On heat equations associated with fractional harmonic oscillators

We establish some fixed-time decay estimates in Lebesgue spaces for the fractional heat propagator e-tHβ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e^{-tH^{\beta }}$$\end{document}, t,β>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t, \beta >0$$\end{document}, associated with the harmonic oscillator H=-Δ+|x|2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H=-\Delta + |x|^2$$\end{document}. We then prove some local and global wellposedness results for nonlinear fractional heat equations.

Strictly speaking, the corresponding fractional heat semigroup e −tH β is defined in terms of the spectral decomposition of the standard Hermite operator H = H 1 = −∆ + |x| 2 .To be precise, recall that where P k stands for the orthogonal projection of L 2 (R d ) onto the eigenspace corresponding to the eigenvalue (2k + d) -see Section 2.1 below for further details.As a consequence of the spectral theorem, we can consider the family of fractional powers of H defined by The heat semigroup e −tH β is then defined accordingly by While there is a wealth of literature on the semigroup e −t(−∆) β (see e.g., [19,31]), stimulated by the very wide range of physics-inspired models involving the fractional Laplacian [11,15], the current research of the semigroup e −tH β is rather limited, even in fundamental settings such as the Lebesgue spaces.This is particularly striking in view of the role played by the Hermite operator H and its fractional powers H β in several aspects of quantum physics and mathematical analysis [17,27].
The purpose of this note is to advance the knowledge of the fractional heat semigroup, in the wake of a research program initiated by the authors in [3].In particular, our main result is a set of fixed-time decay estimates for e −tH β in the Lebesgue space setting.
To the best of our knowledge, the dissipative estimate in Theorem 1.1 is new even for the Hermite operator (β = 1).We also stress that the time decay at infinity in (1.2) is sharp for any choice of Lebesgue exponents.Moreover, since the power of t is never positive for small time, we infer that there is a singularity near the origin for p = q.
It is worth emphasizing that the fractional Hermite propagator e −tH β is not a Fourier multiplier, hence we cannot rely on the arguments typically used to establish L p −L q spacetime estimates for the fractional heat propagator e −t(−∆) β -see for instance [19,Lemma 3.1].In fact, we will resort to techniques of pseudodifferential calculus to deal with the operators e −tH β and e −tH (cf.[20,Section 4.5]), and also to Bochner's subordination formula in order to express the heat semigroup e −tH β , 0 < β ≤ 1, in terms of solutions of the heat equation e −tH (see (3.4)).
As an application of Theorem 1.1, we investigate the wellposedness of First, let us highlight that, due to the occurrence of the quadratic potential |x| 2 , the problem (1.3) has no scaling symmetry.Nevertheless, the companion fractional heat equation is invariant under the following scaling transformation.For λ > 0, set If u(t, x) is a solution of (1.4) with initial datum u 0 (x), then u λ (t, x) is also a solution of (1.4) with initial datum u 0,λ (x).The L p space is invariant under the above scaling only Remark 1.1.Let us discuss some aspects of the previous results.In particular, we highlight some intriguing related problems that we plan to explore in future work.
-The sign in power type non-linearity (focusing or defocusing) will not play any role in our analysis.Therefore, we have chosen to consider the defocusing case for the sake of concreteness.-Using properties of Hermite functions and interpolation, in [32,Theorem 1.6] Wong proved that e −tH f L 2 (R) (sinh t) −1 f L p (R) for t > 0 and 1 ≤ p ≤ 2. We note that Theorem 1.1 recaptures and improves Wong's result.-It is known that (1.4) is ill-posed on Lebesgue spaces in the sub-critical regime, see [14].There is reason to believe that the same conclusion holds for (1.3).However, a thorough analysis of this problem is beyond the scope of this note.-It is expected that Theorem 1.1 could be useful in dealing with other types of non-linearities in (1.3), such as exponential and inhomogeneous type non-linearity (which are also extensively studied in the literature).-In Section 5 we discuss another application of Theorem 1.1, namely Strichartz estimates for the fractional heat semigroup.Our approach here relies on a standard technique (i.e., T T ⋆ method and real interpolation), whereas a refined phase-space analysis of H β is expected to reflect into better estimates.

