We generalize to the RCD(0, N ) setting a family of monotonicity formulas by Colding and Minicozzi for positive harmonic functions in Riemannian manifolds with non-negative Ricci curvature. Rigidity and almost rigidity statements are also proven, the second appearing to be new even in the smooth setting.Motivated by the recent work in [AFM] we also introduce the notion of electrostatic potential in RCD spaces, which also satisfies our monotonicity formulas.Our arguments are mainly based on new estimates for harmonic functions in RCD(K, N ) spaces and on a new functional version of the '(almost) outer volume come implies (almost) outer metric cone' theorem.

We prove that Stein's extension operator preserves Sobolev–Morrey spaces, that is spaces of functions with weak derivatives in Morrey spaces. The analysis concerns classical and generalized Morrey spaces on bounded and unbounded domains with Lipschitz boundaries in the n‐dimensional Euclidean space.

We prove that if M is a closed n-dimensional Riemannian manifold, $$n \ge 3$$ n ≥ 3 , with $$\mathrm{Ric}\ge n-1$$ Ric ≥ n - 1 and for which the optimal constant in the critical Sobolev inequality equals the one of the n-dimensional sphere $$\mathbb {S}^n$$ S n , then M is isometric to $$\mathbb {S}^n$$ S n . An almost-rigidity result is also established, saying that if equality is almost achieved, then M is close in the measure Gromov–Hausdorff sense to a spherical suspension. These statements are obtained in the $$\mathrm {RCD}$$ RCD -setting of (possibly non-smooth) metric measure spaces satisfying synthetic lower Ricci curvature bounds. An independent result of our analysis is the characterization of the best constant in the Sobolev inequality on any compact $$\mathrm {CD}$$ CD space, extending to the non-smooth setting a classical result by Aubin. Our arguments are based on a new concentration compactness result for mGH-converging sequences of $$\mathrm {RCD}$$ RCD spaces and on a Pólya–Szegő inequality of Euclidean-type in $$\mathrm {CD}$$ CD spaces. As an application of the technical tools developed we prove both an existence result for the Yamabe equation and the continuity of the generalized Yamabe constant under measure Gromov–Hausdorff convergence, in the $$\mathrm {RCD}$$ RCD -setting.