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-infinity(beta)({x is an element of Omega : vertical bar I(alpha)f(x)vertical bar > t}) <= Ce-ctq' for all parallel to f parallel to(LN/a,q(Omega)) <= 1 and any beta is an element of (0, N], where Omega subset of R-N, N-infinity(beta) is the Hausdorff content, L-N/(alpha,q) (Omega) is a Lorentz space with q is an element of (1, infinity], q' = q/(q - 1) is the Holder conjugate to q, and I(alpha)f denotes the Riesz potential of f of order alpha is an element of (0, N).