Carleson measures for the generalized Bergman spaces via a T(1)-type theorem

Abstract

In this paper, we give a new characterization of Carleson measures for the generalized Bergman spaces. We show first that this problem is equivalent to a T(1)-type problem. Using an idea of Verdera (see [V]), we introduce a sort of curvature in the unit ball adapted to our kernel. We establish a good λ inequality which then yields the solution of this T(1)-type problem.

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.…”

“…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. However, using different techniques, A.…”

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

“…Among the most profound results in this setting are Carleson's interpolation and corona theorems [39,40] [13,115] and [122]. The reader is referred to these papers for additional details.…”

On Weighted Toeplitz, Big Hankel Operators and Carleson Measures

“…if 0 ≤ n−sp < 1, then µ ∈ CM p s if and only if µ satisfies condition (2.5). This last condition can be expressed in terms of nonisotropic Riesz capacities of open sets in S. In the extreme case p = 2, and n − 2s = 1, which corresponds to the Drury-Arveson space [6,22] have obtained characterizations of the corresponding Carleson measures. Recently [25] (see also [16]) have characterized the Carleson measures for any H 2 s in terms of a T 1-type condition.…”

“…We will denote by CM p s the set of these measures, and by µ CM p s the norm of the embedding B p s ⊂ L p (µ). The Carleson measures for Besov and Hardy-Sobolev spaces have been studied by many authors ( [2,[4][5][6]11,13,14,16,22,25] among others). See Sect.…”