# zbMATH — the first resource for mathematics

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
Full Text: