Abstract
We prove that for antisymmetric vector field \(\Omega \) with small \(L^2\)-norm there exists a gauge \(A \in L^\infty \cap {\dot{W}}^{1/2,2}({\mathbb {R}}^1,GL(N))\) such that
This extends a celebrated theorem by Rivière to the nonlocal case and provides conservation laws for a class of nonlocal equations with antisymmetric potentials, as well as stability under weak convergence.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
In the celebrated work [27] Rivière showed that for two-dimensional disks \(D \subset {\mathbb {R}}^2\) for any \(\Omega \in L^2(D,so(N)\otimes \bigwedge \nolimits ^1 {\mathbb {R}}^2)\), i.e., \(\Omega _{ij}=-\Omega _{ji} \in L^2(D,\bigwedge \nolimits ^1 {\mathbb {R}}^2)\) there exists a GL(N)-gauge, namely a matrix-valued function \(A,A^{-1} \in L^\infty \cap W^{1,2}(D,GL(N))\) such that
These are distortions of the orthonormal Uhlenbeck’s Coulomb gauges, [36], namely \(P \in L^\infty \cap W^{1,2}(D,SO(N))\) which satisfy
As Rivière showed in [27], the GL(N)-gauges have the advantage that they can transform equations of the form
into a conservation law
This is important since (1.1) is the structure of the equation for harmonic maps, H-surfaces, and more generally the Euler–Lagrange equations of a large class of conformally invariant variational functionals. The GL(N)-gauge transform allows for regularity theory and the study of weak convergence [27]; it also is an important tool for energy quantization, see [16].
In recent years a theory of fractional harmonic maps has developed, beginning with the work by Rivière and the first named author, [9, 10]. bubbling analysis was initiated in [6]. Fractional harmonic maps have a variety of applications: they appear as free boundary of minimal surfaces or harmonic maps [8, 21, 24, 31]; they are also related to nonlocal minimal surfaces [22] and to knot energies [2, 3].
We recall that in [10] the first named author and Rivière considered nonlocal Schödinger-type systems of the form
where \(\Omega \) is an antisymmetric potential in \(L^2({\mathbb {R}},so(N))\), \(v\in L^2({\mathbb {R}},{{\mathbb {R}}}^N)\). The main technique to establish the sub-criticality of systems (1.2) is to perform a change of gauge by rewriting them after having multiplied v by a well-chosen rotation-valued map \(P\in {\dot{W}}^{1/2,2}({{\mathbb {R}}},SO(N))\) which is ”integrating” \(\Omega \) in an optimal way. The key point in [9, 10] was the discovery of particular algebraic structures (three-term commutators) that play the role of the Jacobians in the case of local systems in 2-D with an antisymmetric potential and that enjoy suitable integrability by compensations properties. In [17] the second and the third named authors introduced a new approach to fractional harmonic maps by considering nonlocal systems with an antisymmetric potential which is seen itself as a nonlocal operator. As we will explain later, such an approach is similar in the spirit to that introduced by Hélein in [15] in the context of harmonic maps.
It begins with the definition of “nonlocal one forms”. \(F \in L^p(\bigwedge \nolimits _{od}^1 {\mathbb {R}}^n)\) if \(F: {\mathbb {R}}^n \times {\mathbb {R}}^n \rightarrow {\mathbb {R}}\) and
The s-differential, which takes function \(u: {\mathbb {R}}^n \rightarrow {\mathbb {R}}\) into 1-forms, is then given by
The scalar product for two 1-forms, \(F \in L^p(\bigwedge \nolimits _{od}^1 {\mathbb {R}}^n)\) and \(G \in L^{p'}(\bigwedge \nolimits _{od}^1 {\mathbb {R}}^n)\), is then given by
The fractional divergence \({\text {div}}_s\), which takes 1-forms into functions, is then the formal adjoint to \(d_s\), namely
For more details we refer to Sect. 2. With this notation in mind we now consider equations of the form
or in index form
where \(u \in (L^2 + L^\infty )\cap \dot{W}^{\frac{1}{2},2}({\mathbb {R}},{\mathbb {R}}^N)\) and \(\Omega _{ij} = - \Omega _{ji} \in L^2(\bigwedge \nolimits ^1_{od} {\mathbb {R}})\).
The main observation in [17] is that the above notation and the above equation are not merely some random definitions of only analytical interest. Rather it was shown that the role of (1.3) for fractional harmonic maps is similar to the role of (1.1) for harmonic maps. In [17] it was shown that there exists a \(div-curl\) lemma in the spirit of [5], that fractional harmonic maps into spheres satisfy a conservation law in the spirit of [15], and that fractional harmonic maps into spheres essentially satisfy equations of the form (1.3), in the spirit of [27], and that an analogue of Uhlenbeck’s gauge exist. In [20] this argument was further pushed to equations of stationary harmonic map in higher dimensional domains.
We mention that in [7] the authors found quasi conservation laws for nonlocal Schrödinger-type systems of the form
where \(v\in L^2({\mathbb {R}})\), \(\Omega \in L^2({\mathbb {R}},so(N))\), and g is a tempered distribution. As we have already pointed out above, systems (1.4) represent a particular case of systems (1.3) studied in the present paper in the sense that the antisymmetric potential \(\Omega \) in (1.4) is a pointwise function. The conservation laws found in [7] are a consequence of a stability property of some three-term commutators by the multiplication of \(P \in SO(N)\) and also of the regularity results obtained previously for such commutators. The reformulation of (1.4) in terms of conservation laws has permitted to get the quantization in the neck regions of the \(L^2\) norms of the negative part of sequences of solutions to systems of the type (1.4).
The conservation laws that we obtain in the current paper are more similar in the spirit to those found in the paper [27] for harmonic maps and concern nonlocal systems (1.3) where the antisymmetric potential acts in general as a nonlocal operator. We hope this technique to be as useful for the question of concentration compactness and energy quantization for systems as it was in the local case in [16]; a question we will study in a future work.
Applying a gauge \(A \in L^\infty \cap {\dot{W}}^{\frac{1}{2},2}\) to the Eq. (1.3), we find (see Lemma 4.1),
Our main result is then the existence of the nonlocal analogue of Rivière’s GL(N)-Coulomb gauge [27], namely we have
Theorem 1.1
There exists a number \(0<\sigma \ll 1\) such that the following holds.
