Affine fractional $L^p$ Sobolev inequalities

TL;DR

This paper introduces sharp affine fractional $L^p$ Sobolev inequalities that extend classical inequalities, providing stronger bounds and new asymmetric versions, advancing the understanding of fractional Sobolev spaces.

Contribution

The paper develops new sharp affine fractional $L^p$ Sobolev inequalities, including asymmetric variants, which are stronger than existing fractional Sobolev inequalities.

Findings

Established sharp affine fractional $L^p$ Sobolev inequalities.

Derived affine fractional asymmetric $L^p$ Sobolev inequalities.

Showed these inequalities are stronger than classical fractional Sobolev inequalities.

Abstract

Sharp affine fractional $L^p$ Sobolev inequalities for functions on $\mathbb R^n$ are established. The new inequalities are stronger than (and directly imply) the sharp fractional $L^p$ Sobolev inequalities. They are fractional versions of the affine $L^p$ Sobolev inequalities of Lutwak, Yang, and Zhang. In addition, affine fractional asymmetric $L^p$ Sobolev inequalities are established.