On rough Marcinkiewicz integrals along surfaces

In this paper, the authors establish the LP-mapping properties of certain classes of Marcinkiewicz integral operators along surfaces with rough kernels. The results in this paper essentially extend as well as improve previously known results.     

Parabolic Marcinkiewicz integrals along surfaces on product domains

In this paper, parabolic Marcinkiewicz integral operators along surfaces on the product domain ℝ ⁿ × ℝ ^m (n,m ⩾ 2) are introduced. L ^p bounds of such operators are obtained under weak conditions on the kernels.     

Rough Marcinkiewicz integrals with mixed homogeneity on product spaces

In this paper, the parabolic Marcinkiewicz integral operators on the product spaces Rm × Rn (m, n≥ 2) are studied. The Lp -boundedness for such operators are established under rather weak size conditions of the kernels, which essentially improve or extend certain previous results.     

L p boundedness of Marcinkiewicz integrals with H 1 kernels

We establish the L^p boundedness of Marcinkiewicz integral operators with kernels in theHardy space on a compact submanifold of finite type. Previously such boundedness properties were obtained when the submanifold is a sphere. Our result shows that what matters is not the symmetry, but a certain type of curvature property of the submanifold.     

粗糙核带参数的抛物型Marcinkiewicz函数的一个注记

主要研究了带参数的抛物型Marcinkiewicz函数μσΩ,h（f）的L2（（R）n）有界性,用核的分解技术和Fourier变换估计的方法分别在当1＜-γ＜∞,h∈Hγ＇（IR）＋）,Ω∈L（logL）1/γ（Sn-1）条件下和当1〈γ≤∞,h∈△γ（（IR）＋）,Ω∈Llog＋L（Sn-1）条件下,建立了μσΩ,h（f）的L2（（R）n）有界性,并推广了以前学者的结论.     

A new class of weights adapted to rough singular integrals and Marcinkiewicz integrals

In this paper, we introduce a new class of weights Ap （Rn） which retains many fine properties of the classical Muchenhoupt weights Ap （Rn）. While Ap （Rn） is too big a class to obtain the weighted norm inequalities for rough singular integrals and Marcinkiewicz integrals, our new class Ap （Rn） adapts well to these rough operators. As applications, we improve some known weighted estimates.     

On multiple singular integrals along polynomial curves with rough kernels

This paper is devoted to the study of a class of singular integral operators defined by polynomial mappings on product domains. Some rather weak size conditions, which imply the Lp boundedness of these singular integral operators as well as the corresponding maximal truncated singular integral operators for some fixed 1〈p〈 ∞,are given.     

Commutators of Marcinkiewicz integral with rough kernels on Sobolev spaces

In this paper, the authors give the boundedness of the commutator [b, ??_??,?? ] from the homogeneous Sobolev space $\dot L_\gamma ^p \left( {\mathbb{R}^n } \right)$ to the Lebesgue space L ^p (? ⁿ ) for 1 < p < ??, where ??_??,?? denotes the Marcinkiewicz integral with rough hypersingular kernel defined by $\mu _{\Omega ,\gamma } f\left( x \right) = \left( {\int_0^\infty {\left| {\int_{\left| {x - y} \right| \leqslant t} {\frac{{\Omega \left( {x - y} \right)}} {{\left| {x - y} \right|^{n - 1} }}f\left( y \right)dy} } \right|^2 \frac{{dt}} {{t^{3 + 2\gamma } }}} } \right)^{\frac{1} {2}} ,$ , with ?? ?? L ¹(S ^n?1) for $0 < \gamma < min\left\{ {\frac{n} {2},\frac{n} {p}} \right\}$ or ?? ?? L(log⁺ L) ^?? (S ^n?1) for $\left| {1 - \frac{2} {p}} \right| < \beta < 1\left( {0 < \gamma < \frac{n} {2}} \right)$ , respectively.     

A note on the generalized Marcinkiewicz integral operators with rough kernels

In this paper,we study the generalized Marcinkiewicz integral operators MΩ,r on the product space Rn×Rm.Under the condition that Ω is a function in certain block spaces,which is optimal in some senses,the boundedness of such operators from Triebel-Lizorkin spaces to Lp spaces is obtained.     

