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

  • Joachim ToftEmail author
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Abstract

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 \({\text {Op}}(a)\) is continuous from \(M(\omega _0\omega ,{\mathscr {B}})\) to \(M(\omega ,{\mathscr {B}})\).

Keywords

Pseudo-differential operators Modulation spaces BF-spaces Gelfand–Shilov spaces 

Mathematics Subject Classification

35S05 47B37 47G30 42B35 

1 Introduction

Several problems in physics, engineering, partial differential equations, time–frequency analysis and signal processing are ill-posed in the framework of the classical function and distribution spaces
$$\begin{aligned} C_0^\infty ,\quad {\mathscr {S}},\quad C^\infty ,\quad {\mathscr {D}}',\quad {\mathscr {S}}',\quad \text {and}\quad {\mathscr {E}}' \end{aligned}$$
(1.1)
(see [21] or Sect. 2 for notations). For example, the Euler–Tricomi equation \(D^2_{t}f+tD^2_{x}f=0\), useful in the study of transonic flow, is ill-posed in such classical setting. On the other hand, the Euler–Tricomi equation is well-posed in the framework of suitable Gelfand–Shilov or Gevrey spaces and their spaces of ultra-distributions in place of (1.1). (See [4, 26].)
An other classical example concerns the heat problem
$$\begin{aligned} \partial _tf =\Delta _xf,\quad f(0,x)=f_0(x),\quad t\in {\mathbf {R}},\quad x\in \Omega , \end{aligned}$$
where \(\Omega \) 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 \(\Gamma ^{(\omega _0)}_s\) or in \(\Gamma ^{(\omega _0)}_{0,s}\) of Gevrey types which consist of all smooth a defined on the phase space such that
$$\begin{aligned} |\partial ^\alpha a | \le Ch^{|\alpha |}\alpha !^s \omega _0 \end{aligned}$$
(1.2)
holds for some constant \(h>0\) or every constant \(h>0\), respectively. Here \(\omega _0\) 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 \(\omega _0\lesssim e^{r|\, \cdot \, |^{\frac{1}{s}}}\) for some \(r>0\) and \(a\in \Gamma ^{(\omega _0)}_{0,s}\), then corresponding pseudo-differential operators \({\text {Op}}(a)\) is continuous on the Gelfand–Shilov space \(\Sigma _s\) of Beurling type, and its distribution space \(\Sigma _s'\). If instead \(\omega _0\lesssim e^{r|\, \cdot \, |^{\frac{1}{s}}}\) holds for every \(r>0\) and \(a\in \Gamma ^{(\omega _0)}_{s}\), then \({\text {Op}}(a)\) is continuous on the Gelfand–Shilov space \({\mathcal {S}}_s\) of Roumieu type, and its distribution space \({\mathcal {S}}_s'\) (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 \(\omega _0\) be as above, and suppose that the weight \(\omega \) satisfies
$$\begin{aligned} 0<\omega (X+Y)\lesssim \omega (X)e^{r|Y|^{\frac{1}{s}}},\qquad X,Y\in {\mathbf {R}}^{2d} \end{aligned}$$
(1.3)
for every \(r>0\) when \(a\in \Gamma ^{(\omega )}_{s}\) and satisfies (1.3) for some \(r>0\) when \(a\in \Gamma ^{(\omega )}_{0,s}\). Also let \({\mathscr {B}}\) be a suitable invariant quasi-Banach-function space (QBF-space) and let \(M(\omega ,{\mathscr {B}})\) be the modulation space with respect to \(\omega \) and \({\mathscr {B}}\). Then we prove that \({\text {Op}}(a)\) is continuous from \(M(\omega _0\omega ,{\mathscr {B}})\) to \(M(\omega ,{\mathscr {B}})\) (cf. Theorems 3.53.83.10 and Corollary 3.11). In the case when \({\mathscr {B}}\) is a Banach space, then the restrictions on \({\mathscr {B}}\) are given in Definition 2.2, while if \({\mathscr {B}}\) fails to be a Banach space, then suitable mixed Lebesgue quasi-norm estimates are imposed on the elements in \({\mathscr {B}}\).

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 \(S^{(\omega _0)}\), the set of all smooth a which satisfies
$$\begin{aligned} |\partial ^\alpha a | \le C _\alpha \omega _0. \end{aligned}$$
(1.4)
More precisely, let \({\mathscr {B}}\) be a translation invariant BF-space, and that the condition (1.3) for \(\omega \) and \(\omega _0\) are replaced by the stronger estimate
$$\begin{aligned} 0<\omega (X+Y)\lesssim \omega (X)(1+|Y|)^N,\quad X,Y\in {\mathbf {R}}^{2d} \end{aligned}$$
(1.5)
for some \(N\ge 0\), and let \(a\in S^{(\omega _0)}\). Then it is proved in [32, Theorem 3.2] that \({\text {Op}}(a)\) is continuous from the modulation space \(M(\omega _0\omega ,{\mathscr {B}})\) to \(M(\omega ,{\mathscr {B}})\). The obtained result in [32] can also be considered as extensions of certain results in the pioneering paper [29] by Tachizawa. For example, for suitable restrictions on \(\omega \), \(\omega _0\) and \({\mathscr {B}}\), it follows that [32, Theorem 3.2] covers [29, Theorem 2.1].

We observe the different conditions between on one hand the symbol classes \(S^{(\omega _0)}\) in [32], and the other hand the classes \(\Gamma _{s}^{(\omega _0)}\) and \(\Gamma _{0,s}^{(\omega _0)}\) in Sect. 3. The symbols in \(\Gamma _{s}^{(\omega _0)}\) and \(\Gamma _{0,s}^{(\omega _0)}\) must obey Gevrey conditions of order s, while for the symbols in \(S^{(\omega _0)}\), it is only required that they should be smooth. On the other hand, the weight \(\omega _0\) in (1.4) in the definition of \(S^{(\omega _0)}\) should satisfies (1.5) for some N, while for the definition of \(\Gamma _{s}^{(\omega _0)}\) and \(\Gamma _{0,s}^{(\omega _0)}\) this condition is relaxed into conditions of the form (1.5). This implies that the symbols in \(\Gamma _{s}^{(\omega _0)}\) and \(\Gamma _{0,s}^{(\omega _0)}\) are allowed to grow subexponentially, while the symbols in \(S^{(\omega _0)}\) are not allowed to grow faster than polynomials. Consequently, more restrictions on growth are imposed on \(S^{(\omega _0)}\) compared to \(\Gamma _{s}^{(\omega _0)}\) and \(\Gamma _{0,s}^{(\omega _0)}\).

In Sect. 4 we present some examples on continuity properties for pseudo-differential operators under considerations. These continuity properties are straight-forward consequences of the main results Theorems 3.5 and 3.8 in Sect. 3. Especially we explain continuity in the framework of Sobolev type spaces and weighted \(L^2\) spaces, with exponential weights, as well as continuity of such operators on \(\Gamma ^{(\omega )}_{0,s}\) spaces.

2 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 [7, 8, 9, 12, 13, 14, 15, 17, 34].

2.1 Weight functions

A weight or weight function on \({\mathbf {R}}^{d}\) is a positive function in \(L^\infty _{loc}({\mathbf {R}}^{d})\). Let \(\omega \) and v be weights on \({\mathbf {R}}^{d}\). Then \(\omega \) is called v-moderate or moderate, if
$$\begin{aligned} \omega (x_1+x_2)\lesssim \omega (x_1) v(x_2),\quad x_1,x_2\in {\mathbf {R}}^{d}. \end{aligned}$$
(2.1)
Here \(f(\theta )\lesssim g(\theta )\) means that \(f(\theta )\le cg(\theta )\) for some constant \(c>0\) which is independent of \(\theta \) in the domain of f and g. If v can be chosen as polynomial, then \(\omega \) is called a weight of polynomial type. The weight function v is called submultiplicative if it is even and (2.1) holds for \(\omega =v\).
We let \({\mathscr {P}}_E({\mathbf {R}}^{d})\) be the set of all moderate weights on \({\mathbf {R}}^{d}\), and \({\mathscr {P}}({\mathbf {R}}^{d})\) be the subset of \({\mathscr {P}}_E({\mathbf {R}}^{d})\) which consists of all polynomially moderate functions on \({\mathbf {R}}^{d}\). We also let \({\mathscr {P}}_{E,s}({\mathbf {R}}^{d})\) (\({\mathscr {P}}_{E,s}^0({\mathbf {R}}^{d})\)) be the set of all weights \(\omega \) in \({\mathbf {R}}^{d}\) such that
$$\begin{aligned} \omega (x_1+x_2)\lesssim \omega (x_1) e^{r|x_2|^{\frac{1}{s}}},\quad x_1,x_2\in {\mathbf {R}}^{d}. \end{aligned}$$
(2.2)
for some \(r>0\) (for every \(r>0\)). We have
$$\begin{aligned} {\mathscr {P}}\subseteq {\mathscr {P}}_{E,s_1}^0\subseteq {\mathscr {P}}_{E,s_1}\subseteq {\mathscr {P}}_{E,s_2}^0\subseteq {\mathscr {P}}_E \quad \text {when} \quad s_2 <s_1 \end{aligned}$$
and
$$\begin{aligned} {\mathscr {P}}_{E,s} = {\mathscr {P}}_E \quad \text {when} \quad s \le 1, \end{aligned}$$
where the last equality follows from the fact that if \(\omega \in {\mathscr {P}}_E({\mathbf {R}}^{d})\) (\(\omega \in {\mathscr {P}}_E^0({\mathbf {R}}^{d})\)), then
$$\begin{aligned} \omega (x+y)\lesssim \omega (x) e^{r|y|} \quad \text {and}\quad e^{-r|x|}\le \omega (x)\lesssim e^{r|x|},\quad x,y\in {\mathbf {R}}^{d} \end{aligned}$$
(2.3)
hold true for some \(r>0\) (for every \(r>0\)) (cf. [18]).

