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 H_∞^β ({x ∈ Ω: |I_αf(x)| > t}) ≤ Ce^−ctq for all ||f||_LN/α,q(Ω₎ ≤ 1 and any β ∈ (0, N], where Ω ⊂ R^N, H_∞^β is the Hausdorff content, L^N/^α,q(Ω) is a Lorentz space with q ∈ (1, ∞], q' = q/(q − 1) is the Hölder conjugate to q, and I_αf denotes the Riesz potential of f of order α ∈ (0, N).