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Lacunary elliptic maximal operator on the Heisenberg group

Joonil Kim, Jeongtae Oh

TL;DR

This work analyzes lacunary maximal operators on the Heisenberg group $\mathbb{H}^{1}$ associated with ellipses determined by a $2\times2$ matrix $A$. The authors prove $L^{p}$ boundedness for lacunary elliptic maximal operators and identify sharp unboundedness in degenerate cases, showing how curvature induced by $A$ governs boundedness. Distinct from spectral methods, the approach combines the group Fourier transform with Littlewood–Paley theory on $\mathbb{H}^{1}$ and oscillatory integral bounds to handle multi-parameter lacunary maximal operators, including both skew-symmetric and general $A$. They establish necessary and sufficient curvature-type conditions on $A$ for boundedness of lacunary circular and elliptic operators, and treat general matrices by separating symmetric and skew parts, culminating in a complete picture of when these maximal operators are $L^{p}$-bounded for $1<p<\infty$. The results extend prior work on lacunary averages by clarifying the role of matrix-induced curvature in noncommutative two-step groups and provide a detailed bootstrap argument that yields sharp decay and vector-valued estimates.

Abstract

In this paper, we prove \( L^p \) boundedness results for lacunary elliptic maximal operators on the Heisenberg group. Furthermore, we extend these \( L^p \) estimates from skew-symmetric matrices, which naturally arise in Heisenberg group operations, to arbitrary matrices \( A \), investigating how the curvature induced by \( A \) governs the \( L^p \) boundedness of lacunary circular and elliptic maximal operators. Specifically, we provide necessary and sufficient conditions on \( A \) that determine whether these operators are bounded or unbounded on \( L^p \).

Lacunary elliptic maximal operator on the Heisenberg group

TL;DR

This work analyzes lacunary maximal operators on the Heisenberg group associated with ellipses determined by a matrix . The authors prove boundedness for lacunary elliptic maximal operators and identify sharp unboundedness in degenerate cases, showing how curvature induced by governs boundedness. Distinct from spectral methods, the approach combines the group Fourier transform with Littlewood–Paley theory on and oscillatory integral bounds to handle multi-parameter lacunary maximal operators, including both skew-symmetric and general . They establish necessary and sufficient curvature-type conditions on for boundedness of lacunary circular and elliptic operators, and treat general matrices by separating symmetric and skew parts, culminating in a complete picture of when these maximal operators are -bounded for . The results extend prior work on lacunary averages by clarifying the role of matrix-induced curvature in noncommutative two-step groups and provide a detailed bootstrap argument that yields sharp decay and vector-valued estimates.

Abstract

In this paper, we prove boundedness results for lacunary elliptic maximal operators on the Heisenberg group. Furthermore, we extend these estimates from skew-symmetric matrices, which naturally arise in Heisenberg group operations, to arbitrary matrices , investigating how the curvature induced by governs the boundedness of lacunary circular and elliptic maximal operators. Specifically, we provide necessary and sufficient conditions on that determine whether these operators are bounded or unbounded on .
Paper Structure (11 sections, 13 theorems, 118 equations)

This paper contains 11 sections, 13 theorems, 118 equations.

Table of Contents

  1. Introduction and statement of results
  2. Group Fourier transform and Littlewood-Paley Theorem on the Heisenberg group
  3. Structure of the proof for Theorem \ref{['3mt1']}
  4. Proof of the Proposition \ref{['3dcy']} for $p=2$
  5. Proof of (\ref{['dcy2']}) for $2^{-\ell_2/4}$ decay
  6. Proof of (\ref{['dcy2']}) for $2^{-\ell_1/2}$ decay
  7. Proof of Proposition \ref{['3dcy']} and Bootstrap argument
  8. Reduction and the Proof of Theorem \ref{['3mt0']}; Unboundedness
  9. Proof of Theorem \ref{['3mt2']}; General matrix $A$
  10. The non-symmetric cases $A_w\ne 0$
  11. Symmetric Case $A_w=0$

Key Result

Theorem 1.1

For $A \in \mathbb{M}_{2}(\mathbb{R})$, if $A = cI$ for some $c \in \mathbb{R}\setminus \{0\}$, then the operator $\mathcal{E}_{A}^1$ is unbounded on $L^p(\mathbb{R}^3)$ for $0<p<\infty$. Furthermore, if $A = $ for $a \in \mathbb{Z}$ and $c \in \mathbb{R}\setminus \{0\}$, then the operator $\mathcal

Theorems & Definitions (23)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.1
  • Proposition 2.1
  • Lemma 2.1
  • Proposition 3.1
  • Proposition 3.2
  • proof : Proof of (\ref{['p4']})
  • Lemma 4.1
  • ...and 13 more