arXiv subjects

Zhendong Xu

Publications and source records attributed to Zhendong Xu.

6 recordsLinked to original sources

arXiv2026

Hypercontractivity for a family of quantum Ornstein-Uhlenbeck semigroups

We show that a family of quantum Ornstein-Uhlenbeck semigroups is hypercontractive. We also obtain the optimal order of the optimal time up to a constant. The main ingredient of our proof is Meixner polynomials. The goal of this paper is twofold: to provide more examples of hypercontractive quantum Markov semigourps on non-tracial von Neumann algebras, and to determine the optimal order of the optimal time for quantum Ornstein-Uhlenbeck semigroups.

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arXiv2026

Dimension-free Riesz transform estimates on biased hypercubes via non-tracial baby Fock models

We study dimension-free Riesz transform estimates on hypercubes equipped with non-uniform product measures. For $2\leq p<\infty$, we prove a dimension-free Meyer inequality for the associated carré du champ, with constants independent of both the dimension and the product measure. In contrast to the uniform case, the carré-du-champ square function does not in general coincide with the unweighted Riesz square function. Under an additional boundedness assumption on the weight parameters, we further obtain dimension-free estimates for the latter. Our proof is based on a transference from the biased hypercube to a non-tracial baby Fock model. We introduce new annihilation operators and the associated Riesz transforms on the biased hypercube, and derive explicit formulas relating them to the Riesz transforms on the non-tracial baby Fock model. The main analytic ingredient is a dimension-free $L_p$-estimate for the resulting noncommutative Riesz transform, proved by a rotation argument in the spirit of Pisier and Lust-Piquard. The same analytic mechanism also yields dimension-free Riesz transform estimates on $q$-Araki--Woods algebras.

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arXiv2026

The Endpoint Fractional Riesz Estimate on the Hamming Cube

Let $1<p<2$ and $D_j$ be the discrete partial derivative on the Hamming cube $Ω_n = \{ -1, 1\}^n$. Let $Δ=\sum_{j=1}^nD_j$ be the discrete Laplacian.We prove the endpoint inequality \[ \left\|\left(\sum_{j=1}^n|D_jf|^2\right)^{1/2}\right\|_{p} \lesssim_p\|Δ^{1/p}f\|_{p}, \quad \forall f:Ω_n \to \mathbb C. \] This result answers the conjecture proposed by Naor, Eskenazis and Ivanisvili (see [BenEfraimLustPiquard, IvanisviliVolberg] or [Remark 45, EskenazisIvanisvili]). The proof relies heavily on the noncommutative semigroup BMO theory [JungeMei].

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arXiv2024

Best constants in the vector-valued Littlewood-Paley-Stein theory

Let $L$ be a sectorial operator of type $α$ ($0 \leq α< π/2$) on $L^2(\mathbb{R}^d)$ with the kernels of $\{e^{-tL}\}_{t>0}$ satisfying certain size and regularity conditions. Define $$ S_{q,L}(f)(x) = \left(\int_0^{\infty}\int_{|y-x| < t} \|tL{e^{-tL}} (f)(y) \|_X^q \,\frac{{\rm d} y{\rm d} t}{t^{d+1}} \right)^{\frac{1}{q}},$$ $$G_{q,{L}}(f)=\left( \int_0^{\infty} \left\|t{L}{e^{-t{L}}} (f)(y) \right\|_X^q \,\frac{{\rm d} t}{t}\right)^{\frac{1}{q}}.$$ We show that for $\underline{\mathrm{any}}$ Banach space $X$, $1 \leq p < \infty$ and $1 < q < \infty$ and $f\in C_c(\mathbb R^d)\otimes X$, there hold \begin{align*} p^{-\frac{1}{q}}\| S_{q,{\sqrtΔ}}(f) \|_p \lesssim_{d, γ, β} \| S_{q,L}(f) \|_p \lesssim_{d, γ, β} p^{\frac{1}{q}}\| S_{q,{\sqrtΔ}}(f) \|_p, \end{align*} \begin{align*} p^{-\frac{1}{q}}\| S_{q,L}(f) \|_p \lesssim_{d, γ, β} \| G_{q,L}(f) \|_p \lesssim_{d, γ, β} p^{\frac{1}{q}}\| S_{q,L}(f) \|_p, \end{align*} where $Δ$ is the standard Laplacian; moreover all the orders appeared above are {\it optimal} as $p\rightarrow1$. This, combined with the existing results in [29, 33], allows us to resolve partially Problem 1.8, Problem A.1 and Conjecture A.4 regarding the optimal Lusin type constant and the characterization of martingale type in a recent remarkable work due to Xu [48]. Several difficulties originate from the arbitrariness of $X$, which excludes the use of vector-valued Calderón-Zygmund theory. To surmount the obstacles, we introduce the novel vector-valued Hardy and BMO spaces associated with sectorial operators; in addition to Mei's duality techniques and Wilson's intrinsic square functions developed in this setting, the key new input is the vector-valued tent space theory and its unexpected amalgamation with these `old' techniques.

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arXiv2023

Multiplication between elements in Martingale Hardy spaces and their duals

In this paper, we establish continuous bilinear decompositions that arise in the study of products between elements in martingale Hardy spaces $ H^p\ (0<p\leqslant 1) $ and functions in their dual spaces. Our decompositions are based on martingale paraproducts. As a consequence of our work, we also obtain analogous results for dyadic martingales on spaces of homogeneous type equipped with a doubling measure.

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arXiv2021

From the Littlewood-Paley-Stein Inequality to the Burkholder-Gundy Inequality

Let $\{\mathsf{T}_t\}_{t>0}$ be a symmetric diffusion semigroup on a $σ$-finite measure space $(Ω, \mathscr{A}, μ)$ and $G^{\mathsf{T}}$ the associated Littlewood-Paley $g$-function operator: $$G^{\mathsf{T}}(f)=\Big(\int_0^\infty \left|t\frac{\partial}{\partial t} \mathsf{T}_t(f)\right|^2\frac{\mathrm{d}t}{t}\Big)^{\frac12}.$$ The classical Littlewood-Paley-Stein inequality asserts that for any $1 0}$ of $L_p(Ω)$. Recently, Xu proved that $ \mathsf{L}^{\mathsf{T}}_{ p}\lesssim p$ as $p\rightarrow\infty$, and raised the problem abut the optimal order of $ \mathsf{L}^{\mathsf{T}}_{ p}$ as $p\rightarrow\infty$. We solve Xu's open problem by showing that this upper estimate of $\mathsf{L}^{\mathsf{T}}_{ p}$ is in fact optimal. Our argument is based on the construction of a special symmetric diffusion semigroup associated to any given martingale such that its square function $G^{\mathsf{T}}(f)$ for any $f\in L_p(Ω)$ is pointwise comparable with the martingale square function of $f$. Our method also extends to the vector-valued and noncommutative setting.

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