Dyadic harmonic analysis beyond doubling measures

We characterize the Borel measures μ   on RR for which the associated dyadic Hilbert transform, or its adjoint, is of weak-type (1,1)(1,1) and/or strong-type (p,p)(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)(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.