Parabolic martingales and non-symmetric Fourier multipliers

• Autorzy: Bogdan K. , Wojciechowski Ł.
• Języki publikacji: EN

Abstrakty

EN

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

Słowa kluczowe

EN

non-symmetric Fourier multiplier; martingale transform

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2012
• Tom: Vol. 32, Fasc. 2
• Strony: 241--253
• Opis fizyczny: Bibliogr. 15 poz.

Twórcy

autor

Bogdan K.

• Institute of Mathematics, of the Polish Academy of Sciences, Institute of Mathematics and Computer Science of the Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland

autor

Wojciechowski Ł.

• Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4,50-384 Wrocław, Poland

Bibliografia

• [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 Lp-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.