Estimates for parametric Marcinkiewicz integrals in BMO and Campanato spaces

Abstract

In this paper, the authors consider the behaviors of a class of parametric Marcinkiewicz integrals μΩρ, μΩ,λ*,ρ and μΩ,Sρ on BMO(ℝn) and Campanato spaces with complex parameter ρ and the kernel Ω in Llog+L(Sn−1). Here μΩ,λ*,ρ and μΩ,Sρ are parametric Marcinkiewicz functions corresponding to the Littlewood-Paley gλ*-function and the Lusin area function S, respectively. Under certain weak regularity condition on Ω, the authors prove that if f belongs to BMO(ℝn) or to a certain Campanato space, then [μΩ,λ*,ρ(f)]2, [μΩ,Sρ(f)]2 and [μΩρ(f)]2 are either infinite everywhere or finite almost everywhere, and in the latter case, some kind of boundedness are also established.