Existence of extremal functions in higher-order affine Sobolev inequalities

TL;DR

This paper proves the existence of extremal functions in higher-order affine Sobolev inequalities using concentration-compactness methods, with tools applicable to spaces of various regularities.

Contribution

It establishes the existence of extremal functions in higher-order affine Sobolev inequalities, expanding the mathematical understanding of these inequalities.

Findings

Existence of extremal functions proven for higher-order affine Sobolev inequalities.

Methods rely on concentration-compactness in spaces of integer or fractional regularity.

Tools developed are applicable to spaces of arbitrary regularity.

Abstract

In this article, we prove the existence of extremal functions in higher-order affine Sobolev inequalities. Proofs rely on concentration-compactness methods in spaces of integer or fractional regularity. The tools we use, available in spaces of arbitrary regularity, might be of independent interest.