Characterizations of a class of Pilipović spaces by powers of harmonic oscillator

Abstract

We show that a smooth function f on ${\mathbf {R}}^{d}$ belongs to the Pilipović space ${\mathcal {H}}_{\flat _\sigma }({\mathbf {R}}^{d})$ or the Pilipović space ${\mathcal {H}}_{0,\flat _\sigma }({\mathbf {R}}^{d})$, if and only if the $L^p$ norm of $H_d^Nf$ for $N\ge 0$, satisfy certain types of estimates. Here $H_d=|x|^2-\Delta _x$ is the harmonic oscillator.

1 Introduction

In the paper we characterize Pilipović spaces of the form ${\mathcal {H}}_{\flat _\sigma }({\mathbf {R}}^{d})$ and ${\mathcal {H}}_{0,\flat _\sigma } ({\mathbf {R}}^{d})$, considered in [3, 11], in terms of estimates of powers of the harmonic oscillator, on the involved functions.

The set of Pilipović spaces is a family of Fourier invariant spaces, containing any Fourier invariant (standard) Gelfand-Shilov space. The (standard) Pilipović spaces ${\mathcal {H}}_s({\mathbf {R}}^{d})$ and ${\mathcal {H}}_{0,s}({\mathbf {R}}^{d})$ with respect to $s\in \mathbf{R}_+$, are the sets of all formal Hermite series expansions

$$\begin{aligned} f(x) = \sum _{\alpha \in {\mathbf {N}}^{d}}c_\alpha (f)h_\alpha (x) \end{aligned}$$

(0.1)

such that

$$\begin{aligned} |c_\alpha (f)| \lesssim e^{-r|\alpha |^{\frac{1}{2s}}} \end{aligned}$$

(0.2)

holds true for some $r>0$ respective for every $r>0$. Here $f(\theta )\lesssim g(\theta )$ means that $f(\theta )\le cg(\theta )$ for some constant $c>0$ which is independent of $\theta$ in the domain of f and g (see also [6] and Sect. 1 for notations). Evidently, ${\mathcal {H}}_s({\mathbf {R}}^{d})$ and ${\mathcal {H}}_{0,s}({\mathbf {R}}^{d})$ increase with s. It is proved in [7] that if ${\mathcal {S}}_s({\mathbf {R}}^{d})$ and $\Sigma _s({\mathbf {R}}^{d})$ are the Gelfand-Shilov spaces of Roumieu respective Beurling type of order s, then

$$\begin{aligned} {\mathcal {H}}_s({\mathbf {R}}^{d})&= {\mathcal {S}}_s({\mathbf {R}}^{d}), \quad s \ge \frac{1}{2}, \end{aligned}$$

(0.3)

$$\begin{aligned} {\mathcal {H}}_{0,s}({\mathbf {R}}^{d})&= \Sigma _s({\mathbf {R}}^{d}), \quad s > \frac{1}{2}, \end{aligned}$$

(0.4)

and

$$\begin{aligned} {\mathcal {H}}_{0,s}({\mathbf {R}}^{d})\ne \Sigma _s({\mathbf {R}}^{d})=\{0\},\quad s=\frac{1}{2}. \end{aligned}$$

It is also well-known that ${\mathcal {S}}_s({\mathbf {R}}^{d})=\{0\}$ when $s<\frac{1}{2}$ and $\Sigma _s({\mathbf {R}}^{d})=\{0\}$ when $s\le \frac{1}{2}$. These relationships are completed in [11] by the relations

$$\begin{aligned} {\mathcal {H}}_s({\mathbf {R}}^{d}) \ne {\mathcal {S}}_s({\mathbf {R}}^{d}) = \{0\}, \quad s <\frac{1}{2} \end{aligned}$$

and

$$\begin{aligned} {\mathcal {H}}_{0,s}({\mathbf {R}}^{d}) \ne \Sigma _s({\mathbf {R}}^{d})=\{0\}, \quad s \le \frac{1}{2}. \end{aligned}$$

In particular, each Pilipović space is contained in the Schwartz space ${\mathscr {S}}({\mathbf {R}}^{d})$.

For ${\mathcal {H}}_s({\mathbf {R}}^{d})$ (${\mathcal {H}}_{0,s}({\mathbf {R}}^{d})$) we also have the characterizations

$$\begin{aligned} f\in {\mathcal {H}}_s({\mathbf {R}}^{d})\quad (f\in {\mathcal {H}}_{0,s}({\mathbf {R}}^{d}))\quad \Leftrightarrow \quad \Vert H^{N}_{d} f\Vert _{L^\infty }\lesssim r^N N!^{2s} \end{aligned}$$

(0.5)

for some $r>0$ (for every $r>0$) concerning estimates of powers of the harmonic oscillator

$$\begin{aligned} H_d=|x|^2-\Delta _x,\qquad x\in {\mathbf {R}}^{d}, \end{aligned}$$

acting on the involved functions. These relations were obtained in [7] for $s\ge \frac{1}{2}$, and in [11] in the general case $s>0$.

In [3, 11] characterizations of ${\mathcal {H}}_s({\mathbf {R}}^{d})$ and ${\mathcal {H}}_{0,s}({\mathbf {R}}^{d})$ were also obtained by certain spaces of analytic functions on $\mathbf {C}^{d}$, via the Bargmann transform. From these map** properties it follows that near $s=\frac{1}{2}$ there is a jump concerning these Bargmann images. More precisely, if $s=\frac{1}{2}$, then the Bargmann image of ${\mathcal {H}}_{s}({\mathbf {R}}^{d})$ (of ${\mathcal {H}}_{0,s}({\mathbf {R}}^{d})$) is the set of all entire functions F on $\mathbf {C}^{d}$ such that F obeys the condition

$$\begin{aligned} |F(z)|\lesssim e^{(\frac{1}{2}-r)|z|^2} \qquad (\, |F(z)|\lesssim e^{r|z|^2}\, ) \end{aligned}$$

(0.6)

for some $r>0$ (for every $r>0$). For $s<\frac{1}{2}$, this estimate is replaced by

$$\begin{aligned} |F(z)| \lesssim e^{r(\log (1+|z|))^{\frac{1}{1-2s}}} \end{aligned}$$

(0.7)

for some $r>0$ (for every $r>0$), which is indeed a stronger condition compared to the case $s=\frac{1}{2}$.

