Some properties of singular integral operators over an open curve Dedicated to the 85th birthday of Prof. Lu Jianke

In this paper, the stability properties, the endpoint behavior and the invertible relations of Cauchy-type singular integral operators over an open curve are discussed. If the endpoints of the curve are not special, this type of operators are proved to be stable. At the endpoints, either the singularity or smoothness of the operators are exactly described. And the function sets or spaces on which the operators are invertible as well as the corresponding inverted operators are given. Meanwhile, some applications for the solution of Cauchy-type singular integral equations are illustrated.     

Some properties of singular integral operators over an open curve Dedicated to the 85th birthday of Prof.Lu Jinake

In this paper, the stability properties, the endpoint behavior and the invertible relations of Cauchy-type singular integral operators over an open curve are discussed. If the endpoints of the curve are not special, this type of operators are proved to be stable. At the endpoints, either the singularity or smoothness of the operators are exactly described. And the function sets or spaces on which the operators are invertible as well as the corresponding inverted operators are given. Meanwhile, some applications for the solution of Cauchy-type singular integral equations are illustrated.     

Some BMO estimates for vector-valued multilinear singular integral operators

In this paper, we prove some BMO end-point estimates for some vector-valued multilinear operators related to certain singular integral operators.     

On the monotonicity of some singular integral operators

The paper deals with the monotonicity of singular integral operators of the form where is the Cauchy singular integral operator on the interval (0,1) of the real axis and q is a power or logarithmic function. Under suitable assumptions, such singular integral operators are proved to be monotone and maximal monotone in spaces with power weights. Moreover, two related integral equations with weakly singular kernels of logarithmic type are studied. Copyright © 2012 John Wiley & Sons, Ltd.     

A new BLO estimate for maximal singular integral operators

In this paper, we extend Hu and Zhang’s results in [2] to different case.     

Lack of natural weighted estimates for some singular integral operators

We show that the classical Hörmander condition, or analogously the -Hörmander condition, for singular integral operators is not sufficient to derive Coifman's inequality

where , is the Hardy-Littlewood maximal operator, is any weight and is a constant depending upon and the constant of . This estimate is well known to hold when is a Calderón-Zygmund operator.

As a consequence we deduce that the following estimate does not hold:

where and where is an arbitrary weight. However, by a recent result due to A. Lerner, this inequality is satisfied whenever is a Calderón-Zygmund operator.

One of the main ingredients of the proof is a very general extrapolation theorem for weights.

     

Estimates for the maximal bilinear singular integral operators

In this paper, the behavior on the product of Lebesgue spaces is considered for the maximal operators associated with the bilinear singular integral operators whose kernels satisfy certain minimal regularity conditions.     

Weighted estimates for commutators of singular integral operators with non-doubling measures

Letμbe a nonnegative Radon measure on R~d which only satisfiesμ(B(x,r))≤C_0r~n for all x∈R~d,r>0,and some fixed constants C_0>0 and n∈(0,d].In this paper,some weighted weak type estimates with A_(p,(log L)~σ)~ρ(μ) weights are established for the commutators generated by Calder■n-Zygmund singular integral operators with RBMO(μ) functions.     

Boundedness of a class of super singular integral operators and the associated commutators

In this paper we give the (Lα p, Lp) boundedness of the maximal operator of a class of super singular integrals defined bywhich improves and extends the known result. Moreover, by applying an off-Diagonal T1 Theorem, we also obtain the (Lp, Lq) boundedness of the commutator defined by     

Estimates for multilinear singular integral operators with nonsmooth kernels

Let (, d, μ) be a space of homogeneous type in the sense of Coifman and Weiss. In this paper, we consider the behavior on × ⋯ × for the m-linear singular integral operators with nonsmooth kernels which were first introduced by Duong, Grafakos and Yan.     

