Square function estimates for conical regions

2022/03/23 by Gan, Shengwen, Wu, Shukun
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2203.12155

Abstract

We prove square function estimates for certain conical regions. Specifically, let \Δ_j\ be regions of the unit sphere \mathbbS^n-1 and let S_j f be the smooth Fourier restriction of f to the conical region \ξ∈ℝⁿ:ξ/|ξ|∈Δ_j\. We are interested in the following estimate ‖(∑_j|S_jf|²)^1/2‖_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 ℝ³, when \Δ_j\ is a set of disjoint δ×δ^1/2-rectangles contained in the band \mathbbS²∩ N_δ(\ξ₁²+ξ₂²=ξ₃²\) and \rmsupp\widehat f⊂ \ξ∈ℝ³:ξ/|ξ|∈\mathbbS²∩ N_δ(\ξ₁²+ξ₂²=ξ₃²\)\, then the estimate holds for p=8. The two estimates are sharp.

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