Boundedness of Gevrey and Gelfand–Shilov kernels of positive semi-definite operators

Abstract

We show that the strongest Gevrey irregularity of kernels to positive semi-definite operators appear at the diagonals. We also prove that positive elements with respect to the twisted convolution, belonging to a Gevrey class of certain order at the origin, belong to the Gelfand–Shilov space of the same order. In the end we apply these results to positive semi-definite pseudo-differential operators.

1 Introduction

Positivity is an important property in operator theory. A positive operator on a Hilbert space is self-adjoint with positive spectrum, which leads to several requested properties, e.g. on invertibility. When considering partial differential equations it is common to link questions on ellipticity and hypoellipticity with questions on positive semi-definite operators (with respect to the $L^2$-form). Positive operators are also often important when establishing existence and uniqueness for solutions to partial differential equations (see e.g. [5, 6]). We also remark that several operators which appear frequently are positive or positive semi-definite. For example, the negative Laplace operator $-\Delta$ is positive semi-definite, and the harmonic oscillator $|x|^2-\Delta$ is positive (on $L^2(\mathbf R^{d})$).

Positivity in operator theory can also be related to positivity within other fields, e.g. the twisted convolution. In fact, let $T_1$ and $T_2$ be two continuous and linear operators on $L^2(\mathbf R^{d})$ with kernels $K_1$ and $K_2$, and let $*_\sigma$ be the twisted convolution. (See Sect. 2 for notations). Then there is a unique map $A$ which is linear and bijective on ${\fancyscript{S'}}(\mathbf R^{2d})$, and such that

$$\begin{aligned} A(a_1*_\sigma a_2) = (Aa_1)\circ (Aa_2),\quad \text {when}\quad K_j=Aa_j, \quad j=1,2. \end{aligned}$$

[See (2.2) below, or e.g. [3, 11].] Here we have identified operators with their kernels. Then it follows that a linear operator $T$ with kernel $K$ is positive (semi-definite), if and only if $A^{-1}K$ is positive (semi-definite) with respect to the twisted convolution. Similar links between positivity in operator theory and in the Weyl calculus of pseudo-differential operators are given in [11].

In several situations, it is convenient to consider linear operators in background from the point of view of Gelfand-triples $(\mathcal U ,\mathcal H,\mathcal U')$. Here $\mathcal H$ is a Hilbert space, $\mathcal U\subseteq \mathcal H$ is a topological vector space which is dense in $\mathcal H$, $\mathcal U'$ is the dual space of $\mathcal U$, and the duality between $\mathcal U$ and $\mathcal U'$ can be described by a unique extension of the $\mathcal H$-form on $\mathcal U\times \mathcal U$ into $\mathcal U' \times \mathcal U$.

We recall that a linear and continuous operator $T$ on $\mathcal H$ is called positive semi-definite whenever $(Tf,f)_{\mathcal H}\ge 0$ for every $f\in \mathcal H$. More generally, a linear operator $T$ from $\mathcal U$ to $\mathcal U'$ is called positive semi-definite if $(Tf,f)_{\mathcal H}\ge 0$ for every $f\in \mathcal U$, and then we write $T\ge 0$. Since $\mathcal U$ is dense in $\mathcal H$, it follows that a linear and continuous operator $T$ on $\mathcal H$ is positive semi-definite, if and only if $T$ is positive semi-definite as an operator from $\mathcal U$ to $\mathcal U'$.

In our situation, $\mathcal H=L^2(\mathbf R^{d})$, and $\mathcal U$ is the Gelfand–Shilov space $\mathcal S_s(\mathbf R^{d})$ of Roumieu type, or the Gelfand–Shilov space $\Sigma _s(\mathbf R^{d})$ of Beurling type. We recall that such Gelfand–Shilov spaces are dense subsets of the Schwartz space $\fancyscript{S}(\mathbf R^{d})$, which are both invariant under Fourier transformation, and locally the same as certain Gevrey classes. Hence their distribution spaces $\mathcal S'_s(\mathbf R^{d})$ and $\Sigma _s'(\mathbf R^{d})$, which are sets of ultra-distributions of Gelfand–Shilov type, contain $\fancyscript{S}'(\mathbf R^{d})$, the set of tempered distributions.

In the paper we deduce boundedness and detailed regularity properties for kernels of positive semi-definite operators and elements with respect to non-commutative convolutions. Especially we consider such questions in background of Gevrey and Gelfand–Shilov spaces of ultra-differentiable functions, and their dual spaces of ultra-distributions. For example, consider an ultra-distribution of Roumieu type (i.e. an element in the dual space of compactly supported elements in the corresponding Gevrey class), which at the same time is a kernel to a positive semi-definite operator. Then it is proved that the kernel is a Gelfand–Shilov distribution of a certain degree, if and only if its restriction to the diagonal is also a Gelfand–Shilov distribution of the same degree. (see Theorem 3.1 in Sect. 3).

A consequence of this result is that a Roumieu or Beurling distribution (i.e. an ultra-distribution of Roumieu or Beurling type, respectively) which is positive with respect to a non-commutative convolution algebra, belongs to corresponding space of Gelfand–Shilov distributions (see Theorem 3.5 in Sect. 3). (We will see that the usual convolution as well as the twisted convolution are special cases of these non-commutative convolutions.)

For the twisted convolution we perform further investigations when the Roumieu (or Beurling) distributions possess stronger regularity. More precisely, if a Roumieu or Gelfand–Shilov distribution is positive with respect to the twisted convolution, and is of Gevrey class of certain degree near the origin, then we prove that the distribution is a Gelfand–Shilov function of the same degree.

More specifically, let $s>0$ and let $a$ be positive semi-definite with respect to the twisted convolution, and smooth near origin and satisfies

$$\begin{aligned} |\partial ^\alpha a(0)|\le Ch^{|\alpha |}(\alpha !)^s, \end{aligned}$$

(1.1)

for some positive constants $C$ and $h$, which are independent of $\alpha$. Then we prove that $a$ belongs to the Gelfand–Shilov space $\mathcal S_s$. (See Theorem 5.1 in Sect. 5.) As an immediate consequence, it follows that any $a$ which is positive semi-definite with respect to the twisted convolution and satisfies (1.1) for some $s<1/2$, must be identically equal to $0$, since $\mathcal S_s$ is trivial for such choices of $s$ (cf. Theorem 5.2 in Sect. 5).

In Sect. 5 we also apply the results on twisted convolution to deduce equivalent results in the calculus of pseudo-differential operators, using the fact that ${\text {Op}}^w(a)= A(\fancyscript{F}_\sigma a)$, when ${\text {Op}}^w(a)$ is the Weyl quantization of $a$ and $\fancyscript{F}_\sigma$ is the symplectic Fourier transform. For example, as a consequence of Theorem 5.1 it follows that $a\in \mathcal S_s$, when ${\text {Op}}^w(a)$ is positive semi-definite, $X^\alpha a\in L^1$ for every multi-index $\alpha$, and

$$\begin{aligned} \left| \int _{\mathbf R^{2d}}X^\alpha a(X)\, dX\right| \le Ch^{|\alpha |}(\alpha !)^s, \end{aligned}$$

for some positive constants $C$ and $h$, which are independent of $\alpha$. (See Theorem 5.9.)

These results are analogous to results in [11], where similar properties were deduced in the background of the usual distribution theory. However, more comprehensive and complicated analyses are in general required when spaces of Gelfand–Shilov functions and distributions are involved instead of Schwartz spaces and their distribution spaces. Therefore we need to develop new techniques to handle the technical problems that arise when passing from situations involving the usual distribution and test function spaces to ultra-distribution spaces and their test function spaces.

An important reason for using Gelfand–Shilov spaces instead of the Schwartz space is that stronger regularity is required in the former case. For example, if $f\in \mathcal S_s$, then $f$ and its Fourier transform $\widehat{f}$ are bounded by the exponential type functions $x\mapsto Ce^{-c|x|^{1/s}}$ for some positive constants $c$ and $C$. If in addition $s<1$, then $f$ and $\widehat{f}$ are extendable to entire functions. Evidently, there are functions in $\fancyscript{S}$ which do not obey such strong regularity and boundedness properties. Furthermore, it might be more confident to formulate problems in the framework of Gelfand–Shilov distributions instead of tempered distributions, since the former distribution spaces are significantly larger than the latter one. For example, any measurable function $f$ such that $x\mapsto f(x) e^{-\varepsilon |x|^{1/s}}$ is bounded for every $\varepsilon >0$ belongs to $\mathcal S'_s(\mathbf R^{d})$, when $s\ge 1/2$.

Finally we remark that we also investigate local regularity of kernels to positive semi-definite operators. Especially we show that a kernel to a positive semi-definite operator which is Gevrey regular at $(x,x)$ and $(y,y)$ of certain degree obeys the same Gevrey regularity at $(x,y)$ and $(y,x)$ (cf. Proposition 4.3 in Sect. 4).

2 Preliminaries

In this section we recall some basic results. In the first part we recall some facts about Gelfand–Shilov spaces and Gevrey classes. Thereafter we introduce some notions and recall some facts on kernels to positive semi-definite operators and positive elements in non-commutative convolution algebras.

We begin to define Gelfand–Shilov spaces. Let $s \ge 1/2$ and $h>0$. Then $\mathcal S_{s,h} ({\mathbf R ^d})$ is the set of all $\varphi \in C ^{\infty } ({\mathbf R ^d})$ such that the norm

$$\begin{aligned} \Vert \varphi \Vert _{\mathcal S_{s,h}} \equiv \sup _{\alpha ,\beta \in \mathbf N ^d} \sup _{x \in \mathbf R ^d} \frac{|x ^{\alpha } D ^{\beta } \varphi (x)|}{(\alpha ! \beta !) ^s h ^{|\alpha +\beta |}} \end{aligned}$$

is finite. Then the Gelfand–Shilov spaces $\mathcal S_s({\mathbf R ^d})$ and $\Sigma _s({ \mathbf R ^d})$ are given by

$$\begin{aligned} \mathcal S_s({\mathbf R ^d}) =\bigcup \mathcal S_{s,h}({\mathbf R ^d}),\quad \Sigma _s({\mathbf R ^d}) =\bigcap \mathcal S_{s,h}({\mathbf R ^d}). \end{aligned}$$

The topologies for $\mathcal S_s({\mathbf R ^d})$ and $\Sigma _s({\mathbf R ^d})$ are given by the inductive limit and projective limit, respectively, i. e.

$$\begin{aligned} \mathcal S_s({\mathbf R ^d}) = \underset{h}{\text {ind lim}}~\mathcal S_{s,h}({\mathbf R ^d}),\quad \Sigma _s({\mathbf R ^d}) =\projlim _h \mathcal S_{s,h}({\mathbf R ^d}). \end{aligned}$$

The dual space of $\mathcal S_{s,h}({\mathbf R ^d})$ is denoted by $\mathcal S_{s,h} ^{\prime }({\mathbf R ^d})$. The $L^2$-form on $\mathcal S_{s,h}({\mathbf R ^d})$ extends in a usual way uniquely to a duality between $\mathcal S_{s,h} ^{\prime }({\mathbf R ^d}) \times \mathcal S_{s,h}({\mathbf R ^d})$, and in this setting we have $\mathcal S_{s,h}({\mathbf R ^d}) \subseteq \fancyscript{S}^{\prime }(\mathbf R ^d) \subseteq \mathcal S_{s,h} ^{\prime }({\mathbf R ^d})$.

The sets

$$\begin{aligned} \mathcal S_s ^{\prime }({\mathbf R ^d})= \bigcap \mathcal S_{s,h} ^{\prime }({\mathbf R ^d}) \quad \text {and} \quad \Sigma _ s ^{\prime }({\mathbf R ^d})= \bigcup \mathcal S_{s,h} ^{\prime }({\mathbf R ^d}), \end{aligned}$$

equipped by projective and inductive limit topologies, respectively, are called the spaces of tempered ultra-distributions of Roumieu and Beurling types, respectively. They are also called Gelfand–Shilov distribution spaces of order $s$. We have that $\mathcal S_s^{\prime }({\mathbf R^d})$ and $\Sigma _s^{\prime }({\mathbf R^d})$ are the dual spaces of $\mathcal S_s({\mathbf R^d})$ and $\Sigma _s ({\mathbf R^d})$ (also in topological sense) in view of [4].

The space $\mathcal S_s$ is non-trivial, if and only if $s\ge 1/2$, and $\Sigma _s$ is non-trivial, if and only if $s> 1/2$. Furthermore, if $s>1/2$ and $\varepsilon >0$, then

$$\begin{aligned} \mathcal S_{1/2} \subseteq \Sigma _s \subseteq \mathcal S_s \subseteq \fancyscript{S}\subseteq \fancyscript{S}' \subseteq \mathcal S_s' \subseteq \Sigma _s' \subseteq \mathcal S_{1/2}' . \end{aligned}$$

(2.1)

Furthermore, $\mathcal S_{1/2}$ is dense in all test function spaces $\mathcal S_s$, $\Sigma _s$ and $\fancyscript{S}$, and weakly dense in any of the distribution spaces in (2.1). In particular, the first four inclusions in (2.1) are dense. Here and in what follows it is understood that all inclusions are continuous.

Gelfand–Shilov spaces are invariant under several transformations, for example the Fourier transforms. More precisely, for any $f\in \fancyscript{S}'(\mathbf R^{d})$, let $\fancyscript{F}f$ be the Fourier transform of $f$, which takes the form

$$\begin{aligned} \widehat{f}(\xi ) = (\fancyscript{F}f)(\xi ) = (2\pi )^{-d/2}\int f(x)e^{-i\langle x,\xi \rangle }\, dx \end{aligned}$$

when $f\in \fancyscript{S}(\mathbf R^{d})$. We note that $\fancyscript{F}$ restricts to a homeomorphism on $\mathcal S_{1/2}(\mathbf R^{d})$, and then extends uniquely to a homeomorphism on any of the spaces in (2.1).

An other type of operator possessing similar invariance properties concerns the operator $A$, defined by

$$\begin{aligned} (Aa)(x,y) = (2 \pi ) ^{-d/2} \int a((y-x)/2,\xi ) e ^{-i\langle x+y,\xi \rangle }\,d\xi , \end{aligned}$$

(2.2)

when $a\in \fancyscript{S}(\mathbf R^{2d})$. In fact, by straight-forward computations it follows that $A = \fancyscript{F}_2^{-1}\circ U$, where $(\fancyscript{F}_2F)(x,\xi )$ is the partial Fourier transform of $F(x,y)$ with respect to the $y$-variable, and $U$ is the linear map on ${\mathcal {S}}_{1/2}^{\prime }(\mathbf R^{2d})$, given by

$$\begin{aligned} (UF)(x, y) = F((y-x)/2,-(x+y)),\quad x,y\in \mathbf R^{d}, F \in {\mathcal {S}}_{1/2}^{\prime }(\mathbf R^{2d}). \end{aligned}$$

It follows that $A$ is a homeomorphism on $\mathcal S_{1/2}(\mathbf R^{2d})$, which is uniquely extendable to homeomorphisms on any of the spaces in (2.1), since the similar is true for $\fancyscript{F}_2$ and $U$.

