Multilinear Setting Research Articles

Sharp maximal function estimates for linear and multilinear pseudo-differential operators

In this paper, we study pointwise estimates for linear and multilinear pseudo-differential operators with exotic symbols in terms of the Fefferman-Stein sharp maximal function and Hardy-Littlewood type maximal function. Especially in the multilinear case, we use a multi-sublinear variant of the classical Hardy-Littlewood maximal function introduced by Lerner, Ombrosi, Pérez, Torres, and Trujillo-González [16], which provides more elaborate and natural weighted estimates in the multilinear setting.

Weighted Hardy factorizations in terms of Multilinear Calderón-Zygmund and fractional integral operators

The foundational paper of Coifman-Rochberg-Weiss provided a constructive proof of the weak factorizations of the classical Hardy space in terms of Riesz transforms. In this paper, we establish the factorization theorems for the weighted Hardy spaces. The results are also extended to the multilinear setting. Furthermore, we show that the weighted BMO space and weighted Lipschitz spaces are necessary for the weighted boundedness of commutators of the multilinear Calderón-Zygmund and fractional integral operators, respectively.

Linear to multi-linear algebra and systems using tensors

In the past few decades, multi-linear algebra also known as tensor algebra has been adapted and employed as a tool for various engineering applications. Recent developments in tensor algebra have indicated that several well-known concepts from linear algebra can be extended to a multi-linear setting with the help of a special form of tensor contracted product, known as the Einstein product. Thus, the tensor contracted product and its properties can be harnessed to define the notions of multi-linear system theory where the input, output signals, and the system are inherently multi-domain or multi-modal. This study provides an overview of tensor algebra tools which can be seen as an extension of linear algebra, at the same time highlighting the differences and advantages that the multi-linear setting brings forth. In particular, the notions of tensor inversion, tensor singular value, and tensor eigenvalue decomposition using the Einstein product are explained. In addition, this study also introduces the notion of contracted convolution for both discrete and continuous multi-linear system tensors. Tensor network representation of various tensor operations is also presented. In addition, application of tensor tools in developing transceiver schemes for multi-domain communication systems, with an example of MIMO CDMA system, is presented. This study provides a foundation for professionals whose research involves multi-domain or multi-modal signals and systems.

Quadratic Multilinear Discriminant Analysis for Tensorial Data Classification

Over the past decades, there has been an increase of attention to adapting machine learning methods to fully exploit the higher order structure of tensorial data. One problem of great interest is tensor classification, and in particular the extension of linear discriminant analysis to the multilinear setting. We propose a novel method for multilinear discriminant analysis that is radically different from the ones considered so far, and it is the first extension to tensors of quadratic discriminant analysis. Our proposed approach uses invariant theory to extend the nearest Mahalanobis distance classifier to the higher-order setting, and to formulate a well-behaved optimization problem. We extensively test our method on a variety of synthetic data, outperforming previously proposed MDA techniques. We also show how to leverage multi-lead ECG data by constructing tensors via taut string, and use our method to classify healthy signals versus unhealthy ones; our method outperforms state-of-the-art MDA methods, especially after adding significant levels of noise to the signals. Our approach reached an AUC of 0.95(0.03) on clean signals—where the second best method reached 0.91(0.03)—and an AUC of 0.89(0.03) after adding noise to the signals (with a signal-to-noise-ratio of −30)—where the second best method reached 0.85(0.05). Our approach is fundamentally different than previous work in this direction, and proves to be faster, more stable, and more accurate on the tests we performed.

Resonant decompositions and global well-posedness for 2D Zakharov-Kuznetsov equation in Sobolev spaces of negative indices

The Cauchy problem for Zakharov-Kuznetsov equation on R2 is shown to be global well-posed for the initial data in Hs provided s>−113. As conservation laws are invalid in Sobolev spaces below L2, we construct an almost conserved quantity by multilinear correction term following the I-method introduced by Colliander, Keel, Staffilani, Takaoka and Tao. In contrast to the KdV equation, the main difficulty is to handle the resonant interactions due to the multidimensional and multilinear setting of the problem. The proof relies upon the bilinear Strichartz estimate and the nonlinear Loomis-Whitney inequality.

{text {UMD}}-Extensions of Calderón–Zygmund Operators with Mild Kernel Regularity

Armed with new methods we revisit a result of Figiel concerning the {text {UMD}}-extensions of linear Calderón–Zygmund operators with mild kernel regularity and then extend our new proof to the multilinear setting improving recent {text {UMD}}-valued estimates of multilinear singular integrals.

Endpoint Sobolev boundedness and continuity of multilinear fractional maximal functions

In this paper, we explore the endpoint Sobolev regularity of the fractional maximal operators in the multilinear setting. Some new bounds and continuity for the derivatives of the multilinear fractional maximal functions are established, both in the centered and in the uncentered cases.

