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.