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]).
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© 1999 Birkhäuser Boston
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Kohn, J.J. (1999). Quantitative Estimates for Global Regularity. In: Komatsu, G., Kuranishi, M. (eds) Analysis and Geometry in Several Complex Variables. Trends in Mathematics. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-2166-1_6
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DOI: https://doi.org/10.1007/978-1-4612-2166-1_6
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