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.     

Boundedness for parabolic singular integral with rough kernels and its commutators on Triebel-Lizorkin spaces

In this paper, the authors give the boundedness on Triebel-Lizorkin spaces for the parabolic singular integral with rough kernel and its commutator.     

Boundedness of oscillatory singular integral with rough kernels on Triebel-Lizorkin spaces

In this paper, we will prove the Triebel-Lizorkin boundedness for some oscillatory singular integrals with the kernel (x) satisfying a condition introduced by Grafakos and Stefanov. Our theorems will be proved under various conditions on the phase function, radial and nonradial. Since the L p boundedness of these operators is not complete yet, the theorems extend many known results.     

带振荡因子的粗双曲奇异积分算子

The singular integral operator J Ω,α, and the Marcinkiewicz integral operator (~μ)Ω,α are studied. The kernels of the operators behave like |y|-n-α(α＞0) near the origin, and contain an oscillating factor ei|y|-β(β＞0) and a distribution Ω on the unit sphere Sn-1 It is proved that, if Ω is in the Hardy space Hr (Sn-1) with 0＜r= (n-1)/(n-1 )(＞0), and satisfies certain cancellation condition,then J Ω,α and uΩ,α extend the bounded operator from Sobolev space Lpγ to Lebesgue space Lp for some p. The result improves and extends some known results.     

Vector-valued singular integral operators with rough kernels

In this paper, we establish a weak-type (1,1) boundedness criterion for vector-valued singular integral operators with rough kernels. As applications, we obtain weak-type (1,1) bounds for the convolution singular integral operator taking value in the Banach space Y with a rough kernel, the maximal operator taking vector value in Y with a rough kernel and several square functions with rough kernels. Here, $Y={[H,X]}_{\theta }$ is a complex interpolation space between a Hilbert space H and a UMD space X.     

Boundedness of the commutator of Marcinkiewicz integral with rough variable kernel

In this paper the author proves that the commutator of the Marcinkiewicz integral operator with rough variable kernel is bounded from the homogeneous Sobolev space Lγ^2（R^n） to the Lebesgue space L^2（R^n）, which is a substantial improvement and extension of some known results.     

A rough hypersingular integral operator with an oscillating factor on function space

The singular integral operator Tα,βf(x) = p.v.∫Rnei|y|-βΩ(y') /|y|n αf(x-y)dy,defined for all test functions f is studied, where Ω(y') is a distribution on the unit sphere Sn-1 satisfying certain cancellation condition. It is proved that Tα,β is a bounded operator from the Triebel-Lizorkin space Fs,qp to the Triebel-Lizorkin space Fs γ,qp, provided that Ω(y') is a distribution in the Hardy space Hr(Sn-1) with r = (n - 1)/(n - 1 γ).     

Commutators of integral operators with variable kernels on Hardy spaces

LetT _Ω,α (0 ≤ α< n) be the singular and fractional integrals with variable kernel Ω(x, z), and [b, T_Ω,α] be the commutator generated by T_Ω,α and a Lipschitz functionb. In this paper, the authors study the boundedness of [b, T_Ω,α] on the Hardy spaces, under some assumptions such as theL ^r -Dini condition. Similar results and the weak type estimates at the end-point cases are also given for the homogeneous convolution operators . The smoothness conditions imposed on are weaker than the corresponding known results.     

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.      

Operators with rough singular kernels

For α>0, we study the singular integral operators TΩ,α and the Marcinkiewicz integral operator μΩ,α. The kernels of these operators behave like |y|^−n−α near y=0, and contain a distribution Ω on the unit sphere Sn−1. We prove that if Ω∈Hr(Sn−1)(r=(n−1)/(n−1+α)) satisfying certain cancellation condition, then both TΩ,α and μΩ,α can be extend to be the bounded operators from the Sobolev space to the Lebesgue space Lp(Rn). The result improves and extends some known results.     

Herz型Hardy空间上的一类交换子

证明一大类次线性算子与ＢＭＯ函数构成的交换子在Ｈａｒｚ型Ｈａｒｄｙ空间上的有界性。     

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.     

