On the Growth of the Bergman Kernel near an Infinite-type Point

We study diagonal estimates for the Bergman kernels of certain model domains in C near boundary points that are of infinite type. To do so, we need a mild structural condition on the defining functions of interest that facilitates optimal upper and lower bounds. This is a mild condition; unlike earlier studies of this sort, we are able to make estimates for non-convex pseudoconvex domains as well. This condition quantifies, in some sense, how flat a domain is at an infinite-type boundary point. In this scheme of quantification, the model domains considered below range — roughly speaking — from being “mildly infinite-type” to very flat at the infinite-type points. 1. Statement of Results Let Ω ⊂ C2 be a pseudoconvex domain (not necessarily bounded) having a smooth boundary. Let p ∈ ∂Ω be a point of infinite type: by this we mean that for each N ∈ Z+, there exists a germ of a 1-dimensional complex-analytic variety through p whose order of contact with ∂Ω at p is at least N . If ∂Ω is not Levi-flat around p, there exist holomorphic coordinates (z,w;Vp) centered at p such that (1.1) Ω ⋂ Vp = {(z,w) ∈ Vp : Imw > F (z) +R(z,Rew)}, where F is a smooth, subharmonic, non-harmonic function defined in a neighbourhood of z = 0, that vanishes to infinite order at z = 0; R(· , 0) vanishes to infinite order at z = 0; and R is O(|z||Rew|, |Rew|2). Given the infinite order of vanishing of F at z = 0, how does one find estimates for the Bergman kernel of Ω near p ? In many cases — for instance: when ∂Ω∩Vp is pseudoconvex of strict type, in the sense of [5], away from p ∈ ∂Ω — the function F in (1.1) can be extended to a global subharmonic function. In such situations, the model domain (1.2) ΩF := {(z,w) ∈ C 2 : Imw > F (z)}, approximates ∂Ω to infinite order along the complex-tangential directions at p. One is thus motivated to investigate estimates for the Bergman kernel for domains of the form (1.2). In this paper, we shall find estimates for the Bergman kernel of ΩF on the diagonal as one approaches (0, 0) ∈ ∂ΩF . More specifically: • We shall derive estimates that hold not just in a non-tangential interior cone with vertex at (0, 0), but for a family of much larger approach regions that comprises regions with arbitrarily high orders of contact at (0, 0); and 1991 Mathematics Subject Classification. Primary: 32A25, 32A36.