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A local Hilbert transform, Hardy’s inequality and molecular characterization of Goldberg’s local Hardy space

  • Galia DafniEmail author
  • Elijah Liflyand
RESEARCH
  • 25 Downloads
Part of the following topical collections:
  1. 2017 Northeast Analysis Network

Abstract

We prove characterizations of Goldberg’s local Hardy space \(h^1({\mathbb {R}})\) by means of a local Hilbert transform and a molecular decomposition. We use this decomposition to prove a version of Hardy’s inequality for the Fourier transform of functions in this space.

Keywords

Hardy space Hardy’s inequality Atomic decomposition Molecules 

Mathematics Subject Classification

42B35 42B30 

Notes

References

  1. 1.
    Chao, J. A., Gilbert, J. E., Tomas, P. A.: Molecular decompositions in \(H^p\)-theory. In: Proceedings of the Seminar on Harmonic Analysis (Pisa, 1980). Rend. Circ. Mat. Palermo 2, 115–119 (1981)Google Scholar
  2. 2.
    Dafni, G.: Hardy Spaces on Strongly Pseudoconvex Domains in \({\mathbb{C}}^{n}\) and Domains of Finite Type in \({\mathbb{C}}^{2}\), Dissertation. Princeton University, Princeton (1993)Google Scholar
  3. 3.
    Dafni, G., Yue, H.: Some characterizations of local bmo and \(h^1\) on metric measure spaces. Anal. Math. Phys. 2, 285–318 (2012)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Garcia-Cuerva, J., de Rubio Francia, J.L.: Weighted Norm Inequalities and Related Topics. Elsevier, Amsterdam (1985)zbMATHGoogle Scholar
  5. 5.
    Goldberg, D.: Local Hardy spaces. In: Harmonic Analysis in Euclidean Spaces (Proceedings of the Symposium on Pure Mathathematics, Williams College, Williamstown, Mass, 1978), Part 1. Proceedings of the Symposium on Pure Mathathematics, vol. XXXV, pp. 245–248. American Mathematical Society, Providence (1979)Google Scholar
  6. 6.
    Goldberg, D.: A local version of real Hardy spaces. Duke Math. J. 46, 27–42 (1979)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Kober, H.: A note on Hilbert’s operator. Bull. Am. Math. Soc. 48, 421–426 (1942)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Krantz, S.G.: Geometric Analysis and Function Spaces, CBMS Regional Conference Series in Mathematics, vol. 81. American Mathematical Society, Providence (1993)CrossRefGoogle Scholar
  9. 9.
    Meyer, Y.: Wavelets and Operators. Cambridge Studies in Advanced Mathematics, vol. 37. Cambridge University Press, Cambridge (1992)Google Scholar
  10. 10.
    Stein, E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton (1993)zbMATHGoogle Scholar
  11. 11.
    Taibleson, M. H., Weiss, G.: The molecular characterization of Hardy spaces. In: Harmonic Analysis in Euclidean Spaces (Proceedings of the Symposium on Pure Mathathematics, Williams College, Williamstown, Mass., 1978), Part 1. Proceedings of the Symposium on Pure Mathathematics, vol. XXXV, pp. 281–287. American Mathemathical Society, Providence (1979)Google Scholar
  12. 12.
    Taibleson, M. H., Weiss, G.: The molecular characterization of certain Hardy spaces. Astérisque, vol. 77, pp. 67–149. Society of Mathemathics, Paris (1980)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsConcordia UniversityMontréalCanada
  2. 2.Department of MathematicsBar-Ilan UniversityRamat-GanIsrael
  3. 3.S.M. Nikol’skii Institute of MathematicsRUDN UniversityMoscowRussia

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