arXiv subjects

Liangchuan Wu

Publications and source records attributed to Liangchuan Wu.

17 recordsLinked to original sources

arXiv2026

Non-radial Dunkl multipliers at the $L^2$-Sobolev threshold

We prove a Hörmander multiplier theorem for the Dunkl transform associated with an arbitrary finite reflection group. Uniform $H^σ(\mathbb{R}^N)$ bounds for the normalised dyadic pieces of a measurable symbol, with $σ>\mathbf N/2$ and $\mathbf N$ the homogeneous dimension, imply $L^p(dω)$ boundedness for $1<p<\infty$ and weak type $(1,1)$. No radiality or reflection-group invariance is assumed, and the dyadic pieces may meet the reflecting hyperplanes. The main new estimate controls the Dunkl kernel $E_κ(iξ,x)$ and its first Euclidean derivatives in $x$, uniformly near arbitrary intersections of reflecting hyperplanes. Here $ξ$ lies in a fixed compact annulus. Writing $w_κ$ for the Dunkl weight and $Ω_κ$ for its inhomogeneous counterpart, it gives the bound $CΩ_κ(x)^{-1/2}w_κ(ξ)^{-1/2}$. The proof decomposes each root contribution into Hermitian two-dimensional blocks, applies oscillatory changes of variables near the rootwise transition points, and uses a commutator identity to cancel the non-integrable phase derivative. Chamber lifting also yields endpoint bounds for the pullbacks of the componentwise Coifman--Weiss atomic $H^1$ space and the corresponding BMO space on a fixed chamber.

math.CA↗

arXiv2026

Polynomial corners in finite fields beyond the distinct-degree case

We prove a quantitative polynomial Roth theorem for corners in $\mathbb F_p^2$ for arbitrary pairs of linearly independent polynomials. More precisely, given a positive integer $d$, there are constants $p_0$ and $C$ (depending only on $p$) so that for every $ p>p_0$, if polynomials $ϕ_1,ϕ_2\in \mathbb \mathbb{F}_p [y]$ are of degree $\leq d$ vanishing at $0$ and are not linearly dependent, then every $A\subset\mathbb F_p^2$ with $ |A|\ge C p^{2-1/14} $ contains a nontrivial corner $$ (x_1,x_2),\qquad (x_1+ϕ_1(y),x_2),\qquad (x_1,x_2+ϕ_2(y)) $$ for some $y\in\mathbb F_p^\times$. This improves the estimate $p^{2-1/16}$ of Han--Lacey--Yang and removes the distinct-degree restriction from their quantitative theorem. The main obstruction is the equal-degree resonant case, where the Jacobian argument of Han--Lacey--Yang degenerates. We adjoin the frequency-independent part of the phase to form an augmented map $\widetilde F:W\to\mathbb A^3$ from the correlation threefold. We prove that this map is generically finite on every top-dimensional geometric component and has no two-dimensional fibre. Using the associated Artin--Schreier sheaf and Katz--Laumon estimates for Fourier transform of perverse sheaves, we obtain square-root cancellation outside an algebraic exceptional set of dimension at most one and uniformly bounded degree. A separate curve-sum argument gives uniform control on the exceptional set. An $\ell^2$ matrix estimate adapted to such sets completes the resonant case.

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arXiv2026

A Fefferman--Stein inequality for the Dunkl Poisson semigroup and its chamber-lifted formulation

We prove a Fefferman--Stein good-$λ$ inequality for the Dunkl Poisson semigroup associated with a finite reflection group and a non-negative multiplicity function. For arbitrary complex-valued $f\in C_c^\infty(\mathbb R^N)$, with no $G$-invariance assumption, it compares the orbit-conical non-tangential maximal function $\mathcal N_P^βf$ with the area function $\mathcal S_Pf$ formed from the full space-time Dunkl carré du champ, including its reflection-difference energy. The main obstruction is that a general cut-off creates wall differences not controlled by the local Euclidean-gradient product identity. The good set $E=\{x:\mathcal N_P^βf(x)\leλ\}$ is $G$-invariant; by the equivariance of the Poisson semigroup, so is $a=φ(P_t \mathbf 1_E)$, and hence all reflection differences of the cut-off vanish. Poisson maximal and tail estimates, together with the $L^2$ Littlewood--Paley estimate for $P_t \mathbf 1_{E^c}$, then yield the desired distribution inequality. Its integrated form gives maximal-to-area estimates for every $0<p<2$ and endpoint $H^1$-to-$L^1$ bounds for the orbit-conical and Euclidean-conical intrinsic area functions. For chamber lifts of globally smooth data, the inequality has an equivalent formulation on a fundamental chamber, where orbit cones become Euclidean cones and the reflection energy becomes a finite wall coupling. Combined with the known semigroup square-function characterization, these bounds characterize the Dunkl Poisson maximal Hardy space among $L^1(dω)$ data.

