Pluriharmonic function

In mathematics, precisely in the theory of functions of several complex variables, a pluriharmonic function is a real valued function which is locally the real part of a holomorphic function of several complex variables. Sometimes such a function is referred to as n-harmonic function, where n ≥ 2 is the dimension of the complex domain where the function is defined.[1] However, in modern expositions of the theory of functions of several complex variables[2] it is preferred to give an equivalent formulation of the concept, by defining pluriharmonic function a complex valued function whose restriction to every complex line is a harmonic function with respect to the real and imaginary part of the complex line parameter.

Formal definition

Definition 1. Let $G \subseteq C n$ be a complex domain and $f : G \to R$ be a $C 2$ (twice continuously differentiable) function. The function $f$ is called pluriharmonic if, for every complex line

\{a+bz\mid z\in \mathbb {C} \}\subset \mathbb {C} ^{n}

formed by using every couple of complex tuples $a, b \in C n$ , the function

z\mapsto f(a+bz)

is a harmonic function on the set

\{z\in \mathbb {C} \mid a+bz\in G\}\subset \mathbb {C} .

Definition 2. Let $M$ be a complex manifold and $f : M \to R$ be a $C 2$ function. The function $f$ is called pluriharmonic if

dd^{c}f=0.

Basic properties

Every pluriharmonic function is a harmonic function, but not the other way around. Further, it can be shown that for holomorphic functions of several complex variables the real (and the imaginary) parts are locally pluriharmonic functions. However a function being harmonic in each variable separately does not imply that it is pluriharmonic.

Notes

See for example (Severi 1958, p. 196) and (Rizza 1955, p. 202). Poincaré (1899, pp. 111–112) calls such functions "fonctions biharmoniques", irrespective of the dimension n ≥ 2 : his paper is perhaps the older one in which the pluriharmonic operator is expressed using the first order partial differential operators now called Wirtinger derivatives.
See for example the popular textbook by Krantz (1992, p. 92) and the advanced (even if a little outdated) monograph by Gunning & Rossi (1965, p. 271).

Historical references

Gunning, Robert C.; Rossi, Hugo (1965), Analytic Functions of Several Complex Variables, Prentice-Hall series in Modern Analysis, Englewood Cliffs, N.J.: Prentice-Hall, pp. xiv+317, ISBN 9780821869536, MR 0180696, Zbl 0141.08601.
Krantz, Steven G. (1992), Function Theory of Several Complex Variables, Wadsworth & Brooks/Cole Mathematics Series (Second ed.), Pacific Grove, California: Wadsworth & Brooks/Cole, pp. xvi+557, ISBN 0-534-17088-9, MR 1162310, Zbl 0776.32001.
Poincaré, H. (1899), "Sur les propriétés du potentiel et sur les fonctions Abéliennes", Acta Mathematica (in French), 22 (1): 89–178, doi:10.1007/BF02417872, JFM 29.0370.02.
Severi, Francesco (1958), Lezioni sulle funzioni analitiche di più variabili complesse – Tenute nel 1956–57 all'Istituto Nazionale di Alta Matematica in Roma (in Italian), Padova: CEDAM – Casa Editrice Dott. Antonio Milani, pp. XIV+255, Zbl 0094.28002. Notes from a course held by Francesco Severi at the Istituto Nazionale di Alta Matematica (which at present bears his name), containing appendices of Enzo Martinelli, Giovanni Battista Rizza and Mario Benedicty. An English translation of the title reads as:-"Lectures on analytic functions of several complex variables – Lectured in 1956–57 at the Istituto Nazionale di Alta Matematica in Rome".

References

Amoroso, Luigi (1912), "Sopra un problema al contorno", Rendiconti del Circolo Matematico di Palermo (in Italian), 33 (1): 75–85, doi:10.1007/BF03015289, JFM 43.0453.03, S2CID 122956910. The first paper where a set of (fairly complicate) necessary and sufficient conditions for the solvability of the Dirichlet problem for holomorphic functions of several variables is given. An English translation of the title reads as:-"About a boundary value problem".
Fichera, Gaetano (1982a), "Problemi al contorno per le funzioni pluriarmoniche", Atti del Convegno celebrativo dell'80° anniversario della nascita di Renato Calapso, Messina–Taormina, 1–4 aprile 1981 (in Italian), Roma: Libreria Eredi Virgilio Veschi, pp. 127–152, MR 0698973, Zbl 0958.32504."Boundary value problems for pluriharmonic functions" (English translation of the title) deals with boundary value problems for pluriharmonic functions: Fichera proves a trace condition for the solvability of the problem and reviews several earlier results of Enzo Martinelli, Giovanni Battista Rizza and Francesco Severi.
Fichera, Gaetano (1982b), "Valori al contorno delle funzioni pluriarmoniche: estensione allo spazio R²ⁿ di un teorema di L. Amoroso", Rendiconti del Seminario Matematico e Fisico di Milano (in Italian), 52 (1): 23–34, doi:10.1007/BF02924996, MR 0802991, S2CID 122147246, Zbl 0569.31006. An English translation of the title reads as:-"Boundary values of pluriharmonic functions: extension to the space R²ⁿ of a theorem of L. Amoroso".
Fichera, Gaetano (1982c), "Su un teorema di L. Amoroso nella teoria delle funzioni analitiche di due variabili complesse", Revue Roumaine de Mathématiques Pures et Appliquées (in Italian), 27: 327–333, MR 0669481, Zbl 0509.31007. An English translation of the title reads as:-"On a theorem of L. Amoroso in the theory of analytic functions of two complex variables".
Matsugu, Yasuo (1982), "Pluriharmonic functions as the real parts of holomorphic functions", Memoirs of the Faculty of Science, Kyushu University, Series A, Mathematics, 36 (2): 157–163, doi:10.2206/kyushumfs.36.157, MR 0676796, Zbl 0501.32008.
Nikliborc, Ladislas (30 March 1925), "Sur les fonctions hyperharmoniques", Comptes rendus hebdomadaires des séances de l'Académie des sciences (in French), 180: 1008–1011, JFM 51.0364.02, available at Gallica
Nikliborc, Ladislas (11 January 1926), "Sur les fonctions hyperharmoniques", Comptes rendus hebdomadaires des séances de l'Académie des sciences (in French), 182: 110–112, JFM 52.0498.02, available at Gallica
Rizza, G. B. (1955), "Dirichlet problem for n-harmonic functions and related geometrical problems", Mathematische Annalen, 130: 202–218, doi:10.1007/BF01343349, MR 0074881, S2CID 121147845, Zbl 0067.33004, available at DigiZeitschirften.

External links

Solomentsev, E. D. (2001) [1994], "Pluriharmonic function", Encyclopedia of Mathematics, EMS Press

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[1] See for example (Severi 1958, p. 196) and (Rizza 1955, p. 202). Poincaré (1899, pp. 111–112) calls such functions "fonctions biharmoniques", irrespective of the dimension n ≥ 2 : his paper is perhaps the older one in which the pluriharmonic operator is expressed using the first order partial differential operators now called Wirtinger derivatives.

[2] See for example the popular textbook by Krantz (1992, p. 92) and the advanced (even if a little outdated) monograph by Gunning & Rossi (1965, p. 271).