On square roots of class $C^m$ of nonnegative functions of one variable

Bony, Jean-Michel; Colombini, Ferruccio; Pernazza, Ludovico

Geodesic

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On square roots of class

C^{m}

of nonnegative functions of one variable

Bony, Jean-Michel ¹ ; Colombini, Ferruccio ² ; Pernazza, Ludovico ³

¹ École Polytechnique, Centre de Mathématiques, 91128 Palaiseau Cedex, France
² Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo, 5, 56127 Pisa, Italia
³ Dipartimento di Matematica, Università di Pavia, Via Ferrata, 1, 27100 Pavia, Italia

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 9 (2010) no. 3, pp. 635-644

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Résumé

We investigate the regularity of functions $g$ such that $g^{2} = f$ , where $f$ is a given nonnegative function of one variable. Assuming that $f$ is of class $C^{2 m}$ ( $m > 1$ ) and vanishes together with its derivatives up to order $2 m - 4$ at all its local minimum points, one can find a $g$ of class $C^{m}$ . Under the same assumption on the minimum points, if $f$ is of class $C^{2 m + 2}$ then $g$ can be chosen such that it admits a derivative of order $m + 1$ everywhere. Counterexamples show that these results are sharp.

MR Zbl

Classification : 26A15, 26A27

Affiliations des auteurs :

Bony, Jean-Michel ¹ ; Colombini, Ferruccio ² ; Pernazza, Ludovico ³

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     title = {On square roots of class $C^m$ of nonnegative functions of one variable},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {635--644},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 9},
     number = {3},
     year = {2010},
     mrnumber = {2722658},
     zbl = {1207.26004},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ASNSP_2010_5_9_3_635_0/}
}

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Bony, Jean-Michel; Colombini, Ferruccio; Pernazza, Ludovico. On square roots of class $C^m$ of nonnegative functions of one variable. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 9 (2010) no. 3, pp. 635-644. http://geodesic.mathdoc.fr/item/ASNSP_2010_5_9_3_635_0/