For $\Delta f_g = g$, can we prove that $f_g \in L^\infty_{\text{per}}(\Omega)$ and
 \begin{align*}
	  \|f_g\|_{L^\infty_{\rm per}} \le c \|g\|_{L^\infty_{\rm per}}
 \end{align*}
 where $c$ does not depend on $g$ ?