arXiv subjects

José L. Ansorena

Publications and source records attributed to José L. Ansorena.

28 records · Page 2Linked to original sources

arXiv2021

Lorentz spaces and embeddings induced by almost greedy bases in superreflexive Banach spaces

The aim of this paper is to show that almost greedy bases induce tighter embeddings in superreflexive Banach spaces than in general Banach spaces. More specifically, we show that an almost greedy basis in a superreflexive Banach space $\mathbb{X}$ induces embeddings that allow squeezing $\mathbb{X}$ between two superreflexive Lorentz sequence spaces that are close to each other in the sense that they have the same fundamental function.

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arXiv2021

Toward an optimal theory of integration for quasi-Banach-space-valued functions

We present a new approach to define a suitable integral for functions with values in quasi-Banach spaces. The integrals of Bochner and Riemann have deficiencies in the non-locally convex setting. The study of an integral for $p$-Banach spaces initiated by Vogt is neither totally satisfactory, since there are quasi-Banach spaces which are $p$-convex for all $0<p<1$, so it is not always possible to choose an optimal $p$ to develop the integration. Our method puts the emphasis on the galb of the space, which permits a precise definition of its convexity. The integration works for all spaces of galbs known in the literature. We finish with a fundamental theorem of calculus for our integral.

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arXiv2019

On certain subspaces of $\ell_p$ for $0<p\le 1$ and their applications to conditional quasi-greedy bases in $p$-Banach spaces

We construct for each $0<p\le 1$ an infinite collection of subspaces of $\ell_p$ that extend the example from [J. Lindenstrauss, On a certain subspace of $\ell_{1}$, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 12 (1964), 539-542] of a subspace of $\ell_{1}$ with no unconditional basis. The structure of this new class of $p$-Banach spaces is analyzed and some applications to the general theory of $\mathcal{L}_{p}$-spaces for $0<p<1$ are provided. The introduction of these spaces serves the purpose to develop the theory of conditional quasi-greedy bases in $p$-Banach spaces for $p<1$. Among the topics we consider are the existence of infinitely many conditional quasi-greedy bases in the spaces $\ell_{p}$ for $p\le 1$ and the careful examination of the conditionality constants of the "natural basis" of these spaces.

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arXiv2018

Asymptotic greediness of the Haar system in the spaces $L_{p}[0,1]$, $1<p<\infty$

Our aim in this paper is to attain a sharp asymptotic estimate for the greedy constant $C_g[\mathcal{H}^{(p)},L_p]$ of the (normalized) Haar system $\mathcal{H}^{(p)}$ in $L_{p}[0,1]$ for $1<p<\infty$. We will show that the superdemocracy constant of $\mathcal{H}^{(p)}$ in $L_{p}[0,1]$ grows as $p^{\ast}=\max\{p,p/(p-1)\}$ as $p^*$ goes to $\infty$. Thus, since the unconditionality constant of $\mathcal{H}^{(p)}$ in $L_{p}[0,1]$ is $p^*-1$, the well-known general estimates for the greedy constant of a greedy basis obtained from the intrinsic features of greediness (namely, democracy and unconditionality) yield that $p^{\ast}\lesssim C_g[\mathcal{H}^{(p)},L_p]\lesssim (p^{\ast})^{2}$. Going further, we develop techniques that allow us to close the gap between those two bounds, establishing that, in fact, $C_g[\mathcal{H}^{(p)},L_p]\approx p^{\ast}$.

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arXiv2017

Building highly conditional quasi-greedy bases in classical Banach spaces

It is known that for a conditional quasi-greedy basis $\mathcal{B}$ in a Banach space $\mathbb{X}$, the associated sequence $(k_{m}[\mathcal{B}])_{m=1}^{\infty}$ of its conditionality constants verifies the estimate $k_{m}[\mathcal{B}]=\mathcal{O}(\log m)$ and that if the reverse inequality $\log m =\mathcal{O}(k_m[\mathcal{B}])$ holds then $\mathbb{X}$ is non-superreflexive. However, in the existing literature one finds very few instances of non-superreflexive spaces possessing quasi-greedy basis with conditionality constants as large as possible. Our goal in this article is to fill this gap. To that end we enhance and exploit a combination of techniques developed independently, on the one hand by Garrigós and Wojtaszczyk in [Conditional quasi-greedy bases in Hilbert and Banach spaces, Indiana Univ. Math. J. 63 (2014), no. 4, 1017-1036] and, on the other hand, by Dilworth et al. in [On the existence of almost greedy bases in Banach spaces, Studia Math. 159 (2003), no. 1, 67-101], and craft a wealth of new examples of non-superreflexive classical Banach spaces having quasi-greedy bases $\mathcal{B}$ with $k_{m}[\mathcal{B}]=\mathcal{O}(\log m)$.

