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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)

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