On the dimension distortion under fractionally smooth mappings

Abstract

We determine the extent to which certain classes of fractionally `smooth' continuous mappings between metric spaces distort various dimensions, including the Hausdorff, upper Minkowski (box-counting), and upper intermediate dimensions. Our intermediate and Minkowski dimension distortion results are new even for continuous (fractional) Sobolev and, more generally, Triebel--Lizorkin and Besov mappings between Euclidean spaces, complementing the work of Hencl-Honz\'ik (2015) and Huynh (2022). Moreover, our results also extend the aforementioned work, as well as the work of Kaufman (2000) and Fraser-Tyson (2025) to certain weighted Euclidean spaces and, more generally, to doubling metric measure spaces. As an application of our main result, we quantify the corresponding dimension distortion properties of quasisymmetric mappings for non-Ahlfors regular subsets of metric measure spaces, strengthening a result of Bishop-Hakobyan-Williams (2016).