Square function estimates for conical regions

Abstract

We prove square function estimates for certain conical regions. Specifically, let $\{{\Delta }_j\}$ be regions of the unit sphere ${\mathbb{S}}^{n-1}$ and let $S_{j}f$ be the smooth Fourier restriction of $f$ to the conical region $\{\xi \in {\mathbb{R}}^n:\xi /|\xi |\in {\Delta }_j\}$. We are interested in the following estimate $$\Vert (\sum _j|S_{j}f{|}^2{)}^{1/2}{\Vert }_p{\lesssim }_{\epsilon }{\delta }^{-\epsilon }\Vert f{\Vert }_p.$$ The first result is: when $\{{\Delta }_j\}$ is a set of disjoint $\delta$-balls, then the estimate holds for $p=4$. The second result is: In ${\mathbb{R}}^3$, when $\{{\Delta }_j\}$ is a set of disjoint $\delta \times {\delta }^{1/2}$-rectangles contained in the band ${\mathbb{S}}^2\cap N_{\delta }(\{{\xi }_1^2+{\xi }_2^2={\xi }_3^2\})$ and $\mathrm{supp}\hat{f}\subset \{\xi \in {\mathbb{R}}^3:\xi /|\xi |\in {\mathbb{S}}^2\cap N_{\delta }(\{{\xi }_1^2+{\xi }_2^2={\xi }_3^2\})\}$, then the estimate holds for $p=8$. The two estimates are sharp.

27 pages, 5 figures, more references added