Strong ultra-regularity properties for positive elements in the twisted convolutions

We show that positive elements with respect to the twisted convolutions, belonging to some ultra-test function space of certain order at origin, belong to the ultra-test function space of the same order everywhere. We apply the result to positive semi-definite Weyl operators.


Introduction
Several issues in operator theory can be studied by means of the twisted convolution. For example, composition and positivity questions can be carried over to related questions for the twisted convolution product by simple manipulations. We notice the simple structure of the twisted convolution, since it essentially consists of a convolution product, disturbed by a (symplectic) Fourier kernel. It is also common that boundedness, regularity and positivity conditions on operator kernels often correspond to convenient conditions on related elements in the twisted convolution. For example, operators with kernels in the Schwartz space S , or in the Gelfand-Shilov spaces S s or s of Roumieu and Beurling types, respectively, can be transformed to twisted convolutions between elements in the same classes. (See Sect. 2 for notations.) In [10] it is shown that various kinds of singularities for positive elements with respect to the twisted convolution are attained at the origin. Furthermore, it is proved that regularity at origin for such elements impose global regularity and bounedness for these elements and their Fourier transforms.
We note that if (1.1) holds true with s < 1/2 in (5), then a is trivially equal to 0, since the Gelfand-Shilov spaces S s and s are trivial for such choices of s.
In this paper we investigate related questions in background of Pilipović spaces, S s and s of Roumieu and Beurling type respectively, a family of function spaces which agrees with corresponding Gelfand-Shilov spaces when these are non-trivial (cf. [8,9]). We introduce the so-called twisted Pilipović spaces S σ,s and σ,s which are symplectic analogies of Pilipović spaces, and show that they are homeomorphic to S s and s , respectively. We also show that S σ,s = S s = S s when the right-hand side is non-trivial, and similarly for corresponding spaces of Beurling types.
We consider norm conditions of powers of a second order partial differential operator H σ and its conjugate. These operators are symplectic analogies to certain partial harmonic oscillators [7]. We show that H σ andH σ commute and can be used to characterize S σ,s and σ,s as for some h > 0 (for every h > 0). In Sect. 4 we show that if a is positive semi-definite with respect to the twisted convolution, then the relaxed condition of the right-hand of (1.2) is enough to ensure that a should belong to S σ,s or σ,s .

Preliminaries
In the first part we recall definitions of twisted convolution, the Weyl quantization and positivity in operator theory, and discuss basic properties. The verifications are in general omitted since they can be found in e. g. [10]. Thereafter we recall the definitions of Gelfand-Shilov and Pilipović spaces and discuss some properties.

Operators and positivity
Let a and b belong to S (R 2d ), the set of Schwartz functions on R 2d . Then the twisted convolution of a and b is given by Here σ is the symplectic form on R d × R d R 2d , given by The definition of * σ extends in different ways. For example, the map There are strong links between the twisted convolution, and continuity and composition properties in operator theory. This also include analogous questions in the theory of pseudo-differential operators.
In fact, by straight-forward computations it follows that where A is the map on S (R 2d ) defined by the formula Here and in what follows we identify operators with their kernels.) We note that where F is the Fourier transform on S (R d ) which takes the form Alternatively we may reformulate this identity as where F 2 is the partial Fourier transform of (x, y) with respect to the y-variable. Evidently, the mappings F 2 and the pullback which takes (x, y) into are homeomorphisms on S (R 2d ) and on S (R 2d ), and unitary on L 2 (R 2d ). Hence similar facts hold true for A. From these mapping properties it follows that A on S (R 2d ) extends uniquely to a homeomorphism on S (R 2d ), and if a ∈ S (R 2d ), then Aa is a linear and continuous operator from S (R d ) to S (R d ). Furthermore, by the kernel theorem of Schwartz it follows that any linear and continuous operator from S (R d ) to S (R d ) is given by Aa, for a uniquely determined a ∈ S (R 2d ).
At this stage we also note that (2.1) remains true, if more generally, a ∈ S (R 2d ) and b ∈ S (R 2d ), which follows by straight-forward computations.
The operator A can also in convenient ways be formulated in the framework of the Weyl calculus of pseudo-differential operators. More precisely, the Weyl quantization The definition of Op w (a) extends in continuous and similar ways as for Aa to any S (R 2d ), and then Op w (a) is continuous from S (R d ) to S (R d ). This extension can also be performed by the relation which follows by straight-forward computations. Here F σ is the symplectic Fourier transform on S (R 2d ), which takes the form From these facts it follow that the Weyl product #, defined by is given by which again links the twisted convolution to compositions in operator theory.
There are also strong links between positivity for the twisted convolution and positivity in operator theory. We recall that a continuous and linear operator T from , and then we write T ≥ 0.
Positivity for the twisted convolution is defined in an analogous way. That is, an The following proposition explains the links between positivity in operator theory and positivity for the twisted convolution.
. Then the following conditions are equivalent: (1) a is positive semi-definite with respect to the twisted convolution;

