The exponential calculus of microdifferential operators of infinite order. III (Q791054)

The \(micro(=pseudo)\) differential operator defined by a symbol P(x,\(\xi)\) is denoted by :P(x,\(\xi)\):. A symbol p is said to be of order 1-0 if \(p(x,\xi)/| \xi | \to 0\) as \(| \xi | \to \infty\). Let p and q be symbols of order 1-0. Let a and b be symbols of order \(m_ 1\) and of order \(m_ 2\) respectively. In this paper, the author constructs symbols r and c of order 1-0 and of order \(m_ 1+m_ 2\) respectively satisfying \(:a\cdot\exp(p): :b\cdot\exp(q): = :c\cdot\exp(r):\). This gives a generalization of one of the main resuls of part I and II [cf., the author, Ann. Inst. Fourier 33, N. 4, 227-250 (1983; Zbl 0495.58025)].