Sard properties for polynomial maps in infinite dimension
| Autor: | Lerario, Antonio, Rizzi, Luca, Tiberio, Daniele |
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| Rok vydání: | 2024 |
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| Druh dokumentu: | Working Paper |
| Popis: | Motivated by the Sard conjecture in sub-Riemannian geometry, and more specifically on Carnot groups, we consider the set of maps defined on a Hilbert space $H$: \[\mathscr{P}_{d}^m(H)=\bigg\{f:H\to \mathbb{R}^m\,\bigg|\, \forall\, E\subset H, \,\dim(E)<\infty \implies f|_{E} \textrm{ is a polynomial map of degree $d$}\bigg\}. \] Example of maps belonging to this class are: the map $f:\ell^2\to \mathbb{R}$ constructed by Kukpa as a counterexample for the Sard theorem in infinite dimension; the Endpoint maps of Carnot groups. We prove that, for infinite-dimensional subspaces $V \subset H$ satisfying quantitative assumptions on their Kolmogorov $n$-width, the restriction of $f$ on $V$ has the Sard property, i.e. the set of critical values of $f|_{V}$ has measure zero in $\mathbb{R}^m$. We prove various quantitative versions of this result, allowing to study the Sard property also for the case $V=H$. By constructing suitable examples, we prove the sharpness of our $n$-width assumptions. We apply these results to sub-Riemannian geometry, proving that on Carnot groups the Sard property holds true for the restriction of the Endpoint map on the set of real-analytic controls with large enough radius of convergence. This is a more functional-analytic approach to the Sard conjecture that, unlike previous ones, does not resort to reduction to finite-dimensions. Comment: 39 pages |
| Databáze: | arXiv |
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