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June 2016 On the Separated Bumps Conjecture for Calderón-Zygmund Operators
Michael T. LACEY
Hokkaido Math. J. 45(2): 223-242 (June 2016). DOI: 10.14492/hokmj/1470139402

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.

Citation

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Michael T. LACEY. "On the Separated Bumps Conjecture for Calderón-Zygmund Operators." Hokkaido Math. J. 45 (2) 223 - 242, June 2016. https://doi.org/10.14492/hokmj/1470139402

Information

Published: June 2016
First available in Project Euclid: 2 August 2016

zbMATH: 1345.42015
MathSciNet: MR3532130
Digital Object Identifier: 10.14492/hokmj/1470139402

Subjects:
Primary: 42B20
Secondary: 42B25, 42B35

Rights: Copyright © 2016 Hokkaido University, Department of Mathematics

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Vol.45 • No. 2 • June 2016
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