Continuity of Gevrey-H\"ormander pseudo-differential operators on modulation spaces

Let $s\ge 1$, $\omega ,\omega_0\in \mathscr P_{E,s}^0$, $a\in \Gamma _{s}^{(\omega_0)}$, and let $\mathscr B$ be a suitable invariant quasi-Banach function space, Then we prove that the pseudo-differential operator $\operatorname{Op} (a)$ is continuous from $M(\omega_0\omega ,\mathscr B )$ to $M(\omega ,\mathscr B )$.


Introduction
In the paper we consider continuity properties for a class of pseudodifferential operators introduced in [2] when acting on a broad class of modulation spaces. The symbols of the pseudo-differential operators are smooth, should obey strong ultra-regularity of Gevrey or Gelfand-Shilov types, and are allowed to grow exponentially or subexponentially.
Related questions were considered in the framework of the usual distribution theory in [31], where pseudo-differential operators were considered, with symbols in S (ω 0 ) , the set of all smooth a which satisfies |∂ α a| ≤ C α ω 0 .
(0.1) (See [18] and Section 1 for notations.) In [31,Theorem 3.2] it was deduced that if B is a translation invariant BF-space, ω and ω 0 belong to P, i. e. moderate and polynomially bounded weights, and a ∈ S (ω 0 ) , then corresponding pseudo-differential operator, Op(a) is continuous from the modulation space M(ω 0 ω, B) to M(ω, B). The obtained result in [31] can also be considered as extensions of certain results in the pioneering paper [24] by Tachizawa. For example, for suitable restrictions on ω, ω 0 and B, it follows that [31,Theorem 3.2] covers [24, Theorem 2.1]. Several classical continuity properties follows from [31,Theorem 3.2]. For example, since S and S ′ are suitable intersections and unions, respectively, of modulation spaces at above, it follows that Op(a) is continuous on S and on S ′ when a ∈ S (ω 0 ) with ω 0 ∈ P.
Some further conditions on the symbols in S (ω 0 ) are required if corresponding pseudo-differential operators should be continuous on Gelfand-Shilov spaces, because of the imposed Gevrey regularity on the elements in such spaces. For elements a in Γ (ω 0 ) s and Γ (ω 0 ) 0,s , the condition (0.1) is replaced by refined Gevrey-type conditions of the form |∂ α a| ≤ Ch |α| α! s ω 0 , (0. 2) involving global constants C and h which are independent of the derivatives (cf. [2]). More precisely, Γ (ω 0 ) s consists of all smooth a such that (0.2) holds for some constants C > 0 and h > 0, and a belongs to Γ (ω 0 ) 0,s , whenever for every h > 0 there is a constant for some C > 0 (which depends on both a and h) such that (0.2) holds. In the case s ≥ 1, the set P in [31] of weight functions are essentially replaced by the broader classes P 0 E,s and P E,s in [2]. Here ω 0 ∈ P E,s whenever ω is v r -moderate for some r > 0, where v r = e r| · | 1 s , (0. 3) and ω 0 ∈ P E,s whenever ω is v r -moderate for every r > 0. We notice that Hence, despite that Γ for some ω ∈ P 0 E,s , and Γ (ω) 0,s s for some ω ∈ P E,s . In [2] it is proved that if ω 0 ∈ P E,s and a ∈ Γ (ω 0 ) 0,s , then corresponding pseudo-differential operators Op(a) is continuous on the Gelfand-Shilov space Σ s of Beurling type, and its distribution space Σ ′ s . If instead ω 0 ∈ P 0 E,s and a ∈ Γ (ω 0 ) s , then Op(a) is continuous on the Gelfand-Shilov space S s of Roumieu type, and its distribution space S ′ s . (Cf. Theorems 4.10 and 4.11 in [2].) In Section 2 we complement these continuity properties by deducing continuity properties for such pseudo-differential operators when acting on a broad family of modulation spaces. More precisely, if ω 0 , ω ∈ P 0 E,s , B is a suitable invariant quasi-Banach-Function space (QBFspace), M(ω, B) is the modulation space with respect to ω and B, and a ∈ Γ (ω) s , then we show that Op(a) is continuous from M(ω 0 ω, B) to M(ω, B), and that the same holds true with P E,s and Γ with P and S (ω) , our results in Section 2, when B is a Banach space, take the same form as the the main result Theorem 3.2 in [31]. Some of the results in Section 2 can therefore be considered as analogies of the results in [31] in the framework of ultra-distribution theory. We also remark that using the fact that Gelfand-Shilov spaces and their distribution spaces equal suitable intersections and unions of modulation spaces, the continuity results for pseudo-differential operators in [2] are straight-forward consequences of Theorems 2.5 and 2.8. We also refer to [16,19,20,25,26,32,35] and the references therein for more facts about pseudo-differential operators in framework of Gelfand-Shilov and modulation spaces.
