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The first eighty years of Hartogs’ theorem. (English) Zbl 0657.35018

This is an enlarged version, added with the historical survey, of the author’s lecture on the extension of solutions of systems of convolution equations. Chap. 1 starts with the original Hartogs continuity theory on the extension of holomorphic functions of several variables to compact singularity, and then surveys its early generalizations or re-proofs by Severi, Fubini, Segre, Martinelli. Chap. 2 introduces Ehrenpreis’ approach from the viewpoint of overdetermined systems which was epoch making in this problem. Chap. 3 surveys hyperfunction theory and its contribution to this problem. Chap. 4 treats the author’s original work on the extension of solutions of systems of convolution equations, including systems of differential equations of infinite order as a particular case.

As a whole this is a good survey to the problem of extension of solutions of Hartogs type, although a little more references (especially, contributions of some Russians) should be added in order to be a complete survey including the case of isolated singularity, or the case of non- compact singularity.

The seminar notes including this article seem to be available on request to the University of Bologna.

Reviewer: A.Kaneko

MSC:
35B60Continuation of solutions of PDE
32D20Removable singularities (several complex variables)
35N05Overdetermined systems of PDE with constant coefficients
35-02Research monographs (partial differential equations)
32-02Research monographs (several complex variables)