Carleson measures for analytic Besov spaces

Abstract

We characterize Carleson measures for the analytic Besov spaces. The problem is first reduced to a discrete question involving measures on trees which is then solved. Applications are given to multipliers for the Besov spaces and to the determination of interpolating sequences. The discrete theorem is also applied to analysis of function space on trees.

Carleson Measures for the Bloch Space

“…Luecking [26,29], characterized the q-Carleson measures for the space A p α , 0 < p, q < ∞. A good number of results about p-Carleson measures for Besov spaces and spaces of Dirichlet type of analytic functions have been obtained by different authors (see, e. g., [4], [5], [20], [21], [22], [33], [39], [42], [43], and [44]). …”

“…We shall use an argument which is closely related to that used by Arcozzi, Rochberg and Sawyer in the proof of [5,Theorem 7]. Let p, p and µ be as in Theorem 7.…”

“…Arcozzi, Rochberg and Sawyer [5] obtained a characterization of the p-Carleson measures for the weighted Besov space…”

Hardy's inequalities for monotone functions on partly ordered measure spaces

“…The discrete case N is a particular case of a tree, and can be found in [8]. Weights for a general tree were studied, without the monotonicity condition, in [1,9]. It is easy to prove that a weight satisfying case 3 of corollary 2.3 must necessarily be in B p (N) (uniformly) on each geodesic (see [8]), but the converse is not true in general.…”

Sarason Conjecture on the Bergman Space

“…Now recall that |f | 2 R 2 u = (P + f,g |u|) 2 , and use the previous inequality in (4.9) together with the Cauchy-Schwartz inequality and the estimates (4.5) and (4.6) to obtain…”