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 $$ R_n(f,\Omega)= \frac{\int_\Omega |f(x)|^2 \Phi_n^\alpha(|x|) \, d\tau(x)}{\|f\|^2_{\mathbf{B}^2_\alpha}}, $$ over all $f \in\mathbf{B}^2_\alpha$ and $\Omega \subset \mathbb{B}$ with $\tau(\Omega) = s,$ where $d\tau$ denotes the invariant measure on $\mathbb{B},$ and $\|f\|_{{B}^2_\alpha}^2 = \int_\mathbb{B} |f(x)|^2 \Phi_n^\alpha(|x|) d\tau(x) < \infty.$ 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.