Green function and Fourier transform for o-plus operators - Wanchak Satsanit (Electronic Journal of Differential Equations, 2010)
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 $x=(x_1,x_2,dots,x_n)in mathbbR^n$, $p+q=n$, 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.