Publ. Mat. 52 (2008), 267–287 SHARP GROWTH ESTIMATES FOR DYADIC b-INPUT T (b) THEOREMS

The following deals with the T (b) theorems of David, Journé, and Semmes [7] considered in a dyadic setting. We find sharp growth estimates for a global and a local dyadic T (b) Theorem. We use multiscale analysis and Haar wavelets in the local case. 1. Background The T (1) theorem of David and Journe ́ [6] gives a necessary and sufficient condition for a singular integral operator to be bounded on the space L2(Rn). Both the properties of the operator T and cancellation properties of its associated kernel K are considered. The hypotheses of the T (1) theorem are sometimes difficult to verify directly, but in certain instances this problem can be somewhat alleviated by replacing the constant function 1 by a function b whose mean is bounded away from zero. We consider growth conditions on b and T (b) which are necessary and sufficient for T to be bounded sharply, and find the dependency of the bounds on T by the bounds on b. There have been several reformulations of the T (b) problem. In 1990, Christ [3] gave an L ∞ criterion for L2-boundedness of a singular integral operator and posed questions about the application of the theorems to analytic capacity. In their 2002 paper, Nazarov, Treil, and Volberg [12] proved an equivalent condition to that of Christ, only they considered a nondoubling measure, making their result valid for a nonhomogeous space. Furthermore, they allowed the image of the operator to be in the space of bounded mean oscillation (BMO), rather than in L∞. In 2003, Tolsa [14] used the non-doubling T (b) theorem in [12] in his answer to the Painleve ́ problem and his proof of the semiadditivity of analytic capacity of a compact set E ⊂ C. In 2002, Auscher, Hofmann, Muscalu