The Morse criticality revisited and some new applications to the Morse–Sard theorem

  • Xiaosong Kang
  • Xu XuEmail author
  • Dunmu Zhang


Given a \(C^{r}\) function \(f:U\subset {\mathbf {R}}^{n}\rightarrow {\mathbf {R}} ^{m} \). Inspired by a recent result due to Moreira and Ruas (Manuscr Math 129:401–408, 2009), we show that for any \(x\in U\), there exists a \(\delta (x)>0\), such that for all \(y\in B(x,\delta (x))\cap U\), it holds
$$\begin{aligned} |f(x)-f(y)|\le C_{1}|df(y)||x-y|+C_{2}\sup _{0\le t\le 1}|D^{r}f(x+t(y-x))-D^{r}f(x)||x-y|^{r}, \end{aligned}$$
where \(C_{1},C_{2}\) depends only on nm. This inequality can be thought as a generalized Bochnak–Łojasiewicz inequality for smooth functions, it contains a “polynomial term” and a correction term from the finite differentiability. When \(df(y)=0\), the inequality improves the classical Morse criticality theorem, therefore, our approach unifies and simplifies various results on Morse criticality theorems, and leads to some streamlined proofs of the Morse–Sard type theorems. To showcase the wide applicability of our inequality, we provide two novel Morse–Sard type results. Define \( \Sigma _{f}^{\nu }=\{x\in U\) | rank\((df(x))\le \nu \dot{\}}\). In the first place, if \(f\in C^{k+(\alpha )},k\ge 1,k\in {\mathbb {N}},0<\alpha \le 1 \), ( \(C^{k+(\alpha )}\), Moreira’s class), then the packing dimension \(\dim _{ {\mathcal {P}}}f(\Sigma _{f}^{0})\le \frac{n}{k+\alpha }\). Secondly, we consider \(f\in W^{k+s,p}(U;{\mathbf {R}}^{m}),n>m,k\ge 1,sp>n,0<s\le 1\), \( W^{k+s,p}\) is the (possibly fractional) Sobolev space. We will show that, for \(f\in W^{k+s,p}(U;{\mathbf {R}}^{m})\), \({\mathscr {L}}^{m}(f(\Sigma _{f}^{\nu }))=0\) if \(k+1\ge \frac{n-\nu }{m-\nu },\nu =0,1, \ldots ,m-1\); for \(f\in W^{k+s,p},0<s<1,{\mathscr {L}}^{m}(f(\Sigma _{f}^{0}))=0\), if \(k+s\ge \frac{n}{ m}\). To the best of our knowledge, it’s the first result on the Morse–Sard theorem for fractional Sobolev spaces.

Mathematics Subject Classification

58C25 46T20 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



We thank an anonymous referee for several useful comments, corrections, and suggestions.


  1. 1.
    Alberti, G., Bianchini, S., Grippa, G.: Structure of level sets and Sard-type properties of Lipschitz maps. Ann. Scuda Norm. Sup. Pisa Cl. Sci. (5) 12(4), 863–902 (2013)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bates, S.M.: Toward a precise smoothness hypothesis in Sard’s theorem. Proc. Am. Soc. 117(1), 279–283 (1993)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bochnak, J., Łojasiewicz, S.: A converse of the Kuiper–Kuo theorem. Lecture Notes Math. 192, 254–261 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bourgain, J., Korobkov, M., Kristensen, J.: On the Morse–Sard property and level sets of Sobolev and BV functions. Rev. Math. Iberoam. 29(31), 1–23 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    de Moreira, C.G.T.: Hausdorff measure and the Morse–Sard theorem. Publ. Math. 45(1), 149–162 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    De Pascale, L.: The Morse–Sard theorem in Sobolev spaces. Indiana Univ. Math. J. 50(3), 1371–1386 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Evans, C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton (1992)zbMATHGoogle Scholar
  8. 8.
    Fathi, A., Figalli, A., Rifford, L.: On the Hausdorff dimension of the Mather quotient. Commun. Pure Appl. Math. 6(4), 445–500 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ferry, S.: When \(\varepsilon \)-boundaries are manifolds. Fundam. Math. 90(3), 199–210 (1975–1976)Google Scholar
  10. 10.
    Figalli, A.: A simple proof of the Morse–Sard theorem in Sobolev spaces. Proc. Am. Math. Soc. 136(10), 3675–3681 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hajlasz, P.: Whitney’s example by way of Assouad’s embedding. Proc. Am. Math. Soc. 131(11), 3463–3467 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
  13. 13.
    Mattila, P.: Geometry of Sets and Measure in Euclidean Spaces. Cambridge University Press, Cambridge (1995)CrossRefzbMATHGoogle Scholar
  14. 14.
    Moreira, C., Ruas, M.: The curve selection lemma and the Morse–Sard theorem. Manuscr. Math. 129, 401–408 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Morse, A.P.: The behavior of a function on its critical set. Ann. Math. 40, 62–70 (1939)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Nezza, Di, Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136(5), 521–573 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Norton, A.: A critical set with nonnull image has large Hausdorff dimension. Trans. Am. Math. Soc. 296, 367–376 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Sard, A.: The measure of the critical values of differentiable maps. Bull. Am. Math. Soc. 48, 883–890 (1942)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Stein, E.: Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series, vol. 30. Princeton University Press, Princeton (1970)Google Scholar
  20. 20.
    Swanson, D.: Pointwise inequalities and approximations in fractional Sobolev spaces. Studia Math. 149, 147–174 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Whitney, H.: A function not constant on a connected set of critical points. Duke Math. J. 1, 514–517 (1935)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Whitney, H.: On the extension of differentiable functions. Bull. Am. Math. Soc. 50, 76–81 (1944)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Yomdin, Y.: The geometry of critical and near critical values of differentiable mapping. Math. Ann. 264(4), 495–515 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Yomdin, Y.: The set of zeros of an “almost polynomial”function. Proc. Am. Math. Soc. 90, 538–542 (1984)zbMATHGoogle Scholar
  25. 25.
    Yomdin, Y.: Global bounds for the Betti number of regular fibres of differentiable mappings. Topology 24(2), 145–152 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Yomdin, Y., Comte, G.: Tame geometry with applications in smooth analysis. Lecture Notes in Mathematics, vol. 1834. Springer, Berlin (2004)Google Scholar
  27. 27.
    Ziemer, W.: Weakly Differentiable Functions. Springer, New York (1989)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsWuhan UniversityWuhanChina

Personalised recommendations