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- Chen, Yanping 7 (%)
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- Lu, Shanzhen 5 (%)
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- Mathematics [x] 50 (%)
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## Singular Integrals on Product Homogeneous Groups

### Integral Equations and Operator Theory (2013-05-01) 76: 55-79 , May 01, 2013

We consider singular integral operators with rough kernels on the product space of homogeneous groups. We prove *L*^{p} boundedness of them for
$${p \in (1,\infty)}$$
under a sharp integrability condition of the kernels.

## Commutators of Littlewood-Paley operators

### Science in China Series A: Mathematics (2009-11-01) 52: 2493-2505 , November 01, 2009

Let *b* ∈ *L*_{loc}(ℝ^{n}) and *L* denote the Littlewood-Paley operators including the Littlewood-Paley *g* function, Lusin area integral and *g*_{λ}^{*}
function. In this paper, the authors prove that the *L*^{p} boundedness of commutators [*b*, *L*] implies that *b* ∈ BMO(ℝ^{n}). The authors therefore get a characterization of the *L*^{p}-boundedness of the commutators [*b*, *L*]. Notice that the condition of kernel function of *L* is weaker than the Lipshitz condition and the Littlewood-Paley operators *L* is only sublinear, so the results obtained in the present paper are essential improvement and extension of Uchiyama’s famous result.

## Fractional Integrals on Product Manifolds

### Potential Analysis (2009-03-06) 30: 371-383 , March 06, 2009

In this paper, the authors give a mixed norm estimate for the multi-parameter fractional integrals on product measurable spaces. This estimate is applied to obtain the boundedness for the fractional integrals of Nagel-Stein type on product manifolds, the fractional integral of Folland-Stein type with rough convolution kernels on product homogeneous groups, and the discrete fractional integrals of Stein-Wainger type.

## $$\mathcal {A}_{p, {\mathbb {E}}}$$ A p , E Weights, Maximal Operators, and Hardy Spaces Associated with a Family of General Sets

### Journal of Fourier Analysis and Applications (2014-06-01) 20: 608-667 , June 01, 2014

Suppose that $${\mathbb {E}}:=\{E_r(x)\}_{r\in {\mathcal {I}}, x\in X}$$ is a family of open subsets of a topological space $$X$$ endowed with a nonnegative Borel measure $$\mu $$ satisfying certain basic conditions. We establish an $$\mathcal {A}_{{\mathbb {E}}, p}$$ weights theory with respect to $${\mathbb {E}}$$ and get the characterization of weighted weak type (1,1) and strong type $$(p,p)$$ , $$1<p\le \infty $$ , for the maximal operator $${\mathcal {M}}_{{\mathbb {E}}}$$ associated with $${\mathbb {E}}$$ . As applications, we introduce the weighted atomic Hardy space $$H^1_{{\mathbb {E}}, w}$$ and its dual $$BMO_{{\mathbb {E}},w}$$ , and give a maximal function characterization of $$H^1_{{\mathbb {E}},w}$$ . Our results generalize several well-known results.

## Marcinkiewicz integral on hardy spaces

### Integral Equations and Operator Theory (2002-06-01) 42: 174-182 , June 01, 2002

In this paper we prove that the Marcinkiewicz integral μ_{Ω} is an operator of type (*H*^{1},*L*^{1}) and of type (*H*^{1,∞},*L*^{1,∞}). As a corollary of the results above, we obtain again the the weak type (1,1) boundedness of μ_{Ω}, but the smoothness condition assumed on Ω is weaker than Stein's condition.

## Commutators of Littlewood-Paley Operators on the Generalized Morrey Space

### Journal of Inequalities and Applications (2010-07-28) 2010: 1-20 , July 28, 2010

Let , , and denote the Marcinkiewicz integral, the parameterized area integral, and the parameterized Littlewood-Paley function, respectively. In this paper, the authors give a characterization of BMO space by the boundedness of the commutators of , , and on the generalized Morrey space .

## Parametrized Area Integrals on Hardy Spaces and Weak Hardy Spaces

### Acta Mathematica Sinica, English Series (2007-09-01) 23: 1537-1552 , September 01, 2007

In this paper, the authors prove that if Ω satisfies a class of the integral Dini condition, then the parametrized area integral
$$
\mu ^{\rho }_{{\Omega ,S}}
$$
is a bounded operator from the Hardy space *H*^{1}(ℝ^{n}) to *L*^{1}(ℝ^{n}) and from the weak Hardy space *H*^{1,∞}(ℝ^{n}) to *L*^{1,∞}(ℝ^{n}), respectively. As corollaries of the above results, it is shown that
$$
\mu ^{\rho }_{{\Omega ,S}}
$$
is also an operator of weak type (1, 1) and of type (*p, p*) for 1 < *p* < 2, respectively. These conclusions are substantial improvement and extension of some known results.

## Multi-parameter Triebel-Lizorkin and Besov spaces associated with flag singular integrals

### Acta Mathematica Sinica, English Series (2010-04-01) 26: 603-620 , April 01, 2010

Though the theory of one-parameter Triebel-Lizorkin and Besov spaces has been very well developed in the past decades, the multi-parameter counterpart of such a theory is still absent. The main purpose of this paper is to develop a theory of multi-parameter Triebel-Lizorkin and Besov spaces using the discrete Littlewood-Paley-Stein analysis in the setting of implicit multi-parameter structure. It is motivated by the recent work of Han and Lu in which they established a satisfactory theory of multi-parameter Littlewood-Paley-Stein analysis and Hardy spaces associated with the flag singular integral operators studied by Muller-Ricci-Stein and Nagel-Ricci-Stein. We also prove the boundedness of flag singular integral operators on Triebel-Lizorkin space and Besov space. Our methods here can be applied to develop easily the theory of multi-parameter Triebel-Lizorkin and Besov spaces in the pure product setting.

## Multilinear Singular and Fractional Integrals

### Acta Mathematica Sinica (2006-04-01) 22: 347-356 , April 01, 2006

In this paper, we treat a class of non–standard commutators with higher order remainders in the Lipschitz spaces and give (*L*^{p}, *L*^{q}), (*H*^{p}, *L*^{q}) boundedness and the boundedness in the Triebel– Lizorkin spaces. Our results give simplified proofs of the recent works by Chen, and extend his result.

## Global L 2 $L^{2}$ estimates for a class of maximal operators associated to general dispersive equations

### Journal of Inequalities and Applications (2015-06-17) 2015: 1-20 , June 17, 2015

For a function *ϕ* satisfying some suitable growth conditions, consider the general dispersive equation defined by
$\bigl\{ \scriptsize{ \begin{array}{l} i\partial_{t}u+\phi(\sqrt{-\Delta})u=0,\quad (x,t)\in\mathbb {R}^{n}\times\mathbb{R}, \\ u(x,0)=f(x), \quad f\in\mathcal{S}(\mathbb{R}^{n}). \end{array} }\bigr. $
(∗) In the present paper, we give some global
$L^{2}$
estimate for the maximal operator
$S_{\phi}^{*}$
, which is defined by
$S^{\ast}_{\phi}f(x)= \sup_{0< t<1} |S_{t,\phi}f(x)|$
,
$x\in\mathbb{R}^{n}$
, where
$S_{t,\phi}f$
is a formal solution of the equation (∗). Especially, the estimates obtained in this paper can be applied to discuss the properties of solutions of the fractional Schrödinger equation, the fourth-order Schrödinger equation and the beam equation.