Dyadic harmonic analysis beyond doubling measures

Abstract

We characterize the Borel measures $\mu$ 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 $\mu$. 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.

References in corpus (1)

• Sharp estimates of martingale transforms in higher dimensions and applications to the Ahlfors-Beurling operator

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• Haar bases on quasi-metric measure spaces, and dyadic structure theorems for function spaces on product spaces of homogeneous type
• Quantitative ergodic theorems for actions of groups of polynomial growth