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. Counterexamples were provided for any larger modulus of continuity. In this paper we consider the analogous sum of squares problem for smooth nonnegative matrix functions M that are flat at, and positive away from, the origin. We show that such a matrix function M = [ a k j ] k , j = 1 n can be written as a finite sum of squares of C 2 , δ vector fields if the diagonal entries a k k are ω-monotone for some Hölder modulus of continuity ω, and if the off diagonal entries satisfy certain differential bounds in terms of powers of the diagonal entries. Examples are given to show that in some cases at least, these differential inequalities cannot be relaxed. Various refinements of these results are also given in which one or more of the diagonal entries need not be assumed to have any monotonicity properties at all. These sum of squares decompositions will be applied to hypoellipticity in the infinitely degenerate regime in the third paper in this series.