2.2 Gelfand–Shilov spaces

Let \(0<h,s,\sigma \in {\mathbf {R}}\) be fixed. Then \({\mathcal {S}}_{s,h}^\sigma ({\mathbf {R}}^{d})\) consists of all \(f\in C^\infty ({\mathbf {R}}^{d})\) such that
$$\begin{aligned} \Vert f\Vert _{{\mathcal {S}}_{s,h}^\sigma }\equiv \sup \frac{|x^\alpha \partial ^\beta f(x)|}{h^{|\alpha + \beta |}\alpha !^s\, \beta !^\alpha } \end{aligned}$$
(2.4)
is finite. Here the supremum should be taken over all \(\alpha ,\beta \in {\mathbf {N}}^d\) and \(x\in {\mathbf {R}}^{d}\).

Obviously \({\mathcal {S}}_{s,h}^\sigma \) is a Banach space, contained in \({\mathscr {S}}\), and which increases with h, s and \(\sigma \) and \({\mathcal {S}}_{s,h}^\sigma \hookrightarrow {\mathscr {S}}\). Here and in what follows we use the notation \(A\hookrightarrow B\) when the topological spaces A and B satisfy \(A\subseteq B\) with continuous embeddings.

The Gelfand–Shilov spaces \({\mathcal {S}}_{s}^\sigma ({\mathbf {R}}^{d})\) and \(\Sigma _{s}^\sigma ({\mathbf {R}}^{d})\) are defined as the inductive and projective limits respectively of \({\mathcal {S}}_{s,h}^\sigma ({\mathbf {R}}^{d})\). This implies that
$$\begin{aligned} {\mathcal {S}}_{s}^\sigma ({\mathbf {R}}^{d}) = \bigcup _{h>0}{\mathcal {S}}_{s,h}^\sigma ({\mathbf {R}}^{d}) \quad \text {and}\quad \Sigma _{s}^\sigma ({\mathbf {R}}^{d}) =\bigcap _{h>0} {\mathcal {S}}_{s,h}^\sigma ({\mathbf {R}}^{d}), \end{aligned}$$
(2.5)
and that the topology for \({\mathcal {S}}_{s}^\sigma ({\mathbf {R}}^{d})\) is the strongest possible one such that the inclusion map from \({\mathcal {S}}_{s,h}^\sigma ({\mathbf {R}}^{d})\) to \({\mathcal {S}}_{s}^\sigma ({\mathbf {R}}^{d})\) is continuous, for every choice of \(h>0\). The space \(\Sigma _s^\sigma ({\mathbf {R}}^{d})\) is a Fréchet space with seminorms \(\Vert \, \cdot \, \Vert _{{\mathcal {S}}_{s,h}^\sigma }\), \(h>0\). Moreover, \(\Sigma _s^\sigma ({\mathbf {R}}^{d})\ne \{ 0\}\), if and only if \(s+\sigma \ge 1\) and \((s,\sigma )\ne (\frac{1}{2},\frac{1}{2})\), and \({\mathcal {S}}_s^\sigma ({\mathbf {R}}^{d})\ne \{ 0\}\), if and only if \(s+\sigma \ge 1\). If s and \(\sigma \) are chosen such that \(\Sigma _s^\sigma ({\mathbf {R}}^{d})\ne \{ 0\}\), then \(\Sigma _s^\sigma ({\mathbf {R}}^{d})\) is dense in \({\mathscr {S}}({\mathbf {R}}^{d})\) and in \({\mathcal {S}}_s^\sigma ({\mathbf {R}}^{d})\). The same is true with \({\mathcal {S}}_s^\sigma ({\mathbf {R}}^{d})\) in place of \(\Sigma _s^\sigma ({\mathbf {R}}^{d})\) (cf. [16]).
The Gelfand–Shilov distribution spaces \(({\mathcal {S}}_s^\sigma )'({\mathbf {R}}^{d})\) and \((\Sigma _s^\sigma )'({\mathbf {R}}^{d})\) are the projective and inductive limit respectively of \(({\mathcal {S}}_{s,h}^\sigma )'({\mathbf {R}}^{d})\). This means that
$$\begin{aligned} \qquad \quad \;({\mathcal {S}}_s^\sigma )'({\mathbf {R}}^{d}) = \bigcap _{h>0}({\mathcal {S}}_{s,h}^\sigma )'({\mathbf {R}}^{d})\quad \text {and}\quad (\Sigma _s^\sigma )'({\mathbf {R}}^{d}) =\bigcup _{h>0}({\mathcal {S}}_{s,h}^\sigma )'({\mathbf {R}}^{d}). \quad \quad \quad {(2.5)'} \end{aligned}$$
We remark that in [16] it is proved that \(({\mathcal {S}}_s^\sigma )'({\mathbf {R}}^{d})\) is the dual of \({\mathcal {S}}_s^\sigma ({\mathbf {R}}^{d})\), and \((\Sigma _s^\sigma )'({\mathbf {R}}^{d})\) is the dual of \(\Sigma _s^\sigma ({\mathbf {R}}^{d})\) (also in topological sense). For conveniency we set
$$\begin{aligned} {\mathcal {S}}_s={\mathcal {S}}_s^s,\quad {\mathcal {S}}_s'=({\mathcal {S}}_s^s)',\quad \Sigma _s=\Sigma _s^s \quad \text {and}\quad \Sigma _s'=(\Sigma _s^s)'. \end{aligned}$$
For every admissible \(s,\sigma >0\) and \(\varepsilon >0\) we have
$$\begin{aligned} \Sigma _s^\sigma ({\mathbf {R}}^{d})&\hookrightarrow&{\mathcal {S}}_s^\sigma ({\mathbf {R}}^{d}) \hookrightarrow \Sigma _{s+\varepsilon }^{\sigma +\varepsilon }({\mathbf {R}}^{d}) \hookrightarrow {\mathscr {S}}({\mathbf {R}}^{d})\nonumber \\&\hookrightarrow&{\mathscr {S}}'({\mathbf {R}}^{d})\hookrightarrow (\Sigma _{s+\varepsilon }^{\sigma +\varepsilon })' ({\mathbf {R}}^{d}) \hookrightarrow ({\mathcal {S}}_s^\sigma )'({\mathbf {R}}^{d}) \hookrightarrow (\Sigma _s^\sigma )'({\mathbf {R}}^{d}). \end{aligned}$$
(2.6)
From now on we let \({\mathscr {F}}\) be the Fourier transform which takes the form
$$\begin{aligned} ({\mathscr {F}}{f})(\xi )= {\widehat{f}}(\xi ) \equiv (2\pi )^{-\frac{d}{2}}\int _{{\mathbf {R}}^{d}} f(x)e^{-i\langle x,\xi \rangle }\, dx \end{aligned}$$
when \(f\in L^1({\mathbf {R}}^{d})\). Here \(\langle \, \cdot \, ,\, \cdot \, \rangle \) denotes the usual scalar product on \({\mathbf {R}}^{d}\). The map \({\mathscr {F}}\) extends uniquely to homeomorphisms on \({\mathscr {S}}'({\mathbf {R}}^{d})\), from \(({\mathcal {S}}_s^\sigma )'({\mathbf {R}}^{d})\) to \(({\mathcal {S}}_\sigma ^s)'({\mathbf {R}}^{d})\) and from \((\Sigma _s^\sigma )'({\mathbf {R}}^{d})\) to \((\Sigma _\sigma ^s)'({\mathbf {R}}^{d})\). Furthermore, \({\mathscr {F}}\) restricts to homeomorphisms on \({\mathscr {S}}({\mathbf {R}}^{d})\), from \({\mathcal {S}}_s^\sigma ({\mathbf {R}}^{d})\) to \({\mathcal {S}}_\sigma ^s({\mathbf {R}}^{d})\) and from \(\Sigma _s^\sigma ({\mathbf {R}}^{d})\) to \(\Sigma _\sigma ^s({\mathbf {R}}^{d})\), and to a unitary operator on \(L^2({\mathbf {R}}^{d})\). Similar facts hold true when \(s=\sigma \) and the Fourier transform is replaced by a partial Fourier transform.
Let \(\phi \in {\mathcal {S}}_s^\sigma ({\mathbf {R}}^{d})\) be fixed. Then the short-time Fourier transform \(V_\phi f\) of \(f\in ({\mathcal {S}}_s ^\sigma )' ({\mathbf {R}}^{d})\) with respect to the window function \(\phi \) is the Gelfand–Shilov distribution on \({\mathbf {R}}^{2d}\), defined by
$$\begin{aligned} V_\phi f(x,\xi ) \equiv (f,\phi (\, \cdot \, -x)e^{i\langle \, \cdot \, ,\xi \rangle }). \end{aligned}$$
If in addition f is an integrable function, then
$$\begin{aligned} V_\phi f(x,\xi ) = (2\pi )^{-\frac{d}{2}}\int f(y)\overline{\phi (y-x)}e^{-i\langle y,\xi \rangle }\, dy. \end{aligned}$$
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 \((f,\phi )\mapsto V_\phi f\) are also given, for example that this map is uniquely extendable to a continuous map from \(({\mathcal {S}}_s^\sigma )'({\mathbf {R}}^{d})\times ({\mathcal {S}}_s^\sigma )'({\mathbf {R}}^{d})\) to \(({\mathcal {S}}_{s,\sigma }^{\sigma ,s})'({\mathbf {R}}^{2d})\) (see also [1] for notations).

