Analysis & PDE

  • Anal. PDE
  • Volume 8, Number 3 (2015), 713-746.

Large BMO spaces vs interpolation

Jose Conde-Alonso, Tao Mei, and Javier Parcet

Full-text: Open access

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.

Article information

Source
Anal. PDE, Volume 8, Number 3 (2015), 713-746.

Dates
Received: 9 July 2014
Revised: 18 January 2015
Accepted: 6 March 2015
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.apde/1510843098

Digital Object Identifier
doi:10.2140/apde.2015.8.713

Mathematical Reviews number (MathSciNet)
MR3353829

Zentralblatt MATH identifier
1316.42027

Subjects
Primary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B35: Function spaces arising in harmonic analysis 46L52: Noncommutative function spaces 60G46: Martingales and classical analysis

Keywords
nondoubling measures BMO spaces interpolation martingales noncommutative harmonic analysis classical harmonic analysis Calderón–Zygmund theory

Citation

Conde-Alonso, Jose; Mei, Tao; Parcet, Javier. Large BMO spaces vs interpolation. Anal. PDE 8 (2015), no. 3, 713--746. doi:10.2140/apde.2015.8.713. https://projecteuclid.org/euclid.apde/1510843098


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