Multipliers on Spaces of Analytic Functions

In the paper we find, for certain values of the parameters, the spaces of multipliers ( H(p, q, α), H(s, t, β) ) and ( H(p, q, α), ls ) , where H(p, q, α) denotes the space of analytic functions on the unit disc such that (1 − r)Mp(f, r) ∈ Lq( dr 1−r ). As corollaries we recover some new results about multipliers on Bergman spaces and Hardy spaces. §0. Introduction. Given two sequence spaces X and Y , we denote by (X,Y ) the space of multipliers from X into Y , that is the space of sequences of complex numbers (λn) such that (λnan) ∈ Y for (an) ∈ X. When dealing with spaces of analytic functions defined on the open unit disc D we associate to each analytic function f(z) = ∑∞ n=0 anz n the corresponding sequence of Taylor coefficients (an). In this sense any space of analytic functions is regarded as a sequence space and it makes sense to study multipliers acting on different classes of spaces such as Hardy spaces, Bergman spaces and so on. During the last decade lots of results were obtained (see [AS, BST, DS1, M, MP1, MP2, S2, SW]). Recently the interest on similar questions has been renewed and some new results on multipliers on Hardy and Bergman spaces have been achieved (see [W, MP3, JP, MZ, V]). The aim of this paper is to study spaces of multipliers acting on certain general classes of analytic functions, denoted by H(p, q, α), which consists of functions on the unit disc such that ( ∫ 1 0 (1 − r)αq−1Mq p (f, r)dr )1/q < ∞. The definition of these classes goes back to the work of Hardy and Littlewood (see [HL1,HL2]) and they were intensively studied for different reasons and by several authors. The reader is referred to the papers [DRS, F1, F2, MP1, S1, Sh] for information and properties on the spaces. There are two different techniques used in the paper. On one hand the use of a general theorem on operators acting on H(p, q, α) for 0 < p ≤ 1 which allows us to find ( H(p, q, α), H(s, t, β) ) and ( H(p, q, α), l ) for the cases 0 < p, q ≤ 1 and 1 ≤ s, t ≤ ∞ and also for 0 < p ≤ 1 ≤ q although only for particular cases of s and t. In particular we can get a proof of the recent theorem, due to M. Mateljevic and 1991 Mathematics Subject Classification. 42A45.