On Gagliardo–Nirenberg Type Inequalities

We present a Gagliardo–Nirenberg inequality which bounds Lorentz norms of a function by Sobolev norms and homogeneous Besov quasinorms with negative smoothness. We prove also other versions involving Besov or Triebel–Lizorkin quasinorms. These inequalities can be considered as refinements of Sobolev...

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Veröffentlicht in:The Journal of fourier analysis and applications 2014-06, Vol.20 (3), p.577-607
Hauptverfasser: Kolyada, V. I., Pérez Lázaro, F. J.
Format: Artikel
Sprache:eng
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Zusammenfassung:We present a Gagliardo–Nirenberg inequality which bounds Lorentz norms of a function by Sobolev norms and homogeneous Besov quasinorms with negative smoothness. We prove also other versions involving Besov or Triebel–Lizorkin quasinorms. These inequalities can be considered as refinements of Sobolev type embeddings. They can also be applied to obtain Gagliardo–Nirenberg inequalities in some limiting cases. Our methods are based on estimates of rearrangements in terms of heat kernels. These methods enable us to cover also the case of Sobolev norms with p = 1 .
ISSN:1069-5869
1531-5851
1531-5851
DOI:10.1007/s00041-014-9320-y