On Cross-Correlation of a Hyperfunction and a Real Analytic Function 1 Introduction

International Journal of Mathematical Analysis Vol. 9, 2015, no. 2, 95 - 100 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2015.411351

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

Mariia Patra and Sergii Sharyn

Vasyl Stefanyk Precarpathian National University, 57 Shevchenka str., 76-018, Ivano-Frankivsk, Ukraine

Copyright c 2014 Mariia Patra and Sergii Sharyn. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract We describe the cross-correlation operator over the space of real- analytic functions and generalize classic Schwartz’s theorem on shift- invariant operators.

Mathematics Subject Classiﬁcation: 46F05, 46H30, 47A60, 46F15

Keywords: hyperfunction, real-analytic function, shift-invariant operator

1 Introduction

By classic Schwartz’s theorem for shift-invariant operators (see [14]) every continuous linear operator L : D(Rn) −→ C∞(Rn) commuting with the shift group n τh : ϕ(·) 7−→ ϕ(· − h), h ∈ R , is the convolution operator with some distribution f ∈ D0(Rn), i.e., Lϕ = f ∗ϕ. H¨ormanderin [4] performed a comprehensive analysis of the boundedness of translation-invariant operators on Lp(Rn). 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 Ba- nach space. For other results and references on the topic we refer the reader to [3, 5, 11]. 96 Mariia Patra and Sergii Sharyn

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 Bc(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

Cf : A(R+) 3 ϕ 7−→ f ? ϕ, f ∈ Bc(R+), where f ? ϕ is deﬁned by the formula (3). In Theorem 3.2 we show that the algebra Bc(R+) can be represented by the isomorphism Bc(R+) 3 f 7−→ Cf ∈ L A(R+) onto the commutant [T ]c of the shift semigroup T (see formula (2)) in space of linear continuous operators on A(R+).

2 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 {ϕn} 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 {ϕn} converges uniformly to ϕ in U (here and subsequently bold symbol like ϕ denotes an analytic continuation 0 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])

B(Ω) = H(V \ Ω)/H(V ), where H(V ) denotes the restriction of H(V ) to V \ Ω. A hyperfunction represented by a holomorphic function F ∈ H(V \ Ω) is denoted as

f = [F ] = F (t + i0) − F (t − i0) or f(t) = [F (z)]z=t.

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 Bc(R+) denote the space of hyperfunctions with a compact support in R+. From K¨oteduality theorem [8] it follows that the isomorphism of ∼ 0 vector spaces Bc(R+) = A(R+) is valid. Moreover, for a ϕ ∈ A(R+) and an On cross-correlation of a hyperfunction and a real analytic function 97

f = [F ] ∈ Bc(R+) with F ∈ H(V \ supp f), the canonical bilinear functional is given by I hf, ϕi = − F (z)ϕ(z)dz, (1) Γ 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 Bc(R+) we deﬁne their convolution as a hyperfunction f ∗ g = [H], where I H(z) = − F (w)G(z − w) dw, Γ and Γ is a closed path in the intersection of the domains of analytic functions w 7−→ F (w) and w 7−→ G(z − w). It is known [8], that the space Bc(R+) is an algebra with respect to the 1 1 convolution with Dirac delta-function δ(x) = − as a unit element. 2πi z z=x 3 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 Th that is deﬁned on the space A(R+) by the formula

Th : ϕ(·) 7−→ ϕ(· + h), ϕ ∈ A(R+). (2)

It is immediate that T := {Th : h ∈ R+} is an one-parameter (C0)-semigroup. Commutant of the semigroup T is deﬁned to be the set

c [T ] := {A ∈ L A(R+) : Th ◦ A = A ◦ Th, ∀h ∈ R+}.

The cross-correlation of a hyperfunction f = [F ] ∈ Bc(R+) and a real- analytic function ϕ ∈ A(R+) is defined to be I (f ? ϕ)(t) := − F (z)ϕ(z + t)dz = hf, Ttϕi, t ∈ R+, (3) Γ 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 ∈ Bc(R+) and ϕ ∈ A(R+) the cross- correlation f ? ϕ is an infinite differentiable function, satisfying the equality

n n D (f ? ϕ) = f ? D (ϕ), n ∈ N, 98 Mariia Patra and Sergii Sharyn where Dn 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 ∈ Bc(R+) and ϕ ∈ A(R+) the cross-correlation f ? ϕ is a real-analytic function, belonging to A(R+).

