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Proof of Poincare Inequalities for Continuously Differentiable Function on 01

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Abstract

Poincaré-Sobolev-type inequalities involving rearrangement-invariant norms on the entire \(\mathbb R^n\) are provided. Namely, inequalities of the type \(\Vert u-P\Vert _{Y(\mathbb R^n)}\le C\Vert \nabla ^m u\Vert _{X(\mathbb R^n)}\), where X and Y are either rearrangement-invariant spaces over \(\mathbb R^n\) or Orlicz spaces over \(\mathbb R^n\), u is a \(m-\)times weakly differentiable function whose gradient is in X, P is a polynomial of order at most \(m-1\), depending on u, and C is a constant independent of u, are studied. In a sense optimal rearrangement-invariant spaces or Orlicz spaces Y in these inequalities when the space X is fixed are found. A variety of particular examples for customary function spaces are also provided.

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Authors and Affiliations

  1. Faculty of Mathematics and Physics, Department of Mathematical Analysis, Charles University, Sokolovská 83, Praha 8, 186 75, Czech Republic

    Zdeněk Mihula

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Correspondence to Zdeněk Mihula.

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This research was supported by the grant P201-18-00580S of the Grant Agency of the Czech Republic, by the grant SVV-2017-260455, and by Charles University Research program No. UNCE/SCI/023.

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Mihula, Z. Poincaré-Sobolev inequalities with rearrangement-invariant norms on the entire space. Math. Z. 298, 1623–1640 (2021). https://doi.org/10.1007/s00209-020-02652-z

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  • DOI : https://doi.org/10.1007/s00209-020-02652-z

Keywords

  • Optimal spaces
  • Poincaré inequality
  • Poincaré-Sobolev inequality
  • Rearrangement-invariant spaces
  • Sobolev spaces

Mathematics Subject Classification

  • 26D10
  • 46E30
  • 46E35
  • 47B38

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