On sharp constants in higher order Adams-Cianchi inequalities

TL;DR

This paper establishes sharp constants for critical Sobolev embeddings, extending Adams and Cianchi's work, and improves related trace inequalities, contributing to the understanding of optimal bounds in functional analysis.

Contribution

It provides new sharp constants for higher order Sobolev embeddings and enhances existing trace inequalities, advancing the theory of critical functional inequalities.

Findings

Sharp constants for higher order Sobolev embeddings established

Trace inequality improved with a new bound

Extension of Adams and Cianchi's foundational work

Abstract

The main results of this paper are the establishment of sharp constants for several families of critical Sobolev embeddings. These inequalities were pioneered by David R. Adams, while the sharp constant in the first order case is due to Andrea Cianchi. We also prove a trace improvement of an inequality obtained independently by K. Hansson and H. Brezis and S. Wainger.