Mathematical Research Letters

Volume 23 (2016)

Number 5

Boundedness of composition operators associated with mixed homogeneities on Lipschitz spaces

Pages: 1387 – 1403

DOI: http://dx.doi.org/10.4310/MRL.2016.v23.n5.a7

Authors

Yanchang Han (School of Mathematic Sciences, South China Normal University, Guangzhou, China)

Yongsheng Han (Department of Mathematics, Auburn University, Auburn, Alabama, U.S.A.)

Abstract

This note is motivated by Phong and Stein’s work in [PS]. We introduce a new class of Lipschitz spaces associated with mixed homogeneities and characterize these spaces via the Littlewood–Paley theory. We prove that the composition of two Calderón–Zygmund singular integral operators with different homogeneities is bounded on these Lipschitz spaces. Our result answers a question arises in the context of Lipschitz spaces due to the constructions in [MR].

Keywords

Lipschitz spaces, singular integral operator, composition operator, isotropic homogeneity, non-isotropic homogeneity, Littlewood–Paley theory

2010 Mathematics Subject Classification

Primary 42B20. Secondary 42B25, 46A35.

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Published 12 January 2017