On pseudo-differential operators and smoothness of special Lie-group representations

Two algebras of global pseudo-differential operators over ⁿ are investigated, with corresponding classes of symbols A⁰=CB (all (x, )-derivatives bounded over ²ⁿ), and A₁ (all finite applications of _xj, _j, and _pq=_pq–_px_p on the symbol are in A₀). The class A₁ consists of classical symbols, i.e., ^x a= 0((1+||)–^||) for x Kc ;ⁿ, K, compact. It is shown that a bounded operator A of 210C=L²(Rⁿ) is a pseudo-differential operator with symbol aA_j if and only if the map AG^–1AG, G g_j is infinitely differentiable, from a certain Lie-group g_j c GL(210C) to (210C) with operator norm. g₀ is the Weyl (or Heisenberg) group. Extensions to operators of arbitrary order are discussed. Applications to follow in a subsequent paper.Dedicated to Hans Lewy and Charles B. Morrey, Jr.