On the local well-posedness of the nonlinear heat equation associated to the fractional Hermite operator in modulation spaces

In this note we consider the nonlinear heat equation associated to the fractional Hermite operator $H^\beta =(-\Delta+|x|^2)^\beta$, $0<\beta\leq 1$. We show the local solvability of the related Cauchy problem in the framework of modulation spaces. The result is obtained by combining tools from microlocal and time-frequency analysis. As a byproduct, we compute the Gabor matrix of pseudodifferential operators with symbols in the H\"ormander class $S^m_{0,0}$, $m\in\mathcal{R}$.


Introduction and results
In this note we study the Cauchy problem for the nonlinear heat equation associated to the fractional Hermite operator with t ∈ [0, T ], T > 0, x ∈ R d , H β = (−∆ + |x| 2 ) β , β > 0, ∆ = ∂ 2 x 1 + . . . ∂ 2 x d , d ≥ 1. F is a scalar function on C, with F (0) = 0. The solution u(t, x) is a complex valued function of (t, x) ∈ R × R d . We will consider the case in which F is an entire analytic function (in the real sense).
The study of the well-posedness for the heat equation have been studied by many authors, see e.g. [18,8] and the many contributions by Wong, for instance [28,29]. In particular, heat equations associated to fractional Hermite operators were recently studied in [3], for related results see also [5]. Hermite multipliers are considered in [4], see also the textbook [22]. Recently the study of Cauchy problems in modulation spaces have been pursued by many authors, see the pioneering works [1,2]. Many deep results in this framework for nonlinear evolution equations have been obtained by B. Wang et al. in [24,25] and are also available in the textbook [27].
Following the spirit of [6,10], we shall prove the local existence and uniqueness of the solutions in modulation spaces to the Cauchy problem (1). The key arguments come from both microlocal and time-frequency analysis. In fact, we shall rely on the results related to spectral theory of globally elliptic operators developed by Helffer [17] to understand the properties of the fractional Hermite operators H β . Namely, they are pseudodifferential operators with Weyl symbols a β positive globally elliptic and in the Shubin classes Γ 2β 1 (see Definition 2.2 and the estimate (13) below).
The spectra decomposition of the Hermite operator H = −∆ + |x| 2 is given by where P k is the orthogonal projection of L 2 (R d ) onto the eigenspace corresponding to the eigenvalue (2k + d). Namely, the range of the operator P k is the space spanned by the Hermite functions Φ α in R d , with α multiindex in N d , such that |α| = k. The solution to the homogeneous Cauchy problem (1) (i.e., F = 0) can be formally written in terms of the heat semigroup related to H β We shall prove that the propagator K β (t) = e −tH β can be represented as a pseudodifferential operator with Weyl symbol in the Shubin class Γ 0 1 , with related semi-norms uniformly bounded with respect to the time variable t ∈ [0, T ], for any fixed T > 0.
After that we shall leave the microlocal techniques to come to time-frequency analysis. We perform a general study concerning the boundedness of Shubin τpseudodifferential operators with symbols in the Hörmander classes S m 0,0 (for τ = 1/2 we recapture the Weyl case). The outcomes are contained in Theorem 2.4 below (see also the subsequent corollary and remark).
The main tool here is to study the decay of their related Gabor matrix representations, which we shall also control by the semi-norms in S m 0,0 . We think that such result is valuable in and of itself.
We then use the special case of Weyl operators to study (1). The integral version of the problem (1) has the form To show that the Cauchy Problem (1) has a unique solution, we use a variant of the contraction mapping theorem (see Proposition 3.3 below).
As already mentioned, the function spaces used for our results are weighted modulation spaces M p,q m , 1 ≤ p, q ≤ ∞, introduced by H. Feichtinger in 1983 [11] (then extended to 0 < p, q ≤ ∞ in [12]). We refer the reader to Section 2 for their definitions and main properties.
The local well-posedness results for modulation spaces read as follows: and an entire real-analytic function on C with F (0) = 0. For every R > 0, there exists T > 0 such that for every u 0 in the ball B R of center 0 and radius R in M p,1 We actually do not know whether it is possible to obtain better results concerning the nonlinearity F (u) = λ|u| 2k u, we refer to the work [5] for a discussion on the topic.
The tools employed follow the pattern of similar Cauchy problems studied for other equations such as the Schrödinger, wave and Klein-Gordon equations [1, 2, 6, 10].

