Continuity of Gevrey–Hörmander pseudo-differential operators on modulation spaces

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where is a cuboid. It is well-posed when moving forward in time (t > 0), but ill-posed when moving backwards in time (t < 0) within the framework of classical function and distribution spaces. On the other hand, by [37,Example 2.16] it follows that the heat problem is well-posed for suitable Gelfand-Shilov distribution spaces and Gevrey classes when t < 0. Furthermore, if t > 0, then more precise continuity descriptions is deduced in the framework of such spaces compared to classical function and distribution spaces.
Pseudo-differential operators appear in natural and several ways when dealing with problems in partial differential equations, e.g. at above. As long as the analyses for such equations stays within the usual functions and distribution spaces, the symbols to related pseudo-differential operators often belong to the classical symbol classes, given in [21]. On the other hand, when discussing problems within Gelfand-Shilov spaces of functions and distributions, the conditions on the symbols need to be modified, to meet the stronger regularity in corresponding test function spaces.
In the paper we consider continuity properties for a class of pseudo-differential operators introduced in [5] when acting on a broad class of modulation spaces, given in [11,12,15]. 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.
More precisely, we consider pseudo-differential operators with symbols in the spaces (ω 0 ) s or in (ω 0 ) 0,s of Gevrey types which consist of all smooth a defined on the phase space such that |∂ α a| ≤ Ch |α| α! s ω 0 (1.2) holds for some constant h > 0 or every constant h > 0, respectively. Here ω 0 is a suitable weight on the phase space and the constant C > 0 is only depending on h. In [5] it is proved that if ω 0 e r | · | 1 s for some r > 0 and a ∈ (ω 0 ) 0,s , then corresponding pseudo-differential operators Op(a) is continuous on the Gelfand-Shilov space s of Beurling type, and its distribution space s . If instead ω 0 e r | · | 1 s holds for every r > 0 and a ∈ (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 [5]).
In Sect. 3 we enlarge this family of continuity results by deducing continuity properties for such pseudo-differential operators when acting on a broad family of modulation spaces. More precisely, let ω 0 be as above, and suppose that the weight ω satisfies Certain Gelfand-Shilov spaces and their distribution spaces are equal to suitable intersections and unions of modulation spaces (see e.g. [33,35]). This implies that the continuity properties for pseudo-differential operators when acting on Gelfand-Shilov spaces in [5] are straight-forward consequences of the results in Sect. 3. It is expected that the these results will also be useful in other situations. For example, these results are already applied in [2], where lifting properties between the modulation spaces above are established. We refer to [19,24,25,30,31,33,36] and the references therein for more facts about pseudo-differential operators in framework of Gelfand-Shilov and modulation spaces.
Related questions were considered in the framework of the usual distribution theory in [32], where the pseudo-differential operators should have symbols in Hörmander classes of the form S (ω 0 ) , the set of all smooth a which satisfies More precisely, let B be a translation invariant BF-space, and that the condition (1. 3) for ω and ω 0 are replaced by the stronger estimate for some N ≥ 0, and let a ∈ S (ω 0 ) . Then it is proved in [32,Theorem 3.2] that Op(a) is continuous from the modulation space M(ω 0 ω, B) to M(ω, B). The obtained result in [32] can also be considered as extensions of certain results in the pioneering paper [29]  We observe the different conditions between on one hand the symbol classes S (ω 0 ) in [32], and the other hand the classes

Weight functions
A weight or weight function on R d is a positive function in L ∞ loc (R d ).
Let ω and v be weights on R d . Then ω is called v-moderate or moderate, if 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 weight function v is called submultiplicative if it is even and (2.1) 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 functions on R d . We also let 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. [18]).
and that the topology for S σ are the projective and inductive limit respectively of (S σ s,h ) (R d ). This means that We remark that in [16] it is proved that For every admissible s, σ > 0 and ε > 0 we have 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 = σ and the Fourier transform is replaced by a partial Fourier transform.
If in addition f is an integrable function, then Gelfand-Shilov spaces and their distribution spaces can in convenient ways be characterized by means of estimates of Fourier and short-time Fourier transforms (see e.g. [6,20,33,35]). Here some extension of the map ( f , φ) → V φ f are also given, for example that this map is uniquely extendable to a continuous map from [1] for notations).

