R. Smith

will speak on

Lipschitz-free spaces and integral representation I

Time: 3:00PM
Date: Tue 28th February 2023
Location: Seminar Room SCN 1.25 [map]

Further information

Abstract: Given a complete metric space $(M,d)$ with base point $0$, let
$\mathrm{Lip}_0(M)$ denote the real Banach space of Lipschitz functions that vanish at $0$. This space has an isometric predual,denoted $\mathcal{F}(M)$, which is the norm-closure in the dual $\mathrm{Lip}_0(M)^*$ of the set of elementary molecules $m_{x,y}:\mathrm{Lip}_0(M) \to \mathbb{R}$, $x \neq y \in M$, defined by $m_{x,y}(f) = (f(x)-f(y))/d(x,y)$. This predual is sometimes called the Lipschitz-free space of $M$; these spaces have close links with optimal transport theory.

Given $m \in \mathcal{F}(M)$ and $\varepsilon>0$, we can always express $m$ as a sum of the form $m=\sum_{i=1}^\infty a_n m_{x_i,y_i}$, where $\sum_{i=1}^\infty |a_n| \leq \|m\|+\varepsilon$. Distinguished among the elements of $\mathcal{F}(M)$ are those for which we can set $\varepsilon = 0$ in the sum above; equivalently, these elements can be
viewed as Bochner integrals in $\mathcal{F}(M)$ with respect to discrete measures concentrated on the set of elementary molecules.

In the first of two talks, we analyse the properties of elements that can be represented as Bochner integrals with respect to more general measures and derive some consequences, e.g. applications to the extremal structure of $\mathcal{F}(M)$. This analysis will be done indirectly,
using the integral representation of elements of $\mathrm{Lip}_0(M)^*$ established by de Leeuw.

This is joint work with R. J. Aliaga (Valencia Polytechnic University) and E. Pernecká (Czech Technical University, Prague).

https://ucd-ie.zoom.us/j/63806071354

(This talk is part of the Analysis series.)

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