Estimates for derivatives of holomorphic functions in pseudoconvex domains

Abstract

In this paper we will prove several Sobolev type inequalities for holomorphic functions in strictly pseudoconvex domains. To a large extent, this work was motivated by the following problem. Let D be a strictly pseudoconvex domain and let M be the intersection with D of a submanifold of a neighborhood o f / ) which intersects 0D transversally. Let f be a holomorphic function on M lying in some L p class (e.g. a Hardy or Bergman class). We want to find a holomorphic extension of f to D satisfying an optimal L p estimate on D. The case p --c~ was solved by Henkin [14] who showed that a bounded holomorphic function on M is necessarily the restriction of a bounded holomorphic function on D. For certain weighted Bergman spaces on M, optimal L p estimates for extensions were obtained by Cumenge [9] in the case l < p < o o and by Beatrous [4] in the case 0 < p < c~. More precisely, letting 6(z) denote the distance from z to 0D, the following result was proved. Let f be a holomorphic function on M satisfying