International Congress of Mathematicians

Solvability 713mathematical community was aroused in search of a geometric condition on theprincipal symbol, characterizing local solvability of principal type operators. It isimportant to note that for principal type operators with a real principal symbol,such as a non-vanishing real vector field, or the wave equation, local solvability wasknown after the 1955 paper of L.Hörmander in [6]. In fact these results extend quiteeasily to the pseudo-differential real principal type case. As shown by the Hans Lewycounterexample and the necessary condition (1.4), the matters are quite differentfor complex-valued symbols.1963.It is certainly helpful to look now at some simple models. For t, x £ R, withthe usual notationsD t = -W t , (\Dfi\u)(0 = |£|«(£),where « is the ar-Fourier transform of u, I £ N, let us consider the operators definedbyM, = D t + it l D x , Ni = D t + it l \D x \. (1.5)It is indeed rather easy to prove that, for k £ N, M 2 k, N 2 k, A r 2* fc+1 are solvablewhereas M 2 k+i, N 2 k+i are nonsolvable. In particular, the operators Afi, A r i satisfy(1.3). On the other hand, the operator A r 1* = D t — it\D x \ is indeed solvable sinceits adjoint operator A r i verifies the a priori estimater||A r i«|| L 2( R 2 ) > ||«|| L 2( R 2 ) ,for a smooth compactly supported u vanishing for \t\ > T/2. No such estimate issatisfied by N£u since its a:-Fourier transform is-idf.v — it\Ç\v = (—i)(dt.v + t\Ç\v),where v is the ar-Fourier transform of u. A solution of Nfu = 0 is thus given bythe inverse Fourier transform of e - * ^/ 2 , ruining solvability for the operator A r i.A complete study of solvability properties of the models M/ was done in [23] byL.Nirenberg and F.Treves, who also provided a sufficient condition of solvabilityfor vector fields; the analytic-hypoellipticity properties of these operators were alsostudied in a paper by S.Alizohata [21].1971.The ODE-like examples (1.5) led L.Nirenberg and F.Treves in [24-25-26] toformulate a conjecture and to prove it in a number of cases, providing strong groundsin its favour. To explain this, let us look simply at the operatorL = D t + iq(t,x,D x ), (1.6)where q is a real-valued first-order symbol. The symbol of L is thus r + iq(t,x.fi).The bicharacteristic curves of the real part are oriented straight lines with directiond/dt; now we examine the variations of the imaginary part q(t, x, £) along theselines. It amounts only to check the functions t >-¥ q(t,x,Ç) for fixed (#,£). The