Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2508.01978 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911497423159296 |
|---|---|
| author | Zhao, Yunfan Chen, Xiaojing |
| author_facet | Zhao, Yunfan Chen, Xiaojing |
| contents | 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 necessary and sufficient characterization for the compactness of the immersion $H_{μ_w}^1\left(\mathbb{H}^{N+1}\right) \hookrightarrow L_{μ_w}^2\left(\mathbb{H}^{N+1}\right)$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_01978 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rellich-Kondrachov type theorems on the half-space with general singular weights Zhao, Yunfan Chen, Xiaojing Functional Analysis 46E35, 35A23, 35J70 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 necessary and sufficient characterization for the compactness of the immersion $H_{μ_w}^1\left(\mathbb{H}^{N+1}\right) \hookrightarrow L_{μ_w}^2\left(\mathbb{H}^{N+1}\right)$. 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. |
| title | Rellich-Kondrachov type theorems on the half-space with general singular weights |
| topic | Functional Analysis 46E35, 35A23, 35J70 |
| url | https://arxiv.org/abs/2508.01978 |