Georgia Institute of Technology College of Sciences

For dimensions n greater than or equal to 3, and integers N greater than 1, there is a distribution of points P in a unit cube [0,1]^{n}, of cardinality N, for which the discrepancy function D_N associated with P has an optimal Exponential Orlicz norm. In particular the same distribution will have optimal L^p norms, for 1 < p < \infty. The collection P is a random digit shift of the examples of W.L. Chen and M. Skriganov.

A recent conjecture in harmonic analysis that was exploredin the past 20 years was the A_2 conjecture, that is the sharp bound onthe A_p weight characteristic of a Calderon-Zygmund singular integraloperator on weighted L_p space. The non-sharp bound had been knownsince the 1970's, but interest in the sharpness was spurred recentlyby connections to quasiconformal mappings and PDE. Finally solved infull by Hytonen, the proof is complex, intricate and lengthy. A new "simple" approach using local mean oscillation and positive operatorbounds was published by Lerner. We discuss this and some recent progress in the area, including our new proof for spaces of homogeneoustype, in the style of Lerner (Joint work with Armen Vagharshakyan).

We consider a so-called "One sided bump conjecture", which gives asufficient condition for two weight boundedness of a Calderon-Zygmundoperator. The proof will essentially use the Corona decomposition, which isa main tool for a first proof of $A_2$ (also, $A_p$ and $A_p-A_\infty$)conjecture. We will focus on main difficulty, that does not allow to get afull proof of our one sided bump conjecture.

Weighted norm inequalities for singular integral operators acting on scalar weighted L^p is a classical topic that goes back to the 70's with the seminal work of R. Hunt, B. Muckenhoupt, and R. Wheeden. On the other hand, weighted norm inequalities for singular integral operators with matrix valued kernels acting on matrix weighted L^p are poorly understood and results (obtained by F. Nazarov, S. Treil, and A. Volberg in the late 90's) are only known for the situation when the kernel is essentially scalar valued.In this talk, we discuss matrix weighted norm inequalities for matrix valued dyadic paraproducts and discuss the possibility of using our results and a recent result of T. Hytonen to obtain concrete weighted norm inequalities for singular integral operators with matrix kernels acting on matrix weighted L^p. This is joint work with Hyun-Kyoung Kwon and Sandra Pott.