Gelfand-Shilov spaces
The Gelfand-Shilov spaces S s (R d ) and s (R d ) are the inductive and projective limits respectively of S s,h (R d ) with respect to h > 0. Consequently The If ε > 0 and s > 0, then The Gelfand-Shilov distribution spaces S s (R d ) and s (R d ) are the projective and inductive limits respectively of S s,h (R d ). Hence We note that S s and s are the duals of S s and s , respectively, in view of [8,9]. The Gelfand-Shilov spaces and their duals are invariant under translations, dilations, (partial) Fourier transformations and under several other important transformations. In fact, by straight-forward computations it follows that the properties and results in Sect. 2.1 hold true with S s and S s in place of S and S , respectively, when s ≥ 1/2, or with s and s in place of S and S , respectively, when s > 1/2. We refer to [3] for more facts about Gelfand-Shilov functions and their distributions.

The Pilipović spaces
We start to consider spaces which are obtained by suitable estimates of Gelfand-Shilov or Gevrey type when using powers of the harmonic oscillator H = |x| 2 − , x ∈ R d . In general we omit the arguments, since more thorough exposition is available in e. g. [13].
Let s > 0 and ε > 0. Then it follows from the definitions that Furthermore, in [13] it is proved that S 0 (R d ) consists of all finite linear combinations of Hermite functions, while S 0 (R d ) consists of all formal series The next propositions show that Pilipović spaces can be characterized by estimates on the Hermite coefficients c α in (2.6). The proofs can be found in [2,13] and therefore omitted. Here H 1 U and H 2 U are the partial harmonic oscillators given by Proposition 2 Let s ≥ 0 (s > 0) and f ∈ S 0 (R d ) be given by (2.6). Then the following conditions are equivalent: (2) |c α ( f )| e −r |α| 1 2s for some r > 0 (for every r > 0).

Remark 3 Let the hypothesis in Proposition 3 be fulfilled, and let
for some h > 0 (for every h > 0). The same arguments as in [2,13] imply that these conditions are equivalent, Furthermore, let S s (R 2d ) ( s (R 2d )) be the set of all U ∈ S 0 (R 2d ) such that for some r > 0 (for every r > 0). Then it follows by similar arguments as in [2,13] that For future references we note that S s ⊆ S s ⊆ S 2s with strict inclusions.

Twisted Pilipović spaces and their properties
In this section we introduce twisted Pilipović spaces as the counter images of the operator A on Pilipović spaces, and deduce some basic properties. We also consider their distribution spaces. We begin with some definitions.

Definition 1 The Hermite-Wong function of order
on R 2d is given by The Hermite-Wong functions were studied in different ways by M. W. Wong in [14,15]. By the definition it follows that We observe that the Hermite-Wong functions are eigenfunctions to F σ . More precisely, we have which follows from the fact that F σ (W f,g ) = Wf ,g (see e. g. [5]). for every c > 0 (for some c > 0).
The spaces in Definition 2 are equipped by topologies in similar way as for the Pilipović spaces in [13].
The set S σ,s (R 2d ) ( σ,s (R 2d )) is called the twisted Pilipović space of Roumieu type (Beurling type) of order s. It follows that the sets S σ,s (R 2d ) and σ,s (R 2d ) are corresponding distribution spaces, since similar facts hold true for Pilipović space [13].
We extend the definition of A on S by letting giving by (3.1). It follows that A is a homeomorphism from S σ,s (R 2d ) to S s (R 2d ), from σ,s (R 2d ) to s (R 2d ), and similarly for their duals. Since it is clear that A is a homeomorphism on any Fourier invariant Gelfand-Shilov spaces, we get and similarly for corresponding distribution spaces.
Remark 4 Let a ∈ S σ,0 (R 2d ) be as in (3.1). Since A is a homeomorphism on S (R 2d ) and on S (R 2d ), it follows from [12] that a belongs to S (R 2d ) if and only if c α α −N for every N ≥ 0. In the same way, a ∈ S (R 2d ) if and only if c α α N for some N ≥ 0.
Next we discuss the partial harmonic oscillators H 1 and H 2 in Proposition 3, and their counter images under the operator A. We let H σ be the operator on S (R 2d ), given by and we let T σ = H σ •H σ . Here we note that the conjugateH σ of H σ is given bȳ The following lemma explains some spectral properties of the considered operators.

