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Sparse Bounds for Random Discrete Carleson Theorems

  • Ben Krause
  • Michael T. Lacey
Chapter
Part of the Operator Theory: Advances and Applications book series (OT, volume 261)

Abstract

We study discrete random variants of the Carleson maximal operator. Intriguingly, these questions remain subtle and difficult, even in this setting. Let {Xm} be an independent sequence of {0,1} random variables with expectations EXm = σm = m –α, 0 α 1/2, and Sm =∑mk=1 Xk. Then the maximal operator below almost surely is bounded from lp to lp, provided the Minkowski dimension of Λ [–1/2, 1/2] is strictly less than 1 –α. supλ∈Λ|∑m≠0X|m|e(λm) sgn(m)|m| f(x – m)|. This operator also satisfies a sparse type bound. The form of the sparse bound immediately implies weighted estimates in all l2, which are novel in this setting. Variants and extensions are also considered.

Keywords

Carleson maximal operator arithmetic Minkowski dimension sparse operator arithmetic ergodic theorems strong law of large numbers 

Mathematics Subject Classification (2010)

Primary 42B20 Secondary 42A45 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsThe University of British ColumbiaVancouverCanada
  2. 2.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA

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