsilon _1)> 0 \). See also Lemma 6.13, where we need smallness of \(\delta = \delta (\eta , \varepsilon _1) > 0\). Finally, we fix \(\gamma =\gamma (\varepsilon ) > 0\) by applying Lemma 6.3. This completes the proof of Theorem 1.4.
Remark 6.20
In this section, we presented the proof of Theorem 1.4, where the base Gaussian process is given by the Ornstein–Uhlenbeck loop in (6.0.1). When the base Gaussian process is given by the mean-zero Brownian loop in (1.3), the same but simpler argument gives Theorem 1.4. For example, in the case of the mean-zero Brownian loop, we can omit (6.2.2) and the reduction to the mean-zero case at the beginning of the proof of Lemma 6.3. In the proof of Proposition 6.4, we introduced \(V_{\delta ,x_0,\theta }^{\gamma _0}\) in (6.3.4) in order to use a scaling argument in the non-homogeneous setting. In the case of the mean-zero Brownian loop, we can simply use \(V_{\delta ,x_0,\theta }^{0}\), i.e. \(\gamma _0 = 0\). The rest of the proof remains essentially the same.
Notes
Namely, \({{\,\mathrm{Re}\,}}g_n\) and \({{\,\mathrm{Im}\,}}g_n\) are independent real-valued mean-zero Gaussian random variables with variance \(\frac{1}{2}\).
Recall that the \(L^2\)-norm is conserved under the NLS dynamics.
Up to the symmetries.
In a recent preprint [29], Dodson extended this uniqueness / rigidity result of the minimal mass blowup solution to the \(L^2({\mathbb {R}})\)-setting. We, however, point out that our understanding of the corresponding problem on the torus \({\mathbb {T}}\) is rather poor in this direction, in particular in a low regularity setting. For example, even local well-posedness in \(L^2({\mathbb {T}})\) of the focusing quintic NLS on \({\mathbb {T}}\) remains a challenging open problem after Bourgain’s work [8]. See also [43].
The local well-posedness argument in [28] applies to the range \(\beta > \beta _*\) for some \(\beta _* < 1\) sufficiently close to 1.
In the context of Theorem 2.7 on [25], we set \(x = \frac{X}{{\mathbf {E}}[\Vert X\Vert _B]}\). Then, by Markov’s inequality and choosing \(r\gg 1\), we have
$$\begin{aligned} \log \bigg (\frac{\mu (\Vert x\Vert _B> r)}{\mu (\Vert x\Vert _B \le r)} \bigg ) = \log \bigg (\frac{\mu \big (\Vert X\Vert _B> r{\mathbf {E}}[\Vert X\Vert _B]\big )}{1- \mu \big (\Vert X\Vert _B > r{\mathbf {E}}[\Vert X\Vert _B]\big )} \bigg ) \le \log \frac{1}{r-1} \le -2 \end{aligned}$$without using any fine property of X. Then, (4.12) and (4.13) follow in view of Remark 2.8 in [25].
Recall that we work with mean-zero functions on \({\mathbb {T}}\).
Without the multiplication by (a translate) of the cutoff function \(\rho \).
Recall that we view \(H^k({\mathbb {T}})\) as a Hilbert space over reals.
Hereafter, we suppress the x-dependence of \(w_j = w_j(x)\) and \(t_k = t_k(x)\) when there is no confusion.
Once again, recall that we view \(H^k({\mathbb {T}})\) as a Hilbert space over reals.
In the last step of (6.5.9), we used the decomposition \(\mu ^\perp _{\delta , x_0} = \mu ^\perp _{\delta , x_0, \le N}\otimes \mu ^\perp _{\delta , x_0, > N}\), where \( \mu ^\perp _{\delta , x_0, \le N}\) (and \(\mu ^\perp _{\delta , x_0, > N}\), respectively) denotes the Gaussian measure with \(V_{\delta ,x_0,0} \cap E_N\) (and \(V_{\delta ,x_0,0} \cap \pi _N^\perp H^1({\mathbb {T}})\), respectively) as its Cameron–Martin space.
Recall the convention \( V_{\delta ,0,0} = V_{\delta ,0,0}\cap H^1({\mathbb {T}})\) introduced at the beginning of Sect. 6.5.
If there is a linearly independent solution v to \(B_j v = 0\), then by considering the Wronskian, we obtain that \(v' (x)\not \rightarrow 0 \) as \(|x| \rightarrow \infty \). This in particular implies \(v \notin L^2 ({\mathbb {R}}) \cup \dot{H}^1({\mathbb {R}})\).
This claim follows from Weyl’s criterion ([84, Theorem VII.12]).
