The Kahane--Salem--Zygmund inequality is a probabilistic result that guarantees the existence of special matrices with entries $1$ and $-1$ generating unimodular $m$-linear forms $A_{m,n}:\ell_{p_{1}}^{n}\times \cdots\times\ell_{p_{m}}^{n}\longrightarrow\mathbb{R}$ (or $\mathbb{C}$) with relatively small norms. The optimal asymptotic estimates for the smallest possible norms of $A_{m,n}$ when $\left\{ p_{1},...,p_{m}\right\} \subset\lbrack2,\infty]$ and when $\left\{ p_{1},...,p_{m}\right\} \subset\lbrack1,2)$ are well-known and in this paper we obtain the optimal asymptotic estimates for the remaining case: $\left\{ p_{1},...,p_{m}\right\} $ intercepts both $[2,\infty]$ and $[1,2)$. In particular we prove that a conjecture posed by Albuquerque and Rezende is false and, using a special type of matrices that dates back to the works of Toeplitz, we also answer a problem posed by the same authors.
The existence of unimodular forms with small norms on sequence spaces is crucial in a variety of problems in modern analysis. We prove that the infimum of $\left\Vert A\right\Vert $ over all unimodular $d$-linear (complex or real) forms $A$ on $\ell_{p_1}^{n_{1}} \times \cdots \times \ell_{p_d}^{n_{d}}$, for all $p_1,\dots,p_d \in [2, \infty]$ and all positive integers $n_1,\dots,n_d$, behaves (asymptotically) as $\left(n_{1}^{1/2} + \cdots + n_{d}^{1/2}\right) \prod_{j=1}^{d}n_{j}^{\frac{1}{2} - \frac{1}{p_j}}$. Applications to the theory of the multilinear Hardy--Littlewood inequality are al…
In this paper, weighted norm inequalities with $A_p$ weights are established for the multilinear singular integral operators whose kernels satisfy $L^{r'}$-Hörmander regularity condition. As applications, we recover a weighted estimate for the multilinear Fourier multiplier obtained by Fujita and Tomita, and obtain several new weighted estimates for the multilinear Fourier multiplier as well.
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.
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 …
We establish Hölder type inequalities for Lorentz sequence spaces and their duals. In order to achieve these and some related inequalities, we study diagonal multilinear forms in general sequence spaces, and obtain estimates for their norms. We also consider norms of multilinear forms in different Banach multilinear ideals.
We derive criteria governing two-weight estimates for multilinear fractional integrals and appropriate maximal functions. The two and one weight problems for multi(sub)linear strong fractional maximal operators are also studied; in particular, we derive necessary and sufficient conditions guaranteeing the trace type inequality for this operator. We also establish the Fefferman-Stein type inequality, and obtain one-weight criteria when a weight function is of product type. As a consequence, appropriate results for multilinear Riesz potential operator with product kernels follow.
Very recently, motivated by the result of Bhatia and Šemrl which characterizes the Birkhoff-James orthogonality of operators on a finite dimensional Hilbert space in terms of norm attaining points, the Bhatia-Šemrl property was introduced. The main purpose of this article is to study the denseness of the set of multilinear maps with the Bhatia-Šemrl property which is contained in the set of norm attaining ones. Contrary to the most of previous results which were shown for operators on real Banach spaces, we prove the denseness for multilinear maps on some complex Banach spaces. We also show th…
We characterize the sets of norm one vectors $\mathbf{x}_1,\ldots,\mathbf{x}_k$ in a Hilbert space $\mathcal H$ such that there exists a $k$-linear symmetric form attaining its norm at $(\textbf{x}_1,\ldots,\mathbf{x}_k)$. We prove that in the bilinear case, any two vectors satisfy this property. However, for $k\ge 3$ only collinear vectors satisfy this property in the complex case, while in the real case this is equivalent to $\mathbf{x}_1,\ldots,\mathbf{x}_k$ spanning a subspace of dimension at most 2. We use these results to obtain some applications to symmetric multilinear forms, symmetric…
We study tensor norms that destroy unconditionality in the following sense: for every Banach space $E$ with unconditional basis, the $n$-fold tensor product of $E$ (with the corresponding tensor norm) does not have unconditional basis. We establish an easy criterion to check weather a tensor norm destroys unconditionality or not. Using this test we get that all injective and projective tensor norms different from $\varepsilon$ and $π$ destroy unconditionality, both in full and symmetric tensor products. We present applications to polynomial ideals: we show that many usual polynomial ideals nev…
We give conditions that ensure that an operator satisfying a Piestch domination in a given setting also satisfies a Piestch domination in a different setting. From this we derive that a bounded mutlilinear operator $T$ is Lipschitz $p$-summing if and only if the mapping $ f_T(x_1\otimes\cdots \otimes x_n):=T(x_1,\ldots, x_n)$ is Lipschitz $p$-summing. The results are based on the projective tensor norm. An example with the Hilbert tensor norm is provided to show that the statement may not hold when a reasonable cross-norm other than the projective tensor norm is considered.
