# Yongsheng HanAuburn University | AU

Yongsheng Han

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50

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Citations since 2016

## Publications

Publications (50)

Consider a family of hypersurfaces in Cn+1: Ωm = {(z1, z2, . . . , zn, zn+1) : Im(zn+1) = ( nΣ k=1 |zk|2) m }, m N. In this paper, we establish a Hardy space theory on Mm (the boundary manifold of Ωm) via a new discrete square function constructed from the heat kernel. We prove that a class of singular integral operators are bounded on the Hardy sp...

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 theor...

In this paper, we present a construction of frame without using the Fourier transform. Our methods are based on the Calderón–Zygmund operator theory and Coifman’s decomposition of the identity operator, which also work on homogeneous spaces in the sense of Coifman and Weiss.

It was well known that geometric considerations enter in a decisive way in many questions. The embedding theorem arises in several problems from partial differential equations, analysis, and geometry. The purpose of this paper is to provide a deep understanding of analysis and geometry with a particular focus on embedding theorems for spaces of hom...

In this paper, we present a construction of frames on the Heisenberg group without using the Fourier transform. Our methods are based on the Calderón-Zygmund operator theory and Coifman’s decomposition of the identity operator on the Heisenberg group. These methods are expected to be used in further studies of several complex variables.

In Nagel et al. (J Funct Anal 181:29–118, 2001), Nagel–Ricci–Stein established the relationships between product kernels and flag kernels on the Euclidean space, that is, product kernels are finite sums of flag kernels. The main purpose of this paper is to characterize the product Hardy space as the intersection of flag Hardy spaces, and characteri...

The main purpose of this paper is to study multi-parameter singular integral operators which commute with Zygmund dilations. We develop the theory of the weighted multi-parameter Hardy space H z,wp and prove the boundedness for these operators on H z,wp for certain p ≤ 1, which provide endpoint estimates for those singular integral operators studie...

The main purpose of this paper is to study multi-parameter singular integral operators which commute with Zygmund dilations. Motivated by some explicit examples of singular integral operators studied in Ricci and Stein (Ann Inst Fourier (Grenoble) 42:637–670, 1992), Fefferman and Pipher (Am J Math 11:337–369, 1997), and Nagel and Wainger (Am J Math...

Let (X, d, μ) be a metric measure space with doubling property. The Hardy spaces associated with operators L were introduced and studied by many authors. All these spaces, however, were first defined by L2(X) functions and finally the Hardy spaces were formally defined by the closure of these subspaces of L2(X) with respect to Hardy spaces norms. A...

The Marcinkiewicz multipliers are $L^{p}$ bounded for $1
on the Heisenberg group $\mathbb{H}^{n}\simeq \mathbb{C}^{n}\times \mathbb{R}$ (Müller, Ricci, and Stein). This is surprising in the sense that these multipliers are invariant under a two parameter group of dilations on $\mathbb{C}^{n}\times \mathbb{R}$ , while there is no two parameter group...

It is known that the space of homogeneous type introduced by Coifman and Weiss (1971) provides a very natural setting for establishing a theory of Hardy spaces. This paper concentrates on how the geometrical conditions of the space of homogeneous type play a crucial role in building a theory of Hardy spaces via the Littlewood-Paley functions.

It was well known that geometric considerations enter in a decisive way in many questions of harmonic analysis. The main purpose of this paper is to provide the criterion of the boundedness for singular integrals on the Hardy spaces and as well as on its dual, particularly on $\bmo$ for spaces of homogeneous type $(X, d,\mu)$ in the sense of Coifma...

This note is motivated by Phong and Stein's work in [PS]. We introduce a new class of Lipschitz spaces associated with mixed homogeneities and characterize these spaces via the Littlewood-Paley theory. We prove that the composition of two Calderon-Zygmund singular integral operators with different homogeneities is bounded on these Lipschitz spaces....

The embedding theorem arises in several problems from analysis and geometry. The purpose of this paper is to provide a deeper understanding of analysis and geometry with a particular focus on embedding theorems on spaces of homogeneous type in the sense of Coifman and Weiss. We prove that embedding theorems hold on spaces of homogeneous type if and...

The purpose of this paper is to introduce a class of general singular integral operators on spaces M = M1 × ......... × Mn. Each factor space Mi 1 < i <n, is a space of homogeneous type in the sense of Coifman and Weiss. These operators generalize those studied by Journe on the Euclidean space and include operators studied by Nagel and Stein on Car...

In this paper, we obtain the boundedness of singular integral operators T in Journé’s class on weighted multiparameter Hardy spaces
$H^{p}_{w}$
of arbitrary k number of parameters (k≥3) under the assumption that
$T^{\ast}_{i}(1)=0$
, i=1,…,k, and the kernel of T has a regularity of order ϵ>0, where
$w \in A_{r}(\Bbb{R}^{n_{1}}\times \cdots \t...

