The proof of the Nirenberg-Treves conjecture
Journées équations aux dérivées partielles, Tome (2003) , p. 1-25
doi : 10.5802/jedp.619
URL stable : http://www.numdam.org/item?id=JEDP_2003____A5_0

We prove the Nirenberg-Treves conjecture : that for principal type pseudo-differential operators local solvability 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 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. This makes it possible to reduce to the case when the gradient of the imaginary part is non-vanishing, and then the zeroes form a smooth submanifold. The estimate uses a new type of weight, which measures the change 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 which gives the estimate.

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