# zbMATH — the first resource for mathematics

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.

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