论文标题
$ l^{p} \ rightarrow l^{q} $估计与$ \ mathbb {r}^3 $相关的最大功能与非异体扩张相关的最大功能
$L^{p} \rightarrow L^{q}$ estimates for maximal functions associated with nonisotropic dilations of hypersurfaces in $\mathbb{R}^3$
论文作者
论文摘要
本文的目的是建立$ l^{p} \ rightarrow l^{q} $估计与非异体扩张相关的最大函数的估计值x_ {2}，φ（x_1，x_2））$ in $ \ mathbb {r}^3 $，其中允许Hypersurfaces的高斯曲率消失。当$2α_{2} =α_{3} $时，此问题减少到研究$ l^{p} \ rightarrow l^{q} $估计沿曲线$γ（x）=（x，x，x，x，x，x，x，x^2（1+ϕ））$和相关扩张$ ulivation $ upiration $ ulive $ upiration $ upimut $ uper $ upimud $ upimugunage $ upimimal函数的估计。相应的最大函数显示了与Bourgain圆形最大函数相关的功能，其$ l^{p} \ rightarrow l^{q} $估计已由[Schlag，Jams，1997]，[Schlag-Sogge，MRL，1997]和[Lee，Pams，2003]。但是，在研究与上述曲线$γ（x）$和相关扩张相关的最大功能时，我们得到了$ l^{p} \ rightArrow l^{q} $的规律性属性，适用于相应的傅立叶集成型算子，这些算子无法满足“ Cinematic Curv的状况”，这意味着将经典的局部局部估计不足，这意味着我们无法直接估算出来的问题。更重要的是，对于与HyperSurfaces $（x_ {1}，x_ {2}，x_ {2}，φ（x_1，x_1，x_2））$相关的最大功能，$ l^{p} \ rightArrow l^{q} $估计也是新的。
The goal of this article is to establish $L^{p} \rightarrow L^{q}$ estimates for maximal functions associated with nonisotropic dilations $δ_t(x)=(t^{a_1}x_1,t^{a_2}x_2,t^{a_3}x_3)$ of hypersurfaces $(x_{1}, x_{2},Φ(x_1,x_2))$ in $\mathbb{R}^3$, where the Gaussian curvatures of the hypersurfaces are allowed to vanish. When $2 α_{2} = α_{3}$, this problem is reduced to study of the $L^{p} \rightarrow L^{q}$ estimates for maximal functions along the curve $γ(x)=(x,x^2(1+ϕ(x)))$ and associated dilations $δ_t(x)=(tx_1,t^2x_2)$. The corresponding maximal function shows features related to the Bourgain circular maximal function, whose $L^{p} \rightarrow L^{q}$ estimate has been considered by [Schlag, JAMS, 1997], [Schlag-Sogge, MRL, 1997] and [Lee, PAMS, 2003]. However, in the study of the maximal function related to the mentioned curve $γ(x)$ and associated dilations, we get the $L^{p} \rightarrow L^{q}$ regularity properties for a family of corresponding Fourier integral operators which fail to satisfy the "cinematic curvature condition" uniformly, which means that classical local smoothing estimates could not be directly applied to our problem. What's more, the $L^{p} \rightarrow L^{q}$ estimates are also new for maximal functions associated with isotropic dilations of hypersurfaces $(x_{1}, x_{2},Φ(x_1,x_2))$ mentioned before.
