The Mathematical Intelligencer

, Volume 24, Issue 2, pp 4-12

Extension phenomena in multidimensional complex analysis: correction of the historical record

  • R. Michael RangeAffiliated withDepartment of Mathematics and Statistics, State University of New York at Albany Email author 

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Even though Bochner should not be credited with the proof of any version of theCR extension theorem, his 1943 paper remains a landmark in the history of the Hartogs extension phenomenon. His vision to enlarge his horizon from holomorphic functions to certain harmonic functions set the stage for further generalizations by himself (for example [Bochner 1954]) as well as for Ehrenpreis’s investigations on related problems for solutions of more general elliptic partial-differential operators.

In closing, it should be pointed out that Bochner’s 1943 paper, in an ironic twist, includes an important result for which Bochner did not receive any credit until recently [Range 1986, p. 188]. Bochner proved the solution of ∂ on polydiscs (for (0, l)-forms in the real-analytic case, which was the case of interest to him), via the Cauchy transform with parameters in dimension one, and by induction on the number of differentialsdzj appearing in the given form (Theorem 11,op. cit., p. 665). This result, with essentially the same proof, 10 years later became widely known as the Dolbeault-Grothendieck Lemma. But this is another story….