, Volume 52, Issue 11, pp 2493-2505
Date: 04 Dec 2009

Commutators of Littlewood-Paley operators

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Abstract

Let bL 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.

This work was partially supported by National Natural Science Foundation of China (Grant No. 10931001, 10826046) and Specialized Research Foundation for Doctor Programme (Grant No. 20050027025)