Electron. J. Diff. Equ., Vol. 2010(2010), No. 48, pp. 1-14.

Green function and Fourier transform for o-plus operators

Wanchak Satsanit

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 6, 2010.
Math Subject Classifications: 46F10, 46F12.
Key Words: Fourier transform; diamond operator; tempered distribution.

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Wanchak Satsanit
Faculty of Science, Department of Mathematics
Maejo University
Nong Han, San Sai, Chiang Mai, 50290 Thailand
email: aunphue@live.com

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