Application of the Dual Space of Gelfand-Shilov Spaces of Beurling Type

Using a previously obtained structure theorem of Gelfand-Shilov spaces Σα of Beurling type of ultradistributions, we prove that these ultradistributions can be represented as an initial values of solutions of the heat equation by describing the action of the Gauss-Weierstrass semigroup on the dual space (Σα) ′


Introduction
The theory of generalized functions devised by L. Schwartz was to provide a satisfactory framework for the Fourier transform (see [11]). They are objects which generalize functions, and they extend the concept of derivative to all integrable functions and beyond, and used to formulate generalized solutions of partial differential equations (see [5]).
Gelfand and Shilov have introduced other types of distributions called ultradistributions in the study of the uniqueness of the Cauchy problems of partial differential equations (see [3]). These spaces are invariant under Fourier transform, closed under differentiation and multiplication by polynomials, moreover, it contains Schwartz space of tempered distributions as a subspace. This makes the Gelfand Shilov spaces appropriate domains for quantum field theory. S. Pilipovic obtained structural theorems and defined the convolution for Gelfand-Shilov spaces of Roumieu and Beurling type (see [9], [10], [4]).
In this paper, we use the characterization of Gelfand-Shilov spaces of Beurling type of test functions of tempered ultradistribution in terms of their Fourier transform obtained in [2] and the structure theorem for functionals in dual space (Σ β α ) ′ equipped with the weak topology, to study the action of Gauss-Weierstrass semigroup on the dual space (Σ β α ) ′ . Consequently, we prove that these ultradistributions can be represented as an initial values of solutions of the heat equation u t − Au = 0.
Throughout the paper the symbols C ∞ , C ∞ 0 , L p , etc., denote the usual spaces of functions defined on R n , with complex values. We denote |·| the Euclidean norm on R n , while · p indicates the p-norm in the space L p , where 1 ≤ p ≤ ∞. In general, we work on the Euclidean space R n unless we indicate other than that as appropriate. The Fourier transform of a function f will be denoted by F (f ) or f and it will be defined as R n e −2πixξ f (x) dx. A Fréchet spaces are a locally convex topological vector spaces that are completely metrizable.

Preliminary definitions and results
In this section, we introduce basic notations and recalling some facts concerning Gelfand-Shilov spaces.
In the following theorem we state a symmetric characterization of the Gelfand-Shilov spaces Σ β α in terms of the Fourier transformations. Theorem 2.3. The space Σ β α can be described as a set as well as topologically by The proof of Theorem 2.3 mimics the proof of Theorem 3.1 in [7] so we omit it. In the other hand, we can employ the above theorem to prove the following structure theorem for functionals T ∈ (Σ β α ) ′ . Theorem 2.4. If T ∈ (Σ β α ) ′ , then there exist two regular complex Borel measures µ 1 and µ 2 of finite total variation and k ∈ N 0 such that in the sense of (Σ β α ) ′ . The following Lemma will be useful in the proofs later.
Proof: Fix y ∈ R n and let ϕ ∈ Σ β α . First, let us prove that To do so, we use concavity property of |•| 1/α as follows: This proves that e k|x| 1/α ϕ(x + y) This completes the proof of Lemma 2.5.
Given two functionals T and S that are integrable functions, the classical definition of convolution of T and S is given by T * S, φ = T x , S y , φ(y + x) .
We end this section with the definition of operator semigroup on a Banach space that we will use in application in the next section. Definition 2.1. [8] Let B be a Banach space. An operator semigroup on B is a family (T t : t ∈ R + ) of bounded linear operators on B such that

Applications
In this section, we study some applications on the structure theorem of Σ β α tempered ultradistributions stated in Theorem 2.4 by proving some results on a semi-group acting on the Fréchet space Σ β α and extend it to its dual (Σ β α ) ′ . We start this section by recalling a previously proved result which says that the convolution in Theorem 2.6 coincides with classical definition of convolution of two integrable functionals.
coincides with the functional given by integration against the function ψ(x) = T y , ϕ(x − y) .
Proof: Using (2.1) in Theorem 2.4, we can write for each x So, for all φ ∈ Σ β α . This completes the proof of Theorem 3.1.
Now we employ the above theorem to describe the action of the semi-group defined by the convolution Theorem 3.2. Let T ∈ Σ β α and {P t } t≥0 be a semi-group defined by the convolution kernel t −n T ( x−y t ), where t > 0. Then, the action of P t on (Σ β α ) ′ is given by the integration against the function

1)
where S y ∈ (Σ β α ) ′ and y indicates on which variable the functional S acts. Proof: Using Lemma 2.5 and Theorem 3.1, it is enough to show that T ( · t ) ∈ Σ β α for each t > 0. Note that Now if t ≥ 1, then |ξ| 1/β ≤ |tξ| 1/β and therefore For 0 < t < 1, we have where N is an integer such that N ≥ 1 t . This completes the proof of Theorem 3.2.