On the dimension distortion under fractionally smooth mappings

2025/09/18 by Ryan Alvarado, Alvarado, Ryan, Efstathios Konstantinos Chrontsios Garitsis +1
Mathematics · #Advanced Harmonic Analysis Research #Analytic and geometric function theory #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #Metric Geometry (math.MG) #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2509.15149

openalex publication_date 2025/09/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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-Honzík (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).

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Related

• Sobolev mappings on metric spaces and Minkowski dimension
• Sobolev and quasiconformal distortion of intermediate dimension with applications to conformal dimension
• Intermediate dimensions of measures: Interpolating between Hausdorff and Minkowski dimensions
• Some elementary aspects of Hausdorff measure and dimension
• Qusisymmetric dimension distortion of Ahlfors regular subsets of a metric space