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.

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 2m (m>1) and vanishes together with its derivatives up to order 2m-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 2m+2 then g can be chosen such that it admits a derivative of order m+1 everywhere. Counterexamples show that these results are sharp.

Classification : 26A15, 26A27
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
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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://www.numdam.org/item/ASNSP_2010_5_9_3_635_0/

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