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Guixiang Hong

Publications and source records attributed to Guixiang Hong.

At least 19 recordsLinked to original sources
arXiv2026

New matrix-weighted norm inequalities and extrapolation

In this paper, we propose a new type of matrix-weighted theory and show the corresponding Rubio de Francia extrapolation theorems by means of noncommutative maximal inequalities. We introduce a new matrix $A_p$ condition following Volberg's geometric formulation, and characterize it via a noncommutative Hardy--Littlewood maximal inequality. We then establish a noncommutative analogue of Rubio de Francia's abstract extrapolation theorem for trace-finite von Neumann algebras. The argument extends to $L_\infty(\mathbb R)\bar\otimes M_d(\mathbb C)$ through pointwise matrix operations. This yields the desired extrapolation results for individual operators and maximal families, with applications to matrix-weighted boundedness of Calderón--Zygmund operators.

math.FA↗
arXiv2026

Noncommutative maximal weak type $(1,1)$ estimates for convolution operators under $L_1$-Hörmander conditions

We prove weak type $(1,1)$ estimates for families of scalar convolution operators acting on operator-valued function spaces. A joint $L_1$-Hörmander condition, together with a maximal strong type estimate at one exponent $1<r\leq\infty$, suffices. The kernels may be complex-valued and need not have uniformly bounded $L_1$ norms. Under a size condition, the result also yields a weak endpoint estimate for all sharp truncation radii of singular kernels. This resolves, for convolution kernels, the problem mentioned in \cite[Remark~5.5]{parcet09pseudo} and recalled in Parcet's recent survey \cite[Remark~1.3]{Parcet2026ICM}. The proof involves not only a simultaneous dyadic representation of the operators but also a noncommutative maximal weak type $(1,1)$ estimates of the resulting dyadic operators. This approach seems entirely new even in the commutative setting. Numerous applications are provided. On the one hand, it yields various maximal inequalities and pointwise convergence results for integrable kernels and dyadic approximations of the identity, monotone $L\log L$ majorants, and Poisson integrals on nilpotent groups and symmetric spaces. On the other hand, via various transference techniques, it implies maximal inequalities for Fourier multipliers, Schur multipliers and double operator integrals, together with bilateral almost uniform convergence of dyadic Fourier summation methods on $L_1$ of group von Neumann algebras.

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arXiv2026

Limits of Besov Norms on Quantum Euclidean Spaces

We extend the Bourgain--Brézis--Mironescu (BBM) and Maz'ya--Shaposhnikova (MS) limit formulae to higher-order Besov norms on quantum Euclidean spaces $\R^d_θ$, a fundamental class of noncompact noncommutative manifolds obtained by deformation quantization of $\R^d$. A central ingredient is a Banach realization of the homogeneous Sobolev endpoint that forms a compatible couple with \(L_p(\R_θ^d)\), together with the equivalence \[ K\bigl(\x,\varepsilon^k; L_p(\R_θ^d),\dot W_{p,0}^k(\R_θ^d)\bigr) \approx ω_p^k(\x,\varepsilon). \] This framework yields a unified proof of both endpoint formulae through the endpoint continuity of real interpolation scales. The principal new ingredient at the MS endpoint is the sharp large-translation asymptotic \[ \lim_{|h|\to\infty}\|T_h\x-\x\|_p = 2^{1/p}\|\x\|_p, \qquad \x\in L_p(\R_θ^d),\quad 1\leq p<\infty, \] whose proof combines oscillatory decay of the non-pure terms in the tracial expansion with weighted spectral-measure arguments adapted to the semifinite trace.

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arXiv2026

A noncommutative weak $L_1$-estimate for rough singular integrals

In this paper, we establish a noncommutative version of Andreas Seeger's weak type $(1,1)$ estimate for singular integral operators with kernels satisfying angular $L\log L$ integrability and an averaged radial Dini condition. The strategy for obtaining the sharp integrability condition involves not only some new estimates of the off-diagonal good and bad functions in Parcet's noncommutative Calderón-Zygmund decomposition but also certain essential modifications of Seeger's kernel decomposition and estimates.

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arXiv2026

Calderón Commutators on Quantum Euclidean Spaces and Quantum Tori

In the classical setting, Calderón's first commutator theorem states that Lipschitz regularity of the symbol characterizes the $L_2$-boundedness of the first commutators. We show that the formally analogous assertion fails on every quantum Euclidean space $\R^d_θ$ with $θ\neq 0$. The same simultaneous failure occurs on $\T_θ^d$ whenever $θ$ has an irrational entry. For rational $θ$, we obtain the optimal logarithmic bound for the corresponding Calderón first commutator. We also discuss the connection between Calderón's first commutator and the $T1$ theorem, and construct an operator-valued counterexample to Calderón's first commutator theorem, which shows that there is no direct analogue of the classical $T1$ theorem in the operator-valued setting. The counterexamples and the optimality argument rely on the construction of sequences of matrices $\mathsf D_n$ and $\mathsf A_n$ such that $\|[\mathsf D_n,\mathsf A_n]\|$ is uniformly bounded, whereas $\|[|\mathsf D_n|,\mathsf A_n]\|$ tends to infinity. This phenomenon reflects the failure of the absolute value function to be operator Lipschitz with respect to the operator norm. The logarithmic estimate, on the other hand, is obtained by combining matrix-valued $T1$ theory with a transference argument.

