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

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.

arXiv comment37 pages, 2 figures, to appear in Comm. Math. Phys

References in corpus (9)

• Noncommutative vector valued $L_p$-spaces and completely $p$-summing maps
• Harmonic analysis on quantum tori
• Quantum differentiability on quantum tori
• Quantum differentiability on noncommutative Euclidean spaces
• Square function estimates for the Bochner-Riesz means
• Noncommutative maximal ergodic inequalities associated with doubling conditions
• Harmonic analysis approach to Gromov--Hausdorff convergence for noncommutative tori
• Complete boundedness of the Heat Semigroups on the von Neumann Algebra of hyperbolic groups
• Pointwise convergence of noncommutative Fourier series