| Popis: |
We study the singularities of the Bergman and Szegő kernels for domains Ω = {(z 1 , z 2 ) ∈ ℂ 2 |Im z 2 > b(Re z 1 )}· 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 A ⊂ ∂Ω x ∂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 x Ω \ Δ, while if a ≥ 1 the kernels are singular at points on Ω x Ω \ Δ. |
