Inclusion Theorems for the Moyal Multiplier Algebras of Generalized Gelfand–Shilov Spaces

Abstract

We prove that the Moyal multiplier algebras of the generalized Gelfand–Shilov spaces of type S contain Palamodov spaces of type ${\mathscr {E}}$ and the inclusion maps are continuous. We also give a direct proof that the Palamodov spaces are algebraically and topologically isomorphic to the strong duals of the spaces of convolutors for the corresponding spaces of type S. The obtained results provide a general and efficient way to describe the algebraic and continuity properties of pseudodifferential operators with symbols having an exponential or super-exponential growth at infinity.

Appendix

The arguments of the proof of Proposition 3.2 are easily adapted to show that the elements of ${\mathcal {E}}^{\{b\}\prime }_{\{a\}}$ are convolutors of $S_{(a)}^{\{b\}}$. To analyse the mapping properties of pseudodifferential operators with symbols in ${\mathcal {E}}^{\{a\}}_{\{a\}}$ we use the isomorphism $F\left[ S_{(a)}^{\{b\}}\right] =S^{(a)}_{\{b\}}$. As in the case of the spaces $S_{\{a\}}^{\{b\}}$ and $S_{(a)}^{(b)}$, this isomorphism is ensured by condition (2.3) and is easily proved by using the inequality

$$\begin{aligned} \int _{{{\mathbb {R}}}^d}|\partial ^\beta x^\alpha |\,|f(x)|\,dx\le \sqrt{2}\int _{{{\mathbb {R}}}^d}|x^\alpha |\, |\partial ^\beta f(x)|\,dx, \end{aligned}$$

which holds for any function f in the Schwartz space $S({{\mathbb {R}}}^d)$ and for any multi-indices $\alpha , \beta \in {{\mathbb {Z}}}^d_+$ (see Lemma A.1 in [38]).

Lemma A.1

If the space $S^{(a)}_{\{a\}}({{\mathbb {R}}})=\bigcap _{B\rightarrow 0}\bigcup _{A\rightarrow \infty }S_{a,A}^{a,B}({{\mathbb {R}}})$ is nontrivial, then there is a nonnegative even function $\varphi \in S^{(a)}_{\{a\}}({{\mathbb {R}}})$ satisfying

$$\begin{aligned} \varphi (s)\ge 1/w_a(|s|), \qquad s\in {{\mathbb {R}}}. \end{aligned}$$

(A.1)

Proof

It is readily seen that $S^{(a)}_{\{a\}}$ is an algebra under multiplication and is invariant under translations. Therefore, there is a nonnegative even function $\omega \in S^{(a)}_{\{a\}}({{\mathbb {R}}})$ such that $\int _0^1\omega (s)ds=1$. We set

$$\begin{aligned} \varphi (s)= \int \limits _{-\infty }^\infty \frac{\omega (s-t)}{w_a(|t|/2)}dt. \end{aligned}$$

The norm $\Vert \omega \Vert _{A,B}$ is finite for any $B>0$ and some A depending on B. Applying (2.13) and (2.11) gives

$$\begin{aligned} \begin{aligned} |\partial ^n\varphi (s)|&\le \int \limits _{-\infty }^\infty \frac{\Vert \omega \Vert _{A,B}B^na_n}{w_a(|t|/2)w_a(|s-t|/A)} dt \\&\le \int \limits _{-\infty }^\infty \frac{K\Vert \omega \Vert _{A,B}B^na_n}{w_a(|t|/2H)^2\,w_a(|s-t|/A)} dt\le \frac{C\Vert \omega \Vert _{A,B}B^na_n}{w_a(|s|/(2H+A))}, \end{aligned} \end{aligned}$$

where $C=2K\int \limits _0^{\infty }w_a(t/2H)^{-1}dt$. Hence, $\varphi \in S^{(a)}_{\{a\}}$. Using (2.12), we obtain

$$\begin{aligned} \varphi (s)\ge \int \limits _0^1\frac{\omega (t)}{w_a(|s-t|/2)}dt\ge \int \limits _0^1\frac{\omega (t)}{w_a((|s|+1)/2)}dt\ge \frac{1}{w_a(|s|)w_a(1)}=\frac{1}{w_a(|s|)}, \end{aligned}$$

as claimed.

$\square$

Proposition A.2

The space $S_{(a)}^{\{a\}}({{\mathbb {R}}}^{2d})$ is not closed under the Weyl–Moyal product.

