Dual Spaces for Weak Martingale Hardy Spaces Associated with Rearrangement-Invariant Spaces

Given a probability space (Omega, F, P) and a rearrangement-invariant quasi-Banach function space X, the authors of this article first prove the alpha-atomic (alpha is an element of [1, infinity)) characterization of weak martingale Hardy spaces WHX (Omega) associated with X via simple atoms. The authors then introduce the generalized weak martingale BMO spaces which proves to be the dual spaces of WHX (Omega). Consequently, the authors derive a new John-Nirenberg theorem for these weak martingale 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.