Sharp estimates of noncommutative Bochner-Riesz means on two-dimensional quantum tori
Abstract
In this paper, we establish the full Lp boundedness of noncommutative Bochner-Riesz means on two-dimensional quantum tori, which completely resolves an open problem raised in CXY13 in the sense of the Lp 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.
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