For each ideal of multilinear mappings $\cal M$ we explicitly construct a corresponding ideal $^{a}{\cal M}$ such that multilinear forms in $^{a}{\cal M}$ are exactly those which can be approximated, in the uniform norm, by multilinear forms in ${\cal M}$. This construction is then applied to finite type, compact, weakly compact and absolutely summing multilinear mappings. It is also proved that the correspondence ${\cal M} \mapsto ^{a}{\cal M}$ is Aron-Berner stability preserving.
In this paper we prove a characterization of continuity for polynomials on a normed space. Namely, we prove that a polynomial is continuous if and only if it maps compact sets into compact sets. We also provide a partial answer to the question as to whether a polynomial is continuous if and only if it transforms connected sets into connected sets. These results motivate the natural question as to how many non-continuous polynomials there are on an infinite dimensional normed space. A problem on the \emph{lineability} of the sets of non-continuous polynomials and multilinear mappings on infinit…
The Orlicz $\left( \ell_{2},\ell_{1}\right) $-mixed inequality states that $$ \left( \sum_{j_{1}=1}^{n}\left( \sum_{j_{2}=1}^{n}\left\vert A(e_{j_{1} },e_{j_{2}})\right\vert \right) ^{2}\right) ^{\frac{1}{2}}\leq\sqrt {2}\left\Vert A\right\Vert $$ for all bilinear forms $A:\mathbb{K}^{n}\times\mathbb{K}^{n}\rightarrow \mathbb{K}$ and all positive integers $n$, where $\mathbb{K}^{n}$ denotes $\mathbb{R}^{n}$ or $\mathbb{C}^{n}$ endowed with the supremum norm. In this paper we extend this inequality to multilinear forms, with $\mathbb{K}^{n}$ endowed with $\ell_{p}$ norms for all $p\in\lbrack1,\…
Let $G$ be a locally compact unimodular group, and let $ϕ$ be some function of $n$ variables on $G$. To such a $ϕ$, one can associate a multilinear Fourier multiplier, which acts on some $n$-fold product of the non-commutative $L_p$-spaces of the group von Neumann algebra. One may also define an associated Schur multiplier, which acts on an $n$-fold product of Schatten classes $S_p(L_2(G))$. We generalize well-known transference results from the linear case to the multilinear case. In particular, we show that the so-called `multiplicatively bounded $(p_1,\ldots,p_n)$-norm' of a multilinear Sch…
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 $Γ$.
We prove some Sawyer-type characterizations for multilinear fractional maximal function for the upper triangle case. We also provide some two-weight norm estimates for this operator. As one of the main tools, we use an extension of the usual Carleson Embedding that is an analogue of the P. L. Duren extension of the Carleson Embedding for measures.
Among other results we investigate $\left( α,β\right) $-lineability of the set of non-continuous $m$-linear operators defined between normed spaces as a subset of the space of all $m$-linear operators. We also give a partial answer to an open problem on the lineability of the set of non absolutely summing operators.
