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Transactions of the American Mathematical Society

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

 

 

Improved Sobolev inequalities


Author: Robert S. Strichartz
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
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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 $ \vert\{ \xi \in A:\vert\vert\xi \vert\vert \leqslant s\} \vert \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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DOI: https://doi.org/10.1090/S0002-9947-1983-0704623-6
Keywords: Sobolev inequality, symmetric space, Radon transform, wave equation
Article copyright: © Copyright 1983 American Mathematical Society