The propagation of polarization in double refraction
Preparation theorems for systems
The proof of the Nirenberg-Treves conjecture
We prove the Nirenberg-Treves conjecture : that for principal type pseudo-differential operators local solvability is equivalent to condition (). 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)...
The solvability of non solvable operators
Preparation theorems for matrix valued functions
We generalize the Malgrange preparation theorem to matrix valued functions satisfying the condition that vanishes to finite order at . Then we can factor near (0,0), where is inversible and is polynomial function of depending on . The preparation is (essentially) unique, up to functions vanishing to infinite order at , if we impose some additional conditions on . We also have a generalization of the division theorem, and analytic versions generalizing the Weierstrass preparation...
On the solvability of pseudodifferential operators
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