On Dirichlet Spaces With a Class of Superharmonic Weights

Abstract

Abstract. In this paper, we investigate Dirichlet spaces D µ with superharmonic weights induced by positive Borel measures µ on the open unit disk. We establish the AlexanderTaylor-Ullman inequality for D µ spaces and we characterize the cases where equality occurs. We de ne a class of weighted Hardy spaces H µ via the balayage of the measure µ. We show that D µ is equal to H µ if and only if µ is a Carleson measure for D µ . As an application, we obtain the reproducing kernel of D µ when µ is an in nite sum of point mass measures. We consider the boundary behavior and inner-outer factorization of functions in D µ . We also characterize the boundedness and compactness of composition operators on D µ .

A Hardy–Littlewood theorem for Bergman spaces

“…We are going to show that the assumption σ < 2 in the proposition above is best possible. In other words, when σ ≥ 2, we can construct a counterexample of µ such that the induced weight function ω σ does not satisfy condition (3). Thus our condition (3) is sensitive enough to detect the changing behaviour of ω σ at the critical point σ = 2.…”

“…These weights were considered in [3,2] and were used there to study certain weighted Dirichlet spaces and the associated Möbius invariant function spaces.…”

“…Proposition 13. Let ν be a finite positive Borel measure on the unit circle T. Then ω ν satisfies both conditions (1) and (3).…”

“…Thus our condition in Theorem B is fine enough to detect the cut-off point σ = 2 for the weights ω σ . Note that the weight functions ω σ (z) were used in [3,2] to study Dirichlet type spaces and their corresponding Möbius invariant counterparts.…”

Double integral estimates for Besov type spaces and their applications

“…This double integral characterization was also used to study Hankel type operators; see [12,Section 4]. We refer to [6,Section 4] and [29] for some recent results associated with double integral estimates for some analytic function spaces. See A. Reijonen [26] for recent results on Besov type spaces induced by some radial weights.…”