# Luong Dang KyQuy Nhon University · Department of Mathematics

Luong Dang Ky

PhD.

## About

44

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November 2012 - present

October 2009 - October 2012

## Publications

Publications (44)

Let α ∈ ( 0 , n ) and let φ 1 , φ 2 : R n × [ 0 , ∞ ) → [ 0 , ∞ ) be Musielak–Orlicz functions such that φ 1 ( x , · ) , φ 2 ( x , · ) are Orlicz functions and φ 1 ( · , t ) , φ 2 ( · , t ) are Muckenhoupt A ∞ ( R n ) weights. In this paper, we give the necessary and sufficient condition for the boundedness of the fractional integral operator I α f...

Let δ ∈ (0,1] and T be a δ-Calderón–Zygmund operator. Let p ∈ (0,1] be such that p(1 + δ/n) > 1, and assume that w belongs to the Muckenhoupt weight class \(A_{p(1+\delta /n)}(\mathbb {R}^{n})\) with the property \({\int \limits }_{\mathbb {R}^{n}}\frac {w(x)}{(1+|x|)^{np}}dx<\infty \). When \(b\in \text {BMO}(\mathbb {R}^{n})\), it is well-know th...

Let X be a metric space equipped with a measure satisfying the doubling and reverse doubling conditions. In this paper, we develop the theory of new localized Hardy spaces H p ρ (X) for n n+1 < p ≤ 1 associated to critical functions ρ defined on X where n is the doubling order. Our results include the atomic decomposition characterization and the m...

For any p∈(0,1) and α=1/p−1, let Hp(Rⁿ) and Cα(Rⁿ) be the Hardy and the Campanato spaces on the n-dimensional Euclidean space Rⁿ, respectively. In this article, the authors find suitable Musielak–Orlicz functions Φp, defined by setting, for any x∈Rⁿ and t∈[0,∞), [Formula presented] and then establish a bilinear decomposition theorem for multiplicat...

Let X be a space of homogeneous type in the sense of Coifman and Weiss. Let φ : X × [0, ∞) → [0, ∞) be such that φ(x,⋅) is an Orlicz function and φ(⋅, t) is a Muckenhoupt A∞(X) weight uniformly in t. In this paper, we propose John–Nirenberg type inequalities for Musielak–Orlicz Campanato spaces on spaces of homogeneous type. As an application, we s...

In this chapter, we introduce the weak Musielak-Orlicz Hardy space \(W\!H^{\varphi }(\mathbb{R}^{n})\) via the grand maximal function and then obtain its vertical or its non-tangential maximal function characterizations. We also establish other real-variable characterizations of \(W\!H^{\varphi }(\mathbb{R}^{n})\), respectively, by means of the ato...

Let \(s \in \mathbb{R}\), q ∈ (0, ∞], \(\varphi _{1},\ \varphi _{2}:\ \mathbb{R}^{n} \times [0,\infty ) \rightarrow [0,\infty )\) be two Musielak-Orlicz functions that, on the space variable, belong to the Muckenhoupt class \(\mathbb{A}_{\infty }(\mathbb{R}^{n})\) uniformly on the growth variable.

In this chapter, for any α ∈ (0, 1] and \(s \in \mathbb{Z}_{+}\), we establish the s-order intrinsic square function characterizations of \(H^{\varphi }(\mathbb{R}^{n})\) by means of the intrinsic Lusin area function S
α, s
, the intrinsic g-function g
α, s
or the intrinsic g
λ
∗-function g
λ, α, s
∗ with the best known range λ ∈ (2 + 2(α + s)∕n, ∞...

In this chapter, we establish some real-variable characterizations of \(H^{\varphi }(\mathbb{R}^{n})\) in terms of the vertical or the non-tangential maximal functions, via first establishing a Musielak-Orlicz Fefferman-Stein vector-valued inequality.

In this chapter, we introduce a local Musielak-Orlicz Hardy space \(h^{\varphi }(\mathbb{R}^{n})\) by the local grand maximal function, and a local BMO-type space \(\mathrm{bmo}^{\varphi }(\mathbb{R}^{n})\) which is further proved to be the dual space of \(h^{\varphi }(\mathbb{R}^{n})\). As an application, we prove that the class of pointwise multi...

In this chapter, we study the Musielak-Orlicz Campanato space \(\mathcal{L}_{\varphi,q,s}(\mathbb{R}^{n})\) and, as an application, prove that some of them is the dual space of the Musielak-Orlicz Hardy space \(H^{\varphi }(\mathbb{R}^{n})\).

