arXiv subjects

Sijie Luo

Publications and source records attributed to Sijie Luo.

8 recordsLinked to original sources

arXiv2026

Sharp Fractional Riesz Estimates on the Hypercube

Let $\Omega_{n}=\{-1,1\}^n$ be the $n$-dimensional hypercube equipped with the normalized uniform measure, let $\nabla$ be the Walsh gradient and let $\Delta$ be the Walsh Laplacian. For every $1<p\leq 2$ we prove the following estimate \[ \|\nabla f\|_{L_p(\Omega_n;\ell_2^n)} \leq c_{{\rm abs}}(p-1)^{-2}\|\Delta^{1/p}f\|_{L_p(\Omega_n)}. \] The exponent $\frac1p$ is optimal, thus this settles the open problem on the sharp fractional Riesz estimate by Efraim and Lust-Piquard \cite{E-LP2008} which was subsequently highlighted by Ivanisvili and Volberg \cite{I-V2022}. We also establish the higher-order counterpart. As applications of our results, we obtain simpler proofs of the optimal short-time estimate for $\nabla e^{-t\Delta}$, and the Bernstein-Markov type inequality for $d$-bounded degree functions.

math.FA↗

arXiv2026

Large Deviation Inequalities for Noncommutative Martingales

We establish noncommutative analogs of some well-known large deviation inequalities for noncommutative random variables. Firstly, for the noncommutative independent case, we characterize the uniformly exponential integrability of random variables in terms of large deviation inequalities. Secondly, for noncommutative martingale differences, we establish two deviation inequalities according to the exponential integrability and $L_{p}$-boundedness of the martingale differences, respectively. Finally, we establish a noncommutative version of Gordin's decomposition, which enables us to derive a noncommutative ergodic theorem via deviation inequalities for noncommutative martingales.

math.OA↗

arXiv2025

Almost uniform convergence for noncommutative Vilenkin-Fourier series

In the present paper, we study almost uniform convergence for noncommutative Vilenkin-Fourier series. Precisely, we establish several noncommutative (asymmetric) maximal inequalities for the Ces\`{a}ro means of the noncommutative Vilenkin-Fourier series, which in turn give the corresponding almost uniform convergence. The primary strategy in our proof is to explore a noncommutative generalization of Sunouchi square function operator, and the very recent advance of the noncommutative Calder\'{o}n-Zygmund decomposition.

math.FA↗

arXiv2024

Quantum KKL-type Inequalities Revisited

In the present paper, we develop the random restriction method in the quantum framework. By applying this method, we establish the quantum Eldan-Gross inequality, the quantum Talagrand isoperimetric inequality, and related quantum KKL-type inequalities. Our results recover some recent results of Rouzé et al. \cite{RWZ2024} and Jiao et al. \cite{JLZ2025}, which can be viewed as alternative answers to the quantum KKL conjecture proposed by Motanaro and Osborne in \cite{MO2010}.

math.FA↗

arXiv2024

Change-Aware Siamese Network for Surface Defects Segmentation under Complex Background

Despite the eye-catching breakthroughs achieved by deep visual networks in detecting region-level surface defects, the challenge of high-quality pixel-wise defect detection remains due to diverse defect appearances and data scarcity. To avoid over-reliance on defect appearance and achieve accurate defect segmentation, we proposed a change-aware Siamese network that solves the defect segmentation in a change detection framework. A novel multi-class balanced contrastive loss is introduced to guide the Transformer-based encoder, which enables encoding diverse categories of defects as the unified class-agnostic difference between defect and defect-free images. The difference presented by a distance map is then skip-connected to the change-aware decoder to assist in the location of both inter-class and out-of-class pixel-wise defects. In addition, we proposed a synthetic dataset with multi-class liquid crystal display (LCD) defects under a complex and disjointed background context, to demonstrate the advantages of change-based modeling over appearance-based modeling for defect segmentation. In our proposed dataset and two public datasets, our model achieves superior performances than the leading semantic segmentation methods, while maintaining a relatively small model size. Moreover, our model achieves a new state-of-the-art performance compared to the semi-supervised approaches in various supervision settings.

cs.CV↗

arXiv2024

On noncommutative Hölder inequality of Sukochev and Zanin for weak Schatten class

Sukochev and Zanin resolved an open problem due to B. Simon concerning optimal constants in Hölder inequality for the weak Schatten classes of compact operators. In this note we observe that these constants, by introducing the modified weak Schatten quasi-norms, can be renormalised so that the original Simon's conjecture (with optimal constant 1) does hold. We also provide an unexpectedly simple proof for the modified Hölder inequality and its sharpness.

math.FA↗

arXiv2021

On Azuma-type inequalities for Banach space-valued martingales

In this paper, we will study concentration inequalities for Banach space-valued martingales. Firstly, we prove that a Banach space $X$ is linearly isomorphic to a $p$-uniformly smooth space ($1<p\leq 2$) if and only if an Azuma-type inequality holds for $X$-valued martingales. This can be viewed as a generalization of Pinelis' work on Azuma inequality for martingales with values in $2$-uniformly smooth space. Secondly, Azuma-type inequality for self-normalized sums will be presented. Finally, some further inequalities for Banach space-valued martingales, such as moment inequalities for double indexed dyadic martingales and the De la Peña-type inequalities for conditionally symmetric martingales, will also be discussed.

math.FA↗

arXiv2020

On order preserving and order reversing mappings defined on cones of convex functions

In this paper, we first show that for a Banach space $X$ there is a fully order reversing mapping $T$ from ${\rm conv}(X)$ (the cone of all extended real-valued lower semicontinuous proper convex functions defined on $X$) onto itself if and only if $X$ is reflexive and linearly isomorphic to its dual $X^*$. Then we further prove the following generalized ``Artstein-Avidan-Milman'' representation theorem: For every fully order reversing mapping $T:{\rm conv}(X)\rightarrow {\rm conv}(X)$ there exist a linear isomorphism $U:X\rightarrow X^*$, $x_0^*, \;φ_0\in X^*$, $α>0$ and $r_0\in\mathbb R$ so that \begin{equation}\nonumber (Tf)(x)=α(\mathcal Ff)(Ux+x^*_0)+\langleφ_0,x\rangle+r_0,\;\;\forall x\in X, \end{equation} where $\mathcal F: {\rm conv}(X)\rightarrow {\rm conv}(X^*)$ is the Fenchel transform. Hence, these resolve two open questions. We also show several representation theorems of fully order preserving mappings defined on certain cones of convex functions. For example, for every fully order preserving mapping $S:{\rm semn}(X)\rightarrow {\rm semn}(X)$ there is a linear isomorphism $U:X\rightarrow X$ so that \begin{equation}\nonumber (Sf)(x)=f(Ux),\;\;\forall f\in{\rm semn}(X),\;x\in X, \end{equation} where ${\rm semn}(X)$ is the cone of all lower semicontinuous seminorms on $X$.

math.FA↗