Commutators of vector-valued intrinsic square functions on vector-valued generalized weighted Morrey spaces

In this paper, we will obtain the strong type and weak type estimates for vector-valued analogues of intrinsic square functions in the generalized weighted Morrey spaces $M^{\Phi,\varphi}_{w}(\mathbb{R}^n)$. We study the boundedness of intrinsic square functions including the Lusin area integral, Littlewood-Paley $\mathrm{g}$-function and $\mathrm{g}_{\lambda}^{*}$ -function and their commutators on vector-valued generalized weighted Morrey spaces $M^{\Phi,\varphi}_{w}(l_2)$. In all the cases the conditions for the boundedness are given either in terms of Zygmund-type integral inequalities on $\varphi(x,r)$ without assuming any monotonicity property of $\varphi(x,r)$ on $r$.