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Ming-Yi Lee

Publications and source records attributed to Ming-Yi Lee.

12 recordsLinked to original sources
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.

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arXiv2025

Flag Hardy space theory on Heisenberg groups and applications

We establish a complete theory of the flag Hardy space on the Heisenberg group $\mathbb H^{n}$ with characterisations via atomic decompositions, area functions, square functions, maximal functions and singular integrals. We introduce several new techniques to overcome the difficulties caused by the noncommutative Heisenberg group multiplication, and the lack of a suitable Fourier transformation and Cauchy--Riemann type equations. Applications include the boundedness from the flag Hardy space to $L^1(\mathbb H^n)$ of various singular integral operators that arise in complex analysis, a sharp boundedness result on the flag Hardy space of the Marcinkiewicz-type multipliers introduced by Müller, Ricci and Stein, and the decomposition of flag BMO space via singular integrals.

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arXiv2024

Riesz transform and commutators in the Dunkl setting

In this paper we characterise the optimal pointwise size and regularity estimates for the Dunkl Riesz transform kernel involving both the Euclidean metric and the Dunkl metric, where these two metrics are not equivalent. We further establish a suitable version of the pointwise kernel lower bound of the Dunkl Riesz transform via the Euclidean metric only. Then we show that the lower bound of commutator of the Dunkl Riesz transform is with respect to the BMO space associated with the Euclidean metric, and that the upper bound is respect to the BMO space associated with the Dunkl metric. Moreover, the compactness and the two types of VMO are also addressed.

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arXiv2023

Lipschitz and Triebel--Lizorkin spaces, commutators in Dunkl setting

We first study the Lipschitz spaces $Λ_{d}^β$ associated with the Dunkl metric, $β\in(0,1)$, and prove that it is a proper subspace of the classical Lipschitz spaces $Λ^β$ on $\mathbb R^N$, as the Dunkl metric and the Euclidean metric are non-equivalent. Next, we further show that the Lipschitz spaces $Λ^β$ connects to the Triebel--Lizorkin spaces $\dot{ F}^{α,q}_{p,{\rm D}}$ associated with the Dunkl Laplacian $\triangle_{\rm D}$ in $\mathbb R^ N $ and to the commutators of the Dunkl Riesz transform and the fractional Dunkl Laplacian $\triangle_{\rm D}^{-α/2}$, $0<α<\textbf{N}$ (the homogeneous dimension for Dunkl measure), which is represented via the functional calculus of the Dunkl heat semigroup $e^{-t\triangle_{\rm D}}$. The key steps in this paper are a finer decomposition of the underlying space via Dunkl metric and Euclidean metric to bypass the use of Fourier analysis, and a discrete weak-type Calderón reproducing formula in these new Triebel--Lizorkin spaces $\dot{ F}^{α,q}_{p,{\rm D}}$.

math.FA↗
arXiv2023

The Riesz Transform and Fractional Integral Operators in the Bessel Setting

Fix $λ>0$. Consider the Bessel operator $\triangle_λ:=-\frac{d^2}{dx^2}-\frac{2λ}{x} \frac d{dx}$ on $\mathbb{R_+}$, where $\mathbb{R_+}:=(0,\infty)$ and $dm_λ:=x^{2λ}dx$ with $dx$ the Lebesgue measure. We provide a deeper study of the Bessel Riesz transform and fractional integral operator via the related Besov and Triebel--Lizorkin spaces associated with $\triangle_λ$. Moreover, we investigate some possible characterization of the commutator of fractional integral operator, which was missing in the literature of the Bessel setting.

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arXiv2022

The Two Weight Inequality for Poisson Semigroup on Manifold with Ends

Let $M = \mathbb R^m \sharp \mathcal R^n$ be a non-doubling manifold with two ends $\mathbb R^m \sharp \mathcal R^n$, $m > n \ge 3$. Let $Δ$ be the Laplace--Beltrami operator which is non-negative self-adjoint on $L^2(M)$. Then $Δ$ and its square root $\sqrtΔ$ generate the semigroups $e^{-tΔ}$ and $e^{-t\sqrtΔ}$ on $L^2(M)$, respectively. We give testing conditions for the two weight inequality for the Poisson semigroup $e^{-t\sqrtΔ}$ to hold in this setting. In particular, we prove that for a measure $μ$ on $M_{+}:=M\times (0,\infty)$ and $σ$ on $M$: $$ \|\mathsf{P}_σ(f)\|_{L^2(M_{+};μ)} \lesssim \|f\|_{L^2(M;σ)}, $$ with $\mathsf{P}_σ(f)(x,t):= \int_M \mathsf{P}_t(x,y)f(y) \,dσ(y)$ if and only if testing conditions hold for the Poisson semigroup and its adjoint. Further, the norm of the operator is shown to be equivalent to the best constant in these testing conditions.

