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The proof of the Nirenberg-Treves conjecture

Nils Dencker — 2003

Journées équations aux dérivées partielles

We prove the Nirenberg-Treves conjecture : that for principal type pseudo-differential operators local solvability is equivalent to condition ( Ψ ). 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)...

Preparation theorems for matrix valued functions

Nils Dencker — 1993

Annales de l'institut Fourier

We generalize the Malgrange preparation theorem to matrix valued functions F ( t , x ) ∈ C ∞ ( R × R n ) satisfying the condition that t ↦ det F ( t , 0 ) vanishes to finite order at t = 0 . Then we can factor F ( t , x ) = C ( t , x ) P ( t , x ) near (0,0), where C ( t , x ) ∈ C ∞ is inversible and P ( t , x ) is polynomial function of t depending C ∞ on x . The preparation is (essentially) unique, up to functions vanishing to infinite order at x = 0 , if we impose some additional conditions on P ( t , x ) . We also have a generalization of the division theorem, and analytic versions generalizing the Weierstrass preparation...

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