Perron树在几何极大算子中的应用
Application of Perron Trees to Geometric Maximal Operators
摘要：
我们刻画了几何极大算子$M_{a,b}$与基$\mathcal{B}_{a,b}$（$a,b > 0$）相关联的$L^p(\mathbb{R}^2)$有界性，其中基$\mathcal{B}_{a,b}$（$a,b > 0$）由矩形$R$组成，矩形的偏心率和方向对于某些$n \in \mathbb{N}^*$具有$$\left( e_R ,ω_R \right) = \left( \frac{1}{n^a} , \fracπ{4n^b} \right)$$的形式.证明涉及\textit{generalized Perron trees}，如在\cite{KATHRYN JAN}中构造的。
英文摘要：
We characterize the $L^p(\mathbb{R}^2)$ boundeness of the geometric maximal operator $M_{a,b}$ associated to the basis $\mathcal{B}_{a,b}$ ($a,b > 0$) which is composed of rectangles $R$ whose eccentricity and orientation is of the form $$\left( e_R ,ω_R \right) = \left( \frac{1}{n^a} , \fracπ{4n^b} \right)$$ for some $n \in \mathbb{N}^*$. The proof involves \textit{generalized Perron trees}, as constructed in \cite{KATHRYN JAN}.
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