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Register a new userSharp estimates of noncommutative Bochner-Riesz means on two-dimensional quantum tori
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Authors
Xudong Lai
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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 sharper estimates of noncommutative Kakeya maximal functions and geometric estimates in the plain. 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. We point out that even geometric estimates in the plain are different from that in the commutative case.
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We conjecture an explicit formula for the higher dimensional Dirichlet character; the formula is based on the K-theory of the so-called noncommutative tori. It is proved, that our conjecture is true for the two-dimensional and one-dimensional (degenerate) noncommutative tori; in the second case, one gets a noncommutative analog of the Artin reciprocity law.
For $ 0< lambda < frac{1}2$, let $ B_{lambda }$ be the Bochner-Riesz multiplier of index $ lambda $ on the plane. Associated to this multiplier is the critical index $1 < p_lambda = frac{4} {3+2 lambda } < frac{4}3$. We prove a sparse bound for $ B_{lambda }$ with indices $ (p_lambda , q)$, where $ p_lambda < q < 4$. This is a further quantification of the endpoint weak $L^{p_lambda}$ boundedness of $ B_{lambda }$, due to Seeger. Indeed, the sparse bound immediately implies new endpoint weighted weak type estimates for weights in $ A_1 cap RH_{rho }$, where $ rho > frac4 {4 - 3 p_{lambda }}$.
We prove some new $L^p$ estimates for maximal Bochner-Riesz operator in the plane.
We provide a full characterisation of quantum differentiability (in the sense of Connes) on quantum tori. We also prove a quantum integration formula which differs substantially from the commutative case.
Let $A$ be a finite subdiagonal algebra in Arvesons sense. Let $H^p(A)$ be the associated noncommutative Hardy spaces, $0<ple8$. We extend to the case of all positive indices most recent results about these spaces, which include notably the Riesz, Szego and inner-outer type factorizations. One new tool of the paper is the contractivity of the underlying conditional expectation on $H^p(A)$ for $p<1$.
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