Bellman approach to the one-sided bumping for weighted estimates of Calderón--Zygmund operators

Abstract: We give again (see also arXiv:1112.0676) a proof of weighted estimate of any Calder\'on-Zygmund operator. This is under a universal sharp sufficient condition that is weaker than the so-called bump condition. Bump conjecture was recently solved independently and simultaneously by A. Lerner (arXiv:1202.1860) and Nazarov--Reznikov-Treil-Volberg (arXiv:1202.2406). The latter paper uses the Bellman approach. Immediately a very natural and seemingly simple question arises how to strengthen the bump conjecture result by weakening its assumptions in a natural symmetric way. This is what we are dealing with here. However we meet an unexpected and, in our opinion, deep obstacle, that allows us to achieve only a partial result. The result of the present article is slightly stronger than the one in arXiv:1112.0676. Our proof consists of two main parts: reduction to a simple model operator, construction of Bellman function for estimating this simple operator. The newer feature is that the domain of definition of our Bellman function is infinitely dimensional.