A simple proof of the $A_2$ and two-weight conjectures for Calder\'on-Zygmund operators

Andrei K. Lerner

Abstract:
The so-called $A_2$ conjecture was completely solved in 2010
by T. Hyt\"onen. The proof was based on a rather difficult
representation of a general Calder\'on-Zygmund operator in
terms of the Haar shift operators.
Recently we found a different proof of this result completely
avoiding such representations. The method of the proof is
flexible enough, and it allows to obtain a control of an
arbitrary Calder\'on-Zygmund operator  by rather simple
dyadic positive operators. In particular, we solve also
the two-weight bump conjecture.