Higher order Glaeser inequalities and optimal regularity of roots of real functions
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 12 (2013) no. 4, p. 1001-1021

We prove a higher order generalization of the Glaeser inequality, according to which one can estimate the first derivative of a function in terms of the function itself and the Hölder constant of its k-th derivative.

We apply these inequalities in order to obtain pointwise estimates on the derivative of the (k+α)-th root of a function of class C k whose derivative of order k is α-Hölder continuous. Thanks to such estimates, we prove that the root is not just absolutely continuous, but its derivative has a higher summability exponent.

Some examples show that our results are optimal.

Published online : 2019-02-21
Classification:  26A46,  26B30,  26A27
@article{ASNSP_2013_5_12_4_1001_0,
     author = {Ghisi, Marina and Gobbino, Massimo},
     title = {Higher order Glaeser inequalities and optimal regularity of roots of real functions},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 12},
     number = {4},
     year = {2013},
     pages = {1001-1021},
     zbl = {1317.26010},
     mrnumber = {3184577},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2013_5_12_4_1001_0}
}
Ghisi, Marina; Gobbino, Massimo. Higher order Glaeser inequalities and optimal regularity of roots of real functions. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 12 (2013) no. 4, pp. 1001-1021. http://www.numdam.org/item/ASNSP_2013_5_12_4_1001_0/

[1] D. Alekseevsky, A. Kriegl, M. Losik and P. W. Michor, Choosing roots of polynomials smoothly, Israel J. Math. 105 (1998), 203–233. | MR 1639759 | Zbl 0912.26006

[2] J. M. Bony, F. Colombini and L. Pernazza, On square roots of class C m of nonnegative functions of one variable, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 9 (2010), 635–644. | Numdam | MR 2722658 | Zbl 1207.26004

[3] M. D. Bronšteĭn, Smoothness of roots of polynomials depending on parameters, Sibirsk. Mat. Zh. 20 (1979), 493–501, 690 (English translation: Siberian Math. J. 20 (1979), 347–352 (1980)). | MR 537355 | Zbl 0429.30007

[4] F. Colombini, E. Jannelli and S. Spagnolo, Well-posedness in the Gevrey classes of the Cauchy problem for a nonstrictly hyperbolic equation with coefficients depending on time, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 10 (1983), 291–312. | Numdam | MR 728438 | Zbl 0543.35056

[5] F. Colombini and N. Lerner, Une procédure de Calderón-Zygmund pour le problème de la racine k-ième, Ann. Mat. Pura Appl. (4) 182 (2003), 231–246. | MR 1985565 | Zbl 1175.42004

[6] F. Colombini, N. Orrù and L. Pernazza, On the regularity of the roots of hyperbolic polynomials, Israel J. Math. 191 (2012), 923–944. | MR 3011501 | Zbl 1282.26020

[7] G. Glaeser, Racine carrée d’une fonction différentiable, Ann. Inst. Fourier (Grenoble) 13 (1963), 203–210. | Numdam | MR 163995 | Zbl 0128.27903

[8] T. Kato, “A Short Introduction to Perturbation Theory for Linear Operators”, Springer-Verlag, New York-Berlin, 1982. | MR 678094 | Zbl 0493.47008

[9] A. Kriegl, M. Losik and P. W. Michor, Choosing roots of polynomials smoothly. II, Israel J. Math. 139 (2004), 183–188. | MR 2041790 | Zbl 1071.26009

[10] T. Mandai, Smoothness of roots of hyperbolic polynomials with respect to one-dimensional parameter, Bull. Fac. Gen. Ed. Gifu Univ. 21 (1985), 115–118. | MR 840968

[11] A. Rainer, Perturbation of complex polynomials and normal operators, Math. Nachr. 282 (2009), 1623–1636. | MR 2588826 | Zbl 1219.26014

[12] S. Spagnolo, On the absolute continuity of the roots of some algebraic equations, Ann. Univ. Ferrara Sez. VII (N.S.), 45 (1999) suppl., 327–337. | MR 1806507 | Zbl 0993.30004

[13] S. Tarama, On the Lemma of Colombini, Jannelli and Spagnolo, Mem. of the Faculty of Engineering Osaka City Univ. 41 (2000), 111–115.

[14] S. Tarama, Note on the Bronshtein theorem concerning hyperbolic polynomials, Sci. Math. Jap. 63 (2006), 247–285. | MR 2213130 | Zbl 1106.26016

[15] S. Wakabayashi, Remarks on hyperbolic polynomials, Tsukuba J. Math. 10 (1986), 17–28. | MR 846411 | Zbl 0612.35005