On Eigenvalues of Logarithmic Potential Operator in the Hyperbolic Space

Let $\Omega$ be a bounded open set in the Poincar\'e hyperbolic disk, $\mathbb{D}$. In this article, we consider the hyperbolic logarithmic potential operator $\mathcal{L}_h : L^2(\Omega) \to L^2(\Omega)$, defined by \begin{equation*} \mathcal{L}_h u(z)=\frac{1}{2}\int_\Omega \log\frac{1}{[z,w]}\,u(w)\, {\,\rm d}(w), \end{equation*} and the associated eigenvalue problem on $\Omega$ \begin{equation} \mathcal{L}_h u=\tau u. \end{equation} We first extend the notion of polarization with respect to hyperplanes in the Poincar\'e disk and prove the associated properties. Then we establish a reverse Faber-Krahn inequality for the largest eigenvalue, $\tau_{h}$ of $\mathcal{L}_h$, under polarization. Further, we provide a representation formula for the eigenfunctions of $\mathcal{L}_h$. In addition, we show that the operator $\mathcal{L}_h$ is a positive operator on $L^2(\Omega)$.