Rellich-Kondrachov type theorems on the half-space with general singular weights

We prove Rellich-Kondrachov type theorems on the half-space $\mathbb{H}^{N+1}=\{(y, x) \in \left.\mathbb{R} \times \mathbb{R}^N: y>0\right\}$ endowed with the general weighted measure $μ_w:=y^c ϕ(|z|) d z$, where $c \in \mathbb{R}$ and $ϕ$ is a suitable Borel measurable function. We establish a nece