Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Yunfan Zhao, Xiaojing Chen

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 $μ_w:=y^c φ(|z|) d z$, 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.

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

Abstract

We prove Rellich-Kondrachov type theorems on the half-space endowed with the general weighted measure , where and is a suitable Borel measurable function. We establish a necessary and sufficient characterization for the compactness of the immersion . 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 , a weighted Hardy inequality. These results generalize recent work on Gaussian weights to a broader class of radial potentials defined by abstract massvanishing conditions.

Paper Structure

This paper contains 9 sections, 11 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Functional Inequalities and Local Tools
  4. Tail Control and Decay Estimates
  5. Abstract Compactness and Regularity
  6. Necessary and Sufficiency Conditions
  7. Main Embedding Theorems
  8. Conclusion
  9. Acknowledgement

Key Result

Lemma 3.1

Let $\Omega \subset \mathbb{H}^{N+1}$ be any bounded open set strictly separated from the boundary singularity $y=0$ (i.e., $\operatorname{dist}(\Omega,\{y= 0\})>0$). Under Assumption ass:2, the weight $w(z)$ is bounded strictly away from zero and infinity on $\Omega$. Therefore, the weighted Sobole

Theorems & Definitions (42)

  • Remark
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark
  • Definition 2.4
  • Remark
  • Definition 2.5
  • Definition 2.6
  • Remark
  • ...and 32 more