Weighted Norm Inequalities for Some Singular Integral Operators

For bounded Lebesgue measurable functionsα,βon the unit circle,Sα, β=αP++βP−is called a singular integral operator, whereP+is an analytic projection andP−is a co-analytic projection. We study one-weighted norm inequalities ofSα, βonL2(W). We introduce a classHSrof weights withr=|α−β|/‖α−β‖∞in order to characterize those weights. For example, we show thatSα, βis bounded with respect to a weightWif and only ifWbelongs toHSror |α−β|W≡0. Ifris a nonzero constant, thenHSris just a well-known class of weights due to H. Helson and G. Szegő (Ann. Mat. Pura Appl.51(1960), 107–138). Moreover we study the Koosis-type problem of two weights ofSα, βand get very simple necessary and sufficient conditions for such weights.