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

It is known that functions in a Sobolev space with critical exponent embed into the space of functions of bounded mean oscillation, and therefore satisfy the John-Nirenberg inequality and a corresponding exponential integrability estimate. While these inequalities are optimal for general functions of bounded mean oscillation, the main result of this paper is an improvement for functions in a class of critical Sobolev spaces. Precisely, we prove the inequality \[\mathcal{H}^β_{\infty}(\{x\in Ω:|I_αf(x)|>t\})\leq Ce^{-ct^{q'}}\] for all $\|f\|_{L^{N/α,q}(Ω)}\leq 1$ and any $β\in (0,N]$, where $Ω\subset \mathbb{R}^N$, $\mathcal{H}^β_{\infty}$ is the Hausdorff content, $L^{N/α,q}(Ω)$ is a Lorentz space with $q \in (1,\infty]$, $q'=q/(q-1)$ is the Hölder conjugate to $q$, and $I_αf$ denotes the Riesz potential of $f$ of order $α\in (0,N)$.