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 . 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 is only required to satisfy the restriction: , where is the exponent of the Sobolev space and 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.