Tohoku Mathematical Journal
2023
December
SECOND SERIES VOL. 75, NO. 4

Tohoku Math. J.
75 (2023), 533-560

Title MARTINGALE HARDY-LORENTZ SPACES -- A UNIFIED APPROACH

Author Wenfei Fan, Yong Jiao and Lian Wu

(Received January 26, 2022, revised June 2, 2022)
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.

Mathematics Subject Classification. Primary: 60G46; Secondary: 60G42.
Key words and phrases. Martingale Hardy space, weighted Lorentz space, inequality, duality, atomic decomposition.

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