Class Of Multilinear Operators Research Articles

On positive Cohen weakly nuclear multilinear operators

In this article, we establish new relationships involving the class of Cohen positive strongly $p$-summing multilinear operators. Furthermore, we introduce a new class of multilinear operators on Banach lattices, called positive Cohen weakly nuclear multilinear operators. We establish a Pietsch domination-type theorem for this new class of multilinear operators. As an application, we show that every positive Cohen weakly $p$-nuclear multilinear operator is positive Dimant strongly $p$-summing and Cohen positive strongly $p$-summing. We conclude with a tensor representation of our class.

Large structures within the class of summing operators

We investigate lineability/spaceability problems within the setting of multilinear summing operators on quasi-Banach sequence spaces. Furthermore, we deal with the contemporary geometric notions of pointwise-lineability and (α,β)-lineability. Among other results, we prove that the class of multilinear operators taking values on ℓq(0<q≤∞) that are absolutely but not multiple summing is pointwise c-spaceable when non-empty. Spaceability in other classes, such as Dunford-Pettis operators and multilinear operators on general summable scalar families, is also studied.

New Weighted Norm Inequalities for Certain Classes of Multilinear Operators on Morrey-type Spaces

New Weighted Norm Inequalities for Certain Classes of Multilinear Operators on Morrey-type Spaces

On Cohen weakly nuclear multilinear operators

In this paper, we introduce and study a new class of multilinear operators on Banach spaces, called Cohen weakly $p$-nuclear multilinear operators. We establish a Pietsch domination theorem for this new class of multilinear operators. As an application, we show that every Cohen weakly $p$-nuclear multilinear operator is strongly $p$-summing and Cohen strongly $p$-summing.

Product factorable multilinear operators defined on sequence spaces

We prove a factorization theorem for multilinear operators acting in topological products of spaces of (scalar) p-summable sequences through a product. It is shown that this class of multilinear operators called product factorable maps coincides with the well-known class of the zero product preserving operators. Due to the factorization, we obtain compactness and summability properties by using classical functional analysis tools. Besides, we give some isomorphisms between spaces of linear and multilinear operators, and representations of some classes of multilinear maps as n-homogeneous orthogonally additive polynomials.

Factorization of multilinear operators defined on products of function spaces

This paper deals with multilinear operators acting in products of Banach spaces that factor through a canonical mapping. We prove some factorization theorems and characterizations by means of norm inequalities for multilinear operators defined on the n-fold Cartesian product of the space of bounded Borel measurable functions, respectively, products of Banach function spaces. These factorizations allow us to obtain integral dominations and lattice geometric properties and to present integral representations for abstract classes of multilinear operators. Finally, bringing together these ideas, some applications are shown, regarding for example summability properties and representation of multilinear maps as orthogonally additive n-homogeneous polynomials.

Factorization theorems for some new classes of multilinear operators

The research was supported by National Science Centre of Poland, project 2015/17/B/ST1/00064. The research was supported by the Ministerio de Ciencia, Innovacion y Universidades, Agencia Estatal de Investigacion (Spain) and FEDER under project MTM2016-77054-C2-1-P.

Operators with the Maurey–Pietsch multiple splitting property

Let n be a natural number, such that . As a simultaneous extension of the p-summing and multiple p-summing operators we introduce the class of all n-linear operators which are -summing, . If , are such that we introduce the class of the operators with the Maurey–Pietsch -splitting property as the multilinear analogue of the class of -mixing linear operators and, as particular classes, , . We introduce two new classes of multilinear operators both being the multilinear analogues of the class of r-nuclear operators, namely, the class of r-nuclear operators with respect to the -norm, and the class and prove that . By concrete examples, we show that, contrary to the linear case, when , in the multilinear case the inclusions , are not true, which is in contrast to a well-known result shown by us in another paper that.

Techniques to generate hyper-ideals of multilinear operators

The usual techniques to generate ideals of multilinear operators between Banach spaces fail in generating hyper-ideals in general. In this paper, we fill this gap by developing two techniques to generate hyper-ideals of multilinear operators. The techniques we develop generate new classes of multilinear operators and show that some important well-studied classes are Banach or p-Banach hyper-ideals.

New Weighted Norm Inequalities for Certain Classes of Multilinear Operators and their Iterated Commutators

Applying the new class of multiple weights functions and new sharp maximal functions, we obtain the pointwise estimates, strong type and weak end-point estmates for certain classes of multilinear operators and their iterated commutators with new BMO functions.

New class of multiple weights and new weighted inequalities for multilinear operators

Abstract In this paper, we first introduce the new class of multiple weights A p → ∞ ${A^\infty _{\vec{p}}}$ which is larger than the class of multiple weights in [Adv. Math. 220 (2009), 1222–1264]. Then, using this class of weights, we study the weighted norm inequalities for certain classes of multilinear operators and their commutators with new BMO functions introduced in [J. Fourier Anal. Appl. 17 (2011), 115–134]. Finally, we show that some multilinear pseudodifferential operators fall within the scope of the theory obtained in this paper.

Multiple Rademacher means and their applications

We evaluate multiple Rademacher means for various classes of multilinear operators. Under the assumption that the range has cotype q, these results show that the class of multiple ( q , 2 ) -summing operators plays an important role. Further, we get new extensions of Buʼs theorem in the multilinear case.

Fractional Integral Operators on Anisotropic Hardy Spaces

In this paper, we introduce the fractional integral operator T of degree α of order m with respect to a dilation A for 0 < α < 1 and \(m \in {\mathbb{N}}\). First we establish the Hardy-Littlewood-Sobolev inequalities for T on anisotropic Hardy spaces associated with dilation A, which show that T is bounded from H p to H q , or from H p to L q , where 0 < p ≤ 1/(1 + α) and 1/q = 1/p − α. Then we give anisotropic Hardy spaces estimates for a class of multilinear operators formed by fractional integrals or Calderon-Zygmund singular integrals. Finally, we apply the above results to give the boundedness of the commutators of T and a BMO function.

Two-weight norm inequalities for a class of rough multilinear operators

In this paper, we give the (Lp,Lq)-boundedness for a class of multilinear operators with two different weight functions.

Weighted Boundedness for a Class of Rough Multilinear Operators

In this paper we give the (L p (ω p ), L q (ω q )) boundedness for a class of multilinear operators, which is simular to the higher-order commutator for the rough fractional integral. In our results the kernel function is merely assumed on a size condition.

Some classes of multilinear operators on C(K) spaces

The authors obtain in this paper a classification of projective tensor products of C(K) spaces, in terms of the behaviour of certain classes of multilinear operators on the product of the spaces, or the verification of certain Banach space properties of the corresponding tensor product. The main tool used is an improvement of some results of Emmanuele and Hensgen on the reciprocal Dunford-Pettis and Pelczy´nski’s (V) properties of the projective tensor product of Banach spaces. Finally, the paper ends with a study of the relationships between some classes of multilinear operators and their linearizations.