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. 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 a harmonic function with 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 → R be a C² (twice continuously differentiable) function. The function f is called pluriharmonic if, for every complex line

[math]\displaystyle{ \{ a + b z \mid z \in \Complex \} \subset \Complex^n }[/math]

formed by using every couple of complex tuples a, b ∈ Cⁿ, the function

[math]\displaystyle{ z \mapsto f(a + bz) }[/math]

is a harmonic function on the set

[math]\displaystyle{ \{ z \in \Complex \mid a + b z \in G \} \subset \Complex . }[/math]

Definition 2. Let M be a complex manifold and f : M → R be a C² function. The function f is called pluriharmonic if

[math]\displaystyle{ dd^c f = 0. }[/math]

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

• Plurisubharmonic function
• Wirtinger derivatives

Historical references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, https://books.google.com/books?id=L0zJmamx5AAC .
• 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 .
• Poincaré, H. (1899), "Sur les propriétés du potentiel et sur les fonctions Abéliennes" (in fr), Acta Mathematica 22 (1): 89–178, doi:10.1007/BF02417872 .
• Severi, Francesco (1958) (in it), Lezioni sulle funzioni analitiche di più variabili complesse – Tenute nel 1956–57 all'Istituto Nazionale di Alta Matematica in Roma, Padova: CEDAM – Casa Editrice Dott. Antonio Milani, pp. XIV+255 . 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" (in it), Rendiconti del Circolo Matematico di Palermo 33 (1): 75–85, doi:10.1007/BF03015289, https://zenodo.org/record/1914928 . 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" (in it), Atti del Convegno celebrativo dell'80° anniversario della nascita di Renato Calapso, Messina–Taormina, 1–4 aprile 1981, Roma: Libreria Eredi Virgilio Veschi, pp. 127–152 ."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" (in it), Rendiconti del Seminario Matematico e Fisico di Milano 52 (1): 23–34, doi:10.1007/BF02924996 . 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" (in it), Revue Roumaine de Mathématiques Pures et Appliquées 27: 327–333 . 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 .
• Nikliborc, Ladislas (30 March 1925), "Sur les fonctions hyperharmoniques" (in fr), Comptes rendus hebdomadaires des séances de l'Académie des sciences 180: 1008–1011, http://gallica.bnf.fr/ark:/12148/bpt6k3133k/f1008 , available at Gallica
• Nikliborc, Ladislas (11 January 1926), "Sur les fonctions hyperharmoniques" (in fr), Comptes rendus hebdomadaires des séances de l'Académie des sciences 182: 110–112, http://gallica.bnf.fr/ark:/12148/bpt6k31356/f110 , 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, http://www.digizeitschriften.de/dms/img/?PPN=PPN235181684_0130&DMDID=dmdlog33 , available at DigiZeitschirften.

External links

• Hazewinkel, Michiel, ed. (2001), "Pluriharmonic function", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=p/p072920