  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 Poincar\'e 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.
