Quantitative Estimates for Global Regularity

Abstract

Let Ω ⊂ ℂⁿ 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 <Emphasis FontCategory=“NonPropotional”>H</Emphasis>(Ω) the space of square-integrable holomorphic functions on Ω. Let B: L ₂(Ω) → <Emphasis FontCategory=”NonPropotional”>(Ω)</Emphasis> denote the Bergman projection operator, which is the orthogonal projection of L ₂(Ω) onto <Emphasis FontCategory=”NonPropotional”>(Ω)</Emphasis>. 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]).