math.CA↗

arXiv2026

Calderon-type commutators and chamber lifting in the Dunkl setting

Let $G$ be a finite reflection group, and let $T_j$, $Δ_κ$, and $dω$ denote its Dunkl operators, Laplacian, and measure, respectively. Write $\mathcal R_j=-T_j(-Δ_κ)^{-1/2}$ for the $j$th Dunkl Riesz transform. We study the Calderón-type commutator $[M_b,T_i\mathcal R_j]$ on the full space $L^p(\R^N,dω)$, without assuming $G$-invariance of the functions. For $b\in\Lipd$, a full-space Dunkl $T1$ argument for $[M_b,(-Δ_κ)^{1/2}]$, combined with the exact factorization $$ [M_b,T_i\mathcal R_j] =-M_{\partial_i b}\mathcal R_j-\mathcal R_iM_{\partial_jb} +\mathcal R_i[M_b,(-Δ_κ)^{1/2}]\mathcal R_j, $$ implies boundedness on $L^p(dω)$ for every $1<p<\infty$. We also prove that the prescribed heat truncations are uniformly bounded on $L^2(dω)$ and converge jointly to this operator. To control these truncations, we lift all reflected values of an arbitrary function to separate coordinates on a fixed Weyl chamber. This chamber lifting retains all non-$G$-invariant information and places every possible orbit singularity on the ordinary chamber diagonal of a finite matrix operator. Wall-layer estimates and scalar $T1$ bounds for the integrated entries are uniform in both heat endpoints. A dense-core argument then proves joint, path-independent weak-operator convergence to the factorized commutator. The lifted limit is associated, in the separated-support sense, with a finite matrix of scalar Calderón--Zygmund kernels on the chamber.

math.CA↗

arXiv2026

Two-weight commutators for the Bessel Riesz transform with Andersen--Kerman weights

Let $α>-1/2$, $α\ne0$, and let $$ Δ_α=-\frac{d^2}{dx^2}-\frac{2α}{x}\frac{d}{dx} $$ be the Bessel operator on $\mathbb R_+=(0,\infty)$. We characterize boundedness and compactness of commutators of $R_α=\frac{d}{dx}Δ_α^{-1/2}$ on the Andersen--Kerman two-weight setting. For $1<p<\infty$ and $μ,λ\in A_{p,α}$, put $ ν=\left(\fracμλ\right)^{1/p}, \ dρ_α(x)=x^{2α+1}\,dx. $ For real-valued symbols, $$ \|[b,R_α]\|_{L^p(μ\,dx)\to L^p(λ\,dx)} \simeq \|b\|_{\rm{BMO}_{ν,α}}, $$ where the Bloom oscillation is computed with respect to $dρ_α$. Moreover $$ [b,R_α]:L^p(μ\,dx)\to L^p(λ\,dx) \text{ is compact} \quad\Longleftrightarrow\quad b\in\rm{VMO}_{ν,α}. $$ The proof uses the exact conjugation $ U_w(x)=x^{p-2α-1}w(x), \ [U_w]_{A_p(ρ_α)}=[w]_{A_{p,α}}, $ which transfers the problem to the quotient Bessel kernel on $(\mathbb R_+,|x-y|,ρ_α)$. We isolate the required Calderón--Zygmund estimates and one-sided non-degeneracy in this quotient normalization. The measure $dρ_α$ is distinct from the usual Bessel measure $dm_α=x^{2α}\,dx$. It is the Bloom base measure selected by the Andersen--Kerman conjugation and is used throughout the two-weight theory below.

math.CA↗

arXiv2026

Chamber lifting and non-radial Dunkl multipliers

We study non-radial Dunkl multipliers via chamber lifting. For an arbitrary finite reflection group $G$, the chamber lifting records all reflected values of a function and conjugates a multiplier into a finite matrix-valued operator on the chamber. If the dyadic matrix entries admit off-diagonal kernels satisfying the chamber $L^2$ Hörmander condition $\operatorname{CH}^2_{s,η}$ with $s>N_κ/2$, then the original multiplier is bounded on $L^p(\mathbb R^N,dω)$ for every $1 N_κ/2$ therefore imply $L^p$ boundedness, for all $1<p<\infty$, for a genuinely non-radial class of symbols. The order $N_κ/2$ is forced already by the rank-one Bessel transform. The same chamber theorem also applies to non-product examples once the matrix kernel condition is known, including the dihedral groups $I_2(q)$ and hence $A_2\simeq I_2(3)$ and $B_2\simeq I_2(4)$. The scalar Walsh--Sobolev verification is specific to $A_1^N$. In non-product groups such as $A_2$, $A_{N-1}$, and $B_N$, the product parity calculus is absent, so a scalar theorem of the same form would require additional transform estimates.

