Mathematische Zeitschrift

, Volume 287, Issue 3–4, pp 797–816 | Cite as

A method of rotations for Lévy multipliers

Article
First Online:
Received:
Accepted:
  • 156 Downloads

Abstract

We use a method of rotations to study the \(L^p\) boundedness, for \(1<p<\infty \), of Fourier multipliers which arise as the projection of martingale transforms with respect to symmetric \(\alpha \)-stable processes, \(0<\alpha <2\). Our proof does not use the fact that \(0<\alpha <2\), and therefore allows us to obtain a larger class of multipliers which are bounded on \(L^p\). As in the case of the multipliers which arise as the projection of martingale transforms, these new multipliers also have potential applications to the study of the \(L^p\) boundedness of the Beurling-Ahlfors transform.

Keywords

Beurling–Ahlfors transform Lévy processes Martingale transforms Fourier multipliers Method of rotations 

Mathematics Subject Classification

60G46 42A61 
This is a preview of subscription content, log in to check access.

Notes

Acknowledgements

I would like to thank my Ph.D. advisor, Rodrigo Bañuelos, for being an invaluable source of advice throughout the process of writing this paper. I would also like to thank the referee, whose helpful comments greatly improved the quality of this paper and, in particular, helped make the statement of lemma 3.1 more clear.

References

  1. 1.
    Applebaum, D., Bañuleos, R.: Martingale transforms and Lévy processes on Lie groups. Indiana Univ. Math. J. 63, 1109–1138 (2014)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Astala, K., Iwaniec, T., Martin, G.: Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane. Princeton University Press, Princeton (2009)MATHGoogle Scholar
  3. 3.
    Bañuelos, R.: The foundational inequalities of D.L. Burkholder and some of their ramifications. Ill. J. Math. 54, 789–868 (2010)MATHMathSciNetGoogle Scholar
  4. 4.
    Bañuelos, R.: Martingale transforms and related singular integrals. Trans. Am. Math. Soc. 293(2), 547–563 (1986)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Bañuelos, R., Bogdan, K.: Lévy processes and Fourier multipliers. J. Funct. Anal. 250, 197–213 (2007)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Bañuelos, R., Bielaszewski, A., Bogdan, K.: Fourier multipliers for non-symmetric Lévy processes. Marcinkiewicz Centen. Vol. 95, 9–25 (2011)MATHGoogle Scholar
  7. 7.
    Banuelos, R., Janakiraman, P.: \(L^p\)–bounds for the Beurling–Ahlfors transform. Trans. Am. Math. Soc. 360, 3604–3612 (2008)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Bañuelos, R., Méndez-Hernández, P.: Space-time Brownian motion and the Beurling-Ahlfors transform. Indiana Univ. Math. J. 52, 981–990 (2003)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Bañuelos, R., Wang, G.: Sharp inequalities for martingales with applications to the Beurling-Ahlfors and Riesz transforms. Duke Math. J. 80(3), 575–600 (1995)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Bañuelos, R., Yolcu, S.: Heat trace of non-local operators. J. Lond. Math. Soc 87(3), 304–318 (2014)MATHMathSciNetGoogle Scholar
  11. 11.
    Borichev, A., Janakiraman, P., Volberg, A.: Subordination by orthogonal martingales in \(L^{p}\) and zeros of Laguerre polynomials. Duke Math. J. 162(5), 889–924 (2013)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Burkholder, D.L.: Sharp inequalities for martingales and stochastic integrals. Asterique 157–158, 75–94 (1988)MATHMathSciNetGoogle Scholar
  13. 13.
    Chen, Z.Q., Kim, P., Song, R.: Sharp heat kernel estimates for relativistic stable processes on open sets. Ann. Probab. 40(1), 213–244 (2012)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Dragičević, O., Petermichl, S., Volberg, A.: A rotation method which gives linear \(L^p\) estimates for powers of the Ahlfors–Beurling operator. J. Math. Pures Appl. 86(6), 494–509 (2006)MATHMathSciNetGoogle Scholar
  15. 15.
    Grafakos, L.: Classical and Modern Fourier Analysis. Pearson Education, Inc., New Jersey (2004)MATHGoogle Scholar
  16. 16.
    Gundy, R., Varopoulos, N.: Les transformations de Riesz et les integrales stochastiques. CR Acad. Sci. Paris Ser. A 289, 13–16 (1979)MATHMathSciNetGoogle Scholar
  17. 17.
    Iwaniec, T., Martin, G.: The Beurling-Ahlfors transform in \(R^n\) and related singular integrals. J. Reine Agnew. Math. 473, 25–27 (1996)Google Scholar
  18. 18.
    Iwaniec, T.: Extremal inequalities in Sobolev spaces and quasiconfromal mappings. Z. Anal. Anwend. 6, 1–16 (1982)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Lehto, O.: Remarks on the integrability of the derivative of quasiconformal mappings. Ann. Acad. Sci. Fenn. Ser. A. Mat. 371, 3–8 (1965)MATHMathSciNetGoogle Scholar
  20. 20.
    McConnell, T.: On Fourier multiplier transformations of Banach-valued functions. Trans. Am. Math. Soc. 285, 739–757 (1984)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Nazarov, F., Volberg, A.: Heat extension of the Beurling operator and estimates for its norm. St. Petersb. Math. J. 15, 563–573 (2004)MATHMathSciNetGoogle Scholar
  22. 22.
    Perlmutter, M.: On a class of Calderón-Zygmund operators arising from projections of martingale transforms. Potential Anal. 42, 383–401 (2015)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Protter, P.E.: Stochastic Integration and Differential Equations, second ed., Stoch. Model. Appl. Probab., Springer, Berlin (2004)Google Scholar
  24. 24.
    Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1979)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of MathematicsPurdue UniversityWest LafayetteUSA

Personalised recommendations