Sums of squares II: Matrix functions

This paper is the second in a series of three papers devoted to sums of squares and hypoellipticity of infinitely degenerate operators. In the first paper we established a sharp ω-monotonicity criterion for writing a smooth nonnegative function f that is flat at, and positive away from, the origin, as a finite sum of squares of C 2 , δ functions for some δ > 0 , namely that f is ω-monotone for some Hölder modulus of continuity ω. …