Let $\Omega\subset \mathbb R^N$ open bounded be given, I am trying to prove that first any Polynomial can be written as difference of two sub-harmonic functions, and then for any continuous function can be written as difference of two sub-harmonic functions.
I first tried in 1-d. The derivative of 1-d polynomial is of course a bounded variation function and hence I can write it as difference of two monotone functions. Hence by using this two monotone functions I could define, by anti-derivative of each of them, two sub-harmonic function as I wanted.
But this method will not work in multi-dimensions...
Any hint is really welcome!
Equivalent formulation: for every polynomial $p$ there is a subharmonic function $u$ such that $u-p$ is also subharmonic. Indeed, if such $u$ is found, then $p=u-(u-p)$ gives what you want.
As long as $\Delta p$ is bounded from below (which it is for polynomials on a bounded domain) a sufficiently large multiple of $|x|^2$ will work as $u$.
The statement
is false. For example, in one dimension "subharmonic" is equivalent to "convex", and every such function is differentiable almost everywhere. So, a continuous nowhere differentiable function cannot be written as the difference of two subharmonic functions.
The functions that can be written as the difference of two subharmonic functions are called $\delta$-subharmonic in the literature. Their characteristic property is: the distributional Laplacian is a signed measure.