Singular and fractional integrals along variable surfaces

Abstract

We study singular integrals associated with the variable surfaces of revolution. We treat the rough kernel case where the singular integral is defined by an $H^1$ kernel function on the sphere $S^{n-1}$. We prove the $L^p$ boundedness of the singular integral for $1<p\leq 2$ assuming that a certain lower dimensional maximal operator is bounded on $L^s$ for all $s>1$. We also study the $(L^p,L^r)$ boundedness for fractional integrals associated with surfaces of revolution.