On the weighted logarithmic potential operator

For a bounded open set $\Omega \subset \mathbb{R}^N$ with $N\geq 2$, and for positive continuous functions $w,g$ on $\overline{\Omega}$, we consider the weighted eigenvalue problem \begin{equation*} \mathcal{L}_{w} u =\tau gu, \end{equation*} where $\mathcal{L}_{w}$ is the weighted logarithmic potential operator on $L^2(\Omega)$ as defined below: \begin{equation*} \mathcal{L}_{w} u(x)=\int_\Omega \log\left(\frac{w(x)w(y)}{|x-y|}\right)u(y)dy. \end{equation*} We study the monotonicity and continuity of the largest positive eigenvalue $\tau_{w,g}^+(\Omega)$ with respect to $\Omega$, $w$, and $g$. We also establish that $\tau_{w,g}^+(\Omega)$ satisfies a reverse Faber Krahn inequality under polarization. We provide a sufficient condition for the existence of a negative eigenvalue in terms of the weighted transfinite diameter of $\Omega$, under the assumption that $\log w$ is superharmonic. For $\Omega\subset \mathbb{R}^2$, if $\Delta\log w$ is a constant $C$, we show that 0 can be an eigenvalue of $\mathcal{L}_{w}$ only when $C=\frac{2\pi}{|\Omega|}$. For such domains, if $\log w$ is a harmonic function on $\Omega$, we provide a representation formula for the eigenfunctions. Using this representation, we establish variants of the maximum principles that give some insight into the geometry of these eigenfunctions.