An important motivation for considering the spaces ${\mathcal {H}}_{\flat _\sigma }({\mathbf {R}}^{d})$ and ${\mathcal {H}}_{0,\flat _\sigma }({\mathbf {R}}^{d})$ is to make this gap smaller. More precisely, ${\mathcal {H}}_{\flat _\sigma }({\mathbf {R}}^{d})$ and ${\mathcal {H}}_{0,\flat _\sigma }({\mathbf {R}}^{d})$, which are Pilipović spaces of Roumieu respectively Beurling type, is a family of function spaces, which increases with $\sigma$ and such that

$$\begin{aligned} {\mathcal {H}}_{s_1}({\mathbf {R}}^{d}) \subseteq {\mathcal {H}}_{0,\flat _\sigma }({\mathbf {R}}^{d}) \subseteq {\mathcal {H}}_{\flat _\sigma }({\mathbf {R}}^{d}) \subseteq {\mathcal {H}}_{0,s_2}({\mathbf {R}}^{d}), \qquad s_1<\frac{1}{2},\ s_2\ge \frac{1}{2}. \end{aligned}$$

The spaces ${\mathcal {H}}_{\flat _\sigma }({\mathbf {R}}^{d})$ and ${\mathcal {H}}_{0,\flat _\sigma }({\mathbf {R}}^{d})$ consist of all formal Hermite series expansions (0.1) such that

$$\begin{aligned} |c_\alpha (f)|\lesssim r^{|\alpha |}\alpha !^{-\frac{1}{2\sigma }} \end{aligned}$$

(0.8)

hold true for some $r>0$ respectively for every $r>0$. For the Bargmann images of ${\mathcal {H}}_{\flat _\sigma }({\mathbf {R}}^{d})$ and ${\mathcal {H}}_{0,\flat _\sigma }({\mathbf {R}}^{d})$, the conditions (0.6) and (0.7) above are replaced by

$$\begin{aligned} |F(z)|\lesssim e^{r|z|^{\frac{2\sigma }{\sigma +1}}}, \end{aligned}$$

for some $r>0$ respectively for every $r>0$. It follows that the gaps of the Bargmann images of ${\mathcal {H}}_s({\mathbf {R}}^{d})$ and ${\mathcal {H}}_{0,s}({\mathbf {R}}^{d})$ between the cases $s<\frac{1}{2}$ and $s\ge \frac{1}{2}$ are drastically decreased by including the spaces ${\mathcal {H}}_{\flat _\sigma }({\mathbf {R}}^{d})$ and ${\mathcal {H}}_{0,\flat _\sigma }({\mathbf {R}}^{d})$, $\sigma >0$, in the family of Pilipović spaces.

In [3], characterizations of ${\mathcal {H}}_{\flat _1}({\mathbf {R}}^{d})$ and ${\mathcal {H}}_{0,\flat _1}({\mathbf {R}}^{d})$ in terms of estimates of powers of the harmonic oscillator acting on the involved functions which corresponds to (0.5) are deduced. On the other hand, apart from the case $\sigma =1$, it seems that no such characterizations for ${\mathcal {H}}_{\flat _\sigma }({\mathbf {R}}^{d})$ and ${\mathcal {H}}_{0,\flat _\sigma }({\mathbf {R}}^{d})$ have been obtained so far.

In Sect. 2 we fill this gap in the theory, and deduce such characterizations. In particular, as a consequence of our main result, Theorem 2.1 in Sect. 2, we have

$$\begin{aligned}&f\in {\mathcal {H}}_{\flat _\sigma }({\mathbf {R}}^{d}) \quad (f\in {\mathcal {H}}_{0,\flat _\sigma }({\mathbf {R}}^{d})) \\&\qquad \qquad \qquad \qquad \Leftrightarrow \\&\Vert H_d^N f\Vert _{L^\infty } \lesssim 2^N r^{\frac{N}{\log (N\sigma )}} \left( \frac{2N\sigma }{\log (N\sigma )} \right) ^{N ( 1-\frac{1}{\log (N\sigma )} ) } \end{aligned}$$

for some (every) $r > 0$. By choosing $\sigma =1$ we regain the corresponding characterizations in [3] for ${\mathcal {H}}_{\flat _1}({\mathbf {R}}^{d})$ and ${\mathcal {H}}_{0,\flat _1}({\mathbf {R}}^{d})$.

2 Preliminaries

In this section we recall some facts about Gelfand-Shilov spaces, Pilipović spaces and modulation spaces.

Let $s>0$. Then the (Fourier invariant) Gelfand-Shilov spaces ${\mathcal {S}}_s({\mathbf {R}}^{d})$ and $\Sigma _s({\mathbf {R}}^{d})$ of Roumieu and Beurling type, respectively, consist of all $f\in C^\infty ({\mathbf {R}}^{d})$ such that

$$\begin{aligned} \Vert f\Vert _{{\mathcal {S}}_{s;r}}\equiv \sup _{\alpha ,\beta \in {\mathbf {N}}^{d}} \left( \frac{\Vert x^\alpha D^\beta f\Vert _{L^\infty ({\mathbf {R}}^{d})}}{r^{|\alpha +\beta |}(\alpha !\beta !)^s} \right) \end{aligned}$$

(1.1)

is finite, for some $r>0$ respectively for every $r>0$. The topologies of ${\mathcal {S}}_s({\mathbf {R}}^{d})$ and $\Sigma _s ({\mathbf {R}}^{d})$ are the inductive limit topology and the projective limit topology, respectively, supplied by the norms (1.1). We refer to [1, 5] for more facts about Gelfand-Shilov spaces.