If \(\Omega \in L^2(\bigwedge \nolimits _{od}^1{\mathbb {R}})\) is antisymmetric, i.e., \(\Omega _{ij} = -\Omega _{ji}\) and satisfies
then there exists an invertible matrix-valued function \(A \in L^\infty \cap {\dot{W}}^{\frac{1}{2},2}({\mathbb {R}},GL(N))\) such that for \(\Omega ^A :=A\Omega - d_\frac{1}{2} A\) we have
Moreover, we have
As an immediate corollary, we obtain
Corollary 1.2
(Conservation law) Assume \(u\in {\dot{W}}^{\frac{1}{2},2}({\mathbb {R}},{\mathbb {R}}^N)\cap (L^2+L^\infty )({\mathbb {R}},{\mathbb {R}}^N)\) and \(f\in {\dot{W}}^{-\frac{1}{2},2}({\mathbb {R}},{\mathbb {R}}^N)\) satisfy
and \(\Omega \) satisfies the condition of Theorem 1.1. Then there exists a matrix A such that for \(\Omega ^A :=A\Omega - d_\frac{1}{2} A\) we have
where \({(\Omega ^A)^*(x,y):=\Omega ^A(y,x)}\).
Theorem 1.1 is applicable to the half-harmonic map system as derived [17, Proposition 4.2], because of a localization result, see Proposition B.1.
With the methods of Theorem 1.1, we obtain the analogue of [27, Theorem I.5], our second main result.
Theorem 1.3
Assume \(\Omega _\ell \in L^2(\bigwedge \nolimits _{od}^1 {\mathbb {R}})\) is a sequence of antisymmetric vector fields, i.e., \((\Omega _{ij})_\ell = - (\Omega _{ji})_\ell \), weakly convergent in \(L^2\) to an \(\Omega \in L^2(\bigwedge \nolimits _{od}^1 {\mathbb {R}})\). Assume further that \(f_\ell \in {\dot{W}}^{-\frac{1}{2},2}({\mathbb {R}}, {\mathbb {R}}^N)\) converges strongly to f in \({\dot{W}}^{-\frac{1}{2},2}\), and assume that \(u_\ell \in (L^2+L^\infty ({\mathbb {R}}))\cap {\dot{W}}^{\frac{1}{2},2}({\mathbb {R}},{\mathbb {R}}^N)\) is a sequence of solutions to
such that \(\sup _{\ell }\left( \Vert u_\ell \Vert _{L^2+L^\infty ({\mathbb {R}})}+[u_\ell ]_{W^{\frac{1}{2},2}({\mathbb {R}})} \right) <\infty \). Then, up to taking a subsequence \(u_\ell \) converges weakly in \({\dot{W}}^{\frac{1}{2},2}({\mathbb {R}},{\mathbb {R}}^N)\) to some \(u\in {\dot{W}}^{\frac{1}{2},2}({\mathbb {R}},{\mathbb {R}}^N) \cap ((L^2 + L^\infty )({\mathbb {R}},{\mathbb {R}}^N))\), which is a solution to
Here, as usual, we denote
2 Preliminaries and useful tools
We follow the notation of [17] for the nonlocal operators. For readers convenience we recall it here. We write \({\mathcal {M}}({\mathbb {R}}^n)\) for the space of all functions \(f:{\mathbb {R}}^n \rightarrow {\mathbb {R}}\) measurable with respect to the Lebesgue measure \(\,\mathrm {d}x\) and \({\mathcal {M}}(\bigwedge \nolimits _{od}^1{\mathbb {R}}^n)\) for the space of vector fields \(F:{\mathbb {R}}^n \times {\mathbb {R}}^n \rightarrow {\mathbb {R}}\) measurable with respect to the \(\frac{\,\mathrm {d}x\,\mathrm {d}y}{|x-y|^n}\) measure, where “od" stands for “off diagonal”.
For two vector fields \(F,\, G \in {\mathcal {M}}(\bigwedge \nolimits _{od}^1 {\mathbb {R}}^n)\), the scalar product is defined as
For any \(p>1\) the natural \(L^p\)-space on vector fields \(F:{\mathbb {R}}^n\times {\mathbb {R}}^n \rightarrow {\mathbb {R}}\) is induced by the norm
and for \(D\subset {\mathbb {R}}^n\) we define
Let \(s\in (0,1)\). For \(f:{\mathbb {R}}^n \rightarrow {\mathbb {R}}\) we let the s-gradient \(d_s :{\mathcal {M}}({\mathbb {R}}^n)\rightarrow {\mathcal {M}}(\bigwedge \nolimits _{od}^1 {\mathbb {R}}^n)\) to be
Observe that with this notation we have
where
is the Gagliardo–Slobodeckij seminorm.
Let \(s\in (0.1)\) and \(F\in {\mathcal {M}}(\bigwedge \nolimits _{od}^1 {\mathbb {R}}^n)\). We define the fractional s-divergence in the distributional way
whenever the integrals converge.
With this notation we have \({\text {div}}_s d_s = (-\Delta )^s\), i.e.,
where the fractional Laplacian is defined as
A simple observation is the following
Lemma 2.1
Let \(F \in {\mathcal {M}}(\bigwedge \nolimits ^1_{od} {\mathbb {R}}^n)\) then we define
If \({\text {div}}_s F = 0\) then \({\text {div}}_s F^*= 0\).
Moreover, for any \(F \in {\mathcal {M}}(\bigwedge \nolimits ^1_{od} {\mathbb {R}}^n)\) and \(u\in {\mathcal {M}}({\mathbb {R}}^n)\) we have
and
whenever each term is well-defined.
Proof
We have
Thus,
As for the latter term, we have
Combining (2.3) with (2.4), we obtain (2.1). The proof of (2.2) is similar. \(\square \)
We also denote
We will be using the following “Sobolev embedding” theorem.
Theorem 2.2
Let \(s\in (0,1)\), \(t\in (s,1)\), and let \(p,\,p^*>1\) satisfy
where \(q>1\) with \(p^*>\frac{nq}{n+sq}\). Then we have
and for any \(r\in [1,\infty ]\)
For the proof see Appendix C.
We will also need the following Wente’s inequality from [17].
Lemma 2.3
([17, Corollary 2.3]) Let \(s\in (0,1)\), \(p>1\), and let \(p'\) be the Hölder conjugate of p. Assume moreover that \(F'\in L^p(\bigwedge \nolimits _{od}^1 {\mathbb {R}})\) and \(g\in W^{s,p'}({\mathbb {R}})\) with \({\text {div}}_s F=0\). Let R be a linear operator such that for some \(\Lambda >0\) satisfies
where \(L^{(2,\infty )}({\mathbb {R}})\) denote the weak \(L^2\) space. Then any distributional solution \(u\in {\dot{W}}^{\frac{1}{2},2}({\mathbb {R}})\) to
is continuous. Moreover, if \(\lim _{x \rightarrow \pm \infty } |u(x)|=0\), then we have the estimate
Our proof will also be based on the following choice of a good gauge.
