THE EXACT SPECTRAL ASYMPTOTIC OF THE LOGARITHMIC POTENTIAL ON HARMONIC FUNCTION SPACE

Abstract

In this paper we consider the product of the harmonic Bergman projection $P_{h}:L^{2}(D)\rightarrow L_{h}^{2}(D)$ and the operator of logarithmic potential type defined by $Lf(z)=-\frac{1}{2\pi}\int_{D}\ln{|z-\xi|}f(\xi)dA(\xi),$ where $D$ is the unit disc in $\mathbb{C}.$ We describe the asymptotic behaviour of the eigenvalues of the operator $(P_{h}L)^{\ast}(P_{h}L).$ More precisely, we prove that $$\lim_{n\rightarrow+\infty}n^{2}s_{n}(P_{h}L)=\sqrt{\frac{4\pi^{2}}{3}-1}.$$