Weighted Lp Estimates for Multiple Generalized Marcinkiewicz Functions

In this paper we investigate the weighted Lp boundedness of generalized Marcinkiewicz integrals MK(ε) over multiple symmetric domains. Under the conditions K∈Lq(Sm−1×Sn−1), q>1, we establish suitable weighted Lp bounds for the integrals MK(ε). These bounds are combined with Yano’s extrapolation argument to obtain the weighted Lp boundedness of MK(ε) from the Triebel–Lizorkin space F.p0,ε(ω1,ω2) to the space Lp(ω1,ω2) under the weaker conditions K∈Bq(0,2ε−1)(Sm−1×Sn−1)∪L(logL)2/ε(Sm−1×Sn−1). Our findings are essential improvements and extension of several known findings in the literature.