By Robert C. Gunning, Hugo Rossi
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Extra info for Analytic Functions of Several Complex Variables
Set ! D d u. If we can show that ! , then u and v will satisfy the Cauchy–Riemann equations and so f D u C i v will indeed be holomorphic. Since we are working in a polydisc, which is thus a convex domain, we need only check that the differential form ! is closed. By the Poincaré lemma it will then be exact. d! D X j;k X @2 u @2 u dxj ^ dxk C dyj ^ dyk C @xj @xk @yj @xk j;k Ã X Â @2 u @2 u C dxj ^ dyk : C @xj @xk @yj @yk j;k The first sum is zero because @2 u @2 u D @xj @xk @xk @xj while dxj ^ dxk D dxk ^ dxj : The second sum is zero for a similar reason.
1/; for some function u subharmonic on : Theorem 40. If u is plurisubharmonic and > 0; then u is plurisubharmonic, and if u1 and u2 are plurisubharmonic, then so are u1 C u2 and maxfu1 ; u2 g: Proof. Suppose u1 and u2 are plurisubharmonic on a domain ; ` is a complex line and U is a component of ` \ : If u1 and u2 are subharmonic on U; then u1 C u2 is also subharmonic on U: If u1 or u2 is identically 1 on U; then the same is true for u1 C u2 : To see that u1 C u2 is plurisubharmonic on ; there only remains to show that u1 C u2 6Á 1: Now u1 and u2 are both subharmonic on : The sets u1 1 .
Thus d! D 0 and the proof is complete. t u Pluriharmonic functions have the following local representation. Problem 31. A function u defined in an open subset of Cn is pluriharmonic in if and only if it can be locally represented in the form u D f C g; where f and g are holomorphic. 42 8 Plurisubharmonic Functions Recall the definition of a subharmonic function. A function u W G ! Œ 1; C1/; defined on an open subset G Rn ; is said to be subharmonic if it satisfies the following: u is upper semicontinuous, u is not identically 1 on any component of G, and, for each closed ball B G and real-valued continuous function h on B which is harmonic on B; if u Ä h on @B; then u Ä h in B: The following theorem tells us that, for continuous functions u; we don’t need to check all functions h: It is sufficient to check the solution PBu of the Dirichlet problem on B, with boundary values given by u: Theorem 37.
Analytic Functions of Several Complex Variables by Robert C. Gunning, Hugo Rossi