L p estimates for the commutators of Marcinkiewicz integrals with kernels belonging to certain block spaces

This paper is devoted to the study of the L p ‐mapping properties of the higher order commutators μ k Ω, a , μ *, kΩ, λ , a and μ k Ω, S , a , which are formed respectively by a BMO (ℝ n ) function a ( x ) and a class of rough Marcinkiewicz integral operators μ Ω , μ * Ω, λ and μ Ω, S related to the Littlewood–Paley g ‐function, the Littlewood–Paley g * λ ‐function and the Lusin area integral, respectively. By the method of block decomposition for kernel functions and Fourier transforms estimates, some new results about the L p (ℝ n ) boundedness for theses commutators are obtained. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)