Hardy inequalities on metric measure spaces, II: the case p > q

Remarks about Hardy Inequalities on Metric Trees

where ψ > 0 is the so-called Hardy weight and C(Γ, ψ) is a positive constant which might depend on Γ and ψ, but which is independent of u. Evans, Harris and Pick [EHP] found a necessary and sufficient condition such that (1.1) holds for all functions u on Γ such that u(o) = 0 and such that the integral on the right hand side is finite. They consider even the case of non-symmetric Hardy weights....

HARDY–HILBERT’S TYPE INEQUALITIES FOR (p, q)-Hö(0,∞) FUNCTIONS

New inequalities concerning functions of the form f (xy) similar to Hardy-Hilbert’s integral inequality are presented. A new class of functions denoted by (p, q)−-Hö(I) is defined. Many other new inequalities are also given.

Hardy Inequalities in Function Spaces

DIFFERENTIABILITY OF p-HARMONIC FUNCTIONS ON METRIC MEASURE SPACES

We study p-harmonic functions on metric measure spaces, which are formulated as minimizers to certain energy functionals. For spaces supporting a p-Poincaré inequality, we show that such functions satisfy an infinitesmal Lipschitz condition almost everywhere. This result is essentially sharp, since there are examples of metric spaces and p-harmonic functions that fail to be locally Lipschitz co...

Characterizations of Sobolev Inequalities on Metric Spaces

We present isocapacitary characterizations of Sobolev inequalities in very general metric measure spaces.