References
Agrawal, G.P.: Nonlinear Fiber Optics, 5th edn. Academic, San Francisco (2012)
Barashkov, N., Gubinelli, M.: A variational method for \(\Phi ^4_3\). Duke Math. J. 169(17), 3339–3415 (2020)
Bényi, Á., Oh, T.: The Sobolev inequality on the torus revisited. Publ. Math. Debrecen 83(3), 359–374 (2013)
Bényi, Á., Oh, T., Pocovnicu, O.: Wiener randomization on unbounded domains and an application to almost sure well-posedness of NLS. In: Balan, R., Begué, M., Benedetto, J., Czaja, W., Okoudjou, K. (eds.) Excursions in Harmonic Analysis. Applied and Numerical Harmonic Analysis, vol. 4, pp. 3–25. Springer, Cham (2015)
Bényi, Á., Oh, T., Pocovnicu, O.: On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on \({{\mathbb{R}}}^d\), \(d \ge 3\). Trans. Am. Math. Soc. Ser. B 2, 1–50 (2015)
Bényi, Á., Oh, T., Pocovnicu, O.: Higher order expansions for the probabilistic local Cauchy theory of the cubic nonlinear Schrödinger equation on \({{\mathbb{R}}}^3\). Trans. Am. Math. Soc. Ser. B 6, 114–160 (2019)
Bényi, Á., Oh, T., Pocovnicu, O.: On the probabilistic Cauchy theory for nonlinear dispersive PDEs. In: Boggiatto, P., et al. (eds.) Landscapes of Time-Frequency Analysis. Applied and Numerical Harmonic Analysis. Springer, Cham (2019)
Bourgain, J.: Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations. Geom. Funct. Anal. 3(2), 107–156 (1993)
Bourgain, J.: Periodic nonlinear Schrödinger equation and invariant measures. Commun. Math. Phys. 166(1), 1–26 (1994)
Bourgain, J.: Invariant measures for the 2D-defocusing nonlinear Schrödinger equation. Commun. Math. Phys. 176(2), 421–445 (1996)
Bourgain, J.: Invariant measures for the Gross–Piatevskii equation. J. Math. Pures Appl. 76(8), 649–702 (1997)
Bourgain, J., Bulut, A.: Almost sure global well posedness for the radial nonlinear Schrödinger equation on the unit ball I: the 2D case. Ann. Inst. H. Poincaré Anal. Non Linéaire 31(6), 1267–1288 (2014)
Brereton, J.: Invariant measure construction at a fixed mass. Nonlinearity 32(2), 496–558 (2019)
Bringmann, B.: Stable blowup for the focusing energy critical nonlinear wave equation under random perturbations. Commun. Partial Differ. Equ. 45(12), 1755–1777 (2020)
Bringmann, B.: Invariant Gibbs measures for the three-dimensional wave equation with a Hartree nonlinearity II: dynamics. arXiv:2009.04616 [math.AP]
Brydges, D., Slade, G.: Statistical mechanics of the 2-dimensional focusing nonlinear Schrödinger equation. Commun. Math. Phys. 182(2), 485–504 (1996)
Burq, N., Tzvetkov, N.: Random data Cauchy theory for supercritical wave equations. I. Local theory. Invent. Math. 173(3), 449–475 (2008)
Burq, N., Tzvetkov, N.: Probabilistic well-posedness for the cubic wave equation. J. Eur. Math. Soc. 16(1), 1–30 (2014)
Carlen, E., Fröhlich, J., Lebowitz, J.: Exponential relaxation to equilibrium for a one-dimensional focusing non-linear Schrödinger equation with noise. Commun. Math. Phys. 342(1), 303–332 (2016)
Chapouto, A., Kishimoto, N.: Invariance of the Gibbs measures for the periodic generalized KdV equations. arXiv:2104.07382 [math.AP]
Colliander, J., Oh, T.: Almost sure well-posedness of the cubic nonlinear Schrödinger equation below \(L^2({{\mathbb{T}}})\). Duke Math. J. 161(3), 367–414 (2012)
Colliander, J., Raphaël, P.: Rough blowup solutions to the \(L^2\) critical NLS. Math. Ann. 345(2), 307–366 (2009)
Da Prato, G.: An Introduction to Infinite-Dimensional Analysis. Universitext. Springer, Berlin (2006)
Da Prato, G., Debussche, A.: Strong solutions to the stochastic quantization equations. Ann. Probab. 31(4), 1900–1916 (2003)
Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and its Applications, vol. 152, 2nd edn. Cambridge University Press, Cambridge (2014)
Deng, Y., Nahmod, A., Yue, H.: Invariant Gibbs measures and global strong solutions for nonlinear Schrödinger equations in dimension two. arXiv:1910.08492 [math.AP]
