Gaussian Poincaré inequalities on the half-space with singular weights

We prove Rellich-Kondrachov type theorems and weighted Poincaré 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 $\mu :=y^c{e}^{-a\mathrm{\mid }z{\mathrm{\mid }}^2}dz$ 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_μ($\mathbb{R}$^{N+1}+)}\leq C |\nabla u|{L^2_μ($\mathbb{R}$^{N+1}+)},\qquad \forall u\in H^1_μ($\mathbb{R}$^{N+1}_+) \end{align*} where $\bar{u}=\frac{1}{\mu ({\mathbb{R}}_{+}^{N+1})}{\int }_{{\mathbb{R}}_{+}^{N+1}}ud\mu (z)$. Besides this we also consider the local case of bounded domains of ${\mathbb{R}}_{+}^{N+1}$ where the measure $\mu$ is $y^{c}dz$.