The aim of this paper is to start the study of multilinear generalizations of the classical ideals of linear operators of type $p$ and cotype $q$. As a first step in a theory we believe will be long and fruitful, we propose a notion of type and cotype of multilinear operators and the resulting classes of such mappings are studied in the setting of the theory of Banach/quasi-Banach ideals of multilinear operators. Distinctions between the linear and the multilinear theories are pointed out, typical multilinear features of the theory are emphasized and many illustrative examples are provided. The classes we introduce are related to the multi-ideals generated by the linear ideals of operators of some type/cotype and are proved to be maximal and Aron-Berner stable.
Grothendieck's theorem asserts that every continuous linear operator from $\ell_1$ to $\ell_2$ is absolutely $(1,1)$-summing. This kind of result is commonly called coincidence result. In this paper we investigate coincidence results in the multilinear setting, showing how the cotype of the spaces involved affect such results. The special role played by $\ell_1$ spaces is also investigated with relation to interpolation of tensor products. In particular, an open problem on the interpolation of $m$ injective tensor products is solved.
In this paper, we explore the concept of multilinear operators that are multiple almost summing and present a new concept of type and cotype of multilinear operators and investigate the conditions for this new concept to recover the original concept of type and type for multilinear operators. We also show that these classes are Banach multi-ideals and establish the coherence and compatibility of these classes with the generated homogeneous polynomials.
In this paper, we establish multilinear BMO estimates for commutators of multilinear fractional maximal and integral operators both on product generalized Morrey spaces and product generalized vanishing Morrey spaces, respectively. Similar results are still valid for commutators of multilinear maximal and singular integral operators.
We introduce asymptotic analogues of the Rademacher and martingale type and cotype of Banach spaces and operators acting on them. Some classical local theory results related, for example, to the `automatic-type' phenomenon, the type-cotype duality, or the Maurey-Pisier theorem, are extended to the asymptotic setting. We also investigate operator ideals corresponding to the asymptotic subtype/subcotype. As an application of this theory, we provide a sharp version of a result of Brooker and Lancien by showing that any twisted sum of Banach spaces with Szlenk power types $p$ and $q$ has Szlenk po…
We introduce multilinear localization operators in terms of the short-time Fourier transform, and multilinear Weyl pseudodifferential operators. We prove that such localization operators are in fact Weyl pseudodifferential operators whose symbols are given by the convolution between the symbol of the localization operator and the multilinear Wigner transform. For such interpretation we use the kenrel theorem for the Gelfand-Shilov space. Furthermore, we study the continuity properties of the multilinear localization operators on modulation spaces. Our results extend some known results when res…
In this article, we study properties of multilinear Fourier integral operators on weighted modulation spaces. In particular, using the theory of Gabor frames, we study boundedness of multilinear Fourier integral operators on products of weighted modulation spaces. Further, we investigate the periodic multilinear Fourier integral operator. Finally, we study continuity of bilinear pseudo-differential operators on modulation spaces for certain symbol classes, namely $\textbf{SG}$-class.
A standard technique in infinite dimensional holomorphy, which produced several useful results, uses the Borel transform to represent linear functionals on certain spaces of multilinear operators between Banach spaces as multilinear operators. In this paper we develop a technique to represent linear functionals, as linear operators, on spaces of multilinear operators that are beyond the scope of the standard technique. Concrete applications to some well studied classes of multilinear operators, including the class of compact multilinear operators, and to one new class are provided. We can see,…
We construct a general framework that generates classes of multilinear operators between Banach spaces which encompasses, as particular cases, the several classes of summing type multilinear operators that have been studied individually in the literature. Summing operators by blocks in the isotropic and anisotropic cases are taken into account. The classes we create are shown to be Banach ideals of multilinear operators and applications to coherence and coincidence theorems are provided.
In this paper we develop the theory of Fourier multiplier operators $T_{m}:L^{p}(\mathbb{R}^{d};X)\to L^{q}(\mathbb{R}^{d};Y)$, for Banach spaces $X$ and $Y$, $1\leq p\leq q\leq \infty$ and $m:\mathbb{R}^d\to \mathcal{L}(X,Y)$ an operator-valued symbol. The case $p=q$ has been studied extensively since the 1980's, but far less is known for $p<q$. In the scalar setting one can deduce results for $p<q$ from the case $p=q$. However, in the vector-valued setting this leads to restrictions both on the smoothness of the multiplier and on the class of Banach spaces. For example, one often needs that …
We analyze 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.
Unifying several directions of the development of the study of summing multilinear operators between Banach spaces, we construct a general framework that studies, under one single definition, multilinear operators that are summing with respect to sums taken over any number of indices, iterated and non-iterated sums (the isotropic and the anisotropic cases), sums over arbitrary blocks and over several different sequence norms. A large number of special classes of multilinear operators and of methods of generating classes of multilinear operators are recovered as particular instances. Ideal prop…
We apply the geometric approach provided by $Σ$-operators to develop a theory of $p$-summability for multilinear operators. In this way, we introduce the notion of Lipschitz $p$-summing multilinear operators and show that it is consistent with a general panorama of generalization: Namely, they satisfy Pietsch-type domination and factorization theorems and generalizations of the inclusion Theorem, Grothendieck's coincidence Theorems, the weak Dvoretsky-Rogers Theorem and a Lindenstrauss-Pelczyńsky Theorem. We also characterize this new class in tensorial terms by means of a Chevet-Saphar-type t…
We develop the duality theory between ideals of multilinear operators and tensor norms that arises from the geometric approach of $Σ$-operators. To this end, we introduce and develop the notions of $Σ$-ideals of multilinear operators and $Σ$-tensor norms. We establish the foundations of this theory by proving a representation theorem for maximal $Σ$-ideals of multilinear operators by finitely generated $Σ$-tensor norms and a duality theorem for $Σ$-tensor norms. For these notions we also develop basic theory and present concrete examples.