math.CA↗

arXiv2026

Two-weight inequalities for the Dunkl--Poisson integrals

We prove an $L^2$ two-weight testing theorem for the Dunkl--Poisson semigroup. The difficulty is geometric. The Dunkl orbit distance has several reflected diagonals, so a single orbit-box test may mix different chamber components. We avoid this by working on one Weyl chamber and keeping the chamber indices. Under the wall-null assumption the full operator becomes a finite matrix of scalar positive Poisson-type operators. In each entry the orbit diagonal is just the ordinary diagonal in the chamber variables. The scalar proof is then a principal-cube stopping-time argument, with two Dunkl kernel comparisons as the only new estimates. The resulting forward and backward tests are necessary and sufficient for the original Dunkl--Poisson inequality.

math.CA↗

arXiv2026

Haar bases for multi-parameter twisted structures

Motivated by the Cauchy--Szegő projections on a broad class of Siegel domains and the geometric quotient structures of nilpotent Lie groups observed by Nagel, Ricci, and Stein, we develop a martingale and Haar wavelet framework for twisted multi-parameter geometries. We introduce twisted dyadic filtrations and construct adapted Haar bases on Euclidean spaces $\mathbb{R}^{2m}$. Each of the resulting dyadic systems forms a complete orthonormal basis of $L^2(\mathbb R^{2m})$, and their union yields a tight frame with frame bound $3$. We establish $L^p$-equivalences for the associated discrete twisted Littlewood--Paley square functions. Furthermore, we extend this discrete real-variable theory to the non-abelian setting of a nilpotent Lie group of step two, $\mathscr{N}$, which serves as the Shilov boundary of certain fundamental Siegel domains. By projecting product fractal tiles from a lifting group of Heisenberg products, we define twisted dyadic shards and construct twisted nilpotent Haar frames. More precisely, we first introduce raw projected shards that reflect the quotient geometry, and then pass to analytic dyadic shards which are exactly rectifiable and remain uniformly comparable to the raw quotient structure in the relevant scale regimes. This yields a discrete framework adapted to twisted quotient geometries in both the Euclidean and nilpotent settings, providing a basic dyadic infrastructure for further developments in twisted real-variable theory.

math.CA↗

arXiv2025

Endpoint boundedness of singular integrals: CMO space associated to Schrödinger operators

Let $ \mathcal{L} = -Δ+ V $ be a Schrödinger operator acting on $ L^2(\mathbb{R}^n) $, where the nonnegative potential $ V $ belongs to the reverse Hölder class $ RH_q $ for some $ q \geq n/2 $. This article is primarily concerned with the study of endpoint boundedness for classical singular integral operators in the context of the space $ \mathrm{CMO}_{\mathcal{L}}(\mathbb{R}^n) $, consisting of functions of vanishing mean oscillation associated with $ \mathcal{L} $. We establish the following main results: (i) the standard Hardy--Littlewood maximal operator is bounded on $\mathrm{CMO}_{\mathcal{L}}(\mathbb{R}^n) $; (ii) for each $ j = 1, \ldots, n$, the adjoint of the Riesz transform $ \partial_j \mathcal{L}^{-1/2} $ is bounded from $ C_0(\mathbb{R}^n) $ into $ \mathrm{CMO}_{\mathcal{L}}(\mathbb{R}^n) $; and (iii) the approximation to the identity generated by the Poisson and heat semigroups associated with $ \mathcal{L} $ characterizes $ \mathrm{CMO}_{\mathcal{L}}(\mathbb{R}^n) $ appropriately. These results recover the classical analogues corresponding to the Laplacian as a special case. However, the presence of the potential $ V $ introduces substantial analytical challenges, necessitating tools beyond the scope of classical Calderón--Zygmund theory. Our approach leverages precise heat kernel estimates and the structural properties of $ \mathrm{CMO}_{\mathcal{L}}(\mathbb{R}^n) $ established by Song and the third author in [J. Geom. Anal. 32 (2022), no. 4, Paper No. 130, 37 pp].

