We know that \begin{align*} \|u\|_{0,\infty,\Omega}={ess \sup}_{x\in \Omega}|u|. \end{align*} Moreover, the nonconforming Crouzeix-Raviart finite element space is \begin{align*} V_h=\{ v|_K\in P_1{K}, v ~is~continuous~at~midpoints~of~edges~and~v=0~at~the~midpoints~on~boundary~edges\}. \end{align*} For a function $w\in V_h$, we know that $w\notin H_0^1(\Omega)$. My question is what is the definition of $\|w\|_{0,\infty,\Omega}$ and $\|\nabla w\|_{0,\infty,\Omega}$?
In addition, does the inversive inequality \begin{align*} \|\nabla w\|_{0,\infty,\Omega}\leq Ch^{-1}\|w\|_{0,2,\Omega}, w\in V_h \end{align*} hold?

Can anyone give me some advice Could any one give me a hand?