Dual spaces for weak martingale Hardy spaces associated with rearrangement-invariant spaces

Abstract

Given a probability space $(\Omega,\mathcal{F},\mathbb P)$ and a rearrangement-invariant quasi-Banach function space $X$, the authors of this article first prove the $\alpha$-atomic ($\alpha\in [1,\infty)$) characterization of weak martingale Hardy spaces $WH_X(\Omega)$ associated with $X$ via simple atoms. The authors then introduce the generalized weak martingale ${\rm BMO}$ spaces which proves to be the dual spaces of $WH_X(\Omega)$. Consequently, the authors derive a new John--Nirenberg theorem for these weak martingale ${\rm BMO}$ spaces. Finally, the authors apply these results to the generalized grand Lebesgue space and the weighted Lorentz space. Even in these special cases, the results obtained in this article are totally new.