For ${\mathcal {H}}_s({\mathbf {R}}^{d})$ and ${\mathcal {H}}_{0,s}({\mathbf {R}}^{d})$ we consider the norms

$$\begin{aligned} \Vert f\Vert _{{\mathcal {H}}_{s;r}} \equiv \sup \limits _{\alpha \in {\mathbf {N}}^{d}} \left( |c_\alpha (f)|e^{r|\alpha |^{\frac{1}{2s}}}\right)&\quad \text {when} \quad s \in \mathbf{R}_+ \end{aligned}$$

and

$$\begin{aligned} \Vert f\Vert _{{\mathcal {H}}_{s;r}} \equiv \sup \limits _{\alpha \in {\mathbf {N}}^{d}} \left( |c_\alpha (f)| r^{-|\alpha |}\alpha !^{\frac{1}{2\sigma }} \right) \quad \text {when} \quad s = \flat _\sigma , \end{aligned}$$

when $r>0$ is fixed. Then the set ${\mathcal {H}}_{s;r}({\mathbf {R}}^d)$ consists of all $f\in C^\infty ({\mathbf {R}}^{d})$ such that $\Vert f\Vert _{{\mathcal {H}}_{s;r}}$ is finite. It follows that ${\mathcal {H}}_{s;r}({\mathbf {R}}^d)$ is a Banach space.

The Pilipović spaces ${\mathcal {H}}_s({\mathbf {R}}^d)$ and ${\mathcal {H}}_{0,s}({\mathbf {R}}^d)$ are the inductive limit and the projective limit, respectively, of ${\mathcal {H}}_{s;r}({\mathbf {R}}^d)$ with respect to $r>0$. In particular,

$$\begin{aligned} {\mathcal {H}}_s({\mathbf {R}}^d)=\bigcup _{r>0}{\mathcal {H}}_{s;r}({\mathbf {R}}^d)\quad \text {and} \quad {\mathcal {H}}_{0,s}({\mathbf {R}}^d)=\bigcap _{r>0}{\mathcal {H}}_{s;r}({\mathbf {R}}^d) \end{aligned}$$

and it follows that ${\mathcal {H}}_s({\mathbf {R}}^d)$ is complete, and that ${\mathcal {H}}_{0,s}({\mathbf {R}}^d)$ is a Fréchet space. It is well-known that the identities (0.3) and (0.4) also hold in topological sense (cf. [7]).

By extending $\mathbf{R}_+$ into $\mathbf{R}_\flat \equiv \mathbf{R}_+ \cup \{\flat _\sigma \}_{\sigma >0}$ and letting

$$\begin{aligned} s_1< \flat _{\sigma _1}< \flat _{\sigma _2}<s_2 \quad \text {when}\quad s_2 \ge \frac{1}{2}, \ s_1<\frac{1}{2}\ \text {and}\ \sigma _1<\sigma _2, \end{aligned}$$

we have

$$\begin{aligned} {\mathcal {H}}_{s_1}({\mathbf {R}}^{d})\subseteq {\mathcal {H}}_{0,s_2}({\mathbf {R}}^{d}) \subseteq {\mathcal {H}}_{s_2}({\mathbf {R}}^{d}),\quad s_1,s_2\in \mathbf{R}_\flat \; \text {and}\; s_1<s_2. \end{aligned}$$

We also need some facts about weights and modulation spaces, a family of (quasi-)Banach spaces, introduced by Feichtinger in [2]. A weight on ${\mathbf {R}}^{d}$ is a function $\omega \in L^\infty _{loc}({\mathbf {R}}^{d})$ such that $\omega (x)>0$ for every $x\in {\mathbf {R}}^{d}$ and $1/\omega \in L^\infty _{loc}({\mathbf {R}}^{d})$. The weight $\omega$ on ${\mathbf {R}}^{d}$ is called moderate of polynomial type, if there is an integer $N\ge 0$ such that

$$\begin{aligned} \omega (x+y)\lesssim \omega (x)(1+|y|)^N,\qquad x,y\in {\mathbf {R}}^{d} . \end{aligned}$$

The set of moderate weights of polynomial type on ${\mathbf {R}}^{d}$ is denoted by ${\mathscr {P}}({\mathbf {R}}^{d})$.

Let $p,q\in (0,\infty ]$, $\phi \in {\mathscr {S}}({\mathbf {R}}^{d})\setminus 0$ and $\omega \in {\mathscr {P}}({\mathbf {R}}^{2d})$ be fixed. Then the modulation space, $M^{p,q}_{(\omega )}({\mathbf {R}}^{d})$ consists of all $f\in {\mathscr {S}}'({\mathbf {R}}^{d})$ such that

$$\begin{aligned} \Vert f\Vert _{M^{p,q}_{(\omega )}} \equiv \Vert V_\phi f \cdot \omega \Vert _{L^{p,q}} \end{aligned}$$

is finite. Here $V_\phi f$ is the short-time Fourier transform of f with respect to $\phi$, given by

$$\begin{aligned} V_\phi f(x,\xi ) = (2\pi )^{-\frac{d}{2}}\langle f,e^{i\langle \, \cdot \, ,\xi \rangle }\overline{\phi (\, \cdot \, -x)}\rangle \end{aligned}$$

and

$$\begin{aligned} \Vert F\Vert _{L^{p,q}} = \Vert F\Vert _{L^{p,q}({\mathbf {R}}^{2d})} \equiv \Vert g_F\Vert _{L^q({\mathbf {R}}^{d})} \quad \text {when}\quad g_F(\xi ) = \Vert F(\, \cdot \, ,\xi )\Vert _{L^p({\mathbf {R}}^{d})} \end{aligned}$$

and F is measurable on ${\mathbf {R}}^{2d}$.

Modulation spaces possess several convenient properties. For example we have the following proposition (see [2, 4] for proofs).

Proposition 1.1

Let $p,q\in (0,\infty ]$ and $\omega \in {\mathscr {P}}({\mathbf {R}}^{2d})$. Then the following is true:

• (1) $M^{p,q}_{(\omega )}({\mathbf {R}}^{d})$ is a quasi-Banach space under the quasi-norm $\Vert \, \cdot \, \Vert _{M^{p,q}}$ above. If in addition $p,q\ge 1$, then $\Vert \, \cdot \, \Vert _{M^{p,q}}$ is a norm and $M^{p,q}_{(\omega )}({\mathbf {R}}^{d})$ is a Banach space;

• (2) the definition of $M^{p,q}_{(\omega )}({\mathbf {R}}^{d})$ is independent of the choice of $\phi$ above and different $\phi \in {\mathscr {S}}({\mathbf {R}}^{d})\setminus 0$ gives rise to equivalent quasi-norms;

• (3) $M^{p,q}_{(\omega )}({\mathbf {R}}^{d})$ increases with p and q (also in topological sense).

