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I just started learning several complex variables and I'm a little bit confused. I just read that every pluriharmonic function is harmonic and I can't find any proof of that. Please help.

Alem
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1 Answers1

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A function in pluriharmonic if its restriction to every complex line (one-dimensional affine subspace) is harmonic. In particular, a pluriharmonic function is harmonic in each (complex) variable separately, i.e. $$ \frac{\partial^2 f}{\partial z_j\partial \bar z_j} = 0 $$ for each $j$. Hence $$ \sum_{j=1}^n \frac{\partial^2 f}{\partial z_j\partial \bar z_j} = 0 $$ so $f$ is indeed harmonic.

mrf
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  • I don't see the equivalence of the partial derivative definition of pluriharmonic and the restriction to each complex line. –  Jan 28 '17 at 22:09