arXiv subjects

Zhenguo Wei

Publications and source records attributed to Zhenguo Wei.

4 recordsLinked to original sources

arXiv2024

On the boundedness and Schatten class property of noncommutative martingale paraproducts and operator-valued commutators

We study the Schatten class membership of semicommutative martingale paraproducts and use the transference method to describe Schatten class membership of purely noncommutative martingale paraproducts, especially for CAR algebras and $\mathop{\otimes}\limits_{k=1}^{\infty}\mathbb{M}_{d}$ in terms of martingale Besov spaces. Using Hytönen's dyadic martingale technique, we also obtain sufficient conditions on the Schatten class membership and the boundedness of operator-valued commutators involving general singular integral operators. We establish the complex median method, which is applicable to complex-valued functions. We apply it to get the optimal necessary conditions on the Schatten class membership of operator-valued commutators associated with non-degenerate kernels in Hytönen's sense. This resolves the problem on the characterization of Schatten class membership of operator-valued commutators. Our results are new even in the scalar case. Our new approach is built on Hytönen's dyadic martingale technique and the complex median method. Compared with all the previous ones, this new one is more powerful in several aspects: $(a)$ it permits us to deal with more general singular integral operators with little smoothness; $(b)$ it allows us to deal with commutators with complex-valued kernels; $(c)$ it goes much further beyond the scalar case and can be applied to the semicommutative setting. By a weak-factorization type decomposition, we get some necessary but not optimal conditions on the boundedness of operator-valued commutators. In addition, we give a new proof of the boundedness of commutators still involving general singular integral operators concerning $BMO$ spaces in the commutative setting.

math.FA↗

arXiv2024

Boundedness of operator-valued commutators involving martingale paraproducts

Let $1<p<\infty$. We show the boundedness of operator-valued commutators $[π_a,M_b]$ on the noncommutative $L_p(L_\infty(\mathbb{R})\otimes \mathcal{M})$ for any von Neumann algebra $\mathcal{M}$, where $π_a$ is the $d$-adic martingale paraproduct with symbol $a\in BMO^d(\mathbb{R})$ and $M_b$ is the noncommutative left multiplication operator with $b\in BMO^d_\mathcal{M}(\mathbb{R})$. Besides, we consider the extrapolation property of semicommutative $d$-adic martingale paraproducts in terms of the $BMO^d_\mathcal{M}(\mathbb{R})$ space.

math.OA↗

arXiv2024

On the best constants of the noncommutative Littlewood-Paley-Stein inequalities

Let $1 0}$ be a noncommutative symmetric diffusion semigroup on a semifinite von Neumann algebra $\mathcal{M}$, and let $\{P_t\}_{t>0}$ be its associated subordinated Poisson semigroup. The celebrated noncommutative Littlewood-Paley-Stein inequality asserts that for any $x\in L_p(\mathcal{M})$, \begin{equation*} α_p^{-1}\|x\|_{p}\le \|x\|_{p,P}\le β_p \|x\|_{p}, \end{equation*} where $\|\cdot\|_{p,P}$ is the $L_p(\mathcal{M})$-norm of square functions associated with $\{P_t\}_{t>0}$, and $α_p, β_p$ are the best constants only depending on $p$. We show that as $p\to \infty$, $$ β_p\lesssim p, $$ and $p$ is the optimal possible order of $β_p$ as well. We also obtain some lower and upper bounds of $α_p$ and $β_p$ in the other cases.

math.OA↗

arXiv2024

Complex median method and Schatten class membership of commutators

This article is devoted to the study of the Schatten class membership of commutators involving singular integral operators. We utilize martingale paraproducts and Hytönen's dyadic martingale technique to obtain sufficient conditions on the weak-type and strong-type Schatten class membership of commutators in terms of Sobolev spaces and Besov spaces respectively. We also establish the complex median method, which is applicable to complex-valued functions. We apply it to get the optimal necessary conditions on the weak-type and strong-type Schatten class membership of commutators associated with non-degenerate kernels. This resolves the problem on the characterization of the weak-type and strong-type Schatten class membership of commutators. Our new approach is built on Hytönen's dyadic martingale technique and the complex median method. Compared with all the previous ones, this new one is more powerful in several aspects: $(a)$ it permits us to deal with more general singular integral operators with little smoothness; $(b)$ it allows us to deal with commutators with complex-valued kernels; $(c)$ it turns out to be powerful enough to deal with the weak-type and strong-type Schatten class of commutators in a universal way.

math.FA↗