Hartogs's extension theorem: Difference between revisions

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==Historical note==

The original proof was given by [[Friedrich Hartogs]] in 1906, using [[Cauchy's integral formula]] for functions of [[several complex variables]].<ref name="hartogs"/> Today, usual proofs rely on either the [[Bochner–Martinelli–Koppelman formula]] or the solution of the inhomogeneous [[Cauchy–Riemann equations]] with compact support. The latter approach is due to [[Leon Ehrenpreis]] who initiated it in the paper {{Harv|Ehrenpreis|1961}}. Yet another very simple proof of this result was given by [[Gaetano Fichera]] in the paper {{Harv|Fichera|1957}}, by using his solution of the [[Dirichlet problem]] for [[holomorphic function]]s of several variables and the related concept of [[CR-function]]:<ref>Fichera's prof as well as his epoch making paper {{Harv|Fichera|1957}} seem to have been overlooked by many specialists of the [[Several complex variables|theory of functions of several complex variables]]: see {{Harvtxt|Range|2002}} for the correct attribution of many important theorems in this field.</ref> later he extended the theorem to a certain class of [[partial differential operator]]s in the paper {{Harv|Fichera|1983}}, and his ideas were later further explored by Giuliano Bratti.<ref>See {{Harvtxt|Bratti|1986a}} {{Harv|Bratti|1986b}}.</ref> Also the [[Japan|Japanese school]] of the theory of [[partial differential operator]]s worked much on this topic, with notable contributions by Akira Kaneko.<ref>See his paper {{Harv|Kaneko|1973}} and the references therein.</ref> Their approach is to use [[Ehrenpreis' fundamental principle]].

==Hartogs' phenomenon==