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Zbl 1219.46036
Schneider, Cornelia
Traces of Besov and Triebel-Lizorkin spaces on domains.
(English)
[J] Math. Nachr. 284, No. 5-6, 572-586 (2011). ISSN 0025-584X; ISSN 1522-2616/e

Let $0<p,q \le \infty$ and $0<s<r \in \Bbb N$. Then $B^s_{p,q} (\Bbb R^n)$ is the collection of all measurable functions $f \in L_p (\Bbb R^n)$ such that $$\| f \, | B^s_{p,q} (\Bbb R^n) \|_r = \|f \, | L_p (\Bbb R^n) \| + \Big( \int^1_0 t^{-sq} \sup_{|h| \le t} \| \Delta^r_h f \, | L_p (\Bbb R^n) \|^q \frac{dt}{t} \Big)^{1/q}$$ is finite (independent of $r$). These spaces can be characterized by atoms. If $\Omega$ is a bounded domain in $\Bbb R^n$ then $B^s_{p,q} (\Omega)$ is defined by restriction. Via local charts corresponding spaces $B^s_{p,q} (\partial \Omega)$ on the boundary $\partial\Omega$ are introduced. Let $Tr_{\partial \Omega}$ be the trace operator. The main assertion of the paper says that $$Tr_{\partial \Omega} B^s_{p,q} (\Omega) = B^{s- 1/p}_{p,q} (\partial \Omega) \quad \text{if} \quad 0<p,q \le \infty, \ s >1/p.$$ This requires some efforts (diffeomorphisms, pointwise multipliers, atoms). There are similar assertions for $F$-spaces. Furthermore the dichotomy (either trace, or density of smooth functions with compact supports off $\partial \Omega$) is studied.
[Hans Triebel (Jena)]
MSC 2000:
*46E35 Sobolev spaces and generalizations

Keywords: Besov spaces; Triebel-Lizorkin spaces; traces; smooth domains; dichotomy

Cited in: Zbl 1225.42015

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