论文标题
$ l^p（\ mathbb {r}^2）$ bounds用于与同型不变凸相关的几何最大运算符
$L^p(\mathbb{R}^2)$ bounds for geometric maximal operators associated to homothecy invariant convex bases
论文作者
论文摘要
令$ \ mathcal {b} $为$ \ mathbb {r}^2 $中的凸面有限度量集的非空同位构成集合。令$ M_ \ Mathcal {B} $为$$ M_ \ Mathcal {B} F（x）= \ sup_ {x \ in r \ in r \ in \ Mathcal {b}} \ frac {1} {1} {| r | r | int_r | f | $ M_ \ MATHCAL {B} $在$ l^p（\ Mathbb {r}^2）$上限制为每$ 1 <p \ leq \ leq \ infty $或该$ m_ \ mathcal {b} $在$ l^p（\ mathbb {r}^2）上$ l^p <作为推论，我们有任何密度基础是$ \ mathbb {r}^2 $中的同一个凸集集合集合的集合，每个$ 1 <p \ leq \ leq \ infty $都必须区分$ l^p（\ mathbb {r}^2）$。
Let $\mathcal{B}$ be a nonempty homothecy invariant collection of convex sets of positive finite measure in $\mathbb{R}^2$. Let $M_\mathcal{B}$ be the geometric maximal operator defined by $$M_\mathcal{B}f(x) = \sup_{x \in R \in \mathcal{B}}\frac{1}{|R|}\int_R |f|\;.$$ We show that either $M_\mathcal{B}$ is bounded on $L^p(\mathbb{R}^2)$ for every $1 < p \leq \infty$ or that $M_\mathcal{B}$ is unbounded on $L^p(\mathbb{R}^2)$ for every $1 \leq p < \infty$. As a corollary, we have that any density basis that is a homothecy invariant collection of convex sets in $\mathbb{R}^2$ must differentiate $L^p(\mathbb{R}^2)$ for every $1 < p \leq \infty$.
