The optimal constants of the mixed (ℓ1,ℓ2)\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}.Comment: accepted for publication in the Journal of Number Theor

A note on the hypercontractivity of the polynomial Bohnenblust--Hille inequality

For $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$ and $m$ a positive integer, we remark that there is a constant $C$ so that, for all $r\in\lbrack1,\frac {2m}{m+1}],$ the supremum of the ratio between the $\ell_{r}$ norm of the coefficients of any nonzero $m$-homogeneous polynomial $P:\ell_{\infty}^{n}(\mathbb{K}) \rightarrow\mathbb{K}$ and its supremum norm is dominated by $C^{m}\cdot n^{(\frac{m}{r}-\frac{m+1}{2})}$ and, moreover, we prove that the exponent $\frac{m}{r}-\frac{m+1}{2}$ is optimal

Sharp coincidences for absolutely summing multilinear operators

In this note we prove the optimality of a family of known coincidence theorems for absolutely summing multilinear operators. We connect our results with the theory of multiple summing multilinear operators and prove the sharpness of similar results obtained via the complex interpolation method.Comment: This note is part of the author's thesis which is being written for the candidature to the position of Full Professor at Universidade Federal da Para\'{\i}ba, Brazi

Cotype and nonlinear absolutely summing mappings

In this paper we study absolutely summing mappings on Banach spaces by exploring the cotype of their domains and ranges. It is proved that every $n$% -linear mapping from $\mathcal{L}_{\infty}$-spaces into $\mathbb{K}$ is $(2;2,...,2,\infty)$-summing and also shown that every $n$-linear mapping from $\mathcal{L}_{\infty}$-spaces into $F$ is $(q;2,...,2)$-summing whenever $F$ has cotype $q.$ We also give new examples of analytic summing mappings and polynomial and multilinear versions of a linear Extrapolation Theorem

On the multilinear extensions of the concept of absolutely summing operators

In this paper we investigate the connections between the several different extensions of the concept of absolutely summing operators.Comment: 11 page

On Banach spaces whose duals are isomorphic to l_1

In this paper we present new characterizations of Banach spaces whose duals are isomorphic to $l_{1}(\Gamma),$ extending results of Stegall, Lewis-Stegall and Cilia-D'Anna-Guti\'{e}rrez

A remark on absolutely summing multilinear mappings

In this note we obtain new coincidence theorems for absolutely summing multilinear mappings between Banach spaces. We also prove that our results, in general, can not be improved.Comment: 5 page

On ideals of polynomials and their applications

In this paper we obtain some statements concerning ideals of polynomials and apply these results in a number of different situations. Among other results, we present new characterizations of $\mathcal{L}_{\infty}$-spaces, Coincidence theorems, Dvoretzky-Rogers and Extrapolation type theorems for dominated polynomials.Comment: 8 page

Lower bounds for the constants of the Hardy-Littlewood inequalities

Given an integer $m\geq2$, the Hardy--Littlewood inequality (for real scalars) says that for all $2m\leq p\leq\infty$, there exists a constant $C_{m,p}$ such that, for all continuous $m$--linear forms $A:\ell_{p}^{N}\times\cdots\times\ell_{p}^{N}\rightarrow\mathbb{R}$ and all positive integers $N$, \left( \sum_{j_{1},...,j_{m}=1}^{N}\left\vert A(e_{j_{1}},...,e_{j_{m}% })\right\vert ^{\frac{2mp}{mp+p-2m}}\right) ^{\frac{mp+p-2m}{2mp}}\leq C_{m,p}^{\mathbb{R}}\left\Vert A\right\Vert . The limiting case $p=\infty$ is the well-known Bohnenblust--Hille inequality; the behavior of the constants $C_{m,p}^{\mathbb{R}}$ is an open problem. In this note we provide nontrivial lower bounds for these constants

Lineability and uniformly dominated sets of summing nonlinear operators

In this note we prove an abstract version of a result from 2002 due to Delgado and Pi\~{n}ero on absolutely summing operators. Several applications are presented; some of them in the multilinear framework and some in a completely nonlinear setting. In a final section we investigate the size of the set of non uniformly dominated sets of linear operators under the point of view of lineability.Comment: Minor correction