Abstract

One of the major open problems in nonhomoheneous Harmonic analysis is to find the necessary and sufficient conditions for the boundedness of two weights inequality for singular integrals. In this  talk, we will first discuss the famous two weights problem for the Hilbert transform and outline our proof that settles this longstanding open problems. We then discuss some of the main difficulties for higher dimensional singular integrals. Some of the important applications to other fields will be also discussed, in particular an application of our T1 theorem for the Cauchy transform in the complex plane settles the open problem: the embedding problem for the model spaces.