Paper detail

Square function estimates for conical regions

We prove square function estimates for certain conical regions. Specifically, let $\{Δ_j\}$ 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 $\{ξ\in\mathbb{R}^n:ξ/|ξ|\inΔ_j\}$. We are interested in the following estimate $$\Big\|(\sum_j|S_jf|^2)^{1/2}\Big\|_p\lesssim_εδ^{-ε}\|f\|_p.$$ The first result is: when $\{Δ_j\}$ is a set of disjoint $δ$-balls, then the estimate holds for $p=4$. The second result is: In $\mathbb{R}^3$, when $\{Δ_j\}$ is a set of disjoint $δ\timesδ^{1/2}$-rectangles contained in the band $\mathbb{S}^2\cap N_δ(\{ξ_1^2+ξ_2^2=ξ_3^2\})$ and ${\rm{supp}}\widehat f\subset \{ξ\in\mathbb{R}^3:ξ/|ξ|\in\mathbb{S}^2\cap N_δ(\{ξ_1^2+ξ_2^2=ξ_3^2\})\}$, then the estimate holds for $p=8$. The two estimates are sharp.