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arXiv2026

Disproof of Wright's Problems 14 and 15 on lacunary maximal functions

We disprove the proposed weak endpoint bounds in James Wright's Problems 14 and 15 from the 2017 collection of harmonic-analysis problems. For every $d\ge1$, we exhibit a fixed compactly supported singular probability measure with Fourier decay of every order $0<\varepsilon<1/2$, for which the dyadic convolution maximal operator has infinite weak $L^1$ and infinite $H^1$ to weak $L^1$ operator norms. For every finite nonempty set of dyadic scales, both optimal weak constants equal the number of scales, with the Hardy constant measured in the specified atomic norm. Increasingly separated renewal digits provide uniform real-frequency Fourier decay while allowing the disjoint output regions needed for the positive and mean-zero tests. We first establish the fixed measure, then prove the two endpoint estimates using separate normalized tests. Finally we prove the two Fourier lemmas used in the construction.

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arXiv2026

Noncommutative maximal inequalities for polynomial ergodic averages

We prove a noncommutative maximal ergodic inequality for averages along polynomial sequences. Let $γ$ be a trace-preserving automorphism of a semifinite von Neumann algebra $(\mathcal N,τ)$. We show that the associated polynomial averages \begin{equation*} A_Nf:=\frac1N\sum_{n=1}^Nγ^{P(n)}(f), \qquad N\in\mathbb N, \end{equation*} satisfy a strong maximal inequality on $L_p(\mathcal N)$ for every $1<p<\infty$, extending the previously known restricted range of $p$. The proof follows Bourgain's major-arc strategy but requires substantially new ideas and tools for operators that may have further applications in noncommutative analysis. More precisely, we obtain a localized maximal inequality by developing a noncommutative version of Stein's extrapolation, novel combinatorial methods, and a surprising multilinear version of Doob's maximal inequality. For the required decaying $L_2$-approximation, we combine two of Bourgain's constructions in a way that avoids the multi-frequency maximal inequality used in the scalar proof.

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arXiv2026

Vector-valued Littlewood--Paley--Rubio de Francia Inequalities on Vilenkin Systems

We investigate two natural formulations of equal-length Littlewood--Paley--Rubio de Francia (LPR) inequalities on arbitrary Vilenkin systems \(\Gm\). The canonical formulation uses canonical intervals in \(\mathbb N\) of equal cardinality, while Vilenkin's formulation uses translates of a common initial interval under the group operation on the dual of \(\Gm\). In this paper, our main results show that the definitions of these two types of intervals exhibit dramatically different properties. More precisely, for the canonical formulation, we construct a counterexample showing that, for every \(2<p<\infty\), the Schatten class \(S^p\) fails the equal-length LPR property, in sharp contrast to the corresponding result on the torus; for Vilenkin's formulation, we prove that, for every \(2\le p<\infty\), a Banach space \(X\) has the equal-length LPR property if and only if it is UMD and has type~\(2\). The sufficiency proof combines a transference argument from the torus to finite cyclic groups with an embedding of Vilenkin intervals into digital rectangular envelopes, yielding bounds independent of the generating sequence.

math.FA↗
arXiv2026

The weak type $(1,1)$ estimate of Dirichlet Means for unbounded noncommutative Vilenkin systems

Let $\mathcal{R}$ be the hyperfinite $\mathrm{II}_1$ factor. Considering the partial sum operators $(\mathcal{S}_n)_{n\geq 1}$ of the noncommutative Vilenkin-Fourier series associated with an arbitrary admissible Vilenkin group, we prove that there exists a universal constant $c>0$ such that \begin{equation*} \sup_{n\geq1}\|\mathcal{S}_n(f)\|_{L_{1,\infty}(\mathcal{R})} \leq c\|f\|_{L_1(\mathcal{R})},\quad f \in L_1(\mathcal{R}), \end{equation*} and, for every $1<p<\infty$, $$\sup_{n\geq1}\|\mathcal{S}_n(f)\|_{L_p(\mathcal{R})} \leq c\frac{p}{p-1}\|f\|_{L_p(\mathcal{R})},\quad f \in L_p(\mathcal{R}).$$ Besides the transference technique, the main novel ingredient is a modified version of noncommutative Calderón-Zygmund decomposition established in \cite{CCP2022}. Consequently, we resolve the problem of weak type $(1,1)$ estimate communicated to the authors by Fedor Sukochev, and substantially improve the strong type (p,p) estimates obtained in \cite{DFdePS2001} by achieving the optimal order $\frac{p}{p-1}$.