3 Characterizations of ${\mathcal {H}}_{\flat _\sigma }({\mathbf {R}}^{d})$ and ${\mathcal {H}}_{0,\flat _\sigma } ({\mathbf {R}}^{d})$ in terms of powers of the harmonic oscillator

In this section we deduce characterizations of the test function spaces ${\mathcal {H}}_{0,\flat _\sigma }({\mathbf {R}}^{d})$ and ${\mathcal {H}}_{\flat _\sigma }({\mathbf {R}}^{d})$.

More precisely we have the following.

Theorem 2.1

Let $\sigma >0$, $N,N_0\in \mathbf{N}$ be such that $N_0\sigma >1$, $p_0\in [1,\infty ]$, $p,q\in (0,\infty ]$, $\omega \in {\mathscr {P}}({\mathbf {R}}^{2d})$ and let $f\in C^\infty ({\mathbf {R}}^{d})$ be given by (0.1). Then the following conditions are equivalent:

• (1) $f\in {\mathcal {H}}_{\flat _\sigma }({\mathbf {R}}^{d})$ ($f\in {\mathcal {H}}_{0,\flat _\sigma }({\mathbf {R}}^{d})$);

• (2) for some $r > 0$ (for every $r > 0$) it holds

  $$\begin{aligned} \{ c_\alpha (f)r^{-|\alpha |}(\alpha !)^{\frac{1}{2\sigma }} \} _{\alpha \in {\mathbf {N}}^{d}} \in \ell ^q({\mathbf {N}}^{d})\text{; } \end{aligned}$$

• (3) for some $r > 0$ (for every $r > 0$) it holds

  $$\begin{aligned} \Vert H_d^N f\Vert _{L^{p_0}} \lesssim 2^N r^{\frac{N}{\log (N\sigma )}} \left( \frac{2N\sigma }{\log (N\sigma )} \right) ^{N ( 1-\frac{1}{\log (N\sigma )} ) }, \quad N \ge N_0 \text{; } \end{aligned}$$

  (2.1)

• (4) for some $r > 0$ (for every $r > 0$) it holds

  $$\begin{aligned} \Vert H_d^N f\Vert _{M^{p,q}_{(\omega )}} \lesssim 2^N r^{\frac{N}{\log (N\sigma )}} \left( \frac{2N\sigma }{\log (N\sigma )} \right) ^{N ( 1-\frac{1}{\log (N\sigma )} ) }, \quad N \ge N_0. \end{aligned}$$

  (2.2)

We need some preparations for the proof. In the following proposition we treat separately the equivalence between (3) and (4) in Theorem 2.1.

Proposition 2.2

Let $p_0\in [1,\infty ]$, $p,q\in (0,\infty ]$, $\sigma >0$$N_0>\sigma ^{-1}$ be an integer and let $\omega \in {\mathscr {P}}({\mathbf {R}}^{2d})$. Then the following conditions are equivalent:

• (1)  (2.1) holds for some $r>0$ (for every $r>0$);

• (2)  (2.2) holds for some $r>0$ (for every $r>0$).

We need the following lemma for the proof of Proposition 2.2.

Lemma 2.3

Let $R\ge e$, $I = (0,R]$,

$$\begin{aligned} g(r,t_1,t_2) \equiv \frac{r^{\frac{t_2}{\log t_2}}}{r^{\frac{t_1}{\log t_1}}} \quad \text {and}\quad h(t_1,t_2) \equiv \frac{\left( \frac{2t_2}{\log t_2} \right) ^{t_2\left( 1-\frac{1}{\log t_2}\right) } }{\left( \frac{2t_1}{\log t_1} \right) ^{t_1\left( 1-\frac{1}{\log t_1}\right) }}, \end{aligned}$$

when $t_1,t_2>e$ and $r>0$. Then

$$\begin{aligned} 0 \le g(r,t_1,t_2)\le C \quad \text {and} \quad 0\le h(t_1,t_2)\le \left( \frac{2t_1}{\log t_1} \right) ^C \end{aligned}$$

(2.3)

when

$$\begin{aligned} t_1,t_2>R,\ 0\le t_2-t_1 \le R, \ r\in I, \end{aligned}$$

for some constant $C>0$ which only depends on R.

Proof

Since $t\mapsto \frac{t}{\log t}$ is increasing when $t\ge e$, g is upper bounded by one when $r\le 1$, and the boundedness of g follows in this case.

If $r\ge 1$, $t=t_1$, $u =t_2-t_1>0$ and $\rho =\log r$, then

$$\begin{aligned} 0\le \log g(r,t_1,t_2)&= \left( \frac{t+u}{\log (t+u )}-\frac{t}{\log t} \right) \rho \\&= \frac{t}{\log t} \left( \frac{1+\frac{u}{t}}{1+\frac{\log (1+\frac{u}{t})}{\log t}} -1 \right) \rho = \frac{t}{\log t} \left( \frac{\frac{u}{t} - \frac{\log (1+\frac{u}{t})}{\log t}}{1+\frac{\log (1+\frac{u}{t})}{\log t}} \right) \rho \\&\quad < \frac{t}{\log t} \cdot \frac{u}{t} \cdot \rho \\&= \frac{u \rho }{\log t} \le C \end{aligned}$$

for some constant C which only depends on R. This shows the boundedness of g.

Next we show the estimates for $h(t_1,t_2)$ in (2.3). By taking the logarithm of $h(t_1,t_2)=h(t,t_2)$ we get

$$\begin{aligned} \log h(t,t_2)=t_2 \log \left( \frac{2t_2}{\log t_2}\right) -t \log \left( \frac{2t}{\log t} \right) - b(t,t_2), \end{aligned}$$

where

$$\begin{aligned} b(t,t_2) =\left( \frac{t_2}{\log t_2} \log \left( \frac{2t_2}{\log t_2}\right) -\frac{t}{\log t} \log \left( \frac{2t}{\log t}\right) \right) . \end{aligned}$$