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.

This article concerns nonconvolutional type operators (also known as Journéʼs type operators) associated with a multiparameter family of dilations given by (x1,x2,…,xm)→(δ1x1,δ2x2,…,δmxm)(x1,x2,…,xm)→(δ1x1,δ2x2,…,δmxm) where x1∈Rn1,x2∈Rn2,…,xm∈Rnmx1∈Rn1,x2∈Rn2,…,xm∈Rnm and m⩾3m⩾3. We are especially interested in the boundedness of such operators on...

This paper is inspired by the work of Nagel and Stein in which the L p (1 < p < ∞) theory has been developed in the setting of the product Carnot-Carathéodory spaces M = M 1 × · · · × M n formed by vector fields satis-fying Hörmander's finite rank condition. The main purpose of this paper is to provide a unified approach to develop the multiparamet...

It is well known that standard Calderón-Zygmund singular integral operators with isotropic and nonisotropic homogeneities are bounded on the classical Hp(ℝm) and nonisotropic Hhp(ℝm), respectively. In this paper, we develop a new Hardy space theory and prove that the composition of two Calderón-Zygmund singular integral operators with different hom...

In this paper, we study the structure of the Hardy space
${H^p_b}$
associated with a para-accretive function b, which was introduced in Han et al. (J. Geom. Anal. 14:291–318, 2004). The main result is to establish a new atomic decomposition of
${H^p_b}$
. As applications, we obtain criterions for the boundedness of operators on
${H^p_b}$
and...

Nagel and Stein established $L^p$-boundedness for a class of singular
integrals of NIS type, that is, non-isotropic smoothing operators of order 0,
on spaces $\widetilde{M}=M_1\times...\times M_n,$ where each factor space $M_i,
1\leq i\leq n,$ is a smooth manifold on which the basic geometry is given by a
control, or Carnot--Carath\'eodory, metric...

Marcinkiewicz multipliers are L^{p} bounded for 1<p<\infty on the Heisenberg
group H^{n}\simeqC^{n}\timesR (D. Muller, F. Ricci and E. M. Stein) despite the
lack of a two parameter group of automorphic dilations on H^{n}. This lack of
dilations underlies the inability of classical one or two parameter Hardy space
theory to handle Marcinkiewicz mult...

We apply the discrete version of Calderón’s identity and Littlewood–Paley–Stein theory with weights to derive the \((H^p_w, H^p_w)\) and \((H^p_w, L^p_w) (0<p\le 1)\) boundedness for multiparameter singular integral operators. It turns out that even in the one-parameter case, our results substantially improve the known ones in the literature where...

In this article, we establish a new atomic decomposition for f ε L 2w ∩ H pw , where the decomposition converges in L 2w-norm rather than in the distribution sense. As applications of this decomposition, assuming that T is a linear operator bounded on L 2w and 0 < p ≤ 1, we obtain (i) if T is uniformly bounded in L pw -norm for all w-p-atoms, then...

In this paper, we explore a general method to derive Hp → Lp boundedness from Hp → Hp boundedness of linear operators, an idea originated in the work of Han and Lu in dealing with the multiparameter flag singular integrals ([19]). These linear operators include many singular integral operators in one parameter and multiparameter settings. In this p...

In this paper, we introduce the Carleson measure space CMO p on product spaces of homogeneous type in the sense of Coifman and Weiss [4], and prove that it is the dual space of the product Hardy space H p of two homogeneous spaces defined in [15]. Our results thus extend the duality theory of Chang and R. Fefferman [2, 3] on H 1 (R 2 + × R 2 +) wit...

Let T be a product Calderón–Zygmund singular integral introduced by Journé. Using an elegant rectangle atomic decomposition of Hp(Rn×Rm) and Journé's geometric covering lemma, R. Fefferman proved the remarkable Hp(Rn×Rm)−Lp(Rn×Rm) boundedness of T. In this paper we apply vector-valued singular integral, Calderón's identity, Littlewood–Paley theory...

Recently Lacey extended Chanillo's boundedness result of commutators with fractional integrals to a higher parameter setting. In this paper, we extend Lacey's result to higher dimensional spaces by a different method. Our method is in terms of the dual relationship between product BMO and product Hardy space and the maximal function characterizatio...

The main purpose of this paper is to develop a unified approach of multi-parameter Hardy space theory using the discrete Littlewood-Paley-Stein analysis in the setting of implicit multi-parameter structure. It is motivated by the goal to establish and develop the Hardy space theory for the flag singular integral operators studied by Muller-Ricci-St...

We work on RD-spaces $\Xi$ , namely, spaces of homogeneous type in the
sense of Coifman and Weiss with the additional property that a reverse doubling property holds in $\Xi$ . An important example is the Carnot-Carathéodory
space with doubling measure. By constructing an approximation of the identity with bounded support of Coifman type, we develo...