Theorem 2.4
([17, Theorem 4.4]) For \(\Omega _{ij} = -\Omega _{ji} \in L^2(\bigwedge \nolimits ^1_{od}{\mathbb {R}})\), there exists \(P \in {\dot{W}}^{\frac{1}{2}}({\mathbb {R}},SO(N))\) such that
where
and
3 Proof of Theorem 1.1
In this section we prove Theorem 1.1. We will be looking for an A in the form \(A=(I+\varepsilon )P\), where P is chosen to be the good gauge from Theorem 2.4. The idea to take perturbation of rotations of the form \((I+\varepsilon )P\) has been taken from [28] in the context of local Schrödinger equations with antisymmetric potentials. This has been also exploited in [7].
Lemma 3.1
Assume that \(A = (I+\varepsilon ) P\).
Then for
we have
where \(R_\varepsilon \) is given by the formula
Proof
Recall that
Thus, applying this to \(d_\frac{1}{2} ((I+\varepsilon )P)(x,y)\) we get
Next we observe that
That is, plugging in (3.3) into (3.2) we get the claim for
\(\square \)
Lemma 3.2
Assume that we have \(\varepsilon \in L^\infty \cap {\dot{W}}^{1/2,2}({\mathbb {R}}),\, a\in {\dot{W}}^{1/2,2}({\mathbb {R}})\), and \(B\in L^2(\bigwedge \nolimits _{od}^1 {\mathbb {R}})\) satisfying the equations
and
with
Then, for sufficiently small \(\sigma \) we have \(a=const\).
Proof
We multiply (3.4) by \(P^T(y)\) from the right and take the \(\frac{1}{2}\)-divergence on both sides; then subtracting (3.5), we obtain
We use nonlocal Hodge decomposition Lemma A.1 and get the existence of functions \({\tilde{a}} \in {\dot{W}}^{\frac{1}{2},2}({\mathbb {R}})\), \({\tilde{B}}\in L^2(\bigwedge \nolimits _{od}^1 {\mathbb {R}})\) such that
and (recall \(|P|=1\))
Thus, taking the \(\frac{1}{2}\)-divergence in (3.8) we obtain
This gives \((-\Delta )^{\frac{1}{2}}{\tilde{a}} =0\), thus \({\tilde{a}}\) is constant and without loss of generality we can take \({\tilde{a}} = 0\), see also [11, Theorem 1.1]. Thus, (3.8) becomes
That is
Taking the \(\frac{1}{2}\)-divergence, we obtain by Lemma 2.1
since on the right-hand side we have a div-curl term we can apply fractional Wente’s inequality, Lemma 2.3, and obtain from (2.7)
Combining this with (3.9) and (3.6), we get
which implies for sufficiently small \(\sigma \) that
and thus \(a \equiv const\). \(\square \)
Now we will focus on showing that there exists a solution to the Eqs. (3.4) and (3.5). We will do this by using the Banach fixed point theorem.
Proposition 3.3
Let \(\Omega \in L^2(\bigwedge \nolimits _{od}^1 {\mathbb {R}})\) be antisymmetric. There is a number \(0<\sigma \ll 1\) such that the following holds:
Take \(P\in {\dot{W}}^{\frac{1}{2},2}({\mathbb {R}},SO(N))\) and \(\Omega ^P\in L^2(\bigwedge \nolimits _{od}^1{\mathbb {R}})\) from Theorem 2.4. Let us assume that
Then, there exist \(\varepsilon \in L^\infty \cap {\dot{W}}^{1/2,2}({\mathbb {R}})\), \(a\in {\dot{W}}^{1/2,2}({\mathbb {R}})\), and \(B\in L^2(\bigwedge \nolimits _{od}^1 {\mathbb {R}})\) that solve the equations
where \(R_\varepsilon \) is defined in (3.1).
Moreover, \(\varepsilon \) satisfies the estimate
We will need the following remainder terms estimates.
Lemma 3.4
We have the following estimates
and
Proof
We observe that for any \(\varphi \in C_c^\infty ({\mathbb {R}})\) we have
Let \({\mathcal {M}}\) be the Hardy–Littlewood maximal function and let \(\alpha \in (0,1)\). We will use the following fractional counterpart (for the proof see [31, Proposition 6.6])
of the well-known inequality, see [4, 14]
We begin with the estimate of the first term on the right-hand side of (3.16).
We observe that by (3.17) and by the symmetry of the integrals we obtain
Applying Hölder’s inequality (for Lorentz spaces), we obtain
where we used the notation from Sect. 2: for \(s\in (0,1)\) and \(q>1\) we write
Applying Theorem 2.2, (2.6) for \(t=\frac{1}{2}\), we get
Thus, combining (3.18), (3.19), and (3.20) we obtain
As for the second term of (3.16), we have
Applying once again (3.17), we obtain
Using Hölder’s inequality and then Sobolev embedding, we get
where for the estimate of the last term we used again Theorem 2.2, (2.6), with \(t=\frac{1}{2}\).
Combining (3.22), (3.23), and (3.24), we obtain
Finally, from (3.16), (3.21), and (3.25) we get
This finishes the proof of (3.14).
In order to prove (3.15) we observe
Thus, in order to conclude it suffices to apply the estimates (3.21) and (3.25). \(\square \)
Proof of Proposition 3.3
Let \(X = L^\infty \cap {\dot{W}}^{\frac{1}{2},2}({\mathbb {R}})\).
For any \(\varepsilon \in X\) we have \(A=(1+\varepsilon )P\in L^\infty \cap {\dot{W}}^{\frac{1}{2}}({\mathbb {R}})\), which implies \(A\Omega - d_\frac{1}{2} A \in L^2(\bigwedge \nolimits _{od}^1 {\mathbb {R}})\) and thus, from Lemma 3.1, we have
We apply for this term the nonlocal Hodge decomposition, Lemma A.1: given \(\varepsilon \in X\) we find \(a(\varepsilon )\in W^{\frac{1}{2},2}({\mathbb {R}})\) and \(B(\varepsilon )\in L^2(\bigwedge \nolimits _{od}^1{\mathbb {R}})\) with \({\text {div}}_\frac{1}{2} B(\varepsilon )=0\) satisfying
with the estimates
Similarly, if for any two \(\varepsilon _1,\, \varepsilon _2 \in X\) we consider the difference of the corresponding Eq. (3.26) we get
Now we define the mapping \(T:X \rightarrow X\) as the solution to
with \(\lim _{|x|\rightarrow \infty }T(\varepsilon )(x)=0\).
Using Lemma 2.1 Eq. (3.29) can be rewritten as
We used in the second inequality Lemma 2.1.
