Radial Two Weight Inequality for Maximal Bergman Projection Induced by a Regular Weight
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Radial Two Weight Inequality for Maximal Bergman Projection Induced by a Regular Weight Taneli Korhonen1
´ ´ 2 ¨ Jouni Ratty ¨ a¨ 1 ¨ Jose´ Angel Pelaez
Received: 27 May 2019 / Accepted: 10 February 2020 / © The Author(s) 2020
Abstract It is shown in quantitative terms that the maximal Bergman projection ż f pζ q|Bzω pζ q|ωpζ q dApζ q, Pω` pf qpzq “ D
p
p
is bounded from Lν to Lη if and only if ¨ ˛1 ¨ p ¸p1 ˛ p11 ż1˜ żr η p s q ω p s q ˚ ‹ ¯p ds ` 1‚ ˝ ds ‚ ă 8, sup ˝ ´ş 1 1 p 0ăr ă1 0 r ν p s q ω p t q dt s provided ω, ν, η are radial regular weights. A radial weight σ is regular if it satisfies ş1 σ pr q— r σ pt q dt {p1 ´ r q for all 0 ď r ă 1. It is also shown that under an appropriate additional hypothesis involving ω and η, the Bergman projection Pω and Pω` are simultaneously bounded. Keywords Bergman projection ¨ Bergman space ¨ Regular weight ¨ Two weight inequality Mathematics Subject Classification (2010) 30H20 ¨ 46E30 ¨ 47 Dedicated to Fernando P´erez-Gonz´alez on the occasion of his retirement This research was supported in part by Ministerio de Econom´ıa y Competitivivad, Spain, projects PGC2018-096166-B-100 and MTM2017-90584-REDT; La Junta de Andaluc´ıa, project FQM210; Academy of Finland project no. 268009. Taneli Korhonen
[email protected] ´ Jos´e Angel Pel´aez [email protected] Jouni R¨atty¨a [email protected] 1
University of Eastern Finland, P.O.Box 111, 80101 Joensuu, Finland
2
Departamento de An´alisis Matem´atico, Universidad de M´alaga, Campus de Teatinos, 29071 M´alaga, Spain
T. Korhonen et al.
1 Introduction and Main Results A function ω : D Ñ r0, 8q, integrable over the unit disc D, is called a weight. It is radial p if ωpzq “ ωp|z|q for all z P D. For 0 ă p ă 8 and a weight ω, the Lebesgue space Lω consists of complex-valued measurable functions f in D such that
ˆż }f }Lpω “
˙1 p |f pzq|p ωpzq dApzq ă 8,
D
where dApzq “ dxπdy denotes the element of the normalized Lebesgue area measure on D. p p The weighted Bergman space Aω is the space of analytic functions in Lω , and is equipped p with the corresponding Lω -norm. If the norm convergence in the Hilbert space A2ω implies the uniform convergence on compact subsets of D, the point evaluations are bounded linear functionals on A2ω . Therefore there exist reproducing Bergman kernels Bzω P A2ω such that f pzq “
xf, Bzω yA2ω
ż “ D
f pζ qBzω pζ qωpζ q dApζ q,
z P D,
f P A2ω .
The Hilbert space A2ω is a closed subspace of L2ω , and hence the orthogonal projection from L2ω to A2ω is given by ż f pζ qBzω pζ qωpζ q dApζ q, z P D. Pω pf qpzq “ D
The operator Pω is the Bergman projection. In this paper we will characterize the radial two-weight inequality
}Pω` pf q}Lpη ď C }f }Lpν ,
f P Lpν ,
(1.1)
ş for the maximal Bergman projection Pω` pf qpzq “ D f pζ q|Bzω pζ q|ωpζ q dApζ q under certain smoothness requirements on the three radial weights involved. The question of when (1.1) is satisfied is an open problem even in the very particular case ω “ ν “ η if no preliminary hypotheses is imposed on the radial weight. Two
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