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

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.