We remark that the Weyl calculus within the theory of pseudo-differential operators is strongly related to the operator $A$. In fact, for $a \in \fancyscript{S}(\mathbf R ^{2d})$, the Weyl quantization ${\text {Op}}^{w}(a)$, is the operator from $\fancyscript{S}(\mathbf R ^{d})$ to $\fancyscript{S}^{\prime }(\mathbf R ^{d})$, defined by the formula

$$\begin{aligned} ({\text {Op}}^{w}(a)f)(x)= (2 \pi ) ^{-d} \iint a((x+y)/2,\xi )f(y) e ^{i\langle x-y,\xi \rangle } dy \,d\xi . \end{aligned}$$

The integral kernel of ${\text {Op}}^{w}(a)$ is equal to

$$\begin{aligned} (x,y)\mapsto (2 \pi ) ^{-d/2} (Aa)(-x,y). \end{aligned}$$

(2.3)

For arbitrary $a \in \mathcal S_{1/2}' (\mathbf R ^{2d})$, ${\text {Op}}^{w}(a)$ is defined as the operator with kernel given by (2.3).

By straight-forward computations we get

$$\begin{aligned} {\text {Op}}^{w}(a)=(2 \pi ) ^{-d/2} A(\fancyscript{F} _{\sigma } a ), \end{aligned}$$

(2.4)

where $\fancyscript{F} _{\sigma }$ is the symplectic Fourier transform on $\mathcal S_{1/2}^{'}(\mathbf R ^{2d})$, which takes the form

$$\begin{aligned} (\fancyscript{F} _{\sigma } a)(X)=\widehat{a} (X) =\pi ^{-d} \int a(Y)e^{2i \sigma ( X,Y)}dY, \end{aligned}$$

when $a \in \fancyscript{S}(\mathbf R ^{2d})$.

From these continuity properties in combination with Schwartz kernel theorems in [5, 6, 8], it also follows that the mappings

$$\begin{aligned} A : \mathcal S_s'(\mathbf R^{2d})&\rightarrow \fancyscript{L}(\mathcal S_s(\mathbf R^{d}),\mathcal S_s'(\mathbf R^{d})), \quad s \ge 1/2 \\ A : \Sigma _s'(\mathbf R^{2d})&\rightarrow \fancyscript{L}(\Sigma _s(\mathbf R^{d}),\Sigma _s'(\mathbf R^{d})), \quad s > 1/2 \end{aligned}$$

and

$$\begin{aligned} A : \fancyscript{S}'(\mathbf R^{2d}) \rightarrow \fancyscript{L}(\fancyscript{S}(\mathbf R^{d}),\fancyscript{S}'(\mathbf R^{d})), \end{aligned}$$

are bijections, where $\fancyscript{L}(V_1,V_2)$ is the set of all linear and continuous mappings from the topological vector space $V_1$ to the topological vector space $V_2$.

There are several characterizations of Gelfand–Shilov spaces. For example, in [10], Pilipović have an alternative definition of $\mathcal S_s$ and $\Sigma _s$, using the harmonic oscillator in the definition of $\Vert \, \cdot \, \Vert _{\mathcal S_{s,h}}$. It follows that $\mathcal S_s$ and $\Sigma _s$ in [10] are the same as above, except that $\Sigma _{1/2}$ is non-trivial and dense in $\mathcal S_{1/2}$ in [10], while $\Sigma _{1/2}$ above is trivial.

The following two lemmata show that Gelfand–Shilov spaces can be characterized by estimates of the form

$$\begin{aligned} \Vert x^\alpha f\Vert _{L^p} \le Ch^{|\alpha |}(\alpha !)^s \quad \text {and} \quad \Vert \xi ^\alpha \widehat{f}\Vert _{L^p} \le Ch^{|\alpha |}(\alpha !)^s, \end{aligned}$$

(2.5)

and by

$$\begin{aligned} |f(x)| \le Ce^{- |x|^{1/s}/h} \quad \text {and} \quad |\widehat{f} (\xi )| \le Ce^{- |\xi |^{1/s}/h}. \end{aligned}$$

(2.6)

The proofs are omitted, since the results follow from Theorem 2.3 in [2], and its proof.

Lemma 2.1

Let $s>0$, $f\in \mathcal S'_{1/2}(\mathbf R^{d})$ and let $p\in [2,\infty ]$. Then the following conditions are equivalent:

• (1) there are constants $h >0$ and $C>0$ such that (2.5) holds;

• (2) there are constants $h >0$ and $C>0$ such that (2.6) holds;

• (3) $f\in \mathcal S_s(\mathbf R^{d})$.

Lemma 2.2

Let $s>0$, $f\in \mathcal S'_{1/2}(\mathbf R^{d})$ and let $p\in [2,\infty ]$. Then the following conditions are equivalent:

• (1) for every $h>0$, there is a constant $C>0$ such that (2.5) holds;

• (2) for every $h>0$, there is a constant $C>0$ such that (2.6) holds;

• (3) $f\in \Sigma _s(\mathbf R^{d})$.

Next we recall the definition of Gevrey classes and their distribution spaces. Let $s>1$, $h>0$, $\Omega$ be an open set in $\mathbf R^{d}$, and let $K\subset \Omega$ be compact. Then $\mathcal E_{s,h,K}(\Omega )$ is the set of all $\varphi \in C^{\infty }({\Omega })$ such that

$$\begin{aligned} \Vert \varphi \Vert _{\mathcal E_{s,h,K}} \equiv \sup _{\beta \in \mathbf N^{d}}\sup _{x\in K} \frac{|D^{\beta }\varphi (x)|}{(\beta !)^sh^{|\beta |}} \end{aligned}$$

(2.7)

is finite, and $\mathcal D_{s,h}(K)$ consists of all $\varphi \in \mathcal E_{s,h,K}(\Omega )$ such that ${\text {supp}} \varphi \subseteq K$, and the norm (2.7) is finite.

Let $\{ K_n \} _{n\ge 1}$ be an exhaustive sequence of compact sets to $\Omega$. That is $K_n\subseteq K_{n+1}$ for every $n\ge 1$, and for every $x\in \Omega$, there is an open neighborhood $U$ of $x$ and integer $n_0\ge 1$ such that $U\subset K_n$ when $n\ge n_0$. Then the spaces $\mathcal D_s(\Omega )$, $\Delta _s(\Omega )$ and $\mathcal E_s(\Omega )$ are defined by

$$\begin{aligned} \mathcal D_s(\Omega )= & {} \underset{n}{\text {ind lim}} (\underset{h}{\text {ind lim}}~\mathcal D_{s,h}(K_n)),\\ \Delta _s(\Omega )= & {} \underset{n}{\text {ind lim}} (\projlim _{h} \mathcal D_{s,h}(K_n)),\\ \mathcal E_s(\Omega )= & {} \projlim _{n}( \underset{h}{\text {ind lim}}~\mathcal E_{s,h,K_n}(\Omega )), \end{aligned}$$

The dual spaces of $\mathcal D_s(\Omega )$ and $\Delta _s(\Omega )$, denoted by $\mathcal D_s^{\prime }(\Omega )$, $\Delta _s^{\prime }(\Omega )$ respectively, are the sets of ultra-distribution on $\Omega$ of Roumieu type and Beurling type of order $s$, respectively. The dual space of $\mathcal E_s(\Omega )$ is denoted by $\mathcal E_s^{\prime }(\Omega )$, is set of ultra-distribution of compact support of Roumieu type. We have

$$\begin{aligned} \mathcal D_s(\mathbf R^{d})&\subseteq \mathcal S_s(\mathbf R^{d})\subseteq \mathcal E_s(\mathbf R^{d}), \quad \mathcal E_s'(\mathbf R^{d}) \subseteq \mathcal S_s'(\mathbf R^{d}) \subseteq \mathcal D_s'(\mathbf R^{d}), \\ \mathcal D_s(\Omega )&= \mathcal E_s' (\Omega )\cap \mathcal E_s(\Omega ), \quad \mathcal S_s(\mathbf R^{d}) \subseteq \mathcal S_s'(\mathbf R^{d})\subseteq \mathcal D_s'(\mathbf R^{d}), \end{aligned}$$

for admissible $s$, when the dualities between the test function spaces and their distribution spaces are interpreted by the unique extensions of the $L^2$-form on $\mathcal D_s$ or on $\mathcal S_s$. (cf. the analysis in Section 8.4 in [5, 6]).

There are several extensions of definitions of Gelfand–Shilov and Gevrey classes, where the factors $\alpha !^s$ in the definition of the semi-norms are replaced by factors of more general kinds.

We recall that differentiations of Gevrey or Gelfand–Shilov distributions are defined in the usual way, giving that most of the usual rules hold. Especially it follows from Leibniz rule that

$$\begin{aligned} D^{\alpha }x^{\beta } f(x)= \sum _{\alpha _0\le \alpha ,\beta } (-i) ^{|\alpha _0|} {\alpha \atopwithdelims (){\alpha _0}} {\beta \atopwithdelims ()\alpha _0} \alpha _0! x^{\beta -\alpha _0} D^{\alpha -\alpha _0}f(x), \end{aligned}$$

(2.8)

for an admissible distribution $f$. Furthermore, by applying the Fourier transform to this formula we get

$$\begin{aligned} x^{\alpha }D^{\beta }f(x) = \sum _{\alpha _0 \le \alpha ,\beta } i^{|\alpha _0|} {\alpha \atopwithdelims ()\alpha _0} {\beta \atopwithdelims ()\alpha _0}\alpha _0! D^{\beta -\alpha _0}(x^{\alpha -\alpha _0}f(x)). \end{aligned}$$

(2.9)

Since we are especially interested in the behaviour of corresponding Schwartz kernels, the following result is important to us. The proof is omitted since it can be found in [8].

Proposition 2.3

Let $\Omega _j \subset \mathbf R^{d_j}$, $j=1,2$, be open. Then the following is true:

• (1) if $s>1$ and $T$ is a linear and continuous operator from $\mathcal D_s(\Omega _1)$ to $\mathcal D_s^{\prime }(\Omega _2)$, then there is a unique ultra-distribution $K=K_T\in \mathcal D_s^{\prime }(\Omega _2 \times \Omega _1)$ such that

  $$\begin{aligned} (T\varphi _1,\varphi _2)=(K,\varphi _2 \otimes \overline{\varphi }_1), \quad \varphi _1 \in \mathcal D_s(\Omega _1), \varphi _2 \in \mathcal D_s(\Omega _2); \end{aligned}$$

  (2.10)

  Conversely, if $K=K_T\in \mathcal D_s^{\prime } (\Omega _2 \times \Omega _1)$ and $T$ is defined by (2.10), then $T$ is a linear and continuous operator from $\mathcal D_s(\Omega _1)$ to $\mathcal D_s^{\prime }(\Omega _2)$;

• (2) if $s\ge 1/2$ and $T$ is a linear and continous operator from $\mathcal S_s(\mathbf R^{d_1})$ to $\mathcal S_s^{\prime }(\mathbf R^{d_2})$, then there is a unique tempered ultradistribution $K=K_T \in \mathcal S_s^{\prime } (\mathbf R^{d_2} \times \mathbf R^{d_1})$ such that

  $$\begin{aligned} (T\varphi _1,\varphi _2)=(K,\varphi _2 \otimes \overline{\varphi }_1), \quad \varphi _1 \in \mathcal S_s(\mathbf R^{d_1}), \varphi _2 \in \mathcal S_s(\mathbf R^{d_2}); \end{aligned}$$

  (2.11)

  Conversely, if $K=K_T \in \mathcal S_s^{\prime }(\mathbf R^{d_2} \times \mathbf R^{d_1})$ and $T$ is defined by (2.11) , then $T$ is a linear and continuous operator from $\mathcal S_s(\mathbf R^{d_1})$ to $\mathcal S_s^{\prime }(\mathbf R^{d_2})$.

The same is true after $\mathcal D_s$, $\mathcal D_s^{\prime }$, $\mathcal S_s$ and $\mathcal S_s^{\prime }$ are replaced by $\Delta _s$, $\Delta _s^{\prime }$, $\Sigma _s$ and $\Sigma _s^{\prime }$, respectively.

Recall that if $s \ge 1/2$, and $T$ is a linear and continuous operator from $\mathcal S_s(\mathbf R^d)$ to $\mathcal S_s^{\prime }(\mathbf R^d)$, then $T$ is called positive semi-definite if

$$\begin{aligned} (T\varphi ,\varphi )_{L^2(\mathbf R^{d})} \ge 0, \end{aligned}$$

(2.12)

for every $\varphi \in \mathcal S_s (\mathbf R^d)$, and then we write $T\ge 0$. (Here $(\, \cdot \, {\,} , \, \cdot \, )_{L^2}$ is the unique extension of the $L^2$-form from $\mathcal S_s(\mathbf R^{d})$ to $\mathbf C$ into the continuous map from $\mathcal S_s'(\mathbf R^{d}) \times \mathcal S_s(\mathbf R^{d})$ to $\mathbf C$.)

Furthermore, if more restrictive $s>1$, $\Omega \subset \mathbf R^d$ is an open set, and $T$ is a linear and continous operator from $\mathcal D_s(\Omega )$ to $\mathcal D_s^{\prime }(\Omega )$, then $T$ is still called positive semi-definite when (2.12) holds for every $\varphi \in \mathcal D_s(\Omega )$.

Since $\mathcal D_s({\mathbf R^d})$ is dense in $\mathcal S_s({\mathbf R^d})$ when $s>1$, it follows that if $T$ from $\mathcal D_s({\mathbf R^d})$ to $\mathcal D_s^{\prime }({\mathbf R^d})$ is positive semi-definite and extendable to a continuous map from $\mathcal S_s({\mathbf R^d})$ to $\mathcal S_s^{\prime }({\mathbf R^d})$, then this extension is unique and $T$ is positive semi-definite as an operator from $\mathcal S_s({\mathbf R^d})$ to $\mathcal S_s^{\prime }({\mathbf R^d})$.

Definition 2.4

Let $\Omega \subset \mathbf R^d$ be open.

• (1) if $s>1$, then $\mathcal D_{0,s} ^{\prime }(\Omega \times \Omega )$ consists of all $K\in \mathcal D_s^{\prime } (\Omega \times \Omega )$ such that $T_K$ is a positive semi-definite operator from $\mathcal D_s({\Omega })$ to $\mathcal D_s^{\prime }({\Omega })$;

• (2) if $s>1$, then $\Delta _{0,s} ^{\prime }(\Omega \times \Omega )$ consists of all $K\in \Delta _s^{\prime } (\Omega \times \Omega )$ such that $T_K$ is a positive semi-definite operator from $\Delta _s({\Omega })$ to $\Delta _s^{\prime }({\Omega })$;

• (3) if $s\ge 1/2$, then $\mathcal S_{0,s} ^{\prime }(\mathbf R^{2d})$ consists of all $K\in \mathcal S_s^{\prime } (\mathbf R^{2d})$ such that $T_K$ is a positive semi-definite operator from $\mathcal S_s(\mathbf R^{d})$ to $\mathcal S_s^{\prime }(\mathbf R^{d})$;

• (4) if $s>1/2$, then $\Sigma _{0,s} ^{\prime }(\mathbf R^{2d})$ consists of all $K\in \Sigma _s^{\prime } (\mathbf R^{2d})$ such that $T_K$ is a positive semi-definite operator from $\Sigma _s(\mathbf R^{d})$ to $\Sigma _s^{\prime }(\mathbf R^{d})$.

We shall also consider distributions which are positive with respect to a non-commutative convolution. More precisely, let $a$ and $\varphi$ be appropriate (ultra-) distribution respective test function on $\mathbf R^{d}$, and let $B$ be an appropriate smooth function on $\mathbf R^{2d}$. Then the $B$-convolution $a*_B\varphi$ between $a$ and $\varphi$ is defined by

$$\begin{aligned} (a*_B\varphi )(x) \equiv \langle a(x-\, \cdot \, ),\quad B(x,\, \cdot \, ) \varphi \rangle , \end{aligned}$$

provided the right-hand side is well-defined as an (ultra-)distribution.