For any hyperfunction f ∈ Bc(R+), the cross-correlation operator over the space A(R+) is deﬁned to be

Cf : ϕ 7−→ f ? ϕ. Let us show the correctness of the above deﬁnition, i.e. Cf ∈ L A(R+) . The linearity of Cf is clear. Check its continuity. Let {ϕn} be a sequence, converging to zero in the space A(R+). Denote ψn := Cf ϕn. Show that {ψn} 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 I I

sup |ψn(z)| = sup − F (s)ϕn(s + z) ds ≤ sup |F (s)| |ϕn(s + z)| ds z∈U z∈U Γ z∈U Γ (4) ≤ sup sup |ϕn(s + z)| · sup |F (s)| · µ(Γ), z∈U s∈Γ s∈Γ where µ(Γ) denotes the length of the path Γ. Since K is compact, the path Γ is ﬁnite, therefore sups∈Γ |F (s)| · µ(Γ) < ∞. Maximum modulus principle implies that there exists a point s0 ∈ Γ, such that

sup sup |ϕn(s + z)| = sup |ϕn(s0 + z)|. z∈U s∈Γ z∈U

Since ϕn → 0 as n → ∞ in the space A(R+), we get supz∈U |ϕn(s0 + z)| → 0.

Hence, the inequality (4) implies supz∈U Cf [ϕn](z) → 0 as n → ∞, so Cf is continuous operator. The next theorem is a generalization of classic Schwartz’s theorem on shift- invariant operators. Theorem 3.2. The mapping K : Bc(R+) 3 f 7−→ Cf ∈ L A(R+) pro- duces an algebraic isomorphism from the convolution algebra Bc(R+) onto the commutant [T ]c of the shift semigroup T , i.e.

Cf∗g = Cf ◦ Cg, f, g ∈ Bc(R+). (5) In particular, Cδ is the identity in L A(R+) . Proof. The following equalities

(Cf Thϕ)(t) = (Cf ϕ)(t + h) = Th(Cf ϕ)(t) = (ThCf ϕ)(t) On cross-correlation of a hyperfunction and a real analytic function 99

hold for all h ∈ R+ and ϕ ∈ A(R+). Let now A ∈ L A(R+) be an arbitrary operator with the property

(ATh)ϕ(t) = (ThA)ϕ(t), h ∈ R+, ϕ ∈ A(R+). (6)

It is clear that the functional hf0, ϕi := (Aϕ)(0) belongs to Bc(R+). By deﬁni- tion of cross-correlation we get (Cf0 ϕ)(0) = hf0, ϕi, i.e. (Aϕ)(0) = (Cf0 ϕ)(0) for all ϕ ∈ A(R+). Substituting Thϕ instead of ϕ and using the property (6), we get that A = Cf0 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 I I (Cf∗gϕ)(t) = ((f ∗g)?ϕ)(t) = − − F (z)G(ξ−z)dz ϕ(ξ+t)dξ, t ∈ R+,

Γ2 Γ1 where Γ2 is a closed path, surrounding supp f ∗ g and belonging to intersection of domains of functions ξ 7−→ H(ξ) and ξ 7−→ ϕ(ξ + t). Applying Fubini’s theorem and changing variable w = ξ − z in the inner integral we get I I (Cf∗gϕ)(t) = − F (z) − G(ξ − z)ϕ(ξ + t)dξ dz

Γ1 Γ2 I I = − F (z) − G(w)ϕ(w + z + t)dw dz,

Γ1 Γ3 where Γ3 is a shift of a contour Γ2. Let ψ be an analytic continuation of ψ, H where ψ(s) := (g ? ϕ)(s) = − G(w) ϕ(w + s) dω, s ∈ R+. Then Γ3 I (Cf∗gϕ)(t) = − F (z) ψ(z + t) dz = (f ? (g ? ϕ))(t) = (Cf ◦ Cgϕ)(t).

Γ1

In particular, Cf ◦ Cδ = Cf∗δ = Cf = Cδ∗f = Cδ ◦ Cf for all f ∈ Bc(R+). So Cδ is the identity.

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