Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

SOME REMARKS ON VECTOR-VALUED TREE MARTINGALES

  • He, Tong-Jun (College of Mathematics and Computer Science Fuzhou University)
  • Received : 2011.02.12
  • Published : 2012.03.01
Download PDF
⟨ Previous Next ⟩

Abstract

Our first aim of this paper is to define maximal operators a-quadratic variation and of a-conditional quadratic variation for vectorvalued tree martingales and to show that these maximal operators and maximal operators of vector-valued tree martingale transforms are all sublinear operators. The second purpose is to prove that maximal operator inequalities of a-quadratic variation and of a-conditional quadratic variation for vector-valued tree martingales hold provided 2 ${\leq}$ a < $\infty$ by means of Marcinkiewicz interpolation theorem. Based on a result of reference [10] and using Marcinkiewicz interpolation theorem, we also propose a simple proof of maximal operator inequalities for vector-valued tree martingale transforms, under which the vector-valued space is a UMD space.

Keywords

References

  1. T. Ando, Contractive projections in $L^{p}$ spaces, Pacic J. Math. 17 (1966), 391-405. https://doi.org/10.2140/pjm.1966.17.391
  2. J. Bergh and J. Lofstrom, Interpolation Spaces: An Introduction, Vol. 223, Springer-Verlag, Berlin, 1976.
  3. D. L. Burkholder and R. F. Gundy, Extrapolation and interpolation of Quasi-linear operators on martingales, Acta Math. 124 (1970), no. 1, 249-304. https://doi.org/10.1007/BF02394573
  4. R. Cairoli and J. B. Walsh, Stochastic integrals in the plane, Acta Math. 134 (1975), 111-183. https://doi.org/10.1007/BF02392100
  5. L. E. Dor and E. Odell, Monotone bases in $L^{p}$, Pacic J. Math. 60 (1975), no. 2, 51-61. https://doi.org/10.2140/pjm.1975.60.51
  6. S. Fridli and F. Schipp, Tree-martingales, The 5th Pannonian Symposium on Mathematical Statistics, 1985, Grossmann, W. and Pug, G. Ch. and Vincze I. and And Wertz W., 53-63, Visegrad, Hungary, Akademiai Kiado.
  7. G. Gat, On (C, 1) summability for Vilenkin-like systems, Studia Math. 144 (2001), no. 2, 101-120. https://doi.org/10.4064/sm144-2-1
  8. J. Gosselin, Almost everywhere convergence of Vilenkin-Fourier series, Trans. Amer. Math. Soc. 185 (1973), 345-370. https://doi.org/10.1090/S0002-9947-1973-0352883-X
  9. T.-J. He and Y. L Hou, Some inequalities for tree martingales, Acta Math. Appl. Sin. Engl. Ser. 21 (2005), no. 4, 671-682. https://doi.org/10.1007/s10255-005-0274-3
  10. T.-J. He and Y. Shen, Maximal operators of tree martingale transforms and their maximal operator inequalities, Trans. Amer. Math. Soc. 12 (2008), no. 12, 6595-6609.
  11. T.-J. He and Y. Shen, Decomposition and convergence for tree martingales, Stochastic Process. Appl. 119 (2009), no. 8, 2625-2644. https://doi.org/10.1016/j.spa.2009.01.005
  12. T.-J. He, Y. X. Xiao, and Y. L. Hou, Inequalities for vector-valued tree martingales, International Conference on Functional Space Theory and Its Applications, 2004, Liu, P. D., 67-75, Wuhan, Research Information Ltd UK.
  13. D. Khoshnevisan, Multiparameter Processes, Springer-Verlag, New York, 2002.
  14. G. Pisier, Martingales with values in uniformly convex spaces, Israel J. Math. 20 (1975), no. 3-4, 326-350. https://doi.org/10.1007/BF02760337
  15. F. Schipp, Universal contractive projections and a.e. convergence, Probability theory and applications, 221-233, Math. Appl., 80, Kluwer Acad. Publ., Dordrecht, 1992.
  16. F. Schipp and F. Weisz, Tree martingales and a.e. convergence of Vilenkin-Fourier series, Math. Pannon. 8 (1997), no. 1, 17-35.
  17. F. Weisz, Martingale Hardy spaces and their applications in Fourier analysis, Lecture Notes in Mathematics, 1568. Springer-Verlag, Berlin, 1994.
  18. F. Weisz, Almost everywhere convergence of Banach space-valued Vilenkin-Fourier series, Acta Math. Hungar. 116 (2007), no. 1-2, 47-59. https://doi.org/10.1007/s10474-007-5289-1
  19. W.-S. Young, Mean convergence of generalized Walsh-Fourier series, Trans. Amer. Math. Soc. 218 (1976), 311-320. https://doi.org/10.1090/S0002-9947-1976-0394022-8