Oscillation and variation for singular integrals in higher dimensions

Campbell, James; Jones, Roger; Reinhold, Karin; Wierdl, Máté

doi:10.1090/S0002-9947-02-03189-6

Oscillation and variation for singular integrals in higher dimensions
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by James T. Campbell, Roger L. Jones, Karin Reinhold and Máté Wierdl

Trans. Amer. Math. Soc. 355 (2003), 2115-2137

DOI: https://doi.org/10.1090/S0002-9947-02-03189-6

Published electronically: November 14, 2002

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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

Mustafa A. Akcoglu, Roger L. Jones, and Peter O. Schwartz, Variation in probability, ergodic theory and analysis, Illinois J. Math. 42 (1998), no. 1, 154–177. MR 1492045
P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
José Felipe Voloch, Jacobians of curves over finite fields, Rocky Mountain J. Math. 30 (2000), no. 2, 755–759. MR 1787011, DOI 10.1216/rmjm/1022009294
Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
Michael Christ and José Luis Rubio de Francia, Weak type $(1,1)$ bounds for rough operators. II, Invent. Math. 93 (1988), no. 1, 225–237. MR 943929, DOI 10.1007/BF01393693
Robert A. Fefferman, A theory of entropy in Fourier analysis, Adv. in Math. 30 (1978), no. 3, 171–201. MR 520232, DOI 10.1016/0001-8708(78)90036-1
Roger L. Jones, Iosif V. Ostrovskii, and Joseph M. Rosenblatt, Square functions in ergodic theory, Ergodic Theory Dynam. Systems 16 (1996), no. 2, 267–305. MR 1389625, DOI 10.1017/S0143385700008816
Roger L. Jones, Robert Kaufman, Joseph M. Rosenblatt, and Máté Wierdl, Oscillation in ergodic theory, Ergodic Theory Dynam. Systems 18 (1998), no. 4, 889–935. MR 1645330, DOI 10.1017/S0143385798108349
R. Jones, J. Rosenblatt, and M. Wierdl, Oscillation in ergodic theory: higher dimensional results, preprint, to appear in Israel J. of Math.
A. W. Knapp and E. M. Stein, Intertwining operators for semisimple groups, Ann. of Math. (2) 93 (1971), 489–578. MR 460543, DOI 10.2307/1970887
Karl Petersen, Another proof of the existence of the ergodic Hilbert transform, Proc. Amer. Math. Soc. 88 (1983), no. 1, 39–43. MR 691275, DOI 10.1090/S0002-9939-1983-0691275-2
Jinghua Qian, The $p$-variation of partial sum processes and the empirical process, Ann. Probab. 26 (1998), no. 3, 1370–1383. MR 1640349, DOI 10.1214/aop/1022855756
M. Riesz, Sur les fonctions conjugées, Math. Zeit. 27 (1927) , 218-244.
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
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 1056560, DOI 10.1090/S0273-0979-1990-15973-7
Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR 0304972
Stephen Wainger, Discrete analogues of singular and maximal Radon transforms, Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), 1998, pp. 743–753. MR 1648122