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Haar shift operators have served as important models for singular integral operators and have been particularly fruitful in the theory of weighted estimates. A prominent example is the dyadic version of the Hilbert transform. Given a Borel measure mu on R, one can define an adapted Haar system which is orthonormal in L^2(mu). The class of Borel measures for which the associated dyadic Hilbert transform is of weak-type (1,1) is characterized in {LSMP-C}. Surprisingly, the class of such measures is strictly bigger than the doubling class and is strictly contained in the Borel class. The dual class characterizes the weak-type (1,1) of the adjoint of the dyadic Hilbert transform. In higher dimensions, analogous characterizations of the weak-type (1,1) for generalized Haar shift operators are also obtained. Paraproducts and their adjoints arise as important examples.
Secondly, in the operator-valued setting, operators with noncommuting kernels have attracted some interest in recent years. In this situation estimates in L_p fail to hold for p not equal 2 even when underlying measure is doubling. In the doubling scenario weak-type (1,1) estimates are obtained in {HLSMP-NC} in terms of a row/column decomposition of the input function. The row/column parts are given as complementary triangular truncations of the function.
In both problems the main tools at disposal are Calderon-Zygmund decompositions tailored for each context. In this talk we will review these results and, if time allows, sketch the problem in nondoubling-noncommuting setting.