On how Poincaré inequalities imply weighted ones
The present paper is devoted to Poincaré type inequalities that bound the $L_q$-norm of a function $f$ on a domain \(\Omega\) equipped with a non-negative weight $w$ in terms of the $L_p$-norm of its gradient in the form \[ \|f\|_{L^q(\Omega,w)}\le c\|\nabla f\|_{L^p(\Omega,w)} \] where the finite constant $c$ is independent of $f$ and the inequality holds for every $f\in \mathcal{C}$. The space $\mathcal{C}$ considered by the author is a vector space of differentiable (at least in a weak sense) functions $f$ on $\Omega$ satisfying the equality $\int_{\Omega}{fw}=0$ for each $f\in \mathcal{C}$. The main result of this paper is that the Poincaré constant w.r.t. a weight $w$ can be controlled in terms of the Poincaré constants w.r.t. the constant weight 1 on the superlevel sets of $w$. It turns out that it is essentially sufficient that on all superlevel sets of $w$ there hold Poincaré inequalities w.r.t. the constant weight 1 and that the corresponding Poincaré constants satisfy an integrability condition.
scientific article; zbMATH DE number 7045090
scientific article