Pluriharmonic function

<h2 id="formal-definition">Formal definition</h2>
Definition 1. Let G ⊆ Cn be a <a href="/facts/Domain_(mathematical_analysis)/xT44oI3E">complex domain</a> and f : G → R be a C2 (twice <a href="/facts/Differentiable_function/jQqTgLk1">continuously differentiable</a>) function. The function f is called pluriharmonic if, for every <a href="/facts/Complex_number/2w7mqI7e">complex</a> <a href="/facts/Line_(geometry)/gJqsr4Gj">line</a>

{
        a
        +
        b
        z
        ∣
        z
        ∈
        
          C
        
        }
        ⊂
        
          
            C
          
          
            n
          
        
      
    
    {\displaystyle \{a+bz\mid z\in \mathbb {C} \}\subset \mathbb {C} ^{n}}

formed by using every couple of complex <a href="/facts/Tuple/BI1HIxL8">tuples</a> a, b ∈ Cn, the function

z
        ↦
        f
        (
        a
        +
        b
        z
        )
      
    
    {\displaystyle z\mapsto f(a+bz)}

is a <a href="/facts/Harmonic_function/H6pg4dMT">harmonic function</a> on the set

{
        z
        ∈
        
          C
        
        ∣
        a
        +
        b
        z
        ∈
        G
        }
        ⊂
        
          C
        
        .
      
    
    {\displaystyle \{z\in \mathbb {C} \mid a+bz\in G\}\subset \mathbb {C} .}

Definition 2. Let M be a <a href="/facts/Complex_manifold/DlvPnpzm">complex manifold</a> and f : M → R be a C2 function. The function f is called pluriharmonic if

d
        
          d
          
            c
          
        
        f
        =
        0.
      
    
    {\displaystyle dd^{c}f=0.}

<h2 id="basic-properties">Basic properties</h2>
Every pluriharmonic function is a <a href="/facts/Harmonic_function/H6pg4dMT">harmonic function</a>, but not the other way around. Further, it can be shown that for <a href="/facts/Holomorphic_function/kVzthtEf">holomorphic functions</a> 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.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Plurisubharmonic_function/nM5tTVXS">Plurisubharmonic function</a></li>
<li><a href="/facts/Wirtinger_derivatives/n2n6EJ8p">Wirtinger derivatives</a></li></ul>
<h2 id="notes">Notes</h2>

<h2 id="historical-references">Historical references</h2>
<ul><li><a href="/facts/Robert_Gunning_(mathematician)/gKD3DIq4">Gunning, Robert C.</a>; Rossi, Hugo (1965), <a href="https://books.google.com/books?id=L0zJmamx5AAC">Analytic Functions of Several Complex Variables</a>, Prentice-Hall series in Modern Analysis, <a href="/facts/Englewood_Cliffs/gSQUlgvD">Englewood Cliffs</a>, N.J.: <a href="/facts/Prentice-Hall/bynWAuat">Prentice-Hall</a>, pp. xiv+317, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9780821869536, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0180696">0180696</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0141.08601">0141.08601</a>.</li>
<li><a href="/facts/Steven_G._Krantz/GkUkkFTA">Krantz, Steven G.</a> (1992), Function Theory of Several Complex Variables, Wadsworth & Brooks/Cole Mathematics Series (Second ed.), <a href="/facts/Pacific_Grove%2c_California/wfjgM3HR">Pacific Grove, California</a>: Wadsworth & Brooks/Cole, pp. xvi+557, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-534-17088-9, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1162310">1162310</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0776.32001">0776.32001</a>.</li>
<li><a href="/facts/Henri_Poincar%25C3%25A9/AazL2j4X">Poincaré, H.</a> (1899), "Sur les propriétés du potentiel et sur les fonctions Abéliennes", <a href="/facts/Acta_Mathematica/11kbuhDT">Acta Mathematica</a> (in French), 22 (1): 89–178, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF02417872">10.1007/BF02417872</a>, <a href="/facts/JFM_(identifier)/P6rFxKKx">JFM</a> <a href="https://zbmath.org/?format=complete&q=an:29.0370.02">29.0370.02</a>.</li>
<li><a href="/facts/Francesco_Severi/LRqMFxVM">Severi, Francesco</a> (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, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0094.28002">0094.28002</a>. Notes from a course held by Francesco Severi at the <a href="/facts/Istituto_Nazionale_di_Alta_Matematica/u9kyPs1e">Istituto Nazionale di Alta Matematica</a> (which at present bears his name), containing appendices of <a href="/facts/Enzo_Martinelli/8aKLx9wu">Enzo Martinelli</a>, <a href="/facts/Giovanni_Battista_Rizza/wYtUucce">Giovanni Battista Rizza</a> 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".</li></ul>

