Convolutions with Kernels Having Singularities on a Sphere

We prove that convolution with <!-- MATH $(1 - |x{|^2})_ + ^{ - \alpha }$ --> and related convolutions are bounded from to for certain values of p and q. There is a unique choice of p which maximizes the measure of smoothing , in contrast with fractional integration where 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.