The Poincare inequality: A necessary and sufficient condition.

Abstract

Let $\Omega$ be a connected open set in the complex plane and fix a point $W\sb0\in\Omega$. Let $D(\Omega)$ denote the Dirichlet space of analytic functions g on $\Omega$ such that g($w\sb0$) = 0 and whose derivatives are square-integrable with respect to area. Let $B(\Omega$) denote the Bergman space of analytic functions g on $\Omega$ which are square-integrable with respect to area, and let $B\sb0$($\Omega$) denote the subspace of $B(\Omega$) consisting of all functions g in $B(\Omega$) with g ($W\sb0$) = 0. Then $\Omega$ is said to be an analytic Poincare domain if for some constant $\kappa$ the inequality $\Vert g\Vert\sbsp{B}{2}\leq\kappa\ \Vert g\Vert\sbsp{D}{2}$ holds for all $g\in D$ ($\Omega$). This inequality is called the analytic Poincare inequality and the infimum of the values of $\kappa$ is the analytic Poincare constant of $\Omega$ for the distinguished point $w\sb0$. We study the analytic Poincare domains for simply connected domains $\Omega$ with finite area. Let $\phi$ be the Riemann mapping of the unit disc $\rm I\!D$ onto $\Omega$, with $\phi$(0) = $w\sb0$ and $\phi\sp\prime(0) > 0.$ We introduce three integral operators ${\cal A}\sb1, {\cal A}\sb2$ and ${\cal A}\sb3$, where ${\cal A}\sb1:D({\rm I\!D})\to B\sb0({\rm I\!D}), {\cal A}\sb2:B\sb0 ({\rm I\!D})\to D({\rm I\!D})$ and ${\cal A}\sb3:B\sb0 ({\rm I\!D}) \to B\sb0 ({\rm I\!D}).$ Our first three theorems establish that ${\cal A}\sb2$ is the adjoint of ${\cal A}\sb1$, that ${\cal A}\sb3$ = ${\cal A}\sb1{\cal A}\sb2$, and that $\Omega$ is an analytic Poincare domain if and only if ${\cal A}\sb1$ is bounded. Moreover, $\kappa$ = $\Vert{\cal A}\sb1\Vert\sp2$. Using these three theorems, we show that $\Omega$ is an analytic Poincare domain if and only if the operator ${\cal A}\sb3$ is bounded. Moreover, $\kappa$ = $\Vert{\cal A}\sb3\Vert$. Equality in the analytic Poincare inequality is attained if and only if $\Vert{\cal A}\sb3\Vert$ is an eigenvalue of ${\cal A}\sb3$. We next ask when are these operators in the trace class ${\cal J}\sb1$ or in the Hilbert-Schmidt class ${\cal J}\sb2$. Under the assumption that ${\cal A}\sb1$ is bounded on D($\rm I\!D$) we introduce a bounded operator ${\cal A}\sb4$ = ${\cal A}\sb2{\cal A}\sb1$:D($\rm I\!D$) $\to D(\rm I\!D$). We obtain necessary and sufficient conditions in terms of the mapping function $\phi$ for each of the operators ${\cal A}\sb3$ and ${\cal A}\sb4$ to be in ${\cal J}\sb1$. These give us necesary and sufficient conditions for each of the operators ${\cal A}\sb1$ and ${\cal A}\sb2$ to be in ${\cal J}\sb2$. We also show that ${\cal A}\sb3$ and ${\cal A}\sb4$ must satisfy the same necessary and sufficient condition in order to be in ${\cal J}\sb2$. Since an operator that is in ${\cal J}\sb1$ is a compact operator, the above results allow us to show that for two large classes of analytic Poincare domains, namely, the domains $\Omega$ for which $\phi\sp\prime$ is in the Hardy class $H\sp{\rm p}$, p $>$ 1, and for analytic $L\sp1$-averaging domains (which contain convex domains, John domains, quasidisks, Holder domains), the equality in the analytic Poincare inequality is attained.