论文标题
飞机上的Mizohata-Takeuchi估计
Mizohata-Takeuchi estimates in the plane
论文作者
论文摘要
假设$ s $是$ \ bbb r^n $中的平滑紧凑型超表面，而$σ$是$ s $的适当度量。如果$ ef = \ hat {fdσ} $是与$（s，σ）$关联的扩展运算符，则Mizohata-takeuchi猜想断言$ \ \ int | ef（x）|^2 w（x）|^2 w（x）dx \ leq c（\ sup_t w（t）\ sup_t w（t）\ | f \ | _ {l^2（σ）}^2 $ for l^2（σ）$ in l^2（σ）$和权重$ w：\ bbb r^n \至[0，\ infty）$，其中$ \ sup $在$ \ bb r r^n $ of Cross-n $ in of Cross-n $ in of Cross-n $ in Int $ f in Int $ t $ f in the [0，\ infty）$（ DX $。本文研究了如果我们只考虑$ \hatσ$的衰减属性，我们可以在$ \ bbb r^2 $中证明Mizohata-Takeuchi的猜想在多远。由于我们的结果，我们获得了一类凸曲线的新估计，其中包括指数平坦的曲线，例如$（t，e^{ - 1/t^m}）$，$ 0 \ leq t \ leq t \ leq c_m $，$ m \ in \ bbb n $。
Suppose $S$ is a smooth compact hypersurface in $\Bbb R^n$ and $σ$ is an appropriate measure on $S$. If $Ef= \hat{fdσ}$ is the extension operator associated with $(S,σ)$, then the Mizohata-Takeuchi conjecture asserts that $\int |Ef(x)|^2 w(x) dx \leq C (\sup_T w(T)) \| f \|_{L^2(σ)}^2$ for all functions $f \in L^2(σ)$ and weights $w : \Bbb R^n \to [0,\infty)$, where the $\sup$ is taken over all tubes $T$ in $\Bbb R^n$ of cross-section 1, and $w(T)= \int_T w(x) dx$. This paper investigates how far we can go in proving the Mizohata-Takeuchi conjecture in $\Bbb R^2$ if we only take the decay properties of $\hatσ$ into consideration. As a consequence of our results, we obtain new estimates for a class of convex curves that include exponentially flat ones such as $(t,e^{-1/t^m})$, $0 \leq t \leq c_m$, $m \in \Bbb N$.
