You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
abstract: 'We introduce a weaker variant of the concept of three point property, which is equivalent to a non-linear local connectivity condition introduced in [@gkt13], sufficient to guarantee the extendability of a conformal map $f:\mathbb D\to\Omega$ to the entire plane as a homeomorphism of locally exponentially integrable distortion. Sufficient conditions for extendability to a homeomorphism of locally $p$-integrable distortion are also given.'
address: 'Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35, FI-40014 University of Jyväskylä, Finland'
author:
- Changyu Guo
title: Generalized quasidisks and conformality II
---

Introduction {#sec:first}
============

One calls a Jordan domain $\Omega\subset{\ensuremath{\mathbb{R}}}^2$ a quasidisk if it is the image of the unit disk $\mathbb D$ under a quasiconformal mapping $f\colon{\ensuremath{\mathbb{R}}}^2\to{\ensuremath{\mathbb{R}}}^2$ of the entire plane. If $f$ is $K$-quasiconformal, we say that $\Omega$ is a $K$-quasidisk. Another possibility is to require that $f$ is additionally conformal in the unit disk $\mathbb D$. It is essentially due to Kühnau [@k88] that $\Omega$ is a $K$-quasidisk if and only if $\Omega$ is the image of $\mathbb D$ under a $K^2$-quasiconformal mapping $f:{\ensuremath{\mathbb{R}}}^2\to {\ensuremath{\mathbb{R}}}^2$ that is conformal in $\mathbb D$, see [@gh01]. The concept of a quasidisk is central in the theory of planar quasiconformal mappings; see, for example, [@a06; @aim09; @g82; @lv73].

A substantial part of the theory of quasiconformal mappings has recently been shown to extend in a natural form to the setting of mappings of locally exponentially integrable distortion [@agrs10; @aim09; @d88; @hek; @hk03; @iko02; @oz05; @z08]. See Section 2 below for the definition of this class of mappings. Thus one could say that $\Omega\subset{\ensuremath{\mathbb{R}}}^2$ is a generalized quasidisk if it is the image of the unit disk $\mathbb D$ under a homeomorphism $f\colon{\ensuremath{\mathbb{R}}}^2\to{\ensuremath{\mathbb{R}}}^2$ of the entire plane with locally exponentially integrable distortion. However, requiring that $f$ is additionally conformal in the unit disk $\mathbb D$ leads to different classes of domains, see [@gkt13 Theorem 1.1]. In this paper, a Jordan domain $\Omega\subset{\ensuremath{\mathbb{R}}}^2$ is termed a generalized quasidisk if the additional conformality requirement is satisfied.

For a quasidisk $\Omega\subset{\ensuremath{\mathbb{R}}}^2$, there are several equivalent characterizations. One of the simplest is Ahlfors [@a63] three point property. Recall that a Jordan domain $\Omega\subset{\ensuremath{\mathbb{R}}}^2$ has the three point property if there exists a constant $C\geq 1$ such that for each pair of distinct points $P_1,P_2\in{\partial \Omega}$, $$\label{eq:three point property}
\min_{i=1,2}\operatorname{diam}(\gamma_i)\leq C|P_1-P_2|,$$ where $\gamma_1,\gamma_2$ are components of ${\partial \Omega}\backslash\{P_1,P_2\}$. In order to understand the geometry of generalized quasidisks, one naturally has to weaken the three point property. A Jordan domain $\Omega\subset{\ensuremath{\mathbb{R}}}^2$ is said to have the three point property with a control function $\psi$ if there exists a constant $C\geq 1$ and an increasing function $\psi:[0,\infty)\to [0,\infty)$ such that for each pair of distinct points $P_1,P_2\in{\partial \Omega}$, $$\label{eq:three point property with control function}
\min_{i=1,2}\operatorname{diam}(\gamma_i)\leq \psi\Big(C|P_1-P_2|\Big).$$ A closely related concept is the following $\psi$-local connectivity, which was introduced in [@gkt13]. A domain $\Omega\subset{\ensuremath{\mathbb{R}}}^2$ is said to be $\psi$-locally connected if for each $x$ and all $r>0,$

-   each pair of points in $B(x,r)\cap\Omega$ can be joined by an arc in $B(x,\psi^{-1}(r))\cap\Omega$, and

-   each pair of points in $\Omega\backslash B(x,r)$ can be joined by an arc in $\Omega\setminus B(x,\psi(r))$.

If we were to choose $\psi(t)=Ct,$ then this would reduce to the usual linear local connectivity condition. In Lemma \[lemma:equivalence of three point property and LLC\] below, we show that that a Jordan domain $\Omega\subset{\ensuremath{\mathbb{R}}}^2$ has the three point property with a control function $\psi$ if and only if $\Omega$ is $\psi^{-1}$-locally connected.

In [@gkt13 Theorem 1.2], it was proved that if a Jordan domain $\Omega\subset{\ensuremath{\mathbb{R}}}^2$ is $\psi$-locally connected with $\psi(t)=\frac{Ct}{\log^s\log\frac{1}{t}}$ for some positive constant $C$ and some $s\in (0,\frac{1}{4})$, then $\Omega$ is a generalized quasidisk. However, the result is not sharp regarding well-studied examples, see [@gkt13]. In fact, the previous studies in [@gkt13; @kt07; @kt10; @t07] suggest that the critical case should be $\psi(t)=\frac{Ct}{\log\frac{1}{t}}$. Our first main result is the following generalization of [@gkt13 Theorem 1.2].

\[thm:thma\] If a Jordan domain $\Omega\subset{\ensuremath{\mathbb{R}}}^2$ has the three point property with the control function $\psi(t)=Ct\log^{\frac{1}{2}}\frac{1}{t}$ for some positive constant $C$, then $\Omega$ is a generalized quasidisk.

