Pluriharmonic function

Let

$f\colon G\subset {\mathbb {C} }^{n}\to {\mathbb {C} }$

be a $C^{2}$ (twice continuously differentiable) function. $f$ is called pluriharmonic if for every complex line

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

the function

$z\mapsto f(a+bz)$

is a harmonic function on the set

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

Notes

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.

Bibliography

• 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.

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: 1008–10011, 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

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