Sobolev mappings on metric spaces and Minkowski dimension

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.