Gaussian Poincaré inequalities on the half-space with singular weights

Abstract: 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 $y^cdz$.