arXiv subjects

Chian Yeong Chuah

Publications and source records attributed to Chian Yeong Chuah.

9 recordsLinked to original sources

arXiv2026

Sharp Operator Khintchine inequality and $Z_2$ Property

We study the $p=1$ operator Khintchine inequality associated with orthonormal systems and establish two quantitative implications relating its optimal constant $A_1$ to the $Z_2$ property. For an orthonormal system $W$ with finite $Z_2(W)$, we prove $A_1(W)\leq\sqrt{1+Z_2(W)}$. Conversely, there exists an absolute constant $δ>0$ such that, for canonical group unitaries indexed by any subset $V$ of a discrete group, $A_1(V)<\sqrt2+δ$ implies $Z_2(V)\leq2$. The converse is detected by $2\times2$ matrix coefficients supported on at most six group elements. For the classical Rademacher sequence, we determine the sharp operator Khintchine constant $A_1=\sqrt2$, answering the question left open by Haagerup and Musat~\cite{Haagerup2007}. Since this sequence has $Z_2=2$, the conclusion $Z_2(V)\leq2$ is optimal.

math.FA↗

arXiv2026

Extensions of the Furstenberg-Sárközy theorem via the arithmetic level-$d$ inequality

Green and Sawhney recently obtained a quasipolynomial bound in the Furstenberg--Sárközy theorem for square differences by proving an ``arithmetic level-d'' inequality, thereby yielding a greatly improved density increment scheme. We apply their method to treat general intersective polynomials $h\in\mathbb{Z}[x]$. In particular, let \[ D(h(\mathbb{N}),X):= \max{|A|:\ A\subseteq [1,X]\cap\mathbb{N} \text{and}\ (A-A)\cap h(\mathbb{N})\subseteq\{0\}}. \] We prove that for every $0<μ<1/2$ there are constants $c_0, X_{\text{min}}>0$ depending on $h$ and $μ$ such that for every $X>X_{\text{min}}$, \[D(h(\mathbb{N}), X)\leq Xe^{-c_0(\log X)^μ}.\] This is the best quantitative upper bound presently known for sets lacking intersective polynomial differences, improving upon the work of Arala. In order to achieve the admissible exponent range $0<μ<1/2$, we use sieve methods to develop novel exponential sum estimates in the style of Rice, and we use the ``random sparsification'' procedure of Green and Sawhney.

math.NT↗

arXiv2025

A Classification Theorem on Non-compact Embeddings between Besov Spaces

We analyze the embedding properties between Besov spaces, defined on the total space $\mathbb R^n$ and on bounded domains. We give a complete classification on whether or not these embedding maps satisfy certain weak compactness characterized by the so-called strictly and finitely strictly singular condition. The result extends the recent findings on Sobolev embeddings by offering a refined description of the quality of non-compactness in the setting of Besov spaces.

math.FA↗

arXiv2025

A Bourgain-Gromov problem on non-compact Sobolev-Lorentz embeddings

We study the non-compact Sobolev embeddings into the optimal scale of Lorentz spaces, $W_0^mL^{p,q}(Ω) \to L^{\frac{dp}{d - mp},r}(Ω)$, where $Ω\subseteq \mathbb{R}^d$, $1 \le m \le d$ and $0<q<r\le\infty$ with $1<p<\frac dm$ or $p=q=1$. We show that these embeddings are finitely strictly singular with certain upper bounds on the decay rate of the Bernstein numbers. We reduce the Sobolev embeddings to embeddings of Besov spaces and sequence spaces, which simplifies the previous methods by Bourgain-Gromov and Lang-Mihula.

math.FA↗

arXiv2024

Superselection sectors for posets of von Neumann algebras

We study a commutant-closed collection of von Neumann algebras acting on a common Hilbert space indexed by a poset with an order-reversing involution. We give simple geometric axioms for the poset which allow us to construct a braided tensor category of superselection sectors analogous to the construction of Gabbiani and Fröhlich for conformal nets. For cones in $\mathbb{R}^2$, we weaken our conditions to a bounded spread version of Haag duality and obtain similar results. We show that intertwined nets of algebras have isomorphic braided tensor categories of superselection sectors. Finally, we show that the categories constructed here are equivalent to those constructed by Naaijkens and Ogata for certain 2D quantum spin systems.

math.OA↗

arXiv2024

Quality of non-compactness for Sobolev Embedding with one point non-compactness

This paper investigates instances of Sobolev embeddings characterized by local compactness at every point within their domain, except for a single point. We obtain the sharp conditions that distinguish compactness from non-compactness and observe that in the context of Sobolev embeddings, non-compactness occurring at only one point within the domain could give rise to an infinite-dimensional subspace where the embedding is invertible (i.e., not strictly singular). Furthermore, we establish lower bounds for the Bernstein numbers, entropy numbers, and the measure of non-compactness.

math.FA↗

arXiv2023

Boundary algebras of the Kitaev Quantum Double model

The recent article [arXiv:2307.12552] gave local topological order (LTO) axioms for a quantum spin system, showed they held in Kitaev's Toric Code and in Levin-Wen string net models, and gave a bulk boundary correspondence to describe bulk excitations in terms of the boundary net of algebras. In this article, we prove the LTO axioms for Kitaev's Quantum Double model for a finite group $G$. We identify the boundary nets of algebras with fusion categorical nets associated to $(\mathsf{Hilb}(G),\mathbb{C}[G])$ or $(\mathsf{Rep}(G),\mathbb{C}^G)$ depending on whether the boundary cut is rough or smooth respectively. This allows us to make connections to work of Ogata on the type of the cone von Neumann algebras in the algebraic quantum field theory approach to topological superselection sectors. We show that the boundary algebras can also be calculated from a trivial $G$-symmetry protected topological phase ($G$-SPT), and that the gauging map preserves the boundary algebras. Finally, we compute the boundary algebras for the (3+1)D Quantum Double model associated to an abelian group.

cond-mat.str-el↗

arXiv2023

Characterization of positive definite, radial functions on free groups

This article studies the properties of positive definite, radial functions on free groups following the work of Haagerup and Knudby . We obtain characterizations of radial functions with respect to the $\ell^{2}$ length on the free groups with infinite generators and the characterization of the positive definite, radial functions with respect to the $\ell^{p}$ length on the free real line with infinite generators for $0 < p \leq 2$. We obtain the Levi-Khintchine formulas for length-radial conditionally negative functions as well.

math.FA↗

arXiv2019

On Weighted Hardy-Type Inequalities

We revisit weighted Hardy-type inequalities employing an elementary ad hoc approach that yields explicit constants. We also discuss the infinite sequence of power weighted Birman-Hardy-Rellich-type inequalities and derive an operator-valued version thereof.

math.CA↗