具有高度$h<2:$的$\Bbb R^3$中超曲面的极大函数的估计第II部分--一般2-曲面的一个几何猜想及其证明
Estimates for maximal functions associated to hypersurfaces in $\Bbb R^3$ with height $h<2:$ Part II -- A geometric conjecture and its proof for generic 2-surfaces
摘要：
在这篇文章中，我们继续研究的$L^p$ -有界性的最大运营商$\mathcal M_S$与平均沿各向同性膨胀的一个给定的，光滑超曲面$S$在三维欧氏空间。我们在这里集中讨论在给定点$x^0$附近的小曲面片，在这一点上表现出Arnol'd意义下的$\mathcal A$型奇点;这是尚未解决的情况。表示由$p_c$的最小勒贝格指数，使$\mathcal M_S$是$L^p$ -有界的$p>p_c,$，我们能够确定$p_c$的所有分析表面的类型$\mathcal A$（与一个小子类除外），通过数量，可以确定从相关的牛顿多面体。除了众所周知的$x^0,$高度概念外，还有一个新的量，我们称之为有效多重性，它在这里起着至关重要的作用。我们还提出了一个猜想，如何临界指数$p_c$可能是确定的几何测量理论条件，以某种方式措施的顺序接触的任意椭球与$S,$甚至在任意维的超曲面，并表明，这一猜想确实适用于所有类的2-超曲面$S$，我们已经获得了一个基本上完整的理解$\mathcal M_S$到目前为止。我们的结果导致特别是证明的猜想由Iosevich-Sawyer-Seeger任意解析2-曲面。
英文摘要：
In this article, we continue the study of $L^p$-boundedness of the maximal operator $\mathcal M_S$ associated to averages along isotropic dilates of a given, smooth hypersurface $S$ in 3-dimensional Euclidean space. We focus here on small surface-patches near a given point $x^0$ exhibiting singularities of type $\mathcal A$ in the sense of Arnol'd at this point; this is the situation which had yet been left open. Denoting by $p_c$ the minimal Lebesgue exponent such that $\mathcal M_S$ is $L^p$-bounded for $p>p_c,$ we are able to identify $p_c$ for all analytic surfaces of type $\mathcal A$ (with the exception of a small subclass), by means of quantities which can be determined from associated Newton polyhedra. Besides the well-known notion of height at $x^0,$ a new quantity, which we call the effective multiplicity, turns out to play a crucial role here. We also state a conjecture on how the critical exponent $p_c$ might be determined by means of a geometric measure theoretic condition, which measures in some way the order of contact of arbitrary ellipsoids with $S,$ even for hypersurfaces in arbitrary dimension, and show that this conjecture holds indeed true for all classes of 2-hypersurfaces $S$ for which we have gained an essentially complete understanding of $\mathcal M_S$ so far. Our results lead in particular to a proof of a conjecture by Iosevich-Sawyer-Seeger for arbitrary analytic 2-surfaces.
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