The authors consider the Hodge Laplacian Δ on the Heisenberg group H n , endowed with a left-invariant and U(n) -invariant Riemannian metric. For 0≤k≤2n 1 , let Δ k denote the Hodge Laplacian restricted to k -forms.In this paper they address three main, related questions:(1) whether the L 2 and L p -Hodge decompositions, 1(2) whether the Riesz transforms dΔ −12 k are L p -bounded, for 1<<∞ ; (3) how to prove a sharp Mihilin-Hörmander multiplier theorem for Δ k , 0≤k≤2n 1 .
Detlef Muller, Universitat Kiel, Germany.Marco M. Peloso, Universita Degli Studi Di Mila, Milano, Italy.Fulvio Ricci, Scuola Normale Superiore, Pisa, Italy.
IntroductionDifferential forms and the Hodge Laplacian on HnBargmann representations and sections of homogeneous bundlesCores, domains and self-adjoint extensionsFirst properties of Dk exact and closed formsA decomposition of L2 Lkh related to the ¶ and ¶ complexesIntertwining operators and different scalar forms for DkUnitary intertwining operators and projectionsDecomposition of L2 LkLp-multipliersDecomposition of LpLk and boundedness of the Riesz transformsApplicationsAppendixBibliography