arXiv subjects

Runlian Xia

Publications and source records attributed to Runlian Xia.

11 recordsLinked to original sources

arXiv2026

Hilbert transforms on graph products of finite von Neumann algebras

We study Hilbert transforms on graph products of finite von Neumann algebras, with particular interests on their boundedness on the associated noncommutative $L_p$-spaces for $1<p<\infty$. We establish a generalized Cotlar identity for Hilbert transforms, valid on operators whose lengths exceed a constant depending only on the underlying graph. We further prove that graph products of finite von Neumann algebras satisfying a Haagerup-type inequality admit $L_p$-bounded Hilbert transforms, therefore extending the corresponding result of Mei and Ricard for free products of finite von Neumann algebras. In addition, we obtain several equivalent characterizations of this Haagerup-type inequality and show, in particular, that it is equivalent to the graph product being generated by finite-dimensional von Neumann algebras with uniformly bounded dimensions. Our results apply, in particular, to graph products of finite groups, right-angled Hecke von Neumann algebras, and graph products of finite quantum groups. As an application, we provide positive answers to a compactness problem posed by Ozawa in the setting of graph products of finite groups and right-angled Hecke von Neumann algebras.

math.OA↗

arXiv2025

Ergodic Average Dominance for Unimodular Amenable Groups

In this paper we show that the ergodic averages of the action of any unimodular amenable group along certain F{\o}lner sequences can be dominated by the Ces\`aro means of a suitably constructed Markov operator, that is, the ergodic averages of an integer action. Moreover, the restriction on these F{\o}lner sequences are mild enough so that every two-sided F{\o}lner sequence has a subsequence satisfying these conditions. As a consequence of this inequality, we obtain the maximal and pointwise (individual) ergodic theorems for actions of unimodular amenable groups directly from the corresponding ergodic theorems for integer actions. This allows us to deal with the commutative and noncommutative ergodic theorems on an equal footing.

math.DS↗

arXiv2025

Hilbert transforms on Coxeter groups and groups acting on buildings

In this paper, we study Hilbert transforms and their boundedness on $L_p$-spaces associated with Coxeter groups and groups acting on buildings. We establish new models for Hilbert transforms on these groups through the geometric objects they act on, and we show that these Hilbert transforms satisfy a Cotlar identity which was developed in earlier work of Mei and Ricard, and that of Gonz\'alez-P\'erez, Parcet and the second named author, thus conclude the $L_p$-boundedness. We manage to solve the problem firstly for the case when the Coxeter group satisfies a certain nested condition, and then extend it to any finitely generated Coxeter groups and groups that admit actions on buildings satisfying the nested condition. Our results give the first example of groups with property (F$\mathbb{R}$) on which $L_p$-bounded Hilbert transforms can be defined, and generalize the work of Gonz\'alez-P\'erez, Parcet and the second named author for groups acting on simplicial trees.

math.FA↗

arXiv2025

Proper cocycles, measure equivalence and $L_p$-Fourier multipliers

We develop a new transference method for completely bounded $L_p$-Fourier multipliers via proper cocycles arising from probability measure-preserving group actions. This method extends earlier results by Haagerup and Jolissaint, which were limited to the case $p = \infty$. Based on this approach, we present a new and simple proof of the main result in [Hong-Wang-Wang, Mem. Amer. Math. Soc. 2024] regarding the pointwise convergence of noncommutative Fourier series on amenable groups, refining the associated estimate for maximal inequalities. In addition, this framework yields a transference principle for $L_p$-Fourier multipliers from lattices in linear Lie groups to their ambient groups, establishing a noncommutative analogue of Jodeit's theorem. As a further application, we construct a natural analogue of the Hilbert transform on $SL_2(\mathbb{R})$.

math.FA↗

arXiv2023

Asymmetric Burkholder inequalities in noncommutative symmetric spaces

In this paper, we establish noncommutative Burkholder inequalities with asymmetric diagonals in symmetric operator spaces. Our proof mainly relies on a new complex interpolation result on asymmetric vector valued spaces and a duality approach. We include as well the asymmetric versions of noncommutative Johnson-Schechtman inequalities.