We usually assume that $B\in \mathcal E_s(\mathbf R^{2d})$ for some $s\ge 1/2$, and satisfies conditions of the form

$$\begin{aligned} \sup _{\alpha }\sup _{x,y} e^{-\varepsilon (|x|^{1/s}+|y|^{1/s})} \left( \frac{|D^{\alpha }(B(x,y)^{-1})|}{(\alpha !)^sh^{|\alpha |}} +\frac{|D^{\alpha }B(x,y)|}{(\alpha !)^sh^{|\alpha |}}\right) <\infty . \end{aligned}$$

(2.13)

Lemma 2.5

Let $s\ge 1/2$ and let $B\in \mathcal E_s(\mathbf R^{2d})$. Then the following is true:

• (1) if for every $\varepsilon >0$, there is a constant $h>0$ such that (2.13) holds, then

  $$\begin{aligned} \Phi \in \mathcal S_s(\mathbf R^{2d}) \Longleftrightarrow B\cdot \Phi \in \mathcal S_s(\mathbf R^{2d}) \end{aligned}$$

  and

  $$\begin{aligned} \Phi \in \mathcal S_s'(\mathbf R^{2d}) \Longleftrightarrow B\cdot \Phi \in \mathcal S_s'(\mathbf R^{2d})\mathrm{;} \end{aligned}$$

• (2) if (2.13) holds for every $\varepsilon >0$ and $h>0$, then

  $$\begin{aligned} \Phi \in \Sigma _s(\mathbf R^{2d})\Longleftrightarrow B\cdot \Phi \in \Sigma _s(\mathbf R^{2d}) \end{aligned}$$

  and

  $$\begin{aligned} \Phi \in \Sigma _s'(\mathbf R^{2d}) \Longleftrightarrow B\cdot \Phi \in \Sigma _s'(\mathbf R^{2d}). \end{aligned}$$

Proof

The result follows from Theorem A in [10]. $\square$

A consequence of the previous lemma is that $a*_B\varphi$ is well-defined when $B$ is the same as in Lemma 2.5(1), and $s>1$, $a\in \mathcal D_s^{\prime }(\mathbf R^d)$ and $\varphi \in \mathcal D_s(\mathbf R^d)$, or $s\ge 1/2$, $a\in \mathcal S_s^{\prime }(\mathbf R^d)$ and $\varphi \in \mathcal S_s(\mathbf R^d)$. The kernel of the map $\varphi \mapsto a*_B\varphi$ is given by

$$\begin{aligned} K(x,y)=a(x-y)B(x,y), \end{aligned}$$

and we note that $K(x,y) \in \mathcal D_s^{\prime }(\mathbf R^{2d})$, if and only if $a\in \mathcal D_s(\mathbf R^{d})$, and $K(x,y) \in \mathcal S_s^{\prime }(\mathbf R^{2d})$, if and only if $a\in \mathcal S_s(\mathbf R^{d})$. Similar facts hold if instead $B$ is the same as in Lemma 2.5(2), and $\mathcal D_s$, $\mathcal D_s^{\prime }$, $\mathcal S_s$ and $\mathcal S_s^{\prime }$ are replaced by $\Delta _s$, $\Delta _s^{\prime }$, $\Sigma _s$ and $\Sigma _s^{\prime }$, respectively.

Let $\mathcal D_{B,s,+}^{\prime }(\mathbf R^d)$ be the set of all $a\in \mathcal D_s^{\prime }(\mathbf R^d)$ such that the operator defined by the map $\varphi \mapsto a*_B\varphi$ is positive semi-definite. Also let $\mathcal S_{B,s,+}^{\prime }(\mathbf R^d)$ be the set of all $a\in \mathcal S_s^{\prime }(\mathbf R^d)$ such that the operator defined by the map $\varphi \mapsto a*_B\varphi$ is positive semi-definite.

We consider elements $a$ in Gelfand–Shilov classes of distributions which are positive with respect to the twisted convolution. That is, $a$ should fulfill

$$\begin{aligned} (a *_\sigma \varphi ,\varphi ) \ge 0, \end{aligned}$$

(2.14)

for every $\varphi$. For this reason we make the following definition.

Definition 2.6

Let $s\ge 1/2$.

• (1) $\mathcal S_{s,+}^{\prime }(\mathbf R^{2d})$ is the set of all $a\in \mathcal S_s^{\prime }(\mathbf R^{2d})$ such that (2.14) holds for every $\varphi \in \mathcal S_s(\mathbf R^{2d})$.

• (2) If in addition $s>1$, then $\mathcal D_{s,+}^{\prime }(\mathbf R^{2d})$ is the set of all $a\in \mathcal D_s^{\prime } (\mathbf R^{2d})$ such that (2.14) holds for every $\varphi \in \mathcal D_s(\mathbf R^{2d})$.

• (3) $\fancyscript{S}_+^{\prime }(\mathbf R^{2d})$ is the set of all $a\in \fancyscript{S}^{\prime }(\mathbf R^{2d})$ such that (2.14) holds for every $\varphi \in \fancyscript{S}(\mathbf R^{2d})$.

• (4) $\fancyscript{D}_+^{\prime }(\mathbf R^{2d})$ is the set of all $a\in \fancyscript{D}^{\prime }(\mathbf R^{2d})$ such that (2.14) holds for every $\varphi \in C_0^{\infty }(\mathbf R^{2d})$.

• (5) The set $C_+(\mathbf R^{2d})$ consists of all $a\in C(\mathbf R^{2d})$ such that

  $$\begin{aligned} \sum _{j,k}a(X_j-X_k)e^{2i\sigma ( X_j,X_k)} c_j\overline{c_k} \ge 0, \end{aligned}$$

  for every finite sets

  $$\begin{aligned} \{ X_1,X_2,\ldots ,X_N \} \subseteq \mathbf R^{2d} \quad \text {and}\quad \{ c_1,c_2,\ldots , c_N \} \subseteq \mathbf C. \end{aligned}$$

The following result can be found in the Theorem 2.6, Proposition 3.2 and Theorem 3.13 in [11]. Later on we shall prove an analogue in the frame-work of Gelfand–Shilov spaces.

Proposition 2.7

Let $\Omega \subseteq \mathbf R^{2d}$ be a neighborhood of the origin. Then the following is true:

• (1) $\fancyscript{D}_+^{\prime }(\mathbf R^{2d})= \fancyscript{S}_+^{\prime }(\mathbf R^{2d})$;

• (2) $C_+(\mathbf R^{2d})\subseteq \fancyscript{S}_+^{\prime }(\mathbf R^{2d}) \cap L^2(\mathbf R^{2d})\cap L^{\infty }(\mathbf R^{2d}) \cap \fancyscript{F} L^{\infty }(\mathbf R^{2d})$;

• (3) $\fancyscript{S}_+^{\prime }(\mathbf R^{2d}) \cap C(\Omega )=C_+(\mathbf R^{2d})$;

• (4) $\fancyscript{S}_+^{\prime }(\mathbf R^{2d}) \cap C^{\infty }(\Omega )= C_+(\mathbf R^{2d})\cap \fancyscript{S}(\mathbf R^{2d})$.

In the proof of Proposition 2.7, and in proofs of related results in Sect. 5, the following proposition on estimates of the trace-norm is important. The result follows from Proposition 1.5 in [11] and the definitions. Here recall that we identify linear operators with their kernels. We recall that for any linear operator $T$ on $L^2(\mathbf R^{d})$, the trace and Hilbert–Schmidt norms of $T$ are defined by

$$\begin{aligned} \Vert T\Vert _{{\text {Tr}}} \equiv \sup \sum _{j\in J} |(Tf_j,g_j)| \quad \text {and}\quad \Vert T\Vert _{{\text {HS}}} \equiv \sup \left( \sum _{j\in J} |(Tf_j,g_j)|^2\right) ^{1/2}, \end{aligned}$$

where the suprema are taken over all orthonormal sequences $\{ f_j\} _{j\in J}$ and $\{ g_j\} _{j\in J}$ in $L^2(\mathbf R^{d})$. If $\Vert T\Vert _{{\text {Tr}}} <\infty$, then $T$ is called a trace-class operator, and if $\Vert T\Vert _{{\text {HS}}} <\infty$, then $T$ is called a Hilbert–Schmidt operator.

Proposition 2.8

Let $s\ge 1/2$, and let $a\in \mathcal S_s^{\prime } (\mathbf R^{2d})$ be such that $Aa$ is a trace-class operator on $L^2({\mathbf R^d})$. Then $a\in L^{\infty }({\mathbf R^{2d}})$, and

$$\begin{aligned} \Vert a\Vert _ {L^{\infty }} \le (2/\pi )^{d/2} \Vert Aa\Vert _{{\text {Tr}}}. \end{aligned}$$

For future reference we note that if $T$ is a linear and continuous operator on $L^2(\mathbf R^{d})$, then

$$\begin{aligned} \Vert T\Vert _{L^2}=\Vert T\Vert _{{\text {HS}}}\le \Vert T\Vert _{{\text {Tr}}}, \end{aligned}$$

where the equality follows from the definitions and the fact that we identify operators with their kernels. We also note that $T$ is a rank-one operator, if and only if its kernel $T(x,y)=g(x)f(y)$, for some $f,g\in L^2(\mathbf R^{d})$, i. e. $T=g\otimes f$, and then

$$\begin{aligned} \Vert T\Vert _{L^2} = \Vert T\Vert _{{\text {HS}}} =\Vert T\Vert _{{\text {Tr}}} =\Vert f\Vert _{L^2}\Vert g\Vert _{L^2}. \end{aligned}$$

We also note that for a trace-class operator $T$, its trace

$$\begin{aligned} {\text {Tr}}(T) \equiv \sum _{j\in J} (Tf_j,f_j)_{L^2} \end{aligned}$$

is well-defined and independent of the choice of orthonormal basis $\{f_j\} _{j\in J}$.

3 Gelfand–Shilov properties for kernels to positive semi-definite operators

In this section, we study kernels to positive semi-definite operators. Especially we show that the Gelfand–Shilov properties of such kernels completely depends on their behavior along the diagonals. As an application we deduce

$$\begin{aligned} \mathcal D_{B,s,+}^{\prime }(\mathbf R^d)= \mathcal S_{B,s,+}^{\prime }(\mathbf R^d) \quad \text {and}\quad \Delta _{B,s,+}^{\prime }(\mathbf R^d)= \Sigma _{B,s,+}^{\prime }(\mathbf R^d). \end{aligned}$$

More precisely, we have the following result. Here we use the convention

$$\begin{aligned} K_\chi (x,y) \equiv \chi (x-y)K(x,y) \quad \text {and}\quad K_{k,\chi } (x,y) \equiv \chi (x-y)K_k(x,y), \end{aligned}$$

when $K$ and $K_k$ are Gelfand–Shilov distributions or Gevrey distributions on $\mathbf R^{2d}$ and $\chi$ belongs to an appropriate Gelfand–Shilov space. The element $K_\chi$ is called the diagonal localization of K with respect to $\chi$.

Theorem 3.1

Let $s>1$, and let $\chi _1\in \mathcal D_s(\mathbf R^{d})$ and $\chi _2 \in \Delta _s(\mathbf R^{d})$ be such that $\chi _k(0)\ne 0$, $k=1,2$. Also let $K_1\in \mathcal D_{0,s} ^{\prime } (\mathbf R^{2d})$, $K_2\in \Delta _{0,s} ^{\prime } (\mathbf R^{2d})$, and let $K_{k,\chi _k}$ be the diagonal localization of $K_k$ with respect to $\chi _k$, $k=1,2$. Then the following is true:

• (1) if $K_{1,\chi _1}\in \mathcal S_s'(\mathbf R^{2d})$, then $K_1\in \mathcal S_s'(\mathbf R^{2d})$;

• (2) if $K_{2,\chi _2}\in \Sigma _s'(\mathbf R^{2d})$, then $K_2\in \Sigma _s'(\mathbf R^{2d})$.

We need some preparations for the proof of Theorem 3.1.

Let $s$, $h$, $h_1$ and $h_2$ be positive constants, and let $\psi \in C ^{\infty } (\mathbf R^{2d})$. Then we let the semi-norms $\Vert \, \cdot \, \Vert ^{(1)}_{\mathcal S_{s,h}}$ and $\Vert \, \cdot \, \Vert ^{(2)}_{\mathcal S_{s,h_1,h_2}}$ be defined by

$$\begin{aligned} \Vert \psi \Vert ^{(1)}_{\mathcal S_{s,h}}\equiv & {} \sup _{\alpha _1,\alpha _2,\beta _1,\beta _2} \sup _{x,y}\frac{|x^{\alpha _1}y^{\alpha _2} D_x^{\beta _1}D_y^{\beta _2}\psi (x,y)|}{(\alpha _1!\alpha _2!\beta _1!\beta _2!)^s h^{|\alpha _1+\alpha _2+\beta _1+\beta _2|}}, \\ \Vert \psi \Vert ^{(2)}_{\mathcal S_{s,h_1,h_2}}\equiv & {} \sup _{\alpha ,\beta _1,\beta _2} \sup _{x,y}\frac{e^{|y|^{1/s}/{h_2}}| x^{\alpha }D_x^{\beta _1}D_y^{\beta _2}\psi (x,y)|}{(\alpha !\beta _1!\beta _2!)^s h_1^{|\alpha +\beta _1+\beta _2|}}. \end{aligned}$$

In the following lemma we list some relations for binomial coefficients.

Lemma 3.2

Let $s > 0$, $s_0 = \max \{1,s\}$ and let $\alpha ,\beta ,\alpha _0 \in \mathbf N^{d}$. Then the following is true:

• (1) if $\alpha _0 \le \alpha$, then

  $$\begin{aligned} {\alpha \atopwithdelims ()\alpha _0} \frac{1}{(\alpha !\beta !)^s} = {\alpha \atopwithdelims ()\alpha _0}^{1-s}{{\alpha _0+\beta }\atopwithdelims ()\alpha _0}^s \frac{1}{((\alpha -\alpha _0 )!(\alpha _0 +\beta )!)^s}, \end{aligned}$$

  (3.1)

  and

  $$\begin{aligned} \sum _{\alpha _0 \le \alpha } {\alpha \atopwithdelims ()\alpha _0}^{1-s} {{\alpha _0+\beta }\atopwithdelims ()\alpha _0}^s \le 2^{s_0|\alpha |+s|\beta |} \mathrm{;} \end{aligned}$$

  (3.2)

• (2) if $\alpha _0\le \alpha ,\beta$, then

  $$\begin{aligned}&{\alpha \atopwithdelims ()\alpha _0}{\beta \atopwithdelims ()\alpha _0} \left( \frac{(\alpha \!-\!\alpha _0 )!(\beta -\alpha _0 )!}{\alpha !\beta !} \right) ^s \alpha _0 ! ={\alpha \atopwithdelims ()\alpha _0} ^{1-s}{\beta \atopwithdelims ()\alpha _0}^{1-s} (\alpha _0 !)^{1-2s}\mathrm{;}\quad \end{aligned}$$

  (3.3)

• (3) if $s\ge 1/2$, then

  $$\begin{aligned} {\alpha \atopwithdelims ()\alpha _0} ^{1-s}{\beta \atopwithdelims ()\alpha _0}^{1-s}(\alpha _0 !)^{1-2s} \le \frac{1}{2}\left( {\alpha \atopwithdelims ()\alpha _0} + {\beta \atopwithdelims ()\alpha _0} \right) . \end{aligned}$$

  (3.4)

Proof

The formulae (3.1) and (3.3) follow by straight-forward computations and are left for the reader.

