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Luong Dang Ky

Publications and source records attributed to Luong Dang Ky.

At least 19 recordsLinked to original sources
arXiv2026

Endpoint estimates for commutators of singular integral operators associated with admissible functions

Let $(\mathcal X, d, μ)$ be a space of homogeneous type and let $ρ$ be an admissible function on $\mathcal X$. In this paper, we introduce a new class of singular integral operators associated with $ρ$, including a wide range of operators arising in harmonic analysis. For such an operator $T$ and a function $b$ belonging to suitable localized $\mathrm{BMO}$ spaces associated with $ρ$, which are strictly larger than the classical space $\mathrm{BMO}(\mathcal X)$, we establish the boundedness of the commutator $[b, T]$ from the Hardy space $H^1_ρ(\mathcal X)$ into $L^{1,\infty}(\mathcal X)$, $L^1(\mathcal X)$, and $H^1_ρ(\mathcal X)$. Moreover, the boundedness of $[b,T]$ on $H^1_ρ(\mathcal X)$ is characterized by necessary and sufficient conditions. We then apply this to investigate the boundedness of commutators of singular integrals in various settings, including Schrödinger operators on stratified Lie groups and Laguerre operators of convolution type. Our results are new even for Schrödinger operators on $\mathbb R^n$.

math.CA↗
arXiv2026

Endpoint Estimates for Commutators of the Dirichlet Riesz Transform on the Half-Space

Let $\mathfrak R=\nabla(-Δ_D)^{-1/2}$ be the Riesz transform associated with the Dirichlet Laplacian on the upper half-space $\mathbb R^n_+$, and let $H^1_r(\mathbb R^n_+)$ denote the Hardy space of restrictions. We establish an endpoint theory for commutators with $\mathfrak R$ on $H^1_r(\mathbb R^n_+)$. For every $b\in \mathrm{BMO}(\mathbb R^n_+)$, the commutator $[b,\mathfrak R]$ extends boundedly from $H^1_r(\mathbb R^n_+)$ to $L^{1,\infty}(\mathbb R^n_+)$. We then prove the sharp strong-endpoint characterization \[ [b,\mathfrak R]:H^1_r(\mathbb R^n_+)\to H^1_r(\mathbb R^n_+) \quad\Longleftrightarrow\quad b\in \mathrm{BMO}^{\log}(\mathbb R^n_+). \] Moreover, the $\mathrm{BMO}^{\log}$ norm is quantitatively equivalent to the sum of the usual $\mathrm{BMO}$ norm and the $H^1_r$ operator norms of the component commutators $[b,\mathfrak R_j]$. The logarithmic condition records the distance to the boundary and is therefore intrinsic to the Dirichlet geometry.

math.CA↗
arXiv2025

Hausdorff operators on weighted Bergman and Hardy spaces

Let $1\leq p<\infty$, $α>-1$, and let $φ$ be a measurable function on $(0,\infty)$. The main purpose of this paper is to study the Hausdorff operator \[ \mathscr H_φf(z)=\int_0^\infty f\left(\frac{z}{t}\right) \frac{φ(t)}{t} dt, \quad z\in \mathbb C^+, \] on the weighted Bergman space $\mathcal A^p_α(\mathbb C_+)$ and on the power weighted Hardy space $\mathcal H^p_{|\cdot|^α}(\mathbb{C_+})$ of the upper half-plane. Some applications to the real version of $\mathscr H_φ$ are also given.

math.CV↗
arXiv2025

Generalized Calderón-Zygmund operators on the Hardy space $H^1_ρ(\mathcal X)$

Let $(\mathcal X, d,μ)$ be an RD-space, and let $ρ$ be an admissible function on $\mathcal X$. We establish necessary and sufficient conditions for the boundedness of a new class of generalized Calderón-Zygmund operators of log-Dini type on the Hardy space $H^1_ρ(\mathcal X)$, introduced by Yang and Zhou. Our results extend and unify some recent results, providing further insights into the study of singular integral operators in this setting.

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arXiv2017

The multi-parameter Hausdorff operators on $H^1$ and $L^p$

In the present paper, we characterize the nonnegative functions $φ$ for which the multi-parameter Hausdorff operator $\mathcal H_φ$ generated by $φ$ is bounded on the multi-parameter Hardy space $H^1(\mathbb R\times\cdots\times\mathbb R)$ or $L^p(\mathbb R^n)$, $p\in [1,\infty]$. The corresponding operator norms are also obtained. Our results improve some recent results in \cite{FZ, LM2, LM3, We} and give an answer to an open question posted by Liflyand \cite{Li}.

