Hardy Spaces Associated with Monge–Ampère Equation

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Abstract

The main concern of this paper is to study the boundedness of singular integrals related to the Monge–Ampère equation established by Caffarelli and Gutiérrez. They obtained the \(L^2\) boundedness. Since then the \(L^p, 1<p<\infty \), weak (1,1) and the boundedness for these operators on atomic Hardy space were obtained by several authors. It was well known that the geometric conditions on measures play a crucial role in the theory of the Hardy space. In this paper, we establish the Hardy space \(H^p_{\mathcal F}\) via the Littlewood–Paley theory with the Monge–Ampère measure satisfying the doubling property together with the noncollapsing condition, and show the \(H^p_{\mathcal F}\) boundedness of Monge–Ampère singular integrals. The approach is based on the \(L^2\) theory and the main tool is the discrete Calderón reproducing formula associated with the doubling property only.

Keywords

Doubling property Hardy spaces Monge–Ampère equation Singular integral operators 

Mathematics Subject Classification

42B20 42B35 

Notes

Acknowledgements

Ming-Yi Lee and Chin-Cheng Lin are supported by the Ministry of Science and Technology, R.O.C. under Grant Nos. #MOST 106-2115-M-008-003-MY2 and #MOST 106-2115-M-008-004-MY3, respectively, as well as supported by the National Center for Theoretical Sciences of Taiwan.

References

  1. 1.
    Aimar, H., Forzani, L., Toledano, R.: Balls and quasi-metrics: a space of homogeneous type modeling the real analysis related to the Monge–Ampère equation. J. Fourier Anal. Appl. 4, 377–381 (1998)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Caffarelli, L.A., Gutiérrez, C.E.: Real analysis related to the Monge–Ampère equation. Trans. Am. Math. Soc. 348, 1075–1092 (1996)CrossRefMATHGoogle Scholar
  3. 3.
    Caffarelli, L.A., Gutiérrez, C.E.: Singular integrals related to the Monge–Ampère equation. In: D’Atellis, C.A., Fernandez-Berdaguer, E.M. (eds.) Wavelet Theory and Harmonic Analysis in Applied Sciences (Buenos Aires, 1995), pp. 3–13. Birkhäuser, Boston (1997)CrossRefGoogle Scholar
  4. 4.
    Christ, M.: A \(Tb\) theorem with remarks on analytic capacity and the Cauchy integral. Colloq. Math. 60(61), 601–628 (1990)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Coifman, R.R., Weiss, G.: Analyse Harmonique Non-Commutative sur Certains Espaces Homogenes. Lecture Notes in Math. Springer, Berlin/New York (1971)CrossRefMATHGoogle Scholar
  6. 6.
    Coifman, R.R., Weiss, G.: Extensions of Hardy spaces and their use in analysis. Bull. Am. Math. Soc. 83, 569–645 (1977)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    David, G., Journé, J.-L., Semmes, S.: Calderón–Zygmund operators, para-accretive functions and interpolation. Rev. Mat. Iberoam. 1, 1–56 (1985)CrossRefMATHGoogle Scholar
  8. 8.
    Deng, D., Han, Y.: Harmonic Analysis on Spaces of Homogeneous Type (with a Preface by Yves Meyer). Lecture Notes in Math. Springer, Berlin (2009)MATHGoogle Scholar
  9. 9.
    Ding, Y., Lin, C.-C.: Hardy spaces associated to the sections. Tôhoku Math. J. 57, 147–170 (2005)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Frazier, M., Jawerth, B.: A discrete transform and decompositions of distribution spaces. J. Funct. Anal. 93, 34–170 (1990)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Han, Y., Lee, M.-Y., Lin, C.-C.: Boundedness of Monge–Ampère singular integral operators on Besov spaces (2017). arXiv:1709.03278
  12. 12.
    Han, Y., Müller, D., Yang, D.: Littlewood–Paley–Stein characterizations for Hardy spaces on spaces of homogeneous type. Math. Nachr. 279, 1505–1537 (2006)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Han, Y., Müller, D., Yang, D.: A theory of Besov and Triebel–Lizorkin spaces on metric measure spaces modeled on Carnot–Caratheodory spaces. Abstr. Appl. Anal. (2008).  https://doi.org/10.1155/2008/893409
  14. 14.
    Han, Y., Sawyer, E.T.: Littlewood–Paley theory on spaces of homogeneous type and the classical function spaces. Mem. Am. Math. Soc. 110, 530 (1994)MathSciNetMATHGoogle Scholar
  15. 15.
    Incognito, A.: Weak-type \((1,1)\) inequality for the Monge–Ampère SIO’s. J. Fourier Anal. Appl. 7, 41–48 (2001)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Kerkyacharian, G., Petrushev, P.: Heat kernel based decomposition of spaces of distributions in the framework of Dirichlet spaces. Trans. Am. Math. Soc. 367, 121–189 (2015)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Lee, M.-Y.: The boundedness of Monge–Ampère singular integral operators. J. Fourier Anal. Appl. 18, 211–222 (2012)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Lin, C.-C.: Boundedness of Monge–Ampère singular integral operators acting on Hardy spaces and their duals. Trans. Am. Math. Soc. 368, 3075–3104 (2016)CrossRefMATHGoogle Scholar
  19. 19.
    Macías, R.A., Segovia, C.: Lipschitz functions on spaces of homogeneous type. Adv. Math. 33, 257–270 (1979)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Nagel, A., Stein, E.M.: On the product theory of singular integrals. Rev. Mat. Iberoam. 20, 531–561 (2004)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Sawyer, E., Wheeden, R.: Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces. Am. J. Math. 114, 813–874 (1992)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2017

Authors and Affiliations

  1. 1.Department of MathematicsAuburn UniversityAuburnUSA
  2. 2.Department of MathematicsNational Central University320Taiwan, ROC

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