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Guixiang Hong

Publications and source records attributed to Guixiang Hong.

At least 19 recordsLinked to original sources
arXiv2026

Pointwise convergence of noncommutative ergodic averages along the primes

Let $\mathcal N$ be a von Neumann algebra equipped with a normal faithful semifinite trace, and let $\gamma$ be a trace-preserving automorphism of $\mathcal N$. We consider the ergodic averages along the prime numbers \[ A_N(x) := \frac1{|P_N|} \sum_{q\in P_N}\gamma^q(x), \qquad P_N:=\{q\leq N:q\ \text{is prime}\}. \] For every $1<p<\infty$, we prove a strong maximal inequality for $(A_N)_{N\geq2}$ on $L_p(\mathcal N)$ and that $A_N(x)$ converges bilaterally almost uniformly for every $x\in L_p(\mathcal N)$. The proof exploits the circle method and a noncommutative sampling principle. For the convergence result, Bourgain's commutative argument uses pointwise maximal functions and exceptional sets. These tools are not available in the noncommutative setting. Instead, we show that the tails of the ergodic averages tend to zero in $L_2(\mathcal N;\ell_\infty)$ and that the difference from the limit belongs to $L_2(\mathcal N;c_0)$. This gives the desired b.a.u. convergence, and provides a positive answer to one question left open in \cite{ChenHongWang+arXiv2024}.

math.OA↗
arXiv2026

The nonlinear estimates on quantum Besov spaces

The superposition operators have been widely studied in nonlinear analysis, which are essential for the well-posedness theory of nonlinear equations. In this paper, we investigate the boundedness estimates of superposition operators with non-smooth symbols on quantum Besov spaces, which significantly generalize McDonald's results \cite{McNLE} for infinitely differentiable symbols and have rich applications in the well-posedness theory of noncommutative PDEs. The ingredients in the proof involve a novel quantum chain rule and nonlinear interpolation. As a byproduct, we prove the equivalence of the two descriptions of quantum Besov spaces, resolving the conjecture proposed in \cite[Remark 3.16]{McNLE}.

math.FA↗
arXiv2026

Lamperti Operators, Dilation Theory, and Applications in Noncommutative Ergodic Theory

In this paper, we develop a novel framework for quantitative mean ergodic theorems in the noncommutative setting, with a focus on actions of amenable groups and semigroups. We prove square function inequalities for ergodic averages arising from actions of groups of polynomial volume growth on a fixed noncommutative $L_p$-space for $1<p<\8$. To achieve this, we establish two endpoint estimates for a noncommutative square function on non-homogeneous space. Our approach relies on semi-commutative non-homogeneous harmonic analysis, including the non-doubling Calder\'on-Zygmund arguments for non-smooth kernels and $\mathrm{BMO}$ space theory, operator-valued inequalities related to balls and cubes in groups equipped with non-doubling measures, and a noncommutative generalization of the classical transference method for amenable group actions. As an application, we establish a quantitative ergodic theorem for the ergodic averages associated with the positive power of modulus representation arising from a Lamperti representation on noncommutative $L_p$-spaces, extending some results in \cite{Templeman2015}. To obtain quantitative ergodic theorem for semigroups of operators, in this paper, we address the open question of extending dilation theorem of Fackler-Gl\"uck from single operators to commuting tuples on Banach spaces including noncommutative $L_p$-spaces. Indeed our approach provides genuine joint $N$-dilations for commuting families, unifying and extending the classical dilation theorems of Sz.-Nagy--Foia\c{s} and Ak\c{c}oglu--Sucheston for a natural class of commuting tuple of contractions extending the abstract dilation theorem of of Fackler-Gl\"uck for commuting tuple of contractions. This enables us to obtain a quantitative ergodic theorem for a large class of semigroups of operators on $\mathbb{R}^d_{+}$.

