Nondoubling Calderón–Zygmund theory: a dyadic approach
| Autor: | José M. Conde-Alonso, Javier Parcet |
|---|---|
| Přispěvatelé: | Ministerio de Economía y Competitividad (España) |
| Rok vydání: | 2018 |
| Předmět: |
Pointwise
Mathematics::Functional Analysis Pure mathematics Partial differential equation Applied Mathematics General Mathematics 010102 general mathematics Banach space 020206 networking & telecommunications 02 engineering and technology 01 natural sciences Noncommutative geometry Harmonic analysis symbols.namesake Fourier analysis 0202 electrical engineering electronic engineering information engineering symbols 0101 mathematics Martingale (probability theory) Analysis Mathematics |
| Zdroj: | Digital.CSIC. Repositorio Institucional del CSIC instname |
| ISSN: | 1531-5851 1069-5869 |
| Popis: | Given a measure μ of polynomial growth, we refine a deep result by David and Mattila to construct an atomic martingale filtration of supp (μ) which provides the right framework for a dyadic form of nondoubling harmonic analysis. Despite this filtration being highly irregular, its atoms are comparable to balls in the given metric—which in turn are all doubling—and satisfy a weaker but crucial form of regularity. Our dyadic formulation is effective to address three basic questions:(i)A dyadic form of Tolsa’s RBMO space which contains it.(ii)Lerner’s domination and A-type bounds for nondoubling measures.(iii)A noncommutative form of nonhomogeneous Calderón–Zygmund theory. Our martingale RBMO space preserves the crucial properties of Tolsa’s original definition and reveals its interpolation behavior with the L scale in the category of Banach spaces, unknown so far. On the other hand, due to some known obstructions for Haar shifts and related concepts over nondoubling measures, our pointwise domination theorem via sparsity naturally deviates from its doubling analogue. In a different direction, matrix-valued harmonic analysis over noncommutative L spaces has recently produced profound applications. Our analogue for nondoubling measures was expected for quite some time. Finally, we also find a dyadic form of the Calderón–Zygmund decomposition which unifies those by Tolsa and López-Sánchez/Martell/Parcet. The authors thank Xavier Tolsa for bringing to our attention David and Mattila’s construction and for several fruitful discussions on the content of this paper. Both authors were partially supported by CSIC Project PIE 201650E030 and also by ICMAT Severo Ochoa Grant SEV-2015-0554, and the first named author was supported in part by ERC Grant 32501. |
| Databáze: | OpenAIRE |
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