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Decomposition of the weighted Sobolev space $$W^{1,p}(\Omega,d_ M^{\varepsilon})$$ and its traces. (English) Zbl 0832.46027
This paper is a continuation of the previous one [this J. 43, No. 4, 695- 711 (1993; review above)]. Using the same notation, consider also the space $$H^{1, p}(\Omega, d^\varepsilon_M)$$ consisting of all functions $$u$$ with finite norm $\Biggl( \int_\Omega |u(x)|^p d^{\varepsilon- p}_M(x) dx+ \int_\Omega \sum^N_{i= 1} |D_i u(x)|^p d^\varepsilon_M(x) dx\Biggr)^{1/p}.$ If $$\varepsilon\leq k- N$$ or $$\varepsilon> p+ k- N$$ then $$H^{1, p}(\Omega, d^\varepsilon_M)= W^{1,p}(\Omega, d^\varepsilon_M)$$ and the spaces have equivalent norms. In the present paper, if $$k- N< \varepsilon< p+ k- N$$ and $$s_p= 1- {N- k+ \varepsilon\over p}$$, a subspace $$D^p_{\varepsilon, M}(\Omega)$$ of $$W^{1, p}(\Omega, d^\varepsilon_M)$$ is constructed, which is isometrically isomorphic to $$W^{s, p}(M)$$ and for which there is a decomposition $$W^{1, p}(\Omega, d^\varepsilon_M)= H^{1, p}(\Omega, d^\varepsilon_M)\oplus D^p_{\varepsilon, M}(\Omega)$$ with equivalent norms. Functions in $$H^{1, p}(\Omega, d^\varepsilon_M)$$ are those with zero trace on $$M$$. The construction and proof use the trace mappings and their right inverses discussed in the previous review.
Reviewer: A.Pryde (Clayton)

##### MSC:
 4.6e+36 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
##### Keywords:
Sobolev space; decomposition; zero trace; trace mappings
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##### References:
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