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

Summary: The Diameter Space, A Restriction of the
Drury-Arveson-Hardy Space
N. Arcozzi, R. Rochberg, and E. Sawyer
1. Introduction and Summary
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 Cn
; its associated diameter space, Dn, is a space of real analytic
functions on the unit ball in Rn
. 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


Source: Arcozzi, Nicola - Dipartimento di Matematica, UniversitÓ di Bologna


Collections: Mathematics