论文标题
改进的加权限制估计值$ \ bbb r^3 $
Improved weighted restriction estimates in $\Bbb R^3$
论文作者
论文摘要
假设$ 0 <α\ leq n $，$ h：\ bbb r^n \ to [0,1] $是一个可测量的功能，而$a_α（h）$是所有数字$ c $中最多的$ c $，为所有不平等$ \ int_b h（x）dx \ leq c r^α$ bb $ bb bb subs $ bb r^subs $ bb。 \ geq 1 $。在Guth将多项式分区引入傅立叶限制理论之后，形式的加权限制估计值$ \ | ef \ | _ {l^p（b，hdx）} \ leq c r^εa_α（h）^{1/p} \ | f \ | _ {l^q（σ）} $在几篇论文中进行了研究并证明，从而导致了有关测量傅立叶变换的球形手段的衰减特性的新结果，在某些情况下，在几何测量理论中以Falconer的距离群体构想进行了进展。本文在已知的估计值中改进$ e $是与单位抛物面$ {\ Mathcal p} \ subset \ bbb r^3 $相关的扩展运算符，达到$ p，q $指数（Q $ prosponents（最高）$ p \ egq 3 +（α-2）/（α-2）/（α + 1）$和$ 2 <c的完整范围。
Suppose $0 < α\leq n$, $H: \Bbb R^n \to [0,1]$ is a Lebesgue measurable function, and $A_α(H)$ is the infimum of all numbers $C$ for which the inequality $\int_B H(x) dx \leq C R^α$ holds for all balls $B \subset \Bbb R^n$ of radius $R \geq 1$. After Guth introduced polynomial partitioning to Fourier restriction theory, weighted restriction estimates of the form $\| Ef \|_{L^p(B,Hdx)} \leq C R^εA_α(H)^{1/p} \| f \|_{L^q(σ)}$ have been studied and proved in several papers, leading to new results about the decay properties of spherical means of Fourier transforms of measures and, in some cases, to progress on Falconer's distance set conjecture in geometric measure theory. This paper improves on the known estimates when $E$ is the extension operator associated with the unit paraboloid ${\mathcal P} \subset \Bbb R^3$, reaching the full possible range of $p,q$ exponents (up to the sharp line) for $p \geq 3 + (α-2)/(α+1)$ and $2 < α\leq 3$.