math.OA↗

arXiv2022

Noncommutative Cotlar identities for groups acting on tree-like structures

Let $T_m$ be a noncommutative Fourier multiplier. In recent work, Mei and Ricard introduced a noncommutative analogue of Cotlar's identity in order to prove that certain multipliers are bounded on the non-commutative $L_p$-spaces of a free group. Here, we study Cotlar type identities in full generality, giving a closed characterization for them in terms of $m$: \[ \big( m(g h) - m(g) \big) \, \big( m(g^{-1}) - m(h) \big) = 0, \; \forall g \in \mathrm{G} \setminus \{e\}, h \in \mathrm{G}. \] We manage to prove, using a geometric argument, that if $X$ is a tree -- or more generally an $\mathbb{R}$-tree -- on which $\mathrm{G}$ acts and $m$ lifts to a function $\widetilde{m}: X \to \mathbb{C}$ that is constant on the connected subsets of $X \setminus \{x_0\}$, then $m$ satisfies Cotlar's identity and thus $T_m$ is bounded in $L_p$ for $1 < p < \infty$. This result establishes a new connection between groups actions on $\mathbb{R}$-trees and Fourier multipliers. We show that $m$ is trivial when the action has global fixed points. This machinery allows us to simultaneously generalize the free group transforms of Mei and Ricard and the theory of Hilbert transforms in left-orderable groups, which follows from Arveson's subdiagonal algebras. Using Bass-Serre theory, we construct new examples of Fourier multipliers in groups. Those include new families like Baumslag-Solitar groups. We also show that a natural Hilbert transform in $\mathrm{PSL}_2(\mathbb{C})$ satisfies Cotlar's identity when restricted to the Bianchi group $\mathrm{PSL}_2(\mathbb{Z}[\sqrt{-1}])$.

math.FA↗

arXiv2019

Algebraic Calderón-Zygmund theory

Calderón-Zygmund theory has been traditionally developed on metric measure spaces satisfying additional regularity properties. In the lack of good metrics, we introduce a new approach for general measure spaces which admit a Markov semigroup satisfying purely algebraic assumptions. We shall construct an abstract form of "Markov metric" governing the Markov process and the naturally associated BMO spaces, which interpolate with the Lp-scale and admit endpoint inequalities for Calderón-Zygmund operators. Motivated by noncommutative harmonic analysis, this approach gives the first form of Calderón-Zygmund theory for arbitrary von Neumann algebras, but is also valid in classical settings like Riemannian manifolds with nonnegative Ricci curvature or doubling/nondoubling spaces. Other less standard commutative scenarios like fractals or abstract probability spaces are also included. Among our applications in the noncommutative setting, we improve recent results for quantum Euclidean spaces and group von Neumann algebras, respectively linked to noncommutative geometry and geometric group theory.

math.FA↗

arXiv2018

Mapping properties of operator-valued pseudo-differential operators

In this paper, we investigate the mapping properties of pseudo-differential operators with operator-valued symbols. Thanks to the smooth atomic decomposition of the operator-valued Triebel-Lizorkin spaces $F_1^{α,c}(\mathbb{R}^d,\mathcal{M})$ obtained in our previous paper, we establish the $F_1^{α,c}$-regularity of regular symbols for every $α\in \mathbb{R}$, and the $F_1^{α,c}$-regularity of forbidden symbols for $α>0$. As applications, we obtain the same results on the usual and quantum tori.

math.OA↗

arXiv2018

Operator-valued Triebel-Lizorkin spaces

This paper is devoted to the study of operator-valued Triebel-Lizorkin spaces. We develop some Fourier multiplier theorems for square functions as our main tool, and then study the operator-valued Triebel-Lizorkin spaces on $\mathbb{R}^d$. As in the classical case, we connect these spaces with operator-valued local Hardy spaces via Bessel potentials. We show the lifting theorem, and get interpolation results for these spaces. We obtain Littlewood-Paley type, as well as the Lusin type square function characterizations in the general way. Finally, we establish smooth atomic decompositions for the operator-valued Triebel-Lizorkin spaces. These atomic decompositions play a key role in our recent study of mapping properties of pseudo-differential operators with operator-valued symbols.

math.OA↗

arXiv2018

Operator-valued local Hardy spaces

This paper gives a systematic study of operator-valued local Hardy spaces. These spaces are localizations of the Hardy spaces defined by Tao Mei, and share many properties with Mei's Hardy spaces. We prove the ${\rm h}_1$-$\rm bmo$ duality, as well as the ${\rm h}_p$-${\rm h}_q$ duality for any conjugate pair $(p,q)$ when $1<p< \infty$. We show that ${\rm h}_1(\mathbb{R}^d, \mathcal M)$ and ${\rm bmo}(\mathbb{R}^d, \mathcal M)$ are also good endpoints of $L_p(L_\infty(\mathbb{R}^d) \overline{\otimes} \mathcal M)$ for interpolation. We obtain the local version of Calderón-Zygmund theory, and then deduce that the Poisson kernel in our definition of the local Hardy norms can be replaced by any reasonable test function. Finally, we establish the atomic decomposition of the local Hardy space ${\rm h}_1^c(\mathbb{R}^d,\mathcal M)$.

math.FA↗

arXiv2016

Characterizations of operator-valued Hardy spaces and applications to harmonic analysis on quantum tori

This paper deals with the operator-valued Hardy spaces introduced and studied by Tao Mei. Our principal result shows that the Poisson kernel in Mei's definition of these spaces can be replaced by any reasonable test function. As an application, we get a general characterization of Hardy spaces on quantum tori. The latter characterization plays a key role in our recent study of Triebel-Lizorkin spaces on quantum tori.

math.OA↗