$H^{2m}(\Omega)\cap H_{0}^{m}(\Omega)\hookrightarrow W_{0}^{m,p}(\Omega)$?

Question

Consider the following embedding: $$ H^2(\Omega)\cap H_{0}^{1}(\Omega)\hookrightarrow W_{0}^{1,p}(\Omega),\qquad 2\leq p\leq \frac{2N}{N-2},\ \ if\ \ N>2\ \ and\ \ p\geq 2\ \ if\ \ N=1,2. $$ Is it a special case of a generalized form such as $$ H^{2m}(\Omega)\cap H_{0}^{m}(\Omega)\hookrightarrow W_{0}^{m,p}(\Omega),\quad 2\leq p\leq \frac{2N}{N-2m},\ \ if\ \ N>2m\ \ and\ \ p\geq 2\ \ if\ \ N\leq 2m.$$ If so we then have $$ \|D^{m'}u\|_{2}\leq C_{\ast} \|D^{m}u\|_{2}\leq \hat{C_{\ast}}\|(-\Delta )^{m}u\|_{2}?\tag I$$ where $1\leq m'\leq m$ and $u\in H^{2m}(\Omega)\cap H_{0}^{m}(\Omega) $($\Omega$ is open bounded in $R^N$).