From the L1 norms of the complex heat kernels to a Hörmander multiplier theorem for sub-Laplacians on nilpotent Lie groups

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    Abstract

    This paper aims to prove a Hörmander multiplier theorem for sub-Laplacians on nilpotent Lie groups. We investigate the holomorphic functional calculus of the sub-Laplacians, then we link the L1 norm of the complex time heat kernels with the order of differentiability needed in the Hörmander multiplier theorem. As applications, we show that order d/2 + 1 suffices for homogeneous nilpotent groups of homogeneous dimension d, while for generalised Heisenberg groups with underlying space R2n+k and homogeneous dimension 2n + 2k, we show that order n + (k + 5)/2 for k odd and n + 3 + k/2 for k even is enough; this is strictly less than half of the homogeneous dimension when k is sufficiently large.

    Original languageEnglish
    Pages (from-to)413-424
    Number of pages12
    JournalPacific Journal of Mathematics
    Volume173
    Issue number2
    Publication statusPublished - Apr 1996

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