Preliminary results
Notation.The symbol X Y means that the underlying inequality holds with a suitable positive constant factor: 2.1.On the fractional harmonic oscillator H β .Let us briefly review some facts concerning the spectral decomposition of the Hermite operator where In general, given a bounded function m : N → C, the spectral theorem allows us to define the operator m(H) such that In view of the Plancherel theorem for the Hermite expansions, m(H) is bounded on L 2 (R d ).We refer to [27] for further details, in particular for Hörmander multiplier-type results for m(H) on L p (R d ).
2.2.Some relevant function spaces.For the benefit of the reader we review some basic facts of time-frequency analysis -see for instance [1,7,13] for comprehensive treatments.
Recall that the short-time Fourier transform of a temperate distribution f ∈ S ′ (R d ) with respect to a window function 0 = g ∈ S(R d ) (Schwartz space) is defined by where the brackets •, • denote the extension to S ′ (R d ) × S(R d ) of the L 2 inner product.
Modulation spaces, introduced by Feichtinger [9], have proved to be extremely useful in a wide variety of contexts, ranging from analysis of PDEs to mathematical physics -among the most recent contributions, see e.g., [2,8,10,18,21].Modulation spaces are defined as follows.For 1 ≤ p, q ≤ ∞ we have We recall from [3, Theorem 1.1] some bounds for the fractional heat semigroup on modulation spaces.
We briefly recall some properties of the Shubin classes Γ s , which play a central role as symbol classes in the theory of pseudodifferential operators -we refer to [20] for additional details.For s ∈ R we define Γ s as the space of functions a ∈ C ∞ (R 2d ) satisfying the following condition: for every α ∈ N 2d there exists C α > 0 such that This space becomes a Fréchet space endowed with the obvious seminorms.
It is important for our purposes to recall that the fractional Hermite propagator is a pseudodifferential operator with symbol in a suitable Shubin class, as proved in [3, We also recall some facts concerning the so-called Shubin-Sobolev (also known as Hermite-Sobolev) spaces Q s , s ∈ R -see [23], [12, Theorem 2.1] for further details.In particular, Q s is the space of f ∈ S ′ (R d ) such that In view of the characterisation Q s = M 2,2 vs (see for instance [7,Lemma 4.4.19]),Hölder's inequality and the inclusion relations of Shubin-Sobolev spaces (see e.g., [7, Theorem 2.4.17]), it is well known that, for every 1 ≤ p, q ≤ ∞, if s is large enough, 3. Proof of Theorem 1.1

Proof of Part (1). It is known that
L p ֒→ M p,∞ and M q,1 ֒→ L q for 1 ≤ p, q ≤ ∞, see e.g., [7,25].In light of this embedding and Theorem 2.1, for t > 1 we obtain the desired estimate Let us consider now the case where 0 < t ≤ 1.In view of Proposition 2.1 we think of H β as a pseudodifferential operator with Weyl symbol a β ∈ Γ 2β , where for a suitable r ∈ Γ 2β−2 .We may further rewrite Note that the same conclusion holds for the Kohn-Nirenberg symbol of H β (see [20, Proposition 1.2.9]).Therefore, we assume hereinafter that the above functions a(x, ξ), r ′ (x, ξ) denote the Kohn-Nirenberg symbol of the corresponding operators.
It follows from [20, Theorem 4.5.1] that the heat semigroup e −tH β has a Kohn-Nirenberg symbol with the following structure1 : Let us focus now on the symbol By virtue of the Leibniz rule, the chain rule and (3.1), one can verify the estimates x 2β is a finite linear combination of terms of the type Similarly, since a ∈ Γ 2β satisfies (3.1), we have so that, arguing as above, hence we infer The claimed bound thus follows by the Leibniz rule.
To summarize, for every p ∈ (1, ∞) we have On the other hand, we also have and the integral kernel of the operator C t (x, D) given by is readily seen to satisfy K(x, y) ≤ Ct − d 2β .This gives the desired continuity result L 1 → L ∞ , while the remaining bounds follow by interpolation with the above L p → L q estimates.Remark 3.1.Note that some endpoint cases can be obtained in a straightforward way.For instance, from L 1 → L ∞ continuity we also obtain Remark 3.2.Some endpoint cases (e.g., if p, q ∈ {1, ∞}) are not covered in the results above.A deeper investigation of the kernel K(x, y) of C t (x, D) could likely give some result in this connection (for example L 1 → L 1 , L ∞ → L ∞ ), but it will not be essential for the applications to the nonlinear problem in Theorem 1.2.Nevertheless, the dispersive estimate L 1 → L ∞ is covered.(2).In order to prove the second claim in Theorem 1.1, some preparatory work is needed.First, we recast e −tH as the Weyl transform of a function on C d , which allows us to think of e −tH as a pseudodifferential operator.

