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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Besov spaces on closed subsets of $\textbf {R}^ n$
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by Alf Jonsson
Trans. Amer. Math. Soc. 341 (1994), 355-370
DOI: https://doi.org/10.1090/S0002-9947-1994-1132434-6

Abstract:

Motivated by the need in boundary value problems for partial differential equations, classical trace theorems characterize the trace to a subset $F$ of ${\mathbb {R}^n}$ of Sobolev spaces and Besov spaces consisting of functions defined on ${\mathbb {R}^n}$, if $F$ is a linear subvariety ${\mathbb {R}^d}$ of ${\mathbb {R}^n}$ or a $d$-dimensional smooth submanifold of ${\mathbb {R}^n}$. This was generalized in [2] to the case when $F$ is a $d$-dimensional fractal set of a certain type. In this paper, traces are described when $F$ is an arbitrary closed set. The result may also be looked upon as a Whitney extension theorem in ${L^p}$.
References
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  • Alf Jonsson and Hans Wallin, Function spaces on subsets of $\textbf {R}^n$, Math. Rep. 2 (1984), no. 1, xiv+221. MR 820626
  • A. Jonsson and H. Wallin, A trace theorem for generalized Besov spaces with three indexes, Fourier analysis and approximation theory (Proc. Colloq., Budapest, 1976) Colloq. Math. Soc. János Bolyai, vol. 19, North-Holland, Amsterdam-New York, 1978, pp. 429–449. MR 540320
  • A. L. Vol′berg and S. V. Konyagin, A homogeneous measure exists on any compactum in $\textbf {R}^n$, Dokl. Akad. Nauk SSSR 278 (1984), no. 4, 783–786 (Russian). MR 765294
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Bibliographic Information
  • © Copyright 1994 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 341 (1994), 355-370
  • MSC: Primary 46E35
  • DOI: https://doi.org/10.1090/S0002-9947-1994-1132434-6
  • MathSciNet review: 1132434