Harmonic conjugates on Bergman spaces induced by doubling weights
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Harmonic conjugates on Bergman spaces induced by doubling weights José Ángel Peláez1
· Jouni Rättyä2
Received: 24 July 2019 / Revised: 24 July 2019 / Accepted: 18 March 2020 © Springer Nature Switzerland AG 2020
Abstract if there exists C = C(ω) ≥ 1 such that A radial weight ω belongs to the class D 1 1 ω(s) ds for all 0 ≤ r < 1. Write ω ∈ Dˇ if there exist constants r ω(s) ds ≤ C 1+r 2 K = K (ω) > 1 and C = C(ω) > 1 such that ω(r ) ≥ C ω 1 − 1−r K for all 0 ≤ r < 1. These classes of radial weights arise naturally in the operator theory of Bergman spaces induced by radial weights (Peláez and Rättyä in Bergman projection induced by radial weight, 2019. arXiv:1902.09837). Classical results by Hardy and Littlewood (J Reine Angew Math 167:405–423, 1932) and Shields and Williams (Mich Math J 29(1):3– 25, 1982) show that the weighted Bergman space of harmonic functions is not closed \ Dˇ and 0 < p ≤ 1. In this paper we establish by harmonic conjugation if ω ∈ D p \ Dˇ and sharp estimates for the norm of the analytic Bergman space Aω , with ω ∈ D 0 < p < ∞, in terms of quantities depending on the real part of the function. It is also shown that these quantities result equivalent norms for certain classes of radial weights. Keywords Harmonic conjugate · Bergman space · Doubling weights
This research was supported in part by Ministerio de Ciencia Innovación y universidades, Spain, Projects PGC2018-096166-B-100 and MTM2017-90584-REDT; La Junta de Andalucía, Project FQM210; Academy of Finland 286877.
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José Ángel Peláez [email protected] Jouni Rättyä [email protected]
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Departamento de Análisis Matemático, Universidad de Málaga, Campus de Teatinos, 29071 Málaga, Spain
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University of Eastern Finland, P.O. Box 111, 80101 Joensuu, Finland 0123456789().: V,-vol
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J. Á. Peláez, J. Rättyä
1 Introduction and main results Let H(D) and h(D) denote the spaces of analytic and harmonic functions in the unit disc D = {z ∈ C : |z| < 1}, respectively. For 0 < p ≤ ∞, the Hardy space H p consists of f ∈ H(D) for which f H p = sup M p (r , f ) < ∞ 0 1 and C = C(ω) > 1 such ˇ These classes of ∩ D. that ω(r ) ≥ C ω 1 − 1−r for all 0 ≤ r < 1, and set D = D K radial weights arise naturally in the operator theory of weighted Bergman spaces [11]. For instance, the class D describes the radial weights such that the Littlewood-Paley formula p f A p | f (0)| p + | f (z)| p (1 − |z|) p ω(z) d A(z), f ∈ H(D), (1.1) ω
D
holds [11, Theorem 5], and there exists a constant C = C(ω, p) > 0 such that p | f (0)| p + | f (z)| p (1 − |z|) p ω(z) d A(z) ≤ C f A p , f ∈ H(D), (1.2) D
ω
[11, Theorem 6]. These results, together with [11, Theorems 1 if and only if ω ∈ D \ Dˇ and 3] related to bounded Bergman projections on L ∞ , show that weights in D
Harmonic conjugates on Bergman spaces induced by doubling…
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induce in a sense essentially smaller Bergman spaces than the standard radial weights (1 − |z|2 )α with −1 < α < ∞. A classical problem on a space X of harmonic func
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