Lewy's operator and its ramifications
The Cauchy-Kowalewski theorem shows that linear partial differential operators with analytic coefficients map analytic functions locally onto analytic functions. Malgrange and Ehrenpreis proved that local surjectivity also holds in the space of \(C^{\infty}\) functions for operators with constant coefficients. However, Lewy discovered a first order operator L with linear coefficients for which local surjectivity at the \(C^{\infty}\) level does not hold. In fact, \(L=\partial /\partial x+i(\partial /\partial y)-2i(x+iy)\partial /\partial t.\) In this paper two new proofs are presented for the solvability of L. The first proof uses complex Fourier analysis and leads to a theory of linear operators with linear coefficients. The second proof uses the theory of representations of Lie groups and the fact that L can be considered as a left invariant operator on the Heisenberg group.
scientific article; zbMATH DE number 4040373
scientific article