The (classical) modulation spaces M p,q , p, q ∈ [1, ∞], as introduced by Feichtinger in [5], consist of all tempered distributions whose shorttime Fourier transforms (STFT) have finite mixed L p,q norm. It follows that the parameters p and q to some extent quantify the degrees of asymptotic decay and singularity of the distributions in M p,q . The theory of modulation spaces was developed further and generalized in [8][9][10]13], where Feichtinger and Gröchenig established the theory of coorbit spaces. In particular, the modulation space M p,q (ω) , where ω denotes a weight function on phase (or time-frequency shift) space, appears as the set of tempered (ultra-) distributions whose STFT belong to the weighted and mixed Lebesgue space L p,q (ω) .

Preliminaries
In this section we discuss basic properties for modulation spaces and other related spaces. The proofs are in many cases omitted since they can be found in [3-5, 8-10, 14, 27-30]. (1.1) Here f (θ) g(θ) means that f (θ) ≤ cg(θ) for some constant c > 0 which is independent of θ in the domain of f and g. If v can be chosen as polynomial, then ω is called a weight of polynomial type. The function v is called submultiplicative, if it is even and (1.1) holds for ω = v.
We let P E (R d ) be the set of all moderate weights on R d , and P(R d ) be the subset of P E (R d ) which consists of all polynomially moderate 3 functions on R d . We also let P E,s (R d ) (P 0 E,s (R d )) be the set of all weights ω in R d such that for some r > 0 (for every r > 0). We have where the last equality follows from the fact that if hold true for some r > 0 (for every r > 0) (cf. [15]).
is finite. Here the supremum should be taken over all α, β ∈ N d and x ∈ R d . Obviously S t s,h is a Banach space, contained in S , and which increases with h, s and t and S t s,h ֒→ S . Here and in what follows we use the notation A ֒→ B when the topological spaces A and B satisfy A ⊆ B with continuous embeddings. Furthermore, if s, t > 1 2 , or s = t = 1 2 and h is sufficiently large, then S s t,h contains all finite linear combinations of Hermite functions. Since such linear combinations are dense in S and in S s t,h , it follows that the dual ( , for such choices of s and t. The Gelfand-Shilov spaces S s t (R d ) and Σ s t (R d ) are defined as the inductive and projective limits respectively of S s t,h (R d ). This implies that The Gelfand-Shilov distribution spaces (S s t ) ′ (R d ) and (Σ s t ) ′ (R d ) are the projective and inductive limit respectively of (S s t,h ) ′ (R d ). This means that We remark that in [12] it is proved that For every admissible s, t > 0 and ε > 0 we have (1.6) From now on we let F be the Fourier transform which takes the form Here · , · denotes the usual scalar product on R d . The map F extends uniquely to homeomorphisms on and to a unitary operator on L 2 (R d ). Similar facts hold true when s = t and the Fourier transform is replaced by a partial Fourier transform.
Gelfand-Shilov spaces and their distribution spaces can in convenient ways be characterized by means of estimates of short-time Fourier transforms, (see e. g. [17,32,34]). We here recall the details and start by giving the definition of the short-time Fourier transform.
We have now the following characterisations of Gelfand-Shilov functions and their distributions.
Then the following is true: holds for some r > 0; , if and only if (1.7) holds for every r > 0. A proof of Theorem 1.1 can be found in e. g. [17,34] (cf. [17,Theorem 2.7]). The corresponding result for Gelfand-Shilov distributions is the following. We refer to [32,34] for the proof.
Then the following is true: ( (with the obvious modifications when p = ∞ and/or q = ∞). We set M p (ω) = M p,p (ω) , and if ω = 1, then we set M p,q = M p,q (ω) and M p = M p (ω) . The following proposition is a consequence of well-known facts in [5,11,14,33]. Here and in what follows, we let p ′ denotes the conjugate exponent of p, i. e.
Then the following is true: (ω) is a Banach space under the norm in (1.9), and different choices of φ give rise to equivalent norms; (2) if p 1 ≤ p 2 , q 1 ≤ q 2 and ω 2 ≤ Cω 1 for some constant C, then . For example, if C > 0 is a constant and Ω is a subset of Σ ′ 1 , then a M p,q (ω) ≤ C for every a ∈ Ω, means that the inequality holds for some choice of φ ∈ M r (v) \ 0 and every a ∈ Ω. Evidently, for any other choice of φ ∈ M r (v) \ 0, a similar inequality is true although C may have to be replaced by a larger constant, if necessary.
