Characterizations of a class of Pilipovi{\'c} spaces by powers of harmonic oscillator

TL;DR

This paper characterizes Pilipovi{\'c} spaces of smooth functions on \(\mathbf{R}^d\) using growth estimates of powers of the harmonic oscillator applied to the functions.

Contribution

It provides a new characterization of Pilipovi{\'c} spaces via $L^p$ norm estimates of harmonic oscillator powers, linking functional space membership to operator estimates.

Findings

Characterization of Pilipovi{\'c} spaces through harmonic oscillator norms.

Equivalence between space membership and specific $L^p$ estimates.

Provides criteria for smooth functions based on operator growth conditions.

Abstract

We show that a smooth function $f$ on $\mathbf R^d$ belongs to the Pilipovi{\'c} space $\mathcal H_{\flat _\sigma}(\mathbf R^d)$ or the Pilipovi{\'c} 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.

Equations

$$f(x)=\sum _{\alpha \in {\mathbf{N}}^d}{c}_{\alpha }(f)h_{\alpha }(x)$$

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

$${\mathcal{H}}_s({\mathbf{R}}^d)$$

$${\mathcal{H}}_{0,s}({\mathbf{R}}^d)$$

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

$${\mathcal{H}}_s({\mathbf{R}}^d)$$

$${\mathcal{H}}_{0,s}({\mathbf{R}}^d)$$

$$f\in {\mathcal{H}}_s({\mathbf{R}}^d)(f\in {\mathcal{H}}_{0,s}({\mathbf{R}}^d))\iff \Vert H_d^{N}f{\Vert }_{L^{\infty }}\lesssim r^{N}N{!}^{2s}$$

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

$$|F(z)|\lesssim e^{(\frac{1}{2}-r)|z{|}^2}(|F(z)|\lesssim e^{r|z{|}^2})$$

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

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

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

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

$$f\in {\mathcal{H}}_{{♭}_{\sigma }}({\mathbf{R}}^d)(f\in {\mathcal{H}}_{0,{♭}_{\sigma }}({\mathbf{R}}^d))$$

$$\iff$$

$$\Vert H_d^{N}f{\Vert }_{L^{\infty }}\lesssim 2^N{r}^{\frac{N}{\log (N\sigma )}}{\left(\frac{2N\sigma }{\log (N\sigma )}\right)}^{N(1-\frac{1}{\log (N\sigma )})}$$

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

$$\Vert f{\Vert }_{{\mathcal{H}}_{s;r}}$$

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

$$s_1<{♭}_{{\sigma }_1}<{♭}_{{\sigma }_2}<s_2\text{when}{s}_2\ge \frac{1}{2},s_1<\frac{1}{2}\text{and}{\sigma }_1<{\sigma }_2,$$

$${\mathcal{H}}_{s_1}({\mathbf{R}}^d)\subseteq {\mathcal{H}}_{0,s_2}({\mathbf{R}}^d)\subseteq {\mathcal{H}}_{s_2}({\mathbf{R}}^d),s_1,s_2\in {\mathbf{R}}_{♭}\text{and}{s}_1<s_2.$$

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

$$\Vert f{\Vert }_{M_{(\omega )}^{p,q}}\equiv \Vert V_{\varphi }f\cdot \omega {\Vert }_{L^{p,q}}$$

$$V_{\varphi }f(x,\xi )=(2\pi {)}^{-\frac{d}{2}}⟨f,e^{i⟨\cdot ,\xi ⟩}\overline{\varphi (\cdot -x)}⟩$$

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

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

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

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

$$g(r,t_1,t_2)$$

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

$$t_1,t_2>R,0\le t_2-t_1\le R,r\in I,$$

Full text

C. Fernández and A. Galbis were partially supported by the projects MTM2016-76647-P, ACOMP/2015/186 (Spain).

0. 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

[TABLE]

such that

[TABLE]

holds true for some $r>0$ respective for every $r>0$. Here $f(\theta)\lesssim g(\theta)$ means that $f(\theta)\leq cg(\theta)$ for some constant $c>0$ which is independent of $\theta$ in the domain of $f$ and $g$. (See also [6] and Section 1 for notations.) Evidently, $\mathcal{H}_{s}(\mathbf{R}^{d})$ and $\mathcal{H}_{0,s}(\mathbf{R}^{d})$ increases 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

[TABLE]

and

[TABLE]

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 $s\leq\frac{1}{2}$. These relationships are completed in [11] by the relations

[TABLE]

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

[TABLE]

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

[TABLE]

acting on the involved functions. These relations were obtained in [7] for $s\geq\frac{1}{2}$, 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 mapping 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

[TABLE]

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

[TABLE]

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 respective Beurling type, is a family of function spaces, which increases with $\sigma$ and such that

[TABLE]

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

[TABLE]

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

[TABLE]

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 $s\geq\frac{1}{2}$ 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 Section 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 Section 2, we have

[TABLE]

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 $\mathcal{H}_{0,\flat_{1}}(\mathbf{R}^{d})$.

