# Improved fractional Poincar\'e type inequalities on John domains

## TL;DR

This paper establishes enhanced fractional Poincaré inequalities in John domains within metric spaces with doubling measures, providing conditions for their applicability and advancing the understanding of fractional inequalities in irregular domains.

## Contribution

It introduces improved fractional Poincaré inequalities on John domains in metric spaces with doubling measures, under mild regularity conditions, and identifies conditions for domains to support such inequalities.

## Key findings

- Established improved fractional Poincaré inequalities in John domains.
- Provided sufficient conditions for domains to support these inequalities.
- Extended the theory of fractional inequalities to irregular metric spaces.

## Abstract

We obtain improved fractional Poincar\'e inequalities in John domains of a metric space $(X, d)$ endowed with a doubling measure $\mu$ under some mild regularity conditions on the measure $\mu$. We also give sufficient conditions on a bounded domain to support fractional Poincar\'e type inequalities in this setting.