Hi there,
I have the following problems on my hand:
Given arbitrary positive sub-harmonic function $l(x,y)$ on plane, for all analytic functions: $f=a(x,y)+ib(x,y)$ on complex plane, consider the equation:
$|f|^2=l$, i.e. $a^2+b^2=l$
1) for what kind of $l(x,y)$, the equation has $C^2$ solution?
2) for what kind of $l(x,y)$, the number of solutions is finite, after quotient of $S^1$ action: $e^{i\theta}\cdot f$ ?
Thanks!
For this equation to have solutions, it is not enough that $l$ is subharmonic. It must be logarithmically subharmonic which means that $\log l$ is subharmonic. Now if $l$ is logarithmically subharmonic, the nesessary and sufficient condition that $f$ exists in a simply connected domain is that the measure $(2\pi)^{-1}\Delta\log l$ is discrete and integer-valued.
If the solution exists it is always unique up to a constant factor $e^{i\theta}$.
EDIT. To prove sufficiency, one uses the Weierstrass representation. Suppose that $l$ is logarithmically subharmonic and $(2\pi)^{-1}\Delta\log l$ is a discrete integer-valued measure with atoms $m_k$ at the points $a_k$. Consider the Weierstrass product $f$ with zeros $a_k$ of multiplicity $m_k$. This $f$ is an entire function. Now $u:=\log l-\log|f|$ is harmonic, by construction. Let $v$ be such harmonic function that $g:=u+iv$ is entire. Then $l=|f\exp g|.$