2.3 Modulation spaces

We recall that a quasi-norm \(\Vert {\, \cdot \, }\Vert _{{\mathscr {B}}}\) of order \(r \in (0,1]\) on the vector-space \({\mathscr {B}}\) over \({\mathbf {C}}\) is a nonnegative functional on \({\mathscr {B}}\) which satisfies
$$\begin{aligned} \Vert f+g\Vert _{{\mathscr {B}}}\le & {} 2^{\frac{1}{r}-1}(\Vert f\Vert _{{\mathscr {B}}} + \Vert g\Vert _{{\mathscr {B}}}), \quad f,g \in {\mathscr {B}}, \nonumber \\ \Vert \alpha \cdot f\Vert _{{\mathscr {B}}}= & {} |\alpha | \cdot \Vert f\Vert _{{\mathscr {B}}}, \quad \alpha \in {\mathbf {C}}, \quad f \in {\mathscr {B}}\end{aligned}$$
(2.7)
and
$$\begin{aligned} \Vert f\Vert _{{\mathscr {B}}}&= 0\quad \Leftrightarrow \quad f=0. \end{aligned}$$
The vector space \({\mathscr {B}}\) is called a quasi-Banach space if it is a complete quasi-normed space. If \({\mathscr {B}}\) is a quasi-Banach space with quasi-norm satisfying (2.7) then on account of [3, 27] there is an equivalent quasi-norm to \(\Vert \, \cdot \, \Vert _{{\mathscr {B}}}\) which additionally satisfies
$$\begin{aligned} \Vert f+g\Vert _{{\mathscr {B}}}^r \le \Vert f\Vert _{{\mathscr {B}}}^r + \Vert g\Vert _{{\mathscr {B}}}^r, \quad f,g \in {\mathscr {B}}. \end{aligned}$$
(2.8)
From now on we always assume that the quasi-norm of the quasi-Banach space \({\mathscr {B}}\) is chosen in such way that both (2.7) and (2.8) hold.
Let \(\phi \in \Sigma _1({\mathbf {R}}^{d}){\setminus } 0\), \(p,q\in (0,\infty ]\) and \(\omega \in {\mathscr {P}}_E({\mathbf {R}}^{2d})\) be fixed. Then the modulation space \(M^{p,q}_{(\omega )}({\mathbf {R}}^{d})\) consists of all \(f\in \Sigma _1'({\mathbf {R}}^{d})\) such that
$$\begin{aligned} \Vert f\Vert _{M^{p,q}_{(\omega )}} \equiv \Big (\int \Big (\int |V_\phi f(x,\xi )\omega (x,\xi )|^p\, dx\Big )^{q/p} \, d\xi \Big )^{1/q} <\infty \end{aligned}$$
(2.9)
(with the obvious modifications when \(p=\infty \) and/or \(q=\infty \)). We set \(M^p_{(\omega )}=M^{p,p}_{(\omega )}\), and if \(\omega =1\), then we set \(M^{p,q}=M^{p,q}_{(\omega )}\) and \(M^{p}=M^{p}_{(\omega )}\).
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.
$$\begin{aligned} p' = {\left\{ \begin{array}{ll} \infty &{} \text {when}\ p\in (0,1]\\ \displaystyle {\frac{p}{p-1}} &{} \text {when}\ p\in (1,\infty )\\ 1 &{} \text {when}\ p=\infty . \end{array}\right. } \end{aligned}$$

Proposition 2.1

Let \(p,q,p_j,q_j,r\in (0,\infty ]\) be such that \(r\le \min (1,p,q)\), \(j=1,2\), let \(\omega ,\omega _1,\omega _2,v\in {\mathscr {P}}_E({\mathbf {R}}^{2d})\) be such that \(\omega \) is v-moderate, \(\phi \in M^r_{(v)}({\mathbf {R}}^{d}){\setminus } 0\), and let \(f\in \Sigma _1'({\mathbf {R}}^{d})\). Then the following is true:
  1. (1)

    \(f\in M^{p,q}_{(\omega )}({\mathbf {R}}^{d})\) if and only if (2.9) holds, i.e. \(M^{p,q}_{(\omega )}({\mathbf {R}}^{d})\) is independent of the choice of \(\phi \). Moreover, \(M^{p,q}_{(\omega )}\) is a quasi-Banach space under the quasi-norm in (2.9), and different choices of \(\phi \) give rise to equivalent quasi-norms. If in addition \(p,q\ge 1\), then \(M^{p,q}_{(\omega )}({\mathbf {R}}^{d})\) is a Banach space with norm (2.9);

     
  2. (2)
    if \(p_1\le p_2\), \(q_1\le q_2\) and \(\omega _2\lesssim \omega _1\), then
    $$\begin{aligned} \Sigma _1({\mathbf {R}}^{d})\subseteq M^{p_1,q_1}_{(\omega _1)}({\mathbf {R}}^{d}) \subseteq M^{p_2,q_2}_{(\omega _2)}({\mathbf {R}}^{d}) \subseteq \Sigma _1'({\mathbf {R}}^{d}). \end{aligned}$$
     

We refer to [9, 12, 13, 14, 15, 17, 28, 34] for more facts about modulation spaces.

2.4 A broader family of modulation spaces

As announced in the introduction we consider in Sect. 3 mapping properties for pseudo-differential 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].)

Definition 2.2

Let \({\mathscr {B}}\subseteq L^r_{loc}({\mathbf {R}}^{d})\) be a quasi-Banach space of order \(r\in (0,1]\), and let \(v \in {\mathscr {P}}_E({\mathbf {R}}^{d})\). Then \({\mathscr {B}}\) is called a translation invariant Quasi-Banach function space on \({\mathbf {R}}^{d}\) (with respect to v), or invariant QBF space on \({\mathbf {R}}^{d}\), if there is a constant C such that the following conditions are fulfilled:
  1. (1)
    if \(x\in {\mathbf {R}}^{d}\) and \(f\in {\mathscr {B}}\), then \(f(\, \cdot \, -x)\in {\mathscr {B}}\), and
    $$\begin{aligned} \Vert f(\, \cdot \, -x)\Vert _{{\mathscr {B}}}\le Cv(x)\Vert f\Vert _{{\mathscr {B}}}; \end{aligned}$$
    (2.10)
     
  2. (2)
    if \(f,g\in L^r_{loc}({\mathbf {R}}^{d})\) satisfy \(g\in {\mathscr {B}}\) and \(|f| \le |g|\), then \(f\in {\mathscr {B}}\) and
    $$\begin{aligned} \Vert f\Vert _{{\mathscr {B}}}\le C\Vert g\Vert _{{\mathscr {B}}}\text{. } \end{aligned}$$
     

If v belongs to \({\mathscr {P}}_{E,s}({\mathbf {R}}^{d})\,({\mathscr {P}}_{E,s}^0({\mathbf {R}}^{d}))\), then \({\mathscr {B}}\) 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 \(f\in {\mathscr {B}}\) and \(h\in L^\infty \), then \(f\cdot h\in {\mathscr {B}}\), and
$$\begin{aligned} \Vert f\cdot h\Vert _{{\mathscr {B}}}\le C\Vert f\Vert _{{\mathscr {B}}}\Vert h\Vert _{L^\infty }. \end{aligned}$$
(2.11)
If \(r=1\), then \({\mathscr {B}}\) 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 \({\mathscr {B}}\) 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.
$$\begin{aligned} \Vert f*\varphi \Vert _{{\mathscr {B}}}\lesssim \Vert f\Vert _{{\mathscr {B}}}\Vert \varphi \Vert _{L^1_{(v)}}, \qquad f\in {\mathscr {B}},\ \varphi \in C_0^\infty ({\mathbf {R}}^{d}). \end{aligned}$$
(2.12)

Example 2.3

Assume that \(p,q\in [1,\infty ]\), and let \(L^{p,q}_1({\mathbf {R}}^{2d})\) be the set of all \(f\in L^1_{loc}({\mathbf {R}}^{2d})\) such that
$$\begin{aligned} \Vert f\Vert _{L^{p,q}_1} \equiv \Big (\int \Big (\int |f(x,\xi )|^p\, dx\Big )^{q/p}\, d\xi \Big )^{1/q} \end{aligned}$$
if finite. Also let \(L^{p,q}_2({\mathbf {R}}^{2d})\) be the set of all \(f\in L^1_{loc}({\mathbf {R}}^{2d})\) such that
$$\begin{aligned} \Vert f\Vert _{L^{p,q}_2} \equiv \Big (\int \Big (\int |f(x,\xi )|^q\, d\xi \Big )^{p/q}\, dx \Big )^{1/p} \end{aligned}$$
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 2.4

Assume that \(v\in {\mathscr {P}}_E({\mathbf {R}}^{d})\), and that \({\mathscr {B}}\) is an invariant BF-space with respect to v such that (2.12) holds true. Then the convolution mapping \((\varphi ,f)\mapsto \varphi *f\) from \(C_0^\infty ({\mathbf {R}}^{d})\times {\mathscr {B}}\) to \({\mathscr {B}}\) extends uniquely to a continuous mapping from \(L^1_{(v)}({\mathbf {R}}^{d})\times {\mathscr {B}}\) to \({\mathscr {B}}\), and (2.12) holds true for any \(f\in {\mathscr {B}}\) and \(\varphi \in L^1_{(v)}({\mathbf {R}}^{d})\).

The result is a straight-forward consequence of the fact that \(C_0^\infty \) is dense in \(L^1_{(v)}\).

Next we consider the extended class of modulation spaces which we are interested in.

Definition 2.5

Assume that \({\mathscr {B}}\) is a translation invariant QBF-space on \({\mathbf {R}}^{2d}\), \(\omega \in {\mathscr {P}}_E({\mathbf {R}}^{2d})\), and that \(\phi \in \Sigma _1({\mathbf {R}}^{d}){\setminus } 0\). Then the set \(M(\omega ,{\mathscr {B}})\) consists of all \(f\in \Sigma _1'({\mathbf {R}}^{d})\) such that
$$\begin{aligned} \Vert f\Vert _{M(\omega ,{\mathscr {B}})} \equiv \Vert V_\phi f\, \omega \Vert _{{\mathscr {B}}} \end{aligned}$$
is finite.

Obviously, we have \( M^{p,q}_{(\omega )}({\mathbf {R}}^{d})=M(\omega ,{\mathscr {B}})\) when \({\mathscr {B}}=L^{p,q}_1({\mathbf {R}}^{2d})\) (cf. Example 2.3). It follows that many properties which are valid for the classical modulation spaces also hold for the spaces of the form \(M(\omega ,{\mathscr {B}})\). For example we have the following proposition, which shows that the definition of \(M(\omega ,{\mathscr {B}})\) is independent of the choice of \(\phi \) when \({\mathscr {B}}\) 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].

Proposition 2.6

Let \({\mathscr {B}}\) be an invariant BF-space with respect to \(v_0\in {\mathscr {P}}_E({\mathbf {R}}^{2d})\) for \(j=1,2\). Also let \(\omega ,v\in {\mathscr {P}}_E({\mathbf {R}}^{2d})\) be such that \(\omega \) is v-moderate, \(M(\omega ,{\mathscr {B}})\) be the same as in Definition 2.5, and let \(\phi \in M^1_{(v_0v)}({\mathbf {R}}^{d}){\setminus } 0\) and \(f\in \Sigma _1'({\mathbf {R}}^{d})\). Then \(f\in M(\omega ,{\mathscr {B}})\) if and only if \(V_\phi f\, \omega \in {\mathscr {B}}\), and different choices of \(\phi \) gives rise to equivalent norms in \(M(\omega ,{\mathscr {B}})\).