For any j ∈ { 1, …, n}, \(f \in \mathcal{S}(\mathbb{R}^{n})\) and \(x \in \mathbb{R}^{n}\), the j-th Riesz transform
R
j
( f) of f is usually defined by $$\displaystyle{R_{j}(\,f)(x):=\lim _{\epsilon \rightarrow 0^{+}}C_{(n)}\int _{\{y\in \mathbb{R}^{n}:\ \vert y\vert >\epsilon \}} \frac{y_{j}} {\vert y\vert ^{n+1}}f(x - y)\,dy,}$$ where $$\display...

In this chapter, we establish the Littlewood-Paley function and the molecular characterizations of the Musielak-Orlicz Hardy space \(H^{\varphi }(\mathbb{R}^{n})\).

As another application of Musielak-Orlicz Hardy space \(H^{\log }(\mathbb{R}^{n})\), we consider the boundedness of commutators in this chapter. It is well known that the linear commutator [b, T], generated by a BMO function b and a Calderón-Zygmund operator T, may not be bounded from \(H^{1}(\mathbb{R}^{n})\) into \(L^{1}(\mathbb{R}^{n})\).

In this chapter, we first recall the notion of growth functions, establish some technical lemmas and introduce the Musielak-Orlicz Hardy space \(H^{\varphi }(\mathbb{R}^{n})\) which generalize the Orlicz-Hardy space of Janson and the weighted Hardy space of García-Cuerva, Strömberg and Torchinsky.

As an important application of Musielak-Orlicz Hardy spaces, in this chapter.

In the present paper, we characterize the nonnegative functions $\varphi$ for which the multi-parameter Hausdorff operator $\mathcal H_\varphi$ generated by $\varphi$ is bounded on either 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 ob...

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

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

The main purpose of this book is to give a detailed and complete survey of recent progress related to the real-variable theory of Musielak–Orlicz Hardy-type function spaces, and to lay the foundations for further applications.
The real-variable theory of function spaces has always been at the core of harmonic analysis. Recently, motivated by certai...

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...

Let $(\mathcal X, d, \mu)$ be a complete RD-space. Let $\rho$ be an
admissible function on $\mathcal X$, which means that $\rho$ is a positive
function on $\mathcal X$ and there exist positive constants $C_0$ and $k_0$
such that, for any $x,y\in \mathcal X$, $$\rho(y)\leq C_0 [\rho(x)]^{1/(1+k_0)}
[\rho(x)+d(x,y)]^{k_0/(1+k_0)}.$$
In this paper, we...

Let $\delta\in(0,1]$ and $T$ be a $\delta$-Calder\'on-Zygmund operator. Let
$w$ be in the Muckenhoupt class $A_{1+\delta/n}({\mathbb R}^n)$ 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 $L^1(\mathbb R^n)$ i...

The aim of this article is to give the bilinear decompositions of the
products of some Hardy spaces and their duals. The authors establish the
bilinear decompositions of the product spaces
$H^p(\mathbb{R}^n)\times\dot\Lambda_{\alpha}(\mathbb{R}^n)$ and
$H^p(\mathbb{R}^n)\times\Lambda_{\alpha}(\mathbb{R}^n)$, where, for all
$p\in(\frac{n}{n+1},\,1)$...

In this paper, we give a very simple proof of the main result of Dafni (Canad
Math Bull 45:46--59, 2002) concerning with weak$^*$-convergence in the local
Hardy space $h^1(\mathbb R^d)$.

Let $\X$ be a complete space of homogeneous type. In this note, we prove that
the weak$^*$-convergence is true in the Hardy space $H^1(\X)$ of Coifman and
Weiss.

In this paper, we improve a recent result by Li and Peng on products of functions in H-L(1) (R-d) and BMOL (R-d), where L = -Delta + V is a Schrodinger operator with V satisfying an appropriate reverse Holder inequality. More precisely, we prove that such products may be written as the sum of two continuous bilinear operators, one from H-L(1) (R-d)...

This paper deals with 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-Cesáro operators...

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 resu...

Let $T$ be a pseudo-differential operator whose symbol belongs to the
H\"ormander class $S^m_{\rho,\delta}$ with $0\leq \delta<1, 0< \rho\leq 1,
\delta \leq \rho$ and $-(n+1)< m \leq - (n+1)(1-\rho)$. 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|)}{...

