Vol. 8, No. 3, 2015

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ISSN: 1948-206X (e-only)
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Large BMO spaces vs interpolation

Jose M. Conde-Alonso, Tao Mei and Javier Parcet

Vol. 8 (2015), No. 3, 713–746
Abstract

We introduce a class of BMO spaces which interpolate with Lp and are sufficiently large to serve as endpoints for new singular integral operators. More precisely, let (Ω,Σ,μ) be a σ-finite measure space. Consider two filtrations of Σ by successive refinement of two atomic σ-algebras Σa and Σb having trivial intersection. Construct the corresponding truncated martingale BMO spaces. Then, the intersection seminorm only leaves out constants and we provide a quite flexible condition on (Σa,Σb) so that the resulting space interpolates with Lp in the expected way. In the presence of a metric d, we obtain endpoint estimates for Calderón–Zygmund operators on (Ω,μ,d) under additional conditions on (Σa,Σb). These are weak forms of the “isoperimetric” and the “locally doubling” properties of Carbonaro, Mauceri and Meda which admit less concentration at the boundary. Examples of particular interest include densities of the form e±|x|α for any α > 0 or (1 + |x|β)1 for any β n32. A (limited) comparison with Tolsa’s RBMO is also possible. On the other hand, a more intrinsic formulation yields a Calderón–Zygmund theory adapted to regular filtrations over (Σa,Σb) without using a metric. This generalizes well-known estimates for perfect dyadic and Haar shift operators. In contrast to previous approaches, ours extends to matrix-valued functions (via recent results from noncommutative martingale theory) for which only limited results are known and no satisfactory nondoubling theory exists so far.

Keywords
nondoubling measures, BMO spaces, interpolation, martingales, noncommutative harmonic analysis, classical harmonic analysis, Calderón–Zygmund theory
Mathematical Subject Classification 2010
Primary: 42B20, 42B35, 46L52, 60G46
Milestones
Received: 9 July 2014
Revised: 18 January 2015
Accepted: 6 March 2015
Published: 3 June 2015
Authors
Jose M. Conde-Alonso
Instituto de Ciencias Matemáticas
CSIC-UAM-UC3M-UCM Consejo Superior de Investigaciones Científicas
C/ Nicolás Cabrera 13-15
28049 Madrid
Spain
Tao Mei
Department of Mathematics
Wayne State University
656 W. Kirby St.
Detroit, MI 48202
United States
Department of Mathematics
Baylor University
1311 S 5th St.
Waco, TX 77401
United States
Javier Parcet
Instituto de Ciencias Matemáticas
CSIC-UAM-UC3M-UCM Consejo Superior de Investigaciones Cientí{}ficas
C/ Nicolás Cabrera 13-15
28049 Madrid
Spain