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

Abstract

We prove Rellich-Kondrachov type theorems on the half-space ${\mathbb{H}}^{N+1}=\{(y,x)\in \mathbb{R}\times {\mathbb{R}}^N:y>0\}$ endowed with the general weighted measure $\hat{I}{¼}_w:=y^c\ddot{I}•(|z|)dz$, where $c\in \mathbb{R}$ and $\ddot{I}•$ is a suitable Borel measurable function. We establish a necessary and sufficient characterization for the compactness of the immersion $H_{\hat{I}{¼}_w}^1\left({\mathbb{H}}^{N+1}\right)\hookrightarrow L_{\hat{I}{¼}_w}^2\left({\mathbb{H}}^{N+1}\right)$. 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 $c\le -1$, a weighted Hardy inequality. These results generalize recent work on Gaussian weights to a broader class of radial potentials defined by abstract massvanishing conditions.