Note on affine Gagliardo-Nirenberg inequalities

Zhichun Zhai

Note on affine Gagliardo-Nirenberg inequalities

Functional Analysis 2009-08-17 v1 Differential Geometry

Authors: Zhichun Zhai

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.

Keywords

affine Gagliardo-Nirenberg inequalities affine Sobolev inequalities Moser-Trudinger inequalities Morrey-Sobolev inequalities rearrangements Pólya-Szegő principle

Cite

@article{arxiv.0908.2095,
  title  = {Note on affine Gagliardo-Nirenberg inequalities},
  author = {Zhichun Zhai},
  journal= {arXiv preprint arXiv:0908.2095},
  year   = {2009}
}

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10 pages

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