Boundedness of Commutators for Multilinear Marcinkiewicz Integrals with Generalized Campanato Functions on Generalized Morrey Spaces

Abstract

This paper is devoted to exploring the mapping properties for the commutator $\mu_{Ω,\vec{b}}$ generated by multilinear Marcinkiewicz integral operators $\mu_Ω$ with a locally integrable function $\vec{b}= (b_1,···,b_m)$ on the generalized Morrey spaces. $\mu_{Ω,\vec{b}}$ is bounded from $L^{(p_1 ,\varphi_1)} (\mathbb{R}^n )×···×L^{(p_m,\varphi_m)} (\mathbb{R}^n)$ to $L ^{(q,\varphi)} (\mathbb{R}^n),$ where $L^{(p_i ,\varphi_i )} (\mathbb{R}^n),$ $L^{(q,φ)} (\mathbb{R}^n)$ are generalized Morrey spaces with certain variable growth condition, that $b_j(j=1,···,m)$ is a function in generalized Campanato spaces, which contain the BMO$(\mathbb{R}^n)$ and the Lipschitz spaces ${\rm Lip}_α(\mathbb{R}^n) (0<α≤1)$ as special examples.