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A sharp estimate on the norm of the martingale transform. (English) Zbl 0951.42008
While the boundedness of singular integral operators on \( L^2_w\) for \( w\) in the Muckenhoupt class \( A_2\) has been known for many years, the precise dependence of the operator bound on the \( A_2\) constant still is not known. Haar multipliers supply a natural model for singular integral operators on \( \mathbb R\) and techniques developed in the Haar context often can be adapted to prove corresponding estimates on \( \mathbb R\). Here the author proves that for the Haar multipliers – otherwise known as martingale transforms, \( T_r: f\mapsto \sum_{I\in {\mathcal D}[0,1]} r(I)\langle f,h_I\rangle h_I\), one has \( \|T_r f\|_{L^2_w}\leq c\|w\|_{A_2}\|f\|_{L^2_w}\). The constant \( c\) does not depend on \( f\) or \( w\). In other words, the operator norm of \(T_r\) depends linearly on \( \|w\|_{A_2}\). In the definition of \( T_r\), \({\mathcal D}[0,1] \) denotes the dyadic intervals in \([0,1]\), \(\{r_I\}\) is a sequence of \(\pm 1\)’s indexed by dyadic intervals, and the Haar functions are \(h_I= (\chi_{I_l}-\chi_{I_r})/\sqrt{|I|}\) which form an orthonormal basis for \(L^2[0,1]\). Among the main tools are: (i) the Haar product representation of the weight, \( w= \prod(1+c_I h_I)\), (ii) a bilinear weighted Carleson estimate based on related estimates of Nazarov-Treil-Volberg, and (iii) a renormalization of the Haar functions making them orthogonal with respect to the inner product in \(L^2_w\). A related result shows that the dyadic square function is also bounded on \(L^2_w\) with a bound depending linearly on \(\|w\|_{A_2}\). This is better than the known bound depending on \(\|w\|_{A_2}^{3/2}\) for the Littlewood-Paley square function, but in many ways the dyadic square function behaves like the Hardy-Littlewood maximal function whose operator bound depends linearly on \(\|w\|_{A_2}\).

MSC:
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B25 Maximal functions, Littlewood-Paley theory
60G46 Martingales and classical analysis
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