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Fourier analysis methods in operator ergodic theory on super-reflexive Banach spaces
1. | Department of Mathematics, University of Illinois, 1409 W. Green Street, Urbana, IL 61801, United States |
References:
[1] |
D. J. Aldous, Unconditional bases and martingales in $Lp(F)$, Math. Proc. Cambridge Philos. Soc., 85 (1979), 117-123.
doi: 10.1017/S0305004100055559. |
[2] |
B. Beauzamy, "Introduction to Banach Spaces and Their Geometry," North-Holland Math. Studies 68 (Notas de Matemática 86), Elsevier Science, New York, New York, 1982. |
[3] |
E. Berkson, J. Bourgain and T. A. Gillespie, On the almost everywhere convergence of ergodic averages for power-bounded operators on $L^p$-subspaces, Integral Equations and Operator Theory, 14 (1991), 678-715.
doi: 10.1007/BF01200555. |
[4] |
E. Berkson and T. A. Gillespie, AC functions on the circle and spectral families, J. Operator Theory, 13 (1985), 33-47. |
[5] |
E. Berkson and T. A. Gillespie, Fourier series criteria for operator decomposability, Integral Equations and Operator Theory, 9 (1986), 767-789.
doi: 10.1007/BF01202516. |
[6] |
E. Berkson and T. A. Gillespie, Stečkin's theorem, transference, and spectral decompositions, J. Functional Analysis, 70 (1987), 140-170.
doi: 10.1016/0022-1236(87)90128-5. |
[7] |
E. Berkson and T. A. Gillespie, Spectral decompositions and harmonic analysis on UMD spaces, Studia Math., 112 (1994), 13-49. |
[8] |
E. Berkson and T. A. Gillespie, Mean-boundedness and Littlewood-Paley for separation-preserving operators, Trans. Amer. Math. Soc., 349 (1997), 1169-1189.
doi: 10.1090/S0002-9947-97-01896-5. |
[9] |
E. Berkson and T. A. Gillespie, The $q$-variation of functions and spectral integration of Fourier multipliers, Duke Math. J., 88 (1997), 103-132.
doi: 10.1215/S0012-7094-97-08804-9. |
[10] |
E. Berkson and T. A. Gillespie, $Mean_{2}$-bounded operators on Hilbert space and weight sequences of positive operators, Positivity, 3 (1999), 101-133.
doi: 10.1023/A:1009794510984. |
[11] |
E. Berkson and T. A. Gillespie, Spectral decompositions, ergodic averages, and the Hilbert transform, Studia Math., 144 (2001), 39-61.
doi: 10.4064/sm144-1-2. |
[12] |
E. Berkson and T. A. Gillespie, Shifts as models for spectral decomposability on Hilbert space, J. Operator Theory, 50 (2003), 77-106. |
[13] |
E. Berkson and T. A. Gillespie, Operator means and spectral integration of Fourier multipliers, Houston J. Math., 30 (2004), 767-814. |
[14] |
E. Berkson and T. A. Gillespie, The $q$-variation of functions and spectral integration from dominated ergodic estimates, Journal of Fourier Analysis and Applications, 10 (2004), 149-177.
doi: 10.1007/s00041-004-8009-z. |
[15] |
E. Berkson, T. A. Gillespie and P. S. Muhly, Abstract spectral decompositions guaranteed by the Hilbert transform, Proc. London Math. Soc. (3), 53 (1986), 489-517.
doi: 10.1112/plms/s3-53.3.489. |
[16] |
D. Blagojevic, "Spectral Families and Geometry of Banach Spaces," PhD thesis, University of Edinburgh, 2007. |
[17] |
J. Bourgain, Some remarks on Banach spaces in which martingale difference sequences are unconditional, Arkiv för Mat., 21 (1983), 163-168.
doi: 10.1007/BF02384306. |
[18] |
M. M. Day, Reflexive Banach spaces not isomorphic to uniformly convex spaces, Bull. Amer. Math. Soc., 47 (1941), 313-317. |
[19] |
P. Enflo, Banach spaces which can be given an equivalent uniformly convex norm, Israel J. Math., 13 (1972), 281-288. |
[20] |
G. H. Hardy, Weierstrass's non-differentiable function, Trans. Amer. Math. Soc., 17 (1916), 301-325.
doi: 10.2307/1989005. |
[21] |
G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals. I, Math. Zeitschrift, 27 (1928), 565-606.