Modulation spaces
We recall that a quasi-norm · B of order r ∈ (0, 1] on the vector-space B over C is a nonnegative functional on B which satisfies and The vector space B is called a quasi-Banach space if it is a complete quasi-normed space. If B is a quasi-Banach space with quasi-norm satisfying (2.7) then on account of [3,27] there is an equivalent quasi-norm to · B which additionally satisfies (2.8) From now on we always assume that the quasi-norm of the quasi-Banach space B is chosen in such way that both (2.7) and (2.8) hold.
(with the obvious modifications when p = ∞ and/or q = ∞). We set M The following proposition is a consequence of well-known facts in [9,15,17,34].
Here and in what follows, we let p denotes the conjugate exponent of p, i.e.

A broader family of modulation spaces
As announced in the introduction we consider in Sect. 3 mapping properties for pseudodifferential operators when acting on a broader class of modulation spaces, which are defined by imposing certain types of translation invariant solid BF-space norms on the short-time Fourier transforms. (Cf. [9][10][11][12][13].) there is a constant C such that the following conditions are fulfilled: (2.10) (2) if f , g ∈ L r loc (R d ) satisfy g ∈ B and | f | ≤ |g|, then f ∈ B and If v belongs to P E,s (R d ) (P 0 E,s (R d )), then B in Definition 2.2 is called an invariant BF-space of Roumieu type (Beurling type) of order s.
It follows from (2) in Definition 2.2 that if f ∈ B and h ∈ L ∞ , then f · h ∈ B, and (2.11) If r = 1, then B in Definition 2.2 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 [10]. The space B in Definition 2.2 is called an invariant BF-space (with respect to v) if r = 1, and Minkowski's inequality holds true, i.e.
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.
3). 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 [17]. M(ω, B) be the same as in Definition 2.5, and let φ ∈ M 1 Next we recall the following result on completeness for M(ω, B). We refer to [35] for a proof of the first assertion and [22] for the second one.

Proposition 2.7
Let ω be a weight on R 2d , and let B be an invariant QBF-space with respect to the submultiplicative v ∈ P E (R 2d ). Then the following is true:

in addition B is a mixed quasi-norm space of Lebesgue types, then M(ω, B) is a quasi-Banach space; (2) if in addition B an invariant BF-space with respect to v, then M(ω, B) is a Banach space.
Finally we remark that certain modulation spaces without the condition on solidity are considered in [23].
Let E, p and ω be the same as in Definition 2.8. Then the discrete version

Pseudo-differential operators
Next we recall some facts on pseudo-differential operators. For any set , M(d, ) is the set of all d × d-matrices with entries in . Let A ∈ M(d, R) be fixed and let a ∈ 1 (R 2d ). Then the pseudo-differential operator Op A (a) is the linear and continuous operator on 1 (R d ), defined by the formula The definition of Op A (a) extends to any a ∈ 1 (R 2d ), and then Op A (a) is continuous Moreover, for every fixed A ∈ M(d, R), it follows that there is a one to one correspondence between such operators, and pseudo-differential  a(x, D) is obtained. If a 1 , a 2 ∈ 1 (R 2d ) and A 1 , A 2 ∈ M(d, R), then D x a 1 (x, ξ). (2.14) (Cf. [21].) The following special case of [36, Theorem 3.1] is important when discussing continuity of pseudo-differential operators when acting on quasi-Banach modulation spaces.

Proposition 2.10
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 split basis of 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 Sect. 1.)

Definition 2.11
Let ω 0 be a weight on R d , and let s ≥ 0.

Remark 2.12 We have
Hence, despite that for some ω ∈ P 0 E,s , and  (ω 0 , B). This gives an analogy to [32,Theorem 3.2] in the framework of operator theory and Gelfand-Shilov classes.

