# Sobolev trace inequality on $W^{s,q}(\mathbb{R}^n)$

**Authors:** Hee Chul Pak  
**arXiv:** [1909.03230](https://arxiv.org/abs/1909.03230)  
**Date:** 2019-09-07  
**Summary page:** https://www.summarizepaper.com/en/arxiv-id/1909.03230v1/

## Key points

- Paper introduces and establishes Sobolev trace inequalities on nonhomogeneous fractional Sobolev spaces
- Focuses on the space $W^{s,q}(\mathbb{R}^n)$, a function space measuring smoothness of functions
- Extends existing theory of Sobolev spaces to include nonhomogeneous fractional Sobolev spaces
- Main result is establishment of Sobolev trace inequalities on these spaces
- Trace inequalities are fundamental in functional analysis and used to establish properties of functions on bounded domains or manifolds
- Provides new insights into behavior and properties of functions in nonhomogeneous fractional Sobolev spaces
- Potential applications in partial differential equations, harmonic analysis, and mathematical physics
- Contributes to understanding and development of Sobolev spaces

## Summary

The paper titled "Sobolev trace inequality on $W^{s,q}(\mathbb{R}^n)$" by Hee Chul Pak introduces and establishes Sobolev trace inequalities on nonhomogeneous fractional Sobolev spaces. The author focuses on the space $W^{s,q}(\mathbb{R}^n)$, which is a function space equipped with a norm that measures the smoothness of functions. The study of Sobolev spaces is crucial in various areas of mathematics and physics, as they provide a framework for analyzing functions with weak derivatives. In this paper, the author extends the existing theory of Sobolev spaces to include nonhomogeneous fractional Sobolev spaces. The main result of the paper is the establishment of Sobolev trace inequalities on these nonhomogeneous fractional Sobolev spaces. Trace inequalities play a fundamental role in functional analysis and are used to establish important properties of functions defined on bounded domains or manifolds. By establishing these trace inequalities, the author provides new insights into the behavior and properties of functions in nonhomogeneous fractional Sobolev spaces. This result has potential applications in various fields such as partial differential equations, harmonic analysis, and mathematical physics. Overall, this paper contributes to our understanding and development of Sobolev spaces by introducing and studying trace inequalities on nonhomogeneous fractional Sobolev spaces. The results presented in this paper have implications for various areas of mathematics and can potentially lead to further advancements in related fields.