On Eigenvalues of Logarithmic Potential Operator in the Hyperbolic Space

Abstract

Let $\hat{I}\text{c◯}$ be a bounded open set in the Poincaré hyperbolic disk, $\mathbb{D}$. In this article, we consider the hyperbolic logarithmic potential operator ${\mathcal{L}}_h:L^2(\hat{I}\text{c◯})\to L^2(\hat{I}\text{c◯})$, defined by \begin{equation*} \mathcal{L}_h u(z)=\frac{1}{2}\int_Ω\log\frac{1}{[z,w]}\,u(w)\, {\,\rm d}(w), \end{equation*} and the associated eigenvalue problem on $\hat{I}\text{c◯}$ \begin{equation} \mathcal{L}_h u=τu. \end{equation} We first extend the notion of polarization with respect to hyperplanes in the Poincaré disk and prove the associated properties. Then we establish a reverse Faber-Krahn inequality for the largest eigenvalue, $\ddot{I}{„}_h$ 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 $L^2(\hat{I}\text{c◯})$.

26 pages