Deng, Y., Nahmod, A., Yue, H.: Random tensors, propagation of randomness, and nonlinear dispersive equations. arXiv:2006.09285 [math.AP]
Deng, Y., Nahmod, A., Yue, H.: Invariant Gibbs measure and global strong solutions for the Hartree NLS equation in dimension three. J. Math. Phys. 62(3), 031514 (2021)
Dodson, B.: A determination of the blowup solutions to the focusing, quintic NLS with mass equal to the mass of the soliton. arXiv:2104.11690 [math.AP]
Donninger, R., Schörkhuber, B.: Stable self-similar blow up for energy subcritical wave equations. Dyn. Partial Differ. Equ. 9(1), 63–87 (2012)
Driver, B.: Analysis tools with applications, lecture notes (2003). http://www.math.ucsd.edu/~bdriver/240-01-02/Lecture_Notes/anal.pdf
Jentzen, W.E.A., Shen, H.: Renormalized powers of Ornstein–Uhlenbeck processes and well-posedness of stochastic Ginzburg–Landau equations. Nonlinear Anal. 142, 152–193 (2016)
Evans, L.C., Gariepy, R.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics, CRC Press, Boca Raton (1992)
Fan, C., Mendelson, D.: Construction of \(L^2\) log-log blowup solutions for the mass critical nonlinear Schrödinger equation. arXiv:2010.07821 [math.AP]
Fernique, X.: Regularité des trajectoires des fonctions aléatoires gaussiennes, École d’Été de Probabilités de Saint-Flour, IV-1974, 1–96. Lecture Notes in Mathematics, vol. 480. Springer, Berlin (1975)
Frank, R.: Grounds states of semilinear equations, lecture notes from Current topics in Mathematical Physics, Luminy (2013). http://www.mathematik.uni-muenchen.de/~frank/luminy140202.pdf
Gubinelli, M., Imkeller, P., Perkowski, N.: Paracontrolled distributions and singular PDEs. Forum Math. Pi 3, e6 (2015)
Gubinelli, M., Koch, H., Oh, T.: Paracontrolled approach to the three-dimensional stochastic nonlinear wave equation with quadratic nonlinearity. J. Eur. Math. Soc
Hmidi, T., Keraani, S.: Blowup theory for the critical nonlinear Schrödinger equations revisited. Int. Math. Res. Not. 2005, 2815–2828 (2005)
Kamvissis, S., McLaughlin, K., Miller, P.: Semiclassical Soliton Ensembles for the Focusing Nonlinear Schrödinger Equation. Annals of Mathematics Studies, vol. 154. Princeton University Press, Princeton (2003)
Kato, T.: Perturbation theory for linear operators. Reprint of the 1980 edition. Classics in Mathematics. Springer, Berlin (1995)
Killip, R., Vişan, M.: Nonlinear Schrödinger equations at critical regularity. In: Evolution Equations. Clay Mathematics Proceedings, vol. 17, pp. 325–437. American Mathematical Society, Providence (2013)
Kishimoto, N.: Remark on the periodic mass critical nonlinear Schrödinger equation. Proc. Am. Math. Soc. 142(8), 2649–2660 (2014)
Krieger, J., Nakanishi, K., Schlag, W.: Global dynamics above the ground state energy for the one-dimensional NLKG equation. Math. Z. 272(1–2), 297–316 (2012)
Krieger, J., Nakanishi, K., Schlag, W.: Global dynamics of the nonradial energy-critical wave equation above the ground state energy. Discrete Contin. Dyn. Syst. 33(6), 2423–2450 (2013)
Krieger, J., Nakanishi, K., Schlag, W.: Global dynamics away from the ground state for the energy-critical nonlinear wave equation. Am. J. Math. 135(4), 935–965 (2013)
Krieger, J., Nakanishi, K., Schlag, W.: Threshold phenomenon for the quintic wave equation in three dimensions. Commun. Math. Phys. 327(1), 309–332 (2014)
Krieger, J., Nakanishi, K., Schlag, W.: Center-stable manifold of the ground state in the energy space for the critical wave equation. Math. Ann. 361(1–2), 1–50 (2015)
Kwong, M.K.: Uniqueness of positive solutions of \(\Delta u - u + u^p = 0\) in \({\mathbf{R}}^n\). Arch. Ration. Mech. Anal. 105(3), 243–266 (1989)
Lebowitz, J., Mounaix, P., Wang, W.-M.: Approach to equilibrium for the stochastic NLS. Commun. Math. Phys. 321(1), 69–84 (2013)
Lebowitz, J., Rose, H., Speer, E.: Statistical mechanics of the nonlinear Schrödinger equation. J. Stat. Phys. 50(3–4), 657–687 (1988)