We first consider the case $s \le 1$ when proving (3.2). By Hölder’s inequality we get

$$\begin{aligned} \sum _{\alpha _0 \le \alpha } {\alpha \atopwithdelims ()\alpha _0}^{1-s} {{\alpha _0+\beta }\atopwithdelims ()\alpha _0}^s&\le \left( \sum _{\alpha _0 \le \alpha } {\alpha \atopwithdelims ()\alpha _0}\right) ^{1-s} \left( \sum _{\alpha _0 \le \alpha } {{\alpha _0+\beta }\atopwithdelims ()\alpha _0} \right) ^s \\&\le \left( \sum _{\alpha _0 \le \alpha } {\alpha \atopwithdelims ()\alpha _0}\right) ^{1-s} \left( \sum _{\alpha _0 \le \alpha +\beta } {{\alpha +\beta }\atopwithdelims ()\alpha _0} \right) ^s = 2^{|\alpha |+s|\beta |}, \end{aligned}$$

and (3.2) follows in this case.

If instead $s>1$ we get

$$\begin{aligned} \sum _{\alpha _0 \le \alpha } {\alpha \atopwithdelims ()\alpha _0}^{1-s} {{\beta +\alpha _0}\atopwithdelims ()\alpha _0}^s&\le \sum _{\alpha _0 \le \alpha +\beta } {\alpha +\beta \atopwithdelims ()\alpha _0}^s \\&\le \left( {\sum _{\alpha _0 \le \alpha +\beta } {\alpha +\beta \atopwithdelims ()\alpha _0}} \right) ^s = 2^{s|\alpha +\beta |}, \end{aligned}$$

and (3.2) follows.

In the same way, Cauchy Schwartz inequality gives

$$\begin{aligned} {\alpha \atopwithdelims ()\alpha _0} ^{1-s}{\beta \atopwithdelims ()\alpha _0}^{1-s} (\alpha _0 !)^{1-2s}&\le \frac{1}{2}\left( {\alpha \atopwithdelims ()\alpha _0} ^{2-2s} + {\beta \atopwithdelims ()\alpha _0}^{2-2s} \right) \\&\le \frac{1}{2}\left( {\alpha \atopwithdelims ()\alpha _0} + {\beta \atopwithdelims ()\alpha _0} \right) , \end{aligned}$$

which gives (3.4). $\square$

The following lemma deals with semi-norms equivalences. In particular, necessary and sufficient conditions are explained for estimates

$$\begin{aligned} \Vert \psi _0 \Vert ^{(1)}_{\mathcal S_{s,h}}&\le C\Vert \psi _0 \Vert ^{(2)} _{\mathcal S_{s,h_1,h_2}} \end{aligned}$$

(3.5)

and

$$\begin{aligned} \Vert \psi _0\Vert ^{(2)}_{\mathcal S_{s,h_1,h_2}}&\le C\Vert \psi _0\Vert ^{(1)} _{\mathcal S_{s,h}}. \end{aligned}$$

(3.6)

should hold.

Lemma 3.3

Let $s\ge 1/2$, and let $\psi _0 \in C^{\infty }(\mathbf R^{2d})$. Then the following is true:

• (1) for every $h>0$, there are constants $C,h_1,h_2>0$ such that (3.5) holds. Moreover, for every $h>0$, there are constants $C,h_1,h_2>0$ such that (3.6) holds;

• (2) for every $h_1,h_2>0$, there are constants $C,h>0$ such that (3.5) holds. Moreover, for every $h_1,h_2>0$, there are constants $C,h>0$ such that (3.6) holds;

• (3) let $\psi _1$ and $\psi _2$ be given by

  $$\begin{aligned} \psi _1(x,y)=\psi _0(x,x-y), \quad \text {and} \quad \psi _2(x,y)=\psi _0(x+y,x). \end{aligned}$$

  (3.7)

  Then

  $$\begin{aligned} \Vert \psi _j\Vert ^{(1)}_{\mathcal S_{s,h}} \le \Vert \psi _0\Vert ^{(1)}_{\mathcal S_{s,r_0h}}, \quad \text {and} \quad \Vert \psi _0\Vert ^{(1)}_{\mathcal S_{s,h}} \le \Vert \psi _j\Vert ^{(1)}_{\mathcal S_{s,r_0h}} , \end{aligned}$$

  for $j=0,1,2$, where $r_0=\min (2^{-1},2^{-s})$.

We note that Lemma 2.1 guarantees that Gelfand–Shilov spaces and their distribution spaces are invariant under pullbacks of linear bijections on $\mathbf R^{d}$. Lemma 3.3(3) give a more detailed relation between the involved semi-norms, for certain types of these pullbacks.

Proof

The assertions (1) and (2) are related to Lemma 2.1, and follow from Corollary 2.5 in [2] and its proof. The assertion (3) follows by straight-forward elaborations with the semi-norm $\Vert \psi _j\Vert ^{(1)}_{\mathcal S_{s,h}}$, for $j=0,1,2$. (cf, e.g. [2, 4]). In order to be self-contained we here give a proof of the first inequality in the case $j=2$ and $s\le 1$. The other statements follow by similar arguments, and are left for the reader.

By Lemma 3.2 we get

$$\begin{aligned}&\frac{|y^{\alpha _1}(x-y)^{\alpha _2}\psi _2(y,x-y)|}{(\alpha _1! \alpha _2!)^sh^{|\alpha _1+\alpha _2|}} = \frac{|y^{\alpha _1}(x-y)^{\alpha _2}\psi _0(x,y)|}{(\alpha _1! \alpha _2!)^sh^{|\alpha _1+\alpha _2|}} \\&\quad \le \sum _{\alpha _0\le \alpha _2} {\alpha _2 \atopwithdelims ()\alpha _0} \frac{|x^{\alpha _2-\alpha _0} y^{\alpha _1+\alpha _0}\psi _0(x,y)|}{(\alpha _1! \alpha _2!)^sh^{|\alpha _1+\alpha _2|}} \\&\quad =\sum _{\alpha _0\le \alpha _2} {{\alpha _2 }\atopwithdelims ()\alpha _0 }^{1-s} {{\alpha _1+\alpha _0 }\atopwithdelims ()\alpha _0 }^s \frac{|x^{\alpha _2-\alpha _0} y^{\alpha _1+\alpha _0}\psi _0(x,y)|}{((\alpha _1+\alpha _0)!(\alpha _2-\alpha _0)!)^s h^{|\alpha _1+\alpha _2|}} \\&\quad \le 2^{s{|\alpha _1|} +|\alpha _2|} \sup _{\alpha _0 \le \alpha _2} \frac{|x^{\alpha _2-\alpha _0} y^{\alpha _1+\alpha _0}\psi _0(x,y)|}{((\alpha _1+\alpha _0)!(\alpha _2-\alpha _0)!)^s h^{|\alpha _1+\alpha _2|}} \end{aligned}$$

By taking $(y,x-y)$ as new variables, the previous inequalities give

$$\begin{aligned} \frac{|x^{\alpha _1}y^{\alpha _2}\psi _2(x,y)|}{(\alpha _1! \alpha _2!)^sh^{|\alpha _1+\alpha _2|}} \le 2^{|\alpha _1+\alpha _2|} \sup _{\alpha _0\le \alpha _2} \frac{|(x+y)^{\alpha _2-\alpha _0} x^{\alpha _1+\alpha _0}\psi _0(x+y,x)|}{((\alpha _1+\alpha _0)!(\alpha _2-\alpha _0)!)^s h^{|\alpha _1+\alpha _2|}}. \end{aligned}$$

Similarly,

$$\begin{aligned} \frac{|D_x^{\beta _1}D_y^{\beta _2}\psi _2(x,y)|}{(\beta _1! \beta _2!)^sh^{|\beta _1+\beta _2|}}&= \frac{|D_x^{\beta _1}D_y^{\beta _2}(\psi _0(x+y,x))|}{(\beta _1! \beta _2!)^sh^{|\beta _1+\beta _2|}}= \frac{|D_x^{\beta _1}((D_y^{\beta _2}\psi _0)(x+y,x))|}{(\beta _1! \beta _2!)^sh^{|\beta _1+\beta _2|}} \\&\le \sum _{\beta _0\le \beta _1}{\beta _1 \atopwithdelims ()\beta _0} \frac{|(D_x^{\beta _1-\beta _0}D_y^{\beta _2+\beta _0} \psi _0)(x+y,x))|}{(\beta _1! \beta _2!)^sh^{|\beta _1+\beta _2|}} \\&\le 2^{|\beta _1+\beta _2|} \sup _{\beta _0 \le \beta _1} \frac{|(D_x^{\beta _1-\beta _0} D_y^{\beta _2+\beta _0}\psi _0)(x+y,x))|}{((\beta _1-\beta _0)!(\beta _2+\beta _0)!)^s h^{|\beta _1+\beta _2|}}. \end{aligned}$$

A combination of these arguments give

$$\begin{aligned}&\frac{|x^{\alpha _1}y^{\alpha _2} D_x^{\beta _1}D_y^{\beta _2}\psi _2(x,y)|}{(\alpha _1!\alpha _2!\beta _1!\beta _2!)^s h^{|\alpha _1+\alpha _2+\beta _1+\beta _2|}} \\&\quad \le \sup \frac{|(x+y)^{\alpha _2-\alpha _0}x^{\alpha _1+\alpha _0} (D_x^{\beta _1-\beta _0}D_y^{\beta _2+\beta _0}\psi _0)(x+y,x))|}{((\alpha _1+\alpha _0)!(\alpha _2-\alpha _0)! (\beta _1-\beta _0)!(\beta _2+\beta _0)!)^s h_1^{|\alpha _1+\alpha _2+\beta _1+\beta _2|}}, \end{aligned}$$

where $h_1=h/2$ and the last supremum is taken over all multi-indices $\alpha _0$ and $\beta _0$ such that $\alpha _0 \le \alpha _2$ and $\beta _0 \le \beta _1$. By taking the supremum over all $x,y\in \mathbf R^{d}$ and $\alpha ,\beta \in \mathbf N^{d}$, we get

$$\begin{aligned} \Vert \psi _2\Vert _{\mathcal S_{s,h}}^{(1)} \le \Vert \psi _0\Vert _{\mathcal S_{s,h/2}}^{(1)}, \end{aligned}$$

which gives the result for $j=2$. The proof is complete. $\square$

Remark 3.4

The conclusions in Lemma 3.3(1) and (2) can be refined in such way that $h$, $h_1$ and $h_2$ are replaced by certain sequences.

In fact, by Corollary 2.5 in [2] and its proof it follows that there are positive sequences

$$\begin{aligned} \{ h_{1,j} \}_{j=1}^\infty ,\quad \{ h_{2,j} \}_{j=1}^\infty \quad \text {and}\quad \{ C_{j} \}_{j=1}^\infty , \end{aligned}$$

(3.8)

and the estimates (3.5) and (3.6) are replaced by

and

then the following is true:

• (1) for any positive sequence $\{ h_j \}_{j=1}^\infty$ such that $\lim _{j\rightarrow \infty }h_j=\infty$, there are positive sequences (3.8) such that

  $$\begin{aligned} \lim _{j\rightarrow \infty }h_{1,j}=\lim _{j\rightarrow \infty }h_{2,j}=\infty , \end{aligned}$$

  (3.9)

  and (3.5)$^{\prime }$ holds for every $j$. Moreover, for any positive sequence $\{ h_j \}_{j=1}^\infty$ such that $\lim _{j\rightarrow \infty }h_j=\infty$, there are positive sequences (3.8) such that (3.9) and (3.6)$^{\prime }$ holds for every $j$;

• (2) for any positive sequences $\{ h_{1,j} \}_{j=1}^\infty$ and $\{ h_{2,j} \}_{j=1}^\infty$ such that (3.9) holds, there are positive sequences $\{ h_{j} \}_{j=1}^\infty$ and $\{ C_{j} \}_{j=1}^\infty$ such that $\lim _{j\rightarrow \infty }h_j=\infty$ and (3.5)$^{\prime }$ holds for every $j$. Moreover, for any positive sequences $\{ h_{1,j} \}_{j=1}^\infty$ and $\{ h_{2,j} \}_{j=1}^\infty$ such that (3.9) holds, there are positive sequences $\{ h_{j} \}_{j=1}^\infty$ and $\{ C_{j} \}_{j=1}^\infty$ such that $\lim _{j\rightarrow \infty }h_j=\infty$ and (3.5)$^{\prime }$ holds for every $j$.

Consequently, if $\mathcal S_{s,h_1,h_2}(\mathbf R^{2d})$ consists of all $\psi \in C^\infty (\mathbf R^{2d})$ such that $\Vert \psi \Vert ^{(2)}_{\mathcal S_{s,h_1,h_2}}<\infty$, then

$$\begin{aligned} \underset{h_1,h_2}{\text {ind lim}}~\mathcal S_{s,h_1,h_2}(\mathbf R^{2d}) = \mathcal S_s(\mathbf R^{2d}) \quad \text {and}\quad \projlim _{h_1,h_2}\mathcal S_{s,h_1,h_2}(\mathbf R^{2d}) = \Sigma _s(\mathbf R^{2d}). \end{aligned}$$

Proof of Theorem 3.1

We start by proving (1). Let $K=K_1$, $\chi =\chi _1$, $K_\chi =K_{1,\chi _1}$, and let $(\, \cdot \, , \, \cdot \, )_K$ be the semi-scalar product on $\mathcal D_s(\mathbf R^d)$ given by

$$\begin{aligned} (\varphi , \psi )_K=(K,\psi \otimes \overline{\varphi }), \end{aligned}$$

for every $\varphi$, $\psi \in \mathcal D_s(\mathbf R^d)$. Also let $\Vert \, \cdot \, \Vert _K$ be the corresponding semi-norm, i.e, $\Vert \varphi \Vert _K$ is defined by

$$\begin{aligned} \Vert \varphi \Vert _K^2=(\varphi , \varphi )_K= (K,\varphi \otimes \overline{\varphi }), \end{aligned}$$

when $\varphi \in \mathcal D_s(\mathbf R^d)$. Since $\mathcal D_s(\mathbf R^d)$ is dense in $\mathcal S_s(\mathbf R^d)$, the result follows if we prove that for every positive $h$, there is a positive constant $C=C_h$, such that

$$\begin{aligned} |(\varphi , \psi )_K| \le C\Vert \varphi \Vert _{S_{s,h}}^{(1)} \Vert \psi \Vert _{S_{s,h}}^{(1)}, \end{aligned}$$

(3.10)

when $\varphi$, $\psi \in \mathcal D_s(\mathbf R^d) \cap \mathcal S_{s,h}(\mathbf R^d)$.

Since $s>1$, $\chi (0) \ne 0$, and $K_\chi \in \mathcal S_{s}^{\prime } (\mathbf R^{2d} )$, it follows from Theorem 4.1.23 in [7] that $1/\chi \in C_s$ near the origin. Hence $\kappa /\chi \in \mathcal D_s(\mathbf R^d)$ for some $\kappa \in \mathcal D_s(\mathbf R^d)$ which is equal to 1 in a neighborhood $\Omega _0$ of the origin. Since obviously the map $(\varphi ,K)\mapsto \varphi (x-y)K(x,y)$ is continuous from $\mathcal D_s(\mathbf R^d)\times \mathcal S_s^{\prime }(\mathbf R^{2d})$ to $\mathcal S_{s}^{\prime }(\mathbf R^{2d})$, we get

$$\begin{aligned} K_{\kappa } (x,y) = \frac{\kappa (x-y)}{\chi (x-y)} \cdot K_{\chi }(x,y) \in \mathcal S_{s}^{\prime }(\mathbf R^{2d}). \end{aligned}$$

Hence we may assume that $\chi$ in the assumption is equal to 1 in a neighborhood $\Omega$ of the origin.

