Sobolev Space Projections in Strictly Pseudoconvex Domains

The orthogonal projection from a Sobolev space WS(Q) onto the subspace of holomorphic functions is studied. This analogue of the Bergman projection is shown to satisfy regularity estimates in higher Sobolev norms when ß is a smooth bounded strictly pseudoconvex domain in C". The Bergman projection P0: L2(ü) -» L2(S2) n {holomorphic functions}, where S2 c C" is a smooth bounded domain, has proved to be a key element in the study of boundary behavior of holomorphic mappings (see [4, 7, 13] and their references). In the important special case in which Í2 is strictly pseudoconvex, a great deal is known about the projection P0 and the Bergman kernel function K0(w, z) which represents it (see e.g. [14, 16, 19]). In particular the following two regularity theorems are well known consequences of Kohn's theory of the 9-Neumann problem [15, 17]. Theorem A [17]. Let Q, c C" be a smooth bounded strictly pseudoconvex domain. Then the Bergman projection P0 admits both global and local regularity estimates in Sobolev norms: (i) \\PQu\\, < Cr\\u\l, r > 0, and more generally, if Çx, f2 G CX(C") are real-valued cut-off functions with f 2 = 1 in a neighborhood of the support o/f,, then (ii) ïfi*o«l,<Ç(Hr2«Hr + Mo), r>0. Theorem B [16]. Let ß c C" be a smooth bounded strictly pseudoconvex domain. Then the Bergman kernel function KQ(w, z) is smooth up to the boundary off the boundary diagonal, that is, KQ(w,z) G C°°(ñ XÍ2\{z = W G en}). The objects studied in this paper are the analogous projection Ps: W(Q) -» IFs(ß) n {holomorphic functions}, where Q is a smooth bounded strictly pseudoconvex domain and Ws(ü) is the Sobolev space of functions with 5 square-integrable Received by the editors October 24, 1983 and, in revised form, May 29, 1984. 1980 Mathematics Subject Classification. Primary 32A25, 32H10.