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 $$\Big|(\sum_j|S_jf|^2)^{1/2}\Big|p\lesssim\epsilon \delta^{-\epsilon}|f|_p.$$ 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 ${\rm{supp}}\widehat f\subset {\xi\in\mathbb{R}^3:\xi/|\xi|\in\mathbb{S}^2\cap N_\delta({\xi_1^2+\xi_2^2=\xi_3^2})}$, then the estimate holds for $p=8$. The two estimates are sharp.