  In this paper we explore several applications of the recently introduced
spaces of functions of bounded $\beta$-dimensional mean oscillation for $\beta
\in (0,n]$ to regularity theory of critical exponent elliptic equations. One
interesting consequence of these considerations is the observation that for $F
\in L^1$, $\operatorname*{div} F=0$, the solution of \begin{align*}
-\operatorname*{div}( |\nabla U|^{n-2} \nabla U) &= F \mbox{ in }
\Omega,\newline
  U &=0 \mbox{ on }\partial \Omega. \end{align*}
  satisfies
  \begin{align*} \|\nabla U \|_{L^n} \mathbb \leq C \, \|F\|^{1/(n-1)}_{L^1},
\end{align*} where the constant $C$ is independent of $F$.
