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

Abstract

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

23 pages