Convolutions with kernels having singularities on a sphere

Abstract

We prove that convolution with ( 1 − | x | 2 ) + − α (1 - |x{|^2})_ + ^{ - \alpha } and related convolutions are bounded from L p {L^p} to L q {L^q} for certain values of p and q. There is a unique choice of p which maximizes the measure of smoothing 1 / p − 1 / q 1/p - 1/q , in contrast with fractional integration where 1 / p − 1 / q 1/p - 1/q is constant. We apply the results to obtain a priori estimates for solutions of the wave equation in which we sacrifice one derivative but gain more smoothing than in Sobolev’s inequality.