Tohoku Mathematical Journal

Operator-valued martingale transforms

Teresa Martínez and José L. Torrea

Full-text: Open access

Abstract

We develop a general theory of martingale transform operators with operator-valued multiplying sequences. Applications are given to classical operators such as Doob's maximal function and the square function. Some geometric properties of the underlying Banach spaces are also considered.

Article information

Source
Tohoku Math. J. (2), Volume 52, Number 3 (2000), 449-474.

Dates
First available in Project Euclid: 3 May 2007

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1178207823

Digital Object Identifier
doi:10.2748/tmj/1178207823

Mathematical Reviews number (MathSciNet)
MR1772807

Zentralblatt MATH identifier
0967.60048

Subjects
Primary: 60G42: Martingales with discrete parameter
Secondary: 46B09: Probabilistic methods in Banach space theory [See also 60Bxx] 60G46: Martingales and classical analysis

Citation

Martínez, Teresa; Torrea, José L. Operator-valued martingale transforms. Tohoku Math. J. (2) 52 (2000), no. 3, 449--474. doi:10.2748/tmj/1178207823. https://projecteuclid.org/euclid.tmj/1178207823


Export citation

References

  • [1] D G AUSTIN, A sample function property for martingales, Ann. Math Statist 37 (1966), 1396-1397
  • [2] R BANUELOS, Martingale transforms and related singular integrals, Trans Amer Math Soc 293 (1986), 547-563.
  • [3] R BANUELOS AND ARTHUR LINDEMAN II, A martingale study of the Beurling-Ahlfors transform in Rn, Funct Anal 145 (1997), 224-265
  • [4] R. BANUELOS AND GANG WANG, Sharp inequalities for martingales with applications to the Beurling Ahlfors and Riesz transforms, Duke Math J 80(1995), 575-600
  • [5] R BANUELOS AND GANG WANG, Orthogonal martingales under differential subordination and application to Riesz transforms, Illinois J Math 40 (1996), 678-691
  • [6] A BENEDEK, A P CALDERON AND R PANZONE, Convolution operators on Banach space valued functions, Proc Nat Acad Sci USA 48 (1962), 356-365
  • [7] J BOURGAIN, Extension of a result of Benedeck, Calderon and Panzone, Ark Mat 22 (1984), 91-9
  • [8] D L BURKHOLDER, Martingale Transforms, Ann Math Stat 37(1966), 1494-150
  • [9] D L BURKHOLDER, A geometrical characterization of Banach spaces in which martingale difference se quences are unconditional, Ann Probab 9 (1981), 997-1011
  • [10] D L BURKHOLDER AND R F GUNDY, Extrapolation and interpolation of quasilinear operators on martin gales, Acta Math 124 (1970), 249-304
  • [11] D L BURKHOLDER, R F GUNDY AND M SILVERSTEIN, A maximal function characterization of the clas HP, Trans Amer Math Soc 157(1971), 137-153
  • [12] A P CALDERON AND A ZYGMUND, On the existence of certain singular integrals, Acta Math 88(1952), 85-139
  • [13] S D CHATTERJI, Martingale convergence and the Radon-Nikodym theorem in Banach spaces, Math. Scan 22(1968), 21-41
  • [14] D. J. DAVIS, On the integrability of the martingale square function, Israel J Math 8 (1970), 187-19
  • [15] J DIESTEL AND J J UHL, Vector Measures, Math Surveys, No 15, Amer Math Soc, Providence, R I, 1977
  • [16] J L DOOB, Stochastic Processes, John Wiley, New York, 195
  • [17] C FEFFERMAN AND E STEIN, Some maximal inequalities, Amer J Math 93 (1971), 107-11
  • [18] J GARCIA-CUERVA, R MACIAS AND J L TORREA, The Hardy-Littlewood property of Banach lattices, Israeli Math 83 (1993), 177-201
  • [19] R GUNDY, A decomposition for L Abounded martingales, Ann Math Statist 39(1968), 134-13
  • [20] S. KWAPIEN, Isomorphic characterizations of inner product spaces by orthogonal series with vector value coefficients, Studia Math 44 (1972), 583-595
  • [21] J LINDENSTRAUSS AND L. TzAFRiRi, Classical Banach spaces II Function spaces, Ergeb Math Grenzge 97, Springer-Verlag, Berlin, 1979
  • [22] B. MAUREY, Systeme de Haar, Seminaire Maurey-Schwartz 1974-75, Centre Math, Ecole Polytech, Paris, 1975
  • [23] J L RUBIO DE FRANCIA, Martingale and integral transforms of Banach space valued functions, in Confer ence on Probability and Banach spaces (Zaragoza, 1985), 195-222, Lecture Notes in Math 1221, Springer-Verlag, Berlin-New York, 1986
  • [24] F WEISZ, Martingale Hardy spaces and their applications in Fourier Analysis, Lecture Notes in Math 1568, Springer-Verlag, Berlin, 1994