We observe that on the right-hand we have fractional div-curl-terms: \({\text {div}}_\frac{1}{2} {B(\varepsilon )} =0\) and \({\text {div}}_\frac{1}{2} (\Omega ^P)^*=0\). Let us denote
By Lemma 3.4, (3.14), the rest term in (3.30) satisfies
Thus, we may apply the nonlocal Wente’s lemma, i.e., Lemma 2.3 and obtain
Moreover, let \(\varepsilon _1,\, \varepsilon _2 \in X\), then we have
where in the last equality we have used again Lemma 2.1.
Again, we observe that
and from Lemma 3.4, (3.15), we may estimate the reminder term in (3.32)
where
Therefore, we may apply the nonlocal Wente’s Lemma 2.3 for Eq. (3.32) and obtain
Combining (3.35) with (3.28) and (3.34), we get
where in the last inequality we used (3.11).
Thus, taking \(\sigma \) small enough we obtain
for a \(0<\lambda <1\), which implies that T is a contraction. Consequently, by Banach fixed point theorem, there exists a unique \(\varepsilon \in X\), such that \(T(\varepsilon ) = \varepsilon \). That is we have a solution \(T(\varepsilon ) = \varepsilon \), which is a solution to
Moreover, combining (3.31) with (3.27) and (3.11) we obtain the following estimate on \(\varepsilon \)
which gives for sufficiently small \(\sigma \)
\(\square \)
Proof of Theorem 1.1
By Proposition 3.3 we obtain the existence of an \(\varepsilon \in L^\infty \cap {\dot{W}}^{\frac{1}{2},2}({\mathbb {R}})\), \(a\in {\dot{W}}^{\frac{1}{2},2}({\mathbb {R}})\), \(B\in L^2(\bigwedge \nolimits _{od}^1 {\mathbb {R}})\) with \({\text {div}}_\frac{1}{2} B =0\) satisfying the equations solution \(T(\varepsilon ) = \varepsilon \), which is a solution to
where \(P\in {\dot{W}}^{\frac{1}{2},2}({\mathbb {R}},SO(N))\) and \(\Omega ^P \in L^2(\bigwedge \nolimits _{od}^1 {\mathbb {R}})\) are taken from Theorem 2.4 and \([P]_{W^{1/2,2}({\mathbb {R}})} \precsim \Vert \Omega \Vert _{L^2(\bigwedge \nolimits _{od}^1 {\mathbb {R}})} \le \sigma \).
By Lemma 3.2 we have for sufficiently small \(\sigma \)
Thus, defining for \(\varepsilon \) from Proposition 3.3, \(A:=(I+\varepsilon ) P\), we have by Lemma 3.1
The invertibility of A follows from the invertibility of P and \(I+\varepsilon \). Finally, since \(A = (I+\varepsilon )P\), we obtain from (3.13) and (2.8) the estimates
and
This finishes the proof. \(\square \)
4 Weak convergence result: Proof of Theorem 1.3
Using Lemma 2.1, we obtain the following.
Lemma 4.1
Assume that \(\Omega \in L^2(\bigwedge \nolimits _{od}^1 {\mathbb {R}})\). Then \(u\in {\dot{W}}^{\frac{1}{2},2}({\mathbb {R}},{\mathbb {R}}^N)\cap (L^2+L^\infty ({\mathbb {R}}))\) is a solution to
if and only if for any invertible matrix-valued function \(A,A^{-1}\in L^\infty \cap {\dot{W}}^{\frac{1}{2},2}({\mathbb {R}},GL(N))\),
In a first step we prove the “local version” of Theorem 1.3.
Proposition 4.2
Let \(\sigma > 0\) be the number from Theorem 1.1. Let \(\{u_\ell \}_{\ell \in {{\mathbb {N}}}}\) be a sequence as in Theorem 1.3 of solutions to
Additionally, let us assume that for some bounded interval \(D \subset {\mathbb {R}}\) we have
Then
Proof
Let us define \(\Omega _{D,\ell } :=\chi _{D}(x) \chi _{D}(y) \Omega _\ell \in L^2(\bigwedge \nolimits _{od}^1 {\mathbb {R}})\). Then by (4.2) we have
By Theorem 1.1 for \(\Omega _{D,\ell }\) there exists a gauge \(A_\ell \) such that
where \(\Omega _{D,\ell }^{A_\ell }:=A_\ell \Omega _{D,\ell } - d_\frac{1}{2} A_\ell \).
Let \(D_1 \subset \subset D\) be an open set.
By assumption and Lemma 4.1, we have for any \(\psi \in C_c^\infty (D_1)\) and for \(\Omega _\ell ^{A_\ell }=A_\ell \Omega _\ell - d_{\frac{1}{2}}A_\ell \)
Here with a slight abuse of notation we write for the matrix product \(\left( f[A\psi ] \right) ^i :=\sum _{k} f^k[A^{ik} \psi ]\).
Let us denote \(\Omega _{D^c,\ell } :=\Omega _\ell - \Omega _{D,\ell }\). Then we have
By Lemma 2.1 and (4.4), we have \({\text {div}}_\frac{1}{2} \left( \left( \Omega _{D,\ell }^{A_\ell } \right) ^* \right) =0\), thus again by Lemma 2.1 we get \(\Omega _{D,\ell }^{A_\ell }\cdot d_{\frac{1}{2}} u_\ell = {\text {div}}_{\frac{1}{2}}\left( \left( \Omega _{D,\ell }^{A_\ell } \right) ^*u_\ell (x) \right) \). Therefore,
We will pass with \(\ell \rightarrow \infty \) in (4.5). Roughly speaking, the convergence of most of the terms will be a result of a combination of weak–strong convergence. We first observe that by Theorem 1.1 we have
Thus, \(\sup _\ell \Vert A_\ell \Vert _{{\dot{W}}^{\frac{1}{2},2}({\mathbb {R}})} <\infty \) and \(\sup _\ell \Vert A_\ell \Vert _{L^\infty }({\mathbb {R}}) < \infty \). Up to taking a subsequence we obtain
where we used the Rellich–Kondrachov’s compact embedding theorem and \(A\in L^\infty \cap {\dot{W}}^{\frac{1}{2},2}({\mathbb {R}},GL(N))\). By the pointwise a.e. convergence, we have \(\Vert A\Vert _{L^\infty ({\mathbb {R}})}\precsim 1+ \sigma \).
By (4.3) we also have up to a subsequence
where \(\Omega _D\in L^2(\bigwedge \nolimits _{od}^1 {\mathbb {R}})\).
By assumptions of the Theorem we also have, up to a subsequence,
where \(u\in {\dot{W}}^{\frac{1}{2},2}({\mathbb {R}},{\mathbb {R}}^N)\).
Let us choose a large \(R \gg 1\), such that in particular \(D_1 \subset B(R)\). We begin with the first term of (4.5).
