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On functions of bounded $β$-dimensional mean oscillation

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.