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arXiv2025

Navier-Stokes Equations on Quantum Euclidean Spaces

We investigate in the present paper the Navier-Stokes equations on quantum Euclidean spaces $\mathbb{R}^d_{\theta}$ with $\theta$ being a $d\times d$ antisymmetric matrix, which is a standard example of non-compact noncommutative manifolds. The quantum analogues of Ladyzhenskaya and Kato's results are established, that is, we obtain the global well-posedness in the 2D case and the local well-posedness with solution in $L_d(\mathbb{R}^d)$ in higher dimensions. To achieve these optimal results, we develop the related theory of harmonic analysis and function spaces on $\mathbb{R}^d_{\theta}$, and apply the sharp estimates around noncommutative $L_p$-spaces to quantum Navier-Stokes equations. Moreover, our techniques, which are independent of the deformed parameter $\theta$, allow us to conclude some results on the semiclassical limits. This is the first instance of systematical applications to the theory of quantum partial differential equations of the powerful real analysis techniques around noncommutative $L_p$-spaces, which date back to the seminal work \cite{PiXu97} in 1997 on noncommutative martingale inequalities. As in classical case, one may expect numerous similar applications in the future.

math.FA↗
arXiv2024

Campanato spaces via quantum Markov semigroups on finite von Neumann algebras

We study the Campanato spaces associated with quantum Markov semigroups on a finite von Neumann algebra $\mathcal M$. Let $\mathcal T=(T_{t})_{t\geq0}$ be a Markov semigroup, $\mathcal P=(P_{t})_{t\geq0}$ the subordinated Poisson semigroup and $\alpha>0$. The column Campanato space ${\mathcal{L}^{c}_{\alpha}(\mathcal{P})}$ associated to $\mathcal P$ is defined to be the subset of $\mathcal M$ with finite norm which is given by \begin{align*} \|f\|_{\mathcal{L}^{c}_{\alpha}(\mathcal{P})}=\left\|f\right\|_{\infty}+\sup_{t>0}\frac{1}{t^{\alpha}}\left\|P_{t}|(I-P_{t})^{[\alpha]+1}f|^{2}\right\|^{\frac{1}{2}}_{\infty}. \end{align*} The row space ${\mathcal{L}^{r}_{\alpha}(\mathcal{P})}$ is defined in a canonical way. In this article, we will first show the surprising coincidence of these two spaces ${\mathcal{L}^{c}_{\alpha}(\mathcal{P})}$ and ${\mathcal{L}^{r}_{\alpha}(\mathcal{P})}$ for $0<\alpha<2$. This equivalence of column and row norms is generally unexpected in the noncommutative setting. The approach is to identify both of them as the Lipschitz space ${\Lambda_{\alpha}(\mathcal{P})}$. This coincidence passes to the little Campanato spaces $\ell^{c}_{\alpha}(\mathcal{P})$ and $\ell^{r}_{\alpha}(\mathcal{P})$ for $0<\alpha<\frac{1}{2}$ under the condition $\Gamma^{2}\geq0$. We also show that any element in ${\mathcal{L}^{c}_{\alpha}(\mathcal{P})}$ enjoys the higher order cancellation property, that is, the index $[\alpha]+1$ in the definition of the Campanato norm can be replaced by any integer greater than $\alpha$. It is a surprise that this property holds without further condition on the semigroup. Lastly, following Mei's work on BMO, we also introduce the spaces ${\mathcal{L}^{c}_{\alpha}(\mathcal{T})}$ and explore their connection with ${\mathcal{L}^{c}_{\alpha}(\mathcal{P})}$. All the above-mentioned results seem new even in the (semi-)commutative case.

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arXiv2024

A noncommutative maximal inequality for Fej\'{e}r means on totally disconnected non-abelian groups

In this paper, we explore Fourier analysis for noncommutative $L_p$ space-valued functions on $G$, where $G$ is a totally disconnected non-abelian compact group. By additionally assuming that the value of these functions remains invariant within each conjugacy class, we establish a noncommutative maximal inequality for Fej\'er means utilizing the associated character system of $G$. This is an operator-valued version of the classical result due to G\'at. We follow essentially the classical sketch, but due to the noncommutativity, many classical arguments have to be revised. Notably, compared to the classical results. the bounds of our estimates are explicity calculated.

math.FA↗
arXiv2024

Failure of almost uniformly convergence for noncommutative martingales

In this paper, we provide a counterexample to show that in sharp contrast to the classical case, the almost uniform convergence may not happen for truly noncommutative $L_p$-martingales when $1\leq p<2$. The same happens to ergodic averages. The proof consists of some sharp estimates of the distributional function of a sequence of matrices and some non standard transference techniques, which might admit further applications.

