Multilinear Operators Research Articles

Mapping properties of multilinear fractional maximal operators in metric measure spaces

Mapping properties of multilinear fractional maximal operators in metric measure spaces

Pressure-tensor method evaluation of the interfacial tension between Gay-Berne isotropic fluid and a smooth repulsive wall.

The interfacial properties of a confined thermotropic liquid crystalline material are investigated using a molecular dynamics simulation technique. The pairwise interaction among the soft ellipsoidal particles is modeled by the Gay-Berne (GB) potential. The GB ellipsoids are confined by two soft, smooth, repulsive walls defined by the Weeks-Chandler-Andersen (WCA) potential. The aperiodic confinement due to walls makes the system mechanically anisotropic. Hence using the pressure-tensor method, the interfacial tension of an interface between the bulk isotropic (I) phase and WCA wall at various number densities (ρ) is calculated. From the pressure tensor and orientational order profiles, the arrangement of ellipsoids in the bulk and the vicinity of the wall is determined. The effect of system size and the wall-particle interaction strength (εW) on is also analyzed by varying the system size and εW.

Complex structures on jet spaces and bosonic Fock space dynamics for causal variational principles

Based on conservation laws for surface layer integrals for critical points of causal variational principles, it is shown how jet spaces can be endowed with an almost-complex structure. We analyze under which conditions the almost-complex structure can be integrated to a canonical complex structure. Combined with the scalar product expressed by a surface layer integral, we obtain a complex Hilbert space $\mathfrak{h}$. The Euler-Lagrange equations of the causal variational principle describe a nonlinear time evolution on $\mathfrak{h}$. Rewriting multilinear operators on $\mathfrak{h}$ as linear operators on corresponding tensor products and using a conservation law for a nonlinear surface layer integral, we obtain a linear norm-preserving time evolution on bosonic Fock spaces. The so-called holomorphic approximation is introduced, in which the dynamics is described by a unitary time evolution on the bosonic Fock space. The error of this approximation is quantified. Our constructions explain why and under which assumptions critical points of causal variational principles give rise to a second-quantized, unitary dynamics on Fock spaces.

Equivalence of (quasi-)norms on a vector-valued function space and its applications to multilinear operators

In this paper we present (quasi-)norm equivalence on a vector-valued function space $L^p_A(l^q)$ and extend the equivalence to $p=\infty$ and $0<q<\infty$ in the scale of Triebel-Lizorkin space, motivated by Fraizer-Jawerth. By applying the results, we improve the multilinear Hormander's multiplier theorem of Tomita, that of Grafakos-Si, and the boundedness results for bilinear pseudo-differential operators, given by Koezuka-Tomita.

The Method of the Kernel of the Evolution Equation in the Theory of Gravity

Covariant perturbation theory allows one to calculate the nonlocal kernel of the evolution equation on a spin Riemannian manifold. The proposed axiomatic definition of effective action introduces a universal scale parameter with the dimension of the square of the distance into the dimensionless mathematical theory. It is shown that this purely geometric result has the physical meaning of the action of field theory, including gravity. The two lowest tensor orders of this covariant functional are independent of the type of the spin group and are local; they reproduce the action of general relativity with a cosmological constant. The modern value of the universal distance scale can be determined by the measured Hubble constant. This scale parameter, considered a physical variable, makes it possible to construct the cosmological theory axiomatically.

An estimate of average case approximation complexity for tensor degrees of random processes

We consider random fields that are tensor degrees of a random process of second order with continuous covariance function. The average case approximation complexity of a random field is defined as the minimal number of evaluations of linear functionals needed to approximate the field with relative 2-average error not exceeding a given threshold. In the present paper we estimate the growth of average case approximation complexity of random field for arbitrary high its parametric dimension and for arbitrary small error threshold. Under rather weak assumptions on the spectrum of covariance operator of the generating random process, we obtain necessary and sufficient condition that the average case approximation complexity has the upper estimate of a special form. We show that this condition covers a wide class of cases and the order of the estimate of the average case approximation complexity coincides with the order of its asymptotics, which were obtained earlier in the paper by Lifshits and Tulyakova.

On the positive Dimant strongly $p$-summing multilinear operators

In 2003, Dimant V. has defined and studied the interesting class of strongly $p$-summing multilinear operators. In this paper, we introduce and study a new class of operators between two Banach lattices, where we extend the previous notion to the positive framework, and prove, among other results, the domination, inclusion and composition theorems. As consequences, we investigate some connections between our class and other classes of operators, such as duality and linearization.

