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Journal of the American Mathematical Society

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

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Symmetric decreasing rearrangement is sometimes continuous
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by Frederick J. Almgren and Elliott H. Lieb PDF
J. Amer. Math. Soc. 2 (1989), 683-773 Request permission

Abstract:

This paper deals with the operation $\mathcal {R}$ of symmetric decreasing rearrangement which maps ${{\mathbf {W}}^{1,p}}({{\mathbf {R}}^n})$ to ${{\mathbf {W}}^{1,p}}({{\mathbf {R}}^n})$. We show that even though it is norm decreasing, $\mathcal {R}$ is not continuous for $n \geq 2$. The functions at which $\mathcal {R}$ is continuous are precisely characterized by a new property called co-area regularity. Every sufficiently differentiable function is co-area regular, and both the regular and the irregular functions are dense in ${{\mathbf {W}}^{1,p}}({{\mathbf {R}}^n})$. Curiously, $\mathcal {R}$ is always continuous in fractional Sobolev spaces ${{\mathbf {W}}^{\alpha ,p}}({{\mathbf {R}}^n})$ with $0 < \alpha < 1$.
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Additional Information
  • © Copyright 1989 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 2 (1989), 683-773
  • MSC: Primary 49F20; Secondary 46E30, 49A50
  • DOI: https://doi.org/10.1090/S0894-0347-1989-1002633-4
  • MathSciNet review: 1002633