L^p-spectral multipliers for the Hodge Laplacian acting on 1-forms on the Heisenberg group

2005/08/27 by Detlef Müller, Müller, Detlef, Marco M. Peloso +3
Mathematics · #Advanced Algebra and Geometry #Advanced Harmonic Analysis Research #Mathematical Analysis and Transform Methods #math.AP #math.CA #msc:42B15 #msc:43A80

paper · pdf · doi:10.48550/arxiv.math/0508543

34 pages

arxiv created 2005/08/27 · arxiv updated 2009/12/01

Abstract

We prove that, if Δ₁ is the Hodge Laplacian acting on differential 1-forms on the (2n+1)-dimensional Heisenberg group, and if m is a Mihlin-Hörmander multiplier on the positive half-line, with L²-order of smoothness greater than n+1/2, then m(Δ₁) is L^p-bounded for 1<p<∞. Our approach leads to an explicit description of the spectral decomposition of Δ₁ on the space of L²-forms in terms of the spectral analysis of the sub-Laplacian L and the central derivative T, acting on scalar-valued functions.

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