math.CA↗
arXiv2017

Hausdorff operators on holomorphic Hardy spaces and applications

The aim of this paper is to characterize the nonnegative functions $φ$ defined on $(0,\infty)$ for which the Hausdorff operator $$\mathscr H_φf(z)= \int_0^\infty f\left(\frac{z}{t}\right)\frac{φ(t)}{t}dt$$ is bounded on the Hardy spaces of the upper half-plane $\mathcal H_a^p(\mathbb C_+)$, $p\in[1,\infty]$. The corresponding operator norms and their applications are also given.

math.CA↗
arXiv2017

Norm of the Hausdorff operator on the real Hardy space $H^1(\mathbb R)$

Let $φ$ be a nonnegative integrable function on $(0,\infty)$. It is well-known that the Hausdorff operator $\mathcal H_φ$ generated by $φ$ is bounded on the real Hardy space $H^1(\mathbb R)$. The aim of this paper is to give the exact norm of $\mathcal H_φ$. More precisely, we prove that $$\|\mathcal H_φ\|_{H^1(\mathbb R)\to H^1(\mathbb R)}= \int_0^\infty φ(t)dt.$$

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arXiv2016

Bilinear Decompositions of Products of Hardy and Lipschitz Spaces Through Wavelets

The aim of this article is to give a complete solution to the problem of the bilinear decompositions of the products of some Hardy spaces $H^p(\mathbb{R}^n)$ and their duals in the case when $p<1$ and near to $1$, via wavelets, paraproducts and the theory of bilinear Calderón-Zygmund operators. Precisely, the authors establish the bilinear decompositions of the product spaces $H^p(\mathbb{R}^n)\times\dotΛ_α (\mathbb{R}^n)$ and $H^p(\mathbb{R}^n)\timesΛ_α(\mathbb{R}^n)$, where, for all $p\in(\frac{n}{n+1},\,1)$ and $α:=n(\frac{1}{p}-1)$, $H^p(\mathbb{R}^n)$ denotes the classical real Hardy space, and $\dotΛ_α$ and $Λ_α$ denote the homogeneous, respectively, the inhomogeneous Lipschitz spaces. Sharpness of these two bilinear decompositions are also proved. As an application, the authors establish some div-curl lemmas at the endpoint case.

math.CA↗
arXiv2016

Atomic decomposition and weak factorization in generalized Hardy spaces of closed forms

We give an atomic decomposition of closed forms on R n , the coefficients of which belong to some Hardy space of Musielak-Orlicz type. These spaces are natural generalizations of weighted Hardy-Orlicz spaces, when the Orlicz function depends on the space variable. One of them, called H log , appears naturally when considering products of functions in the Hardy space H 1 and in BM O. As a main consequence of the atomic decomposition, we obtain a weak factorization of closed forms whose coefficients are in H log. Namely, a closed form in H log is the infinite sum of the wedge product between an exact form in the Hardy space H 1 and an exact form in BM O. The converse result, which generalizes the classical div-curl lemma, is a consequence of [4]. As a corollary, we prove that the real-valued H log space can be weakly factorized.

math.CA↗
arXiv2015

On weak$^*$-convergence in the localized Hardy spaces $H^1_ρ(\mathcal X)$ and its application

Let $(\mathcal X, d, μ)$ be a complete RD-space. Let $ρ$ be an admissible function on $\mathcal X$, which means that $ρ$ is a positive function on $\mathcal X$ and there exist positive constants $C_0$ and $k_0$ such that, for any $x,y\in \mathcal X$, $$ρ(y)\leq C_0 [ρ(x)]^{1/(1+k_0)} [ρ(x)+d(x,y)]^{k_0/(1+k_0)}.$$ In this paper, we define a space $VMO_ρ(\mathcal X)$ and show that it is the predual of the localized Hardy space $H^1_ρ(\mathcal X)$ introduced by Yang and Zhou \cite{YZ}. Then we prove a version of the classical theorem of Jones and Journé \cite{JJ} on weak$^*$-convergence in $H^1_ρ(\mathcal X)$. As an application, we give an atomic characterization of $H^1_ρ(\mathcal X)$.