Off-Diagonal Estimates for Bilinear Commutators

We find a minimal notion of non-degeneracy for bilinear singular integral operators T and identify testing conditions on the multiplying function b that characterize the Lp × Lq → Lr, 1<p,q<infty and r>frac {1}{2}, boundedness of the bilinear commutator [b, T]1(f, g) = bT(f, g) − T(bf, g). Our arguments cover almost all arrangements of the integrability exponents p, q, r with a single open problem presented in the end. Additionally, the arguments extend to the multilinear setting.

Multilinear commutators in the two‐weight setting

We extend the recently much-studied two-weight commutator estimates to the multilinear setting. In contrast to previous results, our result respects the multilinear nature of the problem fully and is formulated with the genuinely multilinear weights.

Restricted testing conditions for the multilinear maximal operator

Restricted testing conditions were considered recently. For the maximal operator, Hytönen, Li and Sawyer [8] first obtained parental testing condition. Later, they [9] showed that it suffices to restrict testing to doubling cubes. Chen and Lacey [3] gave a similar restricted testing condition. In our paper, we discuss a version of the latter in the multilinear setting.

Two Weight Characterizations for the Multilinear Local Maximal Operators

Let 0 < β < 1 and Ω be a proper open and non-empty subset of Rn. In this paper, the object of our investigation is the multilinear local maximal operator $${{\cal M}_\beta }$$ , defined by $${{\cal M}_\beta }\left( {\overrightarrow f } \right)\left( x \right) = \mathop {\sup }\limits_{\matrix{{Q \ni x} \cr {Q \in {{\cal F}_\beta }} \cr } } \prod\limits_{i = 1}^m {{1 \over {\left| Q \right|}}\int_Q {\left| {{f_i}\left( {{y_i}} \right)} \right|} {\rm{d}}{y_i},} $$ where $${\mathcal{F}_\beta } = \left\{ {Q\left( {x,l} \right):x \in \Omega ,l < \beta d\left( {x,{\Omega ^c}} \right)} \right\},Q = Q\left( {x,l} \right)$$ is denoted as a cube with sides parallel to the axes, and x and l denote its center and half its side length. Two-weight characterizations for the multilinear local maximal operator $${{\cal M}_\beta }$$ are obtained. A formulation of the Carleson embedding theorem in the multilinear setting is proved.

On families between the Hardy–Littlewood and spherical maximal functions

We study a family of maximal operators that provides a continuous link connecting the Hardy-Littlewood maximal function to the spherical maximal function. Our theorems are proved in the multilinear setting but may contain new results even in the linear case. For this family of operators we obtain bounds between Lebesgue spaces in the optimal range of exponents.

Weak factorization of the Hardy space Hp for small values of p, in the multilinear setting

We give a weak factorization proof of the Hardy space Hp(Rn) in the multilinear setting, for nn+1<p<1. As a consequence, we obtain a characterization of the boundedness of the commutator [b,T] from Lr1(Rn) x ... x Lrm(Rn) to Lq′(Rn), where b∈Lipα(Rn), and αn=∑i=1m1ri+1q−1.

On coincidence results for summing multilinear operators: interpolation, $$\ell _1$$-spaces and cotype

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.

Systematics and symmetry in molecular phylogenetic modelling

Probabilistic models of mathematical phylogenetics have been intensively used in recent years in biology as the cornerstone of methods to infer and reconstruct the ancestral relationships between species. We bring a lens of mathematical physics to bear on the formulation of the theoretical models, focussing on the applicability of group and representation theory to guide model specification and to exploit the multilinear setting of the models in the presence of underlying symmetries. We focus on aspects of multipartite entanglement which are shared between descriptions of quantum states on the physics side, and the multi-way tensor probability arrays arising in phylogenetics. We give examples of entanglement invariants (Markov invariants) for the case of quartets, and DNA data; and compare and contrast the rings of quantum/Markov invariants for the three and four qubit case/binary triplet and quartet cases, respectively.

Vector-Valued Extensions of Operators Through Multilinear Limited Range Extrapolation

We give an extension of Rubio de Francia’s extrapolation theorem for functions taking values in {{,mathrm{UMD},}} Banach function spaces to the multilinear limited range setting. In particular we show how boundedness of an m-(sub)linear operator T:Lp1(w1p1)×⋯×Lpm(wmpm)→Lp(wp)\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} T:L^{p_1}(w_1^{p_1})\\times \\cdots \\times L^{p_m}(w_m^{p_m})\\rightarrow L^p(w^p) \\end{aligned}$$\\end{document}for a certain class of Muckenhoupt weights yields an extension of the operator to Bochner spaces L^{p}(w^p;X) for a wide class of Banach function spaces X, which includes certain Lebesgue, Lorentz and Orlicz spaces. We apply the extrapolation result to various operators, which yields new vector-valued bounds. Our examples include the bilinear Hilbert transform, certain Fourier multipliers and various operators satisfying sparse domination results.

