Powers of the Szegö Kernel and Hankel Operators on Hardy Spaces

In this paper we study the action of certain integral operators on spaces of holo-morphic functions on some domains in Cn: These integral operators are defined by using powers of the Szegö kernel as integral kernel. We show that they act like differential operators, or like pseudo-differential operators of not necessarily in-tegral 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 pseudo-convex 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 Szegö kernel. In this case, its powers act like a power of the derivation in the normal di-rection. 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. 1. Basic Notation and Statement of the Main Results Let D D f z: .z / < 0 g be a smoothly bounded domain in Cn; with 2 C1. ND/ and r 6D 0 on @D: For p> 0 letHp.D / denote the Hardy space of holomorphic functions on D; with norm given by kf kpHp.D /:D sup ">0 Z .w/D" jf.w/jp d.w/; where .w /:D ¡.w / is equivalent to the distance to the boundary and d is the surface measure. Let P./S and S denote the Szegö projection and the Szegö ker-nel respectively: