2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/64912We study the horizontal Laplacian $Δ^H$ associated to the Hopf fibration $S^3\to S^2$ with arbitrary Chern number $k$. We use representation theory to calculate the spectrum, describe the heat kernel and obtain the complete heat trace asymptotics of $Δ^H$. We express the Green functions for associated Poisson semigroups and obtain bounds for their contraction properties and Sobolev inequalities for $Δ^H$. The bounds and inequalities improve as $|k|$ increases.19 pagesAnalysis of PDEsDifferential Geometry35K05; 35P10; 58J35text