# A new approach to weighted Sobolev spaces

## TL;DR

This paper introduces a new method for defining weighted Sobolev spaces using a novel weak derivative concept, allowing non-locally integrable functions to be included.

## Contribution

The novelty lies in replacing the distributional derivative with a new weak derivative to handle arbitrary small weight functions.

## Key findings

- A new weak derivative allows non-locally integrable functions to be considered in weighted Sobolev spaces.
- Conditions for unique non-locally integrable solutions to degenerate elliptic PDEs are established.
- Tools like Poincaré inequality, trace operator, and density results for smooth functions are developed.

## Abstract

We present in this paper a new way to define weighted Sobolev spaces when the weight functions are arbitrary small. This new approach can replace the old one consisting in modifying the domain by removing the set of points where at least one of the weight functions is very small. The basic idea is to replace the distributional derivative with a new notion of weak derivative. In this way, non-locally integrable functions can be considered in these spaces. Indeed, assumptions under which a degenerate elliptic partial differential equation has a unique non-locally integrable solution are given. Tools like a Poincaré inequality and a trace operator are developed, and density results of smooth functions are established.