Dyadic harmonic analysis beyond doubling measures

descriptionPublicationkeyboard_double_arrow_right Article , Preprint 01 Dec 2014Embargo end date: 01 Jan 2012 Spain English Publisher:Elsevier BVJournal:Advances in Mathematics, volume 267, pages 44-93 (issn: 0001-8708,

Copyright policy )Funded by:EC | CZOSQP

Authors: Luis Daniel López-Sánchez; José María Martell; Javier Parcet;

doi: 10.1016/j.aim.2014.08.001 , 10.48550/arxiv.1211.6291

arXiv: 1211.6291

handle: 10261/197839

Dyadic harmonic analysis beyond doubling measures

- Summary
- Subjects
- Metrics

Abstract

We characterize the Borel measures $μ$ on $\mathbb{R}$ for which the associated dyadic Hilbert transform, or its adjoint, is of weak-type $(1,1)$ and/or strong-type $(p,p)$ with respect to $μ$. Surprisingly, the class of such measures is strictly bigger than the traditional class of dyadically doubling measures and strictly smaller than the whole Borel class. In higher dimensions, we provide a complete characterization of the weak-type $(1,1)$ for arbitrary Haar shift operators, cancellative or not, written in terms of two generalized Haar systems and these include the dyadic paraproducts. Our main tool is a new Calderón-Zygmund decomposition valid for arbitrary Borel measures which is of independent interest.

Country

Spain

Related Organizations

Spanish National Research Council
Spain
Institute of Mathematical Sciences
Spain

Keywords

Singular and oscillatory integrals (Calderón-Zygmund, etc.), Maximal functions, Littlewood-Paley theory, dyadic paraproducts, non-doubling measures, generalized Haar systems, Non-doubling measures, Dyadic paraproducts, Nontrigonometric harmonic analysis involving wavelets and other special systems, Dyadic cubes, Dyadic Hilbert transform, Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.), dyadic cubes, 42B20, 42B25, 42C40, 42C10, Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type, Haar shift operators, Mathematics - Classical Analysis and ODEs, Classical Analysis and ODEs (math.CA), FOS: Mathematics, Generalized Haar systems, Calderón-Zygmund decomposition, dyadic Hilbert transform

Impact byBIP!

	selected citations These citations are derived from selected sources. This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).	26
	popularity This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.	Top 10%
	influence This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).	Top 10%
	impulse This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.	Top 10%