Since $b(t,t_2)>0$ when $t_2>t$, we get

$$\begin{aligned} \log h(t_1,t_2)&< t_2 \log \left( \frac{2t_2}{\log t_2}\right) -t \log \left( \frac{2t}{\log t}\right) \\&=(t+u ) \left( \log \left( \frac{2t}{\log t} \right) +\log \left( \frac{1+\frac{u}{t}}{1+\frac{\log (1+\frac{u}{t})}{\log t}}\right) \right) -t\log \left( \frac{2t}{\log t}\right) \\&\le u \log \left( \frac{2t}{\log t} \right) +t \log \left( 1+\frac{u}{t} \right) +C \\&\le u \log \left( \frac{2t}{\log t} \right) +u +C \end{aligned}$$

for some constant $C\ge 0$. Here we have used that $t_1,t_2>R\ge e$ and the fact that $t\mapsto \frac{t}{\log t}$ increases for $t\ge R$. $\square$

Proof of Proposition 2.2

First we prove that (2.2) is independent of $N_0>\sigma ^{-1}$ when $p,q\ge 1$. Evidently, if (2.2) is true for $N_0$, then it is true for any larger replacement of $N_0$. On the other hand, the map

$$\begin{aligned} H_d^N \, :\, M^{p,q}_{(v_N\omega )}({\mathbf {R}}^{d})\rightarrow M^{p,q}_{(\omega )}({\mathbf {R}}^{d}), \qquad v_N(x,\xi )=(1+|x|^2+|\xi |^2)^N, \end{aligned}$$

(2.4)

and its inverse are continuous and bijective (cf. e. g. [8, Theorem 3.10]). Hence, if $\sigma ^{-1}< N_1\le N_0$, $N_2=N_0-N_1\ge 0$ and (2.2) holds for $N_0$, then

$$\begin{aligned} \Vert H_d^{N_1}f\Vert _{M^{p,q}_{(\omega )}} \lesssim \Vert H_d^{N_0}f\Vert _{M^{p,q}_{(\omega /v_{N_2})}} \lesssim \Vert H_d^{N_0}f\Vert _{M^{p,q}_{(\omega )}}<\infty , \end{aligned}$$

and a straight-forward combination of these estimates and (2.3) shows that (2.2) holds for $N_1$ in place of $N_0$. This implies that (2.2) is independent of $N_0>\sigma ^{-1}$ when $p,q\ge 1$.

Next we prove that (2.2) is independent of the choice of $\omega \in {\mathscr {P}}({\mathbf {R}}^{2d})$. By the first part of the proof, we may assume that $N_0\sigma >e$. For every $\omega _1,\omega _2\in {\mathscr {P}}({\mathbf {R}}^{2d})$, we may find an integer $N_0 >\sigma ^{-1}e$ such that

$$\begin{aligned} \frac{1}{v_{N_0}}\lesssim \omega _1, \omega _2\lesssim v_{N_0}, \end{aligned}$$

and then

$$\begin{aligned} \Vert f\Vert _{M^{p,q}_{(1/v_{N_0})}}\lesssim \Vert f\Vert _{M^{p,q}_{(\omega _1)}}, \Vert f\Vert _{M^{p,q}_{(\omega _2)}} \lesssim \Vert f\Vert _{M^{p,q}_{(v_{N_0})}}. \end{aligned}$$

(2.5)

Hence the stated invariance follows if we prove that (2.2) holds for $\omega =v_{N_0}$, if it is true for $\omega =1/v_{N_0}$.

Therefore, assume that (2.2) holds for $\omega =1/v_{N_0}$. Let $f_N=H_d^{N}f$, $u =2N_0\sigma$, $t=t_1=N\sigma$, $N_2=N+2N_0$ and $t_2=t_1+u = N_2\sigma$. If $N\ge 2N_0$, then the bijectivity of (2.4) gives

$$\begin{aligned} \frac{\Vert f_N\Vert _{M^{p,q}_{(v_{N_0})}}^\sigma }{2^{N\sigma } r^{\frac{N\sigma }{\log (N\sigma )}} \left( \frac{2N\sigma }{\log (N\sigma )}\right) ^{N\sigma \left( 1-\frac{1}{\log (N\sigma )}\right) }}&=\frac{\Vert f_N\Vert _{M^{p,q}_{(v_{N_0})}}^\sigma }{{2^t r^{\frac{t}{\log t}} \left( \frac{2t}{\log t} \right) ^{t\left( 1-\frac{1}{\log t}\right) }}} \nonumber \\&\lesssim \frac{\Vert f_{N+2N_0}\Vert _{M^{p,q}_{(1/v_{N_0})}}^\sigma }{{2^t r^{\frac{t}{\log t}} \left( \frac{2t}{\log t} \right) ^{t\left( 1-\frac{1}{\log t}\right) }}} \nonumber \\&{=} 2^{u} g(r,t_1,t_2)h(t_1,t_2) \cdot \frac{\Vert f_{N_2}\Vert _{M^{p,q}_{(1/v_{N_0})}}^\sigma }{2^{N_2\sigma }r^{\frac{t_2}{\log (t_2)}} \left( \frac{2t_2}{\log t_2} \right) ^{t_2\left( 1-\frac{1}{\log t_2}\right) }}, \end{aligned}$$

(2.6)

where $g(r,t_1,t_2)$ and $h(t_1,t_2)$ are the same as in Lemma 2.3. A combination of Lemma 2.3, (2.6) and the fact that $N\sigma >e$ shows that (2) is independent of $\omega \in {\mathscr {P}}({\mathbf {R}}^{2d})$. For general $p,q>0$, the invariance of (2.2) with respect to $\omega$, p and q is a consequence of the embeddings

$$\begin{aligned} M^\infty _{(v_N\omega )}({\mathbf {R}}^{d})\subseteq M^{p,q} _{(\omega )}({\mathbf {R}}^{d}) \subseteq M^\infty _{(\omega )}({\mathbf {R}}^{d}), \qquad N> d\left( \frac{1}{p} +\frac{1}{q} \right) \end{aligned}$$

(see e. g. [4, Theorem 3.4] or [10, Proposition 3.5]).

