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.
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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
such that
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
and
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
and
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
for some \(r>0\) (for every \(r>0\)) concerning estimates of powers of the harmonic oscillator
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
for some \(r>0\) (for every \(r>0\)). For \(s<\frac{1}{2}\), this estimate is replaced by
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
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
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
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
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
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
and
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,
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
we have
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
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
is finite. Here \(V_\phi f\) is the short-time Fourier transform of f with respect to \(\phi \), given by
and
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 \(R\ge e\), \(I = (0,R]\),
when \(t_1,t_2>e\) and \(r>0\). Then
when
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
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
where
Since \(b(t,t_2)>0\) when \(t_2>t\), we get
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
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
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
and then
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
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
(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
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
for some \(r>0\) (for every \(r>0\)), then
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
Then
and
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
where \(0\le \sigma _0 \le \sigma \). Then
and
Since
we get
Hence the facts \(\frac{t_1}{\log t_1} \ge 1\) and \(0<r\le 1\) give
A combination of the latter inequality with (2.11) gives
\(\square \)
Lemma 2.7
Let \(s\ge \sigma (e+1)+e^2\)
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
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
where
Obviously, x increases with y, and by function investigations it follows that
giving that \(0<h\le \frac{1}{2e}<1\). Then (2.14) becomes
If \(\rho _1\in \mathbf{R}\) is fixed, then we choose \(\rho _2\in \mathbf{R}\) such that
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
give
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
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
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
and (2.7) we get
By taking the infimum over all \(N\ge 0\), it follows from Lemma 2.7 (2) that
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
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
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
where \(s=|\alpha |\) and \(t=N\sigma \). Since \(0<\rho <1\) we have
This gives
Using (2.18) and Lemma 2.8 we obtain
when
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
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
\(\square \)
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C. Fernández and A. Galbis were partially supported by the projects MTM2016-76647-P, GV Prometeo/2017/102 (Spain). J. Toft was partially supported by Vetenskapsrådet (Sweden) within the project 2019-04890.
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Abdeljawad, A., Fernández, C., Galbis, A. et al. Characterizations of a class of Pilipović spaces by powers of harmonic oscillator. RACSAM 114, 131 (2020). https://doi.org/10.1007/s13398-020-00858-8
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DOI: https://doi.org/10.1007/s13398-020-00858-8