Powers of the Szegő kernel and Hankel operators on Hardy spaces.

Abstract

In this paper we study the action of certain integral operators on spaces of holomorphic functions on some domains in Cn: These integral operators are defined by using powers of the Szego kernel as integral kernel. We show that they act like differential operators, or like pseudo-differential operators of not necessarily integral order. These operators may be used to give equivalent norms for the Besov spaces Bp of holomorphic functions. As a consequence we prove that, when 1 p < 1; the small Hankel operators hf on Hardy and weighted Bergman spaces are in the Schatten class Sp if and only if the symbol f belongs to Bp: The type of domains we deal with are the smoothly bounded strictly pseudoconvex domains in Cn and a class of complex ellipsoids in Cn: Our results for strictly pseudo-convex domains depend on Fefferman's expansion of the Szego kernel. In this case, its powers act like a power of the derivation in the normal direction. The ellipsoids we consider are the simplest examples of domains of finite type. In this case, the symmetries of the domains can be exploited to use methods of harmonic analysis and describe the pseudo-differential operators involved.