On the Separated Bumps Conjecture for Calderón-Zygmund Operators

Abstract

Let $\sigma (dx) = \sigma (x)dx$ and $w (dx)= w (x)dx$ be two weights with non-negative locally finite densities on $\mathbb R^{d}$, and let $1 \lt p \lt \infty$. A sufficient condition for the norm estimate \begin{equation*} \int \lvert T (\sigma f)\rvert^{p} \, w (dx) \le C_{T, \sigma ,w}^{p} \int \lvert f\rvert^{p}\, \sigma (dx) , \end{equation*} valid for all Calder\'on-Zygmund operators $T$ is that the condition below holds. \begin{equation*} \sup_{\textup{$Q$ a cube}} \lVert \sigma^{1/{p'}}\rVert_{L^{A} (Q, {dx}/{\lvert Q\rvert})} \varepsilon \big(\lVert \sigma^{1/{p'}}\rVert_{L^{A} (Q, {dx}/{\lvert Q\rvert})}/ \sigma (Q)^{1/{p'}}\big) \bigg[\frac{w (Q)}{\lvert Q\rvert} \bigg]^{1/{p}} \lt \infty \end{equation*} Here $A$ is Young function, with dual in the P{\'e}rez class $B_{p}$, and the function $\varepsilon (t)$ is increasing on $(1, \infty )$ with $\int^{\infty } \varepsilon (t)^{-p'} ({dt}/ t) \lt \infty$. Moreover, a dual condition holds, with the roles of the weights and $L^{p}$ indices reversed also holds. This is an alternate version of a result of Nazarov, Reznikov and Volberg ($p=2$), one with a simpler formulation, and proof based upon stopping times.