Potential Analysis

, Volume 34, Issue 1, pp 1-12

First online:

Note on Affine Gagliardo–Nirenberg Inequalities

  • Zhichun ZhaiAffiliated withDepartment of Mathematical and Statistical Science, University of Alberta Email author 

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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.


Sobolev spaces Gagliardo–Nirenberg inequalities Sharp constant Rearrangements Pólya–Szegö principle

Mathematics Subject Classifications (2010)

Primary 46E35 46E30