In a research paper https://hal.science/hal-02891557/, the authors used the embedding of $H^{2}_{0}(\Omega)$ in $L^{\infty}(\Omega)$ (see page 16-line 1) to obtain some estimates in their research paper. Here, $\Omega$ is a bounded domain in $\mathbb{R}^{N}$ with smooth boundary.
Is this because of the definition of $H^{2}_{0}(\Omega)$ that is the closure of $C_{0}^{\infty}(\Omega)$ in $H^{2}(\Omega)$ which means that any function in $H^{2}_{0}(\Omega)$ can be approximated by a sequence of smooth functions with compact support in $\Omega$?
Thanks a lot in advance.