Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Variation inequalities for the Fejér and Poisson kernels


Authors: Roger L. Jones and Gang Wang
Translated by:
Journal: Trans. Amer. Math. Soc. 356 (2004), 4493-4518
MSC (2000): Primary 42A24; Secondary 26D05
DOI: https://doi.org/10.1090/S0002-9947-04-03397-5
Published electronically: January 13, 2004
MathSciNet review: 2067131
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we show that the $\varrho$-th order variation operator, for both the Fejér and Poisson kernels, are bounded from $L^p$ to $L^p$, $1<p<\infty$, when $\varrho >2$. Counterexamples are given if $\varrho =2$.


References [Enhancements On Off] (What's this?)

  • 1. Akcoglu, M., Jones, R. and Schwartz, P., Variation in probability, ergodic thoery and analysis, Illinois J. of Math., 42 (1998) 154-177. MR 99a:60048
  • 2. Jones, R., Kaufman, R., Rosenblatt, J., and Wierdl, M., Oscillation in ergodic theory, Ergodic Theory and Dynam. Sys. 18 (1998) 889-935. MR 2000b:28019
  • 3. Jones, R.L. and Rosenblatt, J., Differential and ergodic transforms, Mathematische Annalen, 323 (2002) 525-546. MR 2003g:37003
  • 4. Kaczmarz, S., Über die Konvergenz der Reihen von Orthogonal-funktionen, Math. Z., 23 (1925) 263-270.
  • 5. D. Lepingle, La variation d'order p des semi-martingales, Z. Wahrscheinlichkeitstheorie verw. Gebiete 36 (1976) 295-316. MR 54:8849
  • 6. J. Qian, The $p$ variation of partial sum processes and the emperical process, Ann. of Prob., 26 (1998) 1370-1383. MR 99i:60052
  • 7. E. M. Stein, The development of square funtions in the work of A. Zygmund Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 2, 359-376. MR 83i:42001
  • 8. E. M. Stein, Singular Integrals and Differentiablity Properties of Functions Princeton University Press, Princeton, N.J., 1970. MR 44:7280
  • 9. Zygmund, A., Une remarque sur un théorème de M. Kaczmarz Math. Z., 25 (1926) 297-298.
  • 10. A. Zygmund, Trigonometric series, Vol. 2, second edition, Cambridge University Press, New York, 1977. MR 58:29731

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 42A24, 26D05

Retrieve articles in all journals with MSC (2000): 42A24, 26D05


Additional Information

Roger L. Jones
Affiliation: Department of Mathematics, DePaul University, 2320 N. Kenmore, Chicago, Illinois 60614
Email: rjones@condor.depaul.edu

Gang Wang
Affiliation: Department of Mathematics, DePaul University, 2320 N. Kenmore, Chicago, Illinois 60614
Email: gwang@condor.depaul.edu

DOI: https://doi.org/10.1090/S0002-9947-04-03397-5
Keywords: Fej\'er kernel, Poisson kernel, square functions, variation, jump inequalities
Received by editor(s): August 17, 2001
Received by editor(s) in revised form: May 13, 2003
Published electronically: January 13, 2004
Additional Notes: The first aurthor was partially supported by a grant from the DePaul University Liberal Art and Science research program
The second author was partially supported by NSF grant DMS-0071759
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society