Multilinear Calderon-Zygmund Operators Research Articles

Product generalized local Morrey spaces and commutators of multi-sublinear operators generated by multilinear Calderon-Zygmund operators and local Campanato functions

The aim of this paper is to get the boundedness of the commutators of multi-sublinear operators generated by local campanato functions and multilinear Calder?n-Zygmund operators on the product generalized local Morrey spaces.

Weighted estimates for iterated commutators of multilinear Calderón-Zygmund operators on non-homogeneous metric measure spaces

Let ( $${\cal X}$$ , d,μ) be a non-homogeneous metric measure space satisfying the so-called upper doubling and geometrically doubling conditions, which includes the space of homogeneous type and the Euclidean space with the non-doubling measure as special cases. Let T be a multilinear Calderon-Zygmund operator and $$\vec b: = ({b_1}, \ldots ,{b_m})$$ be a finite family of $$\widetilde{{\rm{RBMO}}}(\mu)$$ functions. In this paper, some weak-type multiple weighted estimates for the iterated commutator $${{\rm{T}}_{\Pi \vec b}}$$ generated by T and $$\vec b$$ are obtained.

Sharp weighted estimates for multi-linear Calderón–Zygmund operators on non-homogeneous spaces

In this article, we address pointwise sparse domination for multilinear Calderon—Zygmund operators on upper doubling, geometrically doubling metric measure spaces. As a consequence, we have obtained sharp quantitative weighted estimates for multilinear Calderon–Zygmund operators.

English

The spaces of multi-Morrey type for positive Radon measures satisfying a growth condition on ℝd are introduced. After defining the spaces, we investigate the multilinear maximal function, the multilinear fractional integral operator and the multilinear Calderon-Zygmund operators, respectively, from multi-Morrey spaces to Morrey spaces.

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 Calderon–Zygmund operators and multilinear Calderon–Zygmund operators with Dini type kernels.

A vector-valued estimate of multilinear Calderón-Zygmund operators in Herz-Morrey spaces with variable exponents

In this paper, we obtain a vector valued inequality of multilinear Calderon-Zygmund operators on products of Herz-Morrey spaces with variable exponents.

Multilinear Marcinkiewicz-Zygmund Inequalities

Fil: Carando, Daniel German. Consejo Nacional de Investigaciones Cientificas y Tecnicas; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matematica; Argentina

Multilinear Calderón-Zygmund operators and their commutators with BMO functions in variable exponent Morrey spaces

The boundedness of multilinear Calderon-Zygmund operators and their commutators with bounded mean oscillation (BMO) functions in variable exponent Morrey spaces are obtained.

Endpoint estimates for the commutators of multilinear Calderón-Zygmund operators with Dini type kernels

Let \(T_{\vec{b}}\) and \(T_{\Pi b}\) be the commutators in the jth entry and iterated commutators of the multilinear Calderon-Zygmund operators, respectively. It was well known that the commutators of linear Calderon-Zygmund operators were not of weak type \((1,1)\) and \((H^{1}, L^{1})\), but they did satisfy certain endpoint \(L\log L\) type estimates. In this paper, our aim is to give more natural sharp endpoint results. We show that \(T_{\vec{b}}\) and \(T_{\Pi b}\) are bounded from the product Hardy space \(H^{1}\times\cdots\times H^{1}\) to weak \(L^{\frac{1}{m},\infty}\) space, whenever the kernel satisfies a class of Dini type condition. This was done by using a key lemma given by Christ, a very complex decomposition of the integrand domains, and carefully splitting the commutators into several terms.

Vector-valued maximal multilinear Calderón–Zygmund operator with nonsmooth kernel on weighted Morrey space

In this paper, vector-valued Maximal multilinear Calderon–Zygmund operator with nonsmooth kernel and its commutators generated by \(\textit{BMO}\) function are studied. The purpose of this paper is to establish that these operators are bounded on certain multiple weighted Morrey space. Moreover, we also get the multiple weighed Morrey estimate for the vector-valued Calderon–Zygmund operator and its commutator generated by \(\textit{BMO}\) function.

Generalized commutators of multilinear Calderón–Zygmund type operators

Let $T$ be an $m$-linear Calderon–Zygmund operator with kernel $K$ and $T^*$ be the maximal operator of $T$. Let $S$ be a finite subset of $Z^+\times \{1,\dots,m\}$ and denote $d\vec{y}=dy_1\cdots dy_m$. Define the commutator $T_{\vec{b},S}$ of $T$, and $T^*_{\vec{b},S}$ of $T^*$ by $T_{\vec{b},S}(\vec{f})(x)= \int_{\mathbb{R}^{nm}}\prod_{(i,j)\in S}(b_i(x)-b_i(y_j)) K(x,y_{1},\dots,y_{m}) \prod_{j=1}^mf_j(y_j)d\vec{y}$ and $T^*_{\vec{b},S}(\vec{f})(x)= \sup_{\delta>0}\big|\int_{{\sum_{j=1}^m|x-y_j|^2>\delta^2}} \prod_{(i,j)\in S}(b_i(x)-b_i(y_j))K(x,y_{1},\dots,y_{m}) \prod_{j=1}^mf_{j}(y_j)d\vec{y}\big|$. These commutators are reflexible enough to generalize several kinds of commutators which already existed. We obtain the weighted strong and endpoint estimates for $T_{\vec{b},S}$ and $T^*_{\vec{b},S}$ with multiple weights. These results are based on an estimate of the Fefferman–Stein sharp maximal function of the commutators, which is proved in a pretty much more organized way than some known proofs. Similar results for the commutators of vector-valued multilinear Calderon–Zygmund operators are also given.

