View source for Multiharmonic function

A [[Harmonic function|harmonic function]] such that the [[Laplace operator|Laplace operator]] acting on separate groups of independent variables vanishes. More precisely: A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065270/m0652701.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065270/m0652702.png" />, of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065270/m0652703.png" /> in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065270/m0652704.png" /> of the Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065270/m0652705.png" /> is called a multiharmonic function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065270/m0652706.png" /> if there exist natural numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065270/m0652707.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065270/m0652708.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065270/m0652709.png" />, such that the following identities hold throughout <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065270/m06527010.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065270/m06527011.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065270/m06527012.png" /></td> </tr></table>

An important proper subclass of the class of multiharmonic functions consists of the pluriharmonic functions (cf. [[Pluriharmonic function|Pluriharmonic function]]), for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065270/m06527013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065270/m06527014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065270/m06527015.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065270/m06527016.png" />, and which also satisfy certain additional conditions.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.M. Stein,   G. Weiss,   "Introduction to harmonic analysis on Euclidean spaces" , Princeton Univ. Press  (1971)</TD></TR></table>



====Comments====
Multiharmonic functions are also called multiply harmonic functions. It was shown by P. Lelong that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065270/m06527017.png" /> is separately harmonic, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065270/m06527018.png" /> is harmonic as a function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065270/m06527019.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065270/m06527020.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065270/m06527021.png" />) while the other variables remain fixed, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065270/m06527022.png" /> is multiharmonic. A different proof is due to J. Siciak. See [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Hervé,   "Analytic and plurisubharmonic functions" , ''Lect. notes in math.'' , '''198''' , Springer  (1971)</TD></TR></table>