Searcharxiv⌕ Search

arXiv subjects

Daniel Pellegrino

Publications and source records attributed to Daniel Pellegrino.

At least 37 records · Page 2Linked to original sources
arXiv2020

On unimodular multilinear forms with small norms on sequence spaces

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.

math.FA↗
arXiv2019

Reflexivity and non-weakly null maximizing sequences

We introduce and explore a new property related to reflexivity that plays an important role in the characterization of norm attaining operators. We also present an application to the theory of compact perturbations of linear operators and characterize norm attaining scalar-valued continuous $2$-homogeneous polynomials on $\ell_{2}$.

math.FA↗
arXiv2019

A unified factorization theorem for Lipschitz summing operators

We prove a general factorization theorem for Lipschitz summing operators in the context of metric spaces which recovers several linear and nonlinear factorization theorems that have been proved recently in different environments. New applications are also given.

math.FA↗
arXiv2018

Lineability and spaceability: a new approach

The area of research called \textquotedblleft Lineability\textquotedblright% \ looks for linear structures inside exotic subsets of vector spaces. In the last decade lineability/spaceability has been investigated in rather general settings; for instance, Set Theory, Probability Theory, Functional Analysis, Measure Theory, etc. It is a common feeling that positive results on lineability/spaceability are quite natural (i.e., in general \textquotedblleft large\textquotedblright subspaces can be found inside exotic subsets of vector spaces, in quite different settings) and more restrictive approaches have been persecuted. In this paper we introduce and explore a new approach in this direction.

math.FA↗
arXiv2016

Bohnenblust--Hille inequality for polynomials whose monomials have uniformly bounded number of variables

In 2015, using an innovative technique, Carando, Defant and Sevilla-Peris succeeded in proving a Bohnenblust--Hille type inequality with constants of polynomial growth in $m$ for a certain family of complex $m$-homogeneous polynomials. In the present paper, using a completely different approach, we prove that the constants of this inequality are uniformly bounded irrespectively of the value of $m$.

math.FA↗
arXiv2016

Some applications of the Hölder inequality for mixed sums

We use the Hölder inequality for mixed exponents to prove some optimal variants of the generalized Hardy--Littlewood inequality for $m$-linear forms on $\ell _{p}$ spaces with mixed exponents. Our results extend recent results of Araujo et al.

math.FA↗
arXiv2016

Towards sharp Bohnenblust--Hille constants

We investigate the optimality problem associated with the best constants in a class of Bohnenblust--Hille type inequalities for $m$--linear forms. While germinal estimates indicated an exponential growth, in this work we provide strong evidences to the conjecture that the sharp constants in the classical Bohnenblust--Hille inequality are universally bounded, irrespectively of the value of $m$; hereafter referred as the \textit{Universality Conjecture}. In our approach, we introduce the {notions of entropy and complexity}, designed to measure, to some extent, the complexity of such optimization problems. We show that the notion of entropy is critically connected to the Universality Conjecture; for instance, that if the entropy grows at most exponentially with respect to $m$, then the optimal constants of the $m$% --linear Bohnenblust--Hille inequality for real scalars are indeed bounded universally in $m$. It is likely that indeed the entropy grows as $4^{m-1}$, and in this scenario, we show that the optimal constants are precisely $2^{1-\frac{1}{m}} $. In the bilinear case, $m=2$, we show that any extremum of the Littlewood's $4/3$-inequality has entropy $4$ and complexity $2$, and thus we are able to classify all extrema of the problem. We also prove that, for any mixed $\left( \ell _{1},\ell _{2}\right) $% --Littlewood inequality, the entropy do grow exponentially and the sharp constants for such a class of inequalities are precisely $(\sqrt{2})^{m-1}$. In addition to the {notions of entropy and complexity}, the approach we develop in this work makes decisive use of a family of strongly non-symmetric $m$--linear forms, which has further consequences to the theory, as we explain herein.

math.FA↗
arXiv2016

Some inclusion results for interpolated summing operator ideals and integrability improvement of vector valued functions

Consider a Banach space valued measurable function $f$ and an operator $u$ from the space where {$f$} takes values. If $f $ is Pettis integrable, a classical result due to J. Diestel shows that composing it with $u$ gives a Bochner integrable function $u \circ f$ whenever $u$ is absolutely summing. In a previous work we have shown that a well-known interpolation technique for operator ideals allows to prove under some requirements that a composition of a $p$-Pettis integrable function with a $q$-summing operator provides an $r$-Bochner integrable function. In this paper a new abstract inclusion theorem for classes of {abstract} summing operators is shown and applied to the class of interpolated operator ideals. Together with the results of the {aforementioned} paper, it provides more results on the relation about the integrability of the function $u \circ f$ and the summability properties of $u$.

math.FA↗
arXiv2016

Regularity principle in sequence spaces and applications

We prove a nonlinear regularity principle in sequence spaces which produces universal estimates for special series defined therein. Some consequences are obtained and, in particular, we establish new inclusion theorems for multiple summing operators. Of independent interest, we settle all Grothendieck's type $(\ell_{1},\ell_{2})$ theorems for multilinear operators. We further employ the new regularity principle to solve the classification problem concerning all pairs of admissible exponents in the anisotropic Hardy--Littlewood inequality.

