On the Inversion of Bessel Ultrahyperbolic Kernel of Marcel Riesz

Darunee Maneetus,Kamsing Nonlaopon

doi:10.1155/2011/419157

Abstract

We define the Bessel ultrahyperbolic Marcel Riesz operator on the functionfbyUαf=RαB*f, whereRαBis Bessel ultrahyperbolic kernel of Marcel Riesz,α…C, the symbol*designates as the convolution, andf∈S,Sis the Schwartz space of functions. Our purpose in this paper is to obtain the operatorEα=Uα-1such that, ifUαf=φ, thenEαφ=f.

Highlights

The n-dimensional ultrahyperbolic operator k iterated k times is defined by ⎛ ⎞k k ⎝ ∂2 ∂x12 ∂2 ∂x22 ∂2 ∂xp2 − −···− ⎠ q1.1 where p q n is the dimension of Rn and k is a nonnegative integer
Let us consider the diamond kernel of Marcel Riesz Kα,β x introduced by Kananthai in 6, which is given by the convolution
Tellez and Kananthai 16 have proved that Kα,β x exists and is in the space of rapidly decreasing distributions

Summary

Introduction

1.1 where p q n is the dimension of Rn and k is a nonnegative integer. Aguirre 15 has defined the ultrahyperbolic Marcel Riesz operator Mα of the function f by. Let us consider the diamond kernel of Marcel Riesz Kα,β x introduced by Kananthai in 6 , which is given by the convolution. Tellez and Kananthai 16 have proved that Kα,β x exists and is in the space of rapidly decreasing distributions They have shown that the convolution of the distributional families Kα,β x relates to the diamond operator. Maneetus and Nonlaopon 17 have defined the diamond Marcel Riesz operator of order α, β of the function f by. We define the Bessel ultrahyperbolic Marcel Riesz operator of order α of the function f by. Before we proceed to our main theorem, the following definitions and concepts require some clarifications

Preliminaries

The Main Theorem