# Local and semilocal Poincar\'e inequalities on metric spaces

## TL;DR

This paper investigates local versions of doubling and Poincaré inequalities in metric spaces, showing they self-improve to semilocal forms in proper connected spaces and exploring their implications for geometric and analytical properties.

## Contribution

It demonstrates that local assumptions on metric spaces lead to significant self-improvement and various properties, advancing understanding of local geometric and analytical structures.

## Key findings

- Local assumptions self-improve to semilocal in proper connected spaces
- Local properties like quasiconvexity and Poincaré inequalities hold under local assumptions
- Many properties of p-harmonic functions can be proved locally, except Liouville theorem

## Abstract

We consider several local versions of the doubling condition and Poincar\'e 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 Poincar\'e 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.