The equivalence between (1) and (2) now follows from these invariance properties and the continuous embeddings

$$\begin{aligned} M^{p_0,q_1}\subseteq L^{p_0}\subseteq M^{p_0,q_2}, \qquad q_1=\min (p_0,p_0'),\quad q_2=\max (p_0,p_0'), \end{aligned}$$

which can be found in e. g. [9, Proposition 1.7]. $\square$

Proposition 2.4

Let $f\in C^\infty ({\mathbf {R}}^{d})$ and $\sigma >0$. If

$$\begin{aligned} \Vert H_d^Nf\Vert _{L^2}\lesssim 2^Nr^{\frac{N}{\log (N\sigma )}} \left( \frac{2N\sigma }{\log (N\sigma )} \right) ^{N(1-\frac{1}{\log (N\sigma )})},\quad N\in \mathbf{N},\ N\sigma \ge e, \end{aligned}$$

(2.7)

for some $r>0$ (for every $r>0$), then

$$\begin{aligned} |c_\alpha (f)| \lesssim r^{|\alpha |}|\alpha |^{-\frac{|\alpha |}{2\sigma }}, \quad \alpha \in {\mathbf {N}}^{d}, \end{aligned}$$

(2.8)

for some $r>0$ (for every $r>0$).

Proposition 2.5

Let $f\in C^\infty ({\mathbf {R}}^{d})$ and $\sigma >0$. If (2.8) holds for some $r>0$ (for every $r>0$), then (2.7) holds for some $r>0$ (for every $r>0$).

For the proofs we need some preparation lemmas.

Lemma 2.6

Let $\sigma >0$, $\sigma _0 \in [0,\sigma ]$ and let

$$\begin{aligned} F(r,t)=\left( \frac{2t}{\log t} \right) ^{t\left( 1-\frac{1}{\log t}\right) } r^{\frac{t}{\log t}}, \qquad r\ge 0, \quad t\ge e\cdot \max (1,\sigma ) . \end{aligned}$$

Then

$$\begin{aligned} F(r,t)&\le F(r,t+\sigma _0), \quad r \in [1,\infty ), \end{aligned}$$

(2.9)

and

$$\begin{aligned} F(r,t) \le F(r^{1-\frac{1}{e}},t+\sigma _0),&\quad r&\in (0,1]. \end{aligned}$$

(2.10)

Proof

If $r\ge 1$, then it follows by straight-forward tests with derivatives that F(r, t) is increasing with respect to $t\ge e$. This gives (2.9).

In order to prove (2.10), let $t_1=t+\sigma _0$ and

$$\begin{aligned} h(t_1,\sigma _0 ) = \frac{1-\frac{\sigma _0}{t_1}}{1+\frac{\log \left( 1-\frac{\sigma _0}{t_1}\right) }{\log t_1}}, \end{aligned}$$

where $0\le \sigma _0 \le \sigma$. Then

$$\begin{aligned} \left( \frac{2t}{\log t} \right) ^{t\left( 1-\frac{1}{\log t}\right) } r^{\frac{t}{\log t}} \le \left( \frac{2t_1}{\log t_1} \right) ^{t_1\left( 1-\frac{1}{\log t_1}\right) } r^{\frac{t}{\log t}} \end{aligned}$$

(2.11)

and

$$\begin{aligned} {\frac{2t}{\log t}}= h(t_1,\sigma _0 ) \cdot \frac{2t_1}{\log t_1}. \end{aligned}$$

Since

$$\begin{aligned} 0\le \frac{\sigma _0}{t_1}\le \frac{1}{e} \quad \text {and}\quad -1<\frac{\log \left( 1-\frac{\sigma _0}{t_1}\right) }{\log t_1}\le 0 \end{aligned}$$

we get

$$\begin{aligned} h(t_1,\sigma _0 ) \ge 1-\frac{\sigma _0}{t_1}\ge 1-\frac{1}{e}. \end{aligned}$$

Hence the facts $\frac{t_1}{\log t_1} \ge 1$ and $0<r\le 1$ give

$$\begin{aligned} r^{\frac{t}{\log t}}= r^{h\left( t_1,\sigma _0 \right) \frac{t_1}{\log t_1}} \le r^{\left( 1-\frac{1}{e}\right) \frac{t_1}{\log t_1}}. \end{aligned}$$

A combination of the latter inequality with (2.11) gives

$$\begin{aligned} F(r,t) \le \left( \frac{2t_1}{\log t_1} \right) ^{t_1\left( 1-\frac{1}{\log t_1}\right) } \left( r^{1-\frac{1}{e} } \right) ^{\frac{t_1}{\log t_1}} = F\left( r^{1-\frac{1}{e}},t_1\right) . \end{aligned}$$

$\square$

Lemma 2.7

Let $s\ge \sigma (e+1)+e^2$

$$\begin{aligned} \Omega _1 = [e,\infty )\cap (\sigma \cdot \mathbf{N}) \quad \text {and}\quad \Omega _2 = [e,\infty ). \end{aligned}$$

Then the following is true:

• (1) for any $r_2>0$, there is an $r_1>0$ such that

  $$\begin{aligned} \inf _{t\in \Omega _j} \left( s^{-t} \left( \frac{2t}{\log t} \right) ^{t\left( 1-\frac{1}{\log t}\right) }r_1^{\frac{t}{\log t}} \right) \lesssim r_2 ^ss^{-\frac{s}{2}},\quad j=1,2\text{; } \end{aligned}$$

  (2.12)

• (2) for any $r_1>0$, there is an $r_2>0$ such that (2.12) holds.

Proof

First prove the result for $j=2$. Let

$$\begin{aligned} x=\log t,\quad y=\log s \ge \log (\sigma (e+1)+e^2) >2, \quad \rho _j=\log r_j,\quad j=1,2. \end{aligned}$$

By applying the logarithm on (2.12), the statements (1) and (2) follow if we prove:

(1)$'$:

for any $\rho _2\in \mathbf{R}$, there is a $\rho _1\in \mathbf{R}$ such that

$$\begin{aligned} \inf _{x\ge x_0} F(x) \le 0, \qquad x_0 = \log (\sigma (e+1)+e^2) \end{aligned}$$

(2.13)

where

$$\begin{aligned} F(x)= -e^xy + e^x \left( 1-\frac{1}{x} \right) (x+\log 2-\log x) + \rho _1 \frac{e^x}{x}-\rho _2e^y+\frac{e^yy}{2} \end{aligned}$$

(2.14)

(2)$'$:

for any $\rho _1\in \mathbf{R}$, there is a $\rho _2\in \mathbf{R}$ such that (2.13) holds.

