Endpoint estimates for commutators of intrinsic square functions in the Morrey type spaces

Abstract: In this paper, the boundedness properties of commutators generated by $b$ and intrinsic square functions in the endpoint case are discussed, where $b\in BMO(\mathbb R^n)$. We first establish the weighted weak $L\log L$-type estimates for these commutator operators. Furthermore, we will prove endpoint estimates of commutators generated by $BMO(\mathbb R^n)$functions and intrinsic square functions in the weighted Morrey spaces $L^{1,\kappa}(w)$ for $0<\kappa<1$ and $w\in A_1$, and in the generalized Morrey spaces $L^{1,\Theta}$, where $\Theta$ is a growth function on $(0,+\infty)$ satisfying the doubling condition.