Each square integrable harmonic function can be written as the sum of an antianalytic and an analytic, square integrable functions.

Question

I want to prove that each square integrable harmonic function with respect to the standard Gaussian measure:
$\gamma ^{{n}}({\mathbb{C}})={\frac {1}{{\sqrt {2\pi }}^{{n}}}}\int _{{\mathbb{C}}}\exp \left(-{\frac {1}{2}}\|x\|_{\mathbb{C}}\right)\,{\mathrm {d}}\lambda ^{{n}}(x)$ , can be written as the sum of two functions, an analytic function and the other antianalytic (i.e. that the conjugate is an analytic function), both square integrables with respect to the Gaussian measure.

I already proved that in a simply connected domain $D$ every complex harmonic function f has the representation $f=h+\bar{g}$ , where h and g are analytic functions on $D$, also this representation is unique.