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

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 \[\mathcal{H}^β_{\infty}(\{x\in Ω:|I_αf(x)|>t\})\leq Ce^{-ct^{q'}}\] for all $\Vert f{\Vert }_{L^{N/\alpha ,q}(\Omega )}\le 1$ and any $\beta \in (0,N]$, where $\Omega \subset {\mathbb{R}}^N$, ${\mathcal{H}}_{\infty }^{\beta }$ is the Hausdorff content, $L^{N/\alpha ,q}(\Omega )$ is a Lorentz space with $q\in (1,\infty ]$, $q^{\prime }=q/(q-1)$ is the Hölder conjugate to $q$, and $I_{\alpha }f$ denotes the Riesz potential of $f$ of order $\alpha \in (0,N)$.

25 pages