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Reduced Sobolev Inequalities

Published online by Cambridge University Press:  20 November 2018

R. A. Adams
R. A. Adams*
Affiliation:
Department of Mathematics, The University of British Columbia, Vancouver, B. C, CanadaV6T 1Y4
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Abstract

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The Sobolev inequality of order m asserts that if p ≧ 1, mp < n and 1/q = 1/p — m/n, then the Lq-norm of a smooth function with compact support in Rn is bounded by a constant times the sum of the Lp-norms of the partial derivatives of order m of that function. In this paper we show that that sum may be reduced to include only the completely mixed partial derivatives or order m, and in some circumstances even fewer partial derivatives.

Keywords

46E35
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 31 , Issue 2 , 01 June 1988 , pp. 159 - 167
Copyright
Copyright © Canadian Mathematical Society 1988

References

1. Adams, R. A., Anisotropic Sobolev inequalities, Research Report CMA-R05-86, Centre for Mathematical Analysis, The Australian National University.Google Scholar
2. Fournier, John J. F., Mixed norms and rearrangements: Sobolev's inequality and Littlewood's inequality, to appear Ann. Mat. Pura Appl.Google Scholar
3. Stein, Elias M., Singular integrals and differentiability properties of functions, Princeton Univ. Press, 1970.Google Scholar