AI 中文总结
本文研究$\boldsymbol{\text{R}}^3$中沿混合齐次超曲面的极大平均算子的$L^p$估计，确定临界勒贝格指数$p_c$，其有界性与黑塞行列式实根重数相关，且关联了FIO锥乘子理论的近期成果。
AI 中文摘要
本文研究$\boldsymbol{\text{R}}^3$中沿超曲面$S$的极大平均算子$\boldsymbol{\text{M}}$的$L^p$估计，其中$S$是在原点外解析的混合齐次函数$\boldsymbol{\text{\textPhi}}$的图像。这类曲面的闭包会经过原点，因此以往许多关于沿超曲面的极大平均的研究所施加的通常横截性条件，即使$\boldsymbol{\text{\textPhi}}$在原点解析也不成立。作为主要结果，在满足例如每个混合齐次多项式$\boldsymbol{\text{\textPhi}}$的温和假设下，我们根据$\boldsymbol{\text{\textPhi}}$的黑塞行列式的实根的重数，确定了临界勒贝格指数$p_c$：当$p>p_c$时，$\boldsymbol{\text{M}}$是$L^p$有界的；当$p<p_c$时，$\boldsymbol{\text{M}}$是无界的。研究某类根邻域的贡献与Dendrinos、Ikromov以及第一作者和第三作者近期的工作密切相关，该工作涉及沿“例外”类横截超曲面的极大平均算子的精确估计，其$L^p$有界性长期以来是一个开放问题，最近通过他们的FIO锥乘子新理论得以确立。
英文摘要
In this paper, we study $L^p$-estimates for maximal averaging operators $\mathcal M$ along hypersurfaces $S$ in $\mathbb{R}^{3}$ which are the graph of a mixed homogeneous function $Φ$ which is analytic away from the origin. The closure of such a surface will pass through the origin, so that the usual transversality condition that had been imposed in many previous works on maximal averages along hypersurfaces will not hold even when $Φ$ is analytic at the origin. As our main result, under mild assumptions which are satisfied for instance for every mixed homogeneous polynomial $Φ,$ we determine the critical Lebesgue exponent $p_c$ for which $\mathcal M$ is $L^p$-bounded for every $p>p_c,$ but unbounded for $p<p_c,$ in terms of multiplicities of the real roots of the Hessian determinant of $Φ.$ It turns out that the study of the contributions by neighborhoods of a certain type of roots is closely related to recent work by Dendrinos, Ikromov and the first and third author on sharp estimates for a maximal averaging operator along a transversal hypersurface of an ``exceptional'' class, whose $L^p$-boundedness had been an open problem for a long time and which has recently been established by means of their new theory of FIO-cone multipliers.
Comments72 pages; The only change compared to the first version is that we have added the Appendix