Let $\phi \in \mathcal D_s(\mathbf R^d)$ be non-negative and chosen such that $\sum _{j\in J}\phi (\, \cdot \, -x_j)=1$, for some lattice $\{x_j\}_{j\in J} \subseteq \mathbf R^d$, and ${\text {supp}}\phi + {\text {supp}}\phi \subset \Omega$. By Cauchy–Schwartz inequality we get

$$\begin{aligned} |(\varphi , \psi )_K| \le \sum _{j,k\in J} |(\varphi _j, \psi _k)_K| \le \sum _{j,k\in J}\Vert \varphi _j\Vert _K\Vert \psi _k\Vert _K, \quad \varphi ,\psi \in \mathcal D_s(\mathbf R^d), \end{aligned}$$

where

$$\begin{aligned} \varphi _j(x)=\varphi (x)\phi (x-x_j), \quad \psi _j(x) =\psi (x)\phi (x-x_j). \end{aligned}$$

Then (3.10) follows if we prove that for every $h>0$, there are $h_1>0$ and $C>0$ such that

$$\begin{aligned} \Vert \varphi _j\Vert _K&\le C \left( \sup _{\alpha ,\beta \in N^d} \sup _{x\in \mathbf R^d} \frac{|x^{\alpha }D^{\beta }\varphi (x)|}{(\alpha ! \beta !)^s h^{|\alpha +\beta |}}\right) \cdot e^{-|x_j|^{1/s}/{h_1}} \nonumber \\&= C\Vert \varphi \Vert ^{(1)}_{\mathcal S_{s,h}} e^{-|x_j|^{1/s}/{h_1}}. \end{aligned}$$

(3.11)

In order to prove (3.11), we note that the support of $\varphi _j(\, \cdot \, +x_j)$ is contained in ${\text {supp}} \phi$. This gives

$$\begin{aligned} \Vert \varphi _j\Vert _K^2=(K_{j,\chi },\varphi _j(\, \cdot \, +x_j) \otimes \overline{\varphi _j(\, \cdot \, +x_j)}), \end{aligned}$$

(3.12)

where

$$\begin{aligned} K_{j,\chi }(x,y)=K(x+x_j,y+x_j)\chi (x-y). \end{aligned}$$

It follows from the definitions that for every $\varepsilon >0$, the sequence

$$\begin{aligned} \{ e^{-{\varepsilon }|x_j|^{1/s}}K_{j,\chi } \} _{j\in J} \end{aligned}$$

is bounded in $\mathcal S_s^{\prime } (\mathbf R^{2d})$ with respect to $j\in J$. It now follows from this fact and (3.12) that for every positive constants $\varepsilon$ and $h$, there is a constant $C_{\varepsilon ,h}$ such that

$$\begin{aligned} \Vert \varphi _j\Vert _K \le C_{\varepsilon ,h} e^{\varepsilon |x_j|^{1/s}/2} \sup _{\alpha ,\beta }\sup _x \frac{|x^{\alpha }D_x^{\beta }\varphi _j(x+x_j)|}{(\alpha ! \beta !)^s h^{|\alpha +\beta |}}. \end{aligned}$$

(3.13)

Let $\psi _0(x,y)=\varphi (x)\phi (y)$, and let $\psi _1$ and $\psi _2$ be as in Lemma 3.3. Then $\psi _2(x,x_j)=\varphi _j(x+x_j)$. By Lemma 3.3 and Remark 3.4, it follows that for every $h>0$, there are constants $C$, $h_1$, $h_2>0$ such that

$$\begin{aligned} \Vert \psi _2\Vert ^{(2)}_{\mathcal S_{s,h_1,h_2}} \le C\Vert \psi _2\Vert ^{(1)}_{\mathcal S_{s,2^{s}h}} \le C\Vert \psi _0\Vert ^{(1)}_{\mathcal S_{s,h}}. \end{aligned}$$

Consequently,

$$\begin{aligned} \sup _{\alpha ,\beta _1,\beta _2} \sup _{x,y}\frac{e^{|y|^{1/s}/{h_2}}|x^{\alpha }D_x^{\beta _1} D_y^{\beta _2}\psi _2(x,y)|}{(\alpha !\beta _1!\beta _2!)^sh_1^{|\alpha +\beta _1+\beta _2|}} \le C\Vert \psi _0\Vert ^{(1)}_{\mathcal S_{s,h}}, \end{aligned}$$

giving that

$$\begin{aligned} |x^{\alpha }D_x^{\beta }\varphi _j(x+x_j)|&= |x^{\alpha }D_x^{\beta }\psi _2(x,x_j)| \nonumber \\&\le C e^{-|x_j|^{1/s}/{h_2}}(\alpha !\beta !)^s h_1^{|\alpha +\beta |}\Vert \psi _0\Vert ^{(1)}_{\mathcal S_{s,h}}. \end{aligned}$$

(3.14)

It now follows from (3.13) and (3.14) that for every $h_3>0$, there is a constant $C_1$ such that

$$\begin{aligned} \Vert \varphi _j\Vert _K&\le C_1 e^{\varepsilon |x_j|^{1/s}/2} e^{-|x_j|^{1/s}/{h_2}} \left( \frac{h_1}{h_3}\right) ^{|\alpha +\beta |} \Vert \psi _0\Vert ^{(1)}_{\mathcal S_{s,h}} \nonumber \\&\le C_{\phi }e^{-(1/h_2-\varepsilon /2)|x_j|^{1/s}} \left( \frac{h_1}{h_3}\right) ^{|\alpha +\beta |} \Vert \varphi \Vert ^{(1)}_{\mathcal S_{s,h}}, \end{aligned}$$

(3.15)

and (3.10) follows if we choose $h_3$ and $\varepsilon$ such that $h_1<h_3$ and $\varepsilon < 2/h_2$. This gives (1).

Next we prove (2). Let $K=K_2$, $\chi =\chi _2$, $K_\chi =K_{2,\chi _2}$, and let $(\, \cdot \, ,\, \cdot \, )_K$ and $\Vert \, \cdot \, \Vert _{K}$ be as above for $\varphi$, $\psi \in \Delta _s(\mathbf R^d)$. Since $\Delta _s(\mathbf R^d)$ is dense in $\Sigma _s(\mathbf R^d)$, the result follows if we prove that there are positive constants $C$ and $h$ such that (3.10) holds when $\varphi$, $\psi \in \Delta _s(\mathbf R^d) \cap \mathcal S_{s,h}(\mathbf R^d)$.

By similar arguments as in the first part of the proof it follows that we may assume that $\chi =1$ in a neighborhood $\Omega$ of the origin. Let $\phi \in \Delta _s(\mathbf R^d)$, $\varphi _j$ and $\psi _k$ be as in the first part of the proof. Evidently, (3.10) follows if we prove that (3.11) holds for some positive constants $C$, $h$ and $h_1$.

By the definitions we have that (3.12) holds, and that for some $\varepsilon >0$, the sequence

$$\begin{aligned} \{ e^{-{\varepsilon }|x_j|^{1/s}}K_{j,\chi } \} _{j\in J} \end{aligned}$$

is bounded in $\Sigma _s^{\prime } (\mathbf R^{2d})$ with respect to $j\in J$. From these facts it follows that for every positive $\varepsilon$, there are positive constants $C_{\varepsilon ,h}$, and $h$ such that (3.13) holds.

Let $\psi _0(x,y)$, $\psi _1(x,y)$ and $\psi _2(x,y)$ be as before. Then for every positive $h_1$ and $h_2$, there are positive constants $C$ and $h$ such that (3.14) holds.

It now follows from (3.13) and (3.14) that for some positive $h_3$ and $C_1$ such that (3.15) holds and (3.10) follows if we choose $h_1$ and $h_2$ such that $h_1<h_3$ and $h_2 < 2/\varepsilon$. This gives (2). $\square$

By choosing $K(x,y)=a(x-y)B(x,y)$ in Theorem 3.1, for suitable $a$ and $B$, we get the following result, which shows that (1) in Proposition 2.7 has an analogue in the framework of Gelfand–Shilov space or Gevrey class.

Theorem 3.5

Let $s\!\!>\!\!1$ and $B\!\in \!\! C_s(\mathbf R^{2d})$ be such that (2.13) holds. Then $\mathcal D_{B,s,+}^{\prime }(\mathbf R^d)= \mathcal S_{B,s,+}^{\prime }(\mathbf R^d)$. In particular, $\mathcal D_{s,+}^{\prime }(\mathbf R^{2d}) = \mathcal S_{s,+}^{\prime }(\mathbf R^{2d})$. The same is true after $\mathcal D_{B,s,+}^{\prime }$ and $\mathcal S_{B,s,+}^{\prime }$ are replaced by $\Delta _{B,s,+}^{\prime }$ and $\Sigma _{B,s,+}^{\prime }$, respectively.

Proof

We only prove the first equality. By straight-forward computation it follows that $T\varphi =a*_B\varphi$ and the kernel

$$\begin{aligned} K(x,y)=a(x-y)B(x,y)\in \mathcal D_{0,s}^{\prime }(\mathbf R^{2d}) \subseteq \mathcal S_{0,s}^{\prime }(\mathbf R^{2d}). \end{aligned}$$

Furthermore, Lemma 2.5 gives

$$\begin{aligned} (x,y) \mapsto \frac{1}{B(x,y)}\cdot K(x,y) = a(x-y)\in \mathcal S_s^{\prime } {(\mathbf R^{2d})}. \end{aligned}$$

This implies that $a\in \mathcal S_s^{\prime } {(\mathbf R^d)}$. $\square$

4 Gevrey properties of kernels to positive semi-definite operators

In this section we deduce smoothness properties of kernels to positive semi-definite operators. In the first part we discuss the results in [12] for distributions in the union $\cup _{t>1}\mathcal D'$. Thereafter we obtain Gevrey regularity properties for such distributions. In contrast to [12], we here allow the Gevrey parameters to be smaller than one.

Proposition 4.1

Let $t>1$ and let $K \in \mathcal D_{0,t}' (\mathbf R^{2d})$. Then $\partial _1^{\alpha }\partial _2^{\alpha }K \in \mathcal D_{0,t}' (\mathbf R^{2d})$. Furthermore, if $K=K_1+iK_2$ is smooth at $(x,x)\in \mathbf R^{2d}$ and $(y,y)\in \mathbf R^{2d}$, where $K_1$ and $K_2$ are real-valued, then

$$\begin{aligned}&(\partial _1^{\alpha }\partial _2^{\alpha }K)(x,x) =(\partial _1^{\alpha }\partial _2^{\alpha }K_1)(x,x) \ge 0,\end{aligned}$$

(4.1)

$$\begin{aligned}&(\partial _1^{\alpha }\partial _2^{\alpha }K_2)(x,x)=0, \end{aligned}$$

(4.2)

and

$$\begin{aligned} |(\partial _1^{\alpha }\partial _2^{\beta }K)(x,y)|^2 \le (\partial _1^{\alpha }\partial _2^{\alpha }K)(x,x) (\partial _1^{\beta }\partial _2^{\beta }K)(y,y). \end{aligned}$$

(4.3)

Remark 4.2

Note that (4.3) is equivalent to

$$\begin{aligned} ((\partial _1^{\alpha }\partial _2^{\beta }K_1)(x,y))^2 +((\partial _1^{\alpha }\partial _2^{\beta }K_2)(x,y))^2 \le (\partial _1^{\alpha }\partial _2^{\alpha }K_1)(x,x) (\partial _1^{\beta }\partial _2^{\beta }K_1)(y,y) \end{aligned}$$

in view of (4.1).

Proof of Proposition 4.1

By a straight-forward computation it follows that

$$\begin{aligned} K_1(x,y)=K_1(y,x), \quad K_2(x,y)=-K_2(y,x). \end{aligned}$$

(4.4)

Moreover,

$$\begin{aligned} (K_1,\phi \otimes \overline{\phi }) =\frac{1}{2}((K,\phi \otimes \overline{\phi })+ \overline{(K, \overline{\phi }\otimes \phi )})\ge 0 \end{aligned}$$

holds for every $\phi \in \mathcal D_t(\mathbf R^{d})$, giving that $K_1\in \mathcal D_{0,t}^{\prime }(\mathbf R^{2d})$.

Let $K_{\alpha }=\partial _1^{\alpha }\partial _2^{\alpha }K$, $K_{j,\alpha }=\partial _1^{\alpha }\partial _2^{\alpha }K_j$ for $j=1,2$, and $\phi \in \mathcal D_t(\mathbf R^d)$. Then

$$\begin{aligned} (K_{\alpha },\phi \otimes \overline{\phi })= (K,(\partial ^{\alpha }\phi )\otimes \overline{(\partial ^{\alpha }\phi )})\ge 0, \end{aligned}$$

and it follows that $K_{\alpha } \in \mathcal D_{0,t}^{\prime }(\mathbf R^{2d})$. By (4.4) we obtain

$$\begin{aligned} K_{2,\alpha }(x,x)=0 \quad \text {and}\quad K_{\alpha }(x,x)=K_{1,\alpha }(x,x). \end{aligned}$$

This gives (4.1) and (4.2).

On the other hand, $K\in \mathcal D_{0,t}^{\prime }(\mathbf R^{2d})$ implies that

$$\begin{aligned} (\phi _1, \phi _2)\mapsto (K, \phi _2\otimes \overline{\phi _1}) \end{aligned}$$

is a semi-scalar product on $\mathcal D_{0,t}(\mathbf R^{2d})$. Hence, Cauchy–Schwartz inequality holds for this product, i.e.

$$\begin{aligned} |(K,\phi _1\otimes \overline{\phi _2})|^2 \le (K,\phi _1\otimes \overline{\phi _1}) (K,\phi _2\otimes \overline{\phi _2}), \end{aligned}$$

for every $\phi _1$, $\phi _2 \in \mathcal D_{t}(\mathbf R^{d})$.

Let $\phi \in \mathcal D_{t}(\mathbf R^{d})$ be real-valued and such that $\int \phi \,dx=1$, and set

$$\begin{aligned} \phi _{x_0,\varepsilon }=\varepsilon ^{-d} \phi ((\, \cdot \, -x_0)/\varepsilon ). \end{aligned}$$

Then

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} (K, \partial ^{\alpha }\phi _{x_0,\varepsilon } \otimes \partial ^{\beta }\phi _{y_0,\varepsilon })&=(-1)^{|\alpha +\beta |} \lim _{\varepsilon \rightarrow 0} (\partial _1^{\alpha } \partial _2^{\beta }K,\phi _{x_0,\varepsilon } \otimes \phi _{y_0,\varepsilon }) \\&= (-1)^{|\alpha +\beta |}\partial _1^{\alpha } \partial _2^{\beta }K(x,y) \end{aligned}$$

(in distribution sense). Hence,

$$\begin{aligned} |(K,\partial ^{\alpha }\phi _{x_0,\varepsilon } \otimes \partial ^{\beta }\phi _{y_0,\varepsilon })|\rightarrow |\partial _1^{\alpha }\partial _2^{\beta }K(x,y)|. \end{aligned}$$

By Cauchy–Schwartz inequality we get

$$\begin{aligned} |(K, \partial ^{\alpha }\phi _{x_0,\varepsilon } \otimes \partial ^{\beta }\phi _{y_0,\varepsilon })|^2 \le (K, \partial ^{\alpha }\phi _{x_0,\varepsilon } \otimes \partial ^{\alpha }\phi _{x_0,\varepsilon }) (K, \partial ^{\beta }\phi _{y_0,\varepsilon } \otimes \partial ^{\beta }\phi _{y_0,\varepsilon }). \end{aligned}$$

Therefore,

$$\begin{aligned} |(\partial _1^{\alpha }\partial _2^{\beta }K)(x,y)|^2 \le (\partial _1^{\alpha }\partial _2^{\alpha }K)(x,x) (\partial _1^{\beta }\partial _2^{\beta }K)(y,y), \end{aligned}$$

(4.5)

and the result follows by combining (4.2) and (4.5). $\square$

Proposition 4.3

Let $s>0$, $t>1$ and let $K \in \mathcal D_{0,t}' (\mathbf R^{2d})$. If $K$ is Gevrey regular of order $s$ at $(x,x)\in \mathbf R^{2d}$ and $(y,y)\in \mathbf R^{2d}$, then $K$ is Gevrey regular of order $s$ at $(x,y)\in \mathbf R^{2d}$ and $(y,x)\in \mathbf R^{2d}$.

