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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Improved Sobolev inequalities
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by Robert S. Strichartz PDF
Trans. Amer. Math. Soc. 279 (1983), 397-409 Request permission

Abstract:

For a function $f$ defined on ${{\mathbf {R}}^n}$, Sobolev’s inequality $\parallel f{\parallel _q} \leqslant c(\parallel f\;{\parallel _{p}} + \parallel \nabla f{\parallel _{p}})$, where $1 < p < q < \infty$ and $1/p - 1/q = 1/n$, can be improved if the Fourier transform $\hat f$ is assumed to have support in a set $A$ which satisfies an estimate $|\{ \xi \in A:||\xi || \leqslant s\} | \leqslant c{s^d}$ for some $d < n$ the improvement being that we can take $1/p - 1/q = 1/d$, provided we also assume $p \leqslant 2 \leqslant q$. Analogous results are proved for other Sobolev inequalities, for embeddings into Lipschitz-Zygmund spaces, and for functions on symmetric spaces whose Fourier expansions are suitably limited. Improved Sobolev inequalities are established locally for solutions of the wave equation. An application to the Radon transform on spheres is given.
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Additional Information
  • © Copyright 1983 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 279 (1983), 397-409
  • MSC: Primary 46E35; Secondary 42B10, 43A77, 43A85
  • DOI: https://doi.org/10.1090/S0002-9947-1983-0704623-6
  • MathSciNet review: 704623