Proof

It suffices to consider the case where $d=1$ and $\hbar =2$. Let $f(q,p)=f_1(q)f_2(p)$ and $f_1,f_2\in S_{(a)}^{\{a\}}({{\mathbb {R}}})$. By the definition (1.1) we have

$$\begin{aligned} \begin{aligned}&(f\star f)(q,p)\\&\quad = (2\pi )^{-2}\! \int \limits _{{{\mathbb {R}}}^4} \!f_1(q-q')f_2(p-p')f_1(q-q'') f_2(p-p'')e^{i(q'p''-q''p')}dq'dq''dp'dp''\\&\quad =(2\pi )^{-1}\!\int \limits _{{{\mathbb {R}}}^2}{\widehat{f}}_1(p'')f_2(p-p'){\widehat{f}}_1(-p')f_2(p-p'')e^{i(qp''-qp')}dp'dp''. \end{aligned} \end{aligned}$$

If $f_1$ and $f_2$ are even, then

$$\begin{aligned} (f\star f)(q,0)=\left[ {\mathcal {F}}^{-1}\left( {\widehat{f}}_1\cdot f_2\right) (q)\right] ^2=(2\pi )^{-1}\left[ \left( f_1*{\widehat{f}}_2\right) (q)\right] ^2. \end{aligned}$$

The function $f_1$ can be taken nonnegative and such that $\int _0^{1}f_1(q)dq=1$. We set ${\widehat{f}}_2(q)=\varphi (q/2)$, where $\varphi$ is the function constructed in Lemma A.1. Then

$$\begin{aligned} \left| \left( f_1*{\widehat{f}}_2\right) (q)\right| \ge \frac{1}{w_a(|q|)} \end{aligned}$$

(A.2)

by the same argument as in deriving (A.1). Assume that $f\star f\in S_{(a)}^{\{a\}}({{\mathbb {R}}}^2)$. Then we have

$$\begin{aligned} |(f\star f)(q)|\le \frac{C_A}{w_a(|q|/A)} \end{aligned}$$

for any $A>0$ and therefore, in view of (2.13),

$$\begin{aligned} \left| \left( f_1*{\widehat{f}}_2\right) (q)\right| \le \frac{C'_A}{w_a(|q|/HA)}. \end{aligned}$$

But this contradicts (A.2) for $H^2A\le 1$ since $w_a(t)/w_a(t/HA)\le K/w_a(t)\rightarrow 0$ as $t\rightarrow \infty$. $\square$

We now turn to the mapping properties of the operators with Weyl symbols in ${\mathcal {E}}^{\{a\}}_{\{a\}}({{\mathbb {R}}}^{2d})$. The symbol class $\Gamma ^\infty _{1,s}$, which coincides with ${\mathcal {E}}^{\{a\}}_{\{a\}}$ for $a_n=n^{sn}$, was studied in [5] using the technique of short time Fourier transform and modulation spaces and the authors came to the conclusion that if $u\in \Gamma ^\infty _{1,s}({\mathbf {R}}^{2d})$, then $\mathop {\mathrm {Op}}\nolimits (u)$ is continuous from $\Sigma _s({\mathbf {R}}^d)$ to ${\mathcal {S}}_s({\mathbf {R}}^d)$, which are respectively $S_{(a)}^{(a)}({{\mathbb {R}}}^d)$ and $S_{\{a\}}^{\{a\}}({{\mathbb {R}}}^d)$ with $a_n=n^{sn}$ in our notation. A similar conclusion was made in [1] for a more general case of anisotropic Gelfand–Shilov spaces. Here we show that these conclusions are incorrect. We use the standard representation of the Weyl system

$$\begin{aligned} \left( T_\zeta ^\hbar \psi \right) (x)= e^{(i\hbar /2)\eta \xi }e^{i\eta x}\psi (x+\hbar \xi ),\quad \psi \in L^2({{\mathbb {R}}}^d), \qquad \zeta =(\eta ,\xi ), \end{aligned}$$

and assume that $S_{(a)}^{(a)}({{\mathbb {R}}}^{2d})$ is nontrivial. If $u\in S_{(a)}^{(a)\prime }({{\mathbb {R}}}^{2d})$, then applying the operator $\mathop {\mathrm {Op}}\nolimits (u)$ to $f\in S_{(a)}^{(a)}({{\mathbb {R}}}^d)$ yields an element of $S_{(a)}^{(a)\prime }({{\mathbb {R}}}^d)$ such that

$$\begin{aligned} \langle \mathop {\mathrm {Op}}\nolimits (u)\,f,g\rangle = (2\pi )^{-d}\bigl \langle {\widehat{u}}, \bigl ({\bar{g}},T^\hbar _{(\cdot )}f\bigr )_{L^2} \bigr \rangle ,\quad g\in S_{(a)}^{(a)}({{\mathbb {R}}}^d). \end{aligned}$$

Explicitly

$$\begin{aligned} \langle \mathop {\mathrm {Op}}\nolimits (u)\,f,g\rangle = (2\pi )^{-d}\Bigl \langle u,\int _{{{\mathbb {R}}}^d} g(q-\hbar \xi /2)\,f(q+\hbar \xi /2) e^{-ip\xi }d\xi \Bigr \rangle . \end{aligned}$$

Setting $\hbar =1$, this can also be expressed by saying that the kernel of $\mathop {\mathrm {Op}}\nolimits (u)$ is given by

$$\begin{aligned} {\mathcal {K}}(x,y)= (2\pi )^{-d/2}\left( F_2^{-1} u\right) \bigl ( (x+y)/2, x-y\bigr ), \end{aligned}$$

which is taken as a starting definition in [1, 5].