Proof

Assume that $K$ is Gevrey regular of order $s$ at $(x_0,x_0)$ and $(y_0,y_0)$. Since $(2\alpha )!\le 2^{|\alpha |}(\alpha !)^2$, it follows that

$$\begin{aligned} 0\le (\partial _1^\alpha \partial _2^\alpha K)(x,x)\le Ch^{2|\alpha |}(\alpha !)^{2s}, \quad x \in B_r(x_0) \end{aligned}$$

and

$$\begin{aligned} 0\le (\partial _1^\alpha \partial _2^\alpha K)(y,y)\le Ch^{2|\alpha |}(\alpha !)^{2s}, \quad y \in B_r (y_0). \end{aligned}$$

for some constants $C,h,r>0$. Hence, if $x\in B_r(x_0)$ and $y\in B_r (y_0)$, then Proposition 4.1 gives

$$\begin{aligned} |(\partial _1^\alpha \partial _2^\beta K)(x,y)| \le Ch^{|\alpha +\beta |}(\alpha ! \beta !)^s, \end{aligned}$$

which implies that $K$ is Gevrey regular of order $s$ at $(x_0,y_0)$. Since $K(x,y)=\overline{K(y,x)}$, it follows that $K$ is Gevrey regular of order $s$ at $(y_0,x_0)$ as well. $\square$

5 Gelfand–Shilov properties for positive elements with respect to the twisted convolution

In this section, we study positive semi-definite elements in $\mathcal S_s^\prime$ and $\Sigma _s^{\prime }$ with respect to the twisted convolution, which have the Gevrey regularity near the origin.

The following theorem is the main result. It shows that (4) in Proposition 2.7 has an analogue in the framework of Gelfand–Shilov spaces or Gevrey classes, provided the smoothness conditions at origin for the involved elements are completed by estimates of the form

$$\begin{aligned} \sup _{\alpha \in \mathbf N^{d}} \left( \frac{|(D_x^\alpha a)(0)|}{(\alpha !) ^s h^{|\alpha |}} \right) < \infty \quad \text {and}\quad \sup _{\alpha \in \mathbf N^{d}} \left( \frac{|(D_\xi ^\alpha a)(0)|}{(\alpha !) ^s h^{|\alpha |}} \right) < \infty . \end{aligned}$$

(5.1)

Theorem 5.1

Let $\Omega \subseteq \mathbf R^{2d}$ be an open neighborhood of origin, $s\ge 1/2$, and assume that $a\in \mathcal S^\prime _{1/2,+} (\mathbf R^{2d}) \cap C^{\infty } (\Omega )$. Then the following is true:

• (1) if (5.1) holds for some $h>0$, then $a\in \mathcal S_s(\mathbf R^{2d})$;

• (2) if (5.1) holds for every $h>0$, then $a\in \Sigma _s(\mathbf R^{2d})$.

Since $\mathcal S_s$ and $\Sigma _t$ are trivial when $s<1/2$ and $t\le 1/2$, the following result is a straight-forward consequence of the previous result.

Theorem 5.2

Let $\Omega \subseteq \mathbf R^{2d}$ be an open neighborhood of origin, and assume that $a\in \mathcal S^\prime _{1/2,+} (\mathbf R^{2d}) \cap C^{\infty } (\Omega )$. Furthermore, assume that one of the following statements hold true:

• (1) $s<1/2$ and (5.1) holds for some $h>0$;

• (2) $s\le 1/2$ and (5.1) holds for every $h>0$.

Then $a=0$.

We need some preparations to prove Theorem 5.1. First we note that the operators

$$\begin{aligned} P_j&= 2 ^{-1}D_{\xi _j} - x_j, \quad \Pi _j = 2^{-1}D_{x_j} + \xi _j, \nonumber \\ T _j&= 2^{-1}D _{\xi _j} + x_j, \quad \Theta _j = 2^{-1}D _{x_j} +\xi _j, \end{aligned}$$

(5.2)

are convenient to use when discussing the operator $a\mapsto (Aa)(x,y)$ in (2.2), because

$$\begin{aligned} A(P_ja)&= x_j Aa,\quad A(\Pi _ja) = -D_{x_j}(Aa), \nonumber \\ A(T_ja)&= y_jAa, \quad A(\Theta _ja) =D_{x_j}(Aa), \end{aligned}$$

(5.3)

which follows by straight-forward computations. As a part of our investigations we show that the operators in (5.2) may replace by the operators $x_j$, $\xi _j$, $D_{x_j}$ and $D_{\xi _j}$ in the definition of Gelfand–Shilov spaces. Here and in what follows we identity the multiplication operator $f\mapsto \varphi \cdot f$ with the function $\varphi$. By (5.2) we get

(5.4)

for $j=1,\ldots ,d$.

We recall that the operator $A$ in Sects. 1 and 2 is continuous on any of the spaces in (2.1).

The following lemma explains commutation rules for the operators in (5.2), and is an immediate consequence of (5.3) and Leibnitz rule.

Lemma 5.3

Let $\alpha , \beta \in \mathbf N^{d}$, and let $P_j$, $T_j$, $\Theta _j$ and $\Pi _j$ be as in (5.2). Then

$$\begin{aligned} P ^{\alpha } \circ T ^{\beta }&= T ^{\beta } \circ P ^{\alpha }, \quad \Pi ^{\alpha }\circ \Theta ^{\beta } = \Theta ^{\beta }\circ \Pi _{j}^{\alpha }, \nonumber \\ P ^{\alpha }\circ \Theta ^{\beta }&= \Theta ^{\beta }\circ P^{\alpha }, \quad T ^{\alpha }\circ \Pi ^{\beta } = \Pi ^{\beta }\circ T ^{\alpha }, \end{aligned}$$

(5.5)

and

$$\begin{aligned} P ^{\alpha } \circ \Pi ^{\beta }&= \sum _{\alpha _0\le \alpha , \beta }(-i) ^{|\alpha _0|} {\alpha \atopwithdelims (){\alpha _0}} {\beta \atopwithdelims ()\alpha _0} \alpha _0!\ \Pi ^{\beta -\alpha _0} \circ P ^{\alpha -\alpha _0},\end{aligned}$$

(5.6)

$$\begin{aligned} \Pi ^{\alpha } \circ P ^{\beta }&= \sum _{\alpha _0\le \alpha ,\beta } i ^{|\alpha _0|} {\alpha \atopwithdelims (){\alpha _0}} {\beta \atopwithdelims ()\alpha _0} \alpha _0! \ P ^{\beta -\alpha _0} \circ \Pi ^{\alpha -\alpha _0},\end{aligned}$$

(5.7)

$$\begin{aligned} T ^{\alpha } \circ \Theta ^{\beta }&= \sum _{\alpha _0\le \alpha ,\beta }i ^{|\alpha _0|} {\alpha \atopwithdelims (){\alpha _0}}{\beta \atopwithdelims ()\alpha _0} \alpha _0! \Theta ^{\beta -\alpha _0} \circ T^{\alpha -\alpha _0}, \end{aligned}$$

(5.8)

$$\begin{aligned} \Theta ^{\alpha } \circ T ^{\beta }&=\sum _{\alpha _0\le \alpha ,\beta }(-i) ^{|\alpha _0|} {\alpha \atopwithdelims (){\alpha _0}} {\beta \atopwithdelims ()\alpha _0} \alpha _0!\ T ^{\beta -\alpha _0} \circ \Theta ^{\alpha -\alpha _0}. \end{aligned}$$

(5.9)

In the next lemma we show that we may interchange the order of differentiation and multiplications with monomials in the definition of Gelfand–Shilov spaces. More precisely, we show that conditions of the form

$$\begin{aligned} |(x^{\alpha _1}D_{\xi }^{\beta _2} (\xi ^{\alpha _2} D_x^{\beta _1}a))(x,\xi )| \le C_1 h_1^{| \alpha _1 +\alpha _2+\beta _1+\beta _2 |} (\alpha _1! \alpha _2 !\beta _1! \beta _2 !)^s \end{aligned}$$

(5.10)

can be used instead of

$$\begin{aligned} |(x^{\alpha _1} \xi ^{\alpha _2} D_x^{\beta _1}D_{\xi }^{\beta _2} a)(x,\xi )| \le C_2 h_2^{| \alpha _1+\alpha _2+\beta _1+\beta _2 |} (\alpha _1! \alpha _2! \beta _1! \beta _2 !)^s \end{aligned}$$

(5.11)

in the definitions of $\mathcal S_s(\mathbf R^{2d})$ and $\Sigma _s(\mathbf R^{2d})$.

Lemma 5.4

Let $s\ge 1/2$ and let $a\in C^\infty (\mathbf R^{2d})$. Then the following is true:

• (1) if (5.10) holds for some $h_1>0$ and $C_1>0$, then (5.11) holds for some $C_2>0$ when

  $$\begin{aligned} h_2\ge \max (32h_1^{1/2},2h_1 ) \mathrm{;} \end{aligned}$$

  (5.12)

• (2) if (5.11) holds for some $h_2>0$ and $C_2>0$, then (5.10) holds for some $C_1>0$ when

  $$\begin{aligned} h_1\ge \max (32 h_2^{1/2},2h_2 ). \end{aligned}$$

Similar facts hold if the operators

$$\begin{aligned} F(x,\xi )&\mapsto x _jF(x,\xi ),\quad F(x,\xi ) \mapsto \xi _jF(x,\xi ),\quad \nonumber \\ F(x,\xi )&\mapsto D_{x_j}F(x,\xi ), \quad F(x,\xi ) \mapsto D_{\xi _j}F(x,\xi ), \end{aligned}$$

are replaced by the operators $P_j$, $T_j$, $\Pi _j$ and $\Theta _j$, respectively, for $j=1,\ldots ,d$.

Proof

By Lemma 5.3 it suffices to prove (1) and (2), and then we only prove (1). The assertion (2) follows by similar arguments and is left for the reader.

We have

$$\begin{aligned}&|(x^{\alpha _1}\xi ^{\alpha _2} D_x^{\beta _1}D_{\xi }^{\beta _2}a)(x,\xi )| \nonumber \\&\quad \le \sum _{\alpha _0 \le \alpha _2,\beta _2} {\alpha _2 \atopwithdelims ()\alpha _0} {\beta _2 \atopwithdelims ()\alpha _0}\alpha _0! |(x^{\alpha _1}D_{\xi }^{\beta _2-\alpha _0} \xi ^{\alpha _2-\alpha _0}D_x^{\beta _1}a)(x,\xi )|. \end{aligned}$$

(5.13)

By the assumptions it follow that

$$\begin{aligned}&|(x^{\alpha _1}D_{\xi }^{\beta _2\!-\!\alpha _0} \xi ^{\alpha _2\!-\!\alpha _0}D_x^{\beta _1}a)(x,\xi )| \le C_1h_1^{|\alpha _1\!+\!\alpha _2\!+\!\beta _1\!+\!\beta _2\!-\!2\alpha _0|}(\alpha _1! (\alpha _2\!-\!\alpha _0)! \beta _1!(\beta _2\!-\!\alpha _0)!)^s. \end{aligned}$$

Inserting this into (5.13) gives

$$\begin{aligned}&|(x^{\alpha _1}\xi ^{\alpha _2} D_x^{\beta _1}D_{\xi }^{\beta _2}a)(x,\xi )| \\&\quad \le C_1\sum _{\alpha _0 \le \alpha _2,\beta _2} {\alpha _2 \atopwithdelims ()\alpha _0} {\beta _2 \atopwithdelims ()\alpha _0}\alpha _0! h_1^{|\alpha _1+\alpha _2+\beta _1+\beta _2-2\alpha _0|} (\alpha _1! (\alpha _2-\alpha _0)! \beta _1!(\beta _2-\alpha _0)!)^s, \end{aligned}$$

or equivalently,

$$\begin{aligned}&|(x^{\alpha _1}\xi ^{\alpha _2} D_x^{\beta _1}D_{\xi }^{\beta _2}a)(x,\xi )| (\alpha _1! \alpha _2 !\beta _1! \beta _2 !)^{-s} \nonumber \\&\quad \le C_1\sum _{\alpha _0 \le \alpha _2,\beta _2} {\alpha _2 \atopwithdelims ()\alpha _0}^{1-s} {\beta _2 \atopwithdelims ()\alpha _0}^{1-s} h_1^{|\alpha _1+\alpha _2+\beta _1+\beta _2-2\alpha _0|}. \end{aligned}$$

(5.14)

First assume that $h_1\ge 1$. Since $s\ge 1/2$, it follows that the sum on the right-hand side of (5.14) can be estimated by

$$\begin{aligned}&h_1^{|\alpha _1+\alpha _2+\beta _1+\beta _2|}\sum _{\alpha _0 \le \alpha _2,\beta _2} {\alpha _2 \atopwithdelims ()\alpha _0}^{1/2} {\beta _2 \atopwithdelims ()\alpha _0}^{1/2} \\&\quad \le h_1^{|\alpha _1+\alpha _2+\beta _1+\beta _2|}\left( \sum _{\alpha _0 \le \alpha _2} {\alpha _2 \atopwithdelims ()\alpha _0}\right) ^{1/2} \left( \sum _{\alpha _0 \le \beta _2} {\beta _2 \atopwithdelims ()\alpha _0}\right) ^{1/2} \\&\quad \le h_1^{|\alpha _1+\alpha _2+\beta _1+\beta _2|}2^{|\alpha _2+\beta _2|/2}\le (2h_1)^{|\alpha _1+\alpha _2+\beta _1+\beta _2|}, \end{aligned}$$

and the result follows in this case.

Next assume that $h_1<1$. We shall consider the cases when $|\beta _2|\le |\alpha _2|/4$, $|\beta _2|\ge 4|\alpha _2|$ and $|\alpha _2|/4\le |\beta _2|\le 4|\alpha _2|$ separately.

First assume that $|\beta _2|\le |\alpha _2|/4$ or $|\beta _2|\ge 4|\alpha _2|$, and let $\alpha _0\le \alpha _2,\beta _2$ be as in the sum in (5.14). Then

$$\begin{aligned} |\alpha _2+\beta _2-2\alpha _0|\ge |\alpha _2+\beta _2|/2, \end{aligned}$$

and (5.14) together with the fact that $h_1<1$ give

$$\begin{aligned}&|(x^{\alpha _1}\xi ^{\alpha _2} D_x^{\beta _1}D_{\xi }^{\beta _2}a)(x,\xi )| \\&\quad \le C_1h_1^{|\alpha _1+\alpha _2+\beta _1+\beta _2|/2}(\alpha _1! \alpha _2! \beta _1! \beta _2 !)^s \sum _{\alpha _0 \le \alpha _2,\beta _2} {\alpha _2 \atopwithdelims ()\alpha _0}^{1/2} {\beta _2 \atopwithdelims ()\alpha _0}^{1/2} \\&\quad \le C_1(2h_1^{1/2})^{|\alpha _1+\alpha _2+\beta _1+\beta _2|}(\alpha _1! \alpha _2! \beta _1! \beta _2 !)^s, \end{aligned}$$

and the result follows in this case.

