2015/05/29 by Piotr Hajłasz, Hajłasz, Piotr, Scott Zimmerman +1
Mathematics · #46E35 #58C25 #Classical Analysis and ODEs (math.CA) #Differential Equations and Boundary Problems #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics #math.CA #msc:46E35 #msc:58C25
paper · pdf · doi:10.48550/arxiv.1506.00025
arxiv created 2015/05/29 · openalex publication_date 2015/05/29 · arxiv updated 2015/06/02 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
The Sard theorem from 1942 requires that a mapping f:ℝn → ℝm is of class Ck, k > max (n-m,0). In 1957 Duvovitski\uı generalized Sard's theorem to the case of Ck mappings for all k. Namely he proved that, for almost all y∈ ℝm, Hℓ(Cf ∩ f-1(y))=0 where ℓ = max(n-m-k+1,0), \mathcal Hℓ denotes the Hausdorff measure, and Cf is the set of critical points of f. In 2001 De Pascale proved that the Sard theorem holds true for Sobolev mappings of the class W\rm lock,p(ℝn,ℝm), k>max(n-m,0) and p>n. We will show that also Dubovitski\uı's theorem can be generalized to the case of W\rm lock,p(ℝn,ℝm) mappings for all k∈ℕ and p>n.