Entropy conditions in two weight inequalities for singular integral operators

A new type of “bumping” of the Muckenhoupt A2A2 condition on weights is introduced. It is based on bumping the entropy integral of the weights. In particular, one gets (assuming mild regularity conditions on the corresponding Young functions) the bump conjecture proved in [20] and [25] as a corollary of entropy bumping. But our entropy bumps cannot be reduced to the bumping with Orlicz norms in the solution of bump conjecture, they are effectively smaller. Henceforth, we get somewhat stronger results than the one that solves the bump conjecture in [20] and [25]. New results concerning one sided bumping conjecture are obtained. All the results hold in the general non-homogeneous situation.