Summability of multilinear mappings: Littlewood, Orlicz and beyond

In this paper we prove new results concerning summability properties of multilinear mappings between Banach spaces, such as an extension of Littlewood’s 4/3 Theorem. The role of the Littlewood–Orlicz property in the theory is established, especially in the question of determining when multilinear mappings are (1; 2, . . . , 2)-summing.     

Almost summing mappings

We introduce a general definition of almost p-summing mappings and give several concrete examples of such mappings. Some known results are considerably generalized and we present various situations in which the space of almost p-summing multilinear mappings coincides with the whole space of continuous multilinear mappings. Received: 17 June 2002     

The Multilinear Littlewood–Paley Operators with Minimal Regularity Conditions

Journal of Fourier Analysis and Applications - We investigate the quantitative weighted estimates for a large class of the multilinear Littlewood–Paley square operators. Our kernels satisfy...     

On bounded multilinear forms on a class of lp spaces

In this paper we generalize an old result of Littlewood and Hardy about bilinear forms defined in a class of sequence spaces. Historically, Littlewood [Quart. J. Math.1 (1930)] first proved a result on bilinear forms on bounded sequences and this result was then generalized by Hardy and Littlewood in a joint paper [Quart. J. Math.5(1934)] to bilinear forms on a class of l^p spaces. Later Davie and Kaijser proved Littlewood's results for multilinear forms. In this paper, Theorems A and B generalize the results to multilinear forms on l^p spaces. All the results are stated at the end of Section 1. Theorems A and B are proved, respectively, in Sections 2 and 3.     

Complexification of Multilinear Mappings and Polynomials

We study various methods of complexifying real normed spaces. We see how the notions of duality and complexification are interchangeable. We obtain estimates for the norms of complexified multilinear mappings and polynomials. We see how polynomials can be complexified without reference to the associated multilinear mappings.     

Polynomials and multilinear mappings in topological modules

Criteria for the equicontinuity of sets of multilinear mappings between topological modules are studied, as well as topological modules of continuous multilinear mappings. As a consequence, criteria for the equicontinuity of sets of homogeneous polynomials between topological modules are also studied, as well as topological modules of continuous homogeneous polynomials.     

A nonlinear Pietsch domination theorem

Pietsch’s domination theorem, which is known for linear, multilinear and polynomial mappings, is extended to a larger class of nonlinear mappings.     

Summability and estimates for polynomials and multilinear mappings

In this paper we extend and generalize several known estimates for homogeneous polynomials and multilinear mappings on Banach spaces. Applying the theory of absolutely summing nonlinear mappings, we prove that estimates which are known for mappings on ?_p spaces in fact hold true for mappings on arbitrary Banach spaces.     

Multilinear extensions of absolutely (p;q;r)-summing operators

In this paper we introduce and study a new class containing the class of absolutely summing multilinear mappings, which we call absolutely (p;q ₁,…,q _m ;r)-summing multilinear mappings. We investigate some interesting properties concerning the absolutely (p;q ₁,…,q _m ;r)-summing m-linear mappings defined on Banach spaces. In particular, we prove a kind of Pietsch’s Domination Theorem and a multilinear version of the Factorization Theorem.     

Factorization of injective ideals by interpolation

We construct a factorization of certain multilinear mappings through linear operators belonging to closed, injective operator ideals using interpolation technique. An extension of the duality theorem for interpolation spaces is also obtained.     

Diagonal multilinear operators on Köthe sequence spaces

We analyse the interplay between maximal/minimal/adjoint ideals of multilinear operators (between sequence spaces) and their associated Köthe sequence spaces. We establish relationships with spaces of multipliers and apply these results to describe diagonal multilinear operators from Lorentz sequence spaces. We also define and study some properties of the ideal of (E, p)-summing multilinear mappings, a natural extension of the linear ideal of absolutely (E, p)-summing operators.     

On factorization of Hilbert-Schmidt mappings

We present some results on factorization of Hilbert-Schmidt multilinear mappings and polynomials through infinite dimensional Banach spaces, L₁ and L_∞ spaces. We conclude this work with a result on factorization of holomorphic mappings of Hilbert-Schmidt type.     

Bornologically isomorphic representations of distributions on manifolds

Distributional tensor fields can be regarded as multilinear mappings on smooth tensor fields with distributional values or as (classical) tensor fields with distributional coefficients. We show that the corresponding isomorphisms hold also in the bornological setting.     

Nicodemi sequences of operators between spaces of multilinear mappings

We obtain results on three aspects of Nicodemi extensions of multilinear mappings between Banach spaces: (i) subspace invariance, (ii) the norms of the extension operators, (iii) when Aron–Berner extensions are Nicodemi extensions.     

Power and potential maps induced by any semivalue: Some algebraic properties and computation by multilinear extensions

The notions of total power and potential, both defined for any semivalue, give rise to two endomorphisms of the vector space of cooperative games on any given player set where the semivalue is defined. Several properties of these linear mappings are stated and the role of unanimity games as eigenvectors is described. We also relate in both cases the multilinear extension of the image game to the multilinear extension of the original game. As a consequence, we derive a method to compute for any semivalue by means of multilinear extensions, in the original game and also in all its subgames, (a) the total power, (b) the potential, and (c) the allocation to each player given by the semivalue.     

Linear algebra of convex sets and the euler characteristic

It is shown that surprisingly many operations over convex sets can be extended to multilinear mappings on appropriate chosen linear spaces if one replaces convex sets by their characteristic functions. Some generalizations of results known in combinatorial geometry of convex sets are given too.     

Linear algebra of convex sets and the euler characteristic

It is shown that surprisingly many operations over convex sets can be extended to multilinear mappings on appropriate chosen linear spaces if one replaces convex sets by their characteristic functions. Some generalizations of results known in combinatorial geometry of convex sets are given too.     

The multisublinear maximal type operators in Banach function lattices

The main aim of this paper is to study a general multisublinear operators generated by quasi-concave functions between weighted Banach function lattices. These operators, in particular, generalize the Hardy–Littlewood and fractional maximal functions playing an important role in harmonic analysis. We prove that under some general geometrical assumptions on Banach function lattices two-weight weak type and also strong type estimates for these operators are true. To derive the main results of this paper we characterize the strong type estimate for a variant of multilinear averaging operators. As special cases we provide boundedness results for fractional maximal operators in concrete function spaces.     

Domination spaces and factorization of linear and multilinear summing operators

It is well known that not every summability property for multilinear operators leads to a factorization theorem. In this paper we undertake a detailed study of factorization schemes for summing linear and nonlinear operators. Our aim is to integrate under the same theory a wide family of classes of mappings for which a Pietsch type factorization theorem holds. Our construction includes the cases of absolutely p-summing linear operators, (p, σ)-absolutely continuous linear operators, factorable strongly p-summing multilinear operators, (p1,?. . .?, pn)-dominated multilinear operators and dominated (p1,?. . .?, pn; σ)-continuous multilinear operators.     

A Moment Characterization of B-Valued Generalized Functionals of White Noise

Kernel theorems are established for Bananch space-valued multilinear mappings, A moment characterization theorem for Banach space-valued generalized functionals of white noise is proved by using the above kernel theorems. A necessary and sufficient condition in terms of moments is given for sequences of Banach space-valued generalized functionals of white noise to converge strongly. The integration is also discussed of functions valued in the space of Banach space-valued generalized functionals.