Multilinear commutators of BMO functions and multilinear singular integrals with non-smooth kernels

This paper is concerned with certain multilinear commutators of BMO functions and multilinear singular integral operators with non-smooth kernels. By the sharp maximal functions estimates, the weighted norm inequalities for this kind of commutators are established.     

Sharp maximal function estimates for multilinear singular integrals with non-smooth kernels

In this article we obtain weighted norm estimates for multilinear singular integrals with non-smooth kernels and the boundedness of certain multilinear commutators by making use of a sharp maximal function.     

Maximal operator for multilinear Calderón-Zygmund singular integral operators on weighted Hardy spaces

In this paper, the maximal operator associated with multilinear Calderón-Zygmund singular integral operators will be studied by using an improved Coltlar's inequality. Moreover, weighted norm inequalities and some estimates on weighted Hardy spaces are obtained for this maximal operator.     

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.     

Weighted inequalities and pointwise estimates for the multilinear fractional integral and maximal operators

In this article we prove weighted norm inequalities and pointwise estimates between the multilinear fractional integral operator and the multilinear fractional maximal. As a consequence of these estimations we obtain weighted weak and strong inequalities for the multilinear fractional maximal operator or function. In particular, we extend some results given in Carro et al. (2005) [7] to the multilinear context. On the other hand we prove weighted pointwise estimates between the multilinear fractional maximal operator Mα,B associated to a Young function B and the multilinear maximal operators Mψ=M0,ψ, ψ(t)=B(t1−α/(nm))^nm/(nm−α). As an application of these estimate we obtain a direct proof of the Lp−Lq boundedness results of Mα,B for the case B(t)=t and Bk(t)=tk(1+log⁺t) when 1/q=1/p−α/n. We also give sufficient conditions on the weights involved in the boundedness results of Mα,B that generalizes those given in Moen (2009) [22] for B(t)=t. Finally, we prove some boundedness results in Banach function spaces for a generalized version of the multilinear fractional maximal operator.     

Commutators of weighted lipschitz functions and multilinear singular integrals with non-smooth kernels

This paper is devoted to studying the commutators of the multilinear singular integral operators with the non-smooth kernels and the weighted Lipschitz functions. Some mapping properties for two types of commutators on the weighted Lebesgue spaces, which extend and generalize some previous results, are obtained.     

Multilinear singular integral operators with generalized kernels and their multilinear commutators

In this paper, the authors study a class of multilinear singular integral operators with generalized kernels and their multilinear commutators with BMO functions. By establishing the sharp maximal estimates, the boundedness on product of weighted Lebesgue spaces and product of variable exponent Lebesgue spaces is obtained, respectively. Moreover, the endpoint estimate of this class of mutilinear singular integral operators is also established. These results can improve the corresponding known results of classical multilinear Calderón–Zygmund operators and multilinear Calder′on–Zygmund operators with Dini type kernels.     

Pseudo-localization of singular integrals and noncommutative Calderón-Zygmund theory

The weak type (1,1) boundedness of singular integrals acting on matrix-valued functions has remained open since the 1980s, mainly because the methods provided by the vector-valued theory are not strong enough. In fact, we can also consider the action of generalized Calderón-Zygmund operators on functions taking values in any other von Neumann algebra. Our main tools for its solution are two. First, the lack of some classical inequalities in the noncommutative setting forces to have a deeper knowledge of how fast a singular integral decreases—L₂ sense—outside of the support of the function on which it acts. This gives rise to a pseudo-localization principle which is of independent interest, even in the classical theory. Second, we construct a noncommutative form of Calderón-Zygmund decomposition by means of the recent theory of noncommutative martingales. This is a corner stone in the theory. As application, we obtain the sharp asymptotic behavior of the constants for the strong Lp inequalities, also unknown up to now. Our methods settle some basics for a systematic study of a noncommutative Calderón-Zygmund theory.     

Maximal commutators of BMO functions and singular integral operators with non-smooth kernels on spaces of homogeneous type

Let X be a space of homogeneous type in the sense of Coifman and Weiss. In this paper, via a new Cotlar type inequality linking commutators and corresponding maximal operators, a weighted Lp(X) estimate with general weights and a weak type endpoint estimate with A₁(X) weights are established for maximal operators corresponding to commutators of BMO(X) functions and singular integral operators with non-smooth kernels.     

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.     

The boundedness of multilinear commutators of singular integrals on Lebesgue spaces with variable exponent

The boundednees of multilinear commutators of Calderón-Zygmund singular integrals on Lebesgue spaces with variable exponent is obtained. The multilinear commutators of generalized Hardy-Littlewood maximal operator are also considered.     

Multilinear singular integrals and commutators in variable exponent Lebesgue spaces

Boundedness of multilinear singular integrals and their commutators from products of variable exponent Lebesgue spaces to variable exponent Lebesgue spaces are obtained. The vector-valued case is also considered.     

Generalized Hörmander's conditions, commutators and weights

We present a general framework to deal with commutators of singular integral operators with BMO functions. Hörmander type conditions associated with Young functions are assumed on the kernels. Coifman type estimates, weighted norm inequalities and two-weight estimates are considered. We give applications to homogeneous singular integrals, Fourier multipliers and one-sided operators.     

Boundedness in Lebesgue spaces for commutators of multilinear singular integrals and RBMO functions with non-doubling measures

The boundedness in Lebesgue spaces for commutators generated by multilinear singular integrals and RBMO(μ) functions of Tolsa with non-doubling measures is obtained, provided that‖μ‖=∞and multilinear singular integrals are bounded from L1(μ)×L1(μ)to L1/2,∞(μ).     

Commutators of BMO functions and singular integral operators with non-smooth kernels on spaces of Homogeneous type of finite measure

Let X be a space of homogeneous type with finite measure. Let T be a singular integral operator which is bounded on Lp (X), 1 ＜ p ＜∞. We give a sufficient condition on the kernel k(x,y) of Tso that when a function b ∈ BMO (X) ,the commutator [b, T] (f) = T (b f) - bT (f) is aounded on spaces Lp for all p, 1 ＜ p ＜∞.     

Multilinear Calderón-Zygmund operators on the product of Lebesgue spaces with non-doubling measures

Under the assumption that μ is a non-doubling measure on Rd, the author proves that for the multilinear Calderón-Zygmund operator, its boundedness from the product of Hardy space H¹(μ)×H¹(μ) into L1/2(μ) implies its boundedness from the product of Lebesgue spaces Lp₁(μ)×Lp₂(μ) into Lp(μ) with 1<p₁,p₂<∞ and p satisfying 1/p=1/p₁+1/p₂.     

Uniform convergence and integrability of Fourier integrals

Firstly, we study the uniform convergence of cosine and sine Fourier transforms. Secondly, we obtain Pitt-Boas type results on Lp-integrability of Fourier transforms with the power weights. The solutions of both problems are written as criteria in terms of general monotone functions.     

The boundedness of maximal operators and singular integrals via Fourier transform estimates

In this paper, the author studies the mapping properties for some general maximal operators and singular integrals on certain function spaces via Fourier transform estimates. Also, some concrete maximal operators and singular integrals are studied as applications.     

On local properties of functions and singular integrals in terms of the mean oscillation

This paper is devoted to research on local properties of functions and multidimensional singular integrals in terms of their mean oscillation. The conditions guaranteeing existence of a derivative in the L ^p -sense at a given point are found. Spaces which remain invariant under singular integral operators are considered.