doi: 10.1007/BF01171116. |
[22] |
G. H. Hardy and J. E. Littlewood, A convergence criterion for Fourier series, Math. Zeitschrift, 28 (1928) 612-634.
doi: 10.1007/BF01181186. |
[23] |
R. C. James, Super-reflexive spaces with bases, Pacific J. Math., 41 (1972), 409-419. |
[24] |
B. Maurey, Système de Haar, Séminaire Maurey-Schwartz, 1974-1975, (École Polytechnique, Paris, 1975), I.1-II.13. |
[25] |
G. Pisier, Un exemple concernant la super-ré flexivité, Séminaire Maurey-Schwartz 1974-1975: Espaces $L^p$ applications radonifiantes et géométrie des espaces de Banach, Annexe No. 2, 12 pp, Centre Math. école Polytech., Paris, 1975. |
[26] |
J. Porter, Helly's selection principle for functions of bounded $p$-variation, Rocky Mountain J. Math., 35 (2005), 675-679.
doi: 10.1216/rmjm/1181069753. |
[27] |
J. L. Rubio de Francia, A Littlewood-Paley inequality for arbitrary intervals, Revista Mat. Iberoamericana, 1 (1985), 1-14. |
[28] |
P. G. Spain, On well-bounded operators of type $B$, Proc. Edinburgh Math. Soc. (2), 18 (1972), 35-48.
doi: 10.1017/S0013091500026134. |
[29] |
S. Treil and A. Volberg, Wavelets and the angle between past and future, Journal of Functional Analysis, 143 (1997), 269-308.
doi: 10.1006/jfan.1996.2986. |
[30] |
L. C. Young, An inequality of the Hölder type, connected with Stieltjes integration, Acta Math., 67 (1936), 251-282.
doi: 10.1007/BF02401743. |
[31] |
A. Zygmund, "Trigonometric Series," 2nd ed., vol. 1, Cambridge Univ. Press, London, 1959. |
show all references
References:
[1] |
D. J. Aldous, Unconditional bases and martingales in $Lp(F)$, Math. Proc. Cambridge Philos. Soc., 85 (1979), 117-123.
doi: 10.1017/S0305004100055559. |
[2] |
B. Beauzamy, "Introduction to Banach Spaces and Their Geometry," North-Holland Math. Studies 68 (Notas de Matemática 86), Elsevier Science, New York, New York, 1982. |
[3] |
E. Berkson, J. Bourgain and T. A. Gillespie, On the almost everywhere convergence of ergodic averages for power-bounded operators on $L^p$-subspaces, Integral Equations and Operator Theory, 14 (1991), 678-715.
doi: 10.1007/BF01200555. |
[4] |
E. Berkson and T. A. Gillespie, AC functions on the circle and spectral families, J. Operator Theory, 13 (1985), 33-47. |
[5] |
E. Berkson and T. A. Gillespie, Fourier series criteria for operator decomposability, Integral Equations and Operator Theory, 9 (1986), 767-789.
doi: 10.1007/BF01202516. |
[6] |
E. Berkson and T. A. Gillespie, Stečkin's theorem, transference, and spectral decompositions, J. Functional Analysis, 70 (1987), 140-170.
doi: 10.1016/0022-1236(87)90128-5. |
[7] |
E. Berkson and T. A. Gillespie, Spectral decompositions and harmonic analysis on UMD spaces, Studia Math., 112 (1994), 13-49. |
[8] |
E. Berkson and T. A. Gillespie, Mean-boundedness and Littlewood-Paley for separation-preserving operators, Trans. Amer. Math. Soc., 349 (1997), 1169-1189.
doi: 10.1090/S0002-9947-97-01896-5. |
[9] |
E. Berkson and T. A. Gillespie, The $q$-variation of functions and spectral integration of Fourier multipliers, Duke Math. J., 88 (1997), 103-132.
