Martingale Hardy-Lorentz spaces -- a unified approach

Abstract

This paper investigates the weighted martingale Hardy-Lorentz spaces $\Lambda_q^s(\omega)$, where $0<q<\infty$, $\omega$ is a weight and $s$ is the conditioned square function. By taking different weights, these spaces reduce to the usual martingale Hardy-Lorentz spaces, martingale Hardy-Lorentz-Karamata spaces, martingale Hardy-Orlicz-Lorentz spaces, etc. We establish various martingale inequalities in the weighted martingale Hardy-Lorentz spaces. We also characterize the dual spaces of $\Lambda_q^s(\omega)$. To this end, we need to introduce two generalized BMO martingale spaces which are defined by stopping times or stopping time sequences. We obtain as well the John-Nirenberg inequalities for these generalized BMO martingale spaces. Our main results strengthen the work [20]. We highlight that our study demonstrates a unified approach to martingale Hardy-Lorentz type spaces. A variety of known and new results on martingale Hardy-Lorentz type spaces can be inferred from our paper.