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….
Bochner, S.: Green’s formula and analytic continuation. In:Contributions to the Theory of Partial Differential Equations, ed. L. Bers et al.,Ann. of Math. Studies33, Princeton Univ. Press, 1954.Google Scholar
Boggess, A.:CR Manifolds and the Tangential Cauchy-Riemann Complex. CRC Press, Boca Raton, FL, 1991.MATHGoogle Scholar
Čirka, E. M.: Analytic representation of CR-functions. Math.USSR Sbornik27 (1975), 526–553.CrossRefGoogle Scholar
Kohn, J. J., and Rossi, H.: On the extension of holomorphic functions from the boundary of a complex manifold.Ann. of Math. (2)81 (1965), 451–472.CrossRefMATHMathSciNetGoogle Scholar
Lewy, H.: On the local character of the solutions of an atypical linear differential equation in three variables and a related theorem for regular functions of two complex variables.Ann. of Math. (2)64 (1956), 514–522.CrossRefMATHMathSciNetGoogle Scholar
Martinelli, E.: Sulla determinazione di una funzione analitica di più variabili complesse in un campo, assegnatane la traccia sulla frontiera.Ann. Matem. Pura e Appl.55 (1961), 191–202.CrossRefMATHMathSciNetGoogle Scholar
[Polking and Wells 1975]
Polking, J., and Wells, R. O.: Hyperfunction boundary values and a generalized Bochner-Hartogs theorem.Proc. Symp. Pure Math.30, 187–194, Amer. Math. Soc., Providence, Rl, 1977.CrossRefMathSciNetGoogle Scholar
Range, R. M.: Holomorphic Functions and Integral Representations in Several Complex Variables. Springer-Verlag, New York 1986, 2nd. corrected printing (1998).CrossRefMATHGoogle Scholar
Serre, J. P.: Quelques problèmes globaux rélatifs aux variétés de Stein. Coll. Plus. Var., Bruxelles, 1953, 57–68.Google Scholar
Severi, F.: Risoluzione générale del problema di Dirichlet per le funzioni biarmoniche.Rend. Reale Accad. Lincei23 (1931), 795–804.Google Scholar
Struppa, D.: The first eighty years of Hartogs’ Theorem.Seminari di Geometria 1987–88, Univ. Bologna, Italy 1988, 127–209.Google Scholar
Trépreau, J. M.: Sur le prolongement holomorphe des fonctiones CR définies sur une hypersurface réelle de classe C2 dans ℂn.Invent. Math.83 (1986), 583–592.CrossRefMATHMathSciNetGoogle Scholar