Abstract
For p≥1 we define L m,p(ℝn) as the set of distributions u on ℝn such that
Here ∇ k u denotes the vector (D α u)|α|=k , \(\mathaccent"7017{L}^{m, p}(\mathbb{R}^{n})\) is the completion of \(C^{\infty}_{0}(\mathbb{R}^{n})\) with respect to the L m,p seminorm. Now let Ω⊂ℝn be an open set and let μ be a nontrivial positive Radon measure on Ω. We will study the space \(H^{m,p}_{\mu}(\varOmega)\), defined as the completion of \(L_{p}(\mu) \cap L^{m,p}(\mathbb{R}^{n}) \cap C^{\infty}_{0}(\varOmega)\) with respect to the norm
The closure of \(C^{\infty}_{0}(\varOmega)\) in \(H^{m, p}_{\mu}(\varOmega)\) is denoted \(\mathaccent"7017{H}^{m, p}_{\mu}(\varOmega)\). Note that if p<n then by Sobolev’s inequality the elements in \(\mathaccent"7017{L}^{m, p}\) can be identified with functions in \(L_{p^{*}}\), where \(p^{*}=\frac{np}{n-p}\). If a domain in the notations of a space is not indicated it is assumed to be ℝn. The elements in \(\mathaccent"7017{H}^{m, p}_{\mu}(\varOmega)\) are naturally identified with elements in \(\mathaccent"7017{L}^{m, p}\). Note also that L m,p⊂L p (ℝn,loc).
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© 2011 Springer-Verlag Berlin Heidelberg
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Maz’ya, V. (2011). Approximation in Weighted Sobolev Spaces. In: Sobolev Spaces. Grundlehren der mathematischen Wissenschaften, vol 342. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15564-2_17
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DOI: https://doi.org/10.1007/978-3-642-15564-2_17
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-15563-5
Online ISBN: 978-3-642-15564-2
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