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Quasisymmetric mappings in b-metric spaces

Evgeniy A. Petrov, Ruslan R. Salimov

TL;DR

The paper extends the theory of quasisymmetric mappings to b-metric and additive metric spaces. It proves a Tukia–Vaisala type diameter-distortion inequality in the b-metric setting, establishing both forward and inverse distortion bounds with coefficients K1 and K2, and derives corollaries linking these bounds to a functional inequality. It then gives sufficient conditions on the distortion function η under which an η-quasisymmetry preserves the b-metric structure, including concrete results for common choices such as power-type η. Finally, it identifies conditions under which additive metrics are preserved by quasisymmetries, providing a criterion that includes the special case η(t)=t. These results advance understanding of distortion and structural preservation in non-metric spaces.

Abstract

Considering quasisymmetric mappings between b-metric spaces we have found a new estimation for the ratio of diameters of two subsets which are images of two bounded subsets. This result generalizes the well-known Tukia-Väisälä inequality. The condition under which the image of a b-metric space under quasisymmetry is also a b-metric space is established. Moreover, the latter question is investigated for additive metric spaces.

Quasisymmetric mappings in b-metric spaces

TL;DR

The paper extends the theory of quasisymmetric mappings to b-metric and additive metric spaces. It proves a Tukia–Vaisala type diameter-distortion inequality in the b-metric setting, establishing both forward and inverse distortion bounds with coefficients K1 and K2, and derives corollaries linking these bounds to a functional inequality. It then gives sufficient conditions on the distortion function η under which an η-quasisymmetry preserves the b-metric structure, including concrete results for common choices such as power-type η. Finally, it identifies conditions under which additive metrics are preserved by quasisymmetries, providing a criterion that includes the special case η(t)=t. These results advance understanding of distortion and structural preservation in non-metric spaces.

Abstract

Considering quasisymmetric mappings between b-metric spaces we have found a new estimation for the ratio of diameters of two subsets which are images of two bounded subsets. This result generalizes the well-known Tukia-Väisälä inequality. The condition under which the image of a b-metric space under quasisymmetry is also a b-metric space is established. Moreover, the latter question is investigated for additive metric spaces.
Paper Structure (4 sections, 12 theorems, 46 equations)

This paper contains 4 sections, 12 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Distortion of diameters
  3. The property to preserve b-metricity
  4. Quasisymmetric mappings preserving additive metrics

Key Result

Proposition 1.1

Let $X, Y$ be metric spaces and let $f$ be $\eta$-quasi-sym-metry. Let $A\subset B\subset X$ with $\mathop{\mathrm{diam}}\nolimits A>0$, $\mathop{\mathrm{diam}}\nolimits B<\infty$. Then $\mathop{\mathrm{diam}}\nolimits f(B)<\infty$ and

Theorems & Definitions (21)

  • Definition 1.1
  • Proposition 1.1
  • Proposition 2.1
  • proof
  • Theorem 2.1
  • proof
  • Corollary 2.1
  • Corollary 2.2
  • Theorem 3.1
  • proof
  • ...and 11 more