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Let $f$ be a plurisubharmonic function, $f < 0$ in $\Omega$. Can we always find two negative plurisubharmonic functions $u$ and $v$ and real numbers $a,b\in(-1,1)$ such that

$$-f=(-u)^{a}+(-v)^{b}$$

in $\Omega$?

Remark. If this is true then we can apply it to the context when we are studying self-bounded gradient functions.

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(Edit).

I think the answer is no. Let $\Omega$ be a region in the plane, and let me use positive superharmonic functions $F=-f,U=-u,V=-v$. Then your question is whether you can write a positive superharmonic function $F$ as a sum $U^a+V^b$. This may not be possible because $$\Delta(U^a)=a(a-1)U^{a-2}(U_x^2+U_y^2)+aU^{a-1}\Delta U.$$ If $a\in(0,1)$ this is strictly negative. If $a\in(-1,0)$ this is strictly positive. Same argument applies to $V^b$. So if $F$ is harmonic but not constant, this representation is not possible with $a,b$ of the same sign.

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  • $\begingroup$ I think the question is to find appropriate $u$, $v$, $a$, $b$ for a given $f$. In your argument, you imposed $a \in (0,1)$. Moreover, even if $a \in (0,1)$ how about $v$, $b$. $\endgroup$
    – Han Ju
    Commented Dec 13, 2017 at 3:17
  • $\begingroup$ My argument really shows that such representation is impossible with $a,b$ of the same sign. I edited the answer. $\endgroup$
    – Alexandre Eremenko
    Commented Dec 13, 2017 at 22:30

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