论文标题
与稀有基础相关的最大运算符的尖锐弱类型估计值
Sharp Weak Type Estimates for Maximal Operators associated to Rare Bases
论文作者
论文摘要
令$ \ MATHCAL {B} $表示$ \ Mathbb {r}^n $的间隔的非空翻译集合（我们认为这是极少数的），并定义了相关的几何最大运算符$ m_ \ Mathcal {b} $ {b} $ {b} $ by $ the $ $ m_ \ mathcal { \ Mathcal {b}} \ frac {1} {| r |} \ int_r | f |。$$我们在$ \ Mathcal {b} $上提供足够的条件α\} | \ leq c_n \ int _ {\ mathbb {r}^{n}}} \ frac {| f |}α\ left（1+\ log^+\ frac {| f | f |}α\ right）作为推论，我们获得了与几类稀有基础相关的最大算子，包括Córdoba，Soria和Zygmund碱基。
Let $\mathcal{B}$ denote a nonempty translation invariant collection of intervals in $\mathbb{R}^n$ (which we regard as a rare basis), and define the associated geometric maximal operator $M_\mathcal{B}$ by $$M_\mathcal{B}f(x) = \sup_{x \in R \in \mathcal{B}} \frac{1}{|R|}\int_R |f|.$$ We provide a sufficient condition on $\mathcal{B}$ so that the estimate $$ |\{x \in \mathbb{R}^n : M_{\mathcal{B}}f(x) > α\}|\leq C_n \int_{\mathbb{R}^{n}} \frac{|f|}α\left(1+\log^+\frac{|f|}α\right)^{n-1} $$ is sharp. As a corollary we obtain sharp weak type estimates for maximal operators associated to several classes of rare bases including Córdoba, Soria and Zygmund bases.
