De nouvelles perspectives sur le théorème de Morse–Sard

doi:10.1016/S0764-4442(00)01723-7

Comptes Rendus de l'Académie des Sciences - Series I - Mathematics

Volume 332, Issue 1, 1 January 2001, Pages 13-17

Comptes Rendus de l'...

https://doi.org/10.1016/S0764-4442(00)01723-7 Get rights and content

Résumé

Soient E un espace vectoriel de dimension finie, F un espace vectoriel normé, k≥1 un entier, f:E→F une application de classe C^k, t>0 un réel, et A⊂E un ensemble borné et de $H^{t}$ -mesure finie, où $H^{t}$ est la mesure de Hausdorff en dimension t. Si la différentielle de f en tout point de A est de rang $≤r$ , nous montrons que l'image f(A) est de $H^{d}$ -mesure nulle, avec $d=r+ t−r k$ . Lorsque de plus D^kf satisfait une condition de Hölder d'exposant α∈(0,1], f(A) est de $H^{d̃}$ -mesure finie, avec $d ̃ =r+ t−r k+α$ .

Lorsque E est un espace de Banach, les résultats précédents sont encore vrais si k≤2. En revanche, nous présentons un polynôme $P:ℓ^{2} → R$ dont l'ensemble critique est de $H^{3}$ -mesure finie et a pour image [0,1].

Abstract

Let E be a vector space of finite dimension, F a normed vector space, k≥1 an integer, f:E→F a C^k function, t>0 a real number and A⊂E a bounded subset with finite $H^{t}$ -measure, where $H^{t}$ is the Hausdorff measure of dimension t. If the differential of f in every point of A has rank $≤r$ , we show that the image f(A) has zero $H^{d}$ -measure, where $d=r+ t−r k$ . Moreover, if D^kf satisfies a Hölder condition of exponent α∈(0,1], then f(A) has finite $H^{d̃}$ -measure, where $d ̃ =r+ t−r k+α$ .

If E is a Banach space, then the preceding results are still true if k≤2. On the other side, we present a polynomial function $P:ℓ^{2} → R$ for which the critical set has finite $H^{3}$ -measure, and whose image is [0,1].

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