On the Fourier transform of the Diamond Kernel of Marcel Riesz

摘要

In this paper, the operator ♢k is introduced and named as the Diamond operator iterated k-times and is defined by ♢k=[(∂2/∂x12+∂2/∂x22+⋯+∂2/∂xp2)2−(∂2/∂xp+12+∂2/∂xp+22+⋯+∂2/∂xp+q2)2]k, where n is the dimension of the Euclidean space Rn,k is a nonnegative integer and p+q=n. The elementary solution of the operator ♢k is called the Diamond Kernel of Marcel Riesz. In this work we study the Fourier transform of the elementary solution and also the Fourier transform of their convolutions.