Next we recall the following result on completeness for \(M(\omega ,{\mathscr {B}})\). We refer to [35] for a proof of the first assertion and [22] for the second one.

Proposition 2.7

Let \(\omega \) be a weight on \({\mathbf {R}}^{2d}\), and let \({\mathscr {B}}\) be an invariant QBF-space with respect to the submultiplicative \(v\in {\mathscr {P}}_E({\mathbf {R}}^{2d})\). Then the following is true:
  1. (1)

    if in addition \({\mathscr {B}}\) is a mixed quasi-norm space of Lebesgue types, then \(M(\omega ,{\mathscr {B}})\) is a quasi-Banach space;

     
  2. (2)

    if in addition \({\mathscr {B}}\) an invariant BF-space with respect to v, then \(M(\omega ,{\mathscr {B}})\) is a Banach space.

     
Finally we remark that certain modulation spaces without the condition on solidity are considered in [23].

2.5 Mixed quasi-normed Lebesgue spaces

In most cases, the quasi-Banach spaces \({\mathscr {B}}\) are mixed quasi-normed Lebesgue space, which are defined next. Let \(E= \{ e_1,\ldots ,e_d \}\) be an orderd basis of \({\mathbf {R}}^{d}\). Then the corresponding lattice is
$$\begin{aligned} \Lambda _E =\{ \, j_1e_1+\cdots +j_de_d;\, (j_1,\ldots ,j_d)\in {\mathbf {Z}}^{d}\, \}, \end{aligned}$$
We define for each \({{\varvec{q}}}=(q_1,\ldots ,q_d)\in (0,\infty ]^d\)
$$\begin{aligned} \max ({{\varvec{q}}}) =\max (q_1,\ldots ,q_d) \quad \text {and}\quad \min ({{\varvec{q}}}) =\min (q_1,\ldots ,q_d). \end{aligned}$$

Definition 2.8

Let \(E = \{ e_1,\ldots ,e_d\}\) be an orderd basis of \({\mathbf {R}}^{d}\), \(\omega \) be a weight on \({\mathbf {R}}^{d}\), \({{\varvec{p}}}=(p_1,\ldots ,p_d)\in (0,\infty ]^{d}\) and \(r=\min (1,{{\varvec{p}}})\). If \(f\in L^r_{loc}({\mathbf {R}}^{d})\), then
$$\begin{aligned} \Vert f\Vert _{L^{{{\varvec{p}}}}_{E,(\omega )}}\equiv \Vert g_{d-1}\Vert _{L^{p_{d}}({\mathbf {R}})}, \end{aligned}$$
where \(g_k(z_k)\), \(z_k\in {\mathbf {R}}^{d-k}\), \(k=0,\ldots ,d-1\), are inductively defined as
$$\begin{aligned} g_0(x_1,\ldots ,x_{d})&\equiv |f(x_1e_1+\cdots +x_{d}e_d) \omega (x_1e_1+\cdots +x_{d}e_d)|, \end{aligned}$$
and
$$\begin{aligned} g_k(z_k)&\equiv \Vert g_{k-1}(\, \cdot \, ,z_k)\Vert _{L^{p_k}({\mathbf {R}})}, \quad k=1,\ldots ,d-1. \end{aligned}$$
The space \(L^{{{\varvec{p}}}}_{E,(\omega )}({\mathbf {R}}^{d})\) consists of all \(f\in L^r_{loc}({\mathbf {R}}^{d})\) such that \(\Vert f\Vert _{L^{{{\varvec{p}}}}_{E,(\omega )}}\) is finite, and is called E-split Lebesgue space (with respect to \({{\varvec{p}}}\) and \(\omega \)).
Let E, \({{\varvec{p}}}\) and \(\omega \) be the same as in Definition 2.8. Then the discrete version \(\ell ^{{{\varvec{p}}}}_{E,(\omega )}(\Lambda _E)\) of \(L^{{{\varvec{p}}}}_{E,(\omega )}({\mathbf {R}}^{d})\) is the set of all sequences \(a=\{ a(j)\} _{j\in \Lambda _E}\) such that the quasi-norm
$$\begin{aligned} \Vert a\Vert _{\ell ^{{{\varvec{p}}}}_{E,(\omega )}} \equiv \Vert f_a\Vert _{L^{{{\varvec{p}}}}_{E,(\omega )}},\qquad f_a=\sum _{j\in \Lambda _E} a(j)\chi _j, \end{aligned}$$
is finite. Here \(\chi _j\) is the characteristic function of \(j+\kappa (E)\), where \(\kappa (E)\) is the parallelepiped spanned by the basis E. We also set \(L^{{{\varvec{p}}}}_{E} = L^{{{\varvec{p}}}}_{E,(\omega )}\) and \(\ell ^{{{\varvec{p}}}}_{E} = \ell ^{{{\varvec{p}}}}_{E,(\omega )}\) when \(\omega =1\).

Definition 2.9

Let E be an ordered basis of the phase space \({\mathbf {R}}^{2d}\). Then E is called phase split if there is a subset \(E_0\subseteq E\) such that the span of \(E_0\) equals \(\{ \, (x,0)\in {\mathbf {R}}^{2d};\, x\in {\mathbf {R}}^{d}\, \}\), and the span of \(E_0\) and \(E{\setminus } E_0\) equals \(\{ \, (x,0)\in {\mathbf {R}}^{2d};\, \xi \in {\mathbf {R}}^{d}\, \}\) respective \(\{ \, (0,\xi )\in {\mathbf {R}}^{2d};\, \xi \in {\mathbf {R}}^{d}\, \}\).

2.6 Pseudo-differential operators

Next we recall some facts on pseudo-differential operators. For any set \(\Omega \), M(d, \(\Omega \)) is the set of all \(d \times d\)-matrices with entries in \(\Omega \). Let \(A\in {\mathbf {M}}(d,{\mathbf {R}})\) be fixed and let \(a\in \Sigma _1({\mathbf {R}}^{2d})\). Then the pseudo-differential operator \({\text {Op}}_A(a)\) is the linear and continuous operator on \(\Sigma _1({\mathbf {R}}^{d})\), defined by the formula
$$\begin{aligned}&({\text {Op}}_A(a)f)(x)\nonumber \\&\quad =(2\pi ) ^{-d}\iint a(x-A(x-y),\xi )f(y)e^{i\langle x-y,\xi \rangle }\, dyd\xi . \end{aligned}$$
(2.13)
The definition of \({\text {Op}}_A(a)\) extends to any \(a\in \Sigma _1'({\mathbf {R}}^{2d})\), and then \({\text {Op}}_A(a)\) is continuous from \(\Sigma _1({\mathbf {R}}^{d})\) to \(\Sigma _1'({\mathbf {R}}^{d})\). Moreover, for every fixed \(A\in {\mathbf {M}}(d,{\mathbf {R}})\), it follows that there is a one to one correspondence between such operators, and pseudo-differential operators of the form \({\text {Op}}_A(a)\). (See e.g. [21].) If \(A=2^{-1}I\), where \(I\in {\mathbf {M}}(d,{\mathbf {R}})\) is the identity matrix, then \({\text {Op}}_A(a)\) is equal to the Weyl operator \({\text {Op}}^w(a)\) of a. If instead \(A=0\), then the standard (Kohn–Nirenberg) representation \({\text {Op}}(a)=a(x,D)\) is obtained.
If \(a_1,a_2\in \Sigma _1'({\mathbf {R}}^{2d})\) and \(A_1,A_2\in {\mathbf {M}}(d,{\mathbf {R}})\), then
$$\begin{aligned} {\text {Op}}_{A_1}(a_1)={\text {Op}}_{A_2}(a_2) \quad \Leftrightarrow \quad a_2(x,\xi )=e^{i\langle (A_1-A_2)D_\xi ,D_x\rangle }a_{1}(x,\xi ). \end{aligned}$$
(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 \(\omega _1,\omega _2 \in {\mathscr {P}}_{E}({\mathbf {R}}^{2d})\) and \(\omega _0\in {\mathscr {P}}_{E}({\mathbf {R}}^{2d}\oplus {\mathbf {R}}^{2d})\) be such that
$$\begin{aligned} \frac{\omega _2(x,\xi )}{\omega _1 (y,\eta )} \lesssim \omega _0(x,\eta ,\xi -\eta ,y-x). \end{aligned}$$
(2.15)
Also let \({{\varvec{p}}}\in (0,\infty ]^{2d}\), E be a phase split basis of \({\mathbf {R}}^{2d}\) and let \(a\in M^{\infty ,1}_{(\omega _0)}({\mathbf {R}}^{2d})\). Then \({\text {Op}}_0(a)\) is continuous from \(M(L^{{{\varvec{p}}},E}, \omega _1)\) to \(M(L^{{{\varvec{p}}},E},\omega _2)\).

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 \(\omega _0\) be a weight on \({\mathbf {R}}^{d}\), and let \(s\ge 0\).
  1. (1)
    The set \(\Gamma ^{(\omega _0)}_{s}({\mathbf {R}}^{d})\) consists of all \(a\in C^\infty ({\mathbf {R}}^{d})\) such that
    $$\begin{aligned} |\partial ^\alpha f(x) | \lesssim h^{|\alpha |}\alpha !^s \omega _0(x),\quad \alpha \in {\mathbf {N}}^{d}, \end{aligned}$$
    (2.16)
    for some constant \(h>0\);
     
  2. (2)

    The set \(\Gamma ^{(\omega _0)}_{0,s}({\mathbf {R}}^{d})\) consists of all \(a\in C^\infty ({\mathbf {R}}^{d})\) such that (2.16) holds for every constant \(h>0\).

