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On certain generalizations of the Gagliardo–Nirenberg inequality and their applications to capacitary estimates and isoperimetric inequalities

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  • Published: 03 March 2013
  • volume 13, pages 271–290 (2013)
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On certain generalizations of the Gagliardo–Nirenberg inequality and their applications to capacitary estimates and isoperimetric inequalities
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  • Agnieszka Kałamajska1 &
  • Jan Peszek1 
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  • 5 Citations

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Abstract

We derive the inequality

$$\int_\mathbb{R}M(|f'(x)|h(f(x))) dx\leq C(M,h)\int_\mathbb{R}M\left({\sqrt{|f''(x)\tau_h(f(x))|}\cdot h(f(x))}\right)dx$$

with a constant C(M, h) independent of f, where f belongs locally to the Sobolev space \({W^{2,1}(\mathbb{R})}\) and f′ has compact support. Here M is an arbitrary N-function satisfying certain assumptions, h is a given function and \({\tau_h(\cdot)}\) is its given transform independent of M. When M(λ) =  λp and \({h \equiv 1}\) we retrieve the well-known inequality \({\int_\mathbb{R}|f'(x)|^{p}dx \leq (\sqrt{p - 1})^{p}\int_\mathbb{R}(\sqrt{|f''(x) f(x)|})^{p}dx}\). We apply our inequality to obtain some generalizations of capacitary estimates and isoperimetric inequalities due to Maz’ya (1985).

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References

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Authors and Affiliations

  1. Faculty of Mathematics, Informatics, and Mechanics, University of Warsaw, ul. Banacha 2, 02–097, Warszawa, Poland

    Agnieszka Kałamajska & Jan Peszek

Authors
  1. Agnieszka Kałamajska
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  2. Jan Peszek
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Correspondence to Agnieszka Kałamajska.

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Kałamajska, A., Peszek, J. On certain generalizations of the Gagliardo–Nirenberg inequality and their applications to capacitary estimates and isoperimetric inequalities. J. Fixed Point Theory Appl. 13, 271–290 (2013). https://doi.org/10.1007/s11784-013-0106-7

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  • Published: 03 March 2013

  • Issue Date: March 2013

  • DOI: https://doi.org/10.1007/s11784-013-0106-7

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Mathematics Subject Classification

  • Primary 46E35
  • Secondary 26D10

Keywords

  • Gagliardo–Nirenberg inequalities
  • interpolation inequalities
  • capacities
  • isoperimetric inequalities
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