Talk Title: Weighted estimates via arbitrarily “smooth” weights
Abstract: Hytönen’s celebrated A_2 theorem states that the norm of any Calderón-Zygmund operator over any weighted Lebesgue space L^2(w) is dominated by the first power of the A_2 characteristic of the weight w. This estimate is known to be sharp for “large” Calderón-Zygmund operators, such as the Hilbert transform. One can then ask whether the exponent 1 in this estimate remains optimal if one replaces the usual A_2 characteristic by a “fattened” variant which instead of averaging over intervals involves averaging against Poisson kernels. In this talk, we discuss how counterexamples involving arbitrarily “smooth” weights show that the answer to this question, and in fact to the analogous question for any p strictly between 1 and infinity, is positive. We rely on ideas originally introduced by J. Bourgain, and further developed by F. Nazarov to disprove Sarason’s conjecture. This is joint work with Professor Sergei Treil (Brown University).