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On Sharp Aperture-Weighted Estimates for Square Functions

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.

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Acknowledgments

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

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Correspondence to Andrei K. Lerner.

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Communicated by Chris Heil.

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Lerner, A.K. On Sharp Aperture-Weighted Estimates for Square Functions. J Fourier Anal Appl 20, 784–800 (2014). https://doi.org/10.1007/s00041-014-9333-6

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  • DOI: https://doi.org/10.1007/s00041-014-9333-6

Keywords

  • Littlewood–Paley operators
  • Sharp weighted inequalities
  • Sharp aperture dependence

Mathematics Subject Classification

  • 42B20
  • 42B25