Cohomology of annuli, duality and L^∞-differential forms on Heisenberg groups

2021/03/03 by Baldi, Annalisa, Franchi, Bruno, Pansu, Pierre
#Classical Analysis and ODEs (math.CA) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2103.02308

Abstract

In the last few years the authors proved Poincaré and Sobolev type inequalities in Heisenberg groups ℍⁿ for differential forms in the Rumin's complex. The need to substitute the usual de Rham complex of differential forms for Euclidean spaces with the Rumin's complex is due to the different stratification of the Lie algebra of Heisenberg groups. The crucial feature of Rumin's complex is that d_c is a differential operator of order 1 or 2 according to the degree of the form. Roughly speaking, Poincaré and Sobolev type inequalities are quantitative formulations of the well known topological problem whether a closed form is exact. More precisely, for suitable p and q, we mean that every exact differential form ω in L^p admits a primitive ϕ in L^q such that‖ϕ‖_L^q≤ C ‖ω‖_L^p. The cases of the norm L^p, p≥ 1 and q

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