Tensor products for Gelfand-Shilov and Pilipovi{\'c} distribution spaces

We show basic properties on tensor products for Gelfand-Shilov distributions and Pilipovi{\'c} distributions. This also includes the Fubbini's property of such tensor products. We also apply the Fubbini property to deduce some properties for short-time Fourier transforms of Gelfand-Shilov and Pilipovi{\'c} distributions.


Introduction
An important issue in mathematics concerns tensor products. When considering the functions f j defined on Ω j ⊆ R d j , j = 1, 2, and with values in C, their tensor product f 1 ⊗ f 2 is the function from Ω 1 × Ω 2 to C given by the formula x j ∈ Ω j , j = 1, 2.
The formulae (0.2) and (0.3) are essential when searching for extention of tensor products to distributions. By the analysis in [7, Chapter V and VII], we have the following.
The existence of a distribution f in the previous theorem which satisfies (0.2) can also be deduced by a general and abstract result on tensor products for nuclear spaces (see [15,Chapter 50]). On the other hand, in order to reach the Fubbini property (0.3), it seems that more structures are needed.
A more specific approach in the lines of the ideas in [15] is indicated in [8,12], where S (R d ) and S ′ (R d ) are described by suitable series expansions of Hermite functions. By following such approach, the situations are essentially reduced to questions on tensor products of weighted ℓ 2 spaces, and both properties (0.2) and (0.3) follows from such approach.
In Sections 2 and 3 we show that Theorem 0.1 holds in the context of Gelfand-Shilov spaces, Pilipović spaces and their distribution (dual) spaces. In particular, we prove that the following results hold true.
The same holds true with H 0,s and H ′ 0,s in place of H s and H ′ s , respectively, at each occurrence.
The distribution f in Theorem 0.1, Theorem 0.2 or in Theorem 0.3 is called the tensor product of f 1 and f 2 and is denoted by f 1 ⊗ f 2 as before.
We remark that Gelfand-Shilov spaces of functions and distributions appear naturally when discussing analyticity and well-posedness of solutions to partial differential equations (cf. [2,3]). Pilipović spaces of functions and distributions often agree with Fourier-invariant Gelfand-Shilov spaces, and possess convenient mapping properties with respect to the Bargmann transform. They therefore seems to be suitable to have in background on problems in partial differential equations which have been transformed by the Bargmann transform (see [5,14] for more details).
Since the spaces in Theorems 0.2 and 0.3 are unions and intersections of nuclear spaces, the existence of f satisfying (0.2) may be deduced by the abstract analogous results in [15]. Some parts of Theorem 0.2 are also proved in [8].
In Section 2 we give a proof of Theorem 0.2, by using the framework in [7] for the proof of Theorem 0.1. In Section 3 we use that Pilipović spaces and their distribution spaces can be described by unions and intersections of Hilbert spaces of Hermite series expansions. In similar ways as in [12], this essentially reduce the situation to deal with questions on tensor products of weighted ℓ 2 spaces.
In the end of Section 2 we also give example on how to apply the Fubbini property (0.3) to deduce certain relations for short-time Fourier transforms (which often called coherent state trasnform in physics) of Gelfand-Shilov distributions (see Example 2.5). In Section 3 we also discuss such questions for Pilipović spaces which are not Gelfand-Shilov distributions (cf. Remark 3.3).

