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We show various sharp Hardy-type inequalities for the linear and quasi-linearLaplacian on non-compact harmonic manifolds with a particular focus on the caseof Damek-Ricci spaces. Our methods make use of the optimality theory developedby Devyver/Fraas/Pinchover and Devyver/Pinchover and are motivated bycorresponding results for hyperbolic spaces by Berchio/Ganguly/Grillo, andBerchio/Ganguly/Grillo/Pinchover.

Sharp Hardy-type inequalities for non-compact harmonic manifolds and Damek-Ricci spaces