Carleson--Buckley measures beyond the scope of $A_\infty$ and their applications

Carleson measures are ubiquitous in Harmonic Analysis. In the paper of Fefferman--Kenig--Pipher in 1991 an interesting class of Carleson measures was introduced for the need of regularity problems of elliptic PDE. These Carleson measures were associated with $A_\infty$ weights. In discrete setting (we need exactly discrete setting here) they were studied by Buckley's, where they were associated with dyadic $A\infty^d$. Our goal here is to show that such Carleson--Buckley measures (in discrete setting) exists for virtually any positive function (weight). Of course some modification is needed, because it is known that Carleson property of Buckley's measure are equivalent to the weight to be in $A_\infty^d$. However a very natural generalization of those facts exist for weights more general $A_\infty$, and of course, in a special case of $A_\infty^d$ it gives Buckley's results. Our generalization of Buckley's inequality beyond the scope of $A_\infty$ allows us to prove the so-called bump conjecture.