Multipliers of Bergman spaces into Lebesgue spaces

Abstract

Let U be the open unit disk in the complex plane endowed with normalized Lebesgue measure m. will denote the usual Lebesgue space with respect to m, with 0<p<+∞. The Bergman space consisting of the analytic functions in will be denoted . Let μ be some positivefinite Borel measure on U. It has been known for some time (see [6] and [9]) what conditions on μ are equivalent to the estimate: There is a constant C such thatprovided 0<p≦q.

Multipliers and integration operators on Dirichlet spaces

“…or, equivalently, if there exists C > 0 such that The q-Carleson measures for the Bergman space A p α have also been characterized by Luecking [24] in the case p ≤ q, and in [25] and [27] in the case q < p (see [35,Theorem D]). …”

Extended Cesàro operators on Bergman spaces

“…In Section 3, we consider the equivalent characterizations, the point evaluation functionals, and the interpolation sequences on L p a,w (B). We generalize some results from [3,4,8,11,14,18,19]. The last section, Section 4, contains the main theorems of the paper, where we will find a sufficient and necessary condition on g ∈ H (B) for which T g is bounded (respectively compact) from L p a,w (B) to L q a,w (B) for all possible 0 < p, q < ∞.…”

“…When w ≡ 1 the assertion (25) is just the main result of [8]. Taking normal weights into account, the proof of (25) uses the same approach with the modification that the lemma on [8, p. 128] should be replaced by the following lemma, Lemma 3.…”

Order bounded weighted composition operators mapping into the dirichlet type spaces

“…The inclusion (1.1) for 1 ≤ p ≤ 2 can be proved by Riesz-Thorin Theorem and the case 0 < p < 1 has been proved by Flett in [5]. The inclusion (1.1) follows by a classical result due to Littlewood and Paley [13], see also the proof by Luecking [14]. It is also well known that, for every p, the Hardy space H p is contained in the Bergman space A 2p.…”