论文标题
锥形区域的方形功能估计
Square function estimates for conical regions
论文作者
论文摘要
我们证明了某些圆锥形区域的平方功能估计。具体来说，让$ \ {δ_j\} $为单位球的区域$ \ mathbb {s}^{n-1} $，让$ s_j f $是$ f $对圆锥形区域$ \ \ \之一的平稳傅立叶限制，in \ in \ n \ in \ m}我们对以下估计感兴趣 $$ \ big \ |（\ sum_j | s_jf |^2）^{1/2} \ big \ | _p \silysim_εδ^{ - ε} \ | | f \ | _p。$$ 第一个结果是：当$ \ {δ_j\} $是一组不相交$δ$ -Balls时，则估算为$ p = 4 $。第二个结果是：在$ \ mathbb {r}^3 $中，当$ \ {δ_j\} $是一组diss $ \timesδ^{1/2} $ - 包含在band $ \ mathbb {s}^2 \ cap cap^2 \ cap cap^2 \ cap cap^2 \ cap^2 \ cap^2 \ cap^2 \ cap^2 \ cap Cap^2 \ cap Cap^2 \timesΔ^{1/2} $时n_δ（\ {ξ_1^2+ξ_2^2 =ξ_3^2 \}）$和$ {\ rm {supp}} \ widehat f \ subset \ subset \ subset \ in \ in \ mathbb {r} n_δ（\ {ξ_1^2+ξ_2^2 =ξ_3^2 \}）\} $，那么估计值以$ p = 8 $保持。这两个估计值很清晰。
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.
