论文标题
魔鬼楼梯的最大功能绝对是连续的
The maximal function of the Devil's staircase is absolutely continuous
论文作者
论文摘要
我们研究了一个奇异函数的中心耐铁的最大功能是否绝对是连续的。对于（0,1）$的参数$ d \ and封闭套件$ e \ subset [0,1] $，让$μ$为$ d $ -ahlfors与$ e $相关的常规措施。我们证明，对于累积分布函数$ f（x）=μ（[0，x]）$其最大函数$ mf $绝对是连续的。 We then adapt our method to the multiparameter case and show that the same is true in the positive cone defined by these functions, i.e., for functions of the form $f(x)=\sum_{i=1}^{n}μ_i([0,x])$ where $\{μ_i\}_{i=1}^{n}$ is any collection of $ d_i $ -ahlfors常规措施，$ d_i \ in（0,1）$，与封闭集$ e_i \ subset [0,1] $相关联。这为以经典的最大运算符的规律性提高了规律性，并且可以看作是Aldaz和PérezLázaro结果的部分类似物，涉及无人最大运算符。
We study the problem of whether the centered Hardy--Littlewood maximal function of a singular function is absolutely continuous. For a parameter $d \in (0,1)$ and a closed set $E\subset [0,1]$, let $μ$ be a $d$-Ahlfors regular measure associated with $E$. We prove that for the cumulative distribution function $f(x)=μ([0,x])$ its maximal function $Mf$ is absolutely continuous. We then adapt our method to the multiparameter case and show that the same is true in the positive cone defined by these functions, i.e., for functions of the form $f(x)=\sum_{i=1}^{n}μ_i([0,x])$ where $\{μ_i\}_{i=1}^{n}$ is any collection of $d_i$-Ahlfors regular measures, $d_i \in (0,1)$, associated with closed sets $E_i\subset [0,1]$. This provides the first improvement of regularity for the classical centered maximal operator, and can be seen as a partial analogue of the result of Aldaz and Pérez Lázaro about the uncentered maximal operator.
