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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

BMO-regularity in lattices of measurable functions on spaces of homogeneous type


Author: D. V. Rutsky
Translated by: the author
Original publication: Algebra i Analiz, tom 23 (2011), nomer 2.
Journal: St. Petersburg Math. J. 23 (2012), 381-412
MSC (2010): Primary 42B35; Secondary 42B20
Published electronically: January 24, 2012
MathSciNet review: 2841677
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Abstract: Let $ X$ be a lattice of measurable functions on a space of homogeneous type $ (S, \nu )$ (for example, $ S = \mathbb{R}^n$ with Lebesgue measure). Suppose that $ X$ has the Fatou property. Let $ T$ be either a Calderón-Zygmund singular integral operator with a singularity nondegenerate in a certain sense, or the Hardy-Littlewood maximal operator. It is proved that $ T$ is bounded on the lattice $ \bigl (X^\alpha \mathrm {L}_1^{1 - \alpha }\bigr )^\beta $ for some $ \beta \in (0, 1)$ and sufficiently small $ \alpha \in (0, 1)$ if and only if $ X$ has the following simple property: for every $ f \in X$ there exists a majorant $ g \in X$ such that $ \log g \in \mathrm {BMO}$ with proper control on the norms. This property is called $ \mathrm {BMO}$-regularity. For the reader's convenience, a self-contained exposition of the $ \mathrm {BMO}$-regularity theory is developed in the new generality, as well as some refinements of the main results.


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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: http://dx.doi.org/10.1090/S1061-0022-2012-01201-X
PII: S 1061-0022(2012)01201-X
Keywords: $\BMO$-regularity, Muckenhoupt weights, singular integral operators, maximal function
Received by editor(s): October 21, 2010
Published electronically: January 24, 2012
Article copyright: © Copyright 2012 American Mathematical Society