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Estimates for Compositions of Maximal Operators with Singular Integrals

Published online by Cambridge University Press:  20 November 2018

Richard Oberlin
Richard Oberlin*
Affiliation:
Mathematics Department, Louisiana State University, Baton Rouge, LAe-mail: oberlin@math.lsu.edu
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Abstract.

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We prove weak-type $\left( 1,\,1 \right)$ estimates for compositions of maximal operators with singular integrals. Our main object of interest is the operator $\Delta *\Psi $ where $\Delta *$ is Bourgain’s maximal multiplier operator and $\Psi $ is the sum of several modulated singular integrals; here our method yields a significantly improved bound for the ${{L}^{q}}$ operator norm when $1\,<\,q\,<\,2$. We also consider associated variation-norm estimates.

Keywords

42A45maximal operatorCalderón–Zygmund
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 56 , Issue 4 , 01 December 2013 , pp. 801 - 813
Copyright
Copyright © Canadian Mathematical Society 2013

References

[1] Bourgain, J., Pointwise ergodic theorems for arithmetic sets. Inst. Hautes Etudes Sci. Publ. Math. 69 (1989), 5–45.Google Scholar
[2] Coifman, Ronald, José Luis Rubio de Francia and Stephen Semmes, Multiplicateurs de Fourier de Lp(R) et estimations quadratiques. C. R. Acad. Sci. Paris Sér. I Math. 306 (1988), 351–354.Google Scholar
[3] Comech, Andrew, Cotlar–Stein almost orthogonality lemma. Unpublished note, http://www.math.tamu.edu/_comech/papers/CotlarStein/CotlarStein.pdf. Google Scholar
[4] Dappa, H. and Trebels, W., On maximal functions generated by Fourier multipliers. Ark. Mat. 23 (1985), 241–259. http://dx.doi.org/10.1007/BF02384428 Google Scholar
[5] Demeter, Ciprian, Improved Range in the Return Times Theorem. Canad. Math. Bull., to appear.Google Scholar
[6] , On some maximal multipliers in Lp. Rev. Mat. Ibero. 26 (2010), 947–964. http://dx.doi.org/10.4171/RMI/622 Google Scholar
[7] Demeter, Ciprian, Lacey, Michael T., Terence Tao and Christoph Thiele, Breaking the duality in the return times theorem. Duke Math. J. 143 (2008), 281–355. http://dx.doi.org/10.1215/00127094-2008-020 Google Scholar
[8] Jones, Roger L., Seeger, Andreas and Wright, James, Strong variational and jump inequalities in harmonic analysis. Trans. Amer. Math. Soc. 360 (2008), 6711–6742. http://dx.doi.org/10.1090/S0002-9947-08-04538-8 Google Scholar
[9] Lacey, Michael T., Issues related to Rubio de Francia's Littlewood–Paley inequality. NYJM Monographs 2, State University of New York University at Albany, Albany, NY, 2007.Google Scholar
[10] Fedor Nazarov, Richard Oberlin and Christoph Thiele, A Calder´on Zygmund decomposition for multiple frequencies and an application to an extension of a lemma of Bourgain. Math. Res. Lett. 17 (2010), 529–545.Google Scholar
[11] Richard Oberlin, Bounds on Walsh model for Mq-Carleson and related operators. Preprint.Google Scholar
[12] Terence Tao and JamesWright, Endpoint multiplier theorems of Marcinkiewicz type. Rev. Mat. Iberoamericana 17 (2001), 521–558.Google Scholar
[13] Titchmarsh, E. C., Introduction to the theory of Fourier integrals. Third edition, Chelsea Publishing Co., New York, 1986 Google Scholar