  We prove Rellich-Kondrachov type theorems and weighted Poincar\'{e}
inequalities on the half-space $\mathbb{R}^{N+1}_+=\{z=(x,y): x \in
\mathbb{R}^N, y>0\}$ endowed with the weighted Gaussian measure $\mu
:=y^ce^{-a|z|^2}dz$ where $c+1>0$ and $a>0$. We prove that for some positive
constant $C>0$ one has \begin{align*}
  \left\|u-\overline u\right\|_{L^2_\mu(\mathbb{R}^{N+1}_+)}\leq C \|\nabla
u\|_{L^2_\mu (\mathbb{R}^{N+1}_+)},\qquad \forall u\in
H^1_\mu(\mathbb{R}^{N+1}_+) \end{align*} where $\overline u=\frac
1{\mu(\mathbb{R}^{N+1}_+)}\int_{\mathbb{R}^{N+1}_+} u\,d\mu(z)$. Besides this
we also consider the local case of bounded domains of $\mathbb{R}^{N+1}_+$
where the measure $\mu$ is $y^cdz$.
