On local solvability of pseudo-differential equations

Abstract

A sufficient condition for the local solvability of the equation u,-A(x, t, D,)u=f(x, t) is proved, where A is a first order pseudo-differential operator with real symbol. This is a special case of the local solvability conjecture of Nirenberg and Treves. Introduction. Let P(x, D) be a linear partial differential operator of principal type with smooth complex value coefficients. The question of when the equation Pu=f is locally solvable has been settled by Nirenberg and Treves [4] and Beals and Fefferman [1]. Local solvability is equivalent to the condition: (P) The imaginary part of P does not change signs on the null bicharacteristics of the real part of P. For a pseudo-differential operator Nirenberg and Treves conjectured that local solvability is equivalent to the condition: (T) On every null bicharacteristic of Re P, if Im P is negative at a point it remains nonpositive from then on. The purpose of this note is to prove the following special case of this conjecture. THEOREM 1. Let P=d/dt-A(t, x, D.) for (t, x) Ec Q where A is a first order pseudo-differential operator with real symbol a(t, x, t). Assume that (TF) if a(to, xo, 40) to, a(t, xo, $0)?O; and if a(to, xo, $%)=O then grad, , a(to, x0, t0)=O. Then P is locally solvable. Theorem 1 is a simple consequence of the following a priori estimate for the adjoint of P. Received by the editors April 9, 1973 and, in revised form, May 4, 1973 and July 2, 1973. AMS (MOS) subject classifications (1970). Primary 35A05; Secondary 35F05, 35A05.