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On the dimension distortion under fractionally smooth mappings

Ryan Alvarado, Efstathios Konstantinos Chrontsios Garitsis

TL;DR

This work analyzes how fractionally smooth mappings distort fractal dimensions between metric spaces, focusing on intermediate dimensions that interpolate between Hausdorff and upper Minkowski dimensions. It introduces a unifying framework via compactly Hölder mappings and Morrey-type embeddings to translate fractional Sobolev–Triebel–Lizorkin–Besov regularity into concrete dimension-distortion bounds, applicable in general metric spaces, weighted Euclidean spaces, and doubling metric measure spaces. The key results include a sharp intermediate-dimension distortion bound for compactly Hölder maps and Morrey-embedded Sobolev mappings (Theorem 1 and Theorem 2 analogs), along with corollaries for Newtonian and quasisymmetric mappings on non-Ahlfors-regular sets. The methods combine a dyadic-cube machinery, a combinatorial graph framework to manage image covers, and Morrey-type control to yield quantitative distortion estimates with broad applicability in analysis on metric spaces and non-smooth geometric contexts.

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).

On the dimension distortion under fractionally smooth mappings

TL;DR

This work analyzes how fractionally smooth mappings distort fractal dimensions between metric spaces, focusing on intermediate dimensions that interpolate between Hausdorff and upper Minkowski dimensions. It introduces a unifying framework via compactly Hölder mappings and Morrey-type embeddings to translate fractional Sobolev–Triebel–Lizorkin–Besov regularity into concrete dimension-distortion bounds, applicable in general metric spaces, weighted Euclidean spaces, and doubling metric measure spaces. The key results include a sharp intermediate-dimension distortion bound for compactly Hölder maps and Morrey-embedded Sobolev mappings (Theorem 1 and Theorem 2 analogs), along with corollaries for Newtonian and quasisymmetric mappings on non-Ahlfors-regular sets. The methods combine a dyadic-cube machinery, a combinatorial graph framework to manage image covers, and Morrey-type control to yield quantitative distortion estimates with broad applicability in analysis on metric spaces and non-smooth geometric contexts.

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).

Paper Structure

This paper contains 10 sections, 13 theorems, 152 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Acknowledgments.
  3. Background
  4. Metric spaces and dimensions.
  5. Mappings between metric spaces
  6. Intermediate dimensions under compactly Hölder mappings
  7. Intermediate dimensions under fractionally smooth Sobolev mappings
  8. Morrey embedding theorem for fractional Hajł asz--Sobolev, Hajł asz--Triebel--Lizorkin, and Hajł asz--Besov spaces
  9. The proof of Theorem \ref{['thm:FractionalIntDim']}
  10. Hausdorff and Minkowski dimension distortion

Key Result

Theorem 1.1

Suppose $(X,d_X)$ is a doubling, uniformly perfect metric space and $(Y,d_Y)$ is a uniformly perfect metric space. For $\theta\in(0,1)$, $p\in(0,\infty)$ and $\alpha\in(0,\infty)$, if $f:X\rightarrow Y$ is $(p,\alpha)$-compactly Hölder and $E\subset X$ is bounded with $\dim_\theta E =d_E(\theta)$, t

Figures (1)

  • Figure 1: The graph $\mathcal{G}(\mathcal{Q}_\delta)$ above includes a few noteworthy characteristics of $\mathcal{Q}_\delta$ and $f$. It is depicted that $\mathcal{Q}_\delta$ has no cubes of level $m_1+2$, and while it has no cubes of level $m_1+4$ either, we need to subdivide red cubes to and past that level for the induced desired covering of $f(E)$. Moreover, there are red cubes of $\mathcal{Q}_\delta$ even at the very last level, which means that $f$ is capable of increasing the diameter of even these small cubes, creating the need to subdivide beyond the $m_{N_\delta}$ level.

Theorems & Definitions (31)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem A: Hytönen, Kairema Hyt:dyadic
  • Remark 2.1
  • Proposition 2.2
  • proof
  • proof : Proof of Claim
  • Remark 2.3
  • ...and 21 more