math.CA↗
arXiv2016

Geometry of the closed unit ball of the space of bilinear forms on $\ell_{\infty}^{2}$

We obtain all extreme and exposed points of the closed unit ball of the space of bilinear forms $T:\ell_{\infty}^{2}\times\ell_{\infty}^{2}\rightarrow \mathbb{R}.$ We also show that any (norm one) bilinear form $T:\ell_{\infty }^{2}\times\ell_{\infty}^{2}\rightarrow\mathbb{R}$ for which the optimal constant of the Littlewood's $4/3$ inequality is achieved is necessarily an extreme point. In the case of complex scalars we combine analytical and numerical evidences supporting that, at least for complex bilinear forms with real coefficients, the optimal constant of Littlewood's $4/3$ inequality seems to the the trivial, i.e., $1.$

math.FA↗
arXiv2016

Optimal constants for a mixed Littlewood type inequality

For $p\in\lbrack2,\infty]$ a mixed Littlewood-type inequality asserts that there is a constant $C_{(m),p}\geq1$ such that \[ \left( \sum_{i_{1}=1}^{\infty}\left( \sum_{i_{2},...,i_{m}=1}^{\infty }|T(e_{i_{1}},...,e_{i_{m}})|^{2}\right) ^{\frac{1}{2}\frac{p}{p-1}}\right) ^{\frac{p-1}{p}}\leq C_{(m),p}\Vert T\Vert \] for all continuous real-valued $m$-linear forms on $\ell_{p}\times c_{0} \times\dots\times c_{0}$ (when $p=\infty$, $\ell_{p}$ is replaced by $c_{0})$. We prove that for $p>2.18006$ the optimal constants $C_{(m),p}$ are $\left( 2^{\frac{1}{2}-\frac{1}{p}}\right) ^{m-1}.$ When $p=\infty,$ we recover the best constants of the mixed $\left( \ell_{1},\ell_{2}\right) $-Littlewood inequality.

math.FA↗
arXiv2016

The optimal constants for the real Hardy--Littlewood inequality for bilinear forms on $c_{0}\times\ell_{p}$

For $p,q\geq2$, the Hardy and Littlewood inequalities for real bilinear forms, in its unified formulation, assert that there is a constant $C_{p,q}\geq1$ such that \begin{equation} \left(\sum\limits_{j=1}^{\infty}\left(\sum\limits_{k=1}^{\infty}\left\vert A(e_{j},e_{k})\right\vert ^{2}\right) ^{\fracλ{2}}\right) ^{\frac {1}λ}\leq C_{p,q}\left\Vert A\right\Vert, \end{equation} with sharp exponent $λ=\frac{pq}{pq-p-q},$ for all continuous bilinear forms $A:\ell_{p}\times\ell_{q}\rightarrow\mathbb{R}$ (as usual, $c_{0}$ replaces $\ell_{p}$ or $\ell_{q}$ when $p=\infty$ or $q=\infty$)$.$ In this note, among other results, we show that the sharp constants $C_{p,\infty}$ are precisely \[ C_{p,\infty}=2^{\frac{1}{2}-\frac{1}{p}}% \] whenever $p\geq\frac{p_{0}}{p_{0}-1}\approx2.18.$ The number $p_{0}\in(1,2)$ is the unique real number satisfying \[ Γ\left(\frac{p_{0}+1}{2}\right) =\frac{\sqrtπ}{2}. \] In the remaining case, i.e., for $2<p<\frac{p_{0}}{p_{0}-1}\approx 2.18,$ we obtain almost optimal constants, with better precision than $4\cdot10^{-4}$. This last result extends a result from Diniz et al. giving the sharp constant of the famous Littlewood's $4/3$ theorem for real scalars.

math.NT↗
arXiv2015

A short communication on the constants of the multilinear Hardy--Littlewood inequality

It was recently proved that for $p>2m^{3}-4m^{2}+2m$ the constants of the Hardy--Littlewood inequality for $m$-linear forms on $\ell_{p}$-spaces are less than or equal to the best known estimates of respective constants of the Bohnenblust--Hille inequality. In this note we obtain upper bounds for opposite side, i.e., the constants when $2m\leq p\leq2m^{3}-4m^{2}+2m.$ For these values of $p$ our result improves previous estimates from 2014 of Araujo \textit{et al}. for all $m\geq3.$

math.NT↗
arXiv2015

The optimal constants of the mixed $\left( \ell_{1},\ell _{2}\right) $-Littlewood inequality

In this note, among other results, we find the optimal constants of the generalized Bohnenblust--Hille inequality for $m$-linear forms over $\mathbb{R}$ and with multiple exponents $\left( 1,2,...,2\right) $, sometimes called mixed $\left( \ell _{1},\ell _{2}\right) $-Littlewood inequality. We show that these optimal constants are precisely $\left( \sqrt{2}\right) ^{m-1}$ and this is somewhat surprising since a series of recent papers have shown that similar constants have a sublinear growth. This result answers a question raised by Albuquerque \textit{et al.} in a paper published in 2014 in the \textit{Journal of Functional Analysis}.

math.FA↗