Step 1. We claim that (up to a subsequence)
Indeed, we observe
By weak convergence of \(d_\frac{1}{2} u_\ell \) in \(L^2(\bigwedge \nolimits _{od}^1 {\mathbb {R}})\), we have
As for the first term on the right-hand side of (4.8), we observe that since \(\mathrm{supp\,}\psi \subset D_1\subset B(R)\),
By strong convergence in \(L^2\) of \(A_\ell \) on compact domains, we have
and (noting once again that \(\mathrm{supp\,}\psi \subset D_1\))
In the last inequality we used the fact that if \(x\in D_1\) and \(y\in {\mathbb {R}}\setminus B(R)\) then \(|x-y|\succsim 1 + |y|\).
For the last term of (4.10), we similarly use that if \(y \in \mathrm{supp\,}\psi \) and \(x \in {\mathbb {R}}\setminus B(R)\), then we have \(|x-y| \succsim 1+|x|\) with a constant independent of R.
So we have
By (4.10), (4.11), (4.12), and (4.13) we obtain the convergence of the first term on the right-hand side of (4.8), i.e.,
Thus, combining (4.8), (4.9), and (4.14) we obtain the claim (4.7).
Step 2. We claim that (up to a subsequence)
where \(\Omega _D^A :=A\Omega _D - d_\frac{1}{2} A\).
Indeed, we write
Now, in order to obtain
we split the integral in two
The first term on the right-hand side of (4.18) converges to zero as \(\ell \rightarrow \infty \). This follows from the weak convergence of \(\Omega _{D,\ell } \rightharpoonup \Omega _D\) in \(L^2(\bigwedge \nolimits ^1_{od}{\mathbb {R}})\), the fact that \(\Omega _{D,\ell } - \Omega _D\) is supported on \(D\times D\), and that \(A(y)u(x)d_\frac{1}{2}\psi (x,y)\chi _{D}(x) \chi _{D}(y) \in L^2(\bigwedge \nolimits ^1_{od}{\mathbb {R}})\) (the easy verification of the latter is left to the reader).
As for the second term on the right-hand side of (4.18), we begin with the observation that
To estimate the first term of the right-hand side of (4.19), we first note that the support of \(\Omega _{D,\ell }\) is \(D\times D\) and then we use Hölder’s inequality
Now we verify the convergence of the second term of the right-hand side of (4.19). Again we use that the support of \(\Omega _{D,\ell }\) is \(D\times D\) and thus by the strong convergence in \(L^2\) of \(u_\ell \) on compact domains we have
We also claim that
To verify this statement, we divide the integral in two
The second term on the right-hand side of (4.23) converges to zero as \(\ell \rightarrow \infty \), because \(d_\frac{1}{2} A\rightharpoonup d_\frac{1}{2} A\) weakly in \(L^2(\bigwedge \nolimits ^1_{od} {\mathbb {R}})\) and \(u(x)d_\frac{1}{2} \psi (x,y) \in L^2(\bigwedge \nolimits ^1_{od}{\mathbb {R}})\).
We verify the convergence of the first term on the right-hand side of (4.23). First we note that by the strong convergence of \(u_\ell \) in \(L^2\) on compact domains we have
and
Finally, we have since \(\mathrm{supp\,}\psi \subset D_1 \subset B(R)\)
This gives
Thus, the convergence of the first term of (4.23) follows from (4.24), (4.25), and (4.27). We proved (4.22).
Now (4.15) follows from (4.16) combined with (4.17) and (4.22).
Step 3. We claim that
That is, we claim that for any \(\varphi \in C_c^\infty ({\mathbb {R}})\) we have
We write
As for the second term of (4.29), we observe that by weak convergence of \(d_\frac{1}{2} A_\ell \) in \(L^2(\bigwedge \nolimits _{od}^1 {\mathbb {R}})\) we have
As for the first term of (4.29), we proceed exactly as in Step 1 and obtain
This finishes the proof of (4.28).
Step 4. We claim that (up to a subsequence)
where \(\Omega _{D^c} = \Omega - \Omega _D\) and \(\Omega \in L^2(\bigwedge \nolimits _{od}^1 {\mathbb {R}})\) is the one given in the assumptions of the theorem.
Indeed, since \(\Omega _{D^c,\ell }(x,y)=0\) whenever both \(x,y \in D\) we have by the support of \(\psi \),
We set
and
We claim that we have the strong convergence
Indeed, we have
For the first term of the right-hand side of (4.33), we take \(R \gg 1\), such that in particular \(\mathrm{supp\,}\psi \subset D_1 \subset \subset D \subset B(R)\) and estimate
Now, for the second term of the right-hand side of (4.34) we have
Thus, by the strong convergence on compact sets of \(u_\ell \) in \(L^2\) we obtain
Now we estimate the first term of the right-hand side of (4.34). We observe that for all large R, whenever \(x \in \mathrm{supp\,}\psi \) and \(y \not \in B(R)\), we have \(|x-y| \succsim 1+|y|\). Therefore,
Thus,
Combining (4.34) with (4.35) and (4.36), we obtain the convergence of the first term of the right-hand side of (4.33)
As for the second term of the right-hand side of (4.33), we observe that since \(A_\ell \rightarrow A\) pointwise almost everywhere, we have
Moreover, we have
and the right-hand side is independent of \(\ell \) and integrable. Thus, by dominated convergence theorem we have
Now, plugging (4.38) and (4.37) into (4.33) we establish (4.32).
Thus, (4.32) and a combination of the weak convergence of \(\Omega _{\ell ,D^c}\) and the strong convergence of \(F_\ell \) implies
This establishes (4.30).
Step 5. We claim that
Indeed, this holds because \(A_\ell \psi \) is uniformly bounded in \({{\dot{W}}}^{\frac{1}{2},2}\), \(A_\ell \psi \) converges weakly to \(A\psi \) in \({\dot{W}}^{\frac{1}{2},2}\), and by assumption \(f_\ell \rightarrow f\) in \({{W}}^{-\frac{1}{2},2}\).
Step 6. Passing to the limit.
Passing with \(\ell \rightarrow \infty \) in (4.5), using (4.7), (4.15), (4.30), and (4.39), we obtain
By (4.15) we know that \(\left( \Omega _{D}^{A} \right) ^*\) is \(\frac{1}{2}\)-divergence free and thus by Lemma 2.1 we have
which combined with (4.40) and formulas \(\Omega _D^A = A\Omega _D - d_\frac{1}{2}A\) and \(\Omega _{D^c}=\Omega -\Omega _D\) gives
This holds for any \(\psi \in C_c^\infty (D_1)\). By density we can invoke Lemma 4.1, which leads to the claim. \(\square \)
Corollary 4.3
Let \(u_\ell \), \(\Omega _\ell \), and \(f_\ell \) be as in Theorem 1.3. Let \(D\subset {\mathbb {R}}\). Then there exists a locally finite \(\Sigma \subset D\) such that
Proof
We follow in spirit the covering argument of Sacks–Uhlenbeck [30, Proposition 4.3 & Theorem 4.4].
