Subunit balls for symbols of pseudodifferential operators

In this work we shall study a definition of subunit ball for non-negative symbols of sub-elliptic pseudodifferential operators, extending in phase-space the one given by Stein, Nagel, and Wainger in the differential-operator case. Using microlocal methods introduced by Fefferman and Phong, we prove that these balls can be straightened, by means of a canonical transformation, to contain and be contained in boxes of certain sizes, which we give in terms of the size of the symbol. After microlocalizing the symbol, in Section 3 we define classes of subunit symbols and study some of their basic properties. Then we define the subunit ball. In the last section the main structure theorems, in the (n + n)-dimensional elliptic case and in the (1 + 1)- and (2 + 2)-dimensional nonelliptic-nondegenerate cases are stated and proved. (C) 1997 Academic Press.