Li, D., Zhang, X.: On the rigidity of minimal mass solutions to the focusing mass-critical NLS for rough initial data. Electron. J. Differ. Equ. 2009(78), 1–19 (2009)
Lührmann, J., Mendelson, D.: Random data Cauchy theory for nonlinear wave equations of power-type on \({\mathbb{R}}^3\). Commun. Partial Differ. Equ. 39(12), 2262–2283 (2014)
Martel, Y., Merle, F.: Blow up in finite time and dynamics of blow up solutions for the \(L^2\)-critical generalized KdV equation. J. Am. Math. Soc. 15(3), 617–664 (2002)
Martel, Y., Merle, F.: Nonexistence of blow-up solution with minimal \(L^2\)-mass for the critical gKdV equation. Duke Math. J. 115(2), 385–408 (2002)
McKean, H.P.: Statistical mechanics of nonlinear wave equations. IV. Cubic Schrödinger. Comm. Math. Phys. 168(3), 479–491 (1995). Erratum: Statistical mechanics of nonlinear wave equations. IV. Cubic Schrödinger, Comm. Math. Phys. 173(3), 675 (1995)
Merle, F.: Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power. Duke Math. J. 69(2), 427–454 (1993)
Merle, F.: Existence of blow-up solutions in the energy space for the critical generalized KdV equation. J. Am. Math. Soc. 14(3), 555–578 (2001)
Merle, F., Raphaël, P.: Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation. Geom. Funct. Anal. 13(3), 591–642 (2003)
Merle, F., Raphaël, P.: On universality of blow-up profile for \(L^2\) critical nonlinear Schrödinger equation. Invent. Math. 156(3), 565–672 (2004)
Merle, F., Raphaël, P.: The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation. Ann. Math. 161(1), 157–222 (2005)
Merle, F., Raphaël, P.: On a sharp lower bound on the blow-up rate for the \(L^2\) critical nonlinear Schrödinger equation. J. Am. Math. Soc. 19(1), 37–90 (2006)
Merle, F., Raphaël, P., Szeftel, J.: The instability of Bourgain–Wang solutions for the \(L^2\) critical NLS. Am. J. Math. 135(4), 967–1017 (2013)
Nagy, B.VSz.: Über Integralgleichungen zwischen einer Funktion und ihrer Ableitung. Acta Univ. Szeged. Sect. Sci. Math 10, 64–74 (1941)
Nakanishi, K., Schlag, W.: Global dynamics above the ground state energy for the focusing nonlinear Klein–Gordon equation. J. Differ. Equ. 250(5), 2299–2333 (2011)
Nakanishi, K., Schlag, W.: Invariant Manifolds and Dispersive Hamiltonian Evolution Equations. Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich (2011)
Nakanishi, K., Schlag, W.: Global dynamics above the ground state energy for the cubic NLS equation in 3D. Calc. Var. Partial Differ. Equ. 44(1–2), 1–45 (2012)
Nakanishi, K., Schlag, W.: Invariant manifolds around soliton manifolds for the nonlinear Klein–Gordon equation. SIAM J. Math. Anal. 44(2), 1175–1210 (2012)
Nakanishi, K., Schlag, W.: Global dynamics above the ground state for the nonlinear Klein–Gordon equation without a radial assumption. Arch. Ration. Mech. Anal. 203(3), 809–851 (2012)
Ogawa, T., Tsutsumi, Y.: Blow-up of solutions for the nonlinear Schrödinger equation with quartic potential and periodic boundary condition. In: Functional-Analytic Methods for Partial Differential Equations (Tokyo, 1989). Lecture Notes in Mathematics, vol. 1450, pp. 236–251. Springer, Berlin (1990)
Oh, T., Okamoto, M., Pocovnicu, O.: On the probabilistic well-posedness of the nonlinear Schrödinger equations with non-algebraic nonlinearities. Discrete Contin. Dyn. Syst. A. 39(6), 3479–3520 (2019)
Oh, T., Okamoto, M., Tolomeo, L.: Focusing \(\Phi ^4_3\)-model with a Hartree-type nonlinearity. arXiv:2009.03251 [math.PR]
Oh, T., Okamoto, M., Tolomeo, L.: Stochastic quantization of the \(\Phi ^3_3\)-model. arXiv:2108.06777 [math.PR]
Oh, T., Pocovnicu, O.: Probabilistic global well-posedness of the energy-critical defocusing quintic nonlinear wave equation on \({{\mathbb{R}}}^3\). J. Math. Pures Appl. 105, 342–366 (2016)
Oh, T., Quastel, J.: On invariant Gibbs measures conditioned on mass and momentum. J. Math. Soc. Jpn. 65(1), 13–35 (2013)
