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Singular integrals and weighted Triebel-Lizorkin and Besov spaces of arbitrary number of parameters. (English) Zbl 1261.42030

Summary: Though the theory of Triebel-Lizorkin and Besov spaces in one parameter has been developed satisfactorily, not so much has been achieved for the multiparameter counterpart of this theory. In this paper, we introduce the weighted Triebel-Lizorkin and Besov spaces with an arbitrary number of parameters and prove the boundedness of singular integral operators on these spaces using discrete Littlewood-Paley theory and Calderón’s identity. This is inspired by the work of discrete Littlewood-Paley analysis with two parameters of implicit dilations associated with the flag singular integrals recently developed by Y. Han and G. Lu [“Discrete Littlewood-Paley-Stein theory and multi-parameter Hardy spaces associated with flag singular integrals”, preprint, arxiv:0801.1701]. Our approach of derivation of the boundedness of singular integrals on these spaces is substantially different from those used in the literature in which atomic decomposition on the one-parameter Triebel-Lizorkin and Besov spaces playes a crucial role. The discrete Littlewood-Paley analysis allows us to avoid using the atomic decomposition or Journe’s covering lemma in the multiparameter setting.

MSC:

42B25 Maximal functions, Littlewood-Paley theory
42B35 Function spaces arising in harmonic analysis
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