An $A_2$ Theorem for One-Sided Calderón-Zygmund Operators

TL;DR

This work establishes the first quantitative one-sided $A_2$ bound for upward-mapping Calderón-Zygmund operators in dimension one, proving a norm bound of $\|T\|_{L^2(w)\to L^2(w)} \lesssim [w]_{A_2^{\uparrow}}\,(1+\log [w]_{A_2^{\uparrow}})$ and highlighting the role of directionality in weighted theory. The authors develop a two-weight testing framework, implement a dyadic/disbalanced-Haar decomposition, and analyze the hard-term via a causal pivotal condition and sparse paraproducts, culminating in a localized weighted weak-type theory that supports extrapolation. The result hinges on a careful blend of local testing, endpoint estimates, and a one-dimensional geometric structure, and it opens the door to a localized weighted theory for one-sided operators with potential future removal of the logarithmic loss. The work also demonstrates that Nazarov–Pérez–Treil–Volberg machinery can yield sharp quantitative bounds beyond the reach of dyadic domination methods in asymmetric settings. Overall, it provides a foundational quantitative framework for one-sided CZOs, with explicit bounds and a localized theory that may inform higher-dimensional extensions and sharpened endpoint results.

Abstract

We present a proof of the one-sided $A_2$ theorem in dimension one, with a logarithmic loss. This theorem concerns one-sided Calderón-Zygmund operators (CZOs) whose kernels $K(x,y)$ vanish whenever $x < y$. These operators are bounded on $L^2(w)$ provided that the weight $w$ belongs to the one-sided class $A_2^{\uparrow}$. The argument reduces the norm estimate to testing on indicator functions via a two-weight testing theorem. By combining this with the weak-type $(1,1)$ estimate in the one-sided setting and an extrapolation theorem, we obtain the one-sided $A_2^{\uparrow}$ theorem with a logarithmic loss. We develop a localized theory on fixed intervals by introducing adapted weight classes and showing that the same quantitative bound holds locally for one-sided operators.

Theorems & Definitions (28)

• Theorem 1
• Definition 1
• Definition 2
• Theorem 2
• Lemma 1
• Lemma 2
• proof
• Lemma 3
• proof
• Proposition 1