article

Title: Weighted Poincare inequalities on symmetric convex domains

Authors: Seng-Kee Chua and Huo-Yuan Duan

Issue: Volume 58 (2009), Issue 5, 2103-2114

Abstract:

$Let $\alpha \ge 0$, $\beta \in \mathbb{R}$, $1 \le p \le q < \infty$ with \[ 1 - \frac{n}{p} + \frac{n}{q}, \quad 1 - \frac{n+\beta}{p} + \frac{n+\alpha}{q} \ge 0. \] Let $\Omega$ be a bounded convex domain in $\mathbb{R}^n$ that is symmetric with respect to its center. Define $\rho(x) = \dist(x,\Omega^c) = \inf\{|x - y| : y \in \Omega^c\}$ and $\rho^{\alpha}(E) = \int_{E} \rho(x)^{\alpha} \mathrm{d}x$. Let $f$ be a Lipschitz continuous function on $\Omega$ and \[ f_{\Omega, \rho^{\alpha}} = \int_{\Omega} f(x) \rho(x)^{\alpha} \mathrm{d}x/\rho^{\alpha}(\Omega). \] We obtain the following weighted Poincar\'e inequality: \begin{align*}{}&\|f - f_{\Omega,\rho^{\alpha}}\|_{L^q_{\rho^{\alpha}}(\Omega)}\\ &\quad\mathrel\le C\eta^{\beta/p-\alpha/q} |\Omega|^{1/q-1/p} \diam(\Omega)^{1-\beta/p+\alpha/q} \|\nabla f\|_{L^p_{\rho^{\beta}}(\Omega)} \end{align*} where $\eta$ is the eccentricity of $\Omega$ and $C$ is a constant depending only on $p$, $q$, $\alpha$, $\beta$, and the dimension $n$. Moreover, the exponent of $\eta $ is sharp.$