Poisson equation, $L^2$ bounds

Question

Consider a bounded domain $\Omega\subset\mathbb{R}^d$ and $u\in H^1_0(\Omega)$.

I know that $$ \|u\|_{H^m}\leq C\|\Delta u\|_{H^{m-2}} $$ for $m\geq 1$. Is the same true for $m=0$, i.e. for the $L^2$ norm of $u$?