论文标题
Lebesgue分化过程沿矩形几乎到处融合
Almost everywhere convergence for Lebesgue differentiation processes along rectangles
论文作者
论文摘要
在本文中，我们研究了沿着矩形$ r_k $在欧几里得飞机上缩小起点的Lebesgue分化过程，以及它们在$ l^p $空间中几乎无处不在的问题。特别是，此类过程的示例类别未能收敛A.E.提供了$ l^\ infty $，为此，已知$ r_k $沿斜率$ k^{ - s} $ for $ s> 0 $，这与与集合$ \ \ \ \ s in \ in \ in \ n \ in \ mathbb {$ lafe in \ n^$ lafe to $ n} $ lafe to $ s}的方向最大运算符相关的事实很有趣。任何$ 1 \ leq p <\ infty $。
In this paper, we study Lebesgue differentiation processes along rectangles $R_k$ shrinking to the origin in the Euclidean plane, and the question of their almost everywhere convergence in $L^p$ spaces. In particular, classes of examples of such processes failing to converge a.e. in $L^\infty$ are provided, for which $R_k$ is known to be oriented along the slope $k^{-s}$ for $s>0$, yielding an interesting counterpart to the fact that the directional maximal operator associated to the set $\{k^{-s}:k\in\mathbb{N}^*\}$ fails to be bounded in $L^p$ for any $1\leq p<\infty$.
