Trace theorems: critical cases and best constants

TL;DR

This paper investigates the critical cases of trace theorems for functions restricted to surfaces, analyzing asymptotic behavior under dilations and determining the optimal constants involved.

Contribution

It provides a detailed analysis of critical cases in trace theorems, including asymptotic norms and best constants, which were previously not fully understood.

Findings

Identified critical cases of trace theorems.

Derived asymptotic formulas for trace norms under surface dilations.

Determined the best constants for trace inequalities.

Abstract

The purpose of this paper is to present the critical cases of the trace theorems for the restriction of functions to closed surfaces, and to give the asymptotics for the norms of the traces under dilations of the surface. We also discuss the best constants for them.