Sobolev spaces on manifolds with lower bounded curvature

Abstract: There are several notions of Sobolev spaces on a Riemannian manifold: from the operator theory viewpoint it is natural to consider Sobolev functions defined by taking the $L^p$ norms of functions and of powers of their Laplacian. Instead, the regularity theory of elliptic equations involves Sobolev functions defined via the $L^p$ norm of all the derivatives up to a certain order. Moreover, Sobolev spaces can be characterized via compactly supported smooth approximations. In this talk, we will focus on non-compact manifolds with lower bounded curvature. We will discuss some results giving the (non)-equivalence between the different Sobolev spaces. In particular, we will highlight the role played in the theory by the Calderon-Zygmund inequality and the Bochner formulas, and we will sketch how to exploit singular metric spaces (e.g. Alexandrov or RCD) as a tool to construct smooth counterexamples.

classical analysis and ODEsdifferential geometryfunctional analysis

Audience: researchers in the topic

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