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 for all and any is the Hausdorff content, ) is a Lorentz space with ∈ (1,∞], = /( − 1) is the Hölder conjugate to , and denotes the Riesz potential of of order ∈ (0, ).