Improved Poincare inequalities with weights

Abstract

In this paper we prove that if $\Omega \in {\mathbb{R}}^n$ is a bounded John domain, the following weighted Poincare-type inequality holds: $$\underset{a\in \mathbb{R}}{\inf }\Vert (f(x)-a)w_1(x){\Vert }_{L^q(\Omega )}\le C\Vert \nabla f(x)d(x{)}^{\alpha }{w}_2(x){\Vert }_{L^p(\Omega )}$$ where $f$ is a locally Lipschitz function on $\Omega$, $d(x)$ denotes the distance of $x$ to the boundary of $\Omega$, the weights $w_1,w_2$ satisfy certain cube conditions, and $\alpha \in [0,1]$ depends on $p,q$ and $n$. This result generalizes previously known weighted inequalities, which can also be obtained with our approach.

Cited by in corpus (3)

• Reconstruction of singular and degenerate inclusions in Calderón's problem
• Geometric rigidity on Sobolev spaces with variable exponent and applications
• Parabolic weighted Sobolev-Poincaré type inequalities