The resolution of the Nirenberg-Treves conjecture

In this paper we give a proof of the Nirenberg-Treves conjecture: that local solvability of principal type pseudo-differential operators is equivalent to condition ($\Psi$). This condition rules out certain sign changes of the imaginary part of the principal symbol along the bicharacteristics of the real part. We obtain local solvability by proving a localizable estimate of 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 non-vanishing, so that the zeroes forms 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 this submanifold. By using condition ($\Psi$) and this weight, we can construct a multiplier giving the estimate.