Sharp Morrey-Sobolev inequalities on complete Riemannian manifolds
The author recalls the well known result that \(W^{1,p}(\mathbb R^n)\), the Sobolev space of order one (such that \(p>n\)), is embedded into the space of bounded measurable functions, this embedding takes the name of Morrey Sobolev Inequality (MSI for short). In the present article, the author enhances the previous result, by giving a new embedding on a complete Riemannian manifold. Precisely, the author considers \((M,g)\), a smooth manifold, simply connected, complete, and endowed with a negative sectional curvature manifold, besides \((M,g)\) satisfies the Cartan-Hadamard conjecture, that is, roughly speaking, any compact domain with smooth boundary satisfies the isoperimetric inequality. Therefore, in the space of Lipschitz functions with compact support in \(M\), the author gives the MSI, besides, up to a multiplicative constant. The constant bound, \(C(p,n)\), is given in terms of the Hausdorff measure of a defined compact set. Also he provides necessary and sufficient conditions for \((M,g)\) to be isometric to the Euclidean space (Theorem 1.1). Concerning the second result, the author considers a smooth complete manifold equipped with a positive Ricci curvature and shows that the volume of a geodesic ball of radius \(\rho\) is lower bounded by a function comprised by a constant \(C\geq C(p,n)\), \(\rho\), and \(\omega_n\) (the volume of the Euclidean unit ball) (Theorem 1.2). Also through an interpolation, he gives versions of the previous results such that the constant bound is \(L_1\)-bound (Theorem 1.3).
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