Oh, T., Quastel, J., Valkó, B.: Interpolation of Gibbs measures and white noise for Hamiltonian PDE. J. Math. Pures Appl. 97(4), 391–410 (2012)
Oh, T., Robert, T., Sosoe, P., Wang, Y.: On the two-dimensional hyperbolic stochastic sine-Gordon equation. Stoch. Partial Differ. Equ. Anal. Comput. 9, 1–32 (2021)
Oh, T., Seong, K., Tolomeo, L.: A remark on Gibbs measures with log-correlated Gaussian fields. arXiv:2012.06729 [math.PR]
Parisi, G., Wu, Y.S.: Perturbation theory without gauge fixing. Sci. Sin. 24(4), 483–496 (1981)
Perelman, G.: On the formation of singularities in solutions of the critical nonlinear Schrödinger equation. Ann. Henri Poincaré 2(4), 605–673 (2001)
Planchon, F., Raphaël, P.: Existence and stability of the log-log blow-up dynamics for the \(L^2\)-critical nonlinear Schrödinger equation in a domain. Ann. Henri Poincaré 8(6), 1177–1219 (2007)
Pocovnicu, O.: Almost sure global well-posedness for the energy-critical defocusing nonlinear wave equation on \({{\mathbb{R}}}^d\), \( d= 4\) and \(5\). J. Eur. Math. Soc. 19(8), 2521–2575 (2017)
Raphaël, P.: Stability of the log–log bound for blow up solutions to the critical non linear Schrödinger equation. Math. Ann. 331(3), 577–609 (2005)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics. I. Functional Analysis, 2nd edn. Academic Press, New York (1980)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics. IV. Analysis of Operators. Academic Press, New York (1978)
Rider, B.: On the \(\infty \)-volume limit of the focusing cubic Schrödinger equation. Commun. Pure Appl. Math. 55(10), 1231–1248 (2002)
Ryang, S., Saito, T., Shigemoto, K.: Canonical stochastic quantization. Progr. Theor. Phys. 73(5), 1295–1298 (1985)
Sulem, C., Sulem, P.-L.: The nonlinear Schrödinger equation. In: Self-Focusing and Wave Collapse. Applied Mathematical Sciences, vol. 139. Springer, New York (1999)
Tao, T.: Nonlinear dispersive equations. Local and global analysis. In: CBMS Regional Conference Series in Mathematics, vol. 106. Published for the Conference Board of the Mathematical Sciences, Washington, DC. American Mathematical Society, Providence (2006)
Tolomeo, L., Weber, H.: Phase transition for invariant measures of the focusing Schrödinger equation (in preparation)
Tzvetkov, N.: Invariant measures for the nonlinear Schrödinger equation on the disc. Dyn. Partial Differ. Equ. 3(2), 111–160 (2006)
Tzvetkov, N.: Invariant measures for the defocusing nonlinear Schrödinger equation. Ann. Inst. Fourier (Grenoble) 58(7), 2543–2604 (2008)
Weinstein, M.: Nonlinear Schrödinger equations and sharp interpolation inequalities. Commun. Math. Phys. 87(4), 567–576 (1983)
Acknowledgements
The authors would like to thank Nikolay Tzvetkov for some interesting discussions. The authors would like to express their gratitude to the anonymous referee for the helpful comments which improved the quality of the paper. T.O. was supported by the European Research Council (Grant No. 637995 “ProbDynDispEq” and Grant No. 864138 “SingStochDispDyn”). P.S. was partially supported by NSF grant DMS-1811093. L.T. was supported by the European Research Council (Grant No. 637995 “ProbDynDispEq”) and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy-EXC-2047/1-390685813, through the Collaborative Research Centre (CRC) 1060. L.T. was also supported by the Maxwell Institute Graduate School in Analysis and its Applications, a Centre for Doctoral Training funded by the UK Engineering and Physical Sciences Research Council (Grant EP/L016508/01), the Scottish Funding Council, Heriot-Watt University, and the University of Edinburgh.
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Oh, T., Sosoe, P. & Tolomeo, L. Optimal integrability threshold for Gibbs measures associated with focusing NLS on the torus. Invent. math. 227, 1323–1429 (2022). https://doi.org/10.1007/s00222-021-01080-y
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DOI: https://doi.org/10.1007/s00222-021-01080-y