math.CA↗

arXiv2025

The fractional Riesz transform and their commutator in Dunkl setting

In this paper, we study the boundedness of the fractional Riesz transforms in the Dunkl setting. Moreover, we establish the necessary and sufficient conditions for the boundedness of their commutator with respect to the central BMO space associated with Euclidean metric and the BMO space associated with Dunkl metric, respectively. Based on this, we further characterize the compactness of the commutator in terms of the corresponding types of VMO spaces.

math.CA↗

arXiv2024

Schatten classes and commutators in the two weight setting, II. Riesz transforms

We characterize the Schatten class $S^p$ of the commutator of Riesz transforms $[b,R_j]$ in $\mathbb R^n$ ($j=1,\ldots, n$) in the two weight setting for $n< p<\infty$, by introducing the condition that the symbol $b$ being in Besov spaces associated with the given two weights. At the critical index $p=n$, the commutator $[b,R_j]$ belongs to Schatten class $S^{n}$ if and only if $b$ is a constant, and to the weak Schatten class $S^{n,\infty}$ if and only if $b$ is in an oscillation sequence space associated with the given two weights. As a direct application, we have the Schatten class estimate for A. Connes' quantised derivative in the two weight setting.

math.CA↗

arXiv2024

Hölder regularity and Liouville Theorem for the Schrödinger equation with certain critical potentials, and applications to Dirichlet problems

Let $(X,d,μ)$ be a metric measure space satisfying a doubling property with the upper/lower dimension $Q\ge n>1$, and admitting an $L^2$-Poincaré inequality. In this article, we establish the Hölder continuity and a Liouville-type theorem for the (elliptic-type) Schrödinger equation $$\mathbb L u(x,t)=-\partial^2_{t}u(x,t)+\mathcal L u(x,t)+V(x)u(x,t)=0,\quad x\in X,\, t\in\mathbb R, $$ where $\mathcal L$ is a non-negative operator generated by a Dirichlet form on $X$, and the non-negative potential $V$ is a Muckenhoupt weight belonging to the reverse Hölder class ${RH}_q(X)$ for some $q>\max\{Q/2,1\}$. Note that $Q/2$ is critical for the regularity theory of $-Δ+V$ on $\mathbb{R}^Q$ ($Q\ge3$) by Shen's work in 1995, which hints the critical index of $V$ for the regularity results above on $X\times \mathbb R$ may be $(Q+1)/2$. Our results show that this critical index is in fact $\max\{Q/2,1\}$. Our approach primarily relies on the controllable growth of $V$ and the elliptic theory for the operator $\mathbb L$/$-\partial^2_{t}+\mathcal{L}$ on $X\times \mathbb R$, rather than the analogs for $\mathcal L+V$/$\mathcal{L}$ on $X$, under the critical index setting. As applications, we further obtain some characterizations for solutions to the Schrödinger equation $-\partial^2_{t}u+\mathcal L u+Vu=0$ in $X\times \mathbb R_+$ with boundary values in BMO/CMO/Morrey spaces related to $V$, improving previous results to the critical index $q>\max\{Q/2,1\}$.

math.AP↗

arXiv2024

On maximal functions generated by Hörmander-type spectral multipliers

Let $(X,d,μ)$ be a metric space with doubling measure and $L$ be a nonnegative self-adjoint operator on $L^2(X)$ whose heat kernel satisfies the Gaussian upper bound. We assume that there exists an $L$-harmonic function $h$ such that the semigroup $\exp(-tL)$, after applying the Doob transform related to $h$, satisfies the upper and lower Gaussian estimates. In this paper we apply the Doob transform and some techniques as in Grafakos-Honzík-Seeger \cite{GHS2006} to obtain an optimal $\sqrt{\log(1+N)}$ bound in $L^p$ for the maximal function $\sup_{1\leq i\leq N}|m_i(L)f|$ for multipliers $m_i,1\leq i\leq N,$ with uniform estimates. Based on this, we establish sufficient conditions on the bounded Borel function $m$ such that the maximal function $M_{m,L}f(x) = \sup_{t>0} |m(tL)f(x)|$ is bounded on $L^p(X)$. The applications include Schrödinger operators with inverse square potential, Scattering operators, Bessel operators and Laplace-Beltrami operators.

math.CA↗

arXiv2024

Global-in-time maximal regularity for the Cauchy problem of the heat equation in BMO and applications

