| The thesis provides answers to two conjectures (in one case partial and in the other final) in the area of weighted inequalities for Singular Integral Operators. It is interesting to understand the mapping properties of these operators. In particular, if the kernel satisfies certain size and smoothness conditions, it is well established that Singular Integral Operators map Lebesgue spaces Lp(dx) into Lp(dx) for 1 < p < infinity. If we want to replace Lebesgue measure by a general weight wdx, where w is a nonnegative locally integrable function, Lp(w) bounds can be obtained if w belongs to the so called Muckenhoupt Ap class. The latest results were established in the early seventies. The novelty of this thesis resides in proving sharp dependence of the operator norm on the Ap constant associated to the weight w. The question was known as the A 2 conjecture. In joint work with my advisor, M. Lacey, and one of his collaborators, S. Petermichl, we were able to prove the conjecture for the special case of dyadic Singular Integral Operators. The full conjecture has been proved by T. Hytonen. Another interesting question considers the end point p = 1. The open problem was known as the Muckenhoupt-Wheeden conjecture. The thesis provides a counterexample to this conjecture in the dyadic setting. The full conjecture has been answered in the negative in a later result with my coauthor C. Thiele, hence closing a problem that has been open since the early seventies. |
