The Chang-Marshall Trace Inequality for Sobolev functions in domains in higher dimensional space $\mathbb{R}^n$

TL;DR

This paper extends the Chang-Marshall trace inequality, originally for holomorphic functions in the unit disk, to Sobolev functions on general domains in higher-dimensional Euclidean spaces, addressing a significant open question.

Contribution

It provides a partial affirmative answer to whether the Chang-Marshall inequality applies to Sobolev functions in higher dimensions, broadening its scope beyond holomorphic functions.

Findings

Established a partial extension of the inequality to higher dimensions

Addressed a question posed by S. Y. Alice Chang

Contributed to the understanding of trace inequalities in Sobolev spaces

Abstract

In their celebrated work [5], Chang and Marshall established a critical trace inequality of Moser-Trudinger type for holomorphic functions with mean value zero on unit disc in the complex plane. The main purpose is to address a question proposed to us by S. Y. Alice Chang who asked whether the Chang-Marshall type inequality for holomorphic functions on unit disk in the complex plane holds for Sobolev functions on general domains in higher dimensional Euclidean space $\mathbb{R}^n$ for all $n\ge 2$. We partially answer her question affirmatively.