By assumptions there is a number \(\Lambda >0\) such that \(\sup _{\ell \in {{\mathbb {N}}}} \Vert \Omega _\ell \Vert _{L^2(\bigwedge \nolimits _{od}^1{\mathbb {R}})}<\Lambda \).
Let \(\alpha \in {{\mathbb {N}}}\) and let \(\mathcal {B}_\alpha :=\{B(x_{i,\alpha },2^{-\alpha }):x_{i,\alpha }\in D\}\) be a family of balls such that \(D\subset \bigcup \mathcal {B}_\alpha \) and each point \(x\in D\) is covered at most \(\lambda \) times, and such that for a smaller radius we still have \(D \subset \bigcup _{i} B(x_{i,\alpha }, 2^{-\alpha -1})\). Then
Now, let \(\sigma >0\) be the number from Theorem 1.1, then there exists at most \(\frac{\Lambda \lambda }{\sigma }\) balls in \(\mathcal {B}_\alpha \) on which
Thus, by Proposition 4.2, we obtain that except for \(K<\frac{\Lambda \lambda }{\sigma } +1\) balls from \(\mathcal {B}_\alpha \) we have
Let us denote those balls by \(B(y_{i,\alpha }, 2^{-\alpha })\) for \(i=1,\dots , K\). Then by (4.42) we get
Since \(\bigcup _{\alpha \in {{\mathbb {N}}}}\left( D\setminus \bigcup _{i=1}^K {\overline{B}}(y_{i,\alpha },2^{-\alpha -1}) \right) = D\setminus \{x_1,\ldots ,x_K\}\), (4.43) holds for any \(\psi \in C_c^\infty (D\setminus \Sigma )\), where \(\Sigma :=\{x_1,\ldots ,x_K\}\). This gives the claim. \(\square \)
In order to conclude we will need a removability of singularities lemma, compare with [18, Proposition 4.7].
Lemma 4.4
Let \(u \in {\dot{W}}^{\frac{1}{2},2}({\mathbb {R}},{\mathbb {R}}^N)\), \(f \in L^1({\mathbb {R}},{\mathbb {R}}^N)\), and \(g \in W^{-\frac{1}{2},2}({\mathbb {R}})\). Assume that for some locally finite set \(\Sigma \subset D\) we have
Then
Proof
For simplicity of presentation let us assume that \(\Sigma =\{x_0\}\). By definition we have for any \(\varphi \in C^\infty _c(D\setminus \{x_0\})\)
Let \(\{\zeta _\ell \}_{\ell \in {{\mathbb {N}}}} \subset C_c^\infty (D,[0,1])\) be the sequence from Lemma D.1, i.e., such that for all \(\ell \in {{\mathbb {N}}}\) we have
for a \(0<\rho _\ell <R_\ell \rightarrow 0\) as \(\ell \rightarrow \infty \).
Now let \(\psi \in C^\infty _c(D)\) and then \(\psi _\ell :=\psi (1-\zeta _\ell )\in C_c^\infty (\Sigma \setminus \{x_0\})\) is an admissible test function and we have
We have
Thus, by (4.44) and by the absolute continuity of the integral we have \(\lim _{\ell \rightarrow \infty } \mathcal {I}_\ell =0\).
Secondly,
by the absolute continuity of the integral.
Thus, passing with \(\ell \rightarrow \infty \) in (4.45) we get for any \(\psi \in C_c^\infty (D)\)
Lastly,
because, by (4.44), we have \([\psi \,\zeta _\ell ]_{W^{\frac{1}{2},2}} \xrightarrow {\ell \rightarrow \infty } 0\).
This finishes the proof. \(\square \)
Proof of Theorem 1.3
Combining Corollary 4.3 and Lemma 4.4, we obtain the claim. \(\square \)
Notes
The decomposition is unique if we normalize a
References
Adams, D.R., Hedberg, L.I.: Function Spaces and Potential Theory. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 314. Springer, Berlin (1996)
Blatt, S., Reiter, P., Schikorra, A.: Harmonic analysis meets critical knots. Critical points of the Möbius energy are smooth. Trans. Am. Math. Soc. 368(9), 6391–6438 (2016)
Blatt, S., Reiter, P., Schikorra, A.: On O’hara knot energies I: regularity for critical knots. J. Differ. Geom. (Accepted) (2019)
Bojarski, B., Hajłasz, P.: Pointwise inequalities for Sobolev functions and some applications. Studia Math. 106(1), 77–92 (1993)
Coifman, R., Lions, P.-L., Meyer, Y., Semmes, S.: Compensated compactness and Hardy spaces. J. Math. Pures Appl. (9) 72(3), 247–286 (1993)
Da Lio, F.: Compactness and bubble analysis for 1/2-harmonic maps. Ann. Inst. H. Poincaré Anal. Non Linéaire 32(1), 201–224 (2015)
Da Lio, F., Laurain, P., Rivière, T.: Neck energies for nonlocal systems with antisymmetric potentials, in preparation
Da Lio, F., Pigati, A.: Free boundary minimal surfaces: a nonlocal approach. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 20(2), 437–489 (2020)
Da Lio, F., Rivière, T.: Sub-criticality of non-local Schrödinger systems with antisymmetric potentials and applications to half-harmonic maps. Adv. Math. 227(3), 1300–1348 (2011)
Da Lio, F., Rivière, T.: Three-term commutator estimates and the regularity of \(\frac{1}{2}\)-harmonic maps into spheres. Anal. PDE 4(1), 149–190 (2011)
Fall, M.M.: Entire \(s\)-harmonic functions are affine. Proc. Am. Math. Soc. 144(6), 2587–2592 (2016)
Fefferman, C.: Inequalities for strongly singular convolution operators. Acta Math. 124, 9–36 (1970)
Gagliardo, E.: Caratterizzazioni delle tracce sulla frontiera relative ad alcune classi di funzioni in \(n\) variabili. Rend. Sem. Mat. Univ. Padova 27, 284–305 (1957)
Hajłasz, P.: Sobolev spaces on an arbitrary metric space. Potential Anal. 5(4), 403–415 (1996)
Hélein, F.: Régularité des applications faiblement harmoniques entre une surface et une sphère. C. R. Acad. Sci. Paris Sér. I Math. 311(9), 519–524 (1990)
Laurain, P., Rivière, T.: Angular energy quantization for linear elliptic systems with antisymmetric potentials and applications. Anal. PDE 7(1), 1–41 (2014)
Mazowiecka, K., Schikorra, A.: Fractional div-curl quantities and applications to nonlocal geometric equations. J. Funct. Anal. 275(1), 1–44 (2018)
Mazowiecka, K., Schikorra, A.: Minimal \(W^{s,\frac{n}{s}}\)-harmonic maps in homotopy classes. arXiv:2006.07138, (2020)
