Sums of squares I: Scalar functions

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 C 3 , 1 nonnegative function on R n can be written as a finite sum of squares of C 1 , 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 …