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Equivalence of analytic and plurisubharmonic Phragmén-Lindelöf conditions. (English) Zbl 0745.32004
Several complex variables and complex geometry, Proc. Summer Res. Inst., Santa Cruz/CA (USA) 1989, Proc. Symp. Pure Math. 52, Part 3, 287-308 (1991).
[For the entire collection see Zbl 0732.00009.]
Let \(P(D)\) be a partial differential operator with constant coefficients. It is known that some interesting properties of \(P(D)\) can be characterized by whether or not satisfying certain estimates of Phragmén-Lindelöf type for plurisubharmonic functions of the form \(u=\log| f|\), \(f\) entire, on the algebraic variety \(V=\{z\in{\mathbb{C}}^ n:\;P(z)=0\}\). In this paper, authors prove that for five Phragmén-Lindelöf conditions that have been appeared in the references the estimate holds for \(u=\log| f|\) if and only if the estimates hold for all plurisubharmonic functions on the variety \(V\). Here the idea of proving the equivalence of the analytic and plurisubharmonic versions of these Phragmén-Lindelöf principles is to write the plurisubharmonic function \(u\) as an upper envelope of functions \(\log| f|\), \(f\) entire. From the above result, it follows that an arbitrary weakly plurisubharmonic function on the variety \(V\) can nearly be written as such an upper envelope except on a small exceptional set.

32U05 Plurisubharmonic functions and generalizations
31C10 Pluriharmonic and plurisubharmonic functions
35E20 General theory of PDEs and systems of PDEs with constant coefficients