Mengesha, T., Schikorra, A., Yeepo, S.: Calderon-Zygmund type estimates for nonlocal PDE with Hölder continuous kernel. arXiv:2001.11944, (2020)
Millot, V., Pegon, M., Schikorra, A.: Partial regularity for fractional harmonic maps into spheres. arXiv: 1909.11466 (2020)
Millot, V., Sire, Y.: On a fractional Ginzburg–Landau equation and 1/2-harmonic maps into spheres. Arch. Ration. Mech. Anal. 215(1), 125–210 (2015)
Millot, V., Sire, Y., Wang, K.: Asymptotics for the fractional Allen–Cahn equation and stationary nonlocal minimal surfaces. Arch. Ration. Mech. Anal. 231(2), 1129–1216 (2019)
Monteil, A., Van Schaftingen, J.: Uniform boundedness principles for Sobolev maps into manifolds. Ann. Inst. H. Poincaré Anal. Non Linéaire 36(2), 417–449 (2019)
Moser, R.: Intrinsic semiharmonic maps. J. Geom. Anal. 21(3), 588–598 (2011)
Prats, M.: Measuring Triebel–Lizorkin fractional smoothness on domains in terms of first-order differences. J. Lond. Math. Soc. (2) 100(2), 692–716 (2019)
Prats, M., Saksman, E.: A \({\rm T}(1)\) theorem for fractional Sobolev spaces on domains. J. Geom. Anal. 27(3), 2490–2538 (2017)
Rivière, T.: Conservation laws for conformally invariant variational problems. Invent. Math. 168(1), 1–22 (2007)
Rivière, T.: The role of conservation laws in the analysis of conformally invariant problems. In: Topics in Modern Regularity Theory, volume 13 of CRM Series, pp. 117–167. Ed. Norm., Pisa (2012)
Runst, T., Sickel, W.: Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations. De Gruyter Series in Nonlinear Analysis and Applications, vol. 3. Walter de Gruyter & Co., Berlin (1996)
Sacks, J., Uhlenbeck, K.: The existence of minimal immersions of \(2\)-spheres. Ann. Math. (2) 113(1), 1–24 (1981)
Schikorra, A.: Boundary equations and regularity theory for geometric variational systems with Neumann data. Arch. Ration. Mech. Anal. 229(2), 709–788 (2018)
Seeger, A.: A note on Triebel-Lizorkin spaces. In: Approximation and Function Spaces (Warsaw, 1986), volume 22 of Banach Center Publ., pp 391–400. PWN, Warsaw (1989)
Stein, E.M.: The characterization of functions arising as potentials. Bull. Am. Math. Soc. 67(1), 102–104 (1961)
Triebel, H.: Theory of Function Spaces. Monographs in Mathematics, Springer, Basel (1983)
Triebel, H.: Local approximation spaces. Z. Anal. Anwendungen 8(3), 261–288 (1989)
Uhlenbeck, K.K.: Connections with \(L^{p}\) bounds on curvature. Commun. Math. Phys. 83(1), 31–42 (1982)
Acknowledgements
Funding is acknowledged as follows
\(\bullet \) (FDL) Swiss National Fund, SNF200020_192062: Variational Analysis in Geometry;
\(\bullet \) (KM) FSR Incoming Post-doctoral Fellowship;
\(\bullet \) (AS) Simons Foundation (579261).
The authors would also like to thank the anonymous referee for helpful comments.
Funding
Open access funding provided by Swiss Federal Institute of Technology Zurich.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendices
Appendix A: Nonlocal Hodge decomposition
Lemma A.1
Let \(p>1\), \(s\in (0,1)\), \(G \in L^p(\bigwedge \nolimits _{od}^1 {\mathbb {R}}^n)\) then there exists a decompositionFootnote 1
where \(a \in {\dot{W}}^{s,p}({\mathbb {R}}^n)\) and \(B \in L^p(\bigwedge \nolimits _{od} ^1{\mathbb {R}}^n)\) with \({\text {div}}_{s} B = 0\). Moreover,
Proof
Since \(G\in L^p(\bigwedge \nolimits _{od}^1 {\mathbb {R}}^n)\) we have \({\text {div}}_s G \in \left( W^{s,p'}({\mathbb {R}}^n) \right) ^*\), namely
Recall that for \(0<s<1\) and \(1\le p <\infty \) we have \({\dot{W}}^{s,p}({\mathbb {R}}^n) = {\dot{F}}^s_{p,p}({\mathbb {R}}^n)\) [34, 2.3.5]. Moreover, \({\text {div}}_s G \in {\dot{F}}^{-s}_{p,p}\), since \((-\Delta )^{-s}: {\dot{F}}^{{s}}_{p,p}({\mathbb {R}}^n) \rightarrow {\dot{F}}^{{-s}}_{p,p}({\mathbb {R}}^n)\) is an isomorphism [29, §2.6.2, Proposition 2, p.95]. In particular, there is a unique solution \(a \in {\dot{F}}^s_{p,p}({\mathbb {R}}^n)\) to the distributional equation
with
We have found \(a \in {\dot{F}}^{s}_{p,p}({\mathbb {R}}^n) = {\dot{W}}^{s,p}({\mathbb {R}}^n)\), and we have
The uniqueness of a up to a normalization assumption would follow by considering a difference of two solutions and an application of nonlocal Liouville theorem [11, Theorem 1.1].
Now define \(B:=G - d_s a\). We have
which finishes the proof. \(\square \)
Appendix B: Localization
The next proposition follows from a relatively straightforward localization results, see, e.g., [19].
Proposition B.1
Assume \(D_1 \subset \subset D_2 \subset \subset D' \subseteq D \subseteq {\mathbb {R}}\) open intervals and let \(u \in L^1({\mathbb {R}},{\mathbb {R}}^N) + L^\infty ({\mathbb {R}},{\mathbb {R}}^N)\cap {\dot{W}}^{{\frac{1}{2}},2}(D,{\mathbb {R}}^N)\) be a solution to
That is, assume
Let \(\eta \in C_c^\infty (D_1)\) and set \(v :=\eta u\) and \({\tilde{\Omega }}_{ij}(x,y) = \chi _{D_2}(x)\chi _{D_2}(y)\Omega _{ij}(x,y)\). Then
where \(\mathcal {G}\) is a bilinear form with the following estimates for any \(s \in (0,\frac{1}{2})\) and \(\varepsilon > 0\)
In particular we have
Proof
Let \(\varphi \in C_c^\infty ({\mathbb {R}})\). We have
Since \(\eta \varphi \in C_c^\infty (D')\) it is an admissible test function and we have from the Eq. (B.1)
Here,
Moreover, we have
where, because \(\mathrm{supp\,}v \subset D_1\),
That is we have
Furthermore, since
and \(\mathrm{supp\,}v\subset D_1\), we have
So if we set
and
then we have shown for any \(\varphi \in C_c^\infty ({\mathbb {R}})\),
It remains to estimate each \(\mathcal {G}_i(u,\varphi )\).