Proof of Part
Recall that the Weyl transform W (F ) of a function F : C d → C is defined by where the symbol b(ξ, η) is the full inverse Fourier transform of F in both variables.In particular, the Weyl transform W (F ) is a pseudodifferential operator in the Weyl calculus with symbol b.
Let us highlight that the Weyl symbol of the Hermite semigroup e −tH is given by the function a t (x, ξ) = C d (cosh t) −d e −(tanh t)(|x| 2 +|ξ| 2 ) , see [28].Thus, In order to bound the above integral I, we first recast the latter expression in terms of convolution.Recall that the Fourier transform of the Gaussian function f (y) = e −πa|y| 2 with a > 0 is given by f (x) = a −d/2 e −π|x| 2 /a , and note that As a result, we have where we set g( for some constant C > 0 that depends only on d. Proof.Using Mehler's formula for the Hermite functions (see e.g., [27]), the kernel K t (x, y) of the semigroup e −tH is explicitly given by For 1 < p < q < ∞, set α = d(1/p − 1/q).Then we have from which we obtain the estimate Since the Riesz potential is bounded from L p to L q for 1 < p < q < ∞, we get To prove the remaining cases, we use the identity (3.3).We consider the case 1 ≤ q ≤ p ≤ ∞ first.Set 1 q = 1 p + 1 q and note that By (3.3) and invoking Hölder and Young's inequalities, we obtain . By (3.3) and Young inequality, we have This completes the proof.
Note that Lemma 3.1 essentially gives the desired fixed-time estimate of Theorem 1.1 (2) for β = 1 -see also Remark 3.3 below.In order to deal with the case 0 < β < 1, Bochner's subordination formula and the property of probability density function (see (3.6)) will play a crucial role.To be precise, Bochner's subordination formula allows us to express the heat semigroup e −t √ H in terms of solutions of the heat equation: The Macdonald function K ν (z) is defined, for z > 0, by A straightforward change of variables shows that Then z ν K ν (z) converges to 2 ν−1 Γ(ν) as z → 0.Moreover, it is known that K ν (z) has exponential decay at infinity (see [16]).Consider now the Gaussian kernel of the form We set p t (x, y) = p t (x − y), where g s is the Gaussian kernel defined above and η t ≥ 0 is the density function of the distribution of the β-stable subordinator at time t, see e.g., [4,5].Therefore, η t (s) = 0 for s ≤ 0 and, for 0 < β < 1, we have The fractional heat semi group e −tH β is thus given in terms of solutions of the heat equation: We are now ready to complete the proof of Theorem 1.1.
Proof of Theorem 1.1 -Part (2).The case t > 1 follows from the proof of Part (1) of Theorem 1.1, as it holds for all p, q ∈ [1, ∞].We then assume 0 < t ≤ 1 from now on.In view of the identity (3.6) and Lemma 3.1 for the case β = 1, we obtain where we set α = d|1/p − 1/q|.Splitting the integral above into two parts, the integral taken over [1, ∞) is bounded by The remaining integral is bounded by Changing the order of integration, and using (3.5), for a suitable constant C > 0 we obtain Finally, the change of variables v = u β gives the estimate This completes the proof for the case 0 < t ≤ 1.
Remark 3.3.We would like to have also a representation in the vein of (3.3) for the fractional heat propagator e −tH β with β > 1 in terms of the Weyl transform.On the other hand, we have a convolution formula for the classical fractional heat propagator e −t(−∆) β .Regretfully, we do not know how to get fixed-time estimates for β > 1 via the Weyl transform at the time.
Remark 3.4.Using the fact that e −tH commutes with the Fourier transform, i.e., e −tH f = e −tH f , one obtains where M > 0 is chosen in such a way that e −tH β u 0 Y T ≤ CM 1 ≤ M. Note that M depends only on u 0 Y T -in particular, it is independent of t.
Consider the mapping Φ : We shall show that in fact Φ is a mapping from B M +1 into B M +1 .Indeed, consider u ∈ B M +1 .By Theorem 1.1, for q ∈ {p, pγ}, we have Since q = p or q = pγ, γ > 1 and p > p β c , we have Therefore, we infer (4.1) If we take q = p or q = pγ in (4.1), then As a result, we conclude that Moreover, for a sufficiently small T > 0, we have This shows that Φ is a mapping from B M +1 into B M +1 , as claimed.
We now prove that Φ : By (4.2) and Hölder inequality, we have In light of the previous computation, for u, v ∈ B M +1 and q ∈ {p, pγ} we thus have for a constant C 3 > 0. By taking q = p or q = pγ in (4.3), we similarly obtain 2pβ > 0, for a sufficiently small T > 0 we have We have thus proved that the mapping Φ is the contraction mapping for a sufficiently small T .By Banach fixed point theorem, there exists a unique fixed point u of the mapping Φ in B M +1 and, in light of Duhamel's principle, the latter is a solution of (1.3).
Remark 4.1.We shall also mention that the result of Part (1) can be alternatively derived from the abstract theorem of Weissler [30,Theorem 1].To this aim, we define ) and e −sH K t = K t+s for t, s > 0. Then (1) follows by [30,Theorem 1].

