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By
Ding, Yong; Sato, Shuichi
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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.
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By
Chen, YanPing; Ding, Yong
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4 Citations
Let b ∈ L_{loc}(ℝ^{n}) and L denote the LittlewoodPaley operators including the LittlewoodPaley 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 LittlewoodPaley operators L is only sublinear, so the results obtained in the present paper are essential improvement and extension of Uchiyama’s famous result.
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By
Ding, Yong; Wu, Xinfeng
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In this paper, the authors give a mixed norm estimate for the multiparameter fractional integrals on product measurable spaces. This estimate is applied to obtain the boundedness for the fractional integrals of NagelStein type on product manifolds, the fractional integral of FollandStein type with rough convolution kernels on product homogeneous groups, and the discrete fractional integrals of SteinWainger type.
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By
Ding, Yong; Lee, MingYi; Lin, ChinCheng
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1 Citations
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 wellknown results.
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By
Ding, Yong; Lu, Shanzhen; Xue, Quigying
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21 Citations
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.
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By
Chen, Yanping; Ding, Yong; Wang, Xinxia
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Let
,
, and
denote the Marcinkiewicz integral, the parameterized area integral, and the parameterized LittlewoodPaley
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
.
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By
Ding, Yong; Lu, Shan Zhen; Xue, Qing Ying
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6 Citations
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.
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By
Ding, Yong; Lu, Guo Zhen; Ma, Bo Lin
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6 Citations
Though the theory of oneparameter TriebelLizorkin and Besov spaces has been very well developed in the past decades, the multiparameter counterpart of such a theory is still absent. The main purpose of this paper is to develop a theory of multiparameter TriebelLizorkin and Besov spaces using the discrete LittlewoodPaleyStein analysis in the setting of implicit multiparameter structure. It is motivated by the recent work of Han and Lu in which they established a satisfactory theory of multiparameter LittlewoodPaleyStein analysis and Hardy spaces associated with the flag singular integral operators studied by MullerRicciStein and NagelRicciStein. We also prove the boundedness of flag singular integral operators on TriebelLizorkin space and Besov space. Our methods here can be applied to develop easily the theory of multiparameter TriebelLizorkin and Besov spaces in the pure product setting.
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By
Ding, Yong; Lu, Shan Zhen; Yabuta, Kôzô
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9 Citations
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.
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By
Ding, Yong; Niu, Yaoming
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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 fourthorder Schrödinger equation and the beam equation.
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