论文标题
$ \ bbb r^3 $与高度$ h <2：$ part ii-几何猜想及其对通用2-Surfaces的证据相关的最大功能的最大功能的估计值
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
论文作者
论文摘要
在本文中，我们继续研究最大运算符$ \ mathcal m_s $的$ l^p $结合度，与给定的3维欧几里得空间中给定的，平滑的超出表面$ s $相关的平均值相关。我们在此处集中在给定点$ x^0 $附近的小型表面斑块上，目前在Arnol的意义上表现出$ \ Mathcal a $ a $的奇异性；这就是尚未打开的情况。用$ p_c $表示最小的lebesgue指数，以至于$ \ mathcal m_s $是$ l^p $ buged of $ p_c，$我们能够识别$ p_c $的所有分析表面，用于所有类型的$ \ \ nathcal a $的分析表面（通过小型子分类量除外），从小可以通过新的数量来确定新的量。除了$ x^0时著名的高度概念外，我们称之为有效多重性的新数量，在这里扮演着重要的角色。 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 远的。我们的结果尤其导致了Iosevich-Sawyer-seeger猜想的证据，以进行任意分析2曲面。
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.
