Weighted estimates for vector-valued intrinsic square functions and commutators in the Morrey type spaces

In this paper, the boundedness properties of vector-valued intrinsic square functions and their vector-valued commutators with $BMO(\mathbb R^n)$ functions are discussed. We first show the weighted strong type and weak type estimates of vector-valued intrinsic square functions in the Morrey type spaces. Then we obtain weighted strong type estimates of vector-valued analogues of commutators in Morrey type spaces. In the endpoint case, we establish the weighted weak $L\log L$-type estimates for these vector-valued commutators in the setting of weighted Lebesgue spaces. Furthermore, we prove weighted endpoint estimates of these commutator operators in Morrey type spaces. In particular, we can obtain strong type and endpoint estimates of vector-valued intrinsic square functions and their commutators in the weighted Morrey spaces and the generalized Morrey spaces.