Carleson measures for the Drury–Arveson Hardy space and other Besov–Sobolev spaces on complex balls

Abstract

For 0 ≤ σ < 1/2 we characterize Carleson measures µ for the analytic Besov-Sobolev spaces B σ 2 on the unit ball Bn in C n by the discrete tree condition

Multipliers and essential norm on the Drury-Arveson space

“…Interpolation problems for multipliers and model theory related to the Drury-Arveson space have been intensely studied over the past decade or so [4,5,10,12]. Recently, Arcozzi, Rochberg and Sawyer gave a characterization of the multipliers in terms of Carleson measures for H 2 n [1]. In [7], Costea, Sawyer and Wick proved a corona theorem for M. More recently, we showed in [9] that for each f ∈ M and each 1…”

“…Because of its connection to various topics in operator theory, e.g. the von Neumann inequality for commuting row contractions, H 2 n has been the subject of intense recent studies [1]- [7], [9], [10], [12].…”

Operator Theory and Function Theory in Drury–Arveson Space and Its Quotients

“…12. Some harmonic analysis in H measures of B σ p (B d ) for ranges of p and σ that include p = 2, σ = 1/2 was obtained in [13], [115] and [122]. The reader is referred to these papers for additional details.…”

“…It is interesting that the value σ = 1/2 seems to play a critical role in some approaches, while in others it does not. For example, the characterization of Carleson measures given in [13,Theorem 23] holds for 0 ≤ σ < 1/2, the case σ = 1/2 is handled differently. On the other hand, the methods of E. Tchoundja [115] work for the range σ ∈ (0, 1/2], but not for σ > 1/2.…”

“…The Drury-Arveson space also fits into a family of function spaces which have been of interest in harmonic analysis (see, e.g., [13,43,122]). In this subsection it is assumed that d < ∞.…”