The John–Nirenberg inequality for Orlicz–Lorentz spaces in a probabilistic setting


  • Libo Li School of Mathematics and Computing Science, Hunan University of Science and Technology, Xiangtan, Hunan 411201, China
  • Zhiwei Hao School of Mathematics and Computing Science, Hunan University of Science and Technology, Xiangtan, Hunan 411201, China



The John–Nirenberg inequality is widely studied in the field of mathematical analysis and probability theory. In this paper we study a new type of the John–Nirenberg inequality for Orlicz–Lorentz spaces in a probabilistic setting. To be precise, let $0 < q \leq \infty$ and $\Phi$ be an $N$-function with some proper restrictions. We prove that if the stochastic basis $\{\mathcal{F}_n\}_{n \geq 0}$ is regular, then $BMO_{\Phi,q}=BMO$, with equivalent (quasi)-norms. The result is new, which improves previous work on martingale Hardy theory.


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