John-Nirenberg Inequalities with Variable Exponents on Probability Spaces

Abstract

In this paper we study the John-Nirenberg inequalities with variable exponents on a probability space. Let $Y$ be a rearrangement invariant Banach function space defined on $(\Omega,\mathcal{F},P)$ and a measurable function $p(\cdot): \Omega\rightarrow \mathbf{R}^+$ be a variable exponent. We prove that if the stochastic basis is regular, then $$BMO_{\phi,Y}=BMO_{\phi,p(\cdot)}\,,\quad \forall 1\leq p(\cdot)<\infty\,,$$ where $\phi(r)=1/r\Phi^{-1}(1/r)$ and $\Phi$ is a concave function with proper condition.