*preprint*

**Inserted:** 19 jul 2020

**Last Updated:** 19 jul 2020

**Year:** 2020

**Abstract:**

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}^{\beta}_{\infty}(\{x\in \Omega:

I_\alpha f(x)

>t\})\leq
Ce^{-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 $I_\alpha
f$ denotes the Riesz potential of $f$ of order $\alpha \in (0,N)$.