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We give a proof of the Nirenberg-Treves conjecture: that local solvability of principal-type pseudo-differential operators is equivalent to condition (Ψ). This condition rules out sign changes from − to + of the imaginary part of the principal symbol along the oriented bicharacteristics of the real part. We obtain local solvability by proving a localizable a priori estimate for the adjoint operator with a loss of two derivatives (compared with the elliptic case). The proof involves a new metric in the Weyl (or Beals-Fefferman) calculus which makes it possible to reduce to the case when the gradient of the imaginary. | Annals of Mathematics The resolution of the Nirenberg-Treves conjecture By Nils Dencker Annals of Mathematics 163 2006 405 444 The resolution of the Nirenberg-Treves conjecture By Nils Dencker Abstract We give a proof of the Nirenberg-Treves conjecture that local solvability of principal-type pseudo-differential operators is equivalent to condition T . This condition rules out sign changes from to of the imaginary part of the principal symbol along the oriented bicharacteristics of the real part. We obtain local solvability by proving a localizable a priori estimate for the adjoint operator with a loss of two derivatives compared with the elliptic case . The proof involves a new metric in the Weyl or Beals-Fefferman calculus which makes it possible to reduce to the case when the gradient of the imaginary part is nonvanishing so that the zeroes form a smooth submanifold. The estimate uses a new type of weight which measures the changes of the distance to the zeroes of the imaginary part along the bicharacteristics of the real part between the minima of the curvature of the zeroes. By using condition T and the weight we can construct a multiplier giving the estimate. 1. Introduction In this paper we shall study the question of local solvability of a classical pseudo-differential operator P G T M on a C 1 manifold M. Thus we assume that the symbol of P is an asymptotic sum of homogeneous terms and that p ơ P is the homogeneous principal symbol of P. We shall also assume that P is of principal type which means that the Hamilton vector field Hp and the radial vector field are linearly independent when p 0 thus dp 0 when p 0. Local solvability of P at a compact set K c M means that the equation 1.1 Pu v has a local solution u G D M in a neighborhood of K for any v G C M in a set of finite codimension. We can also define microlocal solvability at any compactly based cone K c T M see 9 Def. 26.4.3 . Hans Lewy s famous counterexample 19 from 1957 showed that not all smooth
