arXiv subjects

O. Delgado

Publications and source records attributed to O. Delgado.

4 recordsLinked to original sources

arXiv2017

Strong factorizations of operators with applications to Fourier and Cesáro transforms

Consider two continuous linear operators $T\colon X_1(μ)\to Y_1(ν)$ and $S\colon X_2(μ)\to Y_2(ν)$ between Banach function spaces related to different $σ$-finite measures $μ$ and $ν$. We characterize by means of weighted norm inequalities when $T$ can be strongly factored through $S$, that is, when there exist functions $g$ and $h$ such that $T(f)=gS(hf)$ for all $f\in X_1(μ)$. For the case of spaces with Schauder basis our characterization can be improved, as we show when $S$ is for instance the Fourier operator, or the Cesàro operator. Our aim is to study the case when the map $T$ is besides injective. Then we say that it is a~representing operator ---in the sense that it allows to represent each elements of the Banach function space $X(μ)$ by a~sequence of generalized Fourier coefficients---, providing a complete characterization of these maps in terms of weighted norm inequalities. Some examples and applications involving recent results on the Hausdorff-Young and the Hardy-Littlewood inequalities for operators on weighted Banach function spaces are also provided.

math.FA↗

arXiv2015

Optimal extensions for $p$-th power factorable operators

Let $X(μ)$ be a function space related to a measure space $(Ω,Σ,μ)$ with $χ_Ω\in X(μ)$ and let $T\colon X(μ)\to E$ be a Banach space valued operator. It is known that if $T$ is $p$-th power factorable then the largest function space to which $T$ can be extended preserving $p$-th power factorability is given by the space $L^p(m_T)$ of $p$-integrable functions with respect to $m_T$, where $m_T\colonΣ\to E$ is the vector measure associated to $T$ via $m_T(A)=T(χ_A)$. In this paper we extend this result by removing the restriction $χ_Ω\in X(μ)$. In this general case, by considering $m_T$ defined on a certain $δ$-ring, we show that the optimal domain for $T$ is the space $L^p(m_T)\cap L^1(m_T)$. We apply the obtained results to the particular case when $T$ is a map between sequence spaces defined by an infinite matrix.

math.FA↗

arXiv2015

Optimal domain of $q$-concave operators and vector measure representation of $q$-concave Banach lattices

Given a Banach space valued $q$-concave linear operator $T$ defined on a $σ$-order continuous quasi-Banach function space, we provide a description of the optimal domain of $T$ preserving $q$-concavity, that is, the largest $σ$-order continuous quasi-Banach function space to which $T$ can be extended as a $q$-concave operator. We show in this way the existence of maximal extensions for $q$-concave operators. As an application, we show a representation theorem for $q$-concave Banach lattices through spaces of integrable functions with respect to a vector measure. This result culminates a series of representation theorems for Banach lattices using vector measures that have been obtained in the last twenty years.

math.FA↗

arXiv2015

Strong extensions for $q$-summing operators acting in $p$-convex Banach function spaces for $1 \le p \le q$

Let $1\le p\le q<\infty$ and let $X$ be a $p$-convex Banach function space over a $σ$-finite measure $μ$. We combine the structure of the spaces $L^p(μ)$ and $L^q(ξ)$ for constructing the new space $S_{X_p}^{\,q}(ξ)$, where $ξ$ is a probability Radon measure on a certain compact set associated to $X$. We show some of its properties, and the relevant fact that every $q$-summing operator $T$ defined on $X$ can be continuously (strongly) extended to $S_{X_p}^{\,q}(ξ)$. This result turns out to be a mixture of the Pietsch and Maurey-Rosenthal factorization theorems, which provide (strong) factorizations for $q$-summing operators through $L^q$-spaces when $1 \le q \le p$. Thus, our result completes the picture, showing what happens in the complementary case $1\le p\le q$, opening the door to the study of the multilinear versions of $q$-summing operators also in these cases.

math.FA↗