Note on affine Gagliardo-Nirenberg inequalities

TL;DR

This paper establishes sharp affine Gagliardo-Nirenberg inequalities that outperform Euclidean versions, also deriving affine Sobolev, Moser-Trudinger, and Morrey-Sobolev inequalities using rearrangement techniques and Pólya-Szegö principles.

Contribution

It introduces the first sharp affine Gagliardo-Nirenberg inequalities and provides alternative proofs for affine Moser-Trudinger and Morrey-Sobolev inequalities.

Findings

Sharp affine Gagliardo-Nirenberg inequalities proved

Affine $L^{p}$-Sobolev inequalities verified

Alternative proofs for affine Moser-Trudinger and Morrey-Sobolev inequalities provided

Abstract

This note proves sharp affine Gagliardo-Nirenberg inequalities which are stronger than all known sharp Euclidean Gagliardo-Nirenberg inequalities and imply the affine $L^{p}-$Sobolev inequalities. The logarithmic version of affine $L^{p}-$Sobolev inequalities is verified. Moreover, An alternative proof of the affine Moser-Trudinger and Morrey-Sobolev inequalities is given. The main tools are the equimeasurability of rearrangements and the strengthened version of the classical P\'{o}lys-Szeg\"{o} principle.