The boundedness of multi-linear and multi-parameter pseudo-differential operators

In this paper, we establish the boundedness on $ L^r(\mathbb{R}^{n_1}\times\mathbb{R}^{n_2}) $ of bilinear and bi-parameter pseudo-differential operators whose symbols $ \sigma(x,\xi,\eta)\in S^{(0,0)}_{(1,1),(\delta_1,\delta_2)} $ for $ x,\xi,\eta\in\mathbb{R}^{n_1}\times\mathbb{R}^{n_2} $ and $ 0\leq\delta_1,\delta_2&lt;1 $, which extends the result of Dai and Lu in [8].

Domination and Kwapień’s factorization theorems for positive Cohen p–nuclear m–linear operators

Abstract In this paper, we introduce and study the concept of positive Cohen p-nuclear multilinear operators between Banach lattice spaces. We prove a natural analog to the Pietsch domination theorem for this class. Moreover, we give like the Kwapień’s factorization theorem. Finally, we investigate some relations with another known classes.

Regularity for commutators of the local multilinear fractional maximal operators

Abstract In this paper we introduce and study the commutators of the local multilinear fractional maximal operators and a vector-valued function b⃗ = (b 1, …, bm ). Under the condition that each bi belongs to the first order Sobolev spaces, the bounds for the above commutators are established on the first order Sobolev spaces.

High serum neurofilament associates with diffuse white matter damage in MS.

ObjectiveTo evaluate to which extent serum neurofilament light chain (NfL) increase is related to diffusion tensor imaging–MRI measurable diffuse normal-appearing white matter (NAWM) damage in MS.MethodsSeventy-nine patients with MS and 10 healthy controls underwent MRI including diffusion tensor sequences and serum NfL determination by single molecule array (Simoa). Fractional anisotropy and mean, axial, and radial diffusivities were calculated within the whole and segmented (frontal, parietal, temporal, occipital, cingulate, and deep) NAWM. Spearman correlations and multiple regression models were used to assess the associations between diffusion tensor imaging, volumetric MRI data, and NfL.ResultsElevated NfL correlated with decreased fractional anisotropy and increased mean, axial, and radial diffusivities in the entire and segmented NAWM (for entire NAWM ρ = −0.49, p = 0.005; ρ = 0.49, p = 0.005; ρ = 0.43, p = 0.018; and ρ = 0.48, p = 0.006, respectively). A multiple regression model examining the effect of diffusion tensor indices on NfL showed significant associations when adjusted for sex, age, disease type, the expanded disability status scale, treatment, and presence of relapses. In the same model, T2 lesion volume was similarly associated with NfL.ConclusionsOur findings suggest that elevated serum NfL in MS results from neuroaxonal damage both within the NAWM and focal T2 lesions. This pathologic heterogeneity ought to be taken into account when interpreting NfL findings at the individual patient level.

Application of a large-scale molecular dynamics approach to modelling the deposition of TiO2 thin films

The previously developed large-scale molecular dynamics approach is applied to high-performance parallel modelling the deposition of TiO2 thin films. The largest simulated clusters reach 40 nm in size with about two million atoms. The surface roughness and porosity of normally deposited films with a flux of high and low energy atoms for cold and hot substrates are investigated. The formation of separate nanostructures in high-energy glancing angle deposited films is studied. The averaged structural parameters of the elongated pores in these films are determined using the original Monte Carlo based method. The difference between the main components of the refractive index tensor of glancing angle deposited TiO2 thin films is calculated.

On the representation of linear functionals on hyper-ideals of multilinear operators

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, in particular, that sometimes our representations hold under conditions less restrictive than those of the related classical ones.

Positive definiteness for 4th order symmetric tensors and applications

In particle physics, the vacuum stability of scalar potentials is to check positive definiteness (or copositivity) of its coupling tensors, and such a coupling tensor is a 4th order and symmetric tensor. In this paper, we mainly discuss precise expressions of positive definiteness of 4th order tensors. More specifically, two analytically sufficient conditions of positive definiteness for 4th order 2 dimensional symmetric tensors are given by reducing orders of tensors, and applying these conclusions, some sufficient conditions for the positive definiteness of 4th order 3 dimensional symmetric tensors are derived. We also present several other sufficient conditions for the positive definiteness of 4th order 3 dimensional symmetric tensors. Finally, we test and verify the vacuum stability of general scalar potentials of two real singlet scalar fields and the Higgs boson by using these results.

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.

