Generalizations for Functions on Manifolds and Topological Spaces

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.