math.FA↗
arXiv2026

Vector-valued partial sums on unbounded Vilenkin systems

Let \(\Gm=\prod_{k\ge0}\mathbb Z_{m_k}\) be a Vilenkin group that is not necessarily bounded, i.e., \(\sup_k m_k=\infty\). We prove that, for every UMD Banach space \(X\) and every \(1<p<\infty\), the Vilenkin partial-sum operators are uniformly bounded on \(L^p(\Gm;X)\), with a bound depending only on \(p\) and the UMD constant of \(X\), and not on \(\mathbf m\). This resolves an open problem arising from the work of Clément et al.~\cite{ClementDePagterSukochevWitvliet2000} and later recorded explicitly in the book of Hytönen et al.~\cite[p.~362]{HNVWI}. The proof reduces the partial-sum estimate, via a Paley conjugation identity and a tangent-sequence decoupling argument, to a decoupling inequality for Fourier projections on finite cyclic groups, which appears to be new. The same approach also yields \(\mathcal R\)-boundedness for the family of partial-sum operators associated with the finer block decomposition, thereby resolving another related problem communicated to us by Fedor Sukochev.

math.FA↗
arXiv2026

Campanato spaces via quantum semigroups

In this paper, we continue to investigate Campanato spaces via semigroups on von Neumann algebras. One of the main results is their coincidence with Lipschitz spaces for {\it all} regularity indices $α> 0$ and for all {\it analytic} semigroups on von Neumann algebras {\it not necessarily being finite}; the desired self-improving property of Campanato spaces has also been verified. As a consequence, we demonstrate that the column Campanato spaces are isomorphic to the row ones for {\it all} $α> 0$. This not only removes several restrictions in the previous work \cite{HJ24} by the first two authors, but also extends the previous results to any analytic semigroup, and thus resolves several problems left open in \cite{HJ24}. Both the results and the proof are new even in the commutative setting.

math.OA↗
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 $γ$ be a trace-preserving automorphism of $\mathcal N$. We consider the ergodic averages along the prime numbers \[ A_N(x) := \frac1{|P_N|} \sum_{q\in P_N}γ^q(x), \qquad P_N:=\{q\leq N:q\ \text{is prime}\}. \] 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 Calderón-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-Glück 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--Foiaş and Akçoglu--Sucheston for a natural class of commuting tuple of contractions extending the abstract dilation theorem of of Fackler-Glück for commuting tuple of contractions. This enables us to obtain a quantitative ergodic theorem for a large class of semigroups of operators on $\mathbb{R}^d_{+}$.

math.OA↗
arXiv2025

Navier-Stokes Equations on Quantum Euclidean Spaces

We investigate in the present paper the Navier-Stokes equations on quantum Euclidean spaces $\mathbb{R}^d_θ$ with $θ$ 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 $\mathbb{R}^d_θ$, 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 $θ$, 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 $α>0$. The column Campanato space ${\mathcal{L}^{c}_α(\mathcal{P})}$ 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}_α(\mathcal{P})}=\left\|f\right\|_{\infty}+\sup_{t>0}\frac{1}{t^α}\left\|P_{t}|(I-P_{t})^{[α]+1}f|^{2}\right\|^{\frac{1}{2}}_{\infty}. \end{align*} The row space ${\mathcal{L}^{r}_α(\mathcal{P})}$ is defined in a canonical way. In this article, we will first show the surprising coincidence of these two spaces ${\mathcal{L}^{c}_α(\mathcal{P})}$ and ${\mathcal{L}^{r}_α(\mathcal{P})}$ for $0<α<2$. 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 ${Λ_α(\mathcal{P})}$. This coincidence passes to the little Campanato spaces $\ell^{c}_α(\mathcal{P})$ and $\ell^{r}_α(\mathcal{P})$ for $0<α<\frac{1}{2}$ under the condition $Γ^{2}\geq0$. We also show that any element in ${\mathcal{L}^{c}_α(\mathcal{P})}$ enjoys the higher order cancellation property, that is, the index $[α]+1$ in the definition of the Campanato norm can be replaced by any integer greater than $α$. 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 ${\mathcal{L}^{c}_α(\mathcal{T})}$ and explore their connection with ${\mathcal{L}^{c}_α(\mathcal{P})}$. All the above-mentioned results seem new even in the (semi-)commutative case.

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