preprint

Inserted: 10 aug 2024
Last Updated: 10 aug 2024

Year: 2024

Abstract:

Sard’s theorem asserts that the set of critical values of a smooth map from one Euclidean space to another one has measure zero. A version of this result for infinite-dimensional Banach manifolds was proven by Smale for maps with Fredholm differential. It is well–known, however, that when the domain is infinite dimensional and the range is finite dimensional, the result is not true – even under the assumption that the map is “polynomial” – and a general theory is still lacking. Addressing this issue, in this paper, we provide sharp quantitative criteria for the validity of Sard’s theorem in this setting. Our motivation comes from sub–Riemannian geometry and, as an application of our results, we prove the sub–Riemannian Sard conjecture for the restriction of the Endpoint map of Carnot groups to the set of piece–wise real–analytic controls with large enough radius of convergence.