Illinois Journal of Mathematics

Noncommutative extrapolation theorems and applications

Ying Hu

Full-text: Open access

Abstract

In this paper, we prove some noncommutative analogues of Yano’s classical extrapolation theorem. Applying one of them to noncommutative martingales, we obtain a maximal inequality for noncommutative martingales from $L\log^2L$ to $L_1$. Moreover, the exponent $2$ is optimal. We also obtain the noncommutative analogue of the classical theorem of Burkholder and Chow on the iterations of two conditional expectations.

Article information

Source
Illinois J. Math., Volume 53, Number 2 (2009), 463-482.

Dates
First available in Project Euclid: 23 February 2010

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1266934788

Digital Object Identifier
doi:10.1215/ijm/1266934788

Mathematical Reviews number (MathSciNet)
MR2594639

Zentralblatt MATH identifier
1187.46053

Subjects
Primary: 46L53: Noncommutative probability and statistics 65L06: Multistep, Runge-Kutta and extrapolation methods
Secondary: 46L51: Noncommutative measure and integration 46L52: Noncommutative function spaces 60G42: Martingales with discrete parameter

Citation

Hu, Ying. Noncommutative extrapolation theorems and applications. Illinois J. Math. 53 (2009), no. 2, 463--482. doi:10.1215/ijm/1266934788. https://projecteuclid.org/euclid.ijm/1266934788


Export citation

References

  • C. A. Akemann, J. Anderson and G. K. Pedersen, Triangle inequalities in operator algebras, Linear Multilinear Algebra 11 (1982), 167–178.
  • C. Anantharaman-Delaroche, On ergodic theorems for free group actions on noncommutative spaces, Probab. Theory Related Fields 135 (2006), 520–546.
  • D. L. Burkholder and Y. S. Chow, Iterates of conditional expectation operators, Proc. Amer. Math. Soc. 12 (1961), 490–495.
  • A. I. Bufetov, Convergence of spherical averages for actions of free groups, Ann. of Math. (2) 155 (2002), 929–944.
  • M. J. Carro, New extrapolation estimates, J. Funct. Anal. 174 (2000), 155–166.
  • U. Haagerup, $L\sp{p}$-spaces associated with an arbitrary von Neumann algebra, Algèbres d'opérateurs et leurs applications en physique mathématique (Proc. Colloq., Marseille, 1977), Colloq. Internat. CNRS, vol. 274, CNRS, Paris, 1979, pp. 175–184.
  • Y. Hu, Maximal ergodic theorems for some group actions, J. Funct. Anal. 254 (2008), 1282–1306.
  • R. Jajte, Strong limit theorems in noncommutative $L\sb 2$-spaces, Lecture Notes in Mathematics, vol. 1477, Springer-Verlag, Berlin, 1991.
  • M. Junge, C. Le Merdy and Q. Xu, $H\sp\infty$ functional calculus and square functions on noncommutative $L\sp p$-spaces, Astérisque (2006), vi${}+{}$138.
  • B. Jawerth and M. Milman, Extrapolation theory with applications, Mem. Amer. Math. Soc. 89 (1991), iv${}+{}$82.
  • M. Junge, Doob's inequality for non-commutative martingales, J. Reine Angew. Math. 549 (2002), 149–190.
  • M. Junge and Q. Xu, Noncommutative Burkholder/Rosenthal inequalities, Ann. Probab. 31 (2003), 948–995.
  • M. Junge and Q. Xu, On the best constants in some non-commutative martingale inequalities, Bull. London Math. Soc. 37 (2005), 243–253.
  • M. Junge and Q. Xu, Noncommutative maximal ergodic theorems, J. Amer. Math. Soc. 20 (2007), 385–439.
  • S. Kwapień and A. Pełczyński, The main triangle projection in matrix spaces and its applications, Studia Math. 34 (1970), 43–68.
  • A. Nevo and E. M. Stein, A generalization of Birkhoff's pointwise ergodic theorem, Acta Math. 173 (1994), 135–154.
  • G. Pisier and Q. Xu, Non-commutative martingale inequalities, Comm. Math. Phys. 189 (1997), 667–698.
  • N. Randrianantoanina, Non-commutative martingale transforms, J. Funct. Anal. 194 (2002), 181–212.
  • T. Tao, A converse extrapolation theorem for translation-invariant operators, J. Funct. Anal. 180 (2001), 1–10.
  • S. Yano, Notes on Fourier analysis. XXIX. An extrapolation theorem, J. Math. Soc. Japan 3 (1951), 296–305.