We choose

$$\begin{aligned} x=y+\log y -\log 2 \ge \log s \ge x_0 \quad \text {and let}\quad h=g(y), \end{aligned}$$

where

$$\begin{aligned} g(u)=\frac{\log u - \log 2}{u}. \end{aligned}$$

Obviously, x increases with y, and by function investigations it follows that

$$\begin{aligned} 0=g(2)<g(u)\le g(2e)=\frac{1}{2e},\qquad u>2, \end{aligned}$$

giving that $0<h\le \frac{1}{2e}<1$. Then (2.14) becomes

$$\begin{aligned} e^{-y}F(y+\log y-\log 2)&= -\frac{y^2}{2} + \frac{y}{2} \left( 1-\frac{1}{y+\log \frac{y}{2}} \right) \left( y+\log y -\log \left( y+\log \frac{y}{2}\right) \right) \\&\quad +\frac{\rho _1y}{2(y+\log \frac{y}{2})}-\rho _2+\frac{y}{2} \\&= -\frac{y}{2}\log (1+h)+\frac{\log y +\log (1+h)}{2(1+h)} +\frac{\rho _1-\log 2}{2(1+h)}-\rho _2 . \end{aligned}$$

If $\rho _1\in \mathbf{R}$ is fixed, then we choose $\rho _2\in \mathbf{R}$ such that

$$\begin{aligned} \frac{\rho _1-\log 2}{2(1+h)}-\rho _2\le -C_0 \end{aligned}$$

(2.15)

for some large number $C_0>0$. In the same way, if $\rho _2\in \mathbf{R}$ is fixed, then we choose $\rho _1\in \mathbf{R}$ such that (2.15) holds. For such choices and the fact that $0<h<1$, the inequalities

$$\begin{aligned} 0<h-\frac{h^2}{2} \le \log (1+h)\le h \end{aligned}$$

give

$$\begin{aligned} F(y+\log y-\log 2)&\le e^y \left( -\frac{y}{2}\log (1+h)+\frac{\log y +\log (1+h)}{2(1+h)}-C_0 \right) \\&\le e^y \left( -\frac{y}{2}\log (1+h)+\frac{\log y +\log (1+h)}{2} -C_0\right) \\&\le e^y \left( -\frac{\log y-\log 2}{2}+\frac{(\log y-\log 2)^2}{4y}+\frac{1}{2} \left( \log y +h \right) -C_0\right) \\&\le e^y \left( \frac{1}{2}\log 2+\frac{(\log y-\log 2)^2}{4y}+\frac{h}{2} -C_0\right) <0, \end{aligned}$$

provided $C_0$ was chosen large enough. This gives the result in the case $j=2$.

Next we prove the result for $j=1$. Let $r_2>0$. By the first part of the proof, there are $t_1\ge e(\sigma +1) + \sigma$ and $r_0>0$ such that

$$\begin{aligned} s^{-t_1} \left( \frac{2t_1}{\log t_1} \right) ^{t_1\left( 1-\frac{1}{\log t_1}\right) }r_0^{\frac{t_1}{\log t_1}} \le r_2^ss^{-\frac{s}{2}}. \end{aligned}$$

Let $r_1= r_0$ if $r_0\ge 1$ and $r_1=r_0^{\frac{e}{e-1}}$ otherwise. By Lemma 2.6 it follows that

$$\begin{aligned} s^{-t} \left( \frac{2t}{\log t} \right) ^{t\left( 1-\frac{1}{\log t}\right) }r_1^{\frac{t}{\log t}} \le r_2^s s^{-\frac{s}{2}} \end{aligned}$$

holds when $t=N\sigma$ and $N\in \mathbf{N}$ is chosen such that $0\le t_1-N\sigma \le \sigma$. Observe that Lemma 2.6 can be applied since $N\sigma \geqslant e(\sigma +1)$. This gives (1) for $j=1$.

By similar arguments, (2) for $j=1$ follows from (2) in the case $j=2$. The details are left for the reader. $\square$

Proof of Proposition 2.4

Suppose that (2.7) holds for some $r=r_1>0$. By

$$\begin{aligned} c_\alpha (H_d^Nf) = (2|\alpha |+d)^Nc_\alpha (f), \quad |c_\alpha (H_d^Nf)|\le \Vert H_d^Nf\Vert _{L^2} \end{aligned}$$

(2.16)

and (2.7) we get

$$\begin{aligned} |c_\alpha (f)|&= \frac{|c_\alpha (H_d^Nf)|}{(2|\alpha |+d)^N} \\&\lesssim \left( |\alpha |+\frac{d}{2} \right) ^{-N} r_1^{\frac{N}{\log (N\sigma )}} \left( \frac{2N\sigma }{\log (N\sigma )} \right) ^{N\left( 1-\frac{1}{\log (N\sigma )}\right) } \\&\le \left( |\alpha | ^{-N\sigma } r_1^{\frac{N\sigma }{\log (N\sigma )}} \left( \frac{2N\sigma }{\log (N\sigma )} \right) ^{N\sigma \left( 1-\frac{1}{\log (N\sigma )}\right) } \right) ^{\frac{1}{\sigma }}. \end{aligned}$$

By taking the infimum over all $N\ge 0$, it follows from Lemma 2.7 (2) that

$$\begin{aligned} |c_\alpha (f)| \lesssim \left( r_2^{|\alpha |}|\alpha |^{-\frac{|\alpha |}{2}} \right) ^{\frac{1}{\sigma }} = r^{|\alpha |}|\alpha |^{-\frac{|\alpha |}{2\sigma }}, \qquad |\alpha |\ge 2\sigma (e+1)+e^2, \end{aligned}$$

for some $r_2>0$, where $r=r_2^{\frac{1}{\sigma }}$. Hence (2.8) holds for some $r>0$.

By similar arguments, using (1) instead of (2) in Lemma 2.7, it follows that if (2.7) holds for every $r>0$, then (2.8) holds for every $r>0$. $\square$

For the proof of Proposition 2.5 we will use the following result which is essentially a slight clarification of [3, Lemma 2]. The proof is therefore omitted.

