I am deriving the error estimates of one nonconforming finite element method, and for the error of the gradient of true solution, $\mathbf{q}_h$, which is a piecewise constant vector, I have the following results:
$\Omega$ is an open bounded domain, and for any function $v\in H^1(\Omega)$ and $v=0$ on a nontrivial portion of boundary $\Gamma_1$, we have \begin{equation} \sum_{T\in\mathcal{T}}(\mathbf{q}_h,\nabla v)_{0,T}\lesssim C\Vert\nabla v\Vert_0 \end{equation} where $(\cdot,\cdot)_{0,T}$ denotes the $L^2$ inner product on each element $T$ in triangulation $\mathcal{T}$ of $\Omega$.
From this inequality, can we have any estimate about the norm of $\mathbf{q}_h$. For example, can we say that \begin{equation} \Vert \mathbf{q}_h\Vert_0\lesssim C? \end{equation}
Define $C:=\min\{\tilde C>0:(q,\nabla v)\leq \tilde C\,|u|_1\}$.
We know that, for all $|u|_1\neq 0$,
$$\dfrac{(q\,,\nabla u)}{|u|_1}\leq C$$
and, thanks Cauchy--Schwarz inequality,
$$\dfrac{(q\,,\nabla u)}{|u|_1}\leq \|q\|_0$$
for all $q$.
What happen if we suppose that exists $\tilde q$ such that $\|\tilde q\|_0>C$?