A construction of ergodic BMO functions

Petersen, Karl

doi:10.1090/S0002-9939-1980-0572299-X

A construction of ergodic BMO functions
HTML articles powered by AMS MathViewer

by Karl Petersen PDF

Proc. Amer. Math. Soc. 79 (1980), 549-555 Request permission

Abstract:

One can explicitly construct unbounded functions having bounded mean oscillation (and in fact vanishing mean oscillation) with respect to any given ergodic measure-preserving transformation of a Lebesgue space.

References

D. V. Anosov, The additive functional homology equation that is connected with an ergodic rotation of the circles, Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), 1259–1274 (Russian). MR 0352400
R. R. Coifman and R. Rochberg, Another characterization of BMO, Proc. Amer. Math. Soc. 79 (1980), no. 2, 249–254. MR 565349, DOI 10.1090/S0002-9939-1980-0565349-8
Ronald R. Coifman and Guido Weiss, Maximal functions and $H^{P}$ spaces defined by ergodic transformations, Proc. Nat. Acad. Sci. U.S.A. 70 (1973), 1761–1763. MR 344417, DOI 10.1073/pnas.70.6.1761
Jean-Pierre Conze, Équations fonctionnelles et systèmes induits en théorie ergodique, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 23 (1972), 75–82 (French). MR 320272, DOI 10.1007/BF00536692
Mischa Cotlar, A unified theory of Hilbert transforms and ergodic theorems, Rev. Mat. Cuyana 1 (1955), 105–167 (1956) (English, with Spanish summary). MR 84632
Manfred Denker, Christian Grillenberger, and Karl Sigmund, Ergodic theory on compact spaces, Lecture Notes in Mathematics, Vol. 527, Springer-Verlag, Berlin-New York, 1976. MR 0457675
C. Fefferman and E. M. Stein, $H^{p}$ spaces of several variables, Acta Math. 129 (1972), no. 3-4, 137–193. MR 447953, DOI 10.1007/BF02392215
Robert I. Jewett, The prevalence of uniquely ergodic systems, J. Math. Mech. 19 (1969/1970), 717–729. MR 0252604
F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415–426. MR 131498, DOI 10.1002/cpa.3160140317
Shizuo Kakutani, Induced measure preserving transformations, Proc. Imp. Acad. Tokyo 19 (1943), 635–641. MR 14222
Wolfgang Krieger, On unique ergodicity, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971) Univ. California Press, Berkeley, Calif., 1972, pp. 327–346. MR 0393402
John C. Oxtoby, Ergodic sets, Bull. Amer. Math. Soc. 58 (1952), 116–136. MR 47262, DOI 10.1090/S0002-9904-1952-09580-X
Karl Endel Petersen, Brownian motion, Hardy spaces and bounded mean oscillation, London Mathematical Society Lecture Note Series, No. 28, Cambridge University Press, Cambridge-New York-Melbourne, 1977. MR 0651556
V. Rohlin, A “general” measure-preserving transformation is not mixing, Doklady Akad. Nauk SSSR (N.S.) 60 (1948), 349–351 (Russian). MR 0024503
Donald Sarason, Functions of vanishing mean oscillation, Trans. Amer. Math. Soc. 207 (1975), 391–405. MR 377518, DOI 10.1090/S0002-9947-1975-0377518-3