Hardy Type Inequalities via Riccati and Sturm–Liouville Equations

  • Sergey Bobkov
  • Friedrich Götze
Part of the International Mathematical Series book series (IMAT, volume 8)

We discuss integral estimates for domain of solutions to some canonical Riccati and Sturm–Liouville equations on the line. The approach is applied to Hardy and Poincaré type inequalities with weights.


Riccati Equation Type Inequality Liouville Equation Logarithmic Sobolev Inequality Regular Case 
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  1. 1.
    Artola, M.: Unpublished manuscriptGoogle Scholar
  2. 2.
    Bobkov, S.G., Götze, F.: Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J. Funct. Anal.163, no. 1, 1–28 (1999)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Birkhoff, G., Rota, G-C.: Ordinary Differential Equations. Xerox College Publ. (1969)Google Scholar
  4. 4.
    Chen, L.H.Y., Lou, J.-H.: A Characterization of Probability Measures which Admit Poincaré Inequalities. Preprint (1991)Google Scholar
  5. 5.
    Ince, E.L.: Ordinary Differential Equations. Dover Publ. (1956)Google Scholar
  6. 6.
    Jost, J., Li-Jost, X.: Calculus of Variations. Cambridge Univ. Press, Cambridge (1998)MATHGoogle Scholar
  7. 7.
    Kac, I.S., Krein, M.G.: Criteria for the discreteness of the spectrum of a singular string (Russian). Izv. VUZov. Matematika, no. 2(3), 136–153 (1958)MathSciNetGoogle Scholar
  8. 8.
    Levin, A.Yu.: A comparison principle for second-order differential equations (Russian). Dokl. Akad. Nauk SSSR 783–786 (1960); English transl.: Sov. Math., Dokl.1, 1313–1316 (1960)Google Scholar
  9. 9.
    Mazya, V.G.: Sobolev Spaces. Springer-Verlag, Berlin - New York (1985)Google Scholar
  10. 10.
    Miclo, L.: An example of application of discrete Hardy's inequalities. Markov Process. Relat. Fields5, no. 3, 319–330, (1999)MATHMathSciNetGoogle Scholar
  11. 11.
    Muckenhoupt, B.: Hardy's inequality with weights. Stud. Math.44, 31–38 (1972)MATHMathSciNetGoogle Scholar
  12. 12.
    Reid, W.T.: Ordinary Differential Equations. John Wiley & Sons (1971)Google Scholar
  13. 13.
    Ross, S.L.: Differential Equations. Xerox (1974)Google Scholar
  14. 14.
    Talenti, G.: Osservazioni sopra una classe di disuguaglianze. Rend. Sem. Mat. Fis. Milano39, 171–185 (1969)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Tomaselli, G.: A class of inequalities. Boll. Unione Mat. Ital.21, 627–631 (1969)MathSciNetGoogle Scholar

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© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Sergey Bobkov
    • 1
  • Friedrich Götze
    • 2
  1. 1.University of MinnesotaMinneapolisUSA
  2. 2.Bielefeld UniversityBielefeldGermany

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