A Discrete Quadratic Carleson Theorem on with ℓ2 a Restricted Supremum. (24th June 2016)

• Record Type: Journal Article
• Authors: Krause, Ben; Lacey, Michael T.
• Is Part Of: International mathematics research notices. Volume 2017:Number 10(2017)
• Page Start: 3180
• Page End: 3208
• Publication Date: 2016-06-24
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnw116
• Languages: English
• ISSNs: 1073-7928

Abstract

Abstract : Consider the discrete maximal function acting on ℓ 2 ( ℤ ) functions C Λ f ( n ) : = sup λ ∈ Λ | ∑ m ≠ 0 f ( n − m ) e 2 π i λ m 2 m, | where Λ ⊂ [ 0, 1 ] . We give sufficient conditions on Λ, met by certain kinds of Cantor sets, for this to be a bounded sublinear operator. This result is a discrete analogue of E.M.Stein's integral result, that the maximal operator below is bounded on L 2 ( ℝ ) . C 2 f ( x ) : = sup λ ∈ ℝ | ∫ f ( x − y ) e 2 π i λ y 2 y d y | . The proof of our result relies heavily on Bourgain's work on arithmetic ergodic theorems, with novel complexity arising from the oscillatory nature of the question at hand, and difficulties arising from the parameter λ above.