We prove that any pair of reasonable cross norms defined on the tensor product of $n$ Banach spaces induce $(2k)^{n-1}$-Lipschitz equivalent metrics (and thus, a unique topology) on the set $S^k_{X_1,\ldots, X_n}$ of vectors of rank $\leq k$. With this, we define the Segre cone of Banach spaces, $Σ_{X_1,\ldots, X_n},$ and state when $S^k_{X_1,\ldots, X_n}$ is closed. We introduce an auxiliary mapping (a $Σ$-operator) that allows us to study multilinear mappings with a geometrical point of view. We use the isometric correspondence between multilinear mappings and Lipschitz $Σ$-operators, to hav…
We initiate the study of a duality theory which applies to norm inequalities for pointwise weighted geometric means of positive operators. The theory finds its expression in terms of certain pointwise factorisation properties of function spaces which are naturally associated to the norm inequality under consideration. We relate our theory to the Maurey-Nikisin-Stein theory of factorisation of operators, and present a fully multilinear version of Maurey's fundamental theorem on factorisation of operators through $L^1$. The development of the theory involves convex optimisation and minimax theor…
Let $S_α$ be the multilinear square function defined on the cone with aperture $α\geq 1$. In this paper, we investigate several kinds of weighted norm inequalities for $S_α$. We first obtain a sharp weighted estimate in terms of aperture $α$ and $\vec{w} \in A_{\vec{p}}$. By means of some pointwise estimates, we also establish two-weight inequalities including bump and entropy bump estimates, and Fefferman-Stein inequalities with arbitrary weights. Beyond that, we consider the mixed weak type estimates corresponding Sawyer's conjecture, for which a Coifman-Fefferman inequality with the precise…
This note gives a wide-ranging update on the multiplier theorems by Akylzhanov and the second author [J. Funct. Anal., 278 (2020), 108324]. The proofs of the latter crucially rely on $L^p$-$L^q$ norm estimates for spectral projectors of left-invariant weighted subcoercive operators on unimodular Lie groups, such as Laplacians, sub-Laplacians and Rockland operators. By relating spectral projectors to heat kernels, explicit estimates of the $L^p$-$L^q$ norms can be immediately exploited for a much wider range of (connected unimodular) Lie groups and operators than previously known. The compariso…
The paper is concerned with compact bilinear operators on asymmetric normed spaces. The study of multilinear operators on asymmetric normed spaces was initiated by Latreche and Dahia, Colloq. Math. (2020). We go further in this direction and prove a Schauder type theorem on the compactness of the adjoint of a compact bilinear operator and study the ideal properties of spaces of compact bilinear operators. These extend some results of Ramanujan and Schock, Linear and Multilinear Algebra (1985), and Ruch, ibid. (1989), on compact bilinear operators on Banach spaces. On the space of bilinear form…
We prove decoupling inequalities for random polynomials in independent random variables with coefficients in vector space. We use various means of comparison, including rearrangement invariant norms (e.g., Orlicz and Lorentz norms), tail distributions, tightness, hypercontractivity, etc.
We intend to derive the moment and exponential tail estimates for the so-called bivariate or more generally multivariate functional operations, not necessary to be linear or even multilinear. We will show also the strong or at last weak (i.e. up to multiplicative constant) exactness of obtained estimates.
We develop a general framework for the analysis of operator-valued multilinear multipliers acting on Banach-valued functions. Our main result is a Coifman-Meyer type theorem for operator-valued multilinear multipliers acting on suitable tuples of UMD spaces. A concrete case of our theorem is a multilinear generalization of Weis' operator-valued Hörmander-Mihlin linear multiplier theorem. Furthermore, we derive from our main result a wide range of mixed $L^p$-norm estimates for multi-parameter multilinear paraproducts, leading to a novel mixed norm version of the partial fractional Leibniz rule…
We prove that the multiple summing norm of multilinear operators defined on some $n$-dimensional real or complex vector spaces with the $p$-norm may be written as an integral with respect to stables measures. As an application we show inclusion and coincidence results for multiple summing mappings. We also present some contraction properties and compute or estimate the limit orders of this class of operators.
The $n$-linear Bohnenblust-Hille inequality asserts that there is a constant $C_{n}\in\lbrack1,\infty)$ such that the $\ell_{\frac{2n}{n+1}}$-norm of $(U(e_{i_{^{1}}},...,e_{i_{n}}))_{i_{1},...i_{n}=1}^{N}$is bounded above by $C_{n}$ times the supremum norm of $U,$ regardless of the $n$-linear form $U:\mathbb{C}^{N}\times...\times\mathbb{C}^{N}% \rightarrow\mathbb{C}$ and the positive integer $N$ (the same holds for real scalars). The power $2n/(n+1)$ is sharp but the values and asymptotic behavior of the optimal constants remain a mystery. The first estimates for these constants had exponenti…
Perturbation or error bounds of functions have been of great interest for a long time. If the functions are differentiable, then the mean value theorem and Taylor's theorem come handy for this purpose. While the former is useful in estimating $\|f(A+X)-f(A)\|$ in terms of $\|X\|$ and requires the norms of the first derivative of the function, the latter is useful in computing higher order perturbation bounds and needs norms of the higher order derivatives of the function. In the study of matrices, determinant is an important function. Other scalar valued functions like eigenvalues and coeffici…
A Sharp Reverse Minkowski Inequality for the Gaussian Mass of Integral Unimodular Lattices Through Rank $32$
Harvard Business School（哈佛商学院）；Department of Economics and Center of Mathematical Sciences and Applications, Harvard University（哈佛大学经济与数学科学应用中心）；a16z crypto（a16z 加密）
