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 a such function is referred as $n$-harmonic function, where $n$ ≥ 2 is the dimension of the complex domain where the function is defined. However, in modern expositions of the theory of functions of several complex variables 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 an harmonic function respect to the real and imaginary part of the complex line parameter.

Formal definition

Definition 1. Let $G$ ⊆ ℂⁿ be a complex domain and $f$ : $G$ → ℂ 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 $a,b$ ∈ ℂⁿ, the function

$z\mapsto f(a+bz)$

is a harmonic function on the set

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

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.

See also

• Harmonic function
• Plurisubharmonic function
• Wirtinger derivatives

Bibliography

• 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, 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 776.32001.
• 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{{citation}}: CS1 maint: unrecognized language (link).
• Severi, Francesco (1958), Lezioni sulle funzioni analitiche di più variabili complesse – Tenute nel 1956–57 all'Istituto Nazionale di Alta Matematica in Roma (Lectures on analytic functions of several complex variables – Lectured in 1956–57 at the Istituto Nazionale di Alta Matematica in Rome) (in Italian), Padova: CEDAM – Casa Editrice Dott. Antonio Milani, Zbl 0094.28002{{citation}}: CS1 maint: unrecognized language (link). 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.

References

• Amoroso, Luigi (1912), "Sopra un problema al contorno (About a boundary value problem)", Rendiconti del Circolo Matematico di Palermo (in Italian), 33 (1): 75–85, doi:10.1007/BF03015289, JFM 43.0453.03 {{citation}}: External link in |journal= (help)CS1 maint: unrecognized language (link). 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.
• Fichera, Gaetano (1982a), "Problemi al contorno per le funzioni pluriarmoniche (Boundary value problems for pluriharmonic functions)", 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{{citation}}: CS1 maint: unrecognized language (link). A paper where a trace condition for pluriharmonic functions is given.
• Fichera, Gaetano (1982b), "Valori al contorno delle funzioni pluriarmoniche: estensione allo spazio $R^{2n}$ di un teorema di L. Amoroso (Boundary values of pluriharmonic functions: extension to the space $R^{2n}$ of a theorem of L. Amoroso)", Rendiconti del Seminario Matematico e Fisico di Milano (now Milan Journal of Mathematics) (in Italian), 52 (1): 23–34, doi:10.1007/BF02924996, MR 0802991, Zbl 0569.31006 {{citation}}: External link in |journal= (help)CS1 maint: unrecognized language (link).
• Fichera, Gaetano (1982c), "Su un teorema di L. Amoroso nella teoria delle funzioni analitiche di due variabili complesse (On a theorem of L. Amoroso in the theory of analytic functions of two complex variables)", Revue Roumaine de Mathématiques Pures et Appliquées (in Italian), 27: 327–333, MR 0669481, Zbl 0509.31007 {{citation}}: External link in |journal= (help)CS1 maint: unrecognized language (link).
• Nikliborc, Ladislas (30 mars 1925), "Sur les fonctions hyperharmoniques", Comptes rendus hebdomadaires des séances de l'Académie des sciences (in French), 180: 1008–1011, JFM 51.0364.02 {{citation}}: Check date values in: |date= (help)CS1 maint: date and year (link) CS1 maint: unrecognized language (link), available at Gallica
• Nikliborc, Ladislas (11 janvier 1926), "Sur les fonctions hyperharmoniques", Comptes rendus hebdomadaires des séances de l'Académie des sciences (in French), 182: 110–112, JFM 52.0498.02 {{citation}}: Check date values in: |date= (help)CS1 maint: date and year (link) CS1 maint: unrecognized language (link), 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/BF01447872, MR 0074881, Zbl 0067.33004, available at DigiZeitschirften.

External links

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

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