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

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.

The set of Pilipović spaces is a family of Fourier invariant spaces, containing any Fourier invariant (standard) Gelfand-Shilov space. The (standard) Pilipović spaces H s (R d ) and H 0,s (R d ) with respect to s ∈ R + , are the sets of all formal Hermite series expansions holds true for some r > 0 respective for every r > 0. Here f (θ ) g(θ ) means that f (θ ) ≤ cg(θ ) for some constant c > 0 which is independent of θ in the domain of f and g (see also [6] and Sect. 1 for notations). Evidently, H s (R d ) and H 0,s (R d ) increase with s. It is proved in [7] that if S s (R d ) and s (R d ) are the Gelfand-Shilov spaces of Roumieu respective Beurling type of order s, then and It is also well-known that S s (R d ) = {0} when s < 1 2 and s (R d ) = {0} when s ≤ 1 2 . These relationships are completed in [11] by the relations In particular, each Pilipović space is contained in the Schwartz space S (R d ).
For H s (R d ) (H 0,s (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 ≥ 1 2 , and in [11] in the general case s > 0.
In [3,11]  for some r > 0 (for every r > 0), which is indeed a stronger condition compared to the case s = 1 2 . An important motivation for considering the spaces H σ (R d ) and H 0, σ (R d ) is to make this gap smaller. More precisely, H σ (R d ) and H 0, σ (R d ), which are Pilipović spaces of Roumieu respectively Beurling type, is a family of function spaces, which increases with σ and such that In [3], characterizations of H 1 (R d ) and H 0, 1 (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 σ = 1, it seems that no such characterizations for H σ (R d ) and H 0, σ (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 σ = 1 we regain the corresponding characterizations in [3] for H 1 (R d ) and H 0, 1 (R d ).

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 S s (R d ) and s (R d ) of Roumieu and Beurling type, respectively, consist of all f ∈ C ∞ (R d ) such that is finite, for some r > 0 respectively for every r > 0. The topologies of S s (R d ) and s (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 H s (R d ) and H 0,s (R d ) we consider the norms and H 0,s (R d ) are the inductive limit and the projective limit, respectively, of H s;r (R d ) with respect to r > 0. In particular, and it follows that H s (R d ) is complete, and that H 0,s (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 R + into R ≡ R + ∪ { σ } σ >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].
The set of moderate weights of polynomial type on R d is denoted by and and F is measurable on 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 ∈ (0, ∞] and ω ∈ P(R 2d ). Then the following is true: increases with p and q (also in topological sense).

Characterizations of H (R d ) and H 0, (R d ) in terms of powers of the harmonic oscillator
In this section we deduce characterizations of the test function spaces H 0, σ (R d ) and More precisely we have the following.
Then the following conditions are equivalent: ( (3) for some r > 0 (for every r > 0) it holds We need some preparations for the proof. In the following proposition we treat separately the equivalence between (3) and (4) in Theorem 2.1.
We need the following lemma for the proof of Proposition 2.2.
for some constant C > 0 which only depends on R.
Proof Since t → t log t is increasing when t ≥ e, g is upper bounded by one when r ≤ 1, and the boundedness of g follows in this case.
If r ≥ 1, t = t 1 , u = t 2 − t 1 > 0 and ρ = 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 , for some constant C ≥ 0. Here we have used that t 1 , t 2 > R ≥ e and the fact that t → t log t increases for t ≥ R.

Proof of Proposition 2.2. First we prove that (2.2) is independent of N
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 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 > σ −1 when p, q ≥ 1. Next we prove that (2.2) is independent of the choice of ω ∈ P(R 2d ). By the first part of the proof, we may assume that N 0 σ > e. For every ω 1 , ω 2 ∈ P(R 2d ), we may find an integer N 0 > σ −1 e such that Hence the stated invariance follows if we prove that (2.2) holds for 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 σ > e shows that (2) is independent of ω ∈ P(R 2d ). For general p, q > 0, the invariance of (2.2) with respect to ω, 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].
For the proofs we need some preparation lemmas.
We choose Obviously, x increases with y, and by function investigations it follows that giving that 0 < h ≤ 1 2e < 1. Then (2.14) becomes e −y F(y + log y − log 2) = − y 2 2 + y 2 1 − 1 y + log y 2 y + log y − log y + log y 2 If ρ 1 ∈ R is fixed, then we choose ρ 2 ∈ R such that for some large number C 0 > 0. In the same way, if ρ 2 ∈ R is fixed, then we choose ρ 1 ∈ R such that (2.15) holds. For such choices and the fact that 0 < h < 1, the inequalities 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 ≥ e(σ + 1) + σ and r 0 > 0 such that Let r 1 = r 0 if r 0 ≥ 1 and r 1 = r e e−1 0 otherwise. By Lemma 2.6 it follows that holds when t = N σ and N ∈ N is chosen such that 0 ≤ t 1 − N σ ≤ σ . Observe that Lemma 2.6 can be applied since N σ ≥ e(σ + 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.
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. (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 R + .