Journal of Fourier Analysis and Applications

, Volume 20, Issue 4, pp 784–800

# On Sharp Aperture-Weighted Estimates for Square Functions

• Andrei K. Lerner
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

42B20 42B25

## Notes

### Acknowledgments

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

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