     

Remark 2.12

We have
$$\begin{aligned} {\mathscr {P}}\subseteq {\mathscr {P}}_{E,s_1}^0\subseteq {\mathscr {P}}_{E,s_1} \subseteq {\mathscr {P}}_{E,s_2}^0, \qquad s_2<s_1. \end{aligned}$$
Hence, despite that \(\Gamma ^{(\omega _0)}_{0,s}\subseteq \Gamma ^{(\omega _0)}_s\subseteq S^{(\omega _0)}\) holds for every \(\omega _0\), we have
$$\begin{aligned} \Gamma ^{(\omega )}_{0,s}&\nsubseteq \bigcup _{\omega _0\in {\mathscr {P}}} S^{(\omega _0)} \end{aligned}$$
for some \(\omega \in {\mathscr {P}}_{E,s}^0\), and
$$\begin{aligned} \Gamma ^{(\omega )}_{0,s}&\nsubseteq \bigcup _{\omega _0\in {\mathscr {P}}_{E,s}^0} \Gamma ^{(\omega _0)}_{s} \end{aligned}$$
for some \(\omega \in {\mathscr {P}}_{E,s}\).

3 Continuity for pseudo-differential operators with symbols in \(\Gamma ^{(\omega )}_{s}\) and \(\Gamma ^{(\omega )}_{0,s}\)

In this section we discuss continuity for operators in \({\text {Op}}(\Gamma _{s}^{(\omega _0)})\) and \({\text {Op}}(\Gamma _{0,s}^{(\omega _0)})\) when acting on a general class of modulation spaces. In Theorem 3.5 below it is proved that if \(\omega ,\omega _0\in {\mathscr {P}}_{E,s}^0\), \(A\in {\mathbf {M}}(d,{\mathbf {R}})\) and \(a\in \Gamma _{0,s}^{(\omega _0)}\), then \({\text {Op}}_A(a)\) is continuous from \(M(\omega _0\omega ,{\mathscr {B}})\) to \(M(\omega _0,{\mathscr {B}})\). This gives an analogy to [32, 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 \({\mathscr {P}}_E\), there are equivalent weights that satisfy strong Gevrey regularity.

Proposition 3.1

Let \(\omega \in {\mathscr {P}}_E({\mathbf {R}}^{d})\) and \(s >0\). Then there exists a weight \(\omega _0\in {\mathscr {P}}_E({\mathbf {R}}^{d})\cap C^\infty ({\mathbf {R}}^{d})\) such that the following is true:
  1. (1)

    \(\omega _0\asymp \omega \);

     
  2. (2)

    \(|\partial ^{\alpha }\omega _0(x)|\lesssim \omega _0(x) h^{|\alpha |}\alpha !^s \asymp \omega (x) h^{|\alpha |}\alpha !^s\) for every \(h>0\).

     

Proof

We may assume that \(s<\frac{1}{2}\). It suffices to prove that (2) should hold for some \(h>0\). Let \(\phi _0\in \Sigma _{1-s}^s({\mathbf {R}}^{d}){\setminus } 0\), and let \(\phi =|\phi _0|^2\). Then \(\phi \in \Sigma _{1-s}^s({\mathbf {R}}^{d})\) is non-negative. In particular,
$$\begin{aligned} |\partial ^\alpha \phi (x)| \lesssim h^{|\alpha |}\alpha !^s e^{-c|x|^{\frac{1}{1-s}}}, \end{aligned}$$
for every \(h>0\) and \(c>0\). We set \(\omega _0=\omega *\phi \).
Then
$$\begin{aligned} |\partial ^\alpha \omega _0(x)|&= \left| \int \omega (y)(\partial ^\alpha \phi )(x-y) \, dy \right| \\&\lesssim h^{|\alpha |}\alpha !^s \int \omega (y)e^{-c|x-y|^{\frac{1}{1-s}}} \, dy\\&\lesssim h^{|\alpha |}\alpha !^s \int \omega (x+(y-x))e^{-c|x-y|^{\frac{1}{1-s}}} \, dy\\&\lesssim h^{|\alpha |}\alpha !^s \omega (x) \int e^{-\frac{c}{2}|x-y|^{\frac{1}{1-s}}} \, dy \asymp h^{|\alpha |}\alpha !^s \omega (x), \end{aligned}$$
where the last inequality follows (2.2) and the fact that \(\phi \) 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
$$\begin{aligned} \omega _0(x)&= \int \omega (y)\phi (x-y)\, dy = \int \omega (x+(y-x))\phi (x-y)\, dy\\&\lesssim \omega (x) \int e^{c|x-y|}\phi (x-y)\, dy \asymp \omega (x). \end{aligned}$$
In the same way, (2.3) gives
$$\begin{aligned} \omega _0(x)&= \int \omega (y)\phi (x-y)\, dy = \int \omega (x+(y-x))\phi (x-y)\, dy\\&\gtrsim \omega (x) \int e^{-c|x-y|}\phi (x-y)\, dy \asymp \omega (x), \end{aligned}$$
and (1) as well as the second part of (2) follow. \(\square \)

The next result shows that \(\Gamma ^{(\omega )}_{s}\) and \(\Gamma ^{(\omega )}_{0,s}\) can be characterised in terms of estimates of short-time Fourier transforms.

Proposition 3.2

Let \(s\ge 1\), \(\phi \in {\mathcal {S}}_s({\mathbf {R}}^{d}){\setminus } 0\), and let \(f\in {\mathcal {S}}'_{1/2}({\mathbf {R}}^{d})\). Then the following is true:
  1. (1)
    If \(\omega \in {\mathscr {P}}_{E,s}^0({\mathbf {R}}^{d})\), then \(f\in C^\infty ({\mathbf {R}}^{d})\) and satisfies
    $$\begin{aligned} |\partial ^\alpha f(x)|\lesssim \omega (x)h^{|\alpha |}\alpha !^s, \end{aligned}$$
    (3.1)
    for some \(h>0\), if and only if
    $$\begin{aligned} |V_\phi f(x,\xi )|\lesssim \omega (x)e^{-r|\xi |^{\frac{1}{s}}}, \end{aligned}$$
    (3.2)
    for some \(r>0\);
     
  2. (2)

    If \(\omega \in {\mathscr {P}}_{E,s}({\mathbf {R}}^{d})\) and in addition \(\phi \in \Sigma _s({\mathbf {R}}^{d})\), then \(f\in C^\infty ({\mathbf {R}}^{d})\) and satisfies (3.1) for every \(h>0\) (resp. for some \(h>0\)), if and only if (3.2) holds true for every \(r>0\) (resp. for some \(r>0\)).

     

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 \(\varepsilon >0\). The other cases follow by similar arguments and are left for the reader.

Assume that \(\phi \in \Sigma _s({\mathbf {R}}^{d})\), \(\omega \in {\mathscr {P}}_{E,s}({\mathbf {R}}^{d})\) and that (3.1) holds for every \(\varepsilon >0\). Then for every \(x \in {\mathbf {R}}^{d}\) the function
$$\begin{aligned} y\mapsto F_x(y)\equiv f(y+x)\overline{\phi (y)} \end{aligned}$$
belongs to \(\Sigma _s\), and \(\omega (x+y)\lesssim e^{h_0|y|^{\frac{1}{s}}}\omega (x)\) for some \(h_0>0\). By a straight-forward application of Leibnitz formula and the facts that
$$\begin{aligned} |\partial ^\alpha \phi (x)|\lesssim \varepsilon ^{|\alpha |}\alpha !^s e^{-h|x|^{\frac{1}{s}}} \quad \text {and}\quad \omega (x+y)\lesssim \omega (x)e^{h_0|y|^{\frac{1}{s}}} \end{aligned}$$
for some \(h_0>0\) and every \(\varepsilon ,h>0\) we get
$$\begin{aligned} |\partial _y^\alpha F_x(y)| \lesssim \omega (x)e^{-h|y|^{\frac{1}{s}}} \varepsilon ^{|\alpha |}\alpha !^s, \end{aligned}$$
for every \(\varepsilon , h >0\). In particular,
$$\begin{aligned} |F_x(y)|\lesssim \omega (x)e^{-h|y|^{\frac{1}{s}}}\quad \text {and}\quad |{\widehat{F}}_x(\xi )|\lesssim \omega (x)e^{-h|\xi |^{\frac{1}{s}}}, \end{aligned}$$
(3.3)
for every \(h>0\). Since \(|V_\phi f(x,\xi )| = |{\widehat{F}}_x(\xi )|\), the estimate (3.2) follows from the second inequality in (3.3). This shows that if (3.1) holds for every \(\varepsilon >0\), then (3.2) holds for every \(\varepsilon >0\).
Next suppose that (3.2) holds for every \(\varepsilon >0\). By Fourier’s inversion formula we get
$$\begin{aligned} f(x) = (2\pi )^{-\frac{d}{2}} \Vert \phi \Vert _{L^2}^{-2} \iint _{{\mathbf {R}}^{2d}} V_\phi f(y,\eta ) \phi (x-y)e^{i\langle x,\eta \rangle }\, dyd\eta . \end{aligned}$$
By differentiation and the fact that \(\phi \in \Sigma _s\) we get
$$\begin{aligned} |\partial ^\alpha f(x)|&\asymp&\left| \sum _{\beta \le \alpha } {\alpha \atopwithdelims ()\beta } i^{|\beta |} \iint _{{\mathbf {R}}^{2d}} \eta ^\beta V_\phi f(y,\eta ) (\partial ^{\alpha -\beta }\phi )(x-y) e^{i\langle x,\eta \rangle }\, dyd\eta \right| \\\le & {} \sum _{\beta \le \alpha } {\alpha \atopwithdelims ()\beta } \iint _{{\mathbf {R}}^{2d}} |\eta ^\beta V_\phi f(y,\eta ) (\partial ^{\alpha -\beta }\phi )(x-y)| \, dyd\eta \\\lesssim & {} \sum _{\beta \le \alpha } {\alpha \atopwithdelims ()\beta } \iint _{{\mathbf {R}}^{2d}} |\eta ^\beta \omega (y)e^{-\varepsilon _3|\eta |^{\frac{1}{s}}} (\partial ^{\alpha -\beta }\phi )(x-y)| \, dyd\eta \\\lesssim & {} \sum _{\beta \le \alpha } {\alpha \atopwithdelims ()\beta } \varepsilon _2 ^{|\alpha - \beta |} (\alpha -\beta )!^s \iint _{{\mathbf {R}}^{2d}} |\eta ^\beta |\omega (y)e^{-\varepsilon _3|\eta |^{\frac{1}{s}}} e^{-\varepsilon _1|x-y|^{\frac{1}{s}}} \, dyd\eta , \end{aligned}$$
for every \(\varepsilon _1,\varepsilon _2,\varepsilon _3>0\). Since
$$\begin{aligned} |\eta ^\beta e^{-\varepsilon _3|\eta |^{\frac{1}{s}}}|\lesssim \varepsilon _2^{|\beta |}\beta !^s e^{-\varepsilon _3|\eta |^{\frac{1}{s}}/2}, \end{aligned}$$
when \(\varepsilon _3\) is chosen large enough compared to \(\varepsilon _2^{-1}\), we get
$$\begin{aligned} |\partial ^\alpha f(x)|\lesssim & {} \varepsilon _2 ^{|\alpha |} \sum _{\beta \le \alpha } {\alpha \atopwithdelims ()\beta } (\beta ! (\alpha -\beta )!)^s \iint _{{\mathbf {R}}^{2d}} \omega (y)e^{-\varepsilon _3|\eta |^{\frac{1}{s}}/2} e^{-\varepsilon _1|x-y|^{\frac{1}{s}}} \, dyd\eta \\\lesssim & {} (2\varepsilon _2)^{|\alpha |}\alpha !^s \int _{{\mathbf {R}}^{n}} \omega (y) e^{-\varepsilon _1|x-y|^{\frac{1}{s}}} \, dy \end{aligned}$$
Since \(\omega (y)\le \omega (x)e^{h_0|x-y|^{\frac{1}{s}}}\) for some \(h_0\ge 0\) and \(\varepsilon _1\) can be chosen arbitrarily large, it follows from the last estimate that
$$\begin{aligned} |\partial ^\alpha f(x)| \lesssim (2\varepsilon _2)^{|\alpha |}\alpha !^s \omega (x), \end{aligned}$$
for every \(\varepsilon _2>0\), and the result follows. \(\square \)

