Necas theorem gives that
$$\|p\|_{L^2(\Omega)}\lesssim\|p\|_{H^{-1}(\Omega)}+\sum_{i=1}^n\|\frac{\partial p}{\partial x_i}\|_{H^{-1}(\Omega)}\tag{$*$},$$
where $\Omega$ is connected Lipschitz domain and $p\in L^2(\Omega)$.
Now I need to prove that $(*)$ is equavalent to
$$\|p\|_{L^2(\Omega)}\lesssim\sum_{i=1}^n\|\frac{\partial p}{\partial x_i}\|_{H^{-1}(\Omega)},\tag{$**$}$$
where $p\in L_0^2(\Omega)=\{p\in L^2(\Omega):\int_{\Omega}p=0\}$.
I already know how to prove $(*)\Rightarrow(**)$. In order to prove $(**)\Rightarrow(*)$, I think $p\in L^2$ can be decomposed to $p=\bar{p} + p_0$, where $\bar{p}=\frac{1}{|\Omega|}\int_{\Omega}p$ and $p_0=p-\bar{p} \in L_0^2(\Omega)$. Hence, we only need to prove that $\|\bar{p}\|_{L^2}\lesssim\|p\|_{H^{-1}}\quad(***)$.
$\|p\|_{H^{-1}}=sup\frac{\int_\Omega pv}{\|v\|_{H^1}}$ where the supremum is taken for all $v\in H_0^1$. If I could take v as a constant, I could have easily proved $(***)$. I wonder if the inequality $(***)$ holds, and how to prove it if it holds. Maybe I can take $v \in H_0^1$ which is a constant in a large area?
To prove that $(**)$ can lead to $(*)$, notice that $p=p_0+\bar{p}$, and $\|\bar{p}\|_L^2 \lesssim \|\bar{p}\|_H^{-1}$ since $\bar{p}$ is a real number. The $H^{-1}$ norm of $\bar{p}$ can be bounded by $\|p\|_H^{-1}+\|p_0\|_H^{-1}$, and using $\|p_0\|_H^{-1}\lesssim \|p_0\|_L^2 \lesssim \sum_{i=1}^n\|\frac{\partial p_0}{\partial x_i}\|_{H^{-1}}$, we can deduce the desired inequality $(*)$.
$(***)$ may not hold.