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

Abstract

Assume that $Δ_h$ is the hyperbolic Laplacian in the unit ball $\mathbb{B}$ and assume that $Φ_n$ is the unique radial solution of Poisson equation $Δ_h \log Φ_n =-4 (n-1)^2$ satisfying the condition $Φ_n(0)=1$ and $Φ_n(ζ)=0$ for $ζ\in \partial\mathbb{B}$. We explicitly solve the question of maximizing $$ R_n(f,Ω)= \frac{\int_Ω|f(x)|^2 Φ_n^α(|x|) \, dτ(x)}{\|f\|^2_{\mathbf{B}^2_α}}, $$ over all $f \in\mathbf{B}^2_α$ and $Ω\subset \mathbb{B}$ with $τ(Ω) = s,$ where $dτ$ denotes the invariant measure on $\mathbb{B},$ and $\|f\|_{{B}^2_α}^2 = \int_\mathbb{B} |f(x)|^2 Φ_n^α(|x|) dτ(x) < \infty.$ 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 $Φ_n$ 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.