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Almost sharp variational estimates for discrete truncated operators of Carleson type

Jiecheng Chen, Renhui Wan

TL;DR

This work addresses sharp $r$-variational bounds for discrete truncated Carleson-type operators on $\ell^p(\mathbb{Z}^n)$. It integrates circle-method minor-arc controls with a novel multi-frequency variational inequality and a multi-frequency square-function framework, leveraging Ionescu–Wainger-type multipliers and the Mirek–Stein–Trojan transference principle. The authors establish $\ell^p(\mathbb{Z}^n;V^r)$ bounds with a scale loss $R^{\varepsilon/r}$ (and a logarithmic refinement possible), and in the one-dimensional quadratic-phase case remove this scale loss entirely while increasing $p$ slightly; as $r\to\infty$ the bounds converge to Krause–Roos estimates. The results extend the understanding of pointwise convergence and fluctuation control for discrete Carleson-type operators and provide a robust toolkit combining circle-method, frequency analysis, and transference techniques for discrete harmonic analysis.

Abstract

We establish $r$-variational estimates for discrete truncated Carleson-type operators on $\ell^p$ for $1<p<\infty$. Notably, these estimates are sharp and enhance the results obtained by Krause and Roos (J. Eur. Math. Soc. 2022, J. Funct. Anal. 2023), up to a logarithmic loss related to the scale. On the other hand, as $r$ approaches infinity, the consequences align with the estimates proved by Krause and Roos. Moreover, for the case of quadratic phases, we remove this logarithmic loss with respect to the scale, at the cost of increasing $p$ slightly.

Almost sharp variational estimates for discrete truncated operators of Carleson type

TL;DR

This work addresses sharp -variational bounds for discrete truncated Carleson-type operators on . It integrates circle-method minor-arc controls with a novel multi-frequency variational inequality and a multi-frequency square-function framework, leveraging Ionescu–Wainger-type multipliers and the Mirek–Stein–Trojan transference principle. The authors establish bounds with a scale loss (and a logarithmic refinement possible), and in the one-dimensional quadratic-phase case remove this scale loss entirely while increasing slightly; as the bounds converge to Krause–Roos estimates. The results extend the understanding of pointwise convergence and fluctuation control for discrete Carleson-type operators and provide a robust toolkit combining circle-method, frequency analysis, and transference techniques for discrete harmonic analysis.

Abstract

We establish -variational estimates for discrete truncated Carleson-type operators on for . Notably, these estimates are sharp and enhance the results obtained by Krause and Roos (J. Eur. Math. Soc. 2022, J. Funct. Anal. 2023), up to a logarithmic loss related to the scale. On the other hand, as approaches infinity, the consequences align with the estimates proved by Krause and Roos. Moreover, for the case of quadratic phases, we remove this logarithmic loss with respect to the scale, at the cost of increasing slightly.
Paper Structure (38 sections, 23 theorems, 332 equations)

This paper contains 38 sections, 23 theorems, 332 equations.

Table of Contents

  1. Introduction
  2. Motivation and main results
  3. Overview of the proof
  4. Organization
  5. Notation.
  6. Preliminaries
  7. Fourier transforms and Fourier multipliers
  8. $V^r$, $\mathbf{V}^r$ and related inequalities
  9. Ionescu-Wainger-type multiplier theorem
  10. Transference principle by Mirek, Stein and Trojan
  11. Proof of Theorem \ref{['t1']} and reduction of (\ref{['t11']})
  12. General operators and minor arcs estimates
  13. Reduction of (\ref{['long']}) and major arcs estimates I and II
  14. Reduction of long variational inequality (\ref{['long2']})
  15. Reduction of long variational inequality (\ref{['long11']})
  16. ...and 23 more sections

Key Result

Theorem 1.1

Let $n$ and $d$ be positive integers, and let $\lambda(x)$ be an arbitrary function from $\mathbb{Z}^n$ to [0,1]. Suppose $r\in (2,\infty)$ and $p\in (1,\infty)$. Then for any $R\ge 1$ and any $\epsilon>0$, we have with the implicit constant independent of $R$, $f$ and the function $\lambda(x)$.

Theorems & Definitions (36)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • Proposition 3.3
  • Proposition 3.4
  • ...and 26 more