# Muckenhoupt $A_p$-properties of distance functions and applications to Hardy-Sobolev -type inequalities

## TL;DR

This paper characterizes when distance-based weights belong to Muckenhoupt classes in metric spaces and applies these results to establish Hardy-Sobolev inequalities, including fractional variants.

## Contribution

It provides sharp conditions based on Assouad (co)dimension for distance weights to be in Muckenhoupt classes and derives Hardy-Sobolev inequalities in metric spaces.

## Key findings

- Sharp conditions for $A_p$ membership of distance weights.
- Established Hardy-Sobolev inequalities in metric spaces.
- Derived fractional Hardy-Sobolev inequalities.

## Abstract

Let $X$ be a metric space equipped with a doubling measure. We consider weights $w(x)=\operatorname{dist}(x,E)^{-\alpha}$, where $E$ is a closed set in $X$ and $\alpha\in\mathbb R$. We establish sharp conditions, based on the Assouad (co)dimension of $E$, for the inclusion of $w$ in Muckenhoupt's $A_p$ classes of weights, $1\le p<\infty$. With the help of general $A_p$-weighted embedding results, we then prove (global) Hardy-Sobolev inequalities and also fractional versions of such inequalities in the setting of metric spaces.