The problem is
Suppose $\phi\in\mathcal{S}(\mathbb{R}^2)$, $u$ on $\mathbb{R}^3_-=\mathbb{R}^2\times(-\infty,0)$ satisfies
\begin{cases}
\Delta u(x,y,z)=0\\
\lim\limits_{z\to 0^-} u(x,y,z)=\phi(x,y)\\
\lim\limits_{z\to -\infty}\sup\limits_{x,y} |u(x,y,z)|=0
\end{cases}
Show that
(i)For given $n\in \mathbb{Z}^+$, $\exists C$ such that
\begin{equation}
\sum_{|\alpha|\leq n} \int_{\mathbb{R}^3_-} |\partial_{x,y}^{\alpha} u|^2\leq C\sum_{|\alpha|\leq n}\int_{\mathbb{R}^2}\left(|\partial_{x,y}^{\alpha} \phi|^2+|\partial_{x,y}^{\alpha} \phi|\right)
\end{equation}
(ii)For some coefficient that I forgot \begin{equation} \sup_{x,y} |u|\leq C_{?} \frac{1}{|z|^2} \end{equation}
Here $\mathcal{S}$ means the space of Schwartz functions.
The coefficient $C_?$ depends on $\phi$.
The $|\partial_{x,y}^{\alpha} \phi|$ term in (i) seems strange and useless. However that is what the original problem is.