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Journal of Fourier Analysis and Applications

, Volume 20, Issue 4, pp 784–800 | Cite as

On Sharp Aperture-Weighted Estimates for Square Functions

  • Andrei K. LernerEmail author
Article

Abstract

Let \(S_{\alpha ,\psi }(f)\) be the square function defined by means of the cone in \({\mathbb R}^{n+1}_{+}\) of aperture \(\alpha \), and a standard kernel \(\psi \). Let \([w]_{A_p}\) denote the \(A_p\) characteristic of the weight \(w\). We show that for any \(1<p<\infty \) and \(\alpha \ge 1\),
$$\begin{aligned} \Vert S_{\alpha ,\psi }\Vert _{L^p(w)}\lesssim \alpha ^n[w]_{A_p}^{\max \left( \frac{1}{2},\frac{1}{p-1}\right) }. \end{aligned}$$
For each fixed \(\alpha \) the dependence on \([w]_{A_p}\) is sharp. Also, on all class \(A_p\) the result is sharp in \(\alpha \). Previously this estimate was proved in the case \(\alpha =1\) using the intrinsic square function. However, that approach does not allow to get the above estimate with sharp dependence on \(\alpha \). Hence we give a different proof suitable for all \(\alpha \ge 1\) and avoiding the notion of the intrinsic square function.

Keywords

Littlewood–Paley operators Sharp weighted inequalities Sharp aperture dependence 

Mathematics Subject Classification

42B20 42B25 

Notes

Acknowledgments

I am very grateful to the referees for useful remarks and corrections.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of MathematicsBar-Ilan UniversityRamat GanIsrael

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