Electronic Journal of Differential Equations,
Vol. 2010(2010), No. 48, pp. 1-14.

Title: Green function and Fourier transform  for o-plus operators
       
Author: Wanchak Satsanit (Maejo Univ., Chiang Mai, Thailand)

Abstract:
 In this article, we study the o-plus operator defined by
 $$
  \oplus^k =\Big(\Big(\sum^{p}_{i=1}\frac{\partial^2}{\partial
 x^2_i}\Big)^{4}-\Big(\sum^{p+q}_{j=p+1}\frac{\partial^2}{\partial
 x^2_j}\Big)^{4}\Big)^k ,
 $$
 where $x=(x_1,x_2,\dots,x_n)\in \mathbb{R}^n$, $p+q=n$,
 and $k$ is a nonnegative integer. Firstly, we studied the
 elementary solution for the $\oplus^k $ operator
 and then this solution is related to the solution of the wave
 and the Laplacian equations. Finally, we studied the Fourier
 transform of the elementary solution and also the Fourier transform
 of its convolution.

Submitted January 8, 2010. Published April 06, 2010.
Math Subject Classifications: 46F10, 46F12.
Key Words: Fourier transform; diamond operator; tempered distribution.