In this paper we provide a unifying approach to the study of Banach ideals of linear and multilinear operators defined, or characterized, by the transformation of vector-valued sequences. We investigate and apply the linear and multilinear stabilities of some frequently used classes of vector-valued sequences.
We characterize those bounded multilinear operators that factor through a Hilbert space in terms of its behavior in finite sequences. This extends a result, essentially due to S. Kwapień, from the linear to the multilinear setting. We prove that Hilbert-Schmidt and Lipschitz $2$-summing multilinear operators naturally factor through a Hilbert space. It is also proved that the class $Γ$ of all multilinear operators that factor through a Hilbert space is a maximal multi-ideal; moreover, we give an explicit formulation of a finitely generated tensor norm $γ$ which is in duality with $Γ$.
In the nonlinear field of multilinear operators and homogeneous polynomials between Banach spaces, we develop a technique, based on the transformation of vector-valued sequences, to create new examples of hyper-ideals of multilinear operators, polynomial hyper-ideals and polynomial two-sided ideals. Several well studied ideals are shown to be actually hyper-ideals or two-sided ideals.
In this paper, we establish the sharp boundedness of p-adic multilinear Hausdorff operators on the product of Lebesgue and central Morrey spaces associated with both power weights and Muckenhoupt weights. Moreover, the boundedness for the commutators of p-adic multilinear Hausdorff operators on the such spaces with symbols in central BMO space is also obtained.
In this paper we provide an abstract aproach to the study of classes of multiple summing multilinear operators between Banach spaces. The main purpose is unify the study of several known classes and results, for example multiple $(p, q_1,\dots, q_n)$-summing operators, multiple mixing $(s, q, p)$-summing operators and multiple strong $(s, q, p)$-mixing summing operators. We also define new class of multiple summing multilinear operator that are particular cases of our construction and, therefore, satisfy the results proved in the paper.
We introduce a notion of $(p,q)$-dominated multilinear operators which stems from the geometrical approach provided by $Σ$-operators. We prove that $(p,q)$-dominated multilinear operators can be characterized in terms of their behavior on finite sequences and in terms of their relation with a Lapresté tensor norm. We also prove that they verify a generalization of the Pietsch's Domination Theorem and Kwapień's Factorization Theorem. Also, we study the collection $\mathcal{D}_{p,q}$ of all $(p,q)$-dominated multilinear operators showing that $\mathcal{D}_{p,q}$ has a maximal ideal demeanor and …
In this paper, we prove that if a multilinear operator $\mathcal{T}$ and its multilinear commutator $\mathcal{T}_{Σ\vec{b}}$ and iterated commutator $\mathcal{T}_{Π\vec{b}}$ for $\vec{b}\in(\mathbb{R}^n)^m$ are bounded on product weighted Lebesgue space, then $\mathcal{T}$, $\mathcal{T}_{Σ\vec{b}}$ and $\mathcal{T}_{Π\vec{b}}$ are also bounded on product weighted Amalgam space. As its applications, we show that multilinear Littlewood-Paley functions and multilinear Marcinkiewicz integral functions with kernels of convolution type and non-convolution type, and their multilinear commutators and …
The purpose of this paper is to establish some neccessary and sufficient conditions for the boundedness of a general class of multilinear Hausdorff operators that acts on the product of some two weighted function spaces such as the two weighted Morrey, Herz and Morrey-Herz spaces. Moreover, some sufficient conditions for the boundedness of multilinear Hausdorff operators on the such spaces with respect to the Muckenhoupt weights are also given.
Basing on the work of Pellegrino et al. on factorable strongly p-summing multilinear operators, we will continue study this class of operators and borrowing the same idea for the category of p-nuclear operators. Therefore, we will construct other multilinear and polynomial ideals for which the most valuable results of the theory of p-nuclear linear operators are available, namely the Pietsch domination theorem and the well known factorization given by Cohen and Kwapien for p-nuclear operators.
We characterize the positive Schur property in the positive projective tensor products of Banach lattices, we establish the connection with the weak operator topology and we give necessary and sufficient conditions for the space of regular multilinear operators/homogeneous polynomials taking values in a Dedekind complete Banach lattice to have the positive Schur property.
The aim of this paper is to introduce and study the boundedness of a new class of p-adic rough multilinear Hausdorff operators on the product of Herz, central Morrey and Morrey-Herz spaces with power weights and Muckenhoupt weights. We also establish the boundedness for the commutators of p-adic rough multilinear Hausdorff operators on the weighted spaces with symbols in central BMO space.
The theory of $τ$-summing and $σ$-nuclear linear operators on Banach spaces was developed by Pietsch [12, Chapter 23]. Extending the linear case to the range p > 1 and generalizing all cases to the multilinear setting, in this paper we introduce the concept of $σ(p)$-nuclear linear and multilinear operators. In order to develop the duality theory for the spaces of such operators, we introduce the concept of quasi-tau(p)-summing linear/multilinear operators and prove Pietsch-type domination theorems for such operators. The main result of the paper shows that, under usual conditions, linear func…
