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Pointwise inequalities of logarithmic type in Hardy-Hölder spaces

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Abstract

We prove some optimal logarithmic estimates in the Hardy space H ∞(G) with Hölder regularity, where G is the open unit disk or an annular domain of ℂ. These estimates extend the results established by S.Chaabane and I.Feki in the Hardy-Sobolev space H k,∞ of the unit disk and those of I. Feki in the case of an annular domain. The proofs are based on a variant of Hardy-Landau-Littlewood inequality for Hölder functions. As an application of these estimates, we study the stability of both the Cauchy problem for the Laplace operator and the Robin inverse problem.

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References

  1. L. Baratchart, J. Leblond: Harmonic identification and Hardy class trace on an arc of the circle. Optimisation et Contrôle. Proceedings of the Colloquium in Honor of Jean Céa’s sixtieth birthday held in Sophia-Antipolis, 1992 (J.-A. Désidéri, eds.). Cépadu`es Éditions, Toulouse, 1993, pp. 17–29. (In French.)

    Google Scholar 

  2. L. Baratchart, J. Leblond, J. R. Partington: Hardy approximation to L ∞ functions on subsets of the circle. Constr. Approx. 12 (1996), 423–435.

    MATH  MathSciNet  Google Scholar 

  3. L. Baratchart, M. Zerner: On the recovery of functions from pointwise boundary values in a Hardy-Sobolev class of the disk. J. Comput. Appl. Math 46 (1993), 255–269.

    Article  MathSciNet  Google Scholar 

  4. H. Brézis: Functional Analysis. Theory and Applications. Collection of Applied Mathematics for the Master’s Degree, Masson, Paris, 1983. (In French.)

    Google Scholar 

  5. S. Chaabane, I. Feki: Optimal logarithmic estimates in Hardy-Sobolev spaces H k,∞. C. R., Math., Acad. Sci. Paris 347 (2009), 1001–1006.

    Article  MATH  MathSciNet  Google Scholar 

  6. S. Chaabane, I. Fellah, M. Jaoua, J. Leblond: Logarithmic stability estimates for a Robin coefficient in two-dimensional Laplace inverse problems. Inverse Probl. 20 (2004), 47–59.

    Article  MATH  MathSciNet  Google Scholar 

  7. S. Chaabane, J. Ferchichi, K. Kunisch: Differentiability properties of the L 1-tracking functional and application to the Robin inverse problem. Inverse Probl. 20 (2004), 1083–1097.

    Article  MATH  MathSciNet  Google Scholar 

  8. S. Chaabane, M. Jaoua: Identification of Robin coefficients by the means of boundary measurements. Inverse Probl. 15 (1999), 1425–1438.

    Article  MATH  MathSciNet  Google Scholar 

  9. S. Chaabane, M. Jaoua, J. Leblond: Parameter identification for Laplace equation and approximation in Hardy classes. J. Inverse Ill-Posed Probl. 11 (2003), 33–57.

    Article  MATH  MathSciNet  Google Scholar 

  10. I. Chalendar, J. R. Partington: Approximation problems and representations of Hardy spaces in circular domains. Stud. Math. 136 (1999), 255–269.

    MATH  MathSciNet  Google Scholar 

  11. B. Chevreau, C. M. Pearcy, A. L. Shields: Finitely connected domains G, representations of H ∞(G), and invariant subspaces. J. Oper. Theory 6 (1981), 375–405.

    MATH  MathSciNet  Google Scholar 

  12. P. L. Duren: Theory of Hp Spaces. Pure and Applied Mathematics 38, Academic Press, New York, 1970.

    Google Scholar 

  13. I. Feki: Estimates in the Hardy-Sobolev space of the annulus and stability result. Czech. Math. J. 63 (2013), 481–495.

    Article  MATH  MathSciNet  Google Scholar 

  14. I. Feki, H. Nfata, F. Wielonsky: Optimal logarithmic estimates in the Hardy-Sobolev space of the disk and stability results. J. Math. Anal. Appl. 395 (2012), 366–375.

    Article  MATH  MathSciNet  Google Scholar 

  15. E. Gagliardo: Proprietà di alcune classi di funzioni più variabili. Ricerche Mat. 7 (1958), 102–137. (In Italian.)

    MATH  MathSciNet  Google Scholar 

  16. G. H. Hardy, E. Landau, J. E. Littlewood: Some inequalities satisfied by the integrals or derivatives of real or analytic functions. Math. Z. 39 (1935), 677–695.

    Article  MathSciNet  Google Scholar 

  17. M. K. Kwong, A. Zettl: Norm Inequalities for Derivatives and Differences. Lecture Notes in Mathematics 1536, Springer, Berlin, 1992.

    MATH  Google Scholar 

  18. L. Leblond, M. Mahjoub, J. R. Partington: Analytic extensions and Cauchy-type inverse problems on annular domains: stability results. J. Inverse Ill-Posed Probl. 14 (2006), 189–204.

    Article  MATH  MathSciNet  Google Scholar 

  19. L. J. Marangunić, J. Pečarić: On Landau type inequalities for functions with Hölder continuous derivatives. JIPAM, J. Inequal. Pure Appl. Math. (electronic only) 5 (2004), Paper No. 72, 5 pages.

  20. H. Meftahi, F. Wielonsky: Growth estimates in the Hardy-Sobolev space of an annular domain with applications. J. Math. Anal. Appl. 358 (2009), 98–109.

    Article  MATH  MathSciNet  Google Scholar 

  21. D. S. Mitrinović, J. E. Pečarić, A. M. Fink: Inequalities Involving Functions and Their Integrals and Derivatives. Mathematics and Its Applications: East European Series 53, Kluwer Academic Publishers, Dordrecht, 1991.

    Book  MATH  Google Scholar 

  22. C. P. Niculescu, C. Buşe: The Hardy-Landau-Littlewood inequalities with less smoothness. JIPAM, J. Inequal. Pure Appl. Math. (electronic only) 4 (2003), Paper No. 51, 8 pages.

  23. L. Nirenberg: An extended interpolation inequality. Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 20 (1966), 733–737.

    MATH  MathSciNet  Google Scholar 

  24. W. Rudin: Analytic functions of class H p. Trans. Am. Math. Soc. 78 (1955), 46–66.

    MATH  MathSciNet  Google Scholar 

  25. D. Sarason: The H p Spaces of An Annulus. Memoirs of the American Mathematical Society 56, AMS, Providence, 1965.

    Google Scholar 

  26. H.-C. Wang: Real Hardy spaces of an annulus. Bull. Aust. Math. Soc. 27 (1983), 91–105.

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Sciences, Sfax University, B.P 1171, Sfax, 3018, Tunisia

    Slim Chaabane & Imed Feki

Authors
  1. Slim Chaabane
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  2. Imed Feki
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Correspondence to Slim Chaabane.

Additional information

This research has been supported by the Laboratory of Applied Mathematics and Harmonic Analysis: L.A.M.H.A. LR 11ES52

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Chaabane, S., Feki, I. Pointwise inequalities of logarithmic type in Hardy-Hölder spaces. Czech Math J 64, 351–363 (2014). https://doi.org/10.1007/s10587-014-0106-9

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  • DOI: https://doi.org/10.1007/s10587-014-0106-9

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