Boundedness of multilinear fractional type operators on Hardy spaces with variable exponents

In this article, we show that multilinear fractional type operators are bounded from product of Hardy spaces with variable exponents into Lebesgue spaces with variable exponents via the atomic decomposition theory.

Gravitational waves and geometrical optics in scalar-tensor theories

The detection of gravitational waves (GWs) propagating through cosmic structures can provide invaluable information on the geometry and content of our Universe as well as on the fundamental theory of gravity. In order to test possible departures from General Relativity, it is essential to analyze, in a modified gravity setting, how GWs propagate through a perturbed cosmological space-time. Working within the framework of geometrical optics, we develop tools to address this topic for a broad class of scalar-tensor theories, including scenarios with non-minimal, derivative couplings between scalar and tensor modes. By focussing on a set-up where scalar modes propagate with the same speed as tensor degrees of freedom, we determine the corresponding evolution equations for the GW amplitude and polarization tensor. The former satisfies a generalized evolution equation that includes possible effects due to a variation of the effective Planck scale; the latter can fail to be parallel transported along the GW geodesics unless certain conditions are satisfied. We apply our general formulas to specific scalar-tensor theories with unit tensor speed, and then focus on GW propagation on a perturbed space-time. We determine corrections to standard formulas for the GW luminosity distance and for the evolution of the polarization tensor, which depend both on modified gravity and on the effects of cosmological perturbations. Our results can constitute a starting point to disentangle among degeneracies from different sectors that can influence GW propagation through cosmological space-times.

Extension of multilinear fractional integral operators to linear operators on mixed-norm Lebesgue spaces

In Kenig and Stein (Math Res Lett 6(1):1–15, 1999, https://doi.org/10.4310/MRL.1999.v6.n1.a1 ), the following type of multilinear fractional integral $$\begin{aligned} \int _{{\mathbb {R}}^{mn}} \frac{f_1(l_1(x_1,\ldots ,x_m,x))\cdots f_{m+1}(l_{m+1}(x_1,\ldots ,x_m,x))}{(|x_1|+\cdots +|x_m|)^{\lambda }} dx_1\ldots dx_m \end{aligned}$$ was studied, where $$l_i$$ are linear maps from $${\mathbb {R}}^{(m+1)n}$$ to $${\mathbb {R}}^n$$ satisfying certain conditions. They proved the boundedness of such multilinear fractional integral from $$L^{p_1}\times \cdots \times L^{p_{m+1}}$$ to $$L^q$$ when the indices satisfy the homogeneity condition. In this paper, we show that the above multilinear fractional integral extends to a linear operator for functions in the mixed-norm Lebesgue space $$L^{{\mathbf {p}}}$$ which contains $$L^{p_1}\times \cdots \times L^{p_{m+1}}$$ as a subset. Under less restrictions on the linear maps $$l_i$$ , we give a complete characterization of the indices $${\mathbf {p}}$$ , q and $$\lambda $$ for which such an operator is bounded from $$L^{{\mathbf {p}}}$$ to $$L^q$$ . And for $$m=1$$ or $$n=1$$ , we give necessary and sufficient conditions on $$(l_1, \ldots , l_{m+1})$$ , $${\mathbf {p}}=(p_1,\ldots , p_{m+1})$$ , q and $$\lambda $$ such that the operator is bounded.

Symmetries for scalarless scalar theories

We consider theories containing scalar fields interacting with vector or with tensor degrees of freedom, equipped with symmetries that prevent the propagation of linearized scalar excitations around solutions of the equations of motion. We first study the implications of such symmetries for building vector theories that break Abelian gauge invariance without necessarily exciting longitudinal scalar fluctuations in flat space. We then examine scalar-tensor theories in curved space, and relate the symmetries we consider with a non-linear realization of broken space-time symmetries acting on scalar modes. We determine sufficient conditions on the space-time geometry to avoid the propagation of scalar fluctuations. We analyze linearized perturbations around spherically symmetric black holes, proving the absence of scalar excitations, and pointing out modifications in the dynamics of spin-2 fluctuations with respect to Einstein gravity. We then study consequences of this set-up for the dark energy problem, determining scalar constraints on cosmological configurations that can lead to self-accelerating universes whose expansion is insensitive to the value of the bare cosmological constant.

Boundedness of multilinear singular integrals on central Morrey spaces with variable exponents

We prove the boundedness for a class of multi-sublinear singular integral operators on the product of central Morrey spaces with variable exponents. Based on this result, we obtain the boundedness for the multilinear singular integral operators and two kinds of multilinear singular integral commutators on the above spaces.