论文标题
最大运营商连续性的日出策略
Sunrise strategy for the continuity of maximal operators
论文作者
论文摘要
在本文中，我们介绍了几个最大运算符在梯度级别的$ W^{1,1} $ - 连续性。我们全球策略中的一个关键思想是最大运算符的分解，在断开集合中没有严格的局部最大值，将其纳入具有良好单调性和收敛属性的“横向”最大运算符。这种结构在谐波分析中的经典日出引理中灵感。我们的日出策略的模型案例考虑了无hardy-littlewood的最大操作员$ \ widetilde {m} $作用于$ w^{1,1} _ {\ rm rad}（\ rm rad}（\ m mathbb {r}^d）$，$ w^^{1,1,1,1,1，\ bb^r的子空间在$ d \ geq 2 $中，H. luiro最近确定了地图$ f \ mapsto \ nabla \ nabla \ widetilde {m} f $从$ w^{1,1} _ {\ rm rad}（\ rm rad}（\ mathbb {\ mathbb {r}^d）也是连续的。日出策略的进一步应用与$ w^{1,1} $ - 连续性问题有关，包括$ \ mathbb {r}^d $在$ d \ geq 2 $上作用于radial函数上的非区域性最大运算符时，当$ d \ geq 2 $和$ d = 1 $的一般函数以及无材料的硬质量最大$ $ cat $ catt in the n oct ytirtered hardy-litter the n $ catb in catb in catb yath in catb yath catb in catb in catb ythb当$ d \ geq 2 $和常规功能时，当$ d = 1 $时功能。
In this paper we address the $W^{1,1}$-continuity of several maximal operators at the gradient level. A key idea in our global strategy is the decomposition of a maximal operator, with the absence of strict local maxima in the disconnecting set, into "lateral" maximal operators with good monotonicity and convergence properties. This construction is inspired in the classical sunrise lemma in harmonic analysis. A model case for our sunrise strategy considers the uncentered Hardy-Littlewood maximal operator $\widetilde{M}$ acting on $W^{1,1}_{\rm rad}(\mathbb{R}^d)$, the subspace of $W^{1,1}(\mathbb{R}^d)$ consisting of radial functions. In dimension $d\geq 2$ it was recently established by H. Luiro that the map $f \mapsto \nabla \widetilde{M} f$ is bounded from $W^{1,1}_{\rm rad}(\mathbb{R}^d)$ to $L^1(\mathbb{R}^d)$, and we show that such map is also continuous. Further applications of the sunrise strategy in connection with the $W^{1,1}$-continuity problem include non-tangential maximal operators on $\mathbb{R}^d$ acting on radial functions when $d\geq 2$ and general functions when $d=1$, and the uncentered Hardy-Littlewood maximal operator on the sphere $\mathbb{S}^d$ acting on polar functions when $d\geq 2$ and general functions when $d=1$.
