Extremal maps in best constants vector theory. Part I: Duality and compactness

We develop a comprehensive study on sharp potential type Riemannian L2-Sobolev inequalities by means of a local geometric Sobolev inequality of the same kind and suitable De Giorgi–Nash–Moser estimates. In particular we discuss questions like continuous dependence of optimal constants and existence and compactness of extremal maps. The main obstacle arising in the present setting lies at fairly weak conditions of regularity assumed on potential functions.