doi: 10.1215/S0012-7094-97-08804-9. |
[10] |
E. Berkson and T. A. Gillespie, $Mean_{2}$-bounded operators on Hilbert space and weight sequences of positive operators, Positivity, 3 (1999), 101-133.
doi: 10.1023/A:1009794510984. |
[11] |
E. Berkson and T. A. Gillespie, Spectral decompositions, ergodic averages, and the Hilbert transform, Studia Math., 144 (2001), 39-61.
doi: 10.4064/sm144-1-2. |
[12] |
E. Berkson and T. A. Gillespie, Shifts as models for spectral decomposability on Hilbert space, J. Operator Theory, 50 (2003), 77-106. |
[13] |
E. Berkson and T. A. Gillespie, Operator means and spectral integration of Fourier multipliers, Houston J. Math., 30 (2004), 767-814. |
[14] |
E. Berkson and T. A. Gillespie, The $q$-variation of functions and spectral integration from dominated ergodic estimates, Journal of Fourier Analysis and Applications, 10 (2004), 149-177.
doi: 10.1007/s00041-004-8009-z. |
[15] |
E. Berkson, T. A. Gillespie and P. S. Muhly, Abstract spectral decompositions guaranteed by the Hilbert transform, Proc. London Math. Soc. (3), 53 (1986), 489-517.
doi: 10.1112/plms/s3-53.3.489. |
[16] |
D. Blagojevic, "Spectral Families and Geometry of Banach Spaces," PhD thesis, University of Edinburgh, 2007. |
[17] |
J. Bourgain, Some remarks on Banach spaces in which martingale difference sequences are unconditional, Arkiv för Mat., 21 (1983), 163-168.
doi: 10.1007/BF02384306. |
[18] |
M. M. Day, Reflexive Banach spaces not isomorphic to uniformly convex spaces, Bull. Amer. Math. Soc., 47 (1941), 313-317. |
[19] |
P. Enflo, Banach spaces which can be given an equivalent uniformly convex norm, Israel J. Math., 13 (1972), 281-288. |
[20] |
G. H. Hardy, Weierstrass's non-differentiable function, Trans. Amer. Math. Soc., 17 (1916), 301-325.
doi: 10.2307/1989005. |
[21] |
G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals. I, Math. Zeitschrift, 27 (1928), 565-606.
doi: 10.1007/BF01171116. |
[22] |
G. H. Hardy and J. E. Littlewood, A convergence criterion for Fourier series, Math. Zeitschrift, 28 (1928) 612-634.
doi: 10.1007/BF01181186. |
[23] |
R. C. James, Super-reflexive spaces with bases, Pacific J. Math., 41 (1972), 409-419. |
[24] |
B. Maurey, Système de Haar, Séminaire Maurey-Schwartz, 1974-1975, (École Polytechnique, Paris, 1975), I.1-II.13. |
[25] |
G. Pisier, Un exemple concernant la super-ré flexivité, Séminaire Maurey-Schwartz 1974-1975: Espaces $L^p$ applications radonifiantes et géométrie des espaces de Banach, Annexe No. 2, 12 pp, Centre Math. école Polytech., Paris, 1975. |
[26] |
J. Porter, Helly's selection principle for functions of bounded $p$-variation, Rocky Mountain J. Math., 35 (2005), 675-679.
doi: 10.1216/rmjm/1181069753. |
[27] |
J. L. Rubio de Francia, A Littlewood-Paley inequality for arbitrary intervals, Revista Mat. Iberoamericana, 1 (1985), 1-14. |
[28] |
P. G. Spain, On well-bounded operators of type $B$, Proc. Edinburgh Math. Soc. (2), 18 (1972), 35-48.
doi: 10.1017/S0013091500026134. |
[29] |
S. Treil and A. Volberg, Wavelets and the angle between past and future, Journal of Functional Analysis, 143 (1997), 269-308.
doi: 10.1006/jfan.1996.2986. |
[30] |
L. C. Young, An inequality of the Hölder type, connected with Stieltjes integration, Acta Math., 67 (1936), 251-282.
doi: 10.1007/BF02401743. |
[31] |
A. Zygmund, "Trigonometric Series," 2nd ed., vol. 1, Cambridge Univ. Press, London, 1959. |
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