Abstract
According to Corollary 2.3.4, for q≥p>1, the inequality
follows from the isocapacitary inequality
Here and henceforth E is an arbitrary Borel set in ℝn and \(w^{1}_{p}\) is the completion of \(C^{\infty}_{0}\) with respect to the norm \(\|\nabla u\|_{L_{p}}\).
On the other hand, if (11.1.1) is valid for any \(u \in C^{\infty}_{0}\), then
for all E⊂ℝn.
The present chapter contains similar results in which the role of \(w^{1}_{p}\) is played by the spaces \(H^{l}_{p}\), \(h^{l}_{p}\), \(W^{l}_{p}\), \(w^{l}_{p}\), \(B^{l}_{p}\), and \(b^{l}_{p}\).
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© 2011 Springer-Verlag Berlin Heidelberg
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Maz’ya, V. (2011). Capacitary and Trace Inequalities for Functions in ℝn with Derivatives of an Arbitrary Order. In: Sobolev Spaces. Grundlehren der mathematischen Wissenschaften, vol 342. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15564-2_11
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DOI: https://doi.org/10.1007/978-3-642-15564-2_11
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-15563-5
Online ISBN: 978-3-642-15564-2
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