math.CA↗
arXiv2015

Weighted Endpoint Estimates for Commutators of Calderón-Zygmund Operators

Let $δ\in(0,1]$ and $T$ be a $δ$-Calderón-Zygmund operator. Let $w$ be in the Muckenhoupt class $A_{1+δ/n}({\mathbb R}^n)$ satisfying $\int_{{\mathbb R}^n}\frac {w(x)}{1+|x|^n}\,dx<\infty$. When $b\in{\rm BMO}(\mathbb R^n)$, it is well known that the commutator $[b, T]$ is not bounded from $H^1(\mathbb R^n)$ to $L^1(\mathbb R^n)$ if $b$ is not a constant function. In this article, the authors find out a proper subspace ${\mathop\mathcal{BMO}_w({\mathbb R}^n)}$ of $\mathop\mathrm{BMO}(\mathbb R^n)$ such that, if $b\in {\mathop\mathcal{BMO}_w({\mathbb R}^n)}$, then $[b,T]$ is bounded from the weighted Hardy space $H_w^1(\mathbb R^n)$ to the weighted Lebesgue space $L_w^1(\mathbb R^n)$. Conversely, if $b\in{\rm BMO}({\mathbb R}^n)$ and the commutators of the classical Riesz transforms $\{[b,R_j]\}_{j=1}^n$ are bounded from $H^1_w({\mathbb R}^n)$ into $L^1_w({\mathbb R}^n)$, then $b\in {\mathop\mathcal{BMO}_w({\mathbb R}^n)}$.

math.CA↗
arXiv2015

New weighted multilinear operators and commutators of Hardy-Cesàro type

A general class of weighted multilinear Hardy-Cesàro operators that acts on the product of Lebesgue spaces and central Morrey spaces. Their sharp bounds are also obtained. In addition, we obtain sufficient and necessary conditions on weight functions so that the commutators of these weighted multilinear Hardy-Cesàro operators (with symbols in central BMO space) are bounded on the product of central Morrey spaces. These results extends known results on multilinear Hardy operators.

math.CA↗
arXiv2015

Endpoint estimates for commutators of singular integrals related to Schrödinger operators

Let $L= -Δ+ V$ be a Schrödinger operator on $\mathbb R^d$, $d\geq 3$, where $V$ is a nonnegative potential, $V\ne 0$, and belongs to the reverse Hölder class $RH_{d/2}$. In this paper, we study the commutators $[b,T]$ for $T$ in a class $\mathcal K_L$ of sublinear operators containing the fundamental operators in harmonic analysis related to $L$. More precisely, when $T\in \mathcal K_L$, we prove that there exists a bounded subbilinear operator $\mathfrak R= \mathfrak R_T: H^1_L(\mathbb R^d)\times BMO(\mathbb R^d)\to L^1(\mathbb R^d)$ such that $|T(\mathfrak S(f,b))|- \mathfrak R(f,b)\leq |[b,T](f)|\leq \mathfrak R(f,b) + |T(\mathfrak S(f,b))|$, where $\mathfrak S$ is a bounded bilinear operator from $H^1_L(\mathbb R^d)\times BMO(\mathbb R^d)$ into $L^1(\mathbb R^d)$ which does not depend on $T$. The subbilinear decomposition (\ref{abstract 1}) explains why commutators with the fundamental operators are of weak type $(H^1_L,L^1)$, and when a commutator $[b,T]$ is of strong type $(H^1_L,L^1)$. Also, we discuss the $H^1_L$-estimates for commutators of the Riesz transforms associated with the Schrödinger operator $L$.

math.CA↗
arXiv2015

Factorization of some Hardy type spaces of holomorphic functions

We prove that the pointwise product of two holomorphic functions of the upper half-plane, one in the Hardy space $\mathcal H^1$, the other one in its dual, belongs to a Hardy type space. Conversely, every holomorphic function in this space can be written as such a product. This generalizes previous characterization in the context of the unit disc.

math.CA↗
arXiv2015

An Hardy estimate for commutators of pseudo-differential operators

Let $T$ be a pseudo-differential operator whose symbol belongs to the Hörmander class $S^m_{ρ,δ}$ with $0\leq δ<1, 0< ρ\leq 1, δ\leq ρ$ and $-(n+1)< m \leq - (n+1)(1-ρ)$. In present paper, we prove that if $b$ is a locally integrable function satisfying $$\sup_{{\rm balls}\; B\subset \mathbb R^n} \frac{\log(e+ 1/|B|)}{(1+ |B|)^θ} \frac{1}{|B|}\int_{B} \Big|f(x)- \frac{1}{|B|}\int_{B} f(y) dy\Big|dx <\infty$$ for some $θ\in [0,\infty)$, then the commutator $[b,T]$ is bounded on the local Hardy space $h^1(\mathbb R^n)$ introduced by Goldberg \cite{Go}. As a consequence, when $ρ=1$ and $m=0$, we obtain an improvement of a recent result by Yang, Wang and Chen \cite{YWC}.

math.CA↗
arXiv2015

On the product of functions in $BMO$ and $H^1$ over spaces of homogeneous type

Let $\mathcal X$ be an RD-space, which means that $\mathcal X$ is a space of homogeneous type in the sense of Coifman-Weiss with the additional property that a reverse doubling property holds in $\mathcal X$. The aim of the present paper is to study the product of functions in $BMO$ and $H^1$ in this setting. Our results generalize some recent results in \cite{Feu} and \cite{LP}.

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