Nonconvex variational problem with recursive integral functionals in Sobolev spaces: Existence and representation

The purpose of this paper is twofold. First, we present the existence theorem of an optimal trajectory in a nonconvex variational problem with recursive integral functionals by employing the norm-topology of a weighted Sobolev space. We show the continuity of the integral functional and the compactness of the set of admissible trajectories. Second, we show that a recursive integrand is represented by a normal integrand under the conditions guaranteeing the existence of optimal trajectories. We also demonstrate that if the recursive integrand satisfies the convexity conditions, then the normal integrand is a convex function. These results are achieved by the application of the representation theorem in Lp-spaces.     

A rough hypersingular integral operator with an oscillating factor

We study certain hypersingular integrals TΩ,α,βf defined on all test functions f∈S(Rn), where the kernel of the operator TΩ,α,β has a strong singularity |y|^−n−α(α>0) at the origin, an oscillating factor ei|y|^−β(β>0) and a distribution Ω∈Hr(Sn−1), 0<r<1. We show that TΩ,α,β extends to a bounded linear operator from the Sobolev space to the Lebesgue space Lp for β/(β−α)<p<β/α, if the distribution Ω is in the Hardy space Hr(Sn−1) with 0<r=(n−1)/(n−1+γ)(0<γ?α) and β>2α>0.     

Rough singular integral operators on Hardy-Sobolev spaces

The authors study the singular integral operator TΩ,αf(x)=p.v.∫R^nb(|y|Ω(y′)|y|^-n-αf(x-y)dy, defined on all test functions f, where b is a bounded function, α＞0, Ω(y′) is an integrable function on the unit sphere S^n-1 satisfying certain cancellation conditions. It is proved that, for n/(n α)＜p＜∞,TΩ,α is a bounded operator from the Hardy-Sobolev space H^pα to the Hardy space H^p. The results and its applications improve some theorems in a previous paper of the author and they are extensions of the main theorems in Wheeden‘s paper(1969). The proof is based on a new atomic decomposition of the space H^pα by Han, Paluszynski and Weiss(1995). By using the same proof,the singluar integral operators with variable kernels are also studied.     

Marcinkiewicz积分交换子在一些Hardy空间的有界性

讨论了由核函数满足具有某类Dini条件的Marcinkiewicz积分μΩ及函数b∈Lipβ(R~n)生成的交换子μ(_Ω,b)~m的性质.证明了Marcinkiewicz积分交换子μ_(_Ω,b)~m在Hardy型空间H_(bm,s)(R~n)上有界,也在Herz型Hardy空间H_(bm)K_p~(a_q)(R~n)上有界.     

Sobolev spaces on bounded symmetric domains

We obtain a formula for the Sobolev inner product in standard weighted Bergman spaces of holomorphic functions on a bounded symmetric domain in terms of the Peter–Weyl components in the Hua–Schmidt decomposition, and use it to clarify the relationship between the analytic continuation of these standard weighted Bergman spaces and the Sobolev spaces on bounded symmetric domains.     

Boundedness of rough singular integral operators on the Triebel-Lizorkin spaces

We consider the singular integral operator T with kernel K(x)=Ω(x)/n|x| and prove its boundedness on the Triebel-Lizorkin spaces provided that Ω satisfies a size condition which contains the case Ω∈Lr(Sn−1), r>1.     

Weighted estimates for commutators of singular integral operators with non-smooth kernels

In this paper, we establish the boundedness of commutators of singular integral operators with non-smooth kernels on weighted Lipschitz spaces Lipβ,ω. The condition on the kernel in this paper is weaker than the usual pointwise H¨ormander condition.     

Boundedness and compactness characterizations of Cauchy integral commutators on Morrey spaces

Let C_Γ be the Cauchy integral operator on a Lipschitz curve Γ. In this article, the authors show that the commutator [b,C_Γ] is bounded (resp, compact) on the Morrey space for any (or some) p ∈ (1,∞) and λ ∈ (0,1) if and only if (resp, ). As an application, a factorization of the classical Hardy space in terms of C_Γ and its adjoint operator is obtained.