On the Exponential Integrability of Conjugate Functions

We relate the exponential integrability of the conjugate function $${\tilde{f}}$$ f ~ to the size of the gap in the essential range of f. Our main result complements a related theorem of Zygmund.

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