A weighted Poincaré inequality with a doubling weight

Hurri-Syrjānen, Ritva

doi:10.1090/S0002-9939-98-04059-3

A weighted Poincaré inequality with a doubling weight
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by Ritva Hurri-Syrjānen

Proc. Amer. Math. Soc. 126 (1998), 545-552

DOI: https://doi.org/10.1090/S0002-9939-98-04059-3

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Abstract:

We show that unbounded John domains (and even a larger class of domains than John domains) satisfy the weighted Poincaré inequality \begin{equation*}\inf _{a\in \mathbb {R}} \|u(x)-a\|_{L^{q}(D,w_{1})} \le C\|\nabla u(x)\|_{L^{p}(D,w_{2})}\end{equation*} whenever $u$ is a Lipschitz function on $D$, $w_{1}$ is a doubling weight, and weights satisfy certain cube conditions, and $C=C(D,p,q,w_{1},w_{2})$.

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