H p -Functions on Strictly Pseudoconvex Domains

Abstract

Introduction. In this paper we study some questions concerning HP-functions on strictly pseudoconvex domains in CN. For the most part, the results we obtain are analogues of well known theorems in one variable. The first section is devoted to a few preliminaries about the Hardy classes on domains in CN. In the second section we prove an approximation theorem: 0 (D) is dense in HP(D), 1 < p < oo, if D CCN is a strictly pseudoconvex domain with sufficiently smooth boundary. The proof of this theorem is based on the integral formula devised by Henkin and by Ramirez. In the third section we prove that the kernel of the Henkin-Ramirez integral belongs to certain HP spaces, a result parallel to the fact that the one dimensional Cauchy kernel 1/(e 9-z) belongs, for fixed 9, to HP(zA), i the open unit disc, provided p&(0,1). A consequence of this result is that certain multivariate integrals analogous to Cauchy-Stieltjes integrals define elements of HP. The fourth section of the paper is devoted to a result on conjugate functions. We show that if f = u + iv E d (D), D c C CN smoothly bounded but not necessarily strictly pseudoconvex, then provided v (30) = 0 for some fixed 30 c D, there is an estimate II v II p < C II uI p, 1 < p < oo. This result is obtained by integrating in a suitable way the well known theorem of Marcel Riesz on conjugate functions in the disc. It is related to a recent result of Siu and Range. In the final section, we prove that if f= u + iv E& (D) with IuI bounded, then IfIP admits a pluriharmonic majorant for each p c (0, oo).