Illinois Journal of Mathematics

Sharp LlogL inequalities for differentially subordinated martingales and harmonic functions

Adam Oşekowski

Full-text: Access by subscription

Abstract

Let $(x_n)$, $(y_n)$ be two martingales adapted to the same filtration $(\mathcal{F}_n)$ satisfying, with probability $1$, \[ |dx_n|\leq|dy_n|,\quad n=0, 1, 2, \ldots . \] For every $K>0,$ we determine the best constant $L=L(K)$ for which the inequality \[ \mathbb{E} |x_n| \leq K\mathbb{E} |y_n|\log|y_n|+L,\quad n=0, 1, 2, \ldots \] holds true. We also prove a similar inequality for harmonic functions.

Article information

Source
Illinois J. Math., Volume 52, Number 3 (2008), 745-756.

Dates
First available in Project Euclid: 1 October 2009

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1254403712

Digital Object Identifier
doi:10.1215/ijm/1254403712

Mathematical Reviews number (MathSciNet)
MR2546005

Subjects
Primary: 60G42: Martingales with discrete parameter 31B05: Harmonic, subharmonic, superharmonic functions

Citation

Oşekowski, Adam. Sharp LlogL inequalities for differentially subordinated martingales and harmonic functions. Illinois J. Math. 52 (2008), no. 3, 745--756. doi:10.1215/ijm/1254403712. https://projecteuclid.org/euclid.ijm/1254403712


Export citation

References

  • Burkholder, D. L., A geometrical characterization of Banach spaces in which martingale difference sequences are unconditional, Ann. Probab. 9 (1981), 997–1011.
  • Burkholder, D. L., Boundary value problems and sharp inequalities for martingale transforms, Ann. Probab. 12 (1984), 647–702.
  • Burkholder, D. L., Differential subordination of harmonic functions and martingales, Harmonic analysis and partial differential equations (El Escorial, 1987), Lecture Notes in Math., vol. 1384, Springer, Berlin, 1989, pp. 1–23.
  • Burkholder, D. L., Explorations in martingale theory and its applications, École d'Été de Probabilités de Saint-Flour XIX–-1989, Lecture Notes in Math., vol. 1464, Springer, Berlin, 1991, pp. 1–66.
  • Osękowski, A., Inequalities for dominated martingales, Bernoulli 13 (2007), 54–79.
  • Suh, Y., A sharp weak type $(p,p)$ inequality $(p>2)$ for martingale transforms and other subordinate martingales, Trans. Amer. Math. Soc. 357 (2005), 1545–1564 (electronic).
  • Wang, G., Differential subordination and strong differential subordination for continuous-time martingales and related sharp inequalities, Ann. Probab. 23 (1995), 522–551.