Marcinkiewicz integral with rough kernels

In this summary paper, we will introduce some recent progress in the theory of Marcinkiewicz integral and will pay more attention to the case of rough kernels.      

A note on parameterized Marcinkiewicz integrals with variable kernels

In this paper,the parameterized Marcinkiewicz integrals with variable kernels defined by μΩ^ρ（f）（x）=（∫0^∞│∫│1-y│≤t Ω（x,x-y）/│x-y│^n-p f（y）dy│^2dt/t1＋2p）^1/2 are investigated.It is proved that if Ω∈ L∞（R^n） × L^r（S^n-1）（r〉（n-n1p＇/n） is an odd function in the second variable y,then the operator μΩ^ρ is bounded from L^p（R^n） to L^p（R^n） for 1 〈 p ≤ max{（n＋1）/2,2}.It is also proved that,if Ω satisfies the L^1-Dini condition,then μΩ^ρ is of type（p,p） for 1 〈 p ≤ 2,of the weak type（1,1） and bounded from H1 to L1.     

L p bounds for singular integrals with rough kernels on product domains

This paper is concerned with singular integral operators on product domains with rough kernels both along radial direction and on spherical surface.Some rather weaker size conditions,which imply the Lp-boundedness of such operators for certain fixed p(1 p ∞),are given.     

Generalized commutators for Marcinkiewicz type integrals with variable kernels

Let A be a function with derivatives of order m and D γ A ∈■β (0 < β < 1, |γ| = m). The authors in the paper prove that if Ω(x, z) ∈ L ∞ (R n ) × L s (S n 1 ) (s ≥ n/(n β)) is homogenous of degree zero and satisfies the mean value zero condition about the variable z, then both the generalized commutator for Marcinkiewicz type integral μ A Ω and its variation μ A Ω are bounded from L p (R n ) to L q (R n ), where 1 < p < n/β and 1/q = 1/p β/n. The authors also consider the boundedness of μ A Ω and its variation μ A Ω on Hardy spaces.     

Boundedness of Marcinkiewicz integral related to multilinear commutators with variable kernels on Hardy spaces

Let→b=(b1,b2,…,bm),bi∈∧βi(Rn),1≤I≤m,βi＞0,m∑I=1βi=β,0＜β＜1,μΩ→b(f)(x)=(∫∞0|F→b,t(f)(x)|2dt/t3)1/2,F→b,t(f)(x)=∫|x-y|≤t Ω(x,x-y)/|x-y|n-1 mΠi=1[bi(x)-bi(y)dy.We consider the boundedness of μΩ,→b on Hardy type space Hp→b(Rn).     

沿复合子簇的粗糙核极大算子

给出了两类相关于沿复合子簇的粗糙核奇异积分的极大算子的L~p有界性,本质上极大地改进和一般化了已有的结果.作为应用,相关的奇异积分,Marcinkiewicz积分和相应的极大算子的L~p有界性也被建立.     

Maximal operators with rough kernels on product domains

In this paper, we study the Lp boundedness of certain maximal operators on product domains with rough kernels in L(logL). We prove that our operators are bounded on Lp for all 2?p<∞. Moreover, we show that our condition on the kernel is optimal in the sense that the space L(logL) cannot be replaced by Lr(logL) for any r<1. Our results resolve a problem left open in [Y. Ding, A note on a class of rough maximal operators on product domains, J. Math. Anal. Appl. 232 (1999) 222-228].     

Hypersingular parameterized Marcinkiewicz integrals with variable kernels on Sobolev and Hardy-Sobolev spaces

Let α≥ 0 and 0 〈 ρ ≤ n/2, the boundedness of hypersingular parameterized Marcinkiewicz integrals μΩ,α^ρ with variable kernels on Sobolev spaces Lα^ρ and HardySobolev spaces Hα^ρ is established.     

Rough singular integrals associated with surfaces of Van der Corput type简

In this paper,the authors establish the Lp-mapping properties for a class of singular integrals along surfaces in Rn of the form {φ(|u|)u : u ∈ Rn} as well as the related maximal operators provided that the function φ satisfies certain oscillatory integral estimates of Van der Corput type,and the integral kernels are given by the radial function h ∈Δγ(R+) for γ 1 and the sphere function Ω∈ Fβ(Sn.1) for some β 0,which is distinct from H1(Sn.1).