On Sharp Aperture-Weighted Estimates for Square Functions

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Let Sα, ψ (f) be the square function defined by means of the cone in ℝn+1+ of aperture α, and a standard kernel ψ. Let [w]Ap denote the Ap characteristic of the weight w. We show that for any 1<p<∞ and α ≥ 1, (Formula Presented) For each fixed α the dependence on [w]Ap is sharp. Also, on all class Ap the result is sharp in α. Previously this estimate was proved in the case α =1 using the intrinsic square function. However, that approach does not allow to get the above estimate with sharp dependence on α. Hence we give a different proof suitable for all α ≥ 1 and avoiding the notion of the intrinsic square function.

Original languageEnglish
Pages (from-to)784-800
Number of pages17
JournalJournal of Fourier Analysis and Applications
Issue number4
StatePublished - Aug 2014


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


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