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

TL;DR

This paper extends classical harmonic analysis results to noncommutative geometry by establishing a local smoothing estimate on quantum Euclidean space and proving a noncommutative version of Bourgain's circular maximal theorem.

Contribution

It introduces a noncommutative analogue of Bourgain's theorem and a local smoothing estimate on generalized Moyal planes, advancing noncommutative harmonic analysis.

Findings

Established a local smoothing estimate on quantum Euclidean space.

Proved a noncommutative analogue of Bourgain's circular maximal theorem.

Resolved an open problem in noncommutative harmonic 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}.