Lemma 1 Let s ≥ 0. Then the following is true:
(1) the Hermite-Wong functions α are eigenfuctions to H σ ,H σ and T σ , and
For the proof, we shall make use of the operators (see [15,Section 22]). By similar arguments as in the proof of Theorem 22.1 in [15] we get where e 1 , . . . , e d is the standard basis in R d , i. e., e j = (δ 1, j , . . . , δ d, j ), j = 1, . . . , d, and δ i, j is the Kroniker's delta function. In view of (3.3), the operators Z 1, j and Z 2, j can be considered as symplectic analogies of annihilation operators, and Z 1, j and Z 2, j as symplectic analogies of creation operators.
Proof First we prove (1). By straight-forward computations, we obtain
If a ∈ S σ,s (R 2d ) and b ∈ S σ,s (R 2d ). We now let H σ a be defined by as usual, which extends the definitions of H σ andH σ to S σ,s (R 2d ). The extensions of these operators to σ,s (R 2d ) and S (R 2d ) are performed in similar ways. By (3.2), it follows that these extensions are unique.
The next lemma shows important links between the latter operators and partial harmonic oscillators.
Lemma 2 Let H 1 and H 2 be as in Proposition 3, and let a ∈ S σ,s (R 2d ). Then H σ andH σ are commuting to each other, and , and a is given by where the series converge in S (R 2d ).
Proof The commutation between H σ andH σ follows if we prove (3.4). We recall the operators and the relations  (H σ, j a),

Summing up over all j gives
In the same way we get A(H σ a), and the result follows by induction.
From these mapping properties, Proposition 3 can now be carried over to the case of twisted Pilipović spaces as follows. (1) a ∈ S σ,s (R 2d ) (a ∈ σ,s (R 2d )); , when p 1 = min( p, q) and p 2 = max( p, q), we may assume that p = q.
Since A is a homeomorphism on M p (R 2d ), we get and the equivalence between (3) and Proposition 3 (3) follows. The equivalence between (1) and (3) now follows from Proposition 3 and the fact that A is a homeomorphism from S σ,s (R 2d ) to S s (R 2d ). Finally by the embeddings the equivalence between (2) and (3) now follows.
Remark 3 and Lemma 2 show that (3.7) is necessary but not sufficient in order for a ∈ S σ,s (R 2d ) or a ∈ σ,s (R 2d ).

Twisted Pilipović space property for positive elements with respect to the twisted convolution
We study positive elements with respect to twisted convolution in S , having twisted Pilipović space regularities near the origin. We show that such elements are in S σ,s or in σ,s . The following theorem shows that the condition of the form (3.7) at origin is sufficient that the converse of Corollary 1 holds when dealing with positive semidefinite elements with respect to the twisted convolution.
Theorem 1 Let s ≥ 0 and a ∈ S (R 2d ) be positive semi-definite with respect to the twisted convolution. If a is smooth near origin and holds for some h > 0 (for every h > 0), then a ∈ S σ,s (R 2d ) (a ∈ σ,s (R 2d )).
For the proof we recall that the trace of Aa is given by (π/2) d/2 a(0) when a ∈ S (R 2d ).
Proof We have a ∈ S (R 2d ) in view of Theorem 3.13 in [10]. We may write a = k A −1 ( f k ⊗ f k ) for some sequence { f k } ∞ k=0 with convergence in S (R d ). By Lemma 2, we obtain Let K = k f k ⊗ f k be the kernel of Aa. Then Here the second inequality follows from the triangle inequality, the second equality from the fact that (H N 1 f k ) ⊗ (H N 2 f k ) is the kernel of a rank one operator, the third inequality from Cauchy-Schwartz inequality and the last step from the fact that A(T N σ a) is the kernel of a positive semi-definite operator, giving that the trace and the trace norm are the same. Thus for some h > 0 (for every h > 0), giving that K ∈ S s (R 2d ) (K ∈ s (R 2d )) in view of Proposition 3. Hence a ∈ S σ,s (R 2d ) (a ∈ σ,s (R 2d )).