The following result is now a straight-forward consequence of the previous proposition and the definitions.

Proposition 3.3

Let \(s\ge 1\), \(q\in (0,\infty ]\), \(\omega _0\in {\mathscr {P}}_{E,s}({\mathbf {R}}^{d})\) and let \(\omega _r(x,\xi ) = \omega _0(x)e^{-r|\xi |^{\frac{1}{s}}}\) when \(x,\xi \in {\mathbf {R}}^{d}\). Then
$$\begin{aligned} \bigcup _{r>0}M^{\infty ,q}_{(1/\omega _r)}({\mathbf {R}}^{d}) = \Gamma ^{(\omega _0)}_{s}({\mathbf {R}}^{d}) \quad \text {and}\quad \bigcap _{r>0}M^{\infty ,q}_{(1/\omega _r)}({\mathbf {R}}^{d}) = \Gamma ^{(\omega _0)}_{0,s}({\mathbf {R}}^{d}). \end{aligned}$$

The following lemma is a consequence of Theorem 4.6 in [5].

Lemma 3.4

Let \(s\ge 1\) \(\omega \in {\mathscr {P}}_{E}({\mathbf {R}}^{2d})\), \(A_1,A_2\in {\mathbf {M}}(d,{\mathbf {R}})\), and that \(a_1,a_2\in \Sigma _1'({\mathbf {R}}^{2d})\) are such that \({\text {Op}}_{A_1}(a_1)={\text {Op}}_{A_2}(a_2)\). Then
$$\begin{aligned} a_1 \in \Gamma _{s}^{(\omega )}({\mathbf {R}}^{2d}) \qquad \Leftrightarrow \qquad a_2 \in \Gamma _{s}^{(\omega )}({\mathbf {R}}^{2d}) \end{aligned}$$
and
$$\begin{aligned} a_1 \in \Gamma _{0,s}^{(\omega )}({\mathbf {R}}^{2d}) \qquad \Leftrightarrow \qquad a_2 \in \Gamma _{0,s}^{(\omega )}({\mathbf {R}}^{2d}). \end{aligned}$$

We have now the following result.

Theorem 3.5

Let \(A\in {\mathbf {M}}(d,{\mathbf {R}})\), \(s\ge 1\), \(\omega ,\omega _0\in {\mathscr {P}}_{E,s}^0({\mathbf {R}}^{2d})\), \(a\in \Gamma ^{(\omega _0)}_{s}({\mathbf {R}}^{2d})\), and that \({\mathscr {B}}\) is an invariant BF-space on \({\mathbf {R}}^{2d}\). Then \({\text {Op}}_A(a)\) is continuous from \(M(\omega _0\omega ,{\mathscr {B}})\) to \(M(\omega ,{\mathscr {B}})\).

We need some preparations for the proof, and start with the following lemma.

Lemma 3.6

Suppose \(s\ge 1\), \(\omega \in {\mathscr {P}}_E({\mathbf {R}}^{d_0})\) and that \(f\in C^\infty ({\mathbf {R}}^{d+d_0})\) satisfies
$$\begin{aligned} |\partial ^\alpha f(x,y)|\lesssim h^{|\alpha |}\alpha !^se^{-r|x|^{\frac{1}{s}}}\omega (y), \quad \alpha \in {\mathbf {N}}^{d+d_0} \end{aligned}$$
(3.4)
for some \(h>0\) and \(r>0\). Then there are \(f_0\in C^\infty ({\mathbf {R}}^{d+d_0})\) and \(\psi \in {\mathcal {S}}_s({\mathbf {R}}^{d})\) such that (3.4) holds with \(f_0\) in place of f for some \(h>0\) and \(r>0\), and \(f(x,y)= f_0(x,y)\psi (x)\).

Proof

By Proposition 3.1 there is a submultiplicative weight \(v_0\in {\mathscr {P}}_{E,s}({\mathbf {R}}^{d})\cap C^\infty ({\mathbf {R}}^{d})\) such that
$$\begin{aligned} v_0(x)&\asymp e^{\frac{r}{2}|x |^{\frac{1}{s}}} \end{aligned}$$
(3.5)
and
$$\begin{aligned} |\partial ^\alpha v_0(x)|&\lesssim h^{|\alpha |}\alpha !^sv_0(x),\qquad \alpha \in {\mathbf {N}}^{d} \end{aligned}$$
(3.6)
for some \(h>0\). Since \(s\ge 1\), a straight-forward combination of Faà di Bruno’s formula and (3.6) gives
$$\begin{aligned} \left| \partial ^\alpha \left( \frac{1}{v_0(x)}\right) \right|&\lesssim h^{|\alpha |}\alpha !^s\cdot \frac{1}{v_0(x)},\qquad \alpha \in {\mathbf {N}}^{d} \quad \quad \quad \quad \quad {(3.6)'} \end{aligned}$$
for some \(h>0\). It follows from (3.5) and (3.6)\('\) that if \(\psi =1/v\), then \(\psi \in {\mathcal {S}}_s({\mathbf {R}}^{d})\). Furthermore, if \(f_0(x,y)=f(x,y)v_0(x)\), then an application of Leibnitz formula gives
$$\begin{aligned}&|\partial ^{\alpha }_x\partial ^{\alpha _0}_yf_0(x,y)|\lesssim \sum _{\gamma \le \alpha }{{\alpha } \atopwithdelims (){\gamma }} |\partial ^\delta _x \partial ^{\alpha _0}_yf(x,y)| \, |\partial ^{\alpha -\delta }v_0(x)|\\&\quad \lesssim h^{|\alpha |+|\alpha _0|} \sum _{\gamma \le \alpha }{{\alpha } \atopwithdelims (){\gamma }} (\gamma !\alpha _0!)^{s}e^{-r|x|^{\frac{1}{s}}}\omega (y)(\alpha -\gamma )!^sv_0(x)\\&\quad \lesssim (2h)^{|\alpha |+|\alpha _0|}(\alpha !\alpha _0!)^se^{-r|x|^{\frac{1}{s}}}v_0(x)\omega (y)\\&\quad \asymp (2h)^{|\alpha |+|\alpha _0|}(\alpha !\alpha _0!)^se^{-\frac{r}{2}|x|^{\frac{1}{s}}}\omega (y) \end{aligned}$$
for some \(h>0\), which gives the desired estimate on \(f_0\), The result now follows since it is evident that \(f(x,y)= f_0(x,y)\psi (x)\). \(\square \)
For the next lemma we recall that for any \(a\in \Sigma _1'({\mathbf {R}}^{2d})\) there is a unique \(b\in \Sigma _1'({\mathbf {R}}^{2d})\) such that \({\text {Op}}(a)^*={\text {Op}}(b)\), and then \(b(x,\xi )= e^{i\langle D_\xi ,D_x\rangle }\overline{a(x,\xi )}\) in view of [21, Theorem 18.1.7]. Furthermore, by the latter equality and [5, Theorem 4.1] it follows that
$$\begin{aligned} a\in \Gamma ^{(\omega )}_{s}({\mathbf {R}}^{2d}) \quad \Leftrightarrow \quad b\in \Gamma ^{(\omega )}_{s}({\mathbf {R}}^{2d}). \end{aligned}$$