Preliminaries
In this section we recall some basic facts. We start by giving the definition of Gelfand-Shilov spaces. Thereafter we recall some the definition of Pilipović spaces and recall some of their properties.
where the supremum is taken over all α j , β j ∈ N d j , j = 1, . . . , d. Then The Gelfand-Shilov spaces S σ s (R d ) and Σ σ s (R d ) are defined as the inductive and projective limits respectively of S σ s;h (R d ). This implies that and that the topology for S σ s (R d ) is the strongest possible one such that the inclusion map from S σ There are various kinds of characterisations of the spaces S σ s (R d ) and Σ σ s (R d ), e. g. in terms of the exponential decay of their elements. Later on it will be useful that for some h, r > 0 (respectively for every h > 0, ε > 0). If 1 = (1, . . . , 1) ∈ R n and s, σ ∈ R n + , then for every ε > 0. If in addition s j + σ j ≥ 1 for every j, then the last two inclusions in (1.3) are dense, and if in addition (s j , σ j ) = ( 1 2 , 1 2 ) for every j, then the first inclusion in (1.3) is dense.
The Gelfand-Shilov distribution spaces (S σ s (R d ) and (Σ σ s (R d ) are the projective and inductive limit respectively of (S σ s ) ′ (R d ). This means that We remark that the analysis in [10] shows that when s j + σ j ≥ 1, j = 1, . . . , n, and . . , n. The Gelfand-Shilov spaces possess several convenient mapping properties. For example they are invariant under translations, dilations, and to some extent (partial) Fourier transformations. For any f ∈ L 1 (R d ), its Fourier transform is defined by If instead f ∈ L 1 (R d 1 +···+dn ), then the partial Fourier transform of f with respect to k ∈ {1, . . . , n} is given by Then the following follows from the general theory of Schwartz functions and Gelfand-Shilov functions and their distributions (see e. g. [4,7]): (1) the definition of F j extends to a homeomorphism on S ′ (R d ) and restricts to a homeomorphism on S (R d ); (2) the definition of F j extends uniquely to a homeomorphism from 1.2. Pilipović spaces. Next we make a review of Pilipović spaces. These spaces can be defined in terms of Hermite series expansions. We recall that the Hermite function of order α ∈ N d is defined by It is well-known that the set of Hermite functions is a basis for S (R d ) and an orthonormal basis for L 2 (R d ) (cf. [12]). In particular, if f ∈ is the Hermite seriers expansion of f , and In order to define the full scale of Pilipović spaces, their order s should belong to the extended set of R + , with extended inequality relations as [14].) For r > 0 and s ∈ R ♭ we set and (1.7) 6 Definition 1.2. Let s ∈ R ♭ = R ♭ ∪ {0}, and let ϑ r,s and ϑ ′ r,s be as in (1.6) and (1.7).
(1) H 0 (R d ) consists of all Hermite polynomials, and H ′ 0 (R d ) consists of all formal Hermite series expansions in (1.4); holds true for some r ∈ R + (for every r ∈ R + ); consists of all formal Hermite series expansions in (1.4) such that r,s (α) holds true for every r ∈ R + (for some r ∈ R + ).  [14] for notations). Then it is proved in [9,10] that Next we recall the topologies for Pilipović spaces. Let s ∈ R ♭ , r > 0, and let f Hs;r and f H ′ s;r be given by s (R d ) be the inductive respectively projective limit topology of H s;r (R d ) with respect to r > 0. In the same way, the topologies of H ′ s (R d ) and (H 0 s ) ′ (R d ) are the projective respectively inductive limit topology of H ′ s;r (R d ) with respect to r > 0.
Suppose instead s = 0. For any integer N ≥ 0, we set The topology for H ′ 0 (R d ) is defined by the semi-norms · (0,N ) . We also let H 0,N (R d ) be the vector space which consists of all f ∈ H ′ 0 (R d ) such that c α (f ) = 0 when |α| > N, and equip this space with the topology, defined by the norm · (0,N ) . The topology of H 0 (R d ) is then defined as the inductive limit topology of H 0,N (R d ) with respect to N ≥ 0.
It follows that all the spaces in Definition 1.2 are complete, and that H 0 s (R d ) and H ′ s (R d ) are Fréchet space with semi-norms f → f Hs;r and f → f H ′ s;r , respectively. The following characterisations of Pilipović spaces can be found in [14]. The proof is therefore omitted.
, if and only if f ∈ C ∞ (R d ) and satisfies |H N d f (x)| h N N! 2s for every h > 0 (for some h > 0). Finally we remark that the Pilipović spaces of functions and distributions possess convenient mapping properties under the Bargmann transform (cf. [14]).

Tensor product for Gelfand-Shilov spaces
In this section we start by proving Theorem 0.2. Thereafter we deduce a multi-linear version of this result.
Since the right-hand side tends to zero when ε > 0 tends to zero, the stated convergence follows in this case.
The general case follows from the previous case, after writing ϕ = ϕ 1 + iϕ 2 and ψ = ψ 1 + iψ 2 with ϕ j and ψ j being real-valued, j = 1, 2, giving that ϕ * ψ is a superposition of ϕ j 1 * ψ j 2 , j 1 , j 2 ∈ {1, 2}, and using the fact that ϕ j ∈ S σ 1 ,σ 2 s 1 ,s 2 (R d 1 +d 2 ) when ϕ ∈ S σ 1 ,σ 2 s 1 ,s 2 (R d 1 +d 2 ). We may now prove the following result related to [7,Theorem 4.1.2] We have that f * ϕ is smooth, and for some r 0 > 0 we have for every r > 0. This gives for some constant C x which only depends on x and r 0 . It follows that

