Abstract:
The celebrated A_2 condition by Hunt Muckenhoupt and Wheeden in 1973 gives a beautiful characterization for the boundedness of the Hilbert transform H on the weighted space L^2(w). It was quickly realized that in the corresponding two weight situation the nalogous two weight A_2 condition is not sufficient for the boundedness of H on two weighted spaces. It has been believed that the characterization would be a profound extension of the famous T1 nd Tb Theorem of David and Journe. Over the years, partial esults ave been given to the conjecture assuming extra side conditions on the weights, but none of them is a complete characterization. In this alk we will discuss the history of the famous two weight problem and ive an outline of our proof which gives a complete characterization toward the conjecture in terms of Tb language.