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(·,∂Ω)^{βp} is only required to satisfy the restriction: βp>−(n−dimA(∂Ω)), where p is the exponent of the Sobolev space and dimA(∂Ω) 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.