Nondoubling Calder\'on-Zygmund theory -a dyadic approach-

TL;DR

This paper develops a dyadic framework for nondoubling harmonic analysis on measures with polynomial growth, extending classical concepts and addressing key questions in the field.

Contribution

It introduces a dyadic approach to nondoubling harmonic analysis, including a martingale RBMO space, domination results, and a noncommutative Calderón-Zygmund theory.

Findings

Constructed an atomic martingale filtration for nondoubling measures.

Developed a dyadic RBMO space with interpolation properties.

Established a dyadic Calderón-Zygmund decomposition.

Abstract

Given a measure $\mu$ of polynomial growth, we refine a deep result by David and Mattila to construct an atomic martingale filtration of $\mathrm{supp}(\mu)$ 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 $A_2$-type bounds for nondoubling measures. iii) A noncommutative form of nonhomogeneous Calder\'on-Zygmund theory. Our martingale RBMO space preserves the crucial properties of Tolsa's original definition and reveals its interpolation behavior with the $L_p$ 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 $L_p$ 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 Calder\'on-Zygmund decomposition which unifies those by Tolsa and L\'opez-S\'anchez/Martell/Parcet.