Geodesic
On integrability of the Banach indicatrix of a smooth mapping
Izvestiya. Mathematics, Tome 25 (1985) no. 1, pp. 19-44 
S. A. Gulevich. On integrability of the Banach indicatrix of a smooth mapping. Izvestiya. Mathematics, Tome 25 (1985) no. 1, pp. 19-44. http://geodesic.mathdoc.fr/item/IM2_1985_25_1_a1/
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Voir la notice de l'article provenant de la source Math-Net.Ru

Let $f\colon\mathbf R^n\to\mathbf R^n$ be a mapping of class $C^{(l)}$ with compact support, and let $N(f,y)$ be the number of solutions of the equation $f(x)=y$. It is proved that if $p$, then $\int[N(f,y)]^p\,dy\infty$, and an example is given which shows that this integral can diverge if $p>l$. Bibliography: 9 titles.

[1] Gulevich S. A., “O dostatochnom uslovii integriruemosti indikatrisy Banakha gladkogo otobrazheniya”, Izv. AN SSSR. Ser. matem., 47:5 (1983), 1114–1134 | MR | Zbl

[2] Breker T., Lander L., Differentsiruemye rostki i katastrofy, Mir, M., 1977

[3] Van der Varden B. L., Algebra, Nauka, M., 1976 | MR

[4] Ivanov L. D., Variatsii mnozhestv i funktsii, Nauka, M., 1975 | MR

[5] Vitushkin A. G., O mnogomernykh variatsiyakh, Gostekhizdat, M., 1955

[6] Khodzh V., Pido D., Metody algebraicheskoi geometrii, t. 1–3, IL, M., 1954, 1955

[7] Federer H., Geometric measure theory, Springer, Berlin, 1969 | MR

[8] Yomdin Y., Global bounds for the Betti numbers of regular fibers of differentiable mappings, preprint, Max-Planck-Institut für Mathematik, Bonn, 1983 | MR

[9] Merkov A. B., “O nekotorykh svoistvakh gladkikh otobrazhenii”, Vestnik MGU. Ser. matem., 1979, no. 2, 63–69 | MR | Zbl