Variational estimate for the family of discrete averages associated to simplices

Technical Summary: Variational Estimate for the Family of Discrete Averages Associated to Simplices

Problem Statement
This paper addresses the problem of establishing $\ell^2(\mathbb{Z}^{kn})$-boundedness for the long $r$-variational seminorm associated with a family of discrete averages over non-degenerate $k$-simplices. While the $\ell^p$-boundedness of the maximal function associated with discrete spherical averages has been extensively studied (e.g., Magyar-Stein-Wainger [MSW02], Ionescu [Ion04]), and the maximal function for discrete simplicial averages was recently shown to be bounded on $\ell^2$ by Lyall, Magyar, Newman, and Woolfitt [LMNW25], the variational estimates for the latter remain largely unexplored.

Specifically, the paper investigates the operator $V_r(\mathcal{A}^S_L)$, where $\mathcal{A}^S_L = \{A_{\lambda}^S : \lambda \in L\}$ represents discrete linear simplicial averages over a lacunary sequence $L = \{2^l\}_{l=1}^\infty$. The goal is to prove that for $r > 2$, $k \ge 1$, and $n \ge 2k+3$, the variational operator satisfies:
$$\|V_r(\mathcal{A}^S_L f)\|_{\ell^2(\mathbb{Z}^{kn})} \lesssim \|f\|_{\ell^2(\mathbb{Z}^{kn})}.$$
This result is significant because, as noted in the introduction, $\ell^p$-bounds for the long $r$-variations imply bounds for the corresponding lacunary maximal operator and ensure the almost everywhere convergence of the sequence of averages as $\lambda \to \infty$.

Methodology
The proof of the main theorem (Theorem 1.1) relies on a reduction to jump inequalities, following the framework established by Jones, Seeger, and Wright [JSW08]. The authors utilize the following key components:

1. Jump Inequalities: Instead of directly estimating the $r$-variation, the paper establishes the boundedness of the $\lambda$-jump operator $J_\lambda(\mathcal{A}^S_L)$. By Lemma 1.1 (a discrete analogue of [JSW08, Lemma 2.1]), proving $\sup_{\lambda>0} \|\lambda (J_\lambda(\mathcal{A}^S_L f))^{1/2}\|_{\ell^2} \lesssim \|f\|_{\ell^2}$ is sufficient to deduce the variational estimate for $r > 2$.

2. Function Decomposition: The function $f \in \ell^2(\mathbb{Z}^{kn})$ is decomposed into three parts, $f = f_1 + f_2 + f_3$, using a discrete Littlewood-Paley type decomposition adapted to the geometry of the simplex. This involves:

   • A smooth cutoff function $\psi$ and its dilations $\Psi^s_{l,j}$.
   • Differences $\Delta \Psi^s_{l,j} = \Psi^s_{l,j+1} - \Psi^s_{l,j}$ which act as frequency localization operators.
   • The decomposition separates $f$ into a low-frequency part ($f_1$), a middle-frequency part ($f_2$), and a high-frequency remainder ($f_3$).

3. Frequency Localization and Support Analysis: The proof carefully analyzes the Fourier support of the decomposed functions.

   • For $f_2$ and $f_3$, the Fourier supports are shown to lie in the complement of specific sets $\Omega^s_{l,j}$. This allows the application of Proposition 2.2 (derived from [LMNW25]), which provides decay estimates for the maximal function of discrete simplicial averages when the input function has restricted frequency support.
   • For $f_1$, the proof utilizes a decomposition of the lattice $\mathbb{Z}^{kn}$ into dyadic cubes $Q^l_t$. The average is split into a term involving a local average ($E_l f$) and a fluctuation term ($L f$).

4. Square Function Estimates:

   • For the terms $f_2$ and $f_3$, the $\lambda$-jump is dominated by a square function. The authors invoke Minkowski's inequality and the decay estimates from Proposition 2.2 to sum over scales $l$ and frequencies $j$, yielding the desired $\ell^2$ bound.
   • For the term $f_1$, the proof reduces to showing the $\ell^2$-boundedness of a square function associated with the fluctuation term $L f$. This relies on Lemma 2.1, which establishes a decay estimate of the form $\|(\Psi^s_{l,0} * w_{2^l S}) * D_m f - E_l(D_m f)\|_{\ell^2} \lesssim 2^{-\delta|l-m|} \|D_m f\|_{\ell^2}$, where $D_m f$ represents differences of local averages. The proof of this lemma splits into cases ($l \ge m$ and $l < m$), utilizing the mean-value theorem, properties of the Hardy-Littlewood maximal function, and Poisson summation formulas.

Key Contributions and Results

• Theorem 1.1: The primary result establishes the $\ell^2(\mathbb{Z}^{kn})$-boundedness of the long $r$-variational seminorm for discrete simplicial averages for $r > 2$, provided $n \ge 2k+3$.
• Theorem 1.2: The paper proves the corresponding jump inequality, $\sup_{\lambda>0} \|\lambda (J_\lambda(\mathcal{A}^S_L f))^{1/2}\|_{\ell^2} \lesssim \|f\|_{\ell^2}$, which serves as the technical engine for Theorem 1.1.
• Lemma 2.1: A discrete analogue of [JSW08, Lemma 3.2] is proven, providing the necessary decay estimates for the interaction between the simplicial averaging operator and local averages on dyadic cubes.

Significance and Claims
The paper claims to initiate the study of variational estimates for discrete averages over hyper-surfaces that include the sphere, specifically extending the analysis to non-degenerate simplices. The authors position this work as a natural extension of the extensive literature on discrete spherical averages (e.g., [MSW02], [Ion04]) and the recent work on simplicial maximal functions [LMNW25].

The significance of the result lies in its implications for convergence: the $\ell^2$-boundedness of the variational operator immediately implies that the sequence of discrete simplicial averages $\{A_{\lambda}^S f\}$ converges almost everywhere as $\lambda \to \infty$. Furthermore, the paper notes that these variational bounds imply bounds for the lacunary maximal operator associated with simplices. The work does not claim to resolve the $\ell^p$-boundedness for $p \neq 2$ or for the full (non-lacunary) maximal function, noting that these questions remain open in the broader context of discrete harmonic analysis.