Function spaces and preliminaries
We denote by v a continuous, positive, submultiplicative weight function on R d , i.e., v(z 1 + z 2 ) ≤ v(z 1 )v(z 2 ), for all z 1 , z 2 ∈ R d . We say that m ∈ M v (R d ) if m is a positive, continuous weight function on R d v-moderate: m(z 1 + z 2 ) ≤ Cv(z 1 )m(z 2 ) for all z 1 , z 2 ∈ R d (or for all z 1 , z 2 ∈ Z d ). We will mainly work with polynomial weights of the type Observe that, for s < 0, v s is v |s| -moderate. Moreover, we limit to weights m with at most polynomial growth, that is there exists C > 0, s > 0 such that The main characters of time-frequency analysis are T x and M ξ , the so-called translation and modulation operators, defined by T x g(y) = g(y − x) and M ξ g(y) = e 2πiξy g(y). Let g ∈ S(R d ) be a non-zero window function in the Schwartz class and consider the short-time Fourier transform (STFT) V g f of a function/tempered distribution f in S ′ (R d ) with respect to the the window g: i.e., the Fourier transform F applied to f T x g.
For 1 ≤ p, q ≤ ∞ they are Banach spaces, whose norm does not depend on the window g, in the sense that different window functions in S(R d ) yield equivalent norms. Moreover, the window class S(R d ) can be extended to the modulation space Moreover, for m(x, ξ) = (1⊗v s )(x, ξ), we shall symply write, using the standard notation [11], . In our study, we will apply Minkowski's integral inequality to study the operator B in (3). Such inequalities does not hold whenever the indices p < 1 or q < 1, hence we shall limit ourselves to the cases 1 ≤ p, q ≤ ∞.
We do not know whether the local well-posedness is still valid in the quasi-Banach setting.
Recall that for [13,Thm. 11.3.7]). In other words, given any For this work we will use the inversion formula for the STFT (see [13,Proposition 11 and the equality holds in M p,q m (R d ). We also recall their inclusion relations: Other properties and more general definitions of modulation spaces can now be found in textbooks [7,13].
) is a complex vector space for the usual operations of addition and multiplication by complex numbers, and we have The notion of asymptotic expansion of a symbol a ∈ Γ m ρ (R 2d ) (cf. [21], Definition 23.2) reads as follows.
where m r = max j≥r m j we will write a ∼ ∞ j=0 a j and call this relation an asymptotic expansion of the symbol a.
The interest of the asymptotic expansion comes from the fact that every sequence of symbols (a j ) j with a j ∈ Γ m j 1 (R 2d ), the degrees m j being strictly decreasing and such that m j → −∞ determines a symbol in some Γ m 1 (R 2d ), that symbol being unique up to an element of S(R 2d ).
Thanks to the properties of H β above, we can exploit a result by Nicola and Rodino in [19,Theorem 4.5.1] to prove that the operator e −tH β is a pseudodifferential operator with Weyl symbol in the Shubin class Γ 0 1 , with uniform estimates with respect to t ∈ [0, T ], for any fixed T > 0.

2.2.
Gabor analysis of τ -pseudodifferential operators. For τ ∈ [0, 1], f, g ∈ L 2 (R d ), the (cross-)τ -Wigner distribution is defined by It can be used to define the τ -pseudodifferential operator with symbol σ via the formula For τ = 1/2 we recapture the Weyl operator. We want to consider τ -pseudodifferential operators with symbols σ in the Hörmander class S m 0,0 , m ∈ R, consisting of functions σ ∈ C ∞ (R 2d ) such that, for every α ∈ N 2d , The related semi-norms are denote by (19) |σ| N,m := sup We define the Gabor matrix of a linear continuous operator This is a slightly abuse of notation, since originally Gabor matrices we defined for time-frequency shifts π(λ), with λ varying in a lattice Λ ⊂ R 2d . Anyway, the almost diagonalization of Gabor matrices of pseudodifferential operators with symbols in the modulation space M ∞,1 (R 2d ) treated in [14] (and in many subsequent papers on the topic) are valid in both the continuous and discrete case. So we adopt this terminology in the continuous framework. For m = 0 we are reduced to the Hörmander class S 0 0,0 , whose Gabor matrix characterization for Weyl operators was shown in [15, Theorem 6.1], see also [20]. Even though m = 0 is our case of interest, for our goal we need to control such matrix by the semi-norms of S 0 0,0 . Hence, for further references, we shall formulate our result in the case of τ -pseudodifferential operators having symbols in the more general class S m 0,0 , m ∈ R. We shall use the following result for τ -pseudodifferential operators [9, Lemma 4.1] Lemma 2.1. Fix a non-zero window g ∈ S(R d ) and set Φ τ = W τ (g, g) for τ ∈ [0, 1]. Then, for σ ∈ S ′ R 2d , where z = (z 1 , z 2 ), w = (w 1 , w 2 ) and T τ and J are defined as follows: The Gabor matrix for a τ -pseudodifferential operator Op τ (σ) with symbol σ ∈ S m 0,0 enjoys the following decay.