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arXiv2022

Singular integral operators, $T1$ theorem, Littlewood-Paley theory and Hardy spaces in Dunkl Setting

The purpose of this paper is to introduce a new class of singular integral operators in the Dunkl setting involving both the Euclidean metric and the Dunkl metric. Then we provide the $T1$ theorem, the criterion for the boundedness on $L^2$ for these operators. Applying this singular integral operator theory, we establish the Littlewood-Paley theory and the Dunkl-Hardy spaces. As applications, the boundedness of singular integral operators, particularly, the Dunkl-Rieze transforms, on the Dunkl-Hardy spaces is given. The $L^2$ theory and the singular integral operator theory play crucial roles. New tools developed in this paper include the weak-type discrete Calderón reproducing formulae, new test functions, and distributions, the Littlewood-Paley, the wavelet-type decomposition, and molecule characterizations of the Dunkl-Hardy space, Coifman's approximation to the identity and the decomposition of the identity operator on $L^2$, Meyer's commutation Lemma, and new almost orthogonal estimates in the Dunkl setting.

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arXiv2020

Maximal function, Littlewood--Paley theory, Riesz transform and atomic decomposition in the multi-parameter flag setting

In this paper, we develop via real variable methods various characterisations of the Hardy spaces in the multi-parameter flag setting. These characterisations include those via, the non-tangential and radial maximal function, the Littlewood--Paley square function and area integral, Riesz transforms and the atomic decomposition in the multi-parameter flag setting. The novel ingredients in this paper include (1) establishing appropriate discrete Calderón reproducing formulae in the flag setting and a version of the Plancherel--Pólya inequalities for flag quadratic forms; (2) introducing the maximal function and area function via flag Poisson kernels and flag version of harmonic functions; (3) developing an atomic decomposition via the finite speed propagation and area function in terms of flag heat semigroups. As a consequence of these real variable methods, we obtain the full characterisations of the multi-parameter Hardy space with the flag structure.

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arXiv2018

Boundedness of Monge-Ampere singular integral operators on Besov spaces

Let $ϕ: \Bbb R^n \mapsto \Bbb R$ be a strictly convex and smooth function, and $μ= \text{det}\,D^2 ϕ$ be the Monge-Ampère measure generated by $ϕ.$ For $x\in \Bbb R^n$ and $t>0$, let $S(x,t):=\{y\in \Bbb R^n: ϕ(y)<ϕ(x)+\nabla ϕ(x)\cdot(y-x)+t\}$ denote the section. If $μ$ satisfies the doubling property, Caffarelli and Gutiérrez (Trans. AMS 348:1075--1092, 1996) provided a variant of the Calderón-Zygmund decomposition and a John-Nirenberg-type inequality associated with sections. Under a stronger uniform continuity condition on $μ$, they also (Amer. J. Math. 119:423--465, 1997) proved an invariant Harnack's inequality for nonnegative solutions of the Monge-Ampère equations with respect to sections. The purpose of this paper is to establish a theory of Besov spaces associated with sections under only the doubling condition on $μ$ and prove that Monge-Ampère singular integral operators are bounded on these spaces.

math.FA↗
arXiv2016

On weak-star convergence in product Hardy spaces on spaces of homogeneous type

A classical theorem of Jones and Journé on weak-star convergence in the Hardy space $H^1$ was generalised to the multiparameter setting by Pipher and Treil. We prove the analogous result when the underlying space is a product space of homogeneous type. The main tools we use for this setting are from recent work in papers by Chen, Li and Ward and by Han, Li and Ward.

math.FA↗
arXiv2013

$Tb$ theorem on product spaces

In this paper, we prove a $Tb$ theorem on product spaces $\Bbb R^n\times \Bbb R^m$, where $b(x_1,x_2)=b_1(x_1)b_2(x_2)$, $b_1$ and $b_2$ are para-accretive functions on $\Bbb R^n$ and $\Bbb R^m$, respectively.

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