Atomic Decomposition and Carleson Measures for Weighted Mixed Norm Spaces
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Atomic Decomposition and Carleson Measures for Weighted Mixed Norm Spaces José Ángel Peláez1
· Jouni Rättyä2 · Kian Sierra1,2
Received: 14 June 2018 © Mathematica Josephina, Inc. 2019
Abstract The purpose of this paper is to establish an atomic decomposition for functions in the p,q weighted mixed norm space Aω induced by a radial weight ω in the unit disc admitting a two-sided doubling condition. The obtained decomposition is further applied p,q to characterize Carleson measures for Aω , and bounded differentiation operators p,q (n) (n) s acting from Aω to L μ , induced by a positive Borel measure μ, on D (f) = f the full range of parameters 0 < p, q, s < ∞. Keywords Atomic decomposition · Carleson measure · Doubling weight · Mixed norm space Mathematics Subject Classification 46E15 · 47B38
1 Introduction and Main Results Let H(D) denote the space of all analytic functions in the open unit disc D = {z ∈ C : |z| < 1} of the complex plane C. Further, let T stand for the boundary of D and D(a, r ) = {z : |z − a| < r } for the Euclidean disc of center a ∈ C and radius r > 0. For 0 < r < 1 and f ∈ H(D), set
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José Ángel Peláez [email protected] Jouni Rättyä [email protected] Kian Sierra [email protected]
1
Departamento de Análisis Matemático, Universidad de Málaga, Campus de Teatinos, 29071 Málaga, Spain
2
University of Eastern Finland, P.O.Box 111, 80101 Joensuu, Finland
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J. Á. Peláez et al.
1/ p 2π 1 it p M p (r , f ) = | f (r e )| dt , 0 < p < ∞, 2π 0 M∞ (r , f ) = sup | f (z)|. |z|=r
An integrable function ω : D → [0, ∞) is called a weight. It is radial if ω(z) = ω(|z|) 1 for all z ∈ D. For a radial weight ω, write ω(z) = |z| ω(s) ds for all z ∈ D. For 0 < p ≤ ∞, 0 < q < ∞ and a radial weight ω, the weighted mixed norm p,q space Aω consists of f ∈ H(D) such that
q
f A p,q = ω
p,q
1
q
M p (r , f )ω(r ) dr < ∞.
0 p
If q = p, then Aω coincides with the Bergman space Aω induced by the weight p ω. As usual, Aα denotes the weighted Bergman space induced by the standard radial 2 weight (1−|z| )α . Weighted mixed norm spaces arise naturally in operator and function theory, for example, in the study of the boundedness, compactness and Schatten classes 1 of the generalized Hilbert operator Hg ( f )(z) = 0 f (t)g (t z) dt acting on Bergman spaces [16,20]. if there exists a constant C = C(ω) ≥ 1 such A weight ω belongs to the class D ) for all 0 ≤ r < 1. Moreover, if there exist K = K (ω) > 1 and that ω(r ) ≤ C ω( 1+r 2 C = C(ω) > 1 such that 1−r , 0 ≤ r < 1, ω(r ) ≥ C ω 1− K
(1.1)
q Weights ω belonging to D = D q are called doubling. ∩D then we write ω ∈ D. The classes of weights D and D emerge from fundamental questions in operator theory: recently the first two authors showed that the weighted Bergman projection Pω , induced by a radial weight ω, is bounded from L ∞ to the Bloch space B = { f ∈ and further, it is bounded H(D) : supz∈D | f (z)|(1 − |z|) < ∞} if and only if ω ∈ D, and onto if and only if ω ∈ D [21]. The primary aim of this study is to estab
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