论文标题
关于有限$β$维平均振荡的功能
On functions of bounded $β$-dimensional mean oscillation
论文作者
论文摘要
在本文中，我们定义了$β$维的概念$ u：q_0 \ subset \ subset \ mathbb {r}^d \ to \ mathbb {r {r} $，可在$β$ -Dimensional of The Cube $ q_0 $ q_0 $ q_： \ | U \ | _ {bmo^β（q_0）}：= \ sup_ {q \ subset q_0} \ inf_ {c \ in \ in \ mathbb {r}} \ frac {1} {1} {l（q） \; d \ Mathcal {h}^β_\ infty，\ end {align*}，其中将超级人置于所有有限的子橡胶上$ q $ q $ $ q $，平行于$ q_0 $，$ l（q）$是Cube $ q $的长度，以及$ q $，以及$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ restcal {在$β= d $的情况下，我们表明此定义等同于约翰和尼伦贝格的经典概念，而我们的主要结果是，在（0，d] $中，每个$β\ in（0，d] $ in（d，d] $）在约翰·尼伦贝格（John-Nirenberg \ Mathcal {h}^β_\ infty \ left（\ {x \ in q：| u（x）-c_q |> t |> t \} \ right）\ leq c c l（q） \在bmo^β（q_0）$，$ q \ subset q_0 $，和合适的$ c_q \ in \ mathbb {r} $中。
In this paper, we define a notion of $β$-dimensional mean oscillation of functions $u: Q_0 \subset \mathbb{R}^d \to \mathbb{R}$ which are integrable on $β$-dimensional subsets of the cube $Q_0$: \begin{align*} \|u\|_{BMO^β(Q_0)}:= \sup_{Q \subset Q_0} \inf_{c \in \mathbb{R}} \frac{1}{l(Q)^β} \int_{Q} |u-c| \;d\mathcal{H}^β_\infty, \end{align*} where the supremum is taken over all finite subcubes $Q$ parallel to $Q_0$, $l(Q)$ is the length of the side of the cube $Q$, and $\mathcal{H}^β_\infty$ is the Hausdorff content. In the case $β=d$ we show this definition is equivalent to the classical notion of John and Nirenberg, while our main result is that for every $β\in (0,d]$ one has a dimensionally appropriate analogue of the John-Nirenberg inequality for functions with bounded $β$-dimensional mean oscillation: There exist constants $c,C>0$ such that \begin{align*} \mathcal{H}^β_\infty \left(\{x\in Q:|u(x)-c_Q|>t\}\right) \leq C l(Q)^β\exp(-ct/\|u\|_{BMO^β(Q_0)}) \end{align*} for every $t>0$, $u \in BMO^β(Q_0)$, $Q\subset Q_0$, and suitable $c_Q \in \mathbb{R}$. Our proof relies on the establishment of capacitary analogues of standard results in integration theory that may be of independent interest.
