Abstract and Applied Analysis
Volume 2011 (2011), Article ID 419157, 13 pages
http://dx.doi.org/10.1155/2011/419157
Research Article

On the Inversion of Bessel Ultrahyperbolic Kernel of Marcel Riesz

Darunee Maneetus1 and Kamsing Nonlaopon1,2

1Department of Mathematics, Khon Kaen University, Khon Kaen 40002, Thailand
2Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand

Received 26 August 2011; Accepted 8 October 2011

Academic Editor: Chaitan Gupta

Copyright © 2011 Darunee Maneetus and Kamsing Nonlaopon. 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 define the Bessel ultrahyperbolic Marcel Riesz operator on the function 𝑓 by 𝑈 𝛼 ( 𝑓 ) = 𝑅 𝐵 𝛼 ∗ 𝑓 , where 𝑅 𝐵 𝛼 is Bessel ultrahyperbolic kernel of Marcel Riesz, 𝛼 … ℂ , the symbol ∗ designates as the convolution, and 𝑓 ∈ 𝒮 , 𝒮 is the Schwartz space of functions. Our purpose in this paper is to obtain the operator 𝐸 𝛼 = ( 𝑈 𝛼 ) − 1 such that, if 𝑈 𝛼 ( 𝑓 ) = 𝜑 , then 𝐸 𝛼 𝜑 = 𝑓 .