arXiv subjects

Ángeles Prieto

Publications and source records attributed to Ángeles Prieto.

4 recordsLinked to original sources

arXiv2024

Connections between metric differentiability and rectifiability

We combine Kirchheim's metric differentials with Cheeger charts in order to establish a non-embeddability principle for any collection $\mathcal C$ of Banach (or metric) spaces: if a metric measure space $X$ bi-Lipschitz embeds in some element in $\mathcal C$, and if every Lipschitz map $X\to Y\in \mathcal C$ is differentiable, then $X$ is rectifiable. This gives a simple proof of the rectifiability of Lipschitz differentiability spaces that are bi-Lipschitz embeddable in Euclidean space, due to Kell--Mondino. Our principle also implies a converse to Kirchheim's theorem: if all Lipschitz maps from a domain space to arbitrary targets are metrically differentiable, the domain is rectifiable. We moreover establish the compatibility of metric and w$^*$-differentials of maps from metric spaces in the spirit of Ambrosio--Kirchheim.

math.MG↗

arXiv2020

Sobolev spaces of vector-valued functions

We are concerned here with Sobolev-type spaces of vector-valued functions. For an open subset $Ω\subset\mathbb{R}^N$ and a Banach space $V$, we compare the classical Sobolev space $W^{1,p}(Ω, V)$ with the so-called Sobolev-Reshetnyak space $R^{1,p}(Ω, V)$. We see that, in general, $W^{1,p}(Ω, V)$ is a closed subspace of $R^{1,p}(Ω, V)$. As a main result, we obtain that $W^{1,p}(Ω, V)=R^{1,p}(Ω, V)$ if, and only if, the Banach space $V$ has the Radon-Nikodým property

math.FA↗

arXiv2018

The polynomial cluster value problem

The polynomial cluster value problem replaces the role of the continuous linear functionals in the original cluster value problem for the continuous polynomials to describe the corresponding cluster sets and fibers. We prove several polynomial cluster value theorems for uniform algebras $H(B)$ between $A_u(B)$ and $H^{\infty}(B)$, where $B$ is the open unit ball of a complex Banach space $X$. We also obtain new results about the original cluster value problem, especially for $A_{\infty}(B)$. Examples of spaces $X$ considered here are spaces of continuous functions, $\ell_1$ and locally uniformly convex spaces.

math.FA↗

arXiv2016

Ideal structures in vector-valued polynomial spaces

This paper is concerned with the study of geometric structures in spaces of polynomials. More precisely, we discuss for $E$ and $F$ Banach spaces, whether the class of weakly continuous on bounded sets $n$-homogeneous polynomials, $\mathcal P_w(^n E, F)$, is an HB-subspace or an $M(1,C)$-ideal in the space of continuous $n$-homogeneous polynomials, $\mathcal P(^n E, F)$. We establish sufficient conditions under which the problem can be positively solved. Some examples are given. We also study when some ideal structures pass from $\mathcal P_w(^n E, F)$ as an ideal in $\mathcal P(^n E, F)$ to the range space $F$ as an ideal in its bidual $F^{**}$.

math.FA↗