1. 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

[TABLE]

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

[TABLE]

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 $\|f\|_{\mathcal{H}_{s;r}}$ 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,

[TABLE]

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

[TABLE]

we have

[TABLE]

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 $N\geq 0$ such that

[TABLE]

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 $f\in\mathscr{S}^{\prime}(\mathbf{R}^{d})$ such that

[TABLE]

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

[TABLE]

and

[TABLE]

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. (1)

$M^{p,q}_{(\omega)}(\mathbf{R}^{d})$* is a quasi-Banach space under the quasi-norm $\|\,\cdot\,\|_{M^{p,q}}$ above. If in addition $p,q\geq 1$, then $\|\,\cdot\,\|_{M^{p,q}}$ is a norm and $M^{p,q}_{(\omega)}(\mathbf{R}^{d})$ is a Banach space;* 2. (2)

the definition of $M^{p,q}(\mathbf{R}^{d})$ 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. (3)

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

2. 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. (1)

$f\in\mathcal{H}_{\flat_{\sigma}}(\mathbf{R}^{d})$* ($f\in\mathcal{H}_{0,\flat_{\sigma}}(\mathbf{R}^{d})$);* 2. (2)

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

[TABLE] 3. (3)

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

[TABLE] 4. (4)

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

[TABLE]

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. (1)

(2.1) holds for some $r>0$ (for every $r>0$); 2. (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 $R\geq e$, $I=(0,R]$,

[TABLE]

when $t_{1},t_{2}>e$ and $r>0$. Then

[TABLE]

when

[TABLE]

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

Proof.

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

If $r\geq 1$, $t=t_{1}$, $u=t_{2}-t_{1}>0$ and $\rho=\log r$, then

[TABLE]

for some constant $C$ which only depends on $r$ and $R$. 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

[TABLE]

where

[TABLE]

Since $b(t,t_{2})>0$ when $t_{2}>t$, we get

[TABLE]

for some constant $C\geq 0$. Here we have used that $t_{1},t_{2}>R\geq e$ and the fact that $t\mapsto\frac{t}{\log t}$ increases for $t\geq R$. ∎

Proof of Proposition 2.2.

First we prove that (2.2) is independent of $N_{0}>\sigma^{-1}$ when $p,q\geq 1$. 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

[TABLE]

and its inverse are continuous and bijective (cf. e. g. [8, Theorem 3.10]). Hence, if $\sigma^{-1}<N_{1}\leq N_{0}$, $N_{2}=N_{0}-N_{1}\geq 0$ and (2.2) holds for $N_{0}$, then

[TABLE]

and (2.2) holds for $N_{1}$. This implies that (2.2) is independent of $N_{0}>\sigma^{-1}$ when $p,q\geq 1$.

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 $N_{0}\geq 0$ such that

[TABLE]

and then

[TABLE]

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 $N\geq 2N_{0}$, then the bijectivity of (2.4) gives

[TABLE]

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

[TABLE]

(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

[TABLE]

which can be found in e. g. [9, Proposition 1.7]. ∎

Proposition 2.4**.**

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

[TABLE]

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

[TABLE]

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

[TABLE]

Then

[TABLE]

Proof.

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

In order to prove (2.10), let $t_{1}=t+\sigma_{0}$ and

[TABLE]

where $0\leq\sigma_{0}\leq\sigma$. Then

[TABLE]

and

[TABLE]

Since

[TABLE]

we get

[TABLE]

Hence the facts $\frac{t_{1}}{\log t_{1}}\geq 1$ and $0<r\leq 1$ give

[TABLE]

A combination of the latter inequality with (2.11) gives

[TABLE]

Lemma 2.7**.**

Let $s\geq\sigma(e+1)+e^{2}$

[TABLE]

Then the following is true:

1. (1)

for any $r_{2}>0$, there is an $r_{1}>0$ such that

[TABLE] 2. (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

[TABLE]

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

1. (1)′

for any $\rho_{2}\in\mathbf{R}$, there is a $\rho_{1}\in\mathbf{R}$ such that

[TABLE]

where

[TABLE] 2. (2)′

for any $\rho_{1}\in\mathbf{R}$, there is a $\rho_{2}\in\mathbf{R}$ such that (2.13) holds.

We choose

[TABLE]

where

[TABLE]

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

[TABLE]

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

[TABLE]

If $\rho_{1}\in\mathbf{R}$ is fixed, then we choose $\rho_{2}\in\mathbf{R}$ such that

[TABLE]

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 inequality

[TABLE]

gives

[TABLE]

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 $t_{1}\geq e(\sigma+1)+\sigma$ and $r_{0}>0$ such that

[TABLE]

Let $r_{1}=r_{0}$ if $r_{0}\geq 1$ and $r_{1}=r_{0}^{\frac{e}{e-1}}$ otherwise. By Lemma 2.6 it follows that

[TABLE]

holds when $t=N\sigma$ and $N\in\mathbf{N}$ is chosen such that $0\leq t_{1}-N\sigma\leq\sigma$. Observe that Lemma 2.6 can be applied since $N\sigma>e(\sigma+1)$. 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. ∎

Proof of Proposition

2.4.

Suppose that (2.7) holds for some $r=r_{1}>0$. By

[TABLE]

and (2.7) we get

[TABLE]

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

[TABLE]

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 $r>0$. ∎

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

[TABLE]

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

[TABLE]

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 $\rho\in(0,1)$. Suppose that (2.8) holds for some $r>0$ and let $r_{2}>r$. From (2.8) and (2.16) we get

[TABLE]

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

[TABLE]

This gives

[TABLE]

Using (2.18) and Lemma 2.8 we obtain

[TABLE]

when

[TABLE]

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. ∎

Proof of Theorem 2.1.

We have

[TABLE]

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

[TABLE]