An $A_2$ Theorem for One-Sided Calder\'on-Zygmund Operators

Abstract

We present a proof of the one-sided $A_2$ theorem in dimension one, with a logarithmic loss. This theorem concerns one-sided Calder\'on-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.