Lemma 2.8

Let $r>0$ and

$$\begin{aligned} f(s,t,r) = \frac{s^{2t}(2re)^s}{s^s},\qquad s> 1,\ t\ge 0. \end{aligned}$$

Then there exist a positive increasing function $\theta$ on $[0,\infty )$ and a constant $t_0=t_0(r)>e$ which only depends on r such that

$$\begin{aligned} \max _{s>0}f(s,t,r) \le \left( \frac{2t}{\log t} \right) ^{2t\left( 1-\frac{1}{\log t}\right) }(\theta (r)r)^{\frac{2t}{\log t}}, \quad t\ge t_0(r) \text{. } \end{aligned}$$

(2.17)

Remark 2.9

The constants s, t and $t_0(r)$ in Lemma 2.8 are denoted by t, N and $N_0(r)$, respectively in Lemmas 1 and 2 in [3]. In the latter results it is understood that N and $N_0(r)$ are integers. On the other hand, it is evident from the proofs of these results that they also hold when N and $N_0(r)$ are allowed to be in $\mathbf{R}_+$.

Proof of Proposition 2.5

Let $\theta$ be as in Lemma 2.8 and let $\rho \in (0,1)$. Suppose that (2.8) holds for some $r>0$ and let $r_2>r^{\sigma }$. From (2.8) and (2.16) we get

$$\begin{aligned} \Vert H_d^Nf\Vert _{L^2}^2&= \sum \limits _{\alpha \in {\mathbf {N}}^{d}} |(2|\alpha |+d)^Nc_\alpha (f)|^2 \\&\lesssim \sup _{ |\alpha |\ge 1}\left( (2|\alpha |+d)^{2N} r_2^{\frac{2|\alpha |}{\sigma }}|\alpha |^{-\frac{|\alpha |}{\sigma }}\right) \\&=\sup \limits _{s\ge 1}\left( 2^{2t}\left( s+\frac{d}{2}\right) ^{2t}r_2^{2s} s^{-s} \right) ^{\frac{1}{\sigma }}, \end{aligned}$$

where $s=|\alpha |$ and $t=N\sigma$. Since $0<\rho <1$ we have

$$\begin{aligned} s^s&= \left( s-\frac{d}{2}+\frac{d}{2} \right) ^{s-\frac{d}{2}} s^{\frac{d}{2}} = \left( s-\frac{d}{2} \right) ^{s-\frac{d}{2}} s^{\frac{d}{2}} \left( 1+\frac{d}{2s-d} \right) ^{s-\frac{d}{2}} \\&\le \left( s-\frac{d}{2} \right) ^{s-\frac{d}{2}} (se)^{\frac{d}{2}} \lesssim \left( s-\frac{d}{2} \right) ^{s-\frac{d}{2}} \rho ^{-2s}. \end{aligned}$$

This gives

$$\begin{aligned} \Vert H_d^Nf\Vert _{L^2}^2&\lesssim \sup \limits _{s\ge 1} \left( 2^{2t} \left( s+\frac{d}{2} \right) ^{2t} r_2^{2s} s^{-s} \right) ^{\frac{1}{\sigma }} \nonumber \\&= \sup \limits _{s\ge 1+\frac{d}{2}} \left( 2^{2t} s^{2t} r_2^{2s-d} \left( s-\frac{d}{2} \right) ^{-\left( s- \frac{d}{2}\right) } \right) ^{\frac{1}{\sigma }} \nonumber \\&\lesssim \sup \limits _{s\ge 1+\frac{d}{2}} \left( 2^{2t} s^{2t} \left( \frac{r_2}{\rho } \right) ^{2s} s^{-s} \right) ^{\frac{1}{\sigma }}. \end{aligned}$$

(2.18)

Using (2.18) and Lemma 2.8 we obtain

$$\begin{aligned} \Vert H_d^Nf\Vert _{L^2}^2&\lesssim \sup \limits _{s\ge 1+\frac{d}{2}} \left( 2^{2t} s^{2t} \left( \frac{r_2}{\rho } \right) ^{2s} s^{-s} \right) ^{\frac{1}{\sigma }} \\&= \sup \limits _{s\ge 1+\frac{d}{2}} \left( 2^{2t} s^{2t} \left( 2r_3e \right) ^s s^{-s} \right) ^{\frac{1}{\sigma }} \\&\lesssim \left( 2^{2t} \left( \frac{2t}{\log t} \right) ^{2t(1-\frac{1}{\log t})} \left( \theta (r_3)r_3 \right) ^{\frac{2t}{\log t}} \right) ^{\frac{1}{\sigma }} \\&= 2^{2N}(r_3\theta (r_3))^{\frac{2N}{\log (N\sigma )}} \left( \frac{2N\sigma }{\log (N\sigma )} \right) ^{2N\left( 1-\frac{1}{\log (N\sigma )}\right) } \end{aligned}$$

when

$$\begin{aligned} r_3 = \frac{r_2^{2}}{2\rho ^2e} \quad \text {and}\quad N\sigma \ge t_0(r_3). \end{aligned}$$

This gives the result in the Roumieu case.

By similar argument, using the fact that the non-negative function $\theta$ is increasing, it also follows that (2.7) holds for every $r>0$ when (2.8) holds for every $r>0$, and the result follows. $\square$

Proof of Theorem 2.1

We have

$$\begin{aligned} \Vert \{ c_\alpha (f)r^{-|\alpha |}(\alpha !)^{\frac{1}{2\sigma }} \} _{\alpha \in {\mathbf {N}}^{d}}\Vert _{\ell ^\infty ({\mathbf {N}}^{d})}&\le \Vert \{ c_\alpha (f)r^{-|\alpha |}(\alpha !)^{\frac{1}{2\sigma }} \} _{\alpha \in {\mathbf {N}}^{d}}\Vert _{\ell ^q ({\mathbf {N}}^{d})} \\&\lesssim \Vert \{ c_\alpha (f)(cr)^{-|\alpha |}(\alpha !)^{\frac{1}{2\sigma }} \} _{\alpha \in {\mathbf {N}}^{d}}\Vert _{\ell ^\infty ({\mathbf {N}}^{d})} \end{aligned}$$

when $c\in (0,1)$, which shows that (2) is independent of the choice of q. The equivalence between (1) and (2) now follows by the definitions and choosing $q=\infty$ in (2).

By Proposition 2.2 we may assume that $p=2$. The result now follows from Propositions 2.4 and 2.5, together with the fact that

$$\begin{aligned} (d\cdot e)^{-|\alpha |} |\alpha |^{|\alpha |} \le \alpha ! \le |\alpha |^{|\alpha |},\quad \alpha \in {\mathbf {N}}^{d}. \end{aligned}$$

$\square$