[Paper Review] A Discrete Quadratic Carleson Theorem on $ \ell ^2 $ with a Restricted Supremum

Ben Krause, Michael T. Lacey|arXiv (Cornell University)|Dec 22, 2015

Advanced Harmonic Analysis Research18 references3 citations

TL;DR

This paper establishes $\ell^2(\mathbb{Z})$ boundedness of a discrete maximal quadratic Carleson operator with supremum restricted to a parameter set $\Lambda \subset [0,1]$ of arithmetic Minkowski dimension $d < 1$. Using a maximal multiplier approach inspired by Bourgain's ergodic theorems and refined with a novel complexity analysis of oscillatory sums, the authors prove that the operator is bounded when $\Lambda$ satisfies a specific arithmetic-geometric condition, supporting a conjecture by Pierce on the full supremum over $[0,1]$.

ABSTRACT

Consider the discrete maximal function acting on $\ell^2(\mathbb Z)$ functions \[ \mathcal{C}_Λ f( n ) := \sup_{ λ\in Λ} \left| \sum_{m eq 0} f(n-m) \frac{e^{2 πiλm^2}} {m} ight| \] where $Λ\subset [0,1]$. 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 $L^2(\mathbb R)$. \[ \mathcal{C}_2 f(x):= \sup_{λ\in \mathbb R} \left| \int f(x-y) \frac{e^{2πi λy^2}}{y} \ dy ight|.\] 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.

Motivation & Objective

• To address a conjecture by Lillian Pierce on the $\ell^2(\mathbb{Z})$ boundedness of the discrete maximal quadratic Carleson operator over $[0,1]$.
• To establish sufficient conditions on a parameter set $\Lambda \subset [0,1]$ for the restricted supremum of the discrete maximal operator to be bounded on $\ell^2(\mathbb{Z})$.
• To extend Stein's integral Carleson operator result to the discrete setting, where arithmetic structure and oscillatory sums require new techniques beyond standard $TT^*$ methods.
• To analyze the behavior of oscillatory discrete multipliers $M(\lambda, \beta) = \sum_{m \neq 0} \frac{e(\lambda m^2 - \beta m)}{m}$ via the Hardy-Littlewood circle method, separating major and minor arcs.
• To demonstrate that the arithmetic Minkowski dimension condition on $\Lambda$ controls the interaction of minor arcs across varying $\lambda$, enabling $\ell^2$ bounds.

Proposed method

• Treat the discrete maximal operator as a maximal multiplier operator with multipliers $M(\lambda, \beta)$ depending on $\lambda \in \Lambda$.
• Apply the Hardy-Littlewood circle method to decompose the multiplier into major arcs (near rationals with small denominators) and minor arcs (where the multiplier is small).
• Use a novel complexity estimate (Lemma 2.4) to control the minor arcs uniformly across $\lambda \in \Lambda$, leveraging the arithmetic Minkowski dimension condition.
• Approximate the major arc contributions by scaled versions of the continuous Stein operator, enabling comparison to known $L^2$ bounds.
• Combine the major and minor arc estimates via a square function argument and $\ell^2$-boundedness of the maximal function.
• Draw on Bourgain’s polynomial ergodic theorems as a foundational framework, adapting them to the maximal setting with oscillatory phases.

Experimental results

Research questions

• RQ1 Under what conditions on a parameter set $\Lambda \subset [0,1]$ is the discrete maximal quadratic Carleson operator bounded on $\ell^2(\mathbb{Z})$?
• RQ2 Can the conjectured $\ell^2$ boundedness of the full supremum over $[0,1]$ be supported by restricting $\Lambda$ to a set of arithmetic Minkowski dimension $d < 1$?
• RQ3 How can the oscillatory nature of the discrete multiplier $M(\lambda, \beta)$ be controlled uniformly across $\lambda \in \Lambda$ when $\Lambda$ has low arithmetic complexity?
• RQ4 What role does the arithmetic Minkowski dimension play in enabling $\ell^2$ bounds for maximal operators with oscillatory kernels?
• RQ5 How can Bourgain’s methods in arithmetic ergodic theory be adapted to handle maximal functions with parameter-dependent oscillatory multipliers?

Key findings

• The discrete maximal operator $\mathcal{C}_{\Lambda}f(n) = \sup_{\lambda \in \Lambda} \left| \sum_{m \neq 0} f(n-m) \frac{e(\lambda m^2)}{m} \right|$ is bounded on $\ell^2(\mathbb{Z})$ when $\Lambda$ has arithmetic Minkowski dimension $d < 1$.
• The bound is uniform: $\| \mathcal{C}_{\Lambda}f \|_{\ell^2} \leq C_\Lambda \|f\|_{\ell^2}$, where $C_\Lambda$ depends only on the dimension $d$ and the constant $C_\Lambda$ in the definition of arithmetic Minkowski dimension.
• The class of sets $\Lambda$ satisfying the condition includes certain Cantor-like sets, such as $\Lambda = \left\{ \sum_{j \in J} 2^{-D^j} : J \subset \mathbb{N} \right\}$, which have arithmetic Minkowski dimension at most $1/D$.
• The minor arcs are controlled via a uniform smallness estimate derived from the arithmetic Minkowski dimension, which compensates for the lack of Weyl’s Lemma in the discrete setting.
• The major arc contributions are approximated by scaled versions of the continuous Stein operator, and their $\ell^2$-boundedness is inherited from the continuous theory.
• The proof avoids the $TT^*$ method due to the inapplicability of Weyl’s Lemma in the discrete setting, instead relying on a maximal multiplier framework with careful decomposition.