Hence (2.2) holds, and the result follows.
By the previous lemma it is now straight-forward to prove the following.
s and (Σ σ s ) ′ , respectively. Proof. We only prove the result in the Roumieu case. The Beurling case follows by similar arguments and is left for the reader. We use the same notations as in the previous proofs.
For general n ≥ 2, the result follows from the case n = 2 and induction. The details are left for the reader.
Proof of Theorem 0.2. We only prove the result in the Roumieu cases. The Beurling cases follow by similar arguments and are left for the reader.
In order to consider corresponding multi-linear situation of Theorem 0.2, we let S n be the permutation group of {1, . . . , n}, and let inductively when f j for j = 1, . . . , n are suitable distributions and ϕ is a suitable function. Then Theorem 0.2 can be reformulated as follows. It is also convenient to set and d j,τ = d τ (1) + · · · + d τ (j) , (2.5) when j = 1, . . . , n and s, σ ∈ R n + , Theorem 2.4. Let τ ∈ S 2 , d = d 1 + d 2 , s, σ ∈ R 2 + , d j,τ , s j,τ and σ j,τ be as in (2.5), f j ∈ (S σ j s j ) ′ (R d j ), ϕ ∈ S σ s (R d ) and let ϕ j,τ be given by (2.3) and (2.4), j = 1, 2. Then ϕ j,τ ∈ S σ j,τ s jτ (R d j,τ ), and there is a unique distribution f in (S σ s ) ′ (R d ) such that for every ϕ j ∈ S σ j s j (R d j ), j = 1, . . . , n, and ϕ 2 ∈ S σ 2 s 2 (R d 2 ), s and (S σ s ) ′ , respectively, at each occurrence. Here the second equality in (2.6) is the same as the Fubbini property (0.3). The multi-linear version of the previous theorem is the following, and follows by similar arguments as for its proof. The details are left for the reader.
Theorem 2.4 ′ . Let τ ∈ S n , d = d 1 + · · · + d n , s, σ ∈ R n + , d j,τ , s j,τ and σ j,τ be as in (2.5), f j ∈ (S σ j s j ) ′ (R d j ), ϕ ∈ S σ s (R d ) and let ϕ j,τ be given by (2.3) and (2.4), j = 1, . . . , n. Then ϕ j,τ ∈ S σ j,τ s jτ (R d j,τ ), and there is It follows that and for such choices of φ and f . We notice that the right-hand side of (2.7) also makes sense as a smooth function on R 2d if the assumption on f is relaxed into f ∈ (S σ s ) ′ (R d ). For such f we therefore let (2.7) define the short-time Fourier transform of f with respect to φ. Since the map which takes φ into y → φ(y − x)e i y,ξ is continuous and smooth with respect to (x, ξ) from S σ s (R d ) to itself it follows that V φ f is smooth. By [14, Proposition 2.2] it follows that V φ f belongs to (S σ,s s,σ ) ′ (R 2d ). Consequently, . Let U be the operator which takes any F (x, y) into F (y, y − x) and recall that F 2 F is the partial Fourier transform of F (x, y) with respect to the y variable. Then the right-hand side of (2.8) equals (2.9) We notice that the right-hand side makes sense as an element in (S σ,s s,σ ) ′ (R 2d ) for any f, φ ∈ (S σ s ) ′ (R d ) in view of Remark 1.1, which may be used to extend the definition of the short-time Fourier transform to even more general situations.

Tensor product of Pilipović spaces
In this section we discuss the tensor map on Pilipović spaces. Especially we prove Theorem 0.2. Thereafter we deduce a multi-linear version of this result.
First we show that the tensor map possess natural mapping properties on Pilipović spaces. Proposition 3.1. Let s ∈ R ♭ . Then the following is true: restricts to a continuous map from H 0, Proof. We only prove (1) and in the case s > 0. The case s = 0 and (2) follow by similar arguments and are left for the reader. If , α = (α 1 , α 2 ), α j ∈ N d j , j = 1, 2.
If s ∈ R + , then |c α j (f j )| e −c|α j | and it follows that f ∈ H s (R d ).
Proof of Theorem 0.3. Let d = d 1 + d 2 . We shall deal with the Hermite sequence representations of the elements in the Pilipović spaces. Such approach is performed in [12], when deducing tensor product and kernel results for tempered distributions. We only prove the results when f j ∈ H ′ s (R d ) and s > 0. The cases when f j ∈ H ′ 0,s (R d ) or s = 0 follow by similar arguments and are left for the reader.