The Diameter Space, A Restriction of the Drury-Arveson-Hardy Space

We consider Carleson measures, Hankel matrices, and interpolation of values on certain reproducing kernel Hilbert spaces which we call the diameter spaces. We begin by reviewing results for the classical Hardy space which we denote DAH1 and its associated diameter space. Our n−dimensional analog of the Hardy space is the Drury-Arveson-Hardy space, DAHn, a space of holomorphic functions on the unit ball in C; its associated diameter space, Dn, is a space of real analytic functions on the unit ball in R. We will see that, as is true in one dimension, some questions which are difficult for DAHn simplify substantially for Dn. Thus Dn may be a useful starting point for analysis of problems on DAHn. In the next section we recall background and establish notation. In the following section we recall the results for n = 1 and make some comparisons between DAH1 and D2. In Section 4 we consider positive definite Hankel operators on these space. We find that the classical Hardy space results extend to DAHn and also extend to Besov-Sobolev potential spaces intermediate between those spaces and Dirichlet spaces. In the final section we will show how some of the results about interpolating sequences for Dn can be localized and the local versions combined to obtain characterizations of some interpolating sequences for DAHn which accumulate to the boundary with controlled geometry.