math.OA↗
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

Operator-Valued Hardy spaces and BMO Spaces on Spaces of Homogeneous Type

Let $\mathcal{M}$ be a von Neumann algebra equipped with a normal semifinite faithful trace, $(\mathbb{X},\,d,\,\mu)$ be a space of homogeneous type in the sense of Coifman and Weiss, and $\mathcal{N}=L_\infty(\mathbb{X})\overline{\otimes}\mathcal{M}$. In this paper, we introduce and then conduct a systematic study on the operator-valued Hardy space $\mathcal{H}_p(\mathbb{X},\,\mathcal{M})$ for all $1\leq p<\infty$ and operator-valued BMO space $\mathcal{BMO}(\mathbb{X},\,\mathcal{M})$. The main results of this paper include $H_1$--$BMO$ duality theorem, atomic decomposition of $\mathcal{H}_1(\mathbb{X},\,\mathcal{M})$, interpolation between these Hardy spaces and BMO spaces, and equivalence between mixture Hardy spaces and $L_p$-spaces. %Compared with the communcative results, the novelty of this article is that $\mu$ is not assumed to satisfy the reverse double condition. %The approaches we develop bypass the use of harmonicity of infinitesimal generator, which allows us to extend Mei's seminal work \cite{m07} to a broader setting. %Our results extend Mei's seminal work \cite{m07} to a broader setting. In particular, without the use of non-commutative martingale theory as in Mei's seminal work \cite{m07}, we provide a direct proof for the interpolation theory. Moreover, under our assumption on Calder\'{o}n representation formula, these results are even new when going back to the commutative setting for spaces of homogeneous type which fails to satisfy reverse doubling condition. As an application, we obtain the $L_p(\mathcal{N})$-boundedness of operator-valued Calder\'{o}n-Zygmund operators.

math.FA↗
arXiv2023

Sharp endpoint $L_p$ estimates of quantum Schr\"{o}dinger groups

In this article, we establish sharp endpoint $L_p$ estimates of Schr\"odinger groups on general measure spaces which may not be equipped with good metrics but admit submarkovian semigroups satisfying purely algebraic assumptions. One of the key ingredients of our proof is to introduce and investigate a new noncommutative high-cancellation BMO space by constructing an abstract form of P-metric codifying some sort of underlying metric and position. This provides the first form of Schr\"odinger group theory on arbitrary von Neumann algebras and can be applied to many models, including Schr\"odinger groups associated with non-negative self-adjoint operators satisfying purely Gaussian upper bounds on doubling metric spaces, standard Schr\"odinger groups on quantum Euclidean spaces, matrix algebras and group von Neumann algebras with finite dimensional cocycles.

math.AP↗
arXiv2022

Quantitative mean ergodic inequalities: power bounded operators acting on one single noncommutative $L_p$ space

In this paper, we establish the quantitative mean ergodic theorems for two subclasses of power bounded operators on a fixed noncommutative $L_p$-space with $1<p<\infty$, which mainly concerns power bounded invertible operators and Lamperti contractions. Our approach to the quantitative ergodic theorems is the noncommutative square function inequalities. The establishment of the latter involves several new ingredients such as the almost orthogonality and Calder\'on-Zygmund arguments for non-smooth kernels from semi-commutative harmonic analysis, the extension properties of the operators under consideration from operator theory, and a noncommutative version of the classical transference method due to Coifman and Weiss.

math.FA↗
arXiv2022

On Isometric Embeddability of $S_q^m$ into $S_p^n$ as non-commutative Quasi-Banach space

The existence of isometric embedding of $S_q^m$ into $S_p^n$, where $1\leq p\neq q\leq \infty$ and $m,n\geq 2$ has been recently studied in \cite{JFA22}. In this article, we extend the study of isometric embeddability beyond the above mentioned range of $p$ and $q$. More precisely, we show that there is no isometric embedding of the commutative quasi-Banach space $\ell_q^m(\R)$ into $\ell_p^n(\R)$, where $(q,p)\in (0,\infty)\times (0,1)$ and $p\neq q$. As non-commutative quasi-Banach spaces, we show that there is no isometric embedding of $S_q^m$ into $S_p^n$, where $(q,p)\in (0,2)\setminus \{1\}\times (0,1)$ $\cup\, \{1\}\times (0,1)\setminus \{\frac{1}{n}:n\in\mathbb{N}\}$ $\cup\, \{\infty\}\times (0,1)\setminus \{\frac{1}{n}:n\in\mathbb{N}\}$ and $p\neq q$. Moreover, in some restrictive cases, we also show that there is no isometric embedding of $S_q^m$ into $S_p^n$, where $(q,p)\in [2, \infty)\times (0,1)$. A new tool in our paper is the non-commutative Clarkson's inequality for Schatten class operators. Other tools involved are the Kato-Rellich theorem and multiple operator integrals in perturbation theory, followed by intricate computations involving power-series analysis.

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