\({{\textit{Estimate of }\mathcal {G}_1:}}\) By the support of \(\eta \) we have
As for the first term, we have
We observe that for any \(p \in (1,\infty )\) and any \(\varepsilon > 0\) we have
Thus, for any \(\varepsilon >0\) and any \(s\in (0,\frac{1}{2})\) we have
We also have
Combining (B.3) with (B.4) (in which we use Poincarè inequality) and (B.5), we obtain
For the second term of (B.2), observe that for \(x \in D_1\) and \(y \in D \setminus D_2\) we have \(|x-y| \approx 1 + |y|\), so we have
Thus, by (B.2), (B.6), and (B.7) we get
\({{\textit{Estimate of }\mathcal {G}_2:}}\) Similarly as in (B.7), if \(x \in D_1\) and \(y \in {\mathbb {R}}\setminus D\) we have \(|x-y| \approx 1+|y|\), and thus
\({{\textit{Estimate of }\mathcal {G}_3:}}\) Using the support of v, observing again that \(|x-y| \succsim 1+|y|\) if \(y \in {\mathbb {R}}\setminus D_2\) and \(x \in D_1\), we get
This argument works for any \(s \in (0,\frac{1}{2})\).
\({{\textit{Estimate of }\mathcal {G}_4:}}\) We have
Now observe that \(|d_{\frac{1}{2}} \eta (x,y)|^2 \le \Vert \eta \Vert _{\mathrm{Lip\,}}^2 |x-y|\), thus
On the other hand
We also have
and
Thus, combining the estimates on \(\mathcal {G}_4\) we obtain
\(\square \)
Appendix C: A Sobolev inequality
Theorem C.1
Let \(s \in (0,1)\), \(p,q \in (1,\infty )\) and \(f \in L^p({\mathbb {R}}^n)\) then
-
(1)
$$\begin{aligned}{}[f]_{{\dot{F}}^s_{p,q}({\mathbb {R}}^n)} \precsim [f]_{W^{s}_{p,q}({\mathbb {R}}^n)}; \end{aligned}$$
-
(2)
if \(p > \frac{nq}{n+sq}\) then
$$\begin{aligned}{}[f]_{W^{s}_{p,q}({\mathbb {R}}^n)} \precsim [f]_{{\dot{F}}^s_{p,q}({\mathbb {R}}^n)}. \end{aligned}$$
The constants depend on s, p, q, n and are otherwise uniform.
While characterizations such as Theorem C.1 are well known for Besov spaces, for Triebel spaces this seems to have been known only for \(q=p\) (where it follows from the Besov-space characterization), \(q=2\) where it is a result due to Stein and Fefferman, [12, 33]. It was also known “for large s” [34, Section 2.5.10]. Although a conjecture that Theorem C.1 holds is very natural, quite surprisingly, to the best of our knowledge, the first time Theorem C.1 has been proven was recently by Prats and Saksman [26, Theorem 1.2] (see also [25] for further development), but see also [32, 35].
Corollary C.2
Let \(s \in (0,1)\), \(t \in (s,1)\) and \(p,p^*\in (1,\infty )\) where
If \(q \in (1,\infty )\) such that \(p^*> \frac{nq}{n+sq}\) we have
More precisely, in terms of Lorentz spaces we have for any \(r \in [1,\infty ]\),
Proof
From Theorem C.1 we have
We recall the Sobolev-embedding theorem for Triebel–Lizorkin spaces \({\dot{F}}^t_{p,\tilde{q}} \hookrightarrow {\dot{F}}^{{s}}_{{p^*},q}\) for any \(q,\tilde{q} \in (1,\infty )\) and \(s,\, t,\, p,\,p^*\) satisfying (C.1) (see, e.g., [34, Theorem 2.7.1 (ii)]). Thus,
As for the Lorentz-space estimate, we can argue by real interpolation. Indeed, fix \(s,q,p,p^*\). Observe that \(f \mapsto |\mathcal {D}_{s,q} f|\) is a sublinear operator. We can find \(p_1< p < p_2\) such that \(p_1\) and \(p_2\) are still admissible, and thus we have
From real interpolation we now obtain the Lorentz space claim. \(\square \)
Appendix D: A sequence of cut-off functions in the critical Sobolev space
For readers convenience we present here a proof of a well-known result, which essentially says that in the critical Sobolev space a point has zero capacity. See for example [1, Theorem 5.1.9]; compare also with a similar construction [23, Lemma 3.2].
Lemma D.1
There exists a sequence of functions with the following properties:
\(\{\zeta _\ell \}_{\ell \in {{\mathbb {N}}}} \subset C_c^\infty ({\mathbb {R}},[0,1])\) and for all \(\ell \in {{\mathbb {N}}}\) we have
for a sequence of radii \(0<\rho _\ell <R_\ell \rightarrow 0\) as \(\ell \rightarrow \infty \).
Proof
Let \(f(x) = \log \log \left( 1 + \frac{1}{|x|^2} \right) \in W^{1,2}(B^2_1,{\mathbb {R}})\) be an unbounded function. We define
Then,
The support of \(\nabla \tilde{Z}_k \) is the set
where
Now,
which follows from the fact that \(\nabla \tilde{Z}_k\in L^2(B_1^2)\) and that \(|\{x\in B_1^2:A_{k+1}\le |x| \le A_k\}|\) shrinks to zero.
Thus, we obtained a sequence of functions for which
By extending by zero we obtain a sequence \(Z_k\in W^{1,2}({\mathbb {R}}^2_+)\) with the properties
Defining now \(\zeta _k:=Z_k\big |_{{\mathbb {R}}}\) in the trace sense, we obtain by the trace inequality, [13]
Approximating \(\{\zeta _k\}_{k\in {{\mathbb {N}}}}\) by smooth functions, we obtain the desired sequence. \(\square \)
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Da Lio, F., Mazowiecka, K. & Schikorra, A. A fractional version of Rivière’s GL(n)-gauge. Annali di Matematica 201, 1817–1853 (2022). https://doi.org/10.1007/s10231-021-01180-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10231-021-01180-9