A Faber-Krahn type inequality for log-subharmonic functions in the hyperbolic ball

Abstract

Assume that ${\Delta }_h$ is the hyperbolic Laplacian in the unit ball $\mathbb{B}$ and assume that ${\Phi }_n$ is the unique radial solution of Poisson equation ${\Delta }_h\log {\Phi }_n=-4(n-1{)}^2$ satisfying the condition ${\Phi }_n(0)=1$ and ${\Phi }_n(\zeta )=0$ for $\zeta \in \partial \mathbb{B}$. We explicitly solve the question of maximizing $$R_n(f,\Omega )=\frac{{\int }_{\Omega }|f(x){|}^2{\Phi }_n^{\alpha }(|x|)d\tau (x)}{\Vert f{\Vert }_{{\mathbf{B}}_{\alpha }^2}^2},$$ over all $f\in {\mathbf{B}}_{\alpha }^2$ and $\Omega \subset \mathbb{B}$ with $\tau (\Omega )=s,$ where $d\tau$ denotes the invariant measure on $\mathbb{B},$ and $\Vert f{\Vert }_{B_{\alpha }^2}^2={\int }_{\mathbb{B}}|f(x){|}^2{\Phi }_n^{\alpha }(|x|)d\tau (x)<\infty .$ This result extends the main result of Tilli and the second author \cite{ramostilli} to a higher-dimensional context. Our proof relies on a version of the techniques used for the two-dimensional case, with several additional technical difficulties arising from the definition of the weights ${\Phi }_n$ through hypergeometric functions. Additionally, we show that an immediate relationship between a concentration result for log-sunharmonic functions and one for the Wavelet transform is only available in dimension one.

16 pages; several typos corrected