[Paper Review] On some universal Morse-Sard type Theorem

Adele Ferone, Mikhail V. Korobkov+1 more|arXiv (Cornell University)|Jun 15, 2017

Mathematical Analysis and Transform Methods41 references4 citations

TL;DR

This paper establishes a universal Morse-Sard-type theorem for Hölder and Sobolev mappings, proving that for $ v \in C^{k,\alpha} $ or $ W^{k}_p $ with $ p = \max(1, n/k) $, the $ \mathcal{H}^\mu $-measure of the preimage of almost every point under the $ m $-critical set vanishes, where $ \mu = n - m - (k + \alpha)(q - m) $. The result unifies classical and Dubovitskiï´- Federer theorems and extends them to low-regularity settings using $ N $-properties and entropy estimates.

ABSTRACT

The classical Morse--Sard theorem claims that for a mapping $v:\mathbb R^n o\mathbb R^{m+1}$ of class $C^k$ the measure of critical values $v(Z_{v,m})$ is zero under condition $k\ge n-m$. Here the critical set, or $m$-critical set is defined as $Z_{v,m} = \{ x \in \mathbb R^n : \, { m rank}\, abla v(x)\le m \}$. 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 $C^k$ category. Here we formulate and prove a extit{bridge theorem} that includes all the above results as particular cases: namely, if a function $v:\mathbb R^n o\mathbb R^d$ belongs to the Holder class $C^{k,α}$, $0\leα\le1$, then for every $q>m$ the identity $$\mathcal H^μ(Z_{v,m}\cap v^{-1}(y))=0$$ holds for $\mathcal H^q$-almost all $y\in\mathbb R^d$, where $μ=n-m-(k+α)(q-m)$. The result is new even for the classical $C^k$-case (when $α=0$); a similar result is established for the Sobolev classes of mappings $W^k_p(\mathbb R^n,\mathbb R^d)$ 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).

Motivation & Objective

• To unify and generalize classical Morse-Sard, Dubovitskiï´, and Federer theorems into a single framework applicable to less smooth mappings.
• To establish sharp measure estimates for the preimage of critical sets under Hölder and Sobolev mappings with minimal integrability assumptions.
• To demonstrate that the set of nondifferentiability points in Sobolev mappings is negligible in the context of the generalized Morse-Sard theorem.
• To extend the theory to fractional Sobolev spaces and Bessel potential spaces using deep analytic and geometric tools.

Proposed method

• Formulating a bridge theorem that interpolates between classical $ C^k $, Hölder $ C^{k,\alpha} $, and Sobolev $ W^k_p $ settings via a parameter $ \mu = n - m - (k + \alpha)(q - m) $.
• Using $ N $-properties of Sobolev mappings to show that the set of nondifferentiability points has zero $ \mathcal{H}^{\tau_*} $-measure with $ \tau_* = n - k + 1 $.
• Applying Y. Yomdin's entropy estimates for near-critical values of polynomials, relying on algebraic geometry techniques.
• Establishing uniform estimates on the measure of $ Z'_{v} \cap Q $ for $ n $-dimensional intervals $ Q $, using approximation by polynomials and maximal function techniques.
• Proving $ L^1 $-based estimates for $ W^k_1 $-mappings via extension theorems and maximal function bounds on $ \nabla v_Q $.
• Adapting and refining techniques from prior joint work with J. Bourgain and J. Kristensen (2013, 2015) to handle limiting integrability cases.

Experimental results

Research questions

• RQ1 Can the Morse-Sard theorem be extended to mappings with minimal regularity, such as $ C^{k,\alpha} $ or $ W^k_p $ with $ p = \max(1, n/k) $?
• RQ2 What is the sharp Hausdorff dimension of the preimage $ Z_{v,m} \cap v^{-1}(y) $ for $ \mathcal{H}^q $-almost every $ y \in \mathbb{R}^d $?
• RQ3 How can the set of nondifferentiability points in Sobolev mappings be controlled in measure-theoretic terms for critical value analysis?
• RQ4 Is there a unifying framework that includes the classical Morse-Sard, Dubovitskiï´, and Federer theorems as special cases?
• RQ5 Can entropy estimates from algebraic geometry be used to derive sharp measure bounds for critical sets in low-regularity settings?

Key findings

• For $ v \in C^{k,\alpha}({\mathbb{R}}^n, {\mathbb{R}}^d) $, the $ \mathcal{H}^\mu $-measure of $ Z_{v,m} \cap v^{-1}(y) $ is zero for $ \mathcal{H}^q $-almost every $ y \in {\mathbb{R}}^d $, where $ \mu = n - m - (k + \alpha)(q - m) $.
• The result is new even for the classical $ C^k $-case ($ \alpha = 0 $), providing a unified framework that includes both Morse-Sard and Dubovitskiï´- Federer theorems as special cases.
• For $ W^k_1({\mathbb{R}}^n, {\mathbb{R}}^d) $ with $ k \geq n $, the set $ A_v $ of nondifferentiability points satisfies $ \mathcal{H}^{n-k+1}(A_v) = 0 $, making it negligible in the measure-theoretic analysis.
• In the limiting case $ p = 1 $, the estimate $ \Phi(Z'_{v} \cap Q) \leq C \sigma^{q-m} r^{q + \mu + (k-1-n)(q-m)} $ holds for $ k > n $, with $ \sigma = \|\nabla^k v\|_{L_1(Q)} $.
• For $ k = n $, the estimate $ \Phi(Z'_{v} \cap Q) \leq C (\sigma^q r^\mu + \sigma^{q-m} r^{\mu + m}) $ holds under the condition $ q + \mu \geq 1 $.
• The use of Yomdin's entropy estimates for polynomials enables sharp control over the measure of near-critical values, crucial for the proof in low-regularity regimes.