arXiv subjects

Iván Caamaño

Publications and source records attributed to Iván Caamaño.

5 recordsLinked to original sources

arXiv2026

Hardy spaces and quasiregular mappings: averaged derivatives and the $\mathbb{BMO}$ case

We study the Hardy spaces $\mathcal{H}^p$, $0<p<\infty$ of quasiregular mappings on the unit ball $\mathbb{B}^n$ in ${\mathbb{R}}^n$ under the appropriate growth and multiplicity conditions. Our focus is on the averaged derivatives of maps and their Harnack and quantitative Harnack estimates. The averaged derivatives are employed to study the non-tangential limit functions and non-tangential maximal functions of quasiregular mappings and to characterize $\mathcal{H}^p$ in the case of finite multiplicity of $f$. Moreover, we study relations between quasiregular mappings, averaged derivatives, BMO spaces and Carleson measures on $\mathbb{B}^n$ and the role of the multiplicity of a map. We also apply our results to the second order elliptic PDEs and $\mathcal{A}$-harmonic equations. Our paper extends results by Astala and Koskela [AK] and Nolder [No1] to the setting of quasiregular maps.

math.CV↗

arXiv2025

Stepanov theorem for mappings between metric spaces

For Lipschitz maps between a metric measure space and a metric space, combining the ideas of Kirchheim's metric differentiability and Cheeger's differentiable structures leads to a Rademacher-type theorem for a notion of metric differentiability with respect to a rectifiable chart, and in this paper we prove the validity of a Stepanov-type generalization of such result under the same assumptions.

math.MG↗

arXiv2025

Trace and Extension Theorems for Besov Functions in Doubling Metric Measure Spaces

In the setting of a non-complete doubling metric measure space $(Ω,d,μ)$, we construct various bounded linear trace and extension operators for homogeneous and inhomogeneous Besov spaces $B^α_{p,q}$. Equipping the boundary $\partialΩ:=\overlineΩ\setminusΩ$ with a measure which is codimension $θ$ Ahlfors regular with respect to $μ$, these operators take the form \[ T:B^α_{p,q}(Ω)\to B^{α-θ/p}_{p,q}(\partialΩ),\quad E:B^α_{p,q}(\partialΩ)\to B^{α+θ/p}_{p,q}(Ω). \] The trace operators are first constructed under the additional assumption that $Ω$ is a uniform domain in its completion. We then use such results along with the technique of hyperbolic filling to remove this assumption in the case that $Ω$ is bounded. This extends to the doubling setting some earlier results of Marcos and Saksman-Soto proven under the assumption that the ambient measure is Ahlfors regular.

math.MG↗

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↗