Note on Affine Gagliardo–Nirenberg Inequalities
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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 Lp-Sobolev inequalities. The logarithmic version of affine Lp-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.
KeywordsSobolev spaces Gagliardo–Nirenberg inequalities Sharp constant Rearrangements Pólya–Szegö principle
Mathematics Subject Classifications (2010)Primary 46E35 46E30
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