A pointwise estimate for positive dyadic shifts and some applications

Abstract

Abstract. We prove a pointwise estimate for positive dyadic shifts of complexity m which is linear in the complexity. This can be used to give a pointwise estimate for Calderón-Zygmund operators and to answer a question posed by A. Lerner. Several applications to weighted estimates for both multilinear Calderón-Zygmund operators and square functions are discussed.

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.…”

An elementary proof of the A 2 bound

“…The Theorem below is a stronger version of Lerner's inequality [L4], and its extension in [CAR,Corollary A1]. Moreover, it only assumes the Dini condition, while prior approaches [HLP,L4,CAR] require 1/t in the Dini integral be replaced by (log 2/t)/t. To explain the conclusion of the Theorem, we say that D is a dyadic grid of R d , if D is a collection of cubes Q ⊂ R d so that for each integer k ∈ Z, the cubes D k := {Q ∈ D : |Q| = 2 −kd } partition R d , and these collections form an increasing filtration on R d .…”

Some Remarks on the Pointwise Sparse Domination

“…Recall that the family of cubes S is called η-sparse, 0 < η ≤ 1, if for every cube Q ∈ S, there exists a measurable set E Q ⊂ Q such that |E Q | ≥ η|Q|, and the sets {E Q } Q∈S are pairwise disjoint.Localization and sparseness are two main ingredients which make sparse bounds especially effective in quantitative weighted norm inequalities.The literature about sparse bounds is too extensive to be given here in more or less adequate form. We mention only that sparse bounds for Calderón-Zygmund operators can be found in [7,15,16,18,19,20]. Also, there are several general sparse domination principles [5,6,8,19].In [19], a sparse domination principle was obtained in terms of the grand maximal truncated operatordefined for a given operator T .2010 Mathematics Subject Classification.…”