Let (X , d, µ) be a space of homogeneous type in the sense of Coifman and Weiss. Assuming that µ satisfies certain estimates from below and there exists a suitable Calderón reproducing formula in L 2 (X), the authors establish a Lusin-area characterization for the atomic Hardy spaces H p at (X) of Coifman and Weiss for p ∈ (p 0 , 1], where p 0 = n/...

By use of special wavelet bases associated to accretive or pseudo-accretive functions, it was proved that all Calderón-Zygmund
operators satisfying certain conditions form an algebra. In this article, a similar result is proved for more general para-accretive
functions. Since wavelet bases are not available for this general setting, the new idea us...

We extend the well known factorization theorems on the unit disk to product Hardy spaces, which generalizes the previous results obtained by Coifman, Rochberg and Weiss. The basic tools are the boundedness of a certain bilinear form on ℝ+2 × ℝ+2 and the characterization of BMO(ℝ+2 × ℝ+2) recently obtained by Ferguson, Lacey and Sadosky.

It was well known that Calderón–Zygmund operators T are bounded on Hp for nn+εp⩽1 provided T∗(1)=0. A new Hardy space Hbp, where b is a para-accretive function, was introduced in [Y. Han, M. Lee, C. Lin, Hardy spaces and the Tb-theorem, J. Geom. Anal. 14 (2004) 291–318] and the authors proved that Calderón–Zygmund operators T are bounded from the c...

In this paper, we prove the product H
p
boundedness of Calderón- Zygmund operators which were considered by Fefferman and Stein. The methods used in this paper are new even for the classical H
p
boundedness of Calderón- Zygmund operators, namely, using some subtle estimates together with the H
p
−L
p
boundedness of product vector valued Calderón-Zy...

It is well-known that Calderón-Zygmund operators T are bounded on Hp for
$\frac{n}{{n + 1}}< p \leqslant 1$
provided T*(1) = 0. In this article, it is shown that if T*(b) = 0, where b is a para-accretive function, T is bounded from the classical Hardy space Hp to a new Hardy space H
bp. To develop an H
bp theory, a discrete Calderón-type reproduc...

Suppose that μ is a Radon measure on ℝ d , which may be non-doubling. The only condition assumed on μ is a growth condition, namely, there is a constant C 0 >0 such that for all x∈supp(μ) and r>0, μ(B(x,r))≤C 0 r n , where 0<n≤d· The authors provide a theory of Triebel-Lizorkin spaces F ˙ pq s (μ) for 1<p<∞, 1≤q≤∞ and |s|<θ, where θ>0 is a real num...

New norms for some distributions on spaces of homogeneous type which include some fractals are introduced. Using inhomogeneous discrete Calderón reproducing formulae and the Plancherel-Pólya inequalities on spaces of homogeneous type, the authors prove that these norms give a new characterization for the Besov and Triebel-Lizorkin spaces with p, q...

Let d > 0 and θ ε (0, 1].We consider homogeneous type spaces, (X, ρ, μ)d,θ, which are variants of the well known homogeneous type spaces in the sense of Coifman and Weiss. We introduce fractional integrals and derivatives, and prove that the Besov spaces Bspq(X) and Triebel-Lizorkin spaces Fspq(X) have the lifting properties for |s| < θ. Moreover,...

In this article a Littlewood-Paley theorem for a new kind of Littlewood-Paley g-functions over spaces of homogeneous type
is presented. Based on it the authors establish inhomogeneous discrete Calderón reproducing formulas for spaces of homogeneous
type, making use of Calderón-Zygmund operators.

The authors introduce the inhomogeneous Besov space and the inhomogeneous Triebel-Lizorkin space on spaces of homogeneous type: and present their atom and molecule decompositions, their dual spaces and the complex interpolation theorems. They also establishe the relation between the homogoeneous Besov space and the inhomogeneous one, and between th...

In this paaper we use the Calderón-Zygmund operator theory to prove an inhomogenous Calderón reproducing formula on spaces
of homogeneous type with finite or infinite measures. Our formula is new even for classical spaces of homogeneous type such
as the surface of the unit ball and then-torus inR
n, compact Lie groups,C
∞-compact Riemannian manifol...

A generalized paraproduct $$\Pi _b (f)(x) = \int_0^\infty {S_t (b)(x)T_t (f)(x)\frac{{dt}}{t}} $$ is defined, where S
t
, T
t
are non-convolution operator families. The main result is that Πb
(f) is bounded on L
2(R
n
) provided b ε BMO (R
n
).

Typescript. Thesis (Ph. D.)--Washington University, 1984. Dept. of Mathematics. Includes bibliographical references (leaves 169-170).