2017/06/15 by Adele Ferone, Ferone, Adele, Mikhail V. Korobkov +3 · 1 citation
Mathematics · #58C25 (26B35 46E30) #Analysis of PDEs (math.AP) #FOS: Mathematics #Holomorphic and Operator Theory #Mathematical Analysis and Transform Methods #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.1706.05266
openalex publication_date 2017/06/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The classical Morse--Sard theorem claims that for a mapping v:\mathbb Rn→\mathbb Rm+1 of class Ck the measure of critical values v(Zv,m) is zero under condition k≥ n-m. Here the critical set, or m-critical set is defined as Zv,m = \ x ∈ \mathbb Rn : \rm rank ∇ v(x)≤ m \. Further Dubovitski\uı in 1957 and independently Federer and Dubovitski\uı in 1967 found some elegant extensions of this theorem to the case of other (e.g., lower) smoothness assumptions. They also established the sharpness of their results within the Ck category. Here we formulate and prove a bridge theorem that includes all the above results as particular cases: namely, if a function v:\mathbb Rn→\mathbb Rd belongs to the Holder class Ck,α, 0≤α≤1, then for every q>m the identity \mathcal Hμ(Zv,m∩ v-1(y))=0 holds for \mathcal Hq-almost all y∈\mathbb Rd, where μ=n-m-(k+α)(q-m). The result is new even for the classical Ck-case (when α=0); a similar result is established for the Sobolev classes of mappings Wkp(\mathbb Rn,\mathbb Rd) with minimal integrability assumptions p=max(1,n/k), i.e., it guarantees in general only the continuity (not everywhere differentiability) of a mapping. However, using some N-properties for Sobolev mappings, established in our previous paper, we obtained that the sets of nondifferentiability points of Sobolev mappings are fortunately negligible in the above bridge theorem. We cover also the case of fractional Sobolev spaces. The proofs of the most results are based on our previous joint papers with J. Bourgain and J. Kristensen (2013, 2015).