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Ha Duy Hung

Publications and source records attributed to Ha Duy Hung.

10 recordsLinked to original sources
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↗
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 $$\|\mathcal H_φ\|_{H^1(\mathbb R)\to H^1(\mathbb R)}= \int_0^\infty φ(t)dt.$$

math.CA↗
arXiv2016

Multilinear Hardy-Cesàro Operator and Commutator on the product of Morrey-Herz spaces

We obtain sufficient and necessary conditions on weight functions $s_1(t),\ldots,s_m(t)$ and $ψ(t)$ so that the weighted multilinear Hardy-Cesàro operator \[(f_1,\ldots,f_m)\mapsto \int_{[0,1]^n}\left(\prod_{k=1}^nf_k\left(s_k(t) x\right)\right)ψ(t)dt \] is bounded from $\dot{K}^{α_1, p_1}_{q_1}(ω_1)\times \cdots \times\dot{K}^{α_m, p_m}_{q_m}(ω_m)$ to $\dot{K}^{α, p}_{q}(ω)$ and from $M\dot{K}^{α_1, λ_1}_{p_1,q_1}(ω_1)\times \cdots \times M\dot{K}^{α_m, λ_m}_{p_m,q_m}(ω_m)$ to $M\dot{K}^{α, λ}_{p,q}(ω)$. The sharp bounds are also obtained and these results hold for both cases $0<p<1$ and $1\leq p<\infty$. We give a sufficient condition so that if symbols $b_1,\ldots,b_m$ are Lipschitz, then the commutator of the weighted Hardy-Cesàro operator \[ (f_1,\ldots,f_m)\mapsto\int_{[0,1]^n}\left(\prod\limits_{k=1}^mf_k\left(s_k(t)x\right)\right)\left(\prod_{k=1}^m\left(b_k(x)-b_k\left(s_k(t)x\right)\right)\right)ψ(t)dt\] is bounded from $M\dot{K}^{α_1, λ_1}_{p_1, q_1}(ω_1)\times \cdots \times M\dot{K}^{α_m, λ_m}_{p_m, q_m}(ω_m)$ to $M\dot{K}^{α^\prime, λ}_{p, q}(ω)$ for both cases $0<p<1$ and $1\leq p<\infty$. By these we extend and strengthen previous results deu to Tang, Xue, and Zhou [16].

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

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

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↗
arXiv2012

A generalized weighted Hardy-Cesàro operator, and its commutator on weighted $L^p$ and BMO spaces

In this paper, we introduce a new weighted Hardy-Cesàro operator defined by $U_{ψ,s}f(x)=\int\limits_0^1 f(s(t)\cdot x) ψ(t)dt$, which is associated to the parameter curve $s(t,x)=s(t)x$. Under certain conditions on $s(t)$ and on an absolutely homogeneous weight function $ω$, we characterize the weight function $ψ$ such that $U_{ψ,s}$ is bounded on $L^p(ω)$, $BMO(ω)$. The corresponding operator norms are worked out too. These results extend the ones of Jie Xiao \cite{xiao}. We also give a sufficient and a necessary condition on the weight function $ψ$, which ensure the boundedness of the commutators of operator $U_{ψ,s}$ on $L^p(ω)$ with symbols in $BMO(ω)$.

math.CA↗