Searcharxiv⌕ Search

arXiv subjects

Zhendong Xu

Publications and source records attributed to Zhendong Xu.

5 recordsLinked to original sources
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 $\Omega_n = \{ -1, 1\}^n$. Let $\Delta=\sum_{j=1}^nD_j$ be the discrete Laplacian.We prove the endpoint inequality \[ \left\|\left(\sum_{j=1}^n|D_jf|^2\right)^{1/2}\right\|_{p} \lesssim_p\|\Delta^{1/p}f\|_{p}, \quad \forall f:\Omega_n \to \mathbb C. \] 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].

math.FA↗
arXiv2024

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

Let $L$ be a sectorial operator of type $\alpha$ ($0 \leq \alpha < \pi/2$) 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,{\sqrt{\Delta}}}(f) \|_p \lesssim_{d, \gamma, \beta} \| S_{q,L}(f) \|_p \lesssim_{d, \gamma, \beta} p^{\frac{1}{q}}\| S_{q,{\sqrt{\Delta}}}(f) \|_p, \end{align*} \begin{align*} p^{-\frac{1}{q}}\| S_{q,L}(f) \|_p \lesssim_{d, \gamma, \beta} \| G_{q,L}(f) \|_p \lesssim_{d, \gamma, \beta} p^{\frac{1}{q}}\| S_{q,L}(f) \|_p, \end{align*} where $\Delta$ 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 Calder\'on-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.

math.FA↗
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.

math.FA↗
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.

math.FA↗