Abstract
Let G = N ⋊ A, where N is a stratified group and A = ℝ acts on N via automorphic dilations. Homogeneous sub-Laplacians on N and A can be lifted to left-invariant operators on G, and their sum is a sub-Laplacian Δ on G. We prove a theorem of Mihlin–Hörmander type for spectral multipliers of Δ. The proof of the theorem hinges on a Calderón–Zygmund theory adapted to a sub-Riemannian structure of G and on L1-estimates of the gradient of the heat kernel associated to the sub-Laplacian Δ.
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The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
This work was partially supported by the Progetto GNAMPA 2014 “Analisi Armonica e Geometrica su varietà e gruppi di Lie” and by the Progetto PRIN 2010–2011 “Varietà reali e complesse: geometria, topologia e analisi armonica”.
The first-named author gratefully acknowledges the support of the Deutsche Forschungsgemeinschaft (project MA 5222/2-1).
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Martini, A., Ottazzi, A. & Vallarino, M. Spectral multipliers for sub-Laplacians on solvable extensions of stratified groups. JAMA 136, 357–397 (2018). https://doi.org/10.1007/s11854-018-0063-6
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DOI: https://doi.org/10.1007/s11854-018-0063-6