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Mixed norm estimates for certain means


Author: Lennart Börjeson
Journal: Trans. Amer. Math. Soc. 309 (1988), 517-541
MSC: Primary 42B15
DOI: https://doi.org/10.1090/S0002-9947-1988-0929662-6
MathSciNet review: 929662
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Abstract: We obtain estimates of the mean

$\displaystyle F_x^\gamma (t) = {C_\gamma }\int_{\vert y\vert < 1} {{{(1 - \vert y{\vert^2})}^\gamma }f(x - ty)\,dy} $

in mixed Lebesgue and Sobolev spaces. They generalize earlier estimates of the spherical mean $ F_x^{ - 1}(t) = C\;\int_{{S^{n - 1}}} {f(x - ty)\,dS(y)} $ and of solutions of the wave equation $ {\Delta _x}u = {\partial ^2}u/\partial {t^2}$.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1988-0929662-6
Keywords: Mixed norm, spherical mean, wave equation
Article copyright: © Copyright 1988 American Mathematical Society

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