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 Ph: L2(D) ? L2h (D) and the operator of logarithmic potential type defined by L f(z)=- 1/2? ?D ln |z-?|f(?)dA(?), where D is the unit disc in C. We describe the asymptotic behaviour of the eigenvalues of the operator (PhL)+(PhL). More precisely, we prove that limn?+? n2sn(PhL) = ?4?2/3-1.