Lemma 3.7

Let \(s\ge 1\), \(\omega \in {\mathscr {P}}_{E,s}^0({\mathbf {R}}^{2d})\), \(\vartheta \in {\mathscr {P}}_{E,s}^0({\mathbf {R}}^{d})\) and \(v\in {\mathscr {P}}_{E,s}^0({\mathbf {R}}^{d})\) be such that v is submultiplicative, \(\omega \in \Gamma ^{(\omega )}_{0,s}({\mathbf {R}}^{2d})\) is \(v\otimes v\)-moderate, \(\vartheta =v^{-\frac{1}{2}}\) and \(\vartheta \in \Gamma ^{(\vartheta )}_{0,s}({\mathbf {R}}^{d})\). Also let \(a\in \Gamma ^{(\omega )}_{s}({\mathbf {R}}^{2d})\), choose \(b\in \Gamma ^{(\omega )}_{s}({\mathbf {R}}^{2d})\) such that \({\text {Op}}(b)={\text {Op}}(a)^*\), \(f\in {\mathcal {S}}_s({\mathbf {R}}^{d})\), \(\phi \in \Sigma _s({\mathbf {R}}^{d})\), \(\varphi = {\widehat{\phi }} \cdot v\),
$$\begin{aligned} \Phi (x,\xi ,y,\eta )&= \frac{b(y,\xi +\eta )}{\omega (x,\xi )v(x-y)v(\eta )} \end{aligned}$$
(3.7)
and
$$\begin{aligned} H(x,\xi ,y)&= v(x-y)\int \Phi (x,\xi ,y,\eta )\varphi (\eta ) e^{i\langle y-x,\eta \rangle }\, d\eta . \end{aligned}$$
Then
$$\begin{aligned} V_\phi ({\text {Op}}(a)f)(x,\xi ) = (2\pi )^{-d} (f,e^{i\langle \, \cdot \, ,\xi \rangle }H(x,\xi ,\, \cdot \, ))\omega (x,\xi ). \end{aligned}$$
(3.8)
Furthermore the following is true:
  1. (1)
    \(H\in C^\infty ({\mathbf {R}}^{3d})\) and satisfies
    $$\begin{aligned} |\partial ^\alpha H(x,\xi ,y)| \lesssim h_0^{|\alpha |}\alpha !^se^{-r_0|x-y|^{\frac{1}{s}}}, \end{aligned}$$
    (3.9)
    for some \(h_0,r_0>0\);
     
  2. (2)
    there are functions \(H_0\in C^\infty ({\mathbf {R}}^{3d})\) and \(\phi _0\in {\mathcal {S}}_s({\mathbf {R}}^{d})\) such that
    $$\begin{aligned} H(x,\xi ,y) = H_0(x,\xi ,y)\phi _0(y-x), \end{aligned}$$
    (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
$$\begin{aligned} V_\phi ({\text {Op}}(a)f)(x,\xi )= & {} (2\pi )^{-\frac{d}{2}}({\text {Op}}(a)f, \phi (\, \cdot \, -x)\, e^{i\langle \, \cdot \, ,\xi \rangle })\nonumber \\= & {} (2\pi )^{-\frac{d}{2}}(f, {\text {Op}}(b)(\phi (\, \cdot \, -x)\, e^{i\langle \cdot ,\xi \rangle }))\nonumber \\= & {} (2\pi )^{-d} (f,e^{i\langle \, \cdot \, ,\xi \rangle }H_1(x,\xi ,\, \cdot \, ))\omega (x,\xi ), \end{aligned}$$
(3.11)
where
$$\begin{aligned} H_1(x,\xi ,y)= & {} (2\pi )^{\frac{d}{2}}e^{-i\langle y,\xi \rangle }({\text {Op}}(b)(\phi (\, \cdot \, -x) \, e^{i\langle \cdot ,\xi \rangle }))(y)/\omega (x,\xi )\\= & {} \int \frac{b(y,\eta )}{\omega (x,\xi )}{\widehat{\phi }} (\eta -\xi ) e^{-i\langle x-y,\eta -\xi \rangle }\, d\eta \\= & {} v(x-y) \int \Phi (x,\xi ,y,\eta -\xi )\varphi (\eta -\xi )e^{-i\langle x-y,\eta -\xi \rangle }\, d\eta . \end{aligned}$$
If \(\eta -\xi \) are taken as new variables of integrations, it follows that the right-hand side is equal to \(H(x,y,\xi )\). This gives the first part of the lemma.
In order to prove (1), let
$$\begin{aligned} \Phi _0(x,\xi ,y,\eta ) = \Phi (x,\xi ,y,\eta )\varphi (\eta ), \end{aligned}$$
and let \(\Psi ={\mathscr {F}}_4\Phi _0\), where \({\mathscr {F}}_4\Phi _0\) is the partial Fourier transform of \(\Phi _0(x,\xi ,y,\eta )\) with respect to the \(\eta \) variable. Then it follows from the assumptions that
$$\begin{aligned} |\partial ^\alpha \Phi _0(x,\xi ,y,\eta )| \lesssim h_0^{|\alpha |}\alpha !^se^{-r_0|\eta |^{\frac{1}{s}}}, \end{aligned}$$
for some \(h_0,r_0>0\), which shows that \(\eta \mapsto \Phi _0(x,\xi ,y,\eta )\) is an element in \({\mathcal {S}}_s({\mathbf {R}}^{d})\) with values in \(\Gamma ^{(1)}_{s}({\mathbf {R}}^{3d})\). As a consequence, \(\Psi \) satisfies
$$\begin{aligned} |\partial ^\alpha \Psi (x,\xi ,y_1,y_2)| \lesssim h_0^{|\alpha |}\alpha !^se^{-r_0|y_2|^{\frac{1}{s}}}, \end{aligned}$$
for some \(h_0,r_0>0\). The assertion (1) now follows from the latter estimate, Leibnitz rule and the fact that
$$\begin{aligned} H(x,\xi ,y) = v(x-y)\Psi (x,\xi ,x-y). \end{aligned}$$
In order to prove (2) we notice that (3.9) shows that \(y\mapsto H(x,\xi ,x-y)\) is an element in \({\mathcal {S}}_s({\mathbf {R}}^{d})\) with values in \(\Gamma ^{(1)}_{s}({\mathbf {R}}^{2d})\). By Lemma 3.6 there are \(H_2\in C^\infty ({\mathbf {R}}^{3d})\) and \(\phi _0\in {\mathcal {S}}_s({\mathbf {R}}^{d})\) such that (3.9) holds for some \(h_0,r_0>0\) with \(H_2\) in place of H, and
$$\begin{aligned} H(x,\xi ,x-y)= H_2(x,\xi ,x-y)\phi _0(-y). \end{aligned}$$
This is the same as (2), and the result follows. \(\square \)

Proof of Theorem 3.5

We may assume that \(A=0\). Let \(g={\text {Op}}(a)f\). By Lemma 3.7 we have
$$\begin{aligned} V_\phi g(x,\xi )= & {} (2\pi )^{-\frac{d}{2}}{\mathscr {F}}((f\cdot \overline{\phi _0(\, \cdot \, -x)}) \cdot H_0(x,\xi ,\, \cdot \, ))(\xi ) \omega (x,\xi )\\= & {} (2\pi )^{-d}{\mathscr {F}}((f\cdot \overline{\phi _0(\, \cdot \, -x)})) *({\mathscr {F}}(H_0(x,\xi ,\, \cdot \, )))(\xi ) \omega (x,\xi )\\= & {} (2\pi )^{-d} ((V_{\phi _0}f)(x,\, \cdot \, ) * ({\mathscr {F}}(H_0(x,\xi ,\, \cdot \, ))))(\xi ) \omega (x,\xi ) \end{aligned}$$
Since \(\omega \) and \(\omega _0\) belongs to \({\mathscr {P}}_{E,s}^0({\mathbf {R}}^{2d})\), (2) in Lemma 3.7 gives
$$\begin{aligned} |V_\phi g(x,\xi )\omega _0(x,\xi )| \lesssim |V_{\phi _0}f)(x,\, \cdot \, )\omega (x,\, \cdot \, ) \omega _0(x,\, \cdot \, )| * e^{-\frac{r_0}{2}|\, \cdot \, |^{\frac{1}{s}}}. \end{aligned}$$
Here we have used the fact that
$$\begin{aligned} \omega (x,\xi )\omega _0(x,\xi )\lesssim \omega (x,\xi -\eta )\omega _0(x,\xi -\eta ) e^{\frac{r_0}{2}|\eta |^{\frac{1}{s}}}. \end{aligned}$$
By applying the \({\mathscr {B}}\) norm we get for some \(v\in {\mathscr {P}}_{E,s}^0({\mathbf {R}}^{d})\),
$$\begin{aligned}&\Vert g\Vert _{M(\omega _0,{\mathscr {B}})}\lesssim \Vert |V_{\phi _0}f) \cdot \omega \cdot \omega _0| * (e^{-r_0|\, \cdot \, |^{\frac{1}{s}}}\otimes \delta _0)\Vert _{{\mathscr {B}}}\\&\quad \le \Vert V_{\phi _0}f) \cdot \omega \cdot \omega _0\Vert _{{\mathscr {B}}}\Vert e^{-r_0|\, \cdot \, |^{\frac{1}{s}}}v\Vert _{L^1} \asymp \Vert f\Vert _{M(\omega \cdot \omega _0,{\mathscr {B}})}. \end{aligned}$$
This gives the result. \(\square \)

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.

Theorem 3.8

Let \(A\in {\mathbf {M}}(d,{\mathbf {R}})\), \(s\ge 1\), \(\omega ,\omega _0\in {\mathscr {P}}_{E,s}({\mathbf {R}}^{2d})\), \(a\in \Gamma ^{(\omega _0)}_{0,s}({\mathbf {R}}^{2d})\), and let \({\mathscr {B}}\) be an invariant BF-space on \({\mathbf {R}}^{2d}\). Then \({\text {Op}}_A(a)\) is continuous from \(M(\omega _0\omega ,{\mathscr {B}})\) to \(M(\omega ,{\mathscr {B}})\).

Lemma 3.9

Let \(s\ge 1\), \(\omega \in {\mathscr {P}}_{E,s}({\mathbf {R}}^{2d})\), \(\vartheta \in {\mathscr {P}}_{E,s}({\mathbf {R}}^{d})\) and \(v\in {\mathscr {P}}_{E,s}({\mathbf {R}}^{d})\) be such that v is submultiplicative, \(\omega \in \Gamma ^{(\omega )}_{0,s}({\mathbf {R}}^{2d})\) is \(v\otimes v\)-moderate, \(\vartheta =v^{-\frac{1}{2}}\) and \(\vartheta \in \Gamma ^{(\vartheta )}_{0,s}({\mathbf {R}}^{d})\). Also let \(a\in \Gamma ^{(\omega )}_{0,s}({\mathbf {R}}^{2d})\), \(f,\phi \in \Sigma _s({\mathbf {R}}^{d})\), \(\phi _2= \phi v\), and let \(\Phi \) and H be as in Lemma 3.7. Then (3.8) and the following hold true:
  1. (1)

    \(H\in C^\infty ({\mathbf {R}}^{3d})\) and satisfies (3.9) for every \(h_0,r_0>0\);

     
  2. (2)

    there are functions \(H_0\in C^\infty ({\mathbf {R}}^{3d})\) and \(\phi _0\in \Sigma _s({\mathbf {R}}^{d})\) such that (3.10) holds, and such that (3.9) holds for every \(h_0,r_0>0\), with \(H_0\) in place of H.

