Abstract
The results and arguments in Chaps. 2 and 3 based upon isoperimetric and isocapacitary inequalities for sets in the Euclidean space can be readily extended to much more general situtations; the present chapter only touches such opportunities.
Applications to estimates of the principal eigenvalue of the Dirichlet Laplacian on a Riemannian manifold are presented in Sects. 4.2 and 4.3. In Sects. 4.4–4.6 we derive some conductor inequalities for functions defined on a locally compact Hausdorff space \(\cal{X}\). It is worth mentioning that, unlike the Sobolev inequalities, the conductor inequalities do not depend on the dimension of \(\cal{X}\). Furthermore, with a lower estimate for the p-conductivity by a certain measure on \(\cal{X}\), one can readily deduce the Sobolev–Lorentz type inequalities involving this measure.
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© 2011 Springer-Verlag Berlin Heidelberg
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Maz’ya, V. (2011). Generalizations for Functions on Manifolds and Topological Spaces. In: Sobolev Spaces. Grundlehren der mathematischen Wissenschaften, vol 342. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15564-2_4
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DOI: https://doi.org/10.1007/978-3-642-15564-2_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-15563-5
Online ISBN: 978-3-642-15564-2
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