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 $\mu_w:=y^c \phi(|z|) d z$, where $c \in \mathbb{R}$ and $\phi$ is a suitable Borel measurable function. We establish a necessary and sufficient characterization for the compactness of the immersion $H_{\mu_w}^1\left(\mathbb{H}^{N+1}\right) \hookrightarrow L_{\mu_w}^2\left(\mathbb{H}^{N+1}\right)$. We prove that compactness holds if and only if the measure has finite mass and satisfies a "Global Tightness" condition, which we characterize via a coercive tail inequality (Lyapunov condition) and, in the singular case $c \leq-1$, a weighted Hardy inequality. These results generalize recent work on Gaussian weights to a broader class of radial potentials defined by abstract massvanishing conditions.