Gaussian Poincaré Inequalities on the Half-Space with Singular Weights

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