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arXiv2017

1-Greedy renormings of Garling sequence spaces

Garling sequence spaces admit a renorming with respect to which their standard unit vector basis is 1-greedy. We also discuss some additional properties of these Banach spaces related to uniform convexity and superreflexivity. In particular, our approach to the study of the superreflexivity of Garling sequence space provides an example of how essentially non-linear tools from greedy approximation can be used to shed light into the linear structure of the spaces.

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arXiv2017

On Garling sequence spaces

The aim of this paper is to introduce and investigate a new class of separable Banach spaces modeled after an example of Garling from 1968. For each $1\leqslant p<\infty$ and each nonincreasing weight $\textbf{w}\in c_0\setminus\ell_1$ we exhibit an $\ell_p$-saturated, complementably homogeneous, and uniformly subprojective Banach space $g(\textbf{w},p)$. We also show that $g(\textbf{w},p)$ admits a unique subsymmetric basis despite the fact that for a wide class of weights it does not admit a symmetric basis. This provides the first known examples of Banach spaces where those two properties coexist.

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arXiv2017

Conditional quasi-greedy bases in non-superreflexive Banach spaces

For a conditional quasi-greedy basis $\mathcal{B}$ in a Banach space the associated conditionality constants $k_{m}[\mathcal{B}]$ verify the estimate $k_{m}[\mathcal{B}]=\mathcal{O}(\log m)$. Answering a question raised by Temlyakov, Yang, and Ye, several authors have studied whether this bound can be improved when we consider quasi-greedy bases in some special class of spaces. It is known that every quasi-greedy basis in a superreflexive Banach space verifies $k_{m}[\mathcal{B}]=(\log m)^{1-ε}$ for some $0<ε<1$, and this is optimal. Our first goal in this paper will be to fill the gap in between the general case and the superreflexive case and investigate the growth of the conditionality constants in non-superreflexive spaces. Roughly speaking, the moral will be that we can guarantee optimal bounds only for quasi-greedy bases in superreflexive spaces. We prove that if a Banach space $\mathbb{X}$ is not superreflexive then there is a quasi-greedy basis $\mathcal{B}$ in a Banach space $\mathbb{Y}$ finitely representable in $\mathbb{X}$ with $k_{m}[\mathcal{B}] \approx \log m$. As a consequence we obtain that for every $2<q<\infty$ there is a Banach space $\mathbb{X}$ of type $2$ and cotype $q$ possessing a quasi-greedy basis $\mathcal{B}$ with $k_{m}[\mathcal{B}] \approx \log m$. We also tackle the corresponding problem for Schauder bases and show that if a space is non-superreflexive then it possesses a basic sequence $\mathcal{B}$ with $k_m[\mathcal{B}]\approx m$.

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arXiv2015

Unconditional and quasi-greedy bases in $L_p$ with applications to Jacobi polynomials Fourier series

We show that the decreasing rearrangement of the Fourier series with respect to the Jacobi polynomials for functions in $L_p$ does not converge unless $p=2$. As a by-product of our work on quasi-greedy bases in $L_{p}(μ)$, we show that no normalized unconditional basis in $L_p$, $p\not=2$, can be semi-normalized in $L_q$ for $q\not=p$, thus extending a classical theorem of Kadets and Pełczy{ń}ski from 1968.

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arXiv2014

Non-existence of greedy bases in direct sums of mixed $\ell_{p}$ spaces

The fact that finite direct sums of two or more mutually different spaces from the family $\{\ell_{p} : 1\le p<\infty\}\cup c_{0}$ fail to have greedy bases is stated in [Dilworth et al., Greedy bases for Besov spaces, Constr. Approx. 34 (2011), no. 2, 281-296]. However, the concise proof that the authors give of this fundamental result in greedy approximation relies on a fallacious argument, namely the alleged uniqueness of unconditional basis up to permutation of the spaces involved. The main goal of this note is to settle the problem by providing a correct proof. For that we first show that all greedy bases in an $\ell_{p}$ space have fundamental functions of the same order. As a by-product of our work we obtain that every almost greedy basis of a Banach space with unconditional basis and nontrivial type contains a greedy subbasis.

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