论文标题
球形最大功能和扩张集的分形维度
Spherical maximal functions and fractal dimensions of dilation sets
论文作者
论文摘要
对于球形平均运算符$ \ MATHCAL {a} _t $ in $ \ MATHBB {r}^d $，$ d \ ge 2 $，我们考虑了最大函数$ m_ef = \ sup_ {t \ in E} | \ Mathcal {a} _t f | $，with e} |在本文中，我们给出了令人惊讶的封闭凸组表征，这可能会在某些$ e $上封闭$ m_e $的尖锐$ l^p $改善区域。该区域取决于$ e $的Minkowski尺寸，也取决于分形几何形状的其他属性，例如$ e $的Assouad Spectrum和$ e $的子集。关键成分对于一类称为（Quasi-）Assouad常规的套装的$ m_e $的结果基本上是一个鲜明的结果，该集合在二维上是新的。
For the spherical mean operators $\mathcal{A}_t$ in $\mathbb{R}^d$, $d\ge 2$, we consider the maximal functions $M_Ef =\sup_{t\in E} |\mathcal{A}_t f|$, with dilation sets $E\subset [1,2]$. In this paper we give a surprising characterization of the closed convex sets which can occur as closure of the sharp $L^p$ improving region of $M_E$ for some $E$. This region depends on the Minkowski dimension of $E$, but also other properties of the fractal geometry such as the Assouad spectrum of $E$ and subsets of $E$. A key ingredient is an essentially sharp result on $M_E$ for a class of sets called (quasi-)Assouad regular which is new in two dimensions.
