Weighted Poincaré inequalities on convex domains

Let $\Omega$ be a bounded open convex set in $\R^n$. Suppose that $\a\ge 0$, $\beta\in \R$, $1\le p\le q <\infty$, and $$1-\frac{n}{p}+\frac{n}{q}, \ 1-\frac{n+\beta}{p}+\frac{n+\a}{q}\ge 0.$$ Let $\rho(x)=\dist(x,\Omega^c)=\min\{|x-y|:y\in \Omega^c\}$ denote the Euclidean distance to the complement of $\Omega$. Define $\rho^{\a}(\Omega) =\int_{\Omega}\rho(x)^{\a} dx$, and denote $$f_{\Omega,\rho^\a}= \frac{1}{\rho^\a(\Omega)}\int_\Omega f(x) \rho(x)^\a dx \quad \mbox{and } \quad \|f\|\down{L^p_{\rho^\a}(\Omega)} = \left(\int_\Omega |f(x)|^p \rho(x)^\a dx\right)^{\frac{1}{p}}. $$ We derive the following weighted Poincaré inequality for locally Lipschitz continuous functions $f$ on $\Omega$: $$\|f-f_{\Omega,\rho^\a}\|\down{L^q_{\rho^\a}(\Omega)}\le C\eta^{\frac{\beta}{p}-\frac{\a}{q}} |\Omega|^{\frac{1}{q} -\frac{1}{p}} \diam(\Omega)^{1-\frac{\beta}{p}+\frac{\a}{q}} \|\nabla f\|\down{L^p_{\rho^\beta}(\Omega)}, $$ where $\eta$ is the eccentricity of $\Omega$ and $C$ is a constant depending only on $p,q,\a, \beta$ and the dimension $n$. The main point of the estimate is the way the constant depends on $\eta$ for a general convex domain. We also consider the case $1\le q<p<\infty$, where the inequality is valid under the stronger hypothesis $1-(1+\beta)/p+(1+\a)/q >0$. When $q\ge p$, the case of convex domains which are symmetric with respect to a point was settled in \cite{CD}, and our estimate for $q\ge p$ extends that result to nonsymmetric domains. Moreover, the exponent of $\eta$ is sharp and the conditions are necessary.

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Published 1 January 2010