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In this paper we study the family of embeddings Φ t of a compact RCD * (K, N ) space (X, d, m) into L 2 (X, m) via eigenmaps. Extending part of the classical results [B85, BBG94] known for closed Riemannian manifolds, we prove convergence as t ↓ 0 of the rescaled pull-back metrics Φ * t g L 2 in L 2 (X, m) induced by Φ t . Moreover we discuss the behavior of Φ * t g L 2 with respect to measured Gromov-Hausdorff convergence and t. Applications include the quantitative L p -convergence in the noncollapsed setting for all p < ∞, a result new even for closed Riemannian manifolds and Alexandrov spaces.

In this note, we prove global weighted Sobolev inequalities on non-compact CD(0,N) spaces satisfying a suitable growth condition, extending to possibly non-smooth and non-Riemannian structures a previous result from Minerbe stated for Riemannian manifolds with non-negative Ricci curvature. We use this result in the context of RCD(0,N) spaces to get a uniform bound of the corresponding weighted heat kernel via a weighted Nash inequality.

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