Continuity properties for commutators of multilinear type operators on product of certain Hardy spaces

Similar to the property of a linear Calderon-Zygmund operator, a linear fractional type operator Iα associated with a BMO function b fails to satisfy the continuity from the Hardy space Hp into Lp for p ⩽ 1. Thus, an alternative result was given by Y. Ding, S. Lu and P. Zhang, they proved that [b, Iα] is continuous from an atomic Hardy space Hbp into Lp, where Hbp is a subspace of the Hardy space Hp for n/(n+1) < p ⩽ 1. In this paper, we study the commutators of multilinear fractional type operators on product of certain Hardy spaces. The endpoint \((H_{b_1 }^{p_1 } \times \cdots \times H_{bm}^{p_m } ,L^p )\) boundedness for multilinear fractional type operators is obtained. We also give the boundedness for the commutators of multilinear Calderon-Zygmund operators and multilinear fractional type operators on product of certain Hardy spaces when b ∈ (Lipβ)m(ℝn).

Weighted estimates for iterated commutators of multilinear operators with non-smooth kernels

Let T be the multilinear Calderon-Zygmund operator with non-smooth kernels and T∗ be its corresponding maximal function. In this paper, we give the weighted strong type estimates and weak end-point estimates for the iterated commutators of multilinear operator T∗ . Similar results still hold for the operator T . Mathematics subject classification (2010): 42B20, 47G10.

The boundedness of multilinear Calderón–Zygmund operators on Hardy spaces

In this paper, we study the boundedness of the multilinear Calderon–Zygmund operators on products of Hardy spaces.

The boundedness of multilinear operators on generalized Morrey spaces over the quasi-metric space of non-homogeneous type

In this paper we study the multilinear fractional integral operators, the multilinear Calderon-Zygmund operators and the multi-sublinear maximal operators defined on the quasi-metric space with non-doubling measure. We obtain the boundedness of these operators on the generalized Morrey spaces over the quasi-metric space of non-homogeneous type.

Weighted estimates for vector-valued multilinear operators with non-smooth kernels

Let T be the multilinear Calderon-Zygmund operator with non-smooth kernel and let T ∗ be its corresponding maximal operator. In this paper, vector-valued weighted norm inequalities for T and T ∗ are established. As applications, weighted strong type estimates for vector-valued commutators associated with T and T ∗ are deduced respectively.

A T(1) theorem for entangled multilinear dyadic Calderón–Zygmund operators

We prove a boundedness criterion for a class of dyadic multilinear forms acting on two-dimensional functions. Their structure is more general than the one of classical multilinear Calderon–Zygmund operators as several functions can now depend on the same one-dimensional variable. The study of this class is motivated by examples related to the two-dimensional bilinear Hilbert transform and to bilinear ergodic averages. This paper is a sequel to a prior paper by the first author.

Multilinear version of reversed Hölder inequality and its applications to multilinear Calderón-Zygmund operators

In this paper, we give a natural, and generalized reverse Holder inequality, which says that if $\omega_{i}\in A_\infty$, then for every cube $Q$, \begin{equation} \int_Q\prod_{i =1}^m{\omega_i}^{\theta_i} \geq \prod_{i = 1}^m \left( \frac{\int_Q{\omega_i}}{[\omega_i]_{A_\infty}} \right)^{\theta_i}, \end{equation} where $\sum_{i=1}^m\theta_{i}=1$, $0\leq\theta_i\leq1$. As a consequence, we get a more general inequality, which can be viewed as an extension of the reverse Jensen inequality in the theory of weighted inequalities. Based on this inequality (0.1), we then give some results concerning multilinear Calderon-Zygmund operators and maximal operators on weighted Hardy spaces, which improve some known results significantly.

Multilinear Calderón-Zygmund operator on products of Hardy spaces

In this paper, the authors establish the boundedness of the multilinear Calderon-Zygmund operator from products of Hardy spaces into Hardy spaces.

Weighted norm inequalities for the multilinear Calderón-Zygmund operators

In this paper, by proving some suitable weighted endpoint estimates, and then by multilinear interpolation and a new multilinear extrapolation lemma, the author establishes some weighted estimates for the multilinear Calderon-Zygmund operator. Also, the author gives a weighted estimate for the corresponding commutator.