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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Convolutions with kernels having singularities on a sphere

Author: Robert S. Strichartz
Journal: Trans. Amer. Math. Soc. 148 (1970), 461-471
MSC: Primary 47.70; Secondary 46.00
MathSciNet review: 0256219
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Abstract: We prove that convolution with $ (1 - \vert x{\vert^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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Keywords: Convolution, $ {L^p}$ estimate, spherical convolution, wave equation, energy estimate
Article copyright: © Copyright 1970 American Mathematical Society