Define a function space V , $$ V:=\{\mathbf{v} \in \mathbf{L}^{1+\alpha}(\Omega), \mathbf{curl}~\mathbf{v} \in \mathbf{L}^2(\Omega)\}, $$ equipped with graph norm $$ \|\mathbf{v}\|_{V} := \|\mathbf{v}\|_{\mathbf{L}^{1+\alpha}(\Omega)} + \|\mathbf{curl~v}\|_{\mathbf{L}^2(\Omega)}, $$ where, the real number $0<\alpha\le 1$, and $\Omega$ is a bounded domain.
Clearly, when $\alpha = 1$, V becomes to be $\mathbf{H}(\mathbf{curl},\Omega)$. As we all know, $$ \|\mathbf{v}\|_{\mathbf{H}(\mathbf{curl},\Omega)} \le C \big(\|\mathbf{curl}~\mathbf{v}\|_{\mathbf{L}^2(\Omega)} +\|div~\mathbf{v}\|_{L^2(\Omega)}\big), $$ for any $\mathbf{v} \in \mathbf{H}(\mathbf{curl},\Omega)$.
Do functions in V with $0<\alpha<1$ have the similar result, i.e., $$ \|\mathbf{v}\|_{V} \le C \big(\|\mathbf{curl}~\mathbf{v}\|_{\mathbf{L}^2(\Omega)} +\|div~\mathbf{v}\|_{L^2(\Omega)}\big) $$ for any function $\mathbf{v}\in V$?