Nondoubling Calderón-Zygmund theory -a dyadic approach-

Abstract

Given a measure $μ$ of polynomial growth, we refine a deep result by David and Mattila to construct an atomic martingale filtration of $\mathrm{supp}(μ)$ 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 $A_2$-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 $L_p$ 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 Calderón-Zygmund decomposition which unifies those by Tolsa and López-Sánchez/Martell/Parcet.