Potential Analysis

, Volume 34, Issue 1, pp 1–12 | Cite as

Note on Affine Gagliardo–Nirenberg Inequalities

Article

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.

Keywords

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

Mathematics Subject Classifications (2010)

Primary 46E35 46E30 

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© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Department of Mathematical and Statistical ScienceUniversity of AlbertaEdmontonCanada

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