Sums of squares I: Scalar functions

• Published in: Journal of functional analysis 2023-03, Vol.284 (6), p.109827, Article 109827
• Main Authors: Korobenko, Lyudmila, Sawyer, Eric
• Format: Article
• Language: English
• Subjects: Hypoellipticity; Nonnegative functions; Sums of squares; Weak monotonicity
• ISSN: 0022-1236; 1096-0783
• DOI: 10.1016/j.jfa.2022.109827

This is the first in a series of three papers dealing with sums of squares and hypoellipticity in the infinitely degenerate regime. A result of C. Fefferman and D. H. Phong shows that every C3,1 nonnegative function on Rn can be written as a finite sum of squares of C1,1 functions, and was used by them to improve Gårding's inequality, and subsequently by P. Guan to prove regularity for certain degenerate operators. In this paper we investigate sharp criteria sufficient for writing a smooth nonnegative function f on Rn as a finite sum of squares of C2,δ functions for some δ>0, and we denote this property by saying f is SOSregular. The emphasis on C2,δ, as opposed to C1,1, arises because of applications to hypoellipticity for smooth infinitely degenerate operators in the spirit of M. Christ, which are pursued in the third paper of this series. Thus we consider the case where f is smooth and flat at the origin, and positive away from the origin. Our sufficient condition for such an f to be SOSregular is that f is ω-monotone for some modulus of continuity ωs(t)=ts, 00, then there exists a smooth nonnegative function f that is flat at the origin, and positive away from the origin, that is notSOSregular, answering in particular a question left open by Bony. Refinements of these results are given for f∈C4,2δ, and the related problem of extracting smooth positive roots from such smooth functions is also considered.