Higher order Glaeser inequalities and optimal regularity of roots of real functions

Ghisi, Marina; Gobbino, Massimo

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 12 (2013) no. 4, pp. 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 $(k+\alpha )$-th root of a function of class $C^k$ whose derivative of order $k$ is $\alpha$-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.

Classification : 26A46, 26B30, 26A27

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

[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 | Zbl

[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 | Zbl

[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. | EuDML | Numdam | MR | Zbl

[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 | Zbl

[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 | Zbl

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

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

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

[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

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

[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 | Zbl

[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 | Zbl

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