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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.

## Jump and Variational Inequalities for Rough Operators

### Journal of Fourier Analysis and Applications (2016-06-14): 1-33 , June 14, 2016

In this paper, we systematically study jump and variational inequalities for rough operators, whose research have been initiated by Jones *et al*. More precisely, we show some jump and variational inequalities for the families
$$\mathcal T:=\{T_\varepsilon \}_{\varepsilon >0}$$
of truncated singular integrals and
$$\mathcal M:=\{M_t\}_{t>0}$$
of averaging operators with rough kernels, which are defined respectively by
$$\begin{aligned} T_\varepsilon f(x)=\int _{|y|>\varepsilon }\frac{\Omega (y')}{|y|^n}f(x-y)dy \end{aligned}$$
and
$$\begin{aligned} M_t f(x)=\frac{1}{t^n}\int _{|y|<t}\Omega (y')f(x-y)dy, \end{aligned}$$
where the kernel
$$\Omega $$
belongs to
$$L\log ^+\!\!L(\mathbf S^{n-1})$$
or
$$H^1(\mathbf S^{n-1})$$
or
$$\mathcal {G}_\alpha (\mathbf S^{n-1})$$
(the condition introduced by Grafakos and Stefanov). Some of our results are sharp in the sense that the underlying assumptions are the best known conditions for the boundedness of corresponding maximal operators.

## 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.