A reverse Holder inequality for extremal Sobolev functions

Abstract

Let $n \geq 2$, let $\Omega \subset \mathbf{R}^n$ be a bounded domain with smooth boundary, and let $1 \leq p \leq 2$. We prove a reverse-Holder inequality for functions $u$ realizing the best constant in the Sobolev inequality, that is $$\mathcal{C}_p(\Omega) = \inf \left \{ \frac{\int_\Omega |\nabla v|^2}{\left ( \int_\Omega |v|^p \right )^{2/p}} \right \} = \frac{\int_\Omega |\nabla u|^2}{\left ( \int_\Omega |u|^p \right )^{2/p}}.$$ Our inequality has the form $\| u \|_{L^p} \geq K \| u \|_{L^q}$ for any $q > p$, where $K$ depends only on $n$, $p$, $q$, and $\mathcal{C}_p(\Omega)$. This result generalizes work of Chiti, regarding the first Dirichlet eigenfunction of the Laplacian, and of van den Berg, regarding the torsion function.