Commutators of Littlewood-Paley operators

Let b ∈ L _loc(ℝ ⁿ ) and L denote the Littlewood-Paley operators including the Littlewood-Paley g function, Lusin area integral and g _λ^* function. In this paper, the authors prove that the L ^p boundedness of commutators b, L] implies that b ∈ BMO(ℝ ⁿ ). The authors therefore get a characterization of the L ^p -boundedness of the commutators b, L]. Notice that the condition of kernel function of L is weaker than the Lipshitz condition and the Littlewood-Paley operators L is only sublinear, so the results obtained in the present paper are essential improvement and extension of Uchiyama’s famous result.