A link between the asymptotic expansions of the Bergman kernel and the Szegö kernel

Abstract

Introduction Let Ω be a strictly pseudoconvex domain in C. Then the Bergman kernel K and the Szego kernel K of Ω have singularities at the boundary diagonal. These singularities admit asymptotic expansions in powers and log of the defining function of Ω ([3], [2]) and, moreover, the coefficients of which can be expressed in terms of local invariants of the CR structure of the boundary ∂Ω as an application of the parabolic invariant theory developed in [4], [5], [1], [8], [6] and others. While these works provide a geometric algorithm of expressing the expansion of each kernel, it is not easy to read relations between them from this construction — for example, we can say very little about the relation between the log term coefficients of K and K, cf. §2. In this note we present a method of relating these asymptotic expansions. Our strategy is to construct a meromorphic family of kernel functions Ks, s ∈ C, such that K and K are realized as special values of Ks. In the case of the unit ball, {|z| < 1}, such a family is given by Ks(z) = π−nΓ(n− s) (1− |z|2)s−n,