On weak weighted estimates of the martingale transform and a dyadic shift

Abstract

We consider several weak type estimates for singular operators using the Bellman function approach. We disprove the A1 conjecture, which stayed open after Muckenhoupt-Wheeden's conjecture was disproved by Reguera-Thiele.By M w we will denote the dyadic maximal function of w, that is M w(x) = sup{ w J : J ∈ D, J ∋ x}. Then w ∈ A 1 with "norm" Q means that M w ≤ Q · w a. e. , and Q = [w] A 1 is the best constant in this inequality.Recall that a martingale transform is an operator given by T ε ϕ = J∈D ε J ∆ J ϕ . It is convenient to use Haar function h J associated with dyadic interval J,Sometimes it is more convenient to use the Haar functions H J normalized in L ∞ : H J = |J| 1/2 h J . In this notations, the martingale transform ψ of a function ϕ is

Sharp $$A_{1}$$ Weighted Estimates for Vector-Valued Operators

“…However, the fact that ϕ(t) ≃ t was disproved in [26], furthermore, in [18] it was established that ϕ(t) ≃ t log(e + t), and consequently the estimate in [19] is sharp.…”

On the sharp upper bound related to the weak Muckenhoupt-Wheeden conjecture

“…On the other hand, in [7], it was shown for the martingale transform (and explained how to transfer the result to the Hilbert transform case) that the dependence of [w] A 1 in the weighted weak type (1, 1) inequality cannot in general be made better than [w] A 1 log 1/5 (e + [w] A 1 ), thus disproving the weak Muckenhoupt-Wheeden conjecture. Later, in [8], the power of the logarithm was improved to 1/3 (this was again done for the martingale transform).…”

“…Summarizing the results in [5,7,8], if we denote by α H the best possible exponent for which the inequality…”

Martingale transform and Square function: some weak and restricted weak sharp weighted estimates

“…But the question was whether the logarithmic term can be dropped. The Bellman function construction in [16], [17] showed that linear estimate is false, in fact, in these preprints a sequence of weights w, was proved to exist such that [w] A 1 → ∞ but T : L 1 (w) → L 1,∞ (w) ≥ c[w] A 1 (log[w] A 1 ) 1/3 . Operator T was either a martingale transform, or the dyadic shift, or the Hilbert transform.…”