It remains to consider the case when $|\alpha _2|/4\le |\beta _2|\le 4|\alpha _2|$. Let $\gamma =16\beta _2$. Then $|\gamma | \ge 4|\alpha _2|$, and from the first part of the proof we get

$$\begin{aligned} |(x^{\alpha _1}\xi ^{\alpha _2} D_x^{\beta _1}a)(x,\xi )|&\le C_1(2h_1^{1/2})^{|\alpha _1+\alpha _2+\beta _1|}(\alpha _1!\alpha _2! \beta _1!)^s \end{aligned}$$

and

$$\begin{aligned} |(x^{\alpha _1}\xi ^{\alpha _2} D_x^{\beta _1}D_{\xi }^{\gamma }a)(x,\xi )|&\le C_1(2h_1^{1/2})^{|\alpha _1+\alpha _2+\beta _1+\gamma |} (\alpha _1!\alpha _2! \beta _1! \gamma ! )^s. \end{aligned}$$

By Theorem 4.10 in [1] and the last estimates we get

$$\begin{aligned}&|(x^{\alpha _1}\xi ^{\alpha _2} D_x^{\beta _1}D_{\xi }^{\beta _2}a)(x,\xi )| \nonumber \\&\quad \le C_1(2h_1^{1/2})^{|\alpha _1+\alpha _2+\beta _1+\beta _2 |} (\alpha _1! \alpha _2! \beta _1! \beta _2 !)^s\cdot R(\beta _2), \end{aligned}$$

(5.15)

where

$$\begin{aligned} \left( \frac{(16\beta _2)!}{(\beta _2!)^{16}} \right) ^{s/16}\le \big (16^{16|\beta _{2}|}\big )^{s/16}=16^{s|\beta _2|}, \end{aligned}$$

where the last inequality follows by estimating coefficients in suitable multinomial expansions. The result now follows from these inequalities. $\square$

We have now the following results.

Lemma 5.5

Let $a\in C^{\infty }(\mathbf R^{2d})$, $s\ge 1/2$, and let $P_j$, $T_j$, $\Pi _j$ and $\Theta _j$ be as in (5.2). Also let $R_{\alpha , \beta , \gamma ,\delta }$ be a composition of $P^{\alpha }$, $T^{\beta }$, $\Pi ^{\gamma }$ and $\Theta ^{\delta }$. Then the following statements are equivalent:

• (1) there exist positive constants $C$ and $h$ such that

  $$\begin{aligned}&|((P^{\alpha }\circ T^{\beta }\circ \Pi ^{\gamma } \circ \Theta ^{\delta })a)(x,\xi )| \le C h ^{|\alpha +\beta +\gamma +\delta |} (\alpha ! \beta !\gamma ! \delta !)^s, \quad (x,\xi ) \in \mathbf R^{2d}, \end{aligned}$$

  (5.16)

  for every multi-indices $\alpha$, $\beta$, $\gamma$ and $\delta$;

• (2) there exist positive constants $C$ and $h$ such that

  $$\begin{aligned} |(R_{\alpha , \beta , \gamma , \delta }a)(x,\xi )| \le C h^{|\alpha +\beta +\gamma +\delta |}(\alpha ! \beta !\gamma ! \delta !)^s, \quad (x,\xi ) \in \mathbf R^{2d}, \end{aligned}$$

  (5.17)

  for every multi-indices $\alpha$, $\beta$, $\gamma$ and $\delta$;

• (3) $a\in \mathcal S_s (\mathbf R^{2d})$.

Lemma 5.6

Let $a\in C^{\infty }(\mathbf R^{2d})$, $s\ge 1/2$, and let $P_j$, $T_j$, $\Pi _j$ and $\Theta _j$ be defined as in (5.2). Also let $R_{\alpha , \beta , \gamma ,\delta }$ be a composition of $P^{\alpha }$, $T^{\beta }$, $\Pi ^{\gamma }$ and $\Theta ^{\delta }$. Then the following statements are equivalent:

• (1) for every $h>0$, there is a positive constant $C$ such that (5.16) holds for every multi-indices $\alpha$, $\beta$, $\gamma$ and $\delta$;

• (2) for every $h>0$, there is a positive constant $C$ such that (5.17) holds for every multi-indices $\alpha$, $\beta$, $\gamma$ and $\delta$;

• (3) $a\in \Sigma _s (\mathbf R^{2d})$.

Proof of Lemmata 5.5 and 5.6

We only prove the results for $R_{\alpha ,\beta ,\gamma ,\delta }= \Theta ^{\delta } \circ P^{\alpha } \circ T^{\beta } \circ \Pi ^{\gamma }$. The other cases follow by similar arguments, and are left for the reader.

The equivalences between (1) and (2) are immediate consequences of Lemma 5.4.

Next we prove that (2) implies (3). In view of (5.4) and Lemma 5.4 it suffices to prove that if $h>0$ and $C>0$, then

$$\begin{aligned}&\left| ((P-T)^{\alpha }\circ (P+T)^{\beta }\circ (\Pi -\Theta )^{\gamma }\circ (\Pi +\Theta )^{\delta }a)(x,\xi ) \right| \nonumber \\&\quad \le C (8h)^{| \alpha +\beta +\gamma +\delta |} (\alpha !\beta !\gamma !\delta !)^s. \end{aligned}$$

(5.18)

holds for every $\alpha , \beta , \gamma , \delta \in \mathbf N^{d}$, when

$$\begin{aligned} |(\Theta ^{\delta }\circ P ^{\alpha }\circ T ^{\beta }\circ \Pi ^{\gamma }a)(x,\xi ) | \le C h^{|\alpha +\beta +\gamma +\delta |} (\alpha !\beta !\gamma !\delta !)^s. \end{aligned}$$

(5.19)

holds for every $\alpha , \beta , \gamma , \delta \in \mathbf N^{d}$.

Therefore, assume that (5.19) holds. By the binomial theorem and Lemma 5.3, we get

$$\begin{aligned}&|((P-T)^{\alpha }\circ (P+T)^{\beta }\circ (\Pi -\Theta )^{\gamma }\circ (\Pi +\Theta )^{\delta }a)(x,\xi )| \nonumber \\&\quad \le \sum _{J_1}\lambda _0! B^{\alpha ,\beta }_{\alpha _0, \beta _0, \lambda _0} B^{\gamma , \delta }_{\gamma _0, \delta _0, \lambda _0} H_{\alpha _0, \beta _0, \gamma _0, \delta _0, \lambda _0}(x,\xi ), \end{aligned}$$

(5.20)

where

$$\begin{aligned}&H_{\alpha _0, \beta _0, \gamma _0, \delta _0, \lambda _0}(x,\xi ) \\&\quad = | (\Theta ^{\gamma _0+\delta _0-\lambda _0}\circ P^{\alpha _0+\beta _0-\lambda _0} \circ T^{\alpha +\beta -\alpha _0-\beta _0}\circ \Pi ^{\gamma +\delta -\gamma _0-\delta _0} a)(x,\xi )|, \end{aligned}$$

$$\begin{aligned} B^{\alpha ,\beta }_{\alpha _0, \beta _0, \lambda _0} = {\alpha \atopwithdelims ()\alpha _0} {\beta \atopwithdelims ()\beta _0} {\alpha _0+\beta _0 \atopwithdelims ()\lambda _0} \end{aligned}$$

and $J_1$ is the set of all five-tuples $(\alpha _0, \beta _0, \gamma _0,\delta _0,\lambda _0)$ of multi-indices such that

$$\begin{aligned}&\alpha _0\le \alpha ,\quad \beta _0 \le \beta , \quad \gamma _0\le \gamma , \quad \delta _0\le \delta \quad \text {and}\quad \lambda _0 \le \min (\alpha _0+\beta _0, \gamma _0+\delta _0). \end{aligned}$$

(5.21)

By (5.17), we get

$$\begin{aligned}&H_{\alpha _0, \beta _0, \gamma _0, \delta _0, \lambda _0}(x,\xi ) h^{-|\alpha +\beta +\gamma +\delta |}(\alpha !\beta ! \gamma ! \delta !)^{-s} \\&\quad \le C \left( \frac{(\alpha _0+\beta _0-\lambda _0)! (\alpha +\beta -\alpha _0-\beta _0)! }{\alpha !\beta !} \right) ^s \\&\qquad \times \left( \frac{(\gamma _0+\delta _0-\lambda _0)! (\gamma +\delta -\gamma _0-\delta _0)! }{\gamma !\delta !} \right) ^s. \end{aligned}$$

Hence, a combination of this estimate, the fact that $\lambda _0!\le (\lambda _0!)^{2s}$ and (5.20) gives

$$\begin{aligned}&\left| ((P-T)^{\alpha }\circ (P+T)^{\beta }\circ (\Pi -\Theta )^{\gamma }\circ (\Pi +\Theta )^{\delta }a)(x,\xi )\right| \nonumber \\&\quad \le Ch^{|\alpha +\beta +\gamma +\delta |}(\alpha !\beta ! \gamma ! \delta ! )^{s}\sum _{J_1} C^{\alpha ,\beta }_{\alpha _0, \beta _0, \lambda _0} C^{\gamma , \delta }_{\gamma _0, \delta _0, \lambda _0}, \end{aligned}$$

(5.22)

for some constant $C$, where

$$\begin{aligned} C^{\alpha ,\beta }_{\alpha _0, \beta _0, \lambda _0} = B^{\alpha ,\beta }_{\alpha _0, \beta _0, \lambda _0} \cdot \left( \frac{\lambda _0! (\alpha _0+\beta _0-\lambda _0)! (\alpha +\beta -\alpha _0-\beta _0)! }{\alpha !\beta !} \right) ^s. \end{aligned}$$

We need to estimate $C^{\alpha ,\beta }_{\alpha _0, \beta _0, \lambda _0}$. By straight-forward computations, using (3.1) we get

$$\begin{aligned} C^{\alpha ,\beta }_{\alpha _0, \beta _0, \lambda _0}&= {{\alpha } \atopwithdelims (){\alpha _0}} {{\beta } \atopwithdelims (){\beta _0}} {{\alpha _0+\beta _0} \atopwithdelims (){\lambda _0}}^{1-s} {{\alpha +\beta } \atopwithdelims (){\alpha }}^s {{\alpha +\beta } \atopwithdelims (){\alpha _0+\beta _0}} ^{-s}. \nonumber \\&\le {{\alpha } \atopwithdelims (){\alpha _0}} {{\beta } \atopwithdelims (){\beta _0}} {{\alpha _0+\beta _0} \atopwithdelims (){\lambda _0}}^{1/2}2^{|\alpha +\beta |} \end{aligned}$$

(5.23)

Since

$$\begin{aligned}&\sum _{\lambda _0}{{\alpha _0+\beta _0} \atopwithdelims (){\lambda _0}}^{1/2} {{\gamma _0+\delta _0} \atopwithdelims (){\lambda _0}}^{1/2} \\&\quad \le \left( \sum _{\lambda _0\le \alpha _0+\beta _0}{{\alpha _0+\beta _0} \atopwithdelims (){\lambda _0}} \right) ^{1/2} \left( \sum _{\lambda _0\le \gamma _0+\delta _0} {{\gamma _0+\delta _0} \atopwithdelims (){\lambda _0}} \right) ^{1/2} \\&\quad \le 2^{|\alpha _0+\beta _0+\gamma _0+\delta _0|}\le 2^{|\alpha +\beta +\gamma +\delta |}, \end{aligned}$$

(5.22) and (5.23) give

$$\begin{aligned}&\left| ((P-T)^{\alpha }\circ (P+T)^{\beta }\circ (\Pi -\Theta )^{\gamma }\circ (\Pi +\Theta )^{\delta }a)(x,\xi )\right| \nonumber \\&\quad \le C(4h)^{|\alpha +\beta +\gamma +\delta |}(\alpha !\beta ! \gamma ! \delta ! )^{s}\sum _{J_2} {{\alpha } \atopwithdelims (){\alpha _0}} {{\beta } \atopwithdelims (){\beta _0}} {{\gamma } \atopwithdelims (){\gamma _0}} {{\delta } \atopwithdelims (){\delta _0}} \nonumber \\&\quad = C(8h)^{|\alpha +\beta +\gamma +\delta |}(\alpha !\beta ! \gamma ! \delta ! )^{s}. \end{aligned}$$

(5.24)

Here $J_2$ is the set of all four-tuples $(\alpha _0, \beta _0, \gamma _0,\delta _0)$ of multi-indices such that (5.21) holds. This gives (5.18), and (3) follows.

It remains to prove that (3) gives (1). Let $h>0$ and $C>0$. By Lemma 5.4 it suffices to prove that

$$\begin{aligned} |(( P^\alpha \circ T^\beta \circ \Pi ^\gamma \circ \Theta ^\delta )a)(x,\xi )| \le C(4h)^{|\alpha +\beta +\gamma +\delta |}(\alpha ! \beta ! \gamma ! \delta ! )^s \end{aligned}$$

(5.25)

holds for every $\alpha , \beta , \gamma , \delta \in \mathbf N^{d}$, when

$$\begin{aligned} | (x^\alpha D_\xi ^\delta ( \xi ^\beta D_x ^\gamma a))(x,\xi )| \le C h^{|\alpha +\beta +\gamma +\delta |}(\alpha ! \beta ! \gamma ! \delta ! )^s \end{aligned}$$

(5.26)

holds for every $\alpha , \beta , \gamma , \delta \in \mathbf N^{d}$.

By (5.2) and the binomial theorem we get

$$\begin{aligned}&|((P^{\alpha }\circ T^{\beta }\circ \Pi ^{\gamma } \circ \Theta ^{\delta })a)(x,\xi )| \\&\quad =\left| \left( \left( \frac{1}{2}D_{\xi } - x\right) ^{\alpha } \left( \frac{1}{2}D_{\xi } + x\right) ^{\beta } \left( \frac{1}{2}D _{x} +\xi \right) ^{\gamma } \left( \frac{1}{2}D_{x} - \xi \right) ^{\delta }a\right) (x,\xi ) \right| \\&\quad \le \sum _{\alpha _0, \beta _0, \gamma _0, \delta _0} B_{\alpha _0, \beta _0}^{\alpha ,\beta } B_{\gamma _0, \delta _0}^{\gamma ,\delta } |(x^{\alpha +\beta -\alpha _0-\beta _0} D_{\xi }^{\alpha _0+\beta _0} \xi ^{\gamma _0+\delta _0} D_x^{\gamma +\delta -\gamma _0-\delta _0}a)(x,\xi ) |, \end{aligned}$$

where

$$\begin{aligned} B_{\alpha _0, \beta _0}^{\alpha ,\beta } = {\alpha \atopwithdelims ()\alpha _0}{\beta \atopwithdelims ()\beta _0}. \end{aligned}$$

Here the sum is taken over all $\alpha _0$, $\beta _0$, $\gamma _0$, and $\delta _0$ such that $\alpha _0 \le \alpha$, $\beta _0 \le \beta$, $\gamma _0 \le \gamma$, and $\delta _0\le \delta$.

