2024/07/02 by Antonio Lerario, Luca Rizzi, Lerario, Antonio +3
Mathematics · #14P10 #53C17 #Advanced Differential Equations and Dynamical Systems #Advanced Topics in Algebra #Differential Geometry (math.DG) #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Dynamics and Fractals #Metric Geometry (math.MG) #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.2407.02296
openalex publication_date 2024/07/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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, and the strong Sard conjecture for the restriction to the set of piece-wise entire controls.