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It is known that functions in a Sobolev space with critical exponent embedinto the space of functions of bounded mean oscillation, and therefore satisfythe John-Nirenberg inequality and a corresponding exponential integrabilityestimate. While these inequalities are optimal for general functions of boundedmean oscillation, the main result of this paper is an improvement for functionsin a class of critical Sobolev spaces. Precisely, we prove the inequality\[\mathcal{H}^{\beta}_{\infty}(\{x\in \Omega:|I_\alpha f(x)|>t\})\leqCe^{-ct^{q'}}\] for all $\|f\|_{L^{N/\alpha,q}(\Omega)}\leq 1$ and any $\beta\in (0,N]$, where $\Omega \subset \mathbb{R}^N$, $\mathcal{H}^{\beta}_{\infty}$is the Hausdorff content, $L^{N/\alpha,q}(\Omega)$ is a Lorentz space with $q\in (1,\infty]$, $q'=q/(q-1)$ is the H\"older conjugate to $q$, and $I_\alphaf$ denotes the Riesz potential of $f$ of order $\alpha \in (0,N)$.

An improvement to the John-Nirenberg inequality for functions in critical Sobolev spaces