| This dissertation was inspired by T(b) theorems originating in the work of David, Journe, and Semmes [7] and the expository work by Auscher, Hoffmann, Muscalu, Thiele, and Tao [2] who developed these theorems in a dyadic setting.;The T(1) Theorem of David and Journe (1984) [6] gives necessary and sufficient conditions for a singular integral operator to be bounded on the space L2( Rn ). The hypothesis that T(1) is a function of bounded mean oscillation (BMO) is sometimes difficult to verify directly. In certain instances this can be alleviated by replacing the function 1 by a suitably normalized function b whose mean is bounded away from zero by a number gamma called the accretivity constant.;The conclusion of a typical T(b) theorem states that a dyadic singular integral operator is bounded on L 2, where the norm estimate grows with a power of gamma. One goal of this work is to track the power of gamma in the norm estimate, and then to show that the power on gamma is sharp by providing an example, or at least keep the gap between the two power laws minimal.;Each T(b) theorem in this thesis is proven in a global case and a local case, where for each interval I a separate bI is given. In all instances; the local case is much harder to prove; the mean of each function bI is controlled on its support but not on subintervals of its support. To get around this, stopping-time arguments and multiscale analysis must be used.;A second goal of this work is to fill in some of the gaps in the theory of dyadic T(b) theorems. To that end, a b-weighted BMO-norm, a variant of BMO, is formulated. It is defined using b-adapted Haar wavelets, rbI .;Given b ∈ L2 ( R ) such that |[b]I| > 1 for all I, and given a function f, define f BMOb =supI 1I 12 J⊆I &vbm0;&angl0;f,rbJ&angr0;&vbm0; 212 .;We prove both global and local theorems in which we compare the BMO and b-weighted BMO norms. Each comparison theorem has a T( b)-style theorem which follows as a natural corollary. |
