English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Let $S_{\a,\psi}(f)$ be the square function defined by means of the cone in ${\mathbb R}^{n+1}_{+}$ of aperture $\a$, 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 $\a\ge 1$, $$\|S_{\a,\psi}\|_{L^p(w)}\lesssim \a^n[w]_{A_p}^{\max(1/2,\frac{1}{p-1})}.$$ For each fixed $\a$ the dependence on $[w]_{A_p}$ is sharp. Also, on all class $A_p$ the result is sharp in $\a$. Previously this estimate was proved in the case $\a=1$ using the intrinsic square function. However, that approach does not allow to get the above estimate with sharp dependence on $\a$. Hence we give a different proof suitable for all $\a\ge 1$ and avoiding the notion of the intrinsic square function.

On Sharp Aperture-Weighted Estimates for Square Functions