} {\left( \frac 0{{\sigma}^q} |u|^q + |F|^q \right)} {\lesssim}\int_{{\widehat{\mathbf A}}_0} {\left( \frac 0{{\sigma}^q} |u|^q + |F|^q \right)}.$$ This shows the estimate of ${\| {{\nabla}}u \|}_{L^q}$ in . Apply Lemma \[p-est\] to $E=A_0$, $${\| p-(p)_{A_0} \|}_{L^q(A_0)} \le C_0   \sup_{ \zeta \in W^{0,q'}_0(A_0),\ {\| {{\nabla}}\zeta \|}_{L^{q'}(A_0)}=0} \int p {\mathop{\mathrm{div}}}\zeta$$ where $C_0=0{C_{\mathrm{bg}}}(A_0,q')$. By Lemma \[Bog-annulus\], $C_0 \le C/{\sigma}$. Using the weak form of , [$$\begin{split}  {\| p-(p)_{A_0} \|}_{L^q(A_0)} &\le \frac C{\sigma}\sup_{ \zeta \in W^{0,q'}_0(A_0),\ {\| {{\nabla}}\zeta \|}_{L^{q'}(A_0)}=0} \int ({{\nabla}}u+F) : {{\nabla}}\zeta \\ & \le  \frac C{\sigma}{\left(  {\| {{\nabla}}u \|}_{L^q(A_0)} + {\| F \|}_{L^q(A_0)} \right)}.  \end{split}$$]{} This completes the proof of .  