Continuity for pseudo-differential operators with symbols in
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.
such that the following is true: Proof We may assume that s < 1 2 . It suffices to prove that (2) should hold for some for every h > 0 and c > 0. We set ω 0 = ω * φ. Then where the last inequality follows (2.2) 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 e.g. [33]. More precisely, by (2.2) we have In the same way, (2.3) gives and (1) as well as the second part of (2) follow.
The next result shows that (ω) s and (ω) 0,s can be characterised in terms of estimates of short-time Fourier transforms.
, and let f ∈ S 1/2 (R d ). Then the following is true: satisfies

1)
for some h > 0, if and only if

Proof
We shall follow the proof of Proposition 3.1 in [5]. We only prove (2), and then when (3.1) or (3.2) are true for every ε > 0. The other cases follow by similar arguments and are left for the reader. Assume that φ ∈ s (R d ), ω ∈ P E,s (R d ) and that (3.1) holds for every ε > 0. Then for every x ∈ R d the function belongs to 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 for every ε, h > 0. In particular, for every h > 0. Since |V φ f (x, ξ)| = | F x (ξ )|, the estimate (3.2) follows from the second inequality in (3.3). This shows that if (3.1) holds for every ε > 0, then (3.2) holds for every ε > 0. Next suppose that (3.2) holds for every ε > 0. By Fourier's inversion formula we get By differentiation and the fact that φ ∈ 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 [5]. M(d, R), and that a 1 , a 2 ∈ 1 (R 2d ) are such that Op A 1 (a 1 ) = Op A 2 (a 2 ). Then and We have now the following result.
We need some preparations for the proof, and start with the following lemma.
for some h > 0 and r > 0. Then there are f 0 ∈ C ∞ (R d+d 0 ) and ψ ∈ S s (R d ) such that (3.4) holds with f 0 in place of f for some h > 0 and r > 0, and f (x, y) = f 0 (x, y)ψ(x). and for some h > 0. Since s ≥ 1, a straight-forward combination of Faà di Bruno's formula and (3.6) gives for some h > 0. It follows from (3.5) and (3.6) that if ψ = 1/v, then ψ ∈ S s (R d ).
Furthermore, if f 0 (x, y) = f (x, y)v 0 (x), then an application of Leibnitz formula gives for some h > 0, which gives the desired estimate on f 0 , The result now follows since it is evident that f (x, y) = f 0 (x, y)ψ(x).
For the next lemma we recall that for any a and ξ H (x, ξ, · ))ω(x, ξ).
Furthermore the following is true: for some h 0 , r 0 > 0; (3.10) and such that (3.9) holds for some h 0 , r 0 > 0, with H 0 in place of H .
Proof When proving the first part, we shall use some ideas in the proof of [32,Lemma 3.3]. By straight-forward computations we get where If η − ξ are taken as new variables of integrations, it follows that the right-hand side is equal to H (x, y, ξ). This gives the first part of the lemma. In order to prove (1), let 0 (x, ξ, y, η) = (x, ξ, y, η)ϕ(η), and let = F 4 0 , where F 4 0 is the partial Fourier transform of 0 (x, ξ, y, η) with respect to the η variable. Then it follows from the assumptions that for some h 0 , r 0 > 0, which shows that η → 0 (x, ξ, y, η) is an element in S s (R d ) with values in (1) s (R 3d ). As a consequence, satisfies |∂ α (x, ξ, y 1 , y 2 )| h |α| 0 α! s e −r 0 |y 2 | 1 s , for some h 0 , r 0 > 0. The assertion (1) now follows from the latter estimate, Leibnitz rule and the fact that H (x, ξ, y) = v(x − y) (x, ξ, x − y).
In order to prove (2) we notice that (3.9) shows that y → H (x, ξ, x − y) is an element in S s (R d ) with values in (1) s (R 2d ). By Lemma 3.6 there are H 2 ∈ C ∞ (R 3d ) and φ 0 ∈ S s (R d ) such that (3.9) holds for some h 0 , r 0 > 0 with H 2 in place of H , and H (x, ξ, x − y) = H 2 (x, ξ, x − 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.
By similar arguments as in the proof of Theorem 3.5 and Lemma 3.7 we get the following. The details are left for the reader. The asserted continuity now follows from these intersections and (4.2).