, Volume 34, Issue 1, pp 1-12

Note on Affine Gagliardo–Nirenberg Inequalities

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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 Pólya–Szegö principle.

Project supported in part by Natural Science and Engineering Research Council of Canada.