arXiv subjects

Yongsheng Han

Publications and source records attributed to Yongsheng Han.

25 records · Page 2Linked to original sources

arXiv2015

Criterion of the boundedness of singular integrals on spaces of homogeneous type

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 $(X, d,μ)$ in the sense of Coifman and Weiss, where the quasi-metric $d$ may have no regularity and the measure $μ$ satisfies only the doubling property. We make no additional geometric assumptions on the quasi-metric or the doubling measure and thus, the results of this paper extend to the full generality of all related previous ones, in which the extra geometric assumptions were made on both the quasi-metric $d$ and the measure $μ.$ To achieve our goal, we prove that the atomic Hardy spaces introduced by Coifman and Weiss coincide with the Hardy spaces defined in terms of wavelet coefficients and develop the molecule theory for this general setting. The main tools used in this paper are atomic decomposition, the orthonormal wavelet basis constructed recently by Auscher and Hytönen, the discrete Calderón-type reproducing formula, the almost orthogonal estimates, implement various stopping time arguments and the duality of the Hardy spaces with the Carleson measure spaces.

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arXiv2015

Geometric characterizations of embedding theorems

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 only if geometric conditions, namely the measures of all balls have lower bounds, hold. As applications, our results provide new and sharp previous related embedding theorems for the Sobolev, Besov and Triebel-Lizorkin spaces.

math.CA↗

arXiv2015

Hardy space theory on spaces of homogeneous type via orthonormal wavelet bases

In this paper, using the remarkable orthonormal wavelet basis constructed recently by Auscher and Hytönen, we establish the theory of product Hardy spaces on spaces ${\widetilde X} = X_1\times X_2\times\cdot \cdot\cdot\times X_n$, where each factor $X_i$ is a space of homogeneous type in the sense of Coifman and Weiss. The main tool we develop is the Littlewood--Paley theory on $\widetilde X$, which in turn is a consequence of a corresponding theory on each factor space. We define the square function for this theory in terms of the wavelet coefficients. The Hardy space theory developed in this paper includes product~$H^p$, the dual $\cmo^p$ of $H^p$ with the special case $\bmo = \cmo^1$, and the predual $\vmo$ of $H^1$. We also use the wavelet expansion to establish the Calderón--Zygmund decomposition for product $H^p$, and deduce an interpolation theorem. We make no additional assumptions on the quasi-metric or the doubling measure for each factor space, and thus we extend to the full generality of product spaces of homogeneous type the aspects of both one-parameter and multiparameter theory involving the Littlewood--Paley theory and function spaces. Moreover, our methods would be expected to be a powerful tool for developing wavelet analysis on spaces of homogeneous type.

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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.

math.CA↗

arXiv2012

T1 theorem on product Carnot-Caratheodory spaces

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--Carathéodory, metric induced by a collection of vector fields of finite type. In this paper we prove the product $T1$ theorem on $L^2,$ the Hardy space $H^p(\widetilde{M})$ and the space $CMO^p(\widetilde{M})$, the dual of $H^p(\widetilde{M}),$ for a class of product singular integral operators which covers Journé's class and operators studied by Nagel and Stein.

math.FA↗

arXiv2012

Flag Hardy spaces and Marcinkiewicz multipliers on the Heisenberg group: an expanded version

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 multipliers on H^{n} when 0<p\leq1. We address this deficiency by developing a theory of flag Hardy spaces H_{flag}^{p} on the Heisenberg group, 0<p\leq1, that is in a sense `intermediate' between the classical Hardy spaces H^{p} and the product Hardy spaces H_{product}^{p} on C^{n}\timesR. We show that flag singular integral operators, which include the aforementioned Marcinkiewicz multipliers, are bounded on H_{flag}^{p}, as well as from H_{flag}^{p} to L^{p}, for 0<p\leq1. We characterize the dual spaces of H_{flag}^{1} and H_{flag}^{p}, and establish a Calderón-Zygmund decomposition that yields standard interpolation theorems for the flag Hardy spaces H_{flag}^{p}. In particular, this recovers the L^{p} results by interpolating between those for H_{flag}^{p} and L^{2} (but regularity sharpness is lost).

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arXiv2008

Discrete Littlewood-Paley-Stein theory and multi-parameter Hardy spaces associated with flag singular integrals

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-Stein and Nagel-Ricci-Stein. This approach enables us to avoid the use of transference method of Coifman-Weiss as often used in the $L^p$ theory for $p>1$ and establish the Hardy spaces $H^p_F$ and its dual spaces associated with the flag singular integral operators for all $0<p\leq 1$. We also prove the boundedness of flag singular integral operators on $BMO_F$ and $H^p_F$, and from $H^p_F$ to $L^p$ for all $0<p\le 1$ without using the deep atomic decomposition. As a result, it bypasses the use of Journe's type covering lemma in this implicit multi-parameter structure. The method used here provides alternate approaches of those developed by Chang, R. Fefferman, Journe and Pipher in the pure product setting. A Calderon-Zygmund decomposition and interpolation theorem are also proved for the implicit multi-parameter Hardy spaces.

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