Proposition A.3 provides a counterexample to Theorem 4.12 in [5] and to Theorem 3.16 in [1].

Proposition A.3

There is a function $u\in {\mathcal {E}}^{\{a\}}_{\{a\}}({{\mathbb {R}}}^{2d})$ such that the image of $S_{(a)}^{(a)}({{\mathbb {R}}}^d)$ under $\mathop {\mathrm {Op}}\nolimits (u)$ is not contained in $L^2({{\mathbb {R}}}^d)$.

Proof

We set $d=1$, $\hbar =1$, and $u(q,p)=u_1(q)u_2(p)$. Let $\omega _1$ be a nonnegative even function in $S_{(a)}^{(a)}({{\mathbb {R}}})$ with $\int _0^\infty \omega _1(t)dt=1$, and let

$$\begin{aligned} u_1(q)=\int \limits _{-\infty }^\infty \omega _1(q-t)\, w_a(|t|)dt. \end{aligned}$$

The function $u_1$ belongs to ${\mathcal {E}}^{\{a\}}_{\{a\}}({{\mathbb {R}}})$ because $\Vert \omega _1\Vert _{A,B}<\infty$ for any $A,B>0$ and taking $A\le 1/(2H)$ and using (2.13) and (2.12) we obtain

$$\begin{aligned} |\partial ^nu_1(q)|\le & {} \Vert \omega _1\Vert _{A,B}B^na_n \int \limits _{-\infty }^\infty \frac{w_a(|t|)}{w_a(|q-t|/A)} dt \\\le & {} K\Vert \omega _1\Vert _{A,B}B^na_n \int \limits _{-\infty }^\infty \frac{w_a(|q-t|)}{w_a(2|t|)^2} dt\le C\Vert \omega _1\Vert _{A,B}B^na_n w_a(2|q|), \end{aligned}$$

where $C=2K\int _0^\infty w_a(2t)^{-1}dt$. Furthermore,

$$\begin{aligned} u_1(q)\ge w_a(|q|)\qquad \text {for all }q\in {{\mathbb {R}}}. \end{aligned}$$

(A.3)

Indeed, this function is even and for $q\ge 0$, we have

$$\begin{aligned} u_1(q)\ge \int \limits _q^\infty \omega _1(t-q)w_a(t)dt\ge w_a(q)\int \limits _0^\infty \omega _1(t)dt=w_a(q). \end{aligned}$$

Next, let ${\widehat{u}}_2(s)=\varphi (s/\Lambda )$, where $\varphi$ is the function constructed in Lemma A.1 and $\Lambda$ is a positive parameter. Clearly, $u(q,p)=u_1(q)u_2(p)$ belongs to ${\mathcal {E}}^{\{a\}}_{\{a\}}({{\mathbb {R}}}^2)$ and

$$\begin{aligned} (\mathop {\mathrm {Op}}\nolimits (u)\,f)(x)= \frac{1}{\sqrt{2\pi }}\int \limits _{-\infty }^\infty u_1((x+y)/2)\varphi ((x-y)/\Lambda ) f(y)dy. \end{aligned}$$

Let f be an arbitrary nonzero nonnegative function in $S_{(a)}^{(a)}({{\mathbb {R}}})$. From (A.1), (A.3), and (2.12), we obtain

$$\begin{aligned} \begin{aligned} (\mathop {\mathrm {Op}}\nolimits (u)\,f)(x)&\ge \frac{1}{\sqrt{2\pi }}\int \limits _{-\infty }^\infty \frac{w_a(|x+y|/2)f(y)}{w_a(|x-y|/\Lambda )}dy\\&\ge \frac{1}{\sqrt{2\pi }}\frac{w_a(|x|/4)}{w_a(2|x|/\Lambda )}\int \limits _{-\infty }^\infty \frac{f(y)}{w_a(|y|/2)w_a(2|y|/\Lambda )}dy. \end{aligned} \end{aligned}$$

If $\Lambda \ge 8H$, then (2.13) implies $w_a(t/4)/w_a(2t/\Lambda )\ge K^{-1}w_a(t/4H)$, and we conclude that $\mathop {\mathrm {Op}}\nolimits (u)\,f\notin L^2({{\mathbb {R}}})$. $\square$