Analysis Seminar: "Weighted inequalities for t-Haar multipliers"
Abstract: The t-Haar multipliers are dyadic operators analogue to pseudo-differential operators where the trigonometric functions have been replaced by the Haar functions and the symbol is now a function of the space variable and of the dyadic frequency (dyadic intervals) instead of the Fourier frequency. The symbols of the t-Haar multipliers depend on the real number t, a weight w, a choice of signs and the dyadic intervals. We will discuss necessary and sufficient conditions on the triple of weights (u,v,w) (and the parameter t) for the uniform (with respect to the choice of signs ) boundedness of t-Haar multipliers from L^2(u) into L^2(v). Our result recovers all previous known weighted estimates for the martingale transform (t=0) including Wittwer's one-weight estimates as well as the celebrated two-weight estimates of Nazarov, Treil and Volberg. It also recovers and improves upon the known L^2-estimates for the unsigned t-Haar multipliers of Katz and Pereyra.
Host: Walton Green