On Gagliardo-Nirenberg Type Inequalities

  1. Kolyada, V.I. 2
  2. Pérez Lázaro, F.J. 1
  1. 1 Universidad de La Rioja
    info

    Universidad de La Rioja

    Logroño, España

    ROR https://ror.org/0553yr311

  2. 2 Karlstad University
    info

    Karlstad University

    Karlstad, Suecia

    ROR https://ror.org/05s754026

Journal:
Journal of Fourier Analysis and Applications

ISSN: 1069-5869

Year of publication: 2014

Volume: 20

Issue: 3

Pages: 577-607

Type: Article

DOI: 10.1007/S00041-014-9320-Y SCOPUS: 2-s2.0-84902372706 WoS: WOS:000337789300007 arXiv: 1211.1315v1 GOOGLE SCHOLAR lock_openOpen access editor

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Abstract

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. © 2014 Springer Science+Business Media New York.