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Singular integrals and weighted Triebel-Lizorkin and Besov spaces of arbitrary number of parameters

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Abstract

Though the theory of Triebel-Lizorkin and Besov spaces in one-parameter has been developed satisfactorily, not so much has been done for the multiparameter counterpart of such a 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 Han and Lu [12]. Our approach of derivation of the boundedness of singular integrals on these spaces is substantially different from those used in the literature where atomic decomposition on the one-parameter Triebel-Lizorkin and Besov spaces played a crucial role. The discrete Littlewood-Paley analysis allows us to avoid using the atomic decomposition or deep Journe’s covering lemma in multiparameter setting.

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Authors and Affiliations

  1. Department of Mathematics, Wayne State University, Detroit, MI, 48202, USA

    Guo Zhen Lu

  2. School of Science, Nantong University, Nantong, 226007, P. R. China

    Yue Ping Zhu

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  1. Guo Zhen Lu
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  2. Yue Ping Zhu
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Correspondence to Yue Ping Zhu.

Additional information

The first author is partly supported by the NSF of USA (Grant No. DMS0901761); the second author is partly supported by NNSF of China (Grant Nos. 10971228 and 11271209) and Natural Science Foundation of Nantong University (Grant No. 11ZY002)

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Lu, G.Z., Zhu, Y.P. Singular integrals and weighted Triebel-Lizorkin and Besov spaces of arbitrary number of parameters. Acta. Math. Sin.-English Ser. 29, 39–52 (2013). https://doi.org/10.1007/s10114-012-1402-7

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  • DOI: https://doi.org/10.1007/s10114-012-1402-7

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