On Singular Integral Operators with Rough Kernel Along Surface

Abstract

In this note we give a simple method to transfer the effect of the surface to the radial function in the kernel of singular integral along surface. Using this idea, we give some continuity of the singular integrals along surface with Hardy space function kernels on some function spaces, such as ${L^p({\mathbb R}^n),L^p({\mathbb R}^n,\omega)}$, Triebel–Lizorkin spaces ${{\dot F}_{p}^{s,q}({\mathbb R}^n)}$, Besov spaces ${{\dot B}_{p}^{s,q}({\mathbb R}^n)}$, generalized Morrey spaces ${L^{p,\phi}({\mathbb R}^n)}$ and Herz spaces ${\dot K_p^{\alpha, q}({\mathbb R}^n)}$. Our results improve and extend substantially some known results on the singular integral operators along surface.