## Besov spaces on closed subsets of $\textbf {R}^ n$

HTML articles powered by AMS MathViewer

- by Alf Jonsson
- Trans. Amer. Math. Soc.
**341**(1994), 355-370 - DOI: https://doi.org/10.1090/S0002-9947-1994-1132434-6
- PDF | Request permission

## 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

- E. M. Dyn′kin,
*Free interpolation by functions with a derivative from $H^{1}$*, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI)**126**(1983), 77–87 (Russian, with English summary). Investigations on linear operators and the theory of functions, XII. MR**697427** - 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**

## 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