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Remarks on $ \mathrm{\mathbf A}_p$-regular lattices of measurable functions


Author: D. V. Rutsky
Translated by: the author
Original publication: Algebra i Analiz, tom 27 (2015), nomer 5.
Journal: St. Petersburg Math. J. 27 (2016), 813-823
MSC (2010): Primary 42B20; Secondary 46B42
DOI: https://doi.org/10.1090/spmj/1418
Published electronically: July 26, 2016
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Abstract: A Banach lattice $ X$ of measurable functions on a space of homogeneous type is said to be $ \mathrm {A}_p$-regular if every $ f \in X$ admits a majorant $ g \geq \vert f\vert$ belonging to the Muckenhoupt class $ \mathrm {A}_p$ with suitable control on the norm and the constant. It is well known that the $ \mathrm {A}_p$-regularity of the order dual $ X'$ of $ X$ implies the boundedness of the Hardy-Littlewood maximal operator on $ X^{\frac 1 p}$ for $ p > 1$ (equivalently, the $ \mathrm {A}_1$-regularity of this lattice), provided that $ X'$ is norming for $ X$. This result admits a partial converse and an interesting characterization: the $ \mathrm {A}_1$-regularity of $ X^{\frac 1 p}(\ell ^{p})$ implies the $ \mathrm {A}_p$-regularity of $ X'$, and for lattices $ X$ with the Fatou property these conditions are equivalent to the $ \mathrm {A}_1$-regularity of both $ X^{\frac 1 p}$ and $ \big (X^{\frac 1 p}\big )'$. As an application, an exact form of the self-duality of $ \mathrm {BMO}$-regularity is obtained, the $ \mathrm {A}_q$-regularity of the lattices $ \mathrm {L}_{\infty }(\ell ^p)$ for all $ 1 < p,q < \infty $ is established, and in many cases it is shown that the $ \mathrm {A}_1$-regularity of both $ Y$ and $ Y'$ yields the $ \mathrm {A}_1$-regularity of $ Y(\ell ^s)$ for all $ 1 < s < \infty $, which implies the boundedness of the Calderón-Zygmund operators in $ Y(\ell ^s)$.


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Additional Information

D. V. Rutsky
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Fontanka 27, St. Petersburg 191023, Russia
Email: rutsky@pdmi.ras.ru

DOI: https://doi.org/10.1090/spmj/1418
Keywords: $\mathrm{A}_p$-regularity, $\mathrm{BMO}$-regularity, Hardy--Littlewood maximal operator, Calder\'on--Zygmund operators
Received by editor(s): February 10, 2015
Published electronically: July 26, 2016
Article copyright: © Copyright 2016 American Mathematical Society