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Noncommutative Bourgain's circular maximal theorem and a local smoothing estimate on the generalized Moyal planes

Guixiang Hong, Xudong Lai, Liang Wang

TL;DR

This work establishes a local smoothing estimate for the wave equation on two-dimensional quantum Euclidean space $\mathcal{R}_\theta^2$ and resolves the noncommutative analogue of Bourgain's circular maximal theorem. The authors develop a noncommutative transference technique to reduce to operator-valued Fourier multipliers, and prove scale-local $L_p$ bounds via Littlewood-Paley theory, operator-valued Sobolev embeddings, and Kakeya-type maximal inequalities. Central contributions include the first noncommutative 2D local smoothing result and a complete resolution of the noncommutative circular maximal inequality, underpinned by new geometric and analytic tools in the noncommutative setting. The approach combines transference, operator-valued harmonic analysis, and refined geometric estimates in $\mathbb{R}^3$, with potential implications for further noncommutative PDEs on quantum manifolds and related microlocal analysis.

Abstract

In this paper, we establish a local smoothing estimate on two-dimensional quantum Euclidean space. This is the noncommutative analogue of the one due to Mockenhaupt$-$Seeger$-$Sogge \cite{MSS}. As an application and simultaneously one motivation, we obtain the noncommutative analogue of Bourgain's circular maximal theorem, resolving one problem after \cite{Hong}.

Noncommutative Bourgain's circular maximal theorem and a local smoothing estimate on the generalized Moyal planes

TL;DR

This work establishes a local smoothing estimate for the wave equation on two-dimensional quantum Euclidean space and resolves the noncommutative analogue of Bourgain's circular maximal theorem. The authors develop a noncommutative transference technique to reduce to operator-valued Fourier multipliers, and prove scale-local bounds via Littlewood-Paley theory, operator-valued Sobolev embeddings, and Kakeya-type maximal inequalities. Central contributions include the first noncommutative 2D local smoothing result and a complete resolution of the noncommutative circular maximal inequality, underpinned by new geometric and analytic tools in the noncommutative setting. The approach combines transference, operator-valued harmonic analysis, and refined geometric estimates in , with potential implications for further noncommutative PDEs on quantum manifolds and related microlocal analysis.

Abstract

In this paper, we establish a local smoothing estimate on two-dimensional quantum Euclidean space. This is the noncommutative analogue of the one due to MockenhauptSeegerSogge \cite{MSS}. As an application and simultaneously one motivation, we obtain the noncommutative analogue of Bourgain's circular maximal theorem, resolving one problem after \cite{Hong}.
Paper Structure (18 sections, 32 theorems, 268 equations)

This paper contains 18 sections, 32 theorems, 268 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Noncommutative $L_p$ spaces
  4. Noncommutative $\ell_\infty$-valued $L_p$ spaces
  5. Quantum Euclidean spaces
  6. Proof of the main Theorems
  7. Proof of Theorem \ref{['theorem1']}
  8. Proof of Proposition \ref{['theorem2']}
  9. Proof of Theorem \ref{['thmloc']}
  10. Proof of Proposition \ref{['theorem3']}
  11. Proof of Proposition \ref{['key']}
  12. Some lemmas
  13. An $L_4$ estimate
  14. Proofs of Lemma \ref{['geo']} and Lemma \ref{['ka']}
  15. Proof of Lemma \ref{['geo']}
  16. ...and 3 more sections

Key Result

Proposition 1.1

With all the notions above,

Theorems & Definitions (49)

  • Proposition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3: PX03
  • Lemma 2.4: PX03
  • Lemma 2.5
  • ...and 39 more