UNIVERSITY
OF WROC£AW
 
Main Page
Online First
Contents of previous volumes
Forthcoming papers
General Information
Instructions for authors


VOLUMES
39.2 39.1 38.2 38.1 37.2 37.1 36.2
36.1 35.2 35.1 34.2 34.1 33.2 33.1
32.2 32.1 31.2 31.1 30.2 30.1 29.2
29.1 28.2 28.1 27.2 27.1 26.2 26.1
25.2 25.1 24.2 24.1 23.2 23.1 22.2
22.1 21.2 21.1 20.2 20.1 19.2 19.1
18.2 18.1 17.2 17.1 16.2 16.1 15
14.2 14.1 13.2 13.1 12.2 12.1 11.2
11.1 10.2 10.1 9.2 9.1 8 7.2
7.1 6.2 6.1 5.2 5.1 4.2 4.1
3.2 3.1 2.2 2.1 1.2 1.1 .imap
 
 
WROC£AW UNIVERSITY
OF SCIENCE AND
TECHNOLOGY

Contents of PMS, Vol. 32, Fasc. 2,
pages 241 - 253
 

PARABOLIC MARTINGALES AND NON-SYMMETRIC FOURIER MULTIPLIERS

Krzysztof Bogdan
Łukasz Wojciechowski

Abstract: We give a class of Fourier multipliers with non-symmetric symbols and explicit norm bounds on Lp  spaces by using the stochastic calculus of Lévy processes and Burkholder–Wang estimates for differentially subordinate martingales.

2000 AMS Mathematics Subject Classification: Primary: 42B15; Secondary: 60G15, 60G46.

Keywords and phrases: Non-symmetric Fourier multiplier, martingale transform.

References

[1]   R. Bañuelos, The foundational inequalities of D. L. Burkholder and some of their ramifications, Illinois J. Math. 54 (3) (2010), pp. 789–868.

[2]   R. Bañuelos and K. Bogdan, Lévy processes and Fourier multipliers, J. Funct. Anal. 250 (1) (2007), pp. 197–213.

[3]   R. Bañuelos, K. Bogdan, and A. Bielaszewski, Fourier multipliers for non-symmetric Lévy processes, Marcinkiewicz Centenary Volume, Banach Center Publ. 95 (2011), pp. 9–25.

[4]   R. Bañuelos and P. J. Méndez-Hernández, Space-time Brownian motion and the Beurling–Ahlfors transform, Indiana Univ. Math. J. 52 (4) (2003), pp. 981–990.

[5]   R. Bañuelos and A. Osękowski, Martingales and sharp bounds for Fourier multipliers, Ann. Acad. Sci. Fenn. Math. 37 (1) (2012), pp. 251–263.

[6]   J. Bertoin, Lévy Processes, Cambridge Tracts in Math., Vol. 121, Cambridge University Press, 1996.

[7]   K. Bogdan, T. Byczkowski, T. Kulczycki, M. Ryznar, R. Song, and Z. Vondraček, Potential Analysis of Stable Processes and Its Extensions, Lecture Notes in Math., Vol. 1980, P. Graczyk and A. Stos (Eds.), Springer, Berlin 2009.

[8]   C. Dellacherie and P.-A. Meyer, Probabilities and Potential, B. Theory of Martingales, translated from French by J. P. Wilson, North-Holland Math. Stud., Vol. 72, North-Holland Publishing Co., Amsterdam 1982.

[9]   M. Métivier, Semimartingales. A Course on Stochastic Processes, de Gruyter Stud. Math., Vol. 2, Walter de Gruyter & Co., Berlin 1982.

[10]   R. Mikulevičius and H. Pragarauskas, On  p
L  -estimates of some singular integrals related to jump processes, ArXiv e-prints, August 2010.

[11]   P. E. Protter, Stochastic Integration and Differential Equations, second edition, Stoch. Model. Appl. Probab., Vol. 21, Springer, Berlin 2004.

[12]   K. Sato, Lévy Processes and Infinitely Divisible Distributions, translated from the 1990 Japanese original, revised by the author, Cambridge Stud. Adv. Math., Vol. 68, Cambridge University Press, Cambridge 1999.

[13]   E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Math. Ser., No. 30, Princeton University Press, Princeton, N. J., 1970.

[14]   A. Volberg and F. Nazarov, Heat extension of the Beurling operator and estimates for its norm, Algebra i Analiz 15 (4) (2003), pp. 142–158.

[15]   G. Wang, Differential subordination and strong differential subordination for continuous-time martingales and related sharp inequalities, Ann. Probab. 23 (2) (1995), pp. 522–551.

Download:    Abstract    Full text