Analysis & PDE

  • Anal. PDE
  • Volume 11, Number 3 (2018), 583-608.

Square function estimates for discrete Radon transforms

Mariusz Mirek

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Abstract

We show p(d)-boundedness, for p(1,), of discrete singular integrals of Radon type with the aid of appropriate square function estimates, which can be thought of as a discrete counterpart of Littlewood–Paley theory. It is a very robust approach which allows us to proceed as in the continuous case.

Article information

Source
Anal. PDE, Volume 11, Number 3 (2018), 583-608.

Dates
Received: 18 June 2016
Revised: 22 July 2017
Accepted: 20 September 2017
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.apde/1513774529

Digital Object Identifier
doi:10.2140/apde.2018.11.583

Mathematical Reviews number (MathSciNet)
MR3738256

Zentralblatt MATH identifier
1383.42014

Subjects
Primary: 11L07: Estimates on exponential sums 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B25: Maximal functions, Littlewood-Paley theory

Keywords
discrete harmonic analysis Radon operators Hardy–Littlewood circle method exponential sums

Citation

Mirek, Mariusz. Square function estimates for discrete Radon transforms. Anal. PDE 11 (2018), no. 3, 583--608. doi:10.2140/apde.2018.11.583. https://projecteuclid.org/euclid.apde/1513774529


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References

  • J. Bourgain, “On the maximal ergodic theorem for certain subsets of the integers”, Israel J. Math. 61:1 (1988), 39–72.
  • J. Bourgain, “On the pointwise ergodic theorem on $L^p$ for arithmetic sets”, Israel J. Math. 61:1 (1988), 73–84.
  • J. Bourgain, “Pointwise ergodic theorems for arithmetic sets”, Inst. Hautes Études Sci. Publ. Math. 69 (1989), 5–45.
  • Z. Buczolich and R. D. Mauldin, “Divergent square averages”, Ann. of Math. $(2)$ 171:3 (2010), 1479–1530.
  • M. Christ, A. Nagel, E. M. Stein, and S. Wainger, “Singular and maximal Radon transforms: analysis and geometry”, Ann. of Math. $(2)$ 150:2 (1999), 489–577.
  • A. D. Ionescu and S. Wainger, “$L^p$ boundedness of discrete singular Radon transforms”, J. Amer. Math. Soc. 19:2 (2006), 357–383.
  • P. LaVictoire, “Universally $L^1$-bad arithmetic sequences”, J. Anal. Math. 113 (2011), 241–263.
  • M. Mirek, E. M. Stein, and B. Trojan, “$\ell^p(\mathbb{Z}^d) $-estimates for discrete operators of Radon type: maximal functions and vector-valued estimates”, preprint, 2015.
  • M. Mirek, E. M. Stein, and B. Trojan, “$\ell^p(\mathbb{Z}^d) $-estimates for discrete operators of Radon type: variational estimates”, Invent. Math. 209:3 (2017), 665–748.
  • A. Seeger, T. Tao, and J. Wright, “Singular maximal functions and Radon transforms near $L^1$”, Amer. J. Math. 126:3 (2004), 607–647.
  • E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series 43, Princeton University Press, 1993.
  • E. M. Stein and S. Wainger, “Discrete analogues in harmonic analysis, I: $l^2$ estimates for singular Radon transforms”, Amer. J. Math. 121:6 (1999), 1291–1336.
  • E. M. Stein and S. Wainger, “Oscillatory integrals related to Carleson's theorem”, Math. Res. Lett. 8:5-6 (2001), 789–800.