Let s, t ∈ R. Then the weights are common in the applications. It follows that they belong to P(R 2d ) for every s, t ∈ R. If ω ∈ P(R 2d ), then ω is moderated by any of the weights in (1.10) provided s and t are chosen large enough. We refer to [5,[8][9][10][11]14,22,33] for more facts about modulation spaces. (2) if f, g ∈ L r loc (R d ) satisfy g ∈ B and |f | ≤ |g|, then f ∈ B and f B ≤ C g B .
If v belongs to P E,s (R d ) (P 0 E,s (R d )) , then B in Definition 1.4 is called an invariant BF-space of Roumieu type (Beurling type) of order s.
We notice that the quasi-norm · B in Definition 1.4 should satisfy (1.12) By Akira and Rolewić in [1,21] it follows that there is an equivalent quasi-norm to the previous one which additionally satisfies (1.13) From now on we suppose that the quasi-norm of B has been chosen such that both (1.12) and (1.13) hold true. It follows from (2) (1.14) If r = 1, then B in Definition 1.4 is a Banach space, and the condition (2) means that a translation invariant QBF-space is a solid BF-space in the sense of (A.3) in [6]. The space B in Definition 1.4 is called an invariant BF-space (with respect to v) if r = 1, and Minkowski's inequality holds true, i. e.
is finite. Then it follows that L p,q 1 and L p,q 2 are translation invariant BF-spaces with respect to v = 1.
For translation invariant BF-spaces we make the following observation. Proposition 1.6. Assume that v ∈ P E (R d ), and that B is an invariant BF-space with respect to v such that (1.15) holds true. Then the convolution mapping (ϕ, f ) → ϕ * f from C ∞ 0 (R d ) × B to B extends uniquely to a continuous mapping from L 1 (v) (R d ) × B to B, and (1.15) holds true for any f ∈ B and ϕ ∈ L 1 (v) (R d ). The result is a straight-forward consequence of the fact that C ∞ 0 is dense in L 1 (v) . Next we consider the extended class of modulation spaces which we are interested in. B) when B = L p,q 1 (R 2d ) (cf. Example 1.5). It follows that many properties which are valid for the classical modulation spaces also hold for the spaces of the form M(ω, B). For example we have the following proposition, which shows that the definition of M(ω, B) is independent of the choice of φ when B is a Banach space. We omit the proof since it follows by similar arguments as in the proof of Proposition 11.3.2 in [14]. M(ω, B) is the same as in Definition 1.7, and let φ ∈ The quasi-Banach spaces here above is usually a mixed quasi-normed Lebesgue space, given as follows. Let E be a non-degenerate parallelepiped in R d which is spanned by the ordered basis κ(E) = {e 1 , . . . , e d }.
The basis e ′ 1 , . . . , e ′ d is called the dual basis of e 1 , . . . , e d . We observe that there is a matrix T E such that e 1 , . . . , e d and e ′ 1 , . . . , e ′ d are the images of the standard basis under T E and T E ′ = 2π(T −1 E ) t , respectively. In the following we let max q = max(q 1 , . . . , q d ) and min q = min(q 1 , . . . , q d ) when q = (q 1 , . . . , q d ) ∈ (0, ∞] d , and χ Ω be the characteristic function of Ω. is finite, and is called E-split Lebesgue space (with respect to p and κ(E)). Definition 1.10. Let E 0 ⊆ R d be a non-degenerate parallelepiped with dual parallelepiped E ′ 0 , and E ⊆ R 2d be a parallelepiped spanned by the ordered set κ(E) ≡ {e 1 , . . . , e 2d }. Then E is called a phase-shift split parallelepiped (with respect to E 0 ) if E is non-degenerate and d of the vectors {e 1 , . . . , e 2d } spans E 0 and the other d vectors is the corresponding dual basis which spans E ′ 0 . 1.5. Pseudo-differential operators. Next we recall some facts on pseudo-differential operators. Let A ∈ M(d, R) be fixed and let a ∈ Σ 1 (R 2d ). Then the pseudo-differential operator Op A (a) is the linear and continuous operator on Σ 1 (R d ), defined by the formula The definition of Op A (a) extends to any a ∈ Σ ′ 1 (R 2d ), and then Op t (a) is continuous from Σ 1 (R d ) to Σ ′ 1 (R d ). Moreover, for every fixed A ∈ M(d, R), it follows that there is a one to one correspondence between such operators, and pseudo-differential operators of the form Op A (a). (See e. g. [18].) If A = 2 −1 I, where I ∈ M(d, R) is the identity matrix, then Op A (a) is equal to the Weyl operator Op w (a) of a. If instead A = 0, then the standard (Kohn-Nirenberg) representation Op(a) is obtained.
