On Cross-correlation of a Hyperfunction and a Real Analytic Function

We describe the cross-correlation operator over the space of realanalytic functions and generalize classic Schwartz’s theorem on shiftinvariant operators. Mathematics Subject Classification: 46F05, 46H30, 47A60, 46F15


Introduction
By classic Schwartz's theorem for shift-invariant operators (see [14]) every continuous linear operator L : D(R n ) −→ C ∞ (R n ) commuting with the shift group τ h : ϕ(•) −→ ϕ(• − h), h ∈ R n , is the convolution operator with some distribution f ∈ D (R n ), i.e., Lϕ = f * ϕ.Hörmander in [4] performed a comprehensive analysis of the boundedness of translation-invariant operators on L p (R n ).Such operators are of interest and have been considered by several authors in [6,10].The extension of the theory to Besov, Lorents and Hardy spaces was considered in [1], [2] and [15], respectively.The paper [9] is devoted to shift-invariant operators, commuting with contraction multi-parameter semigroups over a Banach space.For other results and references on the topic we refer the reader to [3,5,11].
The purpose of this paper is a generalization of classic Schwartz's theorem on shift-invariant operators.In Theorems 3.2 we describe shift-invariant operators for operator semigroups.
Let B c (R + ) be the space of hyperfunctions with a compact support in the positive semiaxis R + and A(R + ) be the space of germs of real-analytic functions.We consider the cross-correlation operator where f ϕ is defined by the formula (3).In Theorem 3.2 we show that the algebra B c (R + ) can be represented by the isomorphism onto the commutant [T ] c of the shift semigroup T (see formula (2)) in space of linear continuous operators on A(R + ).

Preliminaries and notations
Let A(R + ) be the space of germs of real-analytic functions on neighborhoods of the semiaxis R + := [0, ∞).A restriction of any element of A(R + ) to R + is uniquely defined function.In the sequel we will treat A(R + ) as the space of such restrictions.It is known [7, Prop.1.3.9],that a sequence {ϕ n } converges to ϕ in A(R + ) if and only if for any compact set K ⊂ R + there exists a complex neighborhood U of K, such that {ϕ n } converges uniformly to ϕ in U (here and subsequently bold symbol like ϕ denotes an analytic continuation of a corresponding function ϕ).Let A(R + ) be dual space of A(R + ).Denote H(W ) the vector space of all holomorphic functions on an open set W ⊂ C. Let Ω be an open set in R and V be an open set in C containing Ω as a relatively closed set.The vector space B(Ω) of all hyperfunctions on Ω is defined to be the quotient space (see [8,13]) The representative F is called a defining function of the hyperfunction f .For more details on the theory of hyperfunctions we refer the reader to [8,13].
Let B c (R + ) denote the space of hyperfunctions with a compact support in R + .From Köte duality theorem [8] it follows that the isomorphism of vector spaces B c (R + ) ∼ = A(R + ) is valid.Moreover, for a ϕ ∈ A(R + ) and an where Γ is a closed path in the intersection of the domains of ϕ and F , and surrounding supp f once in the positive orientation.
For any hyperfunctions f = [F ] and g = [G] from B c (R + ) we define their convolution as a hyperfunction f * g = [H], where and Γ is a closed path in the intersection of the domains of analytic functions w −→ F (w) and w −→ G(z − w).
It is known [8], that the space B c (R + ) is an algebra with respect to the convolution with Dirac delta-function δ(x) = − 1 2πi 1 z z=x as a unit element.

Main Result
Denote by L A(R + ) the space of all linear continuous operators on A(R + ).We endow L A(R + ) with the locally convex topology of uniform convergence on bounded subsets of A(R + ).Given an h ∈ R + , consider the shift operator T h that is defined on the space A(R + ) by the formula It is immediate that T := {T h : h ∈ R + } is an one-parameter (C 0 )-semigroup.Commutant of the semigroup T is defined to be the set where Γ is a same path as in (1).The definition of cross-correlation and properties of an integral, depending on a parameter, imply that for any f ∈ B c (R + ) and ϕ ∈ A(R + ) the crosscorrelation f ϕ is an infinite differentiable function, satisfying the equality where D n denotes the n-th derivative operator.Moreover, in [12] it is proved the following assertion, which improves the above result.Proposition 3.1.For any f ∈ B c (R + ) and ϕ ∈ A(R + ) the cross-correlation f ϕ is a real-analytic function, belonging to A(R + ).
For any hyperfunction f ∈ B c (R + ), the cross-correlation operator over the space A(R + ) is defined to be Let us show the correctness of the above definition, i.e.C f ∈ L A(R + ) .The linearity of C f is clear.Check its continuity.Let {ϕ n } be a sequence, converging to zero in the space A(R + ).Denote ψ n := C f ϕ n .Show that {ψ n } converges to zero in the topology of the space A(R + ).For any compact set K ⊂ R + and a complex neighborhood U of K we have where µ(Γ) denotes the length of the path Γ.Since K is compact, the path Γ is finite, therefore sup s∈Γ |F (s)| • µ(Γ) < ∞.
Maximum modulus principle implies that there exists a point s 0 ∈ Γ, such that sup In particular, C δ is the identity in L A(R + ) .
Proof.The following equalities hold for all h ∈ R + and ϕ ∈ A(R + ).Let now A ∈ L A(R + ) be an arbitrary operator with the property It is clear that the functional f 0 , ϕ := (Aϕ)(0) belongs to B c (R + ).By definition of cross-correlation we get (C f 0 ϕ)(0) = f 0 , ϕ , i.e. (Aϕ)(0) = (C f 0 ϕ)(0) for all ϕ ∈ A(R + ).Substituting T h ϕ instead of ϕ and using the property (6), we get that A = C f 0 and hence that image of K coincides with the commutant [T ] c .Check the equality (5).Let f * g = [H].By definition of convolution of two hyperfunctions we have where Γ 2 is a closed path, surrounding supp f * g and belonging to intersection of domains of functions ξ −→ H(ξ) and ξ −→ ϕ(ξ + t).
Applying Fubini's theorem and changing variable w = ξ − z in the inner integral we get (C f * g ϕ)(t) = − In particular, . So C δ is the identity.