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

TL;DR

This paper establishes Rellich-Kondrachov type theorems on the half-space with general singular weights, characterizing compactness of weighted Sobolev embeddings through measure conditions and inequalities.

Contribution

It generalizes Rellich-Kondrachov theorems to weighted measures with singularities, providing necessary and sufficient conditions for compactness involving measure finiteness and tail inequalities.

Findings

Compactness characterized by finite measure and global tightness.

Established coercive tail inequality (Lyapunov condition).

Extended results to broader class of radial weights.

Abstract

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.