On Sharp Aperture-Weighted Estimates for Square Functions

By Andrei K. Lerner. 2013

66 citations recorded by the source corpus.

Let $$S_{\alpha ,\psi }(f)$$Sα,ψ(f) be the square function defined by means of the cone in $${\mathbb R}^{n+1}_{+}$$R+n+1 of aperture $$\alpha $$α, and a standard kernel $$\psi $$ψ. Let $$[w]_{A_p}$$[w]Ap denote the $$A_p$$Ap characteristic of the weight $$w$$w. We show that for any $$1<p<\infty $$1<p<∞ and $$\alpha \ge 1$$α≥1, $$\begin{aligned} \Vert S_{\alpha ,\psi }\Vert _{L^p(w)}\lesssim \alpha ^n[w]_{A_p}^{\max \left( \frac{1}{2},\frac{1}{p-1}\right) }. \end{aligned}$$‖Sα,ψ‖Lp(w)≲αn[w]Apmax12,1p-1. For each fixed $$\alpha $$α the dependence on $$[w]_{A_p}$$[w]Ap is sharp. Also, on all class $$A_p$$Ap the result is sharp in $$\alpha $$α. Previously this estimate was proved in the case $$\alpha =1$$α=1 using the intrinsic square function. However, that approach does not allow to get the above estimate with sharp dependence on $$\alpha $$α. Hence we give a different proof suitable for all $$\alpha \ge 1$$α≥1 and avoiding the notion of the intrinsic square function.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/785560c96db2e800b9f381b8a0f03b77ea06ab0e

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

  1. Dachun Yang
  2. Wen Yuan
  3. Ciqiang Zhuo
  4. Yiyu Liang
  5. Xianjie Yan
  6. José M. Conde-Alonso
  7. G. Rey
  8. Jun Liu

In Computational Theory and Mathematics

  1. Ciqiang Zhuo
  2. José M. Conde-Alonso
  3. Yiyu Liang
  4. Wen Yuan
  5. G. Rey
  6. Dachun Yang

In Mathematical Physics

  1. Dachun Yang
  2. Wen Yuan
  3. Ciqiang Zhuo
  4. Yiyu Liang
  5. Xianjie Yan
  6. José M. Conde-Alonso
  7. G. Rey
  8. Yoshihiro Sawano

Citations in the corpus