By (5.26) we obtain

$$\begin{aligned} |((P^{\alpha }\circ T^{\beta }\circ \Pi ^{\gamma } \circ \Theta ^{\delta })a)(x,\xi )| \le C h^{|\alpha +\beta +\gamma +\delta |} \sum _{\alpha _0, \beta _0, \gamma _0, \delta _0} C_{\alpha _0, \beta _0}^{\alpha ,\beta } C_{\gamma _0, \delta _0}^{\gamma ,\delta }, \end{aligned}$$

where

$$\begin{aligned} C_{\alpha _0, \beta _0}^{\alpha ,\beta }&= B_{\alpha _0, \beta _0}^{\alpha ,\beta } ( (\alpha +\beta -\alpha _0-\beta _0)! (\alpha _0+\beta _0)! )^s \\&=((\alpha +\beta )! )^s {\alpha \atopwithdelims ()\alpha _0}{\beta \atopwithdelims ()\beta _0} {\alpha +\beta \atopwithdelims ()\alpha _0+ \beta _0}^{-s} \le ((\alpha +\beta )!)^s {\alpha \atopwithdelims ()\alpha _0}{\beta \atopwithdelims ()\beta _0}. \end{aligned}$$

This gives

$$\begin{aligned}&|((P^{\alpha }\circ T^{\beta }\circ \Pi ^{\gamma } \circ \Theta ^{\delta })a)(x,\xi )| \\&\quad \le C h^{|\alpha +\beta +\gamma +\delta |} \sum _{\alpha _0, \beta _0, \gamma _0, \delta _0}((\alpha +\beta )!(\gamma +\delta )!)^s B_{\alpha _0, \beta _0}^{\alpha ,\beta } B_{\gamma _0, \delta _0}^{\gamma ,\delta } \\&\quad =C (2h)^{| \alpha +\beta +\gamma +\delta |} ((\alpha +\beta )!(\gamma +\delta )!)^s. \end{aligned}$$

Since $(\alpha +\beta )!\le 2^{|\alpha +\beta |}\alpha ! \beta !$, we get

$$\begin{aligned} |((P^{\alpha }\circ T^{\beta }\circ \Pi ^{\gamma } \circ \Theta ^{\delta })a)(x,\xi )| \le C (4h)^{|\alpha +\beta +\gamma +\delta |} (\alpha ! \beta !\gamma !\delta !)^s, \end{aligned}$$

and (5.25) follows. This gives the result. $\square$

The next lemma is the last step in the proof of Theorem 5.1.

Lemma 5.7

Let $\Omega \subseteq \mathbf R^{2d}$ be an open neighborhood of origin, $h>0$, $s\ge 1/2$, and let $K=Aa$, where $a(x,\xi ) \in C_+(\mathbf R^{2d}) \cap C^{\infty }(\Omega )$ be such that (5.1) holds. Then

$$\begin{aligned} \Vert x^\alpha y^\beta K\Vert _{L^2}&\le C 2^{(s-1)| \alpha +\beta |} h^{|\alpha +\beta |}(\alpha !\beta !)^s \end{aligned}$$

(5.27)

and

$$\begin{aligned} \Vert \xi ^\alpha \eta ^\beta \widehat{K}\Vert _{L^2}&\le C 2^{(s-1)| \alpha +\beta |} h^{|\alpha +\beta |}(\alpha !\beta !)^s, \end{aligned}$$

(5.28)

for some constant $C > 0$ which is independent of $\alpha ,\beta \in \mathbf N^{d}$.

Proof

By Theorems 3.3 and 3.13 in [11] it follows that $K=Aa$ is a positive semi-definite trace-class operator on $L^2(\mathbf R^{d})$, and belongs to $\fancyscript{S} (\mathbf R^{2d})$. Furthermore,

$$\begin{aligned} (Aa)(x,y)=\sum _jf_j(x)\overline{f_j(y)}, \end{aligned}$$

where $(f_j,f_k)=0$ when $j\ne k$, and

$$\begin{aligned} \sum _j \Vert x^{\alpha }D^{\gamma }f_j\Vert _{L^2}^2< \infty , \end{aligned}$$

for every $\alpha ,\gamma \in \mathbf N^{d}$. We also recall that the trace-norm of $Aa$ is given by

$$\begin{aligned} \Vert Aa \Vert _{{\text {Tr}}}=\sum \Vert f_j\Vert _{L^2}^2= (\pi /2)^{d/2}a(0,0) < \infty , \end{aligned}$$

(5.29)

since $f_j$ is orthogonal to $f_k$ when $j\ne k$.

Let

$$\begin{aligned} a_{\alpha ,\gamma }=P^{\alpha }\circ T^{\alpha } \circ \Pi ^{\gamma }\circ \Theta ^{\gamma }a \quad \text {and}\quad a_{\alpha ,\beta ,\gamma ,\delta } = P^{\alpha }\circ T^{\beta } \circ \Pi ^{\gamma }\circ \Theta ^{\delta }a. \end{aligned}$$

Then

$$\begin{aligned} Aa_{\alpha ,\gamma } = (-1)^{|\gamma |} x^\alpha y^\alpha D_x^\gamma D _y^\gamma K&= \sum _j(x^{\alpha }D^{\gamma }f_j)\otimes (\overline{x^{\alpha }D^{\gamma }f_j}) \end{aligned}$$

and

$$\begin{aligned} Aa_{\alpha ,\beta ,\gamma ,\delta } = (-1)^{|\gamma |} x^\alpha y^\beta D _x^\gamma D _y^\delta K&= (-1)^{|\gamma +\delta |}\sum _j(x^{\alpha }D^{\gamma }f_j)\otimes (\overline{x^{\beta }D^{\delta }f_j}). \end{aligned}$$

In particular, $a_{\alpha ,\gamma }\in C_+(\mathbf R^{2d})$, and Proposition 2.8 and (5.29) applied on $Aa_{0,\alpha }$ give

$$\begin{aligned} \Vert Aa_{0,\alpha }\Vert _{L^2}&\le \Vert Aa_{0,\alpha }\Vert _{{\text {Tr}}} = (\pi /2)^{d/2}a_{0,\alpha }(0,0) \nonumber \\&=(\pi /2)^{d/2} 2^{-2|\alpha |}|(D_\xi ^{2\alpha } a)(0,0)| \le C (h/2)^{2|\alpha |}((2\alpha )!)^s \nonumber \\&\le C h^{2|\alpha |}2^{(2s-2)|\alpha |}(\alpha !)^{2s}, \end{aligned}$$

(5.30)

for some constant $C > 0$. Here we have used the fact that $(2\alpha )! \le 2^{2|\alpha |}(\alpha !)^2$.

Since $Aa_{0,\alpha } (x,y)= (-1)^{|\alpha |}D _x^\alpha D _y^\alpha K(x,y)$, we get

$$\begin{aligned} \Vert \xi ^\alpha \eta ^\alpha \widehat{K}\Vert _{L^2}\le Ch^{2|\alpha |}(\alpha !)^{2s} \end{aligned}$$

(5.31)

by combining (5.30) and Parseval’s formula.

Next we consider $Aa_{0,0,\alpha ,\beta } = (-1)^{|\alpha |} D _x^\alpha D _y^\beta K$. By Cauchy–Schwartz inequality and (5.30) we get

$$\begin{aligned} \Vert \xi ^\alpha \eta ^\beta \widehat{K}\Vert _{L^2}^2&= \Vert D_x ^\alpha D_y ^\beta K\Vert _{L^2}^2 \le \Vert Aa_{0,0,\alpha ,\beta }\Vert _{{\text {Tr}}}^2 \\&\le \left( \sum _j \Vert (D^{\alpha }f_j)\otimes (\overline{D^{\beta }f_j})\Vert _{{\text {Tr}}} \right) ^2 = \left( \sum _j \Vert D^{\alpha }f_j\Vert _{L^2} \Vert D^{\beta }f_j\Vert _{L^2} \right) ^2 \\&\le \left( \sum _j \Vert D^{\alpha }f_j\Vert _{L^2}^2\right) \left( \sum _j \Vert D^{\beta }f_j\Vert _{L^2}^2 \right) = \Vert Aa_{0,\alpha }\Vert _{{\text {Tr}}}\Vert Aa_{0,\beta }\Vert _{{\text {Tr}}} \\&\le C h^{2|\alpha +\beta |}2^{(2s-2)|\alpha +\beta |}(\alpha !\beta !)^{2s} \end{aligned}$$

for some constant C > 0. This gives (5.28).

By considering $a_{\alpha ,\beta ,0,0}$ and $a_{\alpha ,0}$ instead of $a_{0,0,\alpha ,\beta }$ and $a_{0,\alpha }$, respectively, we get (5.27). The details are left for the reader. $\square$

Proof of Theorem 5.1

The result follows by combining Lemmata 2.1, 2.2 and 5.7. $\square$

We shall end the section by using Theorems 5.1 and 5.2 to deduce related results for pseudo-differential operators.

Let $t\in \mathbf R$ and $a\in \fancyscript{S}(\mathbf R^{2d})$. Then the pseudo-differential operator ${\text {Op}}_t (a)$ is the linear and continuous operator on $\fancyscript{S}(\mathbf R^{d})$, given by

$$\begin{aligned} ({\text {Op}}_t(a)f)(x) \equiv (2\pi )^{-d}\iint _{\mathbf R^{2d}} a((1-t)x+ty,\xi )f(y)e^{i\langle x-y,\xi \rangle }\, dyd\xi . \end{aligned}$$

By straight-forward computations it follows that the kernel $K_{a,t}$ of ${\text {Op}}_t(a)$ is given by

$$\begin{aligned} K_{a,t}(x,y) = (2\pi )^{-d/2} (Aa)\left( -x+\left( \frac{1}{2} -t\right) (y-x) , y-\left( \frac{1}{2} -t\right) (y-x) \right) \end{aligned}$$

We note that the mappings $A$ and

$$\begin{aligned} F(x,y)\mapsto F\left( -x+\left( \frac{1}{2} -t\right) (y-x) , y-\left( \frac{1}{2} -t\right) (y-x) \right) \end{aligned}$$

are continuous and bijective on any of the spaces in (2.1). This implies that the definition of ${\text {Op}}_t(a)$ extends uniquely to any $a$ in the spaces in (2.1), in a way similar as for $Aa$. In particular, if $a\in \mathcal S_s'(\mathbf R^{2d})$, then ${\text {Op}}_t(a)$ is continuous from $\mathcal S_s(\mathbf R^{d})$ to $\mathcal S_s'(\mathbf R^{d})$. We note that the Weyl quantization is appears when $t=1/2$, and if $t=0$, then the normal representation ${\text {Op}}(a)=a(x,D)$ is obtained.

If $t_t,t_2\in \mathbf R$ and $a_1,a_2\in \mathcal S_s'(\mathbf R^{2d})$, then

$$\begin{aligned} {\text {Op}}_{t_1}(a_1) = {\text {Op}}_{t_2}(a_2) \Longleftrightarrow a_2(x,\xi ) = e^{i(t_1-t_2)\langle D_\xi ,D_x\rangle }a_1(x,\xi ). \end{aligned}$$

(5.32)

Here we note that $e^{it\langle D_\xi ,D_x\rangle }$ is continuous and bijective on any of the spaces in (2.1). A proof of this fact can be found in [13].

Fourier transforms are homeomorphic on any of the spaces in (2.1). We have now the following result related to Theorem 5.1. Here the condition (5.1) should be replaced by

$$\begin{aligned} \sup _{\alpha \in \mathbf N^{d}} \left( \frac{|(D_x^\alpha \widehat{a})(0)|}{(\alpha !) ^s h^{|\alpha |}} \right) < \infty \quad \text {and}\quad \sup _{\alpha \in \mathbf N^{d}} \left( \frac{|(D_\xi ^\alpha \widehat{a})(0)|}{(\alpha !) ^s h^{|\alpha |}} \right) < \infty . \end{aligned}$$

(5.33)

Theorem 5.8

Let $s\ge 1/2$, $t\in \mathbf R$, $\Omega \subseteq \mathbf R^{2d}$ be an open neighborhood of origin, and let $a\in \mathcal S^\prime _{1/2} (\mathbf R^{2d})$ be such that

$$\begin{aligned} \widehat{a}\in C^{\infty } (\Omega ) \quad \text {and}\quad {\text {Op}}_t(a)\ge 0. \end{aligned}$$

(5.34)

Then the following is true:

• (1) if (5.33) holds for some $h>0$, then $a\in \mathcal S_s(\mathbf R^{2d})$;

• (2) if (5.33) holds for every $h>0$, then $a\in \Sigma _s(\mathbf R^{2d})$.

Note here that $\widehat{a}$ in (5.33) and (5.34) can be replaced by $\fancyscript{F}_\sigma a$, since the symplectic Fourier transform is merely a rescaling of the usual Fourier transform.

Proof

The result follows in the case $t=1/2$ by a straight-forward combination of (2.4) and Theorem 5.1.

We need to prove the result for arbitrary $t\in \mathbf R$, and then we only prove (1). The assertion (2) follows by similar arguments and is left for the reader.

Therefore, let $t\in \mathbf R$ be arbitrary, and let $b\in \mathcal S_{1/2}'(\mathbf R^{2d})$ be chosen such that ${\text {Op}}_t(a) = {\text {Op}}^w(b)$. By (5.32) we get

$$\begin{aligned} \widehat{b}(\xi ,x) = e^{i(t-1/2)\langle x,\xi \rangle }\widehat{a}(\xi ,x). \end{aligned}$$

By Leibnitz rule it follows that

$$\begin{aligned} (D_x^\alpha \widehat{b})(0,0) = (D_x^\alpha \widehat{a})(0,0) \quad \text {and}\quad (D_\xi ^\alpha \widehat{b})(0,0) = (D_\xi ^\alpha \widehat{a})(0,0) \end{aligned}$$

for every multi-index $\alpha$. Hence by the assumptions on $a$ it follows that (5.33) holds for some $h>0$ when $a$ is replaced by $b$.

Since ${\text {Op}}^w(b)$ is positive semi-definite, it follows from the first part of the proof that $b\in \mathcal S_s$. The result is now a consequence of the equivalence

$$\begin{aligned} a\in \mathcal S_s(\mathbf R^{2d}) \Longleftrightarrow b\in \mathcal S_s(\mathbf R^{2d}), \end{aligned}$$

remarked above. $\square$

By the previous theorem we get the following result parallel to Theorem 5.2.

Theorem 5.9

Let $s\ge 1/2$, $t\in \mathbf R$, $\Omega \subseteq \mathbf R^{2d}$ be an open neighborhood of origin, and let $a\in \mathcal S^\prime _{1/2} (\mathbf R^{2d})$ be such that (5.34) holds. Furthermore, assume that one of the following statements hold true:

• (1) $s<1/2$ and (5.33) holds for some $h>0$;

• (2) $s\le 1/2$ and (5.33) holds for every $h>0$.

Then $a=0$.

We note that the condition (5.33) in previous results is equivalent to

$$\begin{aligned} \sup _{\alpha \in \mathbf N^{d}} \left| \frac{1}{(\alpha !) ^s h^{|\alpha |}} \iint _{\mathbf R^{2d}} x^\alpha a(x,\xi )\, dxd\xi \right| < \infty \end{aligned}$$

and

$$\begin{aligned} \sup _{\alpha \in \mathbf N^{d}} \left| \frac{1}{(\alpha !) ^s h^{|\alpha |}} \iint _{\mathbf R^{2d}} \xi ^\alpha a(x,\xi )\, dxd\xi \right| < \infty . \end{aligned}$$