Nonnegative functions as squares or sums of squares. (English) Zbl 1093.26007

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

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)

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

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