Acta Mathematica

, Volume 124, Issue 1, pp 249–304 | Cite as

Extrapolation and interpolation of quasi-linear operators on martingales

  • D. L. Burkholder
  • R. F. Gundy
Article

Keywords

Positive Integer Positive Real Number Maximal Function Difference Sequence Integral Inequality 

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References

  1. [1].
    Burkholder, D. L., Martingale transforms.Ann. Math. Statist., 37 (1966), 1494–1504.MATHMathSciNetGoogle Scholar
  2. [2].
    Calderón, A. P. &Zygmund, A., A note on the interpolation of sublinear operations.Amer. J. Math., 78 (1956), 282–288.MATHMathSciNetCrossRefGoogle Scholar
  3. [3].
    Chow, Y. S., Convergence of sums of squares of martingale differences.Ann. Math. Statist., 39 (1968), 123–133.MATHMathSciNetGoogle Scholar
  4. [4].
    Davis, B., Divergence properties of some martingale transforms.Ann. Math. Statist., 40 (1969), 1852–1854.MATHGoogle Scholar
  5. [5].
    Doob, J. L.,Stochastic processes. New York, 1953.Google Scholar
  6. [6].
    Dvoretzky, A., Personal communication.Google Scholar
  7. [7].
    Gundy, R. F., The martingale version of a theorem of Marcinkiewicz and Zygmund.Ann. Math. Statist., 38 (1967), 725–734.MATHMathSciNetGoogle Scholar
  8. [8].
    —, A decomposition forL 1-bounded martingales.Ann. Math. Statist., 39 (1968), 134–138.MATHMathSciNetGoogle Scholar
  9. [9].
    —, On the classL logL, martingales, and singular integrals.Studia Math., 33 (1969), 109–118.MATHMathSciNetGoogle Scholar
  10. [10].
    Hardy, G. H., andLittlewood, J. E., A maximal theorem with function-theoretic applications.Acta Math., 54 (1930), 81–116.MATHGoogle Scholar
  11. [11].
    Marcinkiewicz, J., Quelques théorèmes sur les séries orthogonales.Ann. Soc. Polon. Math., 16 (1937), 84–96 (pages 307–318 of theCollected Papers.)Google Scholar
  12. [12].
    Marcinkiewicz, J. &Zygmund, A., Quelques théorèmes sur les fonctions indépendantes.Studia Math., 7 (1938), 104–120 (pages 374–388 of theCollected Papers of Marcinkiewicz).MATHGoogle Scholar
  13. [13].
    Millar, P. W., Martingale integrals.Trans. Amer. Math. Soc., 133 (1968), 145–166.MATHMathSciNetCrossRefGoogle Scholar
  14. [14].
    Paley, R. E. A. C., A remarkable series of orthogonal functions.Proc. London Math. Soc., 34 (1932), 241–279.MATHGoogle Scholar
  15. [15].
    Stein, E. M.,Topics in harmonic analysis related to the Littlewood-Paley theory. Ann. Math. Studies 63. Princeton, 1970.Google Scholar
  16. [16].
    Tsuchikura, T., Sample properties of martingales and their arithmetic means.Tôhoku Math. J., 20 (1968), 400–415.MATHMathSciNetGoogle Scholar
  17. [17].
    Zygmund, A.,Trigonometric series, 2nd edition. Cambridge, 1959.Google Scholar

Copyright information

© Almqvist & Wiksells Boktryckeri AB 1970

Authors and Affiliations

  • D. L. Burkholder
    • 1
    • 2
    • 3
  • R. F. Gundy
    • 1
    • 2
    • 3
  1. 1.University of IllinoisUrbana
  2. 2.Rutgers UniversityNew Brunswick
  3. 3.Hebrew University of JerusalemJerusalemIsareal

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