论文标题
重排的平均振荡界限
Mean oscillation bounds on rearrangements
论文作者
论文摘要
我们使用几何参数来证明在$ \ mathbb {r}^n $上的两个重要重排的平均振荡上明确界限。为了减少可重排功能的重新安排$ f^*$，在立方体上的有界平均振荡（BMO），我们改善了bennett-devore-- devore-sharpley的经典不平等，$ \ | f^*\ | _ \ | _ {bmo（ \ | f \ | _ {bmo（\ Mathbb {r}^n）} $，通过在dimension $ n $中显示$ c_n $的增长不是指数级，而是$ \ sqrt {n} $的大多数。这是通过将立方体与一个矩形家族进行比较来实现的，该矩形可以证明无尺寸的calderón-zygmund分解。通过将立方体与极地矩形家族进行比较，我们提供了第一个证明，表明类似的不等式适用于对称减少的重排（$ sf $）。
We use geometric arguments to prove explicit bounds on the mean oscillation for two important rearrangements on $\mathbb{R}^n$. For the decreasing rearrangement $f^*$ of a rearrangeable function $f$ of bounded mean oscillation (BMO) on cubes, we improve a classical inequality of Bennett--DeVore--Sharpley, $\|f^*\|_{BMO(\mathbb{R}_+)}\leq C_n \|f\|_{BMO(\mathbb{R}^n)}$, by showing the growth of $C_n$ in the dimension $n$ is not exponential but at most of the order of $\sqrt{n}$. This is achieved by comparing cubes to a family of rectangles for which one can prove a dimension-free Calderón--Zygmund decomposition. By comparing cubes to a family of polar rectangles, we provide a first proof that an analogous inequality holds for the symmetric decreasing rearrangement, $Sf$.
