Bony, Jean-Michel; Broglia, Fabrizio; Colombini, Ferruccio; Pernazza, Ludovico Nonnegative functions as squares or sums of squares. (English) Zbl 1093.26007 J. Funct. Anal. 232, No. 1, 137-147 (2006). In 1978 Fefferman and Phong stated (and sketchily proved) a lemma that any nonnegative \({\mathcal C}^{\infty}\) function in \({\mathbb R}^n\) is a sum of squares \({\mathcal C}^{1,1}\). In the reviewed paper it is proved for \(n\geq 4\) that such a regularity condition is sharp, namely, there exist nonnegative \({\mathcal C}^{\infty}\) functions \(f : {\mathbb R}^n \rightarrow {\mathbb R}\) that are not sums of squares of \({\mathcal C}^{2}\) functions. It is also proved that a general nonnegative \({\mathcal C}^{2}\) function of one variable has a \({\mathcal C}^{1}\) admissible square root, but no better regularity can be assured. If the function is \({\mathcal C}^{4}\) and its values at all its local minima are controlled it has a \({\mathcal C}^{2}\) admissible square root, but no better regularity can be assured. In both cases, increasing the regularity of the nonnegative function up to \({\mathcal C}^{\infty}\) does not provide a better result. Reviewer: Sergei V. Rogosin (Minsk) Cited in 13 Documents MSC: 26B40 Representation and superposition of functions 26C99 Polynomials, rational functions in real analysis 26E10 \(C^\infty\)-functions, quasi-analytic functions Keywords:sums of squares; square roots; nonnegative functions; modulus of continuity; nondifferentiability PDF BibTeX XML Cite \textit{J.-M. Bony} et al., J. Funct. Anal. 232, No. 1, 137--147 (2006; Zbl 1093.26007) Full Text: DOI References: [1] Alekseevski, D.; Kriegl, A.; Michor, P.W.; Losik, M., Choosing roots of polynomials smoothly, Israel J. math., 105, 203-233, (1998) · Zbl 0912.26006 [2] J. Bochnak, M. Coste, M.-F. Roy, Real Algebraic Geometry, Ergeb. Math. (3) 36, Springer, Berlin, Heidelberg, New York, 1998. [3] J.-M. Bony, Sommes de carrés de fonctions dérivables, Bull. Soc. Math. France 133 (2005). [4] Choi, M.-D.; Lam, T.-Y., Extremal positive semidefinite forms, Math. ann., 231, 1-18, (1977) · Zbl 0347.15009 [5] Fefferman, C.; Phong, D., On positivity of pseudo-differential operators, Proc. natl. acad. sci. U.S.A., 75, 4673-4674, (1978) · Zbl 0391.35062 [6] Glaeser, G., Racine carrée d’une fonction différentiable, Ann. inst. Fourier (Grenoble), 13, 203-210, (1963) · Zbl 0128.27903 [7] Guan, P., \(C^2\) a priori estimates for degenerate Monge-Ampère equations, Duke math. J., 86, 323-346, (1997) · Zbl 0879.35059 [8] Hilbert, D., Über die darstellung definiter formen als summe von formenquadraten, Math. ann., 32, 342-350, (1888), see also Gesammelte Abhandlungen, Bd. 2, Chelsea Publishing Company, Bronx, New York, 1965, pp. 154-161 · JFM 20.0198.02 [9] L. Hörmander, The Analysis of Linear Partial Differential Operators, III, Grundl. der math. Wiss. 274, Springer, Berlin, Heidelberg, New York, 1985. [10] Mandai, T., Smoothness of roots of hyperbolic polynomials with respect to one-dimensional parameter, Bull. fac. gen. ed. gifu univ., 21, 115-118, (1985) [11] T.S. Motzkin, The arithmetic-geometric inequality, in: Inequalities, Wright-Patterson Air Force Base 1965, Academic Press, New York, 1967, pp. 205-224. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.