Solvability of Pseudodifferential Operators
speaker: Nils Dencker (Lund University)abstract: In the 50's, Ehrenpreis and Malgrange proved that all constant coefficient linear partial differential equations are solvable, thus it came as a surprise when Hans Lewy in 1957 constructed a non-solvable complex vector field. The vector field is a natural one: it is the Cauchy-Riemann operator on the boundary of a strictly pseudo-convex domain and it is solvable in the analytic category by the Cauchy-Kowalevska Theorem. Hörmander then proved in 1960 that almost all complex vector fields are non-solvable, they are in fact characterized by their images. Nirenberg and Treves formulated in 1970 the conjecture that condition ($\Psi$) is necessary and sufficient for solvability of (pseudo-)differential operators of principal type. This condition determines the sign changes of the imaginary part of the principal symbol, along the bicharacteristics of the real part. It was known that condition (${\Psi}$) was necessary for solvability, but when Lerner in 1994 constructed counterexamples satisfying (${\Psi}$) but not the expected solvability estimates, it was perceived that the conjecture could be wrong. In fact, the estimates would not be localizable and lower order terms could be critical, making condition ($\Psi$) insufficient. In these lectures, we shall present a proof of the Nirenberg-Treves conjecture. We obtain local solvability by proving a localizable estimate of the adjoint operator with a loss of two derivatives (compared with the elliptic case). This is possible since we only estimate with the imaginary part of a real multiplier product, the cut-off terms are symmetric, and lower order terms can be symmetrized.
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