Open AccessNecessary and sufficient conditions for solvability of the Lewy equation
Author(s) -
Peter Greiner,
J. J. Kohn,
Elias M. Stein
Publication year - 1975
Publication title -
proceedings of the national academy of sciences of the united states of america
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 5.011
H-Index - 771
eISSN - 1091-6490
pISSN - 0027-8424
DOI - 10.1073/pnas.72.9.3287
Subject(s) - mathematics , space (punctuation) , mathematical analysis , cauchy distribution , boundary value problem , combinatorics , pure mathematics , computer science , operating system
We find the necessary and sufficient conditions for the local solvability of Lewy's equation, ([unk]/[unk]z +īz [unk]/[unk]t )u =f . If R3 is realized as the boundary of the generalized “upper-half-space” in C2 , then the conditions are, near a pointP [unk] R3 , the analytic continuability of the Cauchy-Szegö integral off pastP . In case the sufficient condition is satisfied, solutions are found that satisfy optimal regularity properties. Various generalizations are also given.