Porosity in the space of H{\"o}lder-functions

TL;DR

This paper investigates the porosity of spaces of Hölder functions, showing that higher regularity spaces are σ-porous within lower regularity spaces under certain metric conditions.

Contribution

It establishes a precise criterion for when spaces of Hölder functions of different orders are σ-porous within each other, linking porosity to the metric's discreteness.

Findings

Lip β 0 (X, Y) is σ-porous in Lip α 0 (X, Y) if and only if the metric space X is non-uniformly discrete.

Porosity depends on the infimum of distances between distinct points in X.

Results extend to more general settings.

Abstract

Let (X, d) be a bounded metric space with a base point 0 X , (Y, $\bullet$) be a Banach space and Lip $\alpha$ 0 (X, Y) be the space of all $\alpha$-H{\"o}lderfunctions that vanish at 0 X , equipped with its natural norm (0 < $\alpha$ $\le$ 1). Let 0 < $\alpha$ < $\beta$ $\le$ 1. We prove that Lip $\beta$ 0 (X, Y) is a $\sigma$-porous subset of Lip $\alpha$ 0 (X, Y), if (and only if) inf{d(x, x ') : x, x ' $\in$ X; x = x ' } = 0 (i.e. d is non-uniformly discrete). A more general result will be given.