Sharp $A_2$ estimates of Haar shifts via Bellman function

Abstract

We use the Bellman function method to give an elementary proof of a sharp weighted estimate for the Haar shifts, which is linear in the $A_2$ norm of the weight and in the complexity of the shift. Together with the representation of a general Calderón--Zygmund operator as a weighted average (over all dyadic lattices) of Haar shifts, (cf. arXiv:1010.0755v2[math.CA], arXiv:1007.4330v1[math.CA]) it gives a significantly simpler proof of the so-called the $A_2$ conjecture. The main estimate is a very general fact about concave functions, which can be very useful in other problems of martingale Harmonic Analysis. Concave functions of such type appear as the Bellman functions for bounds on the bilinear form of martingale multipliers, thus the main estimate allows for the transference of the results for simplest possible martingale multipliers to more general martingale transforms. Note that (although this is not important for the $A_2$ conjecture for general Calderón--Zygmund operators) this elementary proof gives the best known (linear) growth in the complexity of the shift.

23pp

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Cited by in corpus (6)

• Commutators in the Two-Weight Setting
• Weighted estimates for dyadic paraproducts and t-Haar multipiers with complexity (m,n)
• On the A_2 inequality for Calderon-Zygmund Operators
• A Bellman function proof of the $L^2$ bump conjecture
• Sharp bounds and T1 theorem for Calderón-Zygmund operators with matrix kernel on matrix weighted spaces
• Bounds for Calderón-Zygmund operators with matrix $A_2$ weights