Large BMO spaces vs interpolation

Conde-Alonso, Jose; Mei, Tao; Parcet, Javier

doi:10.2140/apde.2015.8.713

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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

DOI: 10.2140/apde.2015.8.713

Abstract

We introduce a class of BMO spaces which interpolate with $L_{p}$ 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 $Σ_{a}$ and $Σ_{b}$ 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 $(Σ_{a}, Σ_{b})$ so that the resulting space interpolates with $L_{p}$ 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 $(Σ_{a}, Σ_{b})$ . 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 $e^{\pm | x |^{α}}$ for any $α > 0$ or ${(1 + | x |^{β})}^{- 1}$ for any $β ≳ n^{3 ∕ 2}$ . 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 $(Σ_{a}, Σ_{b})$ 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

Vol. 8, No. 3, 2015