A Discrete Quadratic Carleson Theorem on ℓ ² with a Restricted Supremum

2015/12/22 by Ben Krause, Michael T. Lacey, Krause, Ben +1
Mathematics · #Advanced Harmonic Analysis Research #Analytic Number Theory Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Analysis and Transform Methods

paper · pdf · doi:10.48550/arxiv.1512.06918

openalex publication_date 2015/12/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Consider the discrete maximal function acting on ℓ²(\mathbb Z) functions C_Λ f( n ) := sup _{λ∈ Λ} | ∑_{m ≠ 0} f(n-m) \frace^{2 πiλm2} 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 L²(\mathbb R). C₂ f(x):= sup_{λ∈ \mathbb R} | ∫ f(x-y) \frace^{2πi λy2}y dy |. 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 the parameter λ above.

Citations

• The Polynomial Carleson Operator
• A polynomial Carleson operator along the paraboloid
• ℓ^p(\mathbb Z^d)-estimates for discrete operators of Radon type: Maximal functions and vector-valued estimates

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