Uniqueness of weighted Sobolev spaces with weakly differentiable weights

TL;DR

This paper establishes the uniqueness of weighted Sobolev spaces with weakly differentiable weights under certain integrability conditions and explores their properties, including alternative definitions and applications to specific weight functions.

Contribution

It proves the uniqueness of weighted Sobolev spaces with weakly differentiable weights and introduces an alternative definition based on integration by parts, extending the theory for specific weight functions.

Findings

Weighted Sobolev spaces are unique under certain integrability conditions.

An alternative definition involving the logarithmic gradient is proposed.

Weights of exponential form satisfy a Poincaré inequality and are p-admissible.

Abstract

We prove that weakly differentiable weights $w$ which, together with their reciprocals, satisfy certain local integrability conditions, admit a unique associated first-order $p$-Sobolev space, that is \[H^{1,p}(\mathbb{R}^d,w\,\d x)=V^{1,p}(\mathbb{R}^d,w\,\d x)=W^{1,p}(\mathbb{R}^d,w\,\d x),\] where $d\in\N$ and $p\in [1,\infty)$. If $w$ admits a (weak) logarithmic gradient $\nabla w/w$ which is in $L^q_{\text{loc}}(w\,\d x;\R^d)$, $q=p/(p-1)$, we propose an alternative definition of the weighted $p$-Sobolev space based on an integration by parts formula involving $\nabla w/w$. We prove that weights of the form $\exp(-\beta |\cdot|^q-W-V)$ are $p$-admissible, in particular, satisfy a Poincar\'e inequality, where $\beta\in (0,\infty)$, $W$, $V$ are convex and bounded below such that $|\nabla W|$ satisfies a growth condition (depending on $\beta$ and $q$) and $V$ is bounded. We apply the uniqueness result to weights of this type. The associated nonlinear degenerate evolution equation is also discussed.