Edgar Lee Stout

H p -Functions on Strictly Pseudoconvex Domains

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).

Radó’s theorem for CR-functions

We establish an analogue of a classical theorem of Rad6 for CRfunctions on certain hypersurfaces in CN .

Holomorphic approximation on compact, holomorphically convex, real-analytic varieties

Every continuous function on a compact, holomorphically convex, real-analytic subset of C N can be approximated uniformly by functions holomorphic on the set.

Totally real imbeddings and the universal covering spaces of domains of holomorphy: Some examples

We begin with a review of the known examples of compact totally realn-dimensional submanifolds of ℂn. We then construct some new families of examples, including some which are simply connected. We conclude by using these examples to construct bounded domains of holomorphy in ℂn whose universal covering spaces are not biholomorphically equivalent to domains in ℂn.

Differentiable interpolation on the polydisc

If f is a function of class Lp on the torus and if the subset E of is the peak set of a function of class Ap (UN ), then there is a function g in A p − l, α (UN for all α between zero and one with g = f on E. The peak sets of functions of class Ap (Un ) lie in locally closed interpolation submanifolds of the torus of class Lp . but an example shows that they need not lie in closed interpolation submanifolds.