Some inequalities on weighted Sobolev spaces, distance weights and the Assouad dimension

Abstract

We study certain inequalities and a related result on weighted Sobolev spaces on bounded John domains in ${\mathbb{R}}^n$. Namely, we prove the existence of a right inverse for the divergence operator, along with the corresponding a priori estimate, the improved and the fractional Poincaré inequalities, the Korn inequality and the local Fefferman-Stein inequality. All these results are obtained on weighted Sobolev spaces, where the weight is a power of the distance to the boundary. In all cases the exponent of the weight $d(\cdot ,\partial \Omega {)}^{\beta p}$ is only required to satisfy the restriction: $\beta p>-(n-{\dim }_A(\partial \Omega ))$, where $p$ is the exponent of the Sobolev space and ${\dim }_A(\partial \Omega )$ is the Assouad dimension of the boundary of the domain. According to our best knowledge, this condition is less restrictive than the ones in the literature.

21 pages, 2 figures