Part (2)
-lower blow-up rate.Let u 0 ∈ L p (R d ) be such that T max < ∞, and let u ∈ C [0, T max ), L p (R d ) be the maximal solution of (1.3).Fix s ∈ [0, T max ) and let with w(0) = u(s).Then, as in the proof of Part (1), we claim that for some constant K > 0. If this were not the case, there would exist M > 0 such that and w would be defined on [0, T max − s] -in particular, u(T max ) would be well defined, a contradiction.Hence, (4.4) is verified, for any t ∈ [0, T max ) fixed and for all M > 0.
Let r be fixed once for all and set .
We observe that Suppose that ρ > 0 and M > 0 satisfy the inequality where K = K(γ, d, r) > 0 is a constant and can explicitly be computed.We claim that if In order to prove our claim, consider It is easy to realize that (X M , d) is a complete metric space.
Consider now the mapping Let u 0 and v 0 satisfy (4.5) and choose u, v ∈ X M .Clearly, we have Using Theorem 1.1 with exponents (p, q) = (r/γ, r), (4.2) and Hölder's inequality, we obtain Using this inequality, we get where Setting v 0 = 0 and v = 0 in (4.8) we have That is, J u 0 maps X M into itself.Letting u 0 = v 0 in (4.8), we note that d(J u 0 u(t), J u 0 v(t)) ≤ K M γ−1 d(u, v).
Since KM γ−1 < 1, then J u 0 is a strict contraction on X M .Therefore, J u 0 has a unique fixed point u in X M , which is a solution of (4.7).
Finally, using Theorem 1.1 with exponents (p, q) = d(γ−1) 2β , r , we see that if u 0 L p β c is sufficiently small then (4.5) is satisfied.

Concluding remarks
In this concluding section we illustrate another application of Theorem 1.1, that is a set of Strichartz estimates for the fractional heat propagator.We emphasize that Strichartz estimates are indispensable tools for a thorough study of the wellposedness theory for nonlinear equations -see e.g., [26,29].Since the proof is based on a standard machinery, via T T ⋆ method and real interpolation (see for instance [19,Lemma 3.2] and [33,Theorem 1.4] and the references therein), we omit the details.
(1) Let (q, p, r) be any admissible triplet and consider f ∈ L r (R d ).Then e −tH β f ∈ L q (I, L p (R d )) ∩ C b (I, L r (R d )) and there exists a constant C > 0 such that e −tH β f L q (I,L p ) ≤ C f L r .