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

Abstract

We considercertain inequalities and a related result on weighted Sobolev spaces on bounded John domains in $R^n$. Namely, we study 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 $d{(·,\partial \Omega )}^{\beta p}$ is only required to satisfy the restriction: $\beta p>-(n-{\dim }_A\left(\partial \Omega \right))$, where $p$ is the exponent of the Sobolev space and ${\dim }_A\left(\partial \Omega \right)$ is the Assouad dimension of the boundary of the domain. To the best of our knowledge, this condition is less restrictive than the ones in the literature.