高度h≤2的$\n\mathbb{R}^3$中超曲面对应卷积算子的精确估计
Sharp estimates for convolution operators associated to hypersurfaces in $\mathbb{R}^3$ with height $h\le2$
AI总结：
本文研究高度≤2的三维空间特征超曲面对应的带振荡核卷积算子，证明其Lp到Lp'有界性的最小指数由对应光滑函数牛顿多边形的离散特征决定，涵盖Arnol'd分类中的几类奇点曲面。
摘要：
本文研究了带振荡核的卷积算子$M_k$，该算子与严格双曲方程柯西问题的解相关。算子$M_k$与方程的特征超曲面$\Sigma\subset \mathbb{R}^3$以及光滑振幅函数相关，该振幅函数在自变量取值较大时为$-k$次齐次函数。我们假设相应振幅函数的支集包含于给定点$v\in \Sigma$的足够小锥形邻域内，且该点处曲面高度小于等于2，在此条件下研究这类卷积算子。该类曲面包含与Arnol'd分类意义下的简单奇点以及$X_9$、$J_{10}$型奇点相关的曲面。记$k_p$为使得当$k>k_p$时$M_k$是$L^p\mapsto L^{p'}$有界的最小指数，我们证明$k_p$的取值取决于在合适坐标系下构造的某光滑函数的牛顿多边形的若干离散特征。
英文摘要：
In this article, we study the convolution operator $M_k$ with oscillatory kernel, which is related with solutions to the Cauchy problem for the strictly hyperbolic equations. The operator $M_k$ is associated to the characteristic hypersurface $Σ\subset \mathbb{R}^3$ of the equation and the smooth amplitude function, which is homogeneous of order $-k$ for large values of the argument. We study the convolution operators assuming that the support of the corresponding amplitude function is contained in a sufficiently small conic neighborhood of a given point $v\in Σ$ at which the height of the surface is less or equal to two. Such class contains surfaces related to simple and the $X_9, \, J_{10}$ type singularities in the sense of Arnol'd's classification. Denoting by $k_p$ the minimal exponent such that $M_k$ is $L^p\mapsto L^{p'}$-bounded for $k>k_p,$ we show that the number $k_p$ depends on some discrete characteristics of the Newton polygon of a smooth function constructed in an appropriate coordinate system.
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