Quantitative Estimates for Global Regularity

Abstract

Let Ω ⊂ ℂ n be a bounded pseudoconvex domain with a smooth boundary. We denote by L 2(Ω) the space of square-integrable functions on Ω and by H (Ω) the space of square-integrable holomorphic functions on Ω. Let B: L 2(Ω) → (Ω) denote the Bergman projection operator, which is the orthogonal projection of L 2(Ω) onto (Ω) . Here we will be concerned with the global regularity of B in terms of Sobolev norms, that is, the question of when B(H S (Ω)) ⊂ H S (Ω) where H s (Ω) denotes the Sobolev space of order s. Of course, if B preserves H S (Ω) locally (i.e., if B(s/loc(Ω)) ⊂ H s loc(Ω)), then B also preserves H s (Ω) globally. Aspects of the local question are very well understood, in particular when Ω is of finite D’Angelo type (see [Cal] and [D’A]). Local regularity can still occur when the D’Angelo type is infinite, as in the examples given in [Chr2] and [K2]. Local regularity fails whenever there is a complex curve V in the boundary of Ω. In that case, if P ∈ V, then for given s there exists an f ∈ L 2(Ω) such that ζf ∈ H S (Ω) for every smooth function ζ with support in a fixed small neighborhood of P and such that ζB(f) ∉ H S (Ω) whenever ζ = 1 in some neighborhood of P. In contrast, global regularity always holds for small s. That is, if Ω is pseudoconvex, then there exists η > 0 such that B(H S (Ω)) ⊂ H S (Ω) for s ⩽ η. Furthermore, there is a series of results showing global regularity under a variety of conditions (see [Ca2], [BC], [Ch], [BS1], and [BS2]).