In this article, we establish global-in-time maximal regularity for the Cauchy problem of the classical heat equation $\partial_t u(x,t)-Δu(x,t)=f(x,t)$ with $u(x,0)=0$ in a certain $\rm BMO$ setting, which improves the local-in-time result initially proposed by Ogawa and Shimizu in \cite{OS, OS2}. In further developing our method originally formulated for the heat equation, we obtain analogous global ${\rm BMO}$-maximal regularity associated to the Schrödinger operator $\mathcal L=-Δ+V$, where the nonnegative potential $V$ belongs to the reverse Hölder class ${\rm RH}_q$ for some $q> n/2$. This extension includes several inhomogeneous estimates as ingredients, such as Carleson-type estimates for the external forces. Our new methodology is to exploit elaborate heat kernel estimates, along with matched space-time decomposition on the involving integral-type structure of maximal operators, as well as some global techniques such as those from de Simon's work and Schur's lemma. One crucial trick is to utilize the mean oscillation therein to contribute a higher and necessary decay order for global-in-time estimates.

math.AP↗

arXiv2021

The CMO-Dirichlet problem for the Schrödinger equation in the upper half-space and characterizations of CMO

Let $\mathcal{L}$ be a Schrödinger operator of the form $\mathcal{L}=-Δ+V$ acting on $L^2(\mathbb R^n)$ where the nonnegative potential $V$ belongs to the reverse Hölder class ${RH}_q$ for some $q\geq (n+1)/2$. Let ${CMO}_{\mathcal{L}}(\mathbb{R}^n)$ denote the function space of vanishing mean oscillation associated to $\mathcal{L}$. In this article we will show that a function $f$ of ${ CMO}_{\mathcal{L}}(\mathbb{R}^n) $ is the trace of the solution to $\mathbb{L}u=-u_{tt}+\mathcal{L} u=0$, $u(x,0)=f(x)$, if and only if, $u$ satisfies a Carleson condition $$ \sup_{B: \ { balls}}\mathcal{C}_{u,B} :=\sup_{B(x_B,r_B): \ { balls}} r_B^{-n}\int_0^{r_B}\int_{B(x_B, r_B)} \big|t \nabla u(x,t)\big|^2\, \frac{ dx\, dt } {t} <\infty, $$ and $$ \lim _{a \rightarrow 0}\sup _{B: r_{B} \leq a} \,\mathcal{C}_{u,B} = \lim _{a \rightarrow \infty}\sup _{B: r_{B} \geq a} \,\mathcal{C}_{u,B} = \lim _{a \rightarrow \infty}\sup _{B: B \subseteq \left(B(0, a)\right)^c} \,\mathcal{C}_{u,B}=0. $$ This continues the lines of the previous characterizations by Duong, Yan and Zhang \cite{DYZ} and Jiang and Li \cite{JL} for the ${ BMO}_{\mathcal{L}}$ spaces, which were founded by Fabes, Johnson and Neri \cite{FJN} for the classical BMO space. For this purpose, we will prove two new characterizations of the ${ CMO}_{\mathcal{L}}(\mathbb{R}^n)$ space, in terms of mean oscillation and the theory of tent spaces, respectively.

math.CA↗

arXiv2019

A q-atomic decomposition of weighted tent spaces on spaces of homogeneous type and its application

The theory of tent spaces on $\mathbb{R}^n$ was introduced by Coifman, Meyer and Stein, including atomic decomposition, duality theory and so on. Russ generalized the atomic decomposition for tent spaces to the case of spaces of homogeneous type $(X,d,μ)$. The main purpose of this paper is to extend the results of Coifman, Meyer, Stein and Russ to weighted version. More precisely, we obtain a $q$-atomic decomposition for the weighted tent spaces $T^p_{2,w}(X)$, where $0<p\leq 1, 1<q<\infty,$ and $w\in A_\infty$. As an application, we give an atomic decomposition for weighted Hardy spaces associated to nonnegative self-adjoint operators on $X$.

math.FA↗

arXiv2016

Exponential-square integrability, weighted inequalities for the square functions associated to operators and applications

Let $X$ be a metric space with a doubling measure. Let $L$ be a nonnegative self-adjoint operator acting on $L^2(X)$, hence $L$ generates an analytic semigroup $e^{-tL}$. Assume that the kernels $p_t(x,y)$ of $e^{-tL}$ satisfy Gaussian upper bounds and Hölder's continuity in $x$ but we do not require the semigroup to satisfy the preservation condition $e^{-tL}1 = 1$. In this article we aim to establish the exponential-square integrability of a function whose square function associated to an operator $L$ is bounded, and the proof is new even for the Laplace operator on the Euclidean spaces ${\mathbb R^n}$. We then apply this result to obtain: (i) estimates of the norm on $L^p$ as $p$ becomes large for operators such as the square functions or spectral multipliers; (ii) weighted norm inequalities for the square functions; and (iii) eigenvalue estimates for Schrödinger operators on ${\mathbb R}^n$ or Lipschitz domains of ${\mathbb R}^n$.

math.AP↗