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Exact imbedding theorems for a certain class of weighted Sobolev spaces. (Russian. English summary) Zbl 0636.46034

Let \(\Omega\subset {\mathbb{R}}^ N \)be a bounded domain, let L q(\(\Omega\) ;w) and \(W^{1,p}(\Omega;v_ 0,v_ 1)\) be normed by \(\| u\|_{q,w}=(\int_{\Omega}| u(x)|\) q w(x)dx)\({}^{1/q}\) and \((\| u\|\) \(p_{p,v_ 0}+\sum^{N}_{k=1}\| \partial u/\partial x_ k\|\) \(p_{p,v_ 1})^{1/p}\), respectively, \(1\leq q<\infty\), \(1\leq p<\infty\), and w, \(v_ 0\), \(v_ 1\) are positive weight functions. Of special interest is the model case \(\Omega =\{x|\) \(0<| x-x_ 0| <R\}\) where the weight functions depend only on \(| x-x_ 0|\). Let \(W_ 0^{1,p}(\Omega;v_ 0,v_ 1)\) be the completion of \(C^{\infty}_ 0(\Omega)\) in \(W^{1,p}(\Omega;v_ 0,v_ 1)\). The paper deals with embeddings and compact embeddings of the type \(W_ 0^{1,p}(\Omega;v_ 0,v_ 1)L\) q(\(\Omega\) ;w) and \(W_ 0^{1,p}(\Omega;v_ 0,v_ 1)L\) q(\(\Omega\) ;w), respectively. The conditions have necessary and sufficient character. Examples are studied: power weights, weights with exponential growth.
Reviewer: H.Triebel

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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