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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Vector-valued boundedness of harmonic analysis operators

Author: D. V. Rutsky
Translated by: the author
Original publication: Algebra i Analiz, tom 28 (2016), nomer 6.
Journal: St. Petersburg Math. J. 28 (2017), 789-805
MSC (2010): Primary 45B20; Secondary 45B25, 46E30
Published electronically: October 2, 2017
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Abstract: Let $ S$ be a space of homogeneous type, $ X$ a Banach lattice of measurable functions on $ S \times \Omega $ with the Fatou property and nontrivial convexity, and $ Y$ some Banach lattice of measurable functions with the Fatou property. Under the assumption that the Hardy-Littlewood maximal operator $ M$ is bounded both in $ X$ and $ X'$, it is proved that the boundedness of $ M$ in $ X (Y)$ is equivalent to its boundedness in  $ \mathrm L_{s}(Y)$ for some (equivalently, for all) $ 1 < s < \infty $. With  $ S = \mathbb{R}^n$, the last condition is known as the Hardy-Littlewood property of $ Y$ and is related to the  $ \mathrm {UMD}$ property. For lattices $ X$ with nontrivial convexity and concavity, the UMD property implies the boundedness of all Calderón-Zygmund operators in $ X (Y)$ and is equivalent to the boundedness of a single nondegenerate Calderón-Zygmund operator. The  $ \mathrm {UMD}$ property of $ Y$ is characterized in terms of the  $ \mathrm A_{p}$-regularity of both  $ \mathrm L_{\infty } (Y)$ and  $ \mathrm L_{\infty } (Y')$. The arguments are based on an improved version of the divisibility property for  $ \mathrm A_{p}$-regularity.

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Additional Information

D. V. Rutsky
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, 191023 St. Petersburg, Russia

Keywords: $\mathrm{A}_p$-regularity, $\BMO$-regularity, Hardy--Littlewood maximal operator, Calder\'on--Zygmund operators
Received by editor(s): July 25, 2016
Published electronically: October 2, 2017
Article copyright: © Copyright 2017 American Mathematical Society

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