Oscar Blasco

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.

Interpolation between vector-valued Hardy spaces

Abstract Let 0 p ⩽ ∞ and let H p h (X) denote the space of X -valued harmonic functions on the half-space with boundary values almost everywhere and Poisson maximal function in L p (R n ), and H p (X) the closure of the X -valued analytic polynomials on the disc under the norm given by sup 0 r f r ∥ p . It is shown that if 0 p 0 , p 1 θ 1 p = (1 − θ) p 0 + θ p 1 , then ( H p0 h (X 0 ), H p1 h (X 1 )) 0 = H p h (X 0 ) . With the restriction p 1 we prove ( H p0 h (X 0 ), H p1 h (X 1 )) 0,p = H h p (X 0, p ) . A counterexample for the case p = 1 is given for the case of real interpolation. It is also proved that H p0 (X 0 ), H p1 (X 1 )) 0 is, in general, smaller than H p (X 0 ) . Finally BMO ( X ) is also considered as the end point for interpolation.

Boundary values of functions in vector-valued hardy-spaces and geometry on banach spaces

Abstract The spaces of boundary values of vector-valued functions in Hardy spaces defined by either holomorphic functions on the disk or harmonic functions with maximal function in L p are characterized in terms of vector-valued measures of bounded p -variation. We extend to the case p = 1 a characterization of the Radon-Nikodym property based on the existence of limits at the boundary for harmonic functions with maximal function in L 1 . In the case 0 p ⩽ 1 we find the UMD property as the necessary and sufficient condition to make the spaces defined by maximal function and by conjugate Poisson kernel coincide.

Theorems of Hardy and Paley for vector-valued analytic functions and related classes of Banach spaces

We investigate the classes of Banach spaces where analogues of the classical Hardy inequality and the Paley gap theorem hold for vector-valued functions. We show that the vector-valued Paley theorem is valid for a large class of Banach spaces (necessarily of cotype 2) which includes all Banach lattices of cotype 2, all Banach spaces whose dual is of type 2 and also the preduals of C*-algebras. For the trace class S1 and the dual of the algebra of all bounded operators on a Hilbert space a stronger result holds; namely, the vector-valued analogue of the Fefferman theorem on multipliers from H into / ; in particular for the latter spaces the vector-valued Hardy inequality holds. This inequality is also true for every Banach space of type > 1 (Bourgain). 0. INTRODUCTION If f = Z Oajeiit is an analytic trigonometric polynomial, then J?O~~~~~~~~~J Elajl ( j 1) 1 < C1 r If (t) I dt ,

Remarks on Vector-valued BMOA and Vector-valued Multipliers

In this paper we consider the vector-valued interpretation of the space BMOA defined in terms of Carleson measures and analyze the relationship with the one defined in terms of oscillation. We study the space of multipliers between Hp and BMOA in the vector-valued setting. This leads us to the consideration of some geometric properties depending upon the validity of certain inequalities due to Littlewood and Paley on the g-function for vector-valued functions.

Operators on weighted Bergman spaces

We describe the boundedness of a linear operator from Bp(ρ) = {f : D → C analytic : (∫ D ρ(1 − |z|) (1 − |z|) |f(z)| dA(z) )1/p < ∞} , for 0 < p ≤ 1 under some conditions on the weight function ρ, into a general Banach space X by means of the growth conditions at the boundary of certain fractional derivatives of a single X-valued analytic function. This, in particular, allows us to characterize the dual of Bp(ρ) for 0 < p < 1 and to give a formulation of generalized Carleson measures in terms of the inclusion B1(ρ) ⊂ L(D,μ). We then apply the result to the study of multipliers, Hankel operators and composition operators acting on Bp(ρ) spaces. 1991 Math. Subject Class. : Primary 47B38, 47B35 Secondary 42A45

(0 < p \leq 1)

We give de Leeuw-type transference theorems for bilinear multipliers. In particular, it is shown that bilinear multipliers arising from regulated functions m ( ξ , η ) in ℝ × ℝ can be transferred to bilinear multipliers acting on 𝕋 × 𝕋 and ℤ × ℤ . The results follow from the description of bilinear multipliers on the discrete real line acting on L p -spaces.

and applications

Abstract We consider spaces of analytic functions depending on a weight p(t)⩾-0, t ϵ [0, 1), defined by certain conditions, namely 1. (1) M p (F′,r) = O(p(1 −r) (1 −r)) , 2. (2) M p (Ft″, r) = O(p(1−r) (1−r) 2 ) , 3. (3) ∝ 1 0 (p(1−r) (1 −r))M p (F,r)dr . We study boundary value problems and duality for these spaces depending on the properties of the weight function.

Bilinear multipliers and transference

Let X be a complex Banach space and let Bp(X) denote the vector-valued Bergman space on the unit disc for 1 ≤ p < ∞ . A sequence (Tn)n of bounded operators between two Banach spaces X and Y defines a multiplier between Bp(X) and Bq(Y ) (respect. Bp(X) and `q(Y )) if for any function f(z) = ∑∞ n=0 xnz n in Bp(X) we have that g(z) = ∑∞ n=0 Tn(xn)z n belongs to Bq(Y ) (respect. (Tn(xn))n ∈ `q(Y )). Several results on these multipliers are obtained, some of them depending upon the Fourier or Rademacher type of the spaces X and Y . New properties defined by the vector-valued version of certain inequalities for Taylor coefficients of functions in Bp(X) are introduced.

Spaces of analytic functions on the disc where the growth of Mp(F, r) depends on a weight

Let 2 p 0s uch thatfHp(X) (� f(0)� p + λ (1 −| z| 2 ) p−1 � f � (z)� p dA(z)) 1/p ,f or all f ∈ H p (X). Applications to embeddings between vector-valued BMOA spaces defined via Poisson integral or Carleson measures are provided.