Sobolev mappings on metric spaces and Minkowski dimension

TL;DR

This paper introduces a new class of metric space mappings called compactly H"older mappings, explores their dimension distortion properties, and connects them to Sobolev spaces, generalizing existing results in Euclidean and metric measure spaces.

Contribution

It defines compactly H"older mappings in metric spaces and establishes their dimension distortion properties, extending Sobolev mapping theory without measure dependence.

Findings

Compactly H"older mappings are a generalization of Sobolev mappings in metric spaces.

Such mappings can distort Minkowski dimension in controlled ways.

Results extend to weighted Euclidean spaces, generalizing prior work.

Abstract

We introduce the class of compactly H\"older mappings between metric spaces and determine the extent to which they distort the Minkowski dimension of a given set. These mappings are defined purely with metric notions and can be seen as a generalization of Sobolev mappings, without the requirement for a measure on the source space. In fact, we show that if $f:X\rightarrow Y$ is a continuous mapping lying in some super-critical Newtonian-Sobolev space $N^{1,p}(X,\mu)$, under standard assumptions on the metric measure space $(X,d,\mu)$, it is then a compactly H\"older mapping. The dimension distortion result we obtain is new even for Sobolev mappings between weighted Euclidean spaces and generalizes previous results of Kaufman and Bishop-Hakobyan-Williams.