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On asymptotics of ring $Q$-homeomorphisms with respect to $p$-modulus near the origin

Ruslan Salimov, Bogdan Klishchuk

TL;DR

The paper addresses how ring $Q$-homeomorphisms with respect to the $p$-modulus ($p>n$) distort distances near the origin in $ obreakmathbb{R}^n$. It develops modulus-based arguments to derive quantitative, sharp lower bounds for the asymptotic distance distortion, revealing Hölder-type control of inverses at $x_0=0$ under growth conditions on $Q$ via $q_{x_0}(t)$. A key contribution is the explicit bound $ limsup_{x o 0} rac{|f(x)|}{|x|^{( alpha+p-n)/(p-n)}} ge ig( rac{p-n}{ alpha+p-n}ig)^{(p-1)/(p-n)} q_0^{1/(n-p)}$, with variants for stronger growth and special cases such as $ alpha=0$; the results are shown sharp by a constructed extremal map $f_0$. These findings enhance understanding of distance distortion in higher-dimensional, non-quasiconformal regimes and provide precise criteria for the regularity of ring $Q$-homeomorphisms near the origin.

Abstract

We consider the class of ring $Q$-homeomorphisms with respect to $p$-modulus in $\mathbb{R}^{n}$ with $p > n$, and obtain lower bounds for limsups of the distance distortions under such mappings. These estimates can be treated as Hölder's continuity of the inverses near the origin. The sharpness is illustrated by example

On asymptotics of ring $Q$-homeomorphisms with respect to $p$-modulus near the origin

TL;DR

The paper addresses how ring -homeomorphisms with respect to the -modulus () distort distances near the origin in . It develops modulus-based arguments to derive quantitative, sharp lower bounds for the asymptotic distance distortion, revealing Hölder-type control of inverses at under growth conditions on via . A key contribution is the explicit bound , with variants for stronger growth and special cases such as ; the results are shown sharp by a constructed extremal map . These findings enhance understanding of distance distortion in higher-dimensional, non-quasiconformal regimes and provide precise criteria for the regularity of ring -homeomorphisms near the origin.

Abstract

We consider the class of ring -homeomorphisms with respect to -modulus in with , and obtain lower bounds for limsups of the distance distortions under such mappings. These estimates can be treated as Hölder's continuity of the inverses near the origin. The sharpness is illustrated by example
Paper Structure (2 sections, 6 theorems, 27 equations)

This paper contains 2 sections, 6 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Main results

Key Result

proposition 1

Let $D$ be a domain in $\mathbb{R}^{n}$, $n\geqslant 2$, $x_{0} \in D$, and let $Q:D\to[0,\infty]$ be a Lebesgue measurable function such that $q_{x_0}(r) \neq \infty$ for a.e. $r\in(0, d_0)$, $d_0 = {\rm dist}(x_{0}, \partial D)$. A homeomorphism $f: D \rightarrow \mathbb{R}^{n}$ is a ring $Q$-hom holds for any $0 < r_{1} < r_{2} < d_0$.

Theorems & Definitions (7)

  • proposition 1
  • theorem 1
  • theorem 2
  • proof
  • corollary 1
  • corollary 2
  • corollary 3