(1.17) (Cf. [18].) The following special case of [35, Theorem 3.1] is important when discussing continuity of pseudo-differential operators when acting on quasi-Banach modulation spaces. Proposition 1.11. Let ω 1 , ω 2 ∈ P E (R 2d ) and ω 0 ∈ P E (R 2d ⊕ R 2d ) be such that Also let p ∈ (0, ∞] 2d , E be a phase-shift split parallelepiped in R 2d and let a ∈ M ∞,1 In the next section we discuss continuity for pseudo-differential operators with symbols in the following definition. (See also the introduction.) Definition 1.12. Let ω 0 be a weight on R d , and let s ≥ 0.
(1) The set Γ for some constant h > 0; (2) The set Γ 2. Continuity for pseudo-differential operators with symbols in Γ This gives an analogy to [31,Theorem 3.2] in the framework of operator theory and Gelfand-Shilov classes.
We need some preparations before discussing these mapping properties. The following result shows that for any weight in P E , there are equivalent weights that satisfy strong Gevrey regularity. Proposition 2.1. Let ω ∈ P E (R d ) and s > 0. Then there exists a weight ω 0 ∈ P E (R d ) ∩ C ∞ (R d ) such that the following is true: Proof. We may assume that s < 1 2 . It suffices to prove that (2) should hold for some h > 0. Let φ 0 ∈ Σ s 1−s (R d ) \ 0, and let φ = |φ 0 | 2 . Then φ ∈ Σ s 1−s (R d ), giving that We have where the last inequality follows (1.2) and the fact that φ is bounded by a super exponential function. This gives the first part of (2). The equivalences in (1) follows in the same way as in [30,32]. More precisely, by (1.2) we have In the same way, (1.3) gives and (1) as well as the second part of (2) follow.
The next result shows that Γ Then the following is true:

1)
for some ε > 0, if and only if
Proof. We shall follow the proof of Proposition 3.1 in [2]. We only prove (2), and then when (2.1) or (2.2) are true for every ε > 0. The other cases follow by similar arguments and are left for the reader. Assume that φ ∈ Σ s (R d ), ω ∈ P E,s (R d ) and that (2.1) holds for every ε > 0. Then for every x ∈ R d the function belongs to Σ s , and ω(x + y) e h 0 |y| 1 s ω(x) for some h 0 > 0. By a straight-forward application of Leibnitz formula and the facts that |∂ α φ(x)| ε |α| α! s e −h|x| 1 s and ω(x + y) ω(x)e h 0 |y| 1 s for some h 0 > 0 and every ε, h > 0 we get for every ε, h > 0. In particular, By differentiation and the fact that φ ∈ Σ s we get for every ε 1 , ε 2 , ε 3 > 0. Since when ε 3 is chosen large enough compared to ε −1 2 , we get Since ω(y) ≤ ω(x)e h 0 |x−y| 1 s for some h 0 ≥ 0 and ε 1 can be chosen arbitrarily large, it follows from the last estimate that for every ε 2 > 0, and the result follows.
The following result is now a straight-forward consequence of the previous proposition and the definitions.
The following lemma is a consequence of Theorem 4.6 in [2].
We need some preparations for the proof, and start by recalling Minkowski's inequality in a somewhat general form. Assume that dµ is a positive measure, and that f ∈ L 1 (dµ; B) for some Banach space B. Then Minkowski's inequality asserts that We also need some lemmas.
for some h > 0. Since s ≥ 1, a straight-forward application of Faà di Bruno's formula on (2.6) gives for some h > 0. It follows from (2.5) and (2.
Furthermore, the following is true: for some h 0 , r 0 > 0; (2) there are functions H 0 ∈ C ∞ (R 3d ) and φ 0 ∈ S s (R d ) such that

10)
and such that (2.9) holds for some h 0 , r 0 > 0, with H 0 in place of H.
In order to prove (2) we notice that (2.12) shows that y → H 2 (x, ξ, y) is an element in S s (R d ) with values in Γ (1) s (R 2d ). By Lemma 2.6 there are H 3 ∈ C ∞ (R 3d ) and φ 0 ∈ S s (R d ) such that (2.12) holds for some h 0 , r 0 > 0 with H 3 in place of H 2 , and H 2 (x, ξ, y) = H 3 (x, ξ, y)φ 0 (−y). This is the same as (2), and the result follows.
By applying the B norm we get for some v ∈ P 0 E,s (R d ), This gives the result.