Convolutions with kernels having singularities on a sphere

Strichartz, Robert S.

doi:10.1090/S0002-9947-1970-0256219-1

Convolutions with kernels having singularities on a sphere
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by Robert S. Strichartz PDF

Trans. Amer. Math. Soc. 148 (1970), 461-471 Request permission

Abstract:

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

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