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)

 
 

 

On singular integral and martingale transforms


Authors: Stefan Geiss, Stephen Montgomery-Smith and Eero Saksman
Journal: Trans. Amer. Math. Soc. 362 (2010), 553-575
MSC (2000): Primary 60G46, 42B15; Secondary 42B20, 46B09, 46B20
DOI: https://doi.org/10.1090/S0002-9947-09-04953-8
Published electronically: September 11, 2009
MathSciNet review: 2551497
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Linear equivalences of norms of vector-valued singular integral operators and vector-valued martingale transforms are studied. In particular, it is shown that the UMD-constant of a Banach space $ X$ equals the norm of the real (or the imaginary) part of the Beurling-Ahlfors singular integral operator, acting on $ L^p_X(\mathbf{R}^2)$ with $ p\in (1,\infty).$ Moreover, replacing equality by a linear equivalence, this is found to be a typical property of even multipliers. A corresponding result for odd multipliers and the Hilbert transform is given. As a corollary we obtain that the norm of the real part of the Beurling-Ahlfors operator equals $ p^*-1$ with $ p^*:= \max \{p, (p/(p-1))\}$, where the novelty is the lower bound.


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

  • 1. A. Baernstein and S. Montgomery-Smith: Some conjectures about integral means of $ \partial f$ and $ \partial^-f$, in: Complex analysis and differential equations, edited by C. Kiselman, Acta Universitatis Upsaliensis C. 64 (1999), 92-109. MR 1758918 (2001i:30002)
  • 2. R. Bañuelos and G. Wang: Sharp inequalities for martingales with applications to the Beurling-Ahlfors and Riesz transforms, Duke Math. J. 80 (1995), 575-600. MR 1370109 (96k:60108)
  • 3. R. Bañuelos and P. J. Méndez-Hernández: Space-time Brownian motion and the Beurling-Ahlfors transform, Indiana Univ. Math. J. 52 (2003), no. 4, 981-990. MR 2001941 (2004h:60067)
  • 4. R. F. Bass: Probabilistic techniques in analysis, Springer, 1995. MR 1329542 (96e:60001)
  • 5. J. Bourgain: Some remarks on Banach spaces in which martingale difference sequences are unconditional, Ark. Mat. 21 (1983), 163-168. MR 727340 (85a:46011)
  • 6. J. Bourgain: Vector-valued singular integrals and the $ H\sp 1$-BMO duality, in: Probability theory and harmonic analysis (Cleveland, Ohio, 1983), 1-19, Monogr. Textbooks Pure Appl. Math., 98, Dekker, New York, 1986. MR 830227 (87j:42049b)
  • 7. D. L. Burkholder: A geometrical characterization of Banach spaces in which martingale difference sequences are unconditional, Ann. Probab. 9 (1981), 997-1011. MR 632972 (83f:60070)
  • 8. D. L. Burkholder: A geometric condition that implies the existence of certain singular integrals of Banach-space-valued functions, in: Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Illinois, 1981), 270-286, Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1983. MR 730072 (85i:42020)
  • 9. D. L. Burkholder: Boundary value problems and sharp inequalities for martingale transforms, Ann. Prob. 12 (1984), 647-702. MR 744226 (86b:60080)
  • 10. D. L. Burkholder: Explorations in martingale theory and its applications, in: Ecole d'Eté de Probabilités de Saint-Flour XIX-1989, 1-66, 1992 Lect. Notes Math., 1464, Springer, 1992. MR 1108183 (92m:60037)
  • 11. D. L. Burkholder: Martingales and singular integrals in Banach spaces, in: Handbook of the geometry of Banach spaces, Vol. I, 233-269, North-Holland, Amsterdam, 2001. MR 1863694 (2003b:46009)
  • 12. R. R. Coifman and G. Weiss: Transference methods in analysis, C.B.M.S. Regional Conference Series in Math. No. 31, Amer. Math. Soc., Providence, RI, 1976. MR 0481928 (58:2019)
  • 13. M. Defant: Zur vektorwertigen Hilberttransformation, PhD thesis, Universität Kiel, 1986.
  • 14. S. Geiss: A counterexample concerning the relation between decoupling constants and UMD-constants, Trans. Amer. Soc. 351 (1999), no. 4, 1355-1375. MR 1458301 (99f:60011)
  • 15. L. Grafakos: Classical and modern Fourier analysis, Pearson, 2004.
  • 16. R. F. Gundy and N. Varopoulos: Les transformations de Riesz et les intégrales stochastiques, C. R. Acad. Sci. Paris Sér. A-B 289 (1979), A13-A16. MR 545671 (82e:60089)
  • 17. T. P. Hytönen: Aspects of probabilistic Littlewood-Paley theory in Banach spaces, in: Banach spaces and their applications in analysis, 343-355, de Gruyter, Berlin, 2007. MR 2374719
  • 18. R. C. James: Super-reflexive Banach spaces, Can. J. Math. 5 (1972), 896-904. MR 0320713 (47:9248)
  • 19. K. de Leeuw: On $ L_p$ multipliers, Ann. of Math 81 (1965), 364-379. MR 0174937 (30:5127)
  • 20. B. Maurey: Système de Haar, Seminaire Maurey-Schwartz, Ecole Polytechnique, Paris, 1974-1975.
  • 21. S. Petermichl and A. Volberg: Heating of the Ahlfors-Beurling operator: weakly quasiregular maps on the plane are quasiregular, Duke Math. J. 112 (2003), no. 2, 281-305. MR 1894362 (2003d:42025)
  • 22. A. Pietsch and J. Wenzel: Orthonormal systems and Banach space geometry, Cambridge University Press, 1998. MR 1646056 (2000d:46010)
  • 23. E. M. Stein: Singular integrals and differentiability properties of functions, Princeton University Press, 1970. MR 0290095 (44:7280)
  • 24. E. M. Stein and G. Weiss: Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, 1971. MR 0304972 (46:4102)
  • 25. A. Volberg and F. Nazarov: Heat extension of the Beurling operator and estimates for its norm, (Russian. Russian summary) Algebra i Analiz 15 (2003), 142-158; translation in St. Petersburg Math. J. 15 (2004), 563-573. MR 2068982 (2005f:30042)
  • 26. L. Weis: Operator-valued Fourier multiplier theorems and maximal $ L\sb p$-regularity, Math. Ann. 319 (2001), 735-758. MR 1825406 (2002c:42016)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 60G46, 42B15, 42B20, 46B09, 46B20

Retrieve articles in all journals with MSC (2000): 60G46, 42B15, 42B20, 46B09, 46B20


Additional Information

Stefan Geiss
Affiliation: Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35 (MaD), 40014 Jyväskylä, Finland
Email: geiss@maths.jyu.fi

Stephen Montgomery-Smith
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
Email: stephen@math.missouri.edu

Eero Saksman
Affiliation: Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68, FIN-00014 Helsinki, Finland
Email: eero.saksman@helsinki.fi

DOI: https://doi.org/10.1090/S0002-9947-09-04953-8
Keywords: UMD property, singular integrals, martingale transforms
Received by editor(s): January 29, 2007
Published electronically: September 11, 2009
Additional Notes: The first and the last author are supported by Project #110599 of the Academy of Finland.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society