An endpoint estimate for maximal multilinear singular integral operators

A weak type endpoint estimate for the maximal multilinear singular integral operator T*Af(x)=supε＞0|(f)(x-y)＞ε (Ω(x-y)/(|x-y|(n 1)))(A(x)-A(y)-▽A(y)(x-y))f(y)dy| is established, where Ω is homogeneous of degree zero, integrable on the unit sphere and has vanishing moment of order one, and A has derivatives of order one in BMO(Rn). A regularity condition on Ω which implies an LlogL type estimate of T*A is given.     

Stability of approximation under the action of singular integral operators

Let T be a singular integral operator, and let 0 < α < 1. If t > 0 and the functions f and Tf are both integrable, then there exists a function $g \in B_{Lip_\alpha } (ct)$ such that $\left\| {f - g} \right\|_{L^1 } \leqslant Cdist_{L^1 } (f,B_{Lip_\alpha } (t))$ and $\left\| {Tf - Tg} \right\|_{L^1 } \leqslant C\left\| {f - g} \right\|_{L^1 } + dist_{L^1 } (Tf,B_{Lip_\alpha } (t)).$ . (Here B _X (τ) is the ball of radius τ and centered at zero in the space X; the constants C and c do not depend on t and f.) The function g is independent of T and is constructed starting with f by a nearly algorithmic procedure resembling the classical Calderón-Zygmund decomposition.     

截口上的奇异积分高阶交换子的加权估计

研究截口上的奇异积分高阶交换子H^mbf（xz）,利用截口F关于核的ki（x,y）估计,在一定假设下得到了H^mbf的Sharp极大函数估计和加权弱（1,1）型估计．     

Boundedness of the anisotropic maximal and anisotropic singular integral operators in generalized Morrey spaces

In this paper we give the conditions on the pair (ω ₁, ω ₂) which ensures the boundedness of the anisotropic maximal operator and anisotropic singular integral operators from one generalized Morrey space M_p,w1 \mathcal{M}_{p,\omega _1 } to another M_p,w2 \mathcal{M}_{p,\omega _2 }, 1 < p < g8, and from the space M_1,w1 \mathcal{M}_{1,\omega _1 } to the weak space WM_1,w2 W\mathcal{M}_{1,\omega _2 }.     

Pointwise estimates for the maximal multilinear singular integral operators

Two pointwise estimates relating the maximal multilinear singular integral operators and some classical maximal operators are established. These pointwise estimates imply the rearrangement estimate and the BLO(Rn) estimate for the maximal multilinear singular integral operators.     

Weighted estimate for the multilinear oscillatory singular integral operators

In this paper, the weighted L^p(R^n) boundedness for the multilinear oscillatory singular integral operators with polynomial phases is studied.     

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.     

A Cotlar type inequality for the multilinear singular integral operators and its applications

A Cotlar type inequality is established for the multilinear singular integral operators. As applications, some two-weight norm inequalities are obtained for the maximal operator corresponding to the multilinear singular integral operators.     

A note on the multilinear singular integral operators

Lp(Rn) boundedness is considered for the multilinear singular integral operator defined by TAf(x) = ∫Rn Ω(x - y)/|x - y|n 1 (A(x) - A(y) - (△)A(y)(x - y))f(y)dy,where Ω is homogeneous of degree zero, integrable on the unit sphere and has vanishing moment of order one. A has derivatives of order one in BMO(Rn). We give a smoothness condition which is fairly weaker than that Ω∈ Lipα(Sn-1) (0 ＜α≤ 1) and implies the Lp(Rn) (1 ＜ p ＜ oo) boundedness for the operator TA. Some endpoint estimates are also established.     

Boundary properties for several singular integral operators in real Clifford analysis

On the basis of the Cauchy integral formulas for regular and biregular functions, we define some Cauchy-type singular integral operators. Then we discuss the Hlder continuous property of some singular integral operators with one integral variable. Then we divide a singular integral operator with two variables into three parts and prove its Hlder continuous property on the boundary.