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.

Let $\mathcal{X}$ be a metric space with doubling measure and $L$ a
one-to-one operator of type $\omega$ having a bounded $H_\infty$-functional
calculus in $L^2(\mathcal{X})$ satisfying the reinforced $(p_L, q_L)$
off-diagonal estimates on balls, where $p_L\in[1,2)$ and $q_L\in(2,\infty]$.
Let $\varphi:\,\mathcal{X}\times[0,\infty)\to[0,\infty)$ be...

Let $L= -\Delta+ V$ be a Schrödinger operator on $\mathbb R^d$, $d\geq 3$, where $V$ is a nonnegative function, $V\ne 0$, and belongs to the reverse Hölder class $RH_{d/2}$. In this paper, we prove a version of the classical theorem of Jones and Journé on weak$^*$-convergence in the Hardy space $H^1_L(\mathbb R^d)$.

Voir à la fin du fichier de thèse

Let L = − Δ + V be a Schrödinger operator on
$\mathbb R^d$
, d ≥ 3, where V is a nonnegative function,
$V\ne 0$
, and belongs to the reverse Hölder class RH
d/2. In this paper, we prove a version of the classical theorem of Jones and Journé on weak*-convergence in the Hardy space
$H^1_L(\mathbb R^d)$
.

In this paper, we improve a recent result by Li and Peng on products of
functions in $H_L^1(\bR^d)$ and $BMO_L(\bR^d)$, where $L=-\Delta+V$ is a
Schr\"odinger operator with $V$ satisfying an appropriate reverse H\"older
inequality. More precisely, we prove that such products may be written as the
sum of two continuous bilinear operators, one from $...

Let $L= -\Delta+ V$ be a Schr\"odinger operator on $\mathbb R^d$, $d\geq 3$,
where $V$ is a nonnegative potential, $V\ne 0$, and belongs to the reverse
H\"older 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...

In this paper, we prove that the product (in the distribution sense) of two functions, which are respectively in and , may be written as the sum of two continuous bilinear operators, one from into , the other one from into a new kind of Hardy–Orlicz space denoted by . More precisely, the space is the set of distributions f whose grand maximal funct...

This paper has been withdrawn by the author due to crucial sign error in
Lemma 8.1.

Let 0 < p ≤ 1 and w in the Muckenhoupt class A
1. Recently, by using the weighted atomic decomposition and molecular characterization, Lee, Lin and Yang[11] established that the Riesz transforms R
j
, j = 1,2, …,n, are bounded on H
w
p
(R
n
). In this note we extend this to the general case of weight w in the Muckenhoupt class A
∞ through molec...

Let $0 < p \leq 1$ and $w$ in the Muckenhoupt class $A_1$. Recently, by using
the weighted atomic decomposition and molecular characterization; Lee, Lin and
Yang \cite{LLY} (J. Math. Anal. Appl. 301 (2005), 394--400) established that
the Riesz transforms $R_j, j=1, 2,...,n$, are bounded on $H^p_w(\mathbb R^n)$.
In this note we extend this to the ge...

Let b be a BMO-function. It is well known that the linear commutator [b, T] of a Calderón-Zygmund operator T does not, in general, map continuously H1) (Rn) into L1) (Rn)). However, Pérez showed that if H1(Rn) is replaced by a suitable atomic subspace H1b (Rn), then the commutator is continuous from H1b (Rn) into L1(Rn). In this paper, we find the...

We introduce a new class of Hardy spaces $H^{\varphi(\cdot,\cdot)}(\mathbb
R^n)$, called Hardy spaces of Musielak-Orlicz type, which generalize the
Hardy-Orlicz spaces of Janson and the weighted Hardy spaces of Garc\'ia-Cuerva,
Str\"omberg, and Torchinsky. Here, $\varphi: \mathbb R^n\times [0,\infty)\to
[0,\infty)$ is a function such that $\varphi(...

In this paper, we prove that the product (in the distribution sense) of
two functions, which are respectively in $ \BMO(\bR^n)$ and
$\H^1(\bR^n)$, may be written as the sum of two continuous bilinear
operators, one from $\H^1(\bR^n)\times \BMO(\bR^n) $ into $L^1(\bR^n)$,
the other one from $\H^1(\bR^n)\times \BMO(\bR^n) $ into a new kind of
Hardy-O...