Let satisfy that , for any given , is an Orlicz function and that is a Muckenhoupt weight uniformly in . The weak Musielak–Orlicz Hardy space is defined to be the set of all tempered distributions such that their grand maximal functions belong to the weak Musielak–Orlicz space . For parameter and measurable function on , the parametric Marcinkiewicz integral related to the Littlewood–Paley -function is defined by setting, for all ,
where is homogeneous of degree zero satisfying the cancellation condition.
In this article, we discuss the boundedness of the parametric Marcinkiewicz integral with rough kernel from weak Musielak–Orlicz Hardy space to weak Musielak–Orlicz space . These results are new even for the classical weighted weak Hardy space of Quek and Yang, and probably new for the classical weak Hardy space of Fefferman and Soria.
"Parametric Marcinkiewicz integrals with rough kernels acting on weak Musielak–Orlicz Hardy spaces." Banach J. Math. Anal. 13 (1) 47 - 63, January 2019. https://doi.org/10.1215/17358787-2018-0015