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)

 
 

 

Oscillation and variation for singular integrals in higher dimensions


Authors: James T. Campbell, Roger L. Jones, Karin Reinhold and Máté Wierdl
Journal: Trans. Amer. Math. Soc. 355 (2003), 2115-2137
MSC (2000): Primary 42B25; Secondary 40A30
DOI: https://doi.org/10.1090/S0002-9947-02-03189-6
Published electronically: November 14, 2002
MathSciNet review: 1953540
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we continue our investigations of square function inequalities in harmonic analysis. Here we investigate oscillation and variation inequalities for singular integral operators in dimensions $d \geq 1$. Our estimates give quantitative information on the speed of convergence of truncations of a singular integral operator, including upcrossing and $\lambda$ jump inequalities.


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

  • 1. M. Akcoglu, R. L. Jones and P. Schwartz, Variation in probability, ergodic theory and analysis, Illinois J. Math. 42 (1998) 154-177. MR 99a:60048
  • 2. A. P. Calderón and A. Zygmund, On the existence of certain singular integrals, Acta. Math. (1952) 88, 85-139. MR 14:637f
  • 3. A. Calderón and A. Zygmund, On singular integrals Amer. J. Math, 78 (1956), 289-309. MR 18:894a
  • 4. J. Campbell, R. Jones, K. Reinhold, and M. Wierdl, Oscillation and Variation for the Hilbert Transform, Duke Math. J. 105 (2000), 59-83. MR 2001h:14021
  • 5. M. Cotlar, A unified theory of Hilbert transforms and ergodic theorems, Rev. Math. Cuyana, 1 (1955), 105 - 167. MR 18:893d
  • 6. M. Christ and J. L. Rubio de Francia Weak-type (1,1) bounds for rough operators II, Invent. Math. 93 (1988), no. 1, 225-237. MR 90d:42021
  • 7. R. Fefferman, A theory of entropy in Fourier analysis, Adv. in Math., 30(1978) 171-201. MR 81g:42022
  • 8. R. Jones, I. Ostrovskii, and J. Rosenblatt, Square functions in ergodic theory, Ergodic Theory and Dyn. Sys. 16 (1996), 267-305. MR 97f:28044
  • 9. R. Jones, R. Kaufmann, J. Rosenblatt and M. Wierdl, Oscillation in ergodic theory, Ergodic Theory and Dyn. Sys. 18 (1998), 889-935. MR 2000b:28019
  • 10. R. Jones, J. Rosenblatt, and M. Wierdl, Oscillation in ergodic theory: higher dimensional results, preprint, to appear in Israel J. of Math.
  • 11. A. W. Knapp and E. M. Stein, Intertwining operators on semi-simple groups, Ann. of Math. 93 (1971), 489 - 578. MR 57:536
  • 12. K. Petersen, Another proof of the existence of the ergodic Hilbert transform, Proc. Amer. Math. Soc. 88, 39-44. MR 84i:28022
  • 13. J. Qian, The p-variation of partial sum processes and the empirical process, Ann. of Prob., (1998), 26, 1370-1383. MR 99i:60052
  • 14. M. Riesz, Sur les fonctions conjugées, Math. Zeit. 27 (1927) , 218-244.
  • 15. E. M. Stein, Singular integrals and differentiability properties of functions, Princeton University Press, Princeton, N. J., 1970. MR 44:7280
  • 16. E. M. Stein and S. Wainger, Discrete analogues of singular Radon transforms, Bull. Amer. Math. Soc. (N.S.) 23 (1990), no. 2, 537-544. MR 92e:42010
  • 17. E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean spaces, Princeton University Press, Princeton, N. J., 1971. MR 46:4102
  • 18. S. Wainger, Discrete analogues of singular and maximal Radon transforms, Doc. Math., Extra Vol. ICM (1998), 743-753. MR 99g:44005

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 42B25, 40A30

Retrieve articles in all journals with MSC (2000): 42B25, 40A30


Additional Information

James T. Campbell
Affiliation: Department of Mathematical Sciences, Dunn Hall 373, University of Memphis, Memphis, Tennessee 38152
Email: jtc@campbeljpc2.msci.memphis.edu

Roger L. Jones
Affiliation: Department of Mathematics, DePaul University, 2219 N. Kenmore, Chicago Illinois 60614
Email: rjones@condor.depaul.edu

Karin Reinhold
Affiliation: Department of Mathematics, University at Albany, SUNY, 1400 Washington Ave., Albany, New York 12222
Email: reinhold@csc.albany.edu

Máté Wierdl
Affiliation: Department of Mathematical Sciences, Dunn Hall 373, University of Memphis, Memphis, Tennessee 38152
Email: mw@moni.msci.memphis.edu

DOI: https://doi.org/10.1090/S0002-9947-02-03189-6
Keywords: Singular integrals, square functions, variation, oscillation, upcrossing inequalities, jump inequalities
Received by editor(s): April 4, 2002
Received by editor(s) in revised form: August 19, 2002
Published electronically: November 14, 2002
Additional Notes: The second author was partially supported by NSF Grant DMS—9302012
The fourth author was partially supported by NSF Grant DMS—9500577
Article copyright: © Copyright 2002 American Mathematical Society

American Mathematical Society