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


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.


Littlewood–Paley operators Sharp weighted inequalities Sharp aperture dependence 

Mathematics Subject Classification

42B20 42B25 



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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