On functions of bounded $\beta$-dimensional mean oscillation

In this paper, we define a notion of $\beta$-dimensional mean oscillation of functions $u:Q_0\subset {\mathbb{R}}^d\to \mathbb{R}$ which are integrable on $\beta$-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 ${\mathcal{H}}_{\mathrm{\infty }}^{\beta }$ is the Hausdorff content. In the case $\beta =d$ we show this definition is equivalent to the classical notion of John and Nirenberg, while our main result is that for every $\beta \in (0,d]$ one has a dimensionally appropriate analogue of the John-Nirenberg inequality for functions with bounded $\beta$-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 $t>0$, $u\in BM{O}^{\beta }(Q_0)$, $Q\subset Q_0$, 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.