Marcinkiewicz multipliers and Lipschitz spaces on Heisenberg groups

Yanchang Han, Yongsheng Han, Ji Li*, Chaoqiang Tan

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    3 Citations (Scopus)

    Abstract

    The Marcinkiewicz multipliers are Lp bounded for 1 < p < ∞ on the Heisenberg group Hn Cn × R (Müller, Ricci, and Stein). This is surprising in the sense that these multipliers are invariant under a two parameter group of dilations on Cn × R, while there is no two parameter group of automorphic dilations on Hn. The purpose of this paper is to establish a theory of the flag Lipschitz space on the Heisenberg group Hn Cn × R that is, in a sense, intermediate between that of the classical Lipschitz space on the Heisenberg group Hn and the product Lipschitz space on Cn × R. We characterize this flag Lipschitz space via the Littlewood–Paley theory and prove that flag singular integral operators, which include the Marcinkiewicz multipliers, are bounded on these flag Lipschitz spaces.

    Original languageEnglish
    Pages (from-to)607-627
    Number of pages21
    JournalCanadian Journal of Mathematics
    Volume71
    Issue number3
    DOIs
    Publication statusPublished - Jun 2019

    Keywords

    • Heisenberg group
    • Marcinkiewicz multiplier
    • flag singular integral
    • flag Lipschitz space
    • reproducing formula
    • discrete Littlewood–Paley analysis

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