Local and semilocal Poincaré inequalities on metric spaces

Abstract

We consider several local versions of the doubling condition and Poincaré inequalities on metric spaces. Our first result is that in proper connected spaces, the weakest local assumptions self-improve to semilocal ones, i.e. holding within every ball. We then study various geometrical and analytical consequences of such local assumptions, such as local quasiconvexity, self-improvement of Poincaré inequalities, existence of Lebesgue points, density of Lipschitz functions and quasicontinuity of Sobolev functions. It turns out that local versions of these properties hold under local assumptions, even though they are not always straightforward. We also conclude that many qualitative, as well as quantitative, properties of p-harmonic functions on metric spaces can be proved in various forms under such local assumptions, with the main exception being the Liouville theorem, which fails without global assumptions.

References in corpus (1)

• The annular decay property and capacity estimates for thin annuli

Cited by in corpus (9)

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• Almost uniform domains and Poincaré inequalities
• Locally $p$-admissible measures on $\mathbb{R}$
• Bounded geometry and $p$-harmonic functions under uniformization and hyperbolization
• Existence and almost uniqueness for $p$-harmonic Green functions on bounded domains in metric spaces
• Volume growth, capacity estimates, $p$-parabolicity and sharp integrability properties of $p$-harmonic Green functions
• Poincaré inequalities and $A_p$ weights on bow-ties
• Potential Theory on Gromov Hyperbolic Spaces
• Condenser capacities and capacitary potentials for unbounded sets, and global $p$-harmonic Green functions on metric spaces