arXiv subjects

Anderson Barbosa

Publications and source records attributed to Anderson Barbosa.

5 recordsLinked to original sources

arXiv2026

Sharp Summability on Supports of Prescribed Combinatorial Dimension

We solve four questions raised by Bayart concerning coefficient summability for multilinear forms with prescribed supports. For every $m\ge 2$ and $d\in[1,m]$, we determine the product-summability exponent and the multilinear summability invariant: \[ \mathrm{prod}(m,d) =\min\left\{\frac{m}{d},\,m-\lceil d\rceil+1\right\}, \qquad \gamma_{\mathrm{mult}}(m,d) =\min\left\{m-\lceil d\rceil+1,\frac{2m}{d+1}\right\}. \] In particular, $\gamma_{\mathrm{mult}}(4,2)=8/3$, showing that the multilinear invariant need not be an integer. We also prove that, for every $d\in[1,m]$, there is a single infinite support of exact combinatorial dimension $d$ on which the dimensional Hardy--Littlewood bound is attained over both scalar fields for every anisotropic parameter $\mathbf p=(p_1,\ldots,p_m)$ with $\sum_j 1/p_j<1$, simultaneously across the two regimes separated by $\sum_j 1/p_j=1/2$.

math.FA↗

arXiv2023

On the set of functions that vanish at infinity and have a unique maximum

In this paper, we show that the set of continuous functions defined on $\mathbb{R}^n$ that approach zero at infinity and attain their maximum at precisely one (and only one) point is $n$-lineable but not $(n+2)$-lineable. This result complements some recent published works on an open question originally posed by Vladimir I. Gurariy (1935--2005) in 2003.

math.FA↗

arXiv2023

Complements of unions: insights on spaceability and applications

This paper presents two general criteria to determine spaceability results in the complements of unions of subspaces. The first criterion applies to countable unions of subspaces under specific conditions and is closely related to the results of Kitson and Timoney in [J. Math. Anal. Appl. \textbf{378} (2011), 680-686]. This criterion extends and recovers some classical results in this theory. The second criterion establishes sufficient conditions for the complement of a union of Lebesgue spaces to be $\left(\alpha,\beta\right)$-spaceable, or not, even when they are not locally convex. We use this result to characterize the measurable subsets having positive measure. Armed with these results, we have improved existing results in environments such as: Lebesgue measurable function sets, spaces of continuous functions, sequence spaces, nowhere H\"{o}lder function sets, Sobolev spaces, non-absolutely summing operator spaces, and even sets of functions of bounded variation.

math.FA↗

arXiv2023

On the spaceability of the set of functions in the Lebesgue space $L_p$ which are in no other $L_q$

In this note we prove that, for $p>0$, $L_{p}[0,1]\smallsetminus\bigcup_{q\in(p,\infty)}L_{q}[0,1]$ is $(\alpha,\mathfrak{c})$-spaceable if, and only if, $\alpha<\aleph_{0}$. Such a problem first appears in [V. F\'avaro, D. Pellegrino, D. Tomaz, Bull. Braz. Math. Soc. \textbf{51} (2020) 27-46], where the authors get the $(1,\mathfrak{c})$-spaceability of $L_{p}[0,1]\smallsetminus\bigcup_{q\in(p,\infty)}L_{q}[0,1]$ for $p>0$. The definitive answer to this problem continued to be sought by other authors, and some partial answers were obtained. The veracity of this result was expected, as a similar result is known for sequence spaces.

math.FA↗

arXiv2023

A general lineability criterion for complements of vector spaces

In 1931, Banach proved that, far from being exceptional objects, the Weierstrass functions form a residual set in the space $\mathcal{C}[0,1]$ of continuous functions. Later on, in 1966, V. I. Gurariy showed that, except for zero, there is an infinite-dimensional linear subspace of Weierstrass functions. This was the first example of \textit{lineability}. Over the last decade, this topic has attracted the continuous attention of the mathematical community, with a steady stream of papers being published, many of them in highly ranked mathematical journals. Several lineability criteria are known and applied to specific topological vector spaces. To paraphrase L. Bernal-Gonz\'alez and M. O. Cabrera in [J. Funct. Anal. \textbf{266} (2014), 3997-4025], ``sometimes, such criteria furnish unified proofs of a number of scattered results in the related literature''. In this article, we provide a general lineability criterion in the context of complements of vector spaces.

math.FA↗