Weighted -Integral Representations of -Functions in

Arman H Karapetyan

doi:10.1155/2012/265092

Abstract

For -functions , given in the complex space , integral representations of the form are obtained. Here, is the orthogonal projector of the space onto its subspace of entire functions and the integral operator appears by means of explicitly constructed kernel Φ which is investigated in detail.

Highlights

Let n ≥ 1, 1 < p < ∞, and 0 < ρ, σ < ∞, γ > −2n
Denote by Lpρ,σ,γ Cn the space of all measurable complex-valued functions f z, z ∈ Cn, satisfying the condition f z pe−σ|z|ρ |z|γ dm z < ∞
Let Hρp,σ,γ Cn be the corresponding subspace of entire functions

Summary

Introduction

Let n ≥ 1, 1 < p < ∞, and 0 < ρ, σ < ∞, γ > −2n. Denote by Lpρ,σ,γ Cn the space of all measurable complex-valued functions f z , z ∈ Cn, satisfying the condition f z pe−σ|z|ρ |z|γ dm z < ∞. Let Hρp,σ,γ Cn be the corresponding subspace of entire functions It was established in 1, 2 when n 1 and 3 when n > 1 that arbitrary function f ∈ Hρp,σ,γ Cn has an integral representation of the form fz ρσμ 2π n f w e−σ|w|ρ |w|γ · Eρn/2 σ2/ρ z, w ; μ dm w , Cn z ∈ Cn, 1.2 where μ γ 2n /ρ and. Note that for p 2, ρ 2, γ 0, Hρp,σ,γ Cn coincides with the well-known Fock space of entire functions and 1.2 takes the form fz σn πn f w · e−σ|w|2 · eσ z,w dm w , Cn z ∈ Cn. In 4 general weighted integral representations were obtained for differential forms. We prove certain important differential and integral properties of this kernel As it will be seen below, in the case of radial weight function, a new approach is requested and significant analytical difficulties are arised. In that formula the operator of orthogonal projection of the space L2 Cn; exp{−φ w }dm w onto its subspace of entire functions does not appear, except of the special case φ w σ|w|2 when we again obtain 1.5

Heuristic Argument

Auxiliary Results

The Main Properties of the Kernel Φ

The Main Integral Representation

The Computable Form of the Kernel Φ