The Bergman and Szegő kernels near points of infinite type

Jennifer Halfpap, Alexander Nagel and Stephen Wainger

Vol. 246 (2010), No. 1, 75–128

DOI: 10.2140/pjm.2010.246.75

Abstract

We study the singularities of the Bergman and Szegő kernels for domains Ω = {(z₁,z₂) ∈ ℂ²∣Imz₂ > b(Rez₁)}. Here b is an even function in C^∞(ℝ) satisfying b(0) = b′(0) = 0,  b′′(r) > 0 for r≠0, and vanishing to infinite order at r = 0. A model example is b(r) = exp(−|r|^−a) for |r| small and b(r) = r^2m for |r| large, with a,m > 0. If Δ ⊂ ∂Ω ×∂Ω is the diagonal of the boundary, our results show in particular that if 0 < a < 1 the Bergman and Szegő kernels extend smoothly to Ω ×Ω ∖ Δ, while if a ≥ 1 the kernels are singular at points on Ω ×Ω ∖ Δ.

Keywords

Bergman kernel, Szegő kernel

Mathematical Subject Classification 2000

Primary: 32T99, 42B20

Milestones

Received: 1 March 2009

Accepted: 3 November 2009

Published: 1 May 2010