Proof. Using the representation in (21) and (observe that the above integral is absolutely convergent since Φ t ∈ S(R 2d )). Now we estimate for every s ≥ 0 since Φ τ ∈ S(R 2d ). Choose s = |m| + 2d + 1. Then the submultiplicativity of · |m| allows us to control from above the right-hand side of the last inequality by Hence, for every N ∈ N we can find C(N) > 0 such that (23) is satisfied. This concludes the proof.
For τ ∈ (0, 1) we observe that T τ (z, w) ≍ z + w , hence the matrix decay can be controlled by a function which does not depend on the τ -quantization. Namely, ∈ (0, 1). Consider a τ -pseudodifferential operator Op τ (σ) with symbol σ ∈ S m 0,0 . Then, for every N ∈ N, there exists C = C(N) > 0 such that Remark 2.2. We may expect that pseudodifferential operators in the Hörmander class S m 0,0 , m ∈ R, can be characterized via the Gabor matrix in (23), extending the case m = 0 already shown in [15]. To be precise, we should have Studying the Gabor matrix decay for M ∞ · m ⊗ · s and following the pattern of the proofs as in the paper [15] one should get the result easily. Since this subject is outside the scope of the paper, we will write the details in a separate work.
The results in the previous yields the boundedness of the Weyl operator e −tH β on modulation spaces.
such that g 2 = 1. Then using the inversion formula for u 0 in (8) we can write e −tH β π(z)g, π(w)g V g u 0 (z) dz ≤ R 2d m(w)| e −tH β π(z)g, π(w)g ||V g u 0 (z)| dz In the previous section we showed that e −tH β is a Weyl operator with symbol b(t, ·) in Γ 0 1 with semi-norms uniformly bounded w.r.t. t ∈ [0, T ]. The continuous embedding in (24) and Theorem 2.4 let us write Since m(w) v(w − z)m(z), and v(z) z s for some s > 0, we can write Choosing N such that 2N − s > 2d + 1 and using the convolution relations L 1 * L p,q ֒→ L p,q we obtain the claim.
Corollary 3.2. Consider 0 < β ≤ 1, s ∈ R. For any fixed T > 0 there exists C = C(T ) > 0 such that The same result holds by replacing the Sobolev space H s with the Shubin-Sobolev space Q s .
As already done in [10], in order to show the local existence of the solution we will make use of the following variant of the contraction mapping theorem (cf., e.g., [23, Proposition 1.38]).
for some C 0 > 0, and suppose to have a nonlinear operator F : T → N with F (0) = 0 and Lipschitz bounds for all u, v in the ball B µ := {u ∈ T : u T ≤ µ}, for some µ > 0. Then, for all u lin ∈ B µ/2 there exists a unique solution u ∈ B µ to the equation u = u lin + BF (u), with the map u lin → u Lipschitz continuous with constant at most 2.
Proof of Theorem 1.1. We apply Theorem 3.1 with T = 1, q = 1 and m(x, ξ) = ξ s . For every 1 ≤ p < ∞, the operator K β (t), in (3) is a bounded operator on M p,1 s (R d ), and there exists a C > 0 such that (29) K β (t)u 0 M p,1 s ≤ C u 0 M p,1 s , t ∈ [0, 1]. Notice that such result provides the uniformity of the constant C, when t varies in [0, 1]. Now the result follows by Proposition 3.3, with T = N = C 0 ([0, T ]; M p,1 s ), the nonlinear operator B in (3), where 0 < T ≤ 1 will be chosen later on. Here u lin := K β (t)u 0 is in the ball B µ/2 ⊂ T by (29), if µ is sufficiently large, depending on R. Using Minkowski's integral inequality and (29), we obtain (27). Namely, The proof of Condition (28) can be found in [6,Theorem 4.1]. Hence, by choosing T small enough we prove the existence, and also the uniqueness among the solution in T with norm O(R). Standard continuity arguments allow to eliminate the last constraint (see, e.g., [23,Proposition 3.8]). For p = ∞, by repeating the argument above, one can obtain well-posedness when the initial datum is in Observe that similar results were obtained in [18, Theorem 1.1]. We conclude this note by addressing the reader to open problems on the topic. First, it is till not clear whether better results can be obtained when considering the nonlinearity (30) F (u) = F k (u) = λ|u| 2k u = λu k+1ūk , λ ∈ C, k ∈ N.
In fact, this was the case for the wave and vibrating place equation, cf. [6,10].
Moreover, another open question is the well-posedness of the Cauchy problem (1) with initial datum u 0 ∈ M p,q s (R d ), 0 < p ≤ ∞, 0 < q ≤ 1. We may expect that the result holds true as well, but the techniques employed so far do not apply in this case.