# A weighted anisotropic Sobolev type inequality and its applications to Hardy inequalities

## TL;DR

This paper establishes a new weighted anisotropic Sobolev inequality involving a general gauge function and applies it to derive refined Hardy inequalities, advancing the understanding of functional embeddings and inequalities.

## Contribution

It introduces a novel weighted anisotropic Sobolev embedding involving a general gauge function and derives refined Hardy inequalities from it.

## Key findings

- Established a new weighted anisotropic Sobolev inequality.
- Derived refined Hardy inequalities based on the Sobolev embedding.
- Extended the class of inequalities applicable to weighted Sobolev spaces.

## Abstract

In this paper we focus our attention on an embedding result for a weighted Sobolev space that involves as weight the distance function from the boundary taken with respect to a general smooth gauge function $F$. Starting from this type of inequalities we prove some refined Hardy-type inequalities.