Weighted boundedness of multilinear maximal function using Dirac deltas

In this article we extend a method of Miguel de Guzman involving boundedness properties of maximal functions using Dirac deltas to multilinear setting. This method involves estimating maximal functions over finite linear combination of Dirac deltas. As an application, we obtain end-point weighted boundedness of the multilinear Hardy–Littlewood fractional maximal function with respect to multilinear weights.

Small-Depth Multilinear Formula Lower Bounds for Iterated Matrix Multiplication with Applications

The complexity of Iterated Matrix Multiplication (IMM) is a central theme in Computational Complexity theory, as the problem is closely related to the problem of separating various complexity classes within ${P}$. In this paper, we study the algebraic formula complexity of multiplying $d$ many $2\times 2$ matrices, denoted ${IMM}_{d}$, and show that the well-known divide-and-conquer algorithm cannot be significantly improved at any depth as long as the formulas are multilinear. Formally, for each depth $\Delta \leq \log d$, we show that any product-depth $\Delta$ multilinear formula for ${IMM}_d$ must have size $\exp(\Omega(\Delta d^{1/\Delta})).$ It also follows from this that any multilinear circuit of product-depth $\Delta$ for the same polynomial of the above form must have a size of $\exp(\Omega(d^{1/\Delta})).$ In particular, any polynomial-sized multilinear formula for ${IMM}_d$ must have depth $\Omega(\log d)$, and any polynomial-sized multilinear circuit for ${IMM}_d$ must have depth $\Omega(\log d/\log \log d).$ Both of these bounds are tight up to constant factors. Our lower bound has the following three consequences for multilinear formula complexity. 1. Depth-reduction: A well-known result of Brent [J. ACM, 21 (1974), pp. 201--206] implies that any formula of size $s$ can be converted to one of size $s^{O(1)}$ and depth $O(\log s)$; further, this reduction continues to hold for multilinear formulas. On the other hand, our lower bound implies that any depth-reduction in the multilinear setting cannot reduce the depth to $o(\log s)$ without a superpolynomial blow-up in size. 2. Circuits vs. formulas: Any circuit of size $s$ and product-depth $\Delta$ can be converted into a formula of product-depth $\Delta$ and size $s^{O(\Delta)}$. In the multilinear setting, we show that it is not possible to improve on this significantly for small depths. Formally, our results imply that for all large enough $s$ and $\Delta = o(\log s/\log \log s)$, there is an explicit multilinear polynomial $P_{s,\Delta}$ that has a syntactic multilinear circuit of size $s$ and is such that any multilinear formula of product-depth $\Delta$ computing $P_{s,\Delta}$ must have size $s^{\Omega(\Delta)}$. 3. Separations from general formulas: Shpilka and Yehudayoff [Found. Trends Theor. Comput. Sci., 5 (2010), pp. 207--388] asked whether general formulas can be more efficient than multilinear formulas for computing multilinear polynomials. Our result, along with a nontrivial upper bound for ${IMM}_{d}$ implied by a result of Gupta et al. [SIAM J. Comput., 45 (2016), pp. 1064--1079], shows that for any size $s$ and product-depth $\Delta = o(\log s),$ general formulas of size $s$ and product-depth $\Delta$ cannot be converted to multilinear formulas of size $s^{O(1)}$ and product-depth $\Delta$ when the underlying field has characteristic zero.

Multilinear operators factoring through Hilbert spaces

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. Kwapie\'{n}, 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 $\Gamma$ 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 $\gamma$ which is in duality with $\Gamma$.

A note on commutator in the multilinear setting

Let $m\in \mathbb{N}$ and $\vec{b}=(b_{1},\cdots,b_{m})$ be a collection of locally integrable functions. It is proved that $b_{1},b_{2},\cdots, b_{m}\in BMO$ if and only if $$\sup_{Q}\frac{1}{|Q|^{m}}\int_{Q^{m}}\Big|\sum_{i=1}^{m}\big(b_{i}(x_{i})-(b_{i})_{Q}\big)\Big|d\vec{x}<\infty,$$ where $(b_{i})_{Q}=\frac{1}{|Q|}\int_{Q}b_{i}(x)dx$. As an application, we show that if the linear commutator of certain multilinear Calder\'{o}n-Zygmund operator $[\Sigma \vec{b},T]$ is bounded from $L^{p_{1}}\times\cdots\times L^{p_{m}}$ to $L^{p}$ with $\sum_{i=1}^{m}1/p_{i}=1/p$ and $1<p,p_{1},\cdots,p_{m}<\infty$, then $b_{1},\cdots,b_{m}\in BMO$. Therefore, the result of Chaffee \cite{C} (or Li and Wick \cite{LW}) is extended to the general case.