arXiv subjects

Chaojie Wen

Publications and source records attributed to Chaojie Wen.

6 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.

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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.

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arXiv2026

Sharp weighted estimates for the Bessel Riesz transform and its commutator with Andersen--Kerman weights

Let $λ>-1/2$, $λ\neq0$, and let $Δ_λ=-\frac{d^2}{dx^2}-\frac{2λ}{x}\frac{d}{dx}$ be the Bessel operator on $\mathbb R_+=(0,\infty)$ studied by Muckenhoupt and Stein (TAMS 1965). Andersen and Kerman (Studia Math. 1981) proved that for $1<p<\infty$, the Bessel Riesz transform $R_λ=\frac{d}{dx}Δ_λ^{-1/2}$ is bounded on $L^p(\mathbb R_+,w(x)\,dx)$ if and only if $w$ is in the intrinsic class $A_{p,λ}$. However, the sharp quantitative weighted bound via $[w]_{A_{p,λ}}$ was not addressed before. In this paper, we give a positive answer to this question by proving the sharp quantitative estimate $$ \|R_λf\|_{L^p(\mathbb R_+,w\,dx)} \le C_{p,λ} [w]_{A_{p,λ}}^{\max\{1,1/(p-1)\}} \|f\|_{L^p(\mathbb R_+,w\,dx)}. $$ Moreover, for a real-valued function $b$ in the Bessel BMO space ${\rm BMO}_λ$, the sharp weighted bound for the Riesz commutator is $$ \|[b,R_λ]f\|_{L^p(\mathbb R_+,w\,dx)} \le C_{p,λ}\|b\|_{{\rm BMO}_λ} [w]_{A_{p,λ}}^{2\cdot\max\{1,1/(p-1)\}} \|f\|_{L^p(\mathbb R_+,w\,dx)}. $$ The argument uses the exact conjugation $U(x)=x^{p-2λ-1}w(x), \ dν_λ=x^{2λ+1}\,dx, \ [U]_{A_p(dν_λ)}=[w]_{A_{p,λ}},$ which reduces the Andersen--Kerman estimate to an $A_p$ weighted estimate for the auxiliary operator $$ \mathcal R_λF(x)=\frac{1}{x}R_λ(yF(y))(x) $$ on the space of homogeneous type $(\mathbb R_+,|x-y|,dν_λ)$, whose kernel is a standard Calderón--Zygmund kernel with respect to $ν_λ$.

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arXiv2026

Maximal Averages on the Affine Group $G_n$ and applications

Let $G_n=\mathbb R^n\rtimes\mathbb R_+$ be equipped with the left Haar measure $ dμ(x,y)=\frac{dx\,dy}{y^{n+1}}. $ We study maximal averages associated with three basic motions on $G_n$: horizontal translations, vertical dilations, and fixed hyperbolic geodesics in the upper half-space model. The translation maximal operator is the Euclidean Hardy--Littlewood maximal operator on each horizontal slice. The Haar-compatible dilation maximal operator is of weak type $(1,1)$ and bounded on $L^p(G_n)$ for $1<p\le\infty$, but it is not strongly bounded on $L^1(G_n)$. By contrast, the unweighted Lebesgue dilation average is unbounded on every finite $L^p(G_n)$ and is not of weak type $(1,1)$. For fixed hyperbolic geodesic averages, the large-time part is strongly bounded on $L^1(G_n)$ because of modular exponential decay. The small-time part is a finite-type parabolic maximal problem. Using the corresponding local finite-type $L\log\log L$ endpoint estimate for the geodesic slice, we prove $ \mathcal M_{γ_ω}:L\log\log L(G_n) \longrightarrow L^{1,\infty}(G_n) $ in weak Orlicz form, together with the strong $L^p(G_n)$ bounds for $1<p\le\infty$. We also show that the strong $L^1$ endpoint fails. Finally, we record a discrete random-walk maximal inequality whose sufficient condition is expressed through the modular drift $$ ρ_p(σ)=\int_{G_n}y(h)^{n/p}\,dσ(h), $$ where $σ$ is the probability measure defining the right random walk.

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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.

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arXiv2026

Maximal functions with twisted structures, distribution inequality and applications

Motivated by the geometric reduction of Cauchy--Szegő projections on quadratic surfaces of higher codimension (Nagel--Ricci--Stein, 2001) and recent developments on the real-variable theory adapted to twisted multiparameter structures (arXiv:2603.26119), we establish the Fefferman--Stein type distribution inequality relating the twisted area function and the twisted non-tangential maximal function over $\mathbb{R}^{2m}$. By deploying a recursive integration-by-parts argument involving the twisted gradient and Laplacian, and constructing smooth, compactly supported weight functions to absorb cross-derivative errors, we obtain the required estimate. As an application, we prove the uniform $L^1$ boundedness of the twisted maximal function on the twisted atoms and complete the maximal function characterization of the twisted Hardy space.

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