     
We finish the section by discussing continuity for pseudo-differential operators with symbols in \(\Gamma ^{(\omega _0)}_{s}\) or in \(\Gamma ^{(\omega _0)}_{0,s}\) when acting on quasi-Banach modulation spaces. More precisely, by straight-forward computations it follows that if \(\omega ,\omega _0 \in {\mathscr {P}}_{E,s}({\mathbf {R}}^{2d})\) (\(\omega ,\omega _0\in {\mathscr {P}}_{E,s}^0({\mathbf {R}}^{2d})\)), then
$$\begin{aligned} \frac{\omega (x,\xi )}{\omega (y,\eta )\omega _0(y,\eta )} \lesssim \frac{e^{r(|\xi -\eta |^{\frac{1}{s}}+|y-x|^{\frac{1}{s}})}}{\omega _0(x,\eta )}. \end{aligned}$$
holds for some \(r>0\) (for every \(r>0\)). Hence the following result is a straight-forward consequence of Propositions 2.10 and 3.3, and Lemma 3.4. (Cf. Definition 2.9 for the definition of phase split basis.)

Theorem 3.10

Let \(A\in {\mathbf {M}}(d,{\mathbf {R}})\), \(s\ge 1\), \(\omega ,\omega _0 \in {\mathscr {P}}_{E,s}^0({\mathbf {R}}^{2d})\), \({{\varvec{p}}}\in (0,\infty ]^{2d}\), E be a phase split basis of \({\mathbf {R}}^{2d}\), and let \(a\in \Gamma ^{(\omega _0)}_{s}({\mathbf {R}}^{2d})\). Then \({\text {Op}}_A(a)\) is continuous from \(M(L^{{{\varvec{p}}}}_E({\mathbf {R}}^{2d}), \omega _0\omega )\) to \(M(L^{{{\varvec{p}}}}_E({\mathbf {R}}^{2d}),\omega )\).

The same holds true with \({\mathscr {P}}_{E,s}\) and \(\Gamma ^{(\omega _0)}_{0,s}\), or with \({\mathscr {P}}\) and \(S^{(\omega _0)}\) in place of \({\mathscr {P}}_{E,s}^0\) and \(\Gamma ^{(\omega _0)}_{s}\), respectively, at each occurence.

Corollary 3.11

Let \(A\in {\mathbf {M}}(d,{\mathbf {R}})\), \(s\ge 1\) and \(\omega ,\omega _0 \in {\mathscr {P}}_{E,s}^0({\mathbf {R}}^{2d})\), \(p,q \in (0,\infty ]\), and let \(a\in \Gamma ^{(\omega _0)}_{s}({\mathbf {R}}^{2d})\). Then \({\text {Op}}_A(a)\) is continuous from \(M^{p,q}_{(\omega _0\omega )}({\mathbf {R}}^{d})\) to \(M^{p,q}_{(\omega )}({\mathbf {R}}^{d})\).

The same holds true with \({\mathscr {P}}_{E,s}\) and \(\Gamma ^{(\omega _0)}_{0,s}\), or with \({\mathscr {P}}\) and \(S^{(\omega _0)}\) in place of \({\mathscr {P}}_{E,s}^0\) and \(\Gamma ^{(\omega _0)}_{s}\), respectively, at each occurence.

4 Examples

In this section we list some examples and show how the continuity results of the pseudo-differential operators in the previous section leads to continuity on certain Sobolev spaces, weighted Lebesgue spaces and on \(\Gamma ^{(\omega )}_{0,s}\) spaces.

In the examples here we consider pseudo-differential operators with symbols in \(\Gamma ^{(\omega )}_{0,s}\) spaces. By some modifications, we may also deduce similar continuity results for operators with symbols in \(\Gamma ^{(\omega )}_{s}\) spaces.

Example 4.1

Let \(s\ge 1\), \(r,r_0\in {\mathbf {R}}\), \(A\in {\mathbf {M}}(d,{\mathbf {R}})\), and let \(H_r^2({\mathbf {R}}^{d})\) be the Sobolev space of all \(f\in \Sigma _1'({\mathbf {R}}^{d})\) such that \({\widehat{f}}\in L^2_{loc}({\mathbf {R}}^{d})\) and
$$\begin{aligned} \Vert f\Vert _{H_r^2}\equiv \left( \int _{{\mathbf {R}}^{d}}|{\widehat{f}}(\xi )e^{r|\xi |^{\frac{1}{s}}}|^2\, d\xi \right) ^{\frac{1}{2}} \end{aligned}$$
is finite. If \(a\in C^\infty ({\mathbf {R}}^{2d})\) satisfies
$$\begin{aligned} |\partial ^\alpha a(x,\xi )|\lesssim h^{|\alpha |}\alpha !^se^{r_0|\xi |^{\frac{1}{s}}} \end{aligned}$$
(4.1)
for every \(h>0\), then \({\text {Op}}_A(a)\) is continuous from \(H^2_r({\mathbf {R}}^{d})\) to \(H^2_{r-r_0}({\mathbf {R}}^{d})\).

In fact, if \(\omega _r(x,\xi )= e^{r|\xi |^{\frac{1}{s}}}\in {\mathscr {P}}_{E,s}({\mathbf {R}}^{2d})\), then it follows that the condition (4.1) holds true for every \(h>0\) is the same as \(a\in \Gamma ^{(\omega _{r_0})}_{0,s}({\mathbf {R}}^{2d})\). By a straight-forward applications of Fourier’s inversion formula we also have \(H_r^2({\mathbf {R}}^{d})=M^{2,2}_{(\omega _r)}({\mathbf {R}}^{d})\) (cf. the proof of [17, Proposition 11.3.1]). The assertion now follows from these observations and letting \({\mathscr {B}}= L^2({\mathbf {R}}^{2d})\) in Theorem 3.8.

Example 4.2

Let \(s\ge 1\), \(r,r_0\in {\mathbf {R}}\), \(A\in {\mathbf {M}}(d,{\mathbf {R}})\), and let \(L_r^2({\mathbf {R}}^{d})\) be the set \(L^2_{r}({\mathbf {R}}^{d})\) which consists of all \(f\in L^2_{loc}({\mathbf {R}}^{d})\) such that
$$\begin{aligned} \Vert f\Vert _{L_r^2}\equiv \left( \int _{{\mathbf {R}}^{d}}|f(x)e^{r|x|^{\frac{1}{s}}}|^2\, dx \right) ^{\frac{1}{2}} \end{aligned}$$
is finite. If \(a\in C^\infty ({\mathbf {R}}^{2d})\) satisfies
$$\begin{aligned} |\partial ^\alpha a(x,\xi )|\lesssim h^{|\alpha |}\alpha !^se^{r_0|x|^{\frac{1}{s}}} \end{aligned}$$
for every \(h>0\), then \({\text {Op}}_A(a)\) is continuous from \(L^2_r({\mathbf {R}}^{d})\) to \(L^2_{r-r_0}({\mathbf {R}}^{d})\).

In fact, if \(\omega _r(x,\xi )= e^{r|x|^{\frac{1}{s}}}\in {\mathscr {P}}_{E,s}({\mathbf {R}}^{2d})\), then it follows that the conditions a is the same as \(a\in \Gamma ^{(\omega _{r_0})}_{0,s}({\mathbf {R}}^{2d})\), and that \(L_r^2({\mathbf {R}}^{d})=M^{2,2}_{(\omega _r)}({\mathbf {R}}^{d})\) (cf. the proof of [17, Proposition 11.3.1]). The assertion now follows from these observations and Theorem 3.8.

Example 4.3

Let \(s\ge 1\), \(\vartheta \in {\mathscr {P}}_{E,s}({\mathbf {R}}^{2d})\), \(\vartheta _0=\vartheta (\, \cdot \, ,0) \in {\mathscr {P}}_{E,s}({\mathbf {R}}^{d})\), \(\omega _r\) and \(\omega _0\) be the same as in Proposition 3.3, \(A\in {\mathbf {M}}(d,{\mathbf {R}})\), and let \(a\in \Gamma ^{(\vartheta )}_{0,s}({\mathbf {R}}^{2d})\). Then \({\text {Op}}_A(a)\) is continuous from \(\Gamma ^{(\omega _0)}_{0,s}({\mathbf {R}}^{d})\) to \(\Gamma ^{(\omega _0\vartheta _0)}_{0,s}({\mathbf {R}}^{d})\).

In fact, by Theorem 3.8 it follows that
$$\begin{aligned} {\text {Op}}_A(a){:}\, M^{\infty ,1}_{(1/\omega _r)}({\mathbf {R}}^{d}) \rightarrow M^{\infty ,1}_{(1/(\omega _r\vartheta ))}({\mathbf {R}}^{d}) \end{aligned}$$
(4.2)
is continuous. Since \(M^{\infty ,1}_{(1/\omega _r)}({\mathbf {R}}^{d})\) is decreasing with respect to r and that
$$\begin{aligned} \omega _{r-r_0}\vartheta _0 \lesssim \omega _r\vartheta \lesssim \omega _{r+r_0}\vartheta _0, \end{aligned}$$
for some fixed \(r_0\ge 0\), Proposition 3.3 shows that
$$\begin{aligned} \bigcap _{r>0} M^{\infty ,1}_{(1/(\omega _r))}({\mathbf {R}}^{d}) = \Gamma ^{(\omega _0)}_{0,s}({\mathbf {R}}^{d}) \quad \text {and}\quad \bigcap _{r>0} M^{\infty ,1}_{(1/(\omega _r\vartheta ))}({\mathbf {R}}^{d}) = \Gamma ^{(\omega _0\vartheta _0)}_{0,s}({\mathbf {R}}^{d}). \end{aligned}$$
The asserted continuity now follows from these intersections and (4.2).

Notes

Acknowledgements

The author is grateful to Ahmed Abdeljawad for careful reading of the manuscript and giving valuable comments, leading to improvements of the content and the style.

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Authors and Affiliations

  1. 1.Department of MathematicsLinnæus UniversityVäxjöSweden

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