L² continuity of the Calderón type commutator for the Littlewood-Paley operator with rough variable kernel

Abstract

For b ∈ Lip(Rⁿ), the Calderón type commutator for the Littlewood-Paley operator with variable kernel is defined by

μ_Ω,1;b(f)(x) = .

By giving a method based on Littlewood-Paley theory, Fourier transform and the spherical harmonic development, we prove the L² norm inequalities for the rough operators μ_Ω,1;b with Ω(x,z′) ∈ L^∞(Rⁿ) × L^q(S^n-1) satisfying certain cancellation conditions.