<ul><li><a href="/facts/Luigi_Amoroso/Gs7xGcn2">Amoroso, Luigi</a> (1912), <a href="https://zenodo.org/record/1914928">"Sopra un problema al contorno"</a>, <a href="/facts/Rendiconti_del_Circolo_Matematico_di_Palermo/b7fCfvG5">Rendiconti del Circolo Matematico di Palermo</a> (in Italian), 33 (1): 75–85, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF03015289">10.1007/BF03015289</a>, <a href="/facts/JFM_(identifier)/P6rFxKKx">JFM</a> <a href="https://zbmath.org/?format=complete&q=an:43.0453.03">43.0453.03</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:122956910">122956910</a>. The first paper where a set of (fairly complicate) necessary and sufficient conditions for the solvability of the <a href="/facts/Dirichlet_problem/uHErgNCb">Dirichlet problem</a> for <a href="/facts/Function_of_several_complex_variables/axwNpV4a">holomorphic functions of several variables</a> is given. An English translation of the title reads as:-"About a boundary value problem".</li>
<li><a href="/facts/Gaetano_Fichera/gMo4k9mu">Fichera, Gaetano</a> (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, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0698973">0698973</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0958.32504">0958.32504</a>."Boundary value problems for pluriharmonic functions" (English translation of the title) deals with <a href="/facts/Boundary_value_problem/QEfmUqmP">boundary value problems</a> for pluriharmonic functions: Fichera proves a <a href="/facts/Trace_operator/7uQ4kSE4">trace condition</a> for the solvability of the problem and reviews several earlier results of Enzo Martinelli, Giovanni Battista Rizza and Francesco Severi.</li>
<li>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, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF02924996">10.1007/BF02924996</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0802991">0802991</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:122147246">122147246</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0569.31006">0569.31006</a>. An English translation of the title reads as:-"Boundary values of pluriharmonic functions: extension to the space R2n of a theorem of L. Amoroso".</li>
<li>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, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0669481">0669481</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0509.31007">0509.31007</a>. An English translation of the title reads as:-"On a theorem of L. Amoroso in the theory of analytic functions of two complex variables".</li>
<li>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, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2206%2Fkyushumfs.36.157">10.2206/kyushumfs.36.157</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0676796">0676796</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0501.32008">0501.32008</a>.</li>
<li>Nikliborc, Ladislas (30 March 1925), <a href="http://gallica.bnf.fr/ark:/12148/bpt6k3133k/f1008">"Sur les fonctions hyperharmoniques"</a>, <a href="/facts/Comptes_rendus_de_l%2527Acad%25C3%25A9mie_des_sciences/9yR9XTN5">Comptes rendus hebdomadaires des séances de l'Académie des sciences</a> (in French), 180: 1008–1011, <a href="/facts/JFM_(identifier)/P6rFxKKx">JFM</a> <a href="https://zbmath.org/?format=complete&q=an:51.0364.02">51.0364.02</a>, available at <a href="/facts/Gallica/4TNyWsxG">Gallica</a></li>
<li>Nikliborc, Ladislas (11 January 1926), <a href="http://gallica.bnf.fr/ark:/12148/bpt6k31356/f110">"Sur les fonctions hyperharmoniques"</a>, <a href="/facts/Comptes_rendus_de_l%2527Acad%25C3%25A9mie_des_sciences/9yR9XTN5">Comptes rendus hebdomadaires des séances de l'Académie des sciences</a> (in French), 182: 110–112, <a href="/facts/JFM_(identifier)/P6rFxKKx">JFM</a> <a href="https://zbmath.org/?format=complete&q=an:52.0498.02">52.0498.02</a>, available at <a href="/facts/Gallica/4TNyWsxG">Gallica</a></li>
<li><a href="/facts/Giovanni_Battista_Rizza/wYtUucce">Rizza, G. B.</a> (1955), <a href="http://www.digizeitschriften.de/dms/img/?PPN=PPN235181684_0130&DMDID=dmdlog33">"Dirichlet problem for n-harmonic functions and related geometrical problems"</a>, <a href="/facts/Mathematische_Annalen/GPhGaonc">Mathematische Annalen</a>, 130: 202–218, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF01343349">10.1007/BF01343349</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0074881">0074881</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:121147845">121147845</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0067.33004">0067.33004</a>, available at <a href="http://www.digizeitschriften.de/">DigiZeitschirften</a>.</li></ul>
<h2 id="external-links">External links</h2>
<ul><li>Solomentsev, E. D. (2001) [1994], <a href="https://www.encyclopediaofmath.org/index.php?title=Pluriharmonic_function">"Pluriharmonic function"</a>, <a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a></li></ul>
This article incorporates material from pluriharmonic function on <a href="/facts/PlanetMath/wphPCtLf">PlanetMath</a>, which is licensed under the Creative Commons Attribution/Share-Alike License.

<h2 id="references">References</h2>

<ol>
<li id="fn: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[citation needed] the older one in which the pluriharmonic operator is expressed using the first order partial differential operators now called Wirtinger derivatives. - 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 <a href="https://zbmath.org/?format=complete&q=an:0094.28002" target="_blank">https://zbmath.org/?format=complete&q=an:0094.28002</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn: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). - 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 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1162310" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=1162310</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
</ol>

Pluriharmonic function open-in-new

Pluriharmonic function