Sharp estimates of noncommutative Bochner-Riesz means on two-dimensional quantum tori

TL;DR

This paper proves the boundedness of noncommutative Bochner-Riesz means on two-dimensional quantum tori, resolving a key open problem by employing sharp Kakeya maximal function estimates and advanced noncommutative analysis techniques.

Contribution

It provides the first complete $L_p$ boundedness results for noncommutative Bochner-Riesz means on 2D quantum tori, advancing noncommutative harmonic analysis.

Findings

Established $L_p$ boundedness of noncommutative Bochner-Riesz means in 2D.

Developed sharp estimates for noncommutative Kakeya maximal functions.

Utilized microlocal decompositions for improved analysis.

Abstract

In this paper, we establish the full $L_p$ boundedness of noncommutative Bochner-Riesz means on two-dimensional quantum tori, which completely resolves an open problem raised in \cite{CXY13} in the sense of the $L_p$ convergence for two dimensions. The main ingredients are sharp estimates of noncommutative Kakeya maximal functions and geometric estimates in the plane. We make the most of noncommutative theories of maximal/square functions, together with microlocal decompositions in both proofs of sharper estimates of Kakeya maximal functions and Bochner-Riesz means.