An elementary proof of the A 2 bound

Abstract

Abstract. A martingale transform T , applied to an integrable locally supported function f, is pointwise dominated by a positive sparse operator applied to |f|, the choice of sparse operator being a function of T and f. As a corollary, one derives the sharp A p bounds for martingale transforms, recently proved by Thiele-Treil-Volberg, as well as a number of new sharp weighted inequalities for martingale transforms. The (very easy) method of proof (a) only depends upon the weak-L 1 norm of maximal truncations of martingale transforms, (b) applies in the vector valued setting, and (c) has an extension to the continuous case, giving a new elementary proof of the A 2 bounds in that setting.

Domination of multilinear singular integrals by positive sparse forms

“…Such a pointwise domination principle, albeit in a slightly weaker sense, appears explicitly for the first time in the proof of the theorem by Lerner . We also point out the recent improvements by Lacey and Lerner , and the analogue for multilinear Calderón‐‐Zygmund operators obtained independently by Conde‐Alonso and Rey and by Lerner and Nazarov . Most recently, Bernicot, Frey and Petermichl extend this approach to nonintegral singular operators associated with a second‐order elliptic operator, lying outside the scope of classical Calderón–Zygmund theory.…”

Borderline weak‐type estimates for singular integrals and square functions

“…With this domination result at hand, we can restrict our attention to sparse operators in proving Theorem 1.6. Concerning square functions, a variant of the argument in [11] shows that the intrinsic square function is dominated by a sum of at most 3 d sparse square functions defined by…”

On Entropy Bumps for Calderón-Zygmund Operators

“…For this Theorem, see Theorem 5.2 of [7]. As a consequence, we see that it su ces to prove our main Theorems for sparse operators.…”

“…With the recent argument of one of us, [7], this reduction now applies more broadly, namely it applies to (a) Calderón-Zygmund operators on Euclidean spaces as stated above; (b) non-homogeneous Calderón-Zygmund operators; and (c) general martingales. See [7] for some details.…”

“…We use the recent inequality [7], which gives pointwise control of a Calderón-Zygmund operator by a sparse operator. S is a sparse operator if Sf = Q∈Q f Q Q , where S is a collection of dyadic cubes for which…”