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 H∞β({x∈Ω:|Iαf(x)|>t})≤Ce−ctq′ $$\mathcal{H}^{\beta}_{\infty}(\{x\in \Omega:|I_\alpha f(x)|>t\})\leq Ce^{-ct^{q'}}$$ for all ∥f∥LN/α,q(Ω)≤1 $\|f\|_{L^{N/\alpha,q}(\Omega)}\leq 1$ and any β∈(0,N],whereΩ⊂RN,H∞β $\beta \in (0,N], \; {\text{where}} \; \Omega \subset \mathbb{R}^N, \mathcal{H}^{\beta}_{\infty}$ 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).

中文翻译：

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临界 Sobolev 空间中函数的 John-Nirenberg 不等式的改进

摘要 已知Sobolev空间中的临界指数函数嵌入到有界平均振荡函数空间中，因此满足John-Nirenberg不等式和相应的指数可积性估计。虽然这些不等式对于有界平均振荡的一般函数是最优的，但本文的主要结果是对一类临界 Sobolev 空间中的函数进行了改进。准确地说，我们证明不等式 H∞β({x∈Ω:|Iαf(x)|>t})≤Ce−ctq′ $$\mathcal{H}^{\beta}_{\infty}(\{ x\in \Omega:|I_\alpha f(x)|>t\})\leq Ce^{-ct^{q'}}$$ 对于所有 ∥f∥LN/α,q(Ω)≤1 $\|f\|_{L^{N/\alpha,q}(\Omega)}\leq 1$ 和任意 β∈(0,N],whereΩ⊂RN,H∞β $\beta \in ( 0,N], \; {\text{where}} \; \Omega \subset \mathbb{R}^N, \mathcal{H}^{\beta}_{\infty}$ 是豪斯多夫内容，LN /α,q(Ω) 是一个洛伦兹空间，其中 q ∈ (1,∞],