A weighted anisotropic Sobolev type inequality and its applications to Hardy inequalities

Abstract

In this paper we focus our attention on an embedding result for a weighted Sobolev space that involves as weight the distance function from the boundary taken with respect to a general smooth gauge function F. Starting from this type of inequalities we prove some refined Hardy-type inequalities.

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Notes

  1. 1.

    Since \(d_F\in \mathrm{BV}(\varOmega )\), its level sets \(\{d_F<r\}\) are of finite perimeter for a.e. \(r\in (0,\infty )\) (see [12, Theorem 1–§5.5])

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Acknowledgements

G. di Blasio and G. Pisante are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the IstitutoNazionale di Alta Matematica (INdAM) whose support through the GNAMPA Project 2019 Pogram (U-UFMBAZ-2019-000477 11-03-2019) is gratefully acknowledged. G Pisante and G. Psaradakis also acknowledge the INdAM-GNAMPA for the Visiting Professor Program 2018 (U-FMBAZ-2018-001525 18-12-2018). G. Psaradakis was supported in part from Università degli Studi della Campania “Luigi Vanvitelli” (D.R. 0950-2017) through a visiting researcher position. The authors are grateful to both referees whose remarks and suggestions helped us to considerably enhance the initial version of the article.

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di Blasio, G., Pisante, G. & Psaradakis, G. A weighted anisotropic Sobolev type inequality and its applications to Hardy inequalities. Math. Ann. (2020). https://doi.org/10.1007/s00208-019-01930-4

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Mathematics Subject Classification

  • 53C60
  • 58J60
  • 26D10
  • 46E35