Inventiones mathematicae

, Volume 209, Issue 3, pp 665–748 | Cite as

\(\ell ^p\left( \mathbb {Z}^d\right) \)-estimates for discrete operators of Radon type: variational estimates

  • Mariusz Mirek
  • Elias M. Stein
  • Bartosz Trojan


We prove \(\ell ^p\left( {\mathbb {Z}}^d\right) \) bounds for \(p\in (1, \infty )\), of r-variations \(r\in (2, \infty )\), for discrete averaging operators and truncated singular integrals of Radon type. We shall present a new powerful method which allows us to deal with these operators in a unified way and obtain the range of parameters of p and r which coincide with the ranges of their continuous counterparts.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Mariusz Mirek
    • 1
    • 2
  • Elias M. Stein
    • 3
  • Bartosz Trojan
    • 4
  1. 1.School of MathematicsInstitute for Advanced StudyPrincetonUSA
  2. 2.Instytut MatematycznyUniwersytet WrocławskiWrocławPoland
  3. 3.Department of MathematicsPrinceton UniversityPrincetonUSA
  4. 4.Wrocław University of TechnologyWrocławPoland

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