Equivalently, Theorem \[thm:thma\] provides a general sufficient condition for extendability of a conformal mapping $f:\mathbb{D}\to \Omega$ to a homeomorphism of locally exponentially integrable distortion. It was pointed out in [@gkt13] that this is essentially equivalent to extending the corresponding conformal welding to the whole plane as a homeomorphism of locally exponentially integrable distortion, see also Section 4 below.

Our second main result asserts that if we relax the control function $\psi$ to be a root in Theorem \[thm:thma\], then we end up with a homeomorphism of the whole plane with locally $p$-integrable distortion.

\[thm:thmb\] Let $\Omega\subset{\ensuremath{\mathbb{R}}}^2$ be a Jordan domain that has the three point property with the control function $\psi(t)=t^{s}$, $0<s<1$. Then any conformal mapping $f\colon \mathbb D\to\Omega$ can be extended to the entire plane as a homeomorphism of locally $p$-integrable distortion for all $p\in (0,\frac{s^2}{2(1-s^2)})$.

As pointed out in [@gkt13], (polynomial) interior cusps are more dangerous than (polynomial) exterior cusps in the locally exponentially integrable distortion case. Thus one expects that this is still the case for extensions with locally $p$-integrable distortion. Our next result confirms this expectation.

\[thm:thmc\] Let $\Omega\subset{\ensuremath{\mathbb{R}}}^2$ be a LLC-1 Jordan domain. Then any conformal mapping $f\colon \mathbb D\to\Omega$ can be extended to the entire plane as a homeomorphism of locally $p$-integrable distortion for some $p>0$.

This paper is organized as follows. Section 2 contains the basic definitions and Section 3 some auxiliary results. In Section 4, we study the relation of extending a Riemann mapping and the corresponding conformal welding. We prove our main results in Section 5. In the final section, Section 6, we make some concluding remarks.

Notation and Definitions {#sec:notdef}
========================

We sometimes associate the plane ${\ensuremath{\mathbb{R}}}^2$ with the complex plane ${\ensuremath{\mathbb{C}}}$ for convenience and denote by $\hat{{\ensuremath{\mathbb{C}}}}$ the extended complex plane. The closure of a set $U\subset{\ensuremath{\mathbb{R}}}^2$ is denoted ${\overline{U}}$ and the boundary ${\partial U}$. The open disk of radius $r>0$ centered at $x\in{\ensuremath{\mathbb{R}}}^2$ is denoted by $B(x,r)$ and we simply write $\mathbb D$ for the unit disk. The boundary of $B(x,r)$ will be denoted by $S(x,r)$ and the boundary of the unit disk $\mathbb D$ is written as $\partial \mathbb D.$ The symbol $\Omega$ always refers to a domain, a connected and open subset of ${\ensuremath{\mathbb{R}}}^2$. We call a homeomorphism $f\colon\Omega\to f(\Omega)\subset{\ensuremath{\mathbb{R}}}^2$ a homeomorphism of finite distortion if $f\in W_{{\mathrm{loc}}}^{1,1}(\Omega;{\ensuremath{\mathbb{R}}}^2)$ and $$\label{eq:disteq}
  \|Df(x)\|^2\leq K(x)J_f(x) \text{ a.e. in } \Omega,$$ for some measurable function $K(x)\geq1$ that is finite almost everywhere. Recall here that $J_f\in L_{loc}^{1}(\Omega)$ for each homeomorphism $f\in W_{loc}^{1,1}(\Omega;{\ensuremath{\mathbb{R}}}^2)$(cf. [@aim09]). In the distortion inequality , $Df(x)$ is the formal differential of $f$ at the point $x$ and $J_f(x):=\det Df(x)$ is the Jacobian. The norm of $Df(x)$ is defined as $$\|Df(x)\|:=\max_{e\in{\partial \mathbb D}} |Df(x)e|.$$ For a homeomorphism of finite distortion it is convenient to write $K_f$ for the optimal distortion function. This is obtained by setting $K_f(x)=\|Df(x)\|^2/J_f(x)$ when $Df(x)$ exists and $J_f(x)>0$, and $K_f(x)=1$ otherwise. The distortion of $f$ is said to be locally $\lambda$-exponentially integrable if $\exp(\lambda K_f(x))\in L_{{\mathrm{loc}}}^1(\Omega)$, for some $\lambda>0$. Note that if we assume $K_f(x)$ to be bounded, $K_f\le K,$ we recover the class of $K$-quasiconformal mappings, see [@aim09] for the theory of quasiconformal mappings. Recall that a domain $\Omega$ is said to be linearly locally connected (LLC) if there is a constant $C\geq 1$ so that

-   (LLC-1) each pair of points in $B(x,r)\cap \Omega$ can be joined by an arc in $B(x,Cr)\cap \Omega$, and

-   (LLC-2) each pair of points in $\Omega\backslash B(x,r)$ can be joined by an arc in $\Omega\setminus B(x,C^{-1}r)$.

We need a weaker version of this condition, defined as follows. We say that $\Omega$ is $(\varphi,\psi)$-locally connected ($(\varphi,\psi)$-LC) if

-   ($\varphi$-LC-1) each pair of points in $B(x,r)\cap\Omega$ can be joined by an arc in $B(x,\varphi(r))\cap\Omega$, and

-   ($\psi$-LC-2) each pair of points in $\Omega\backslash B(x,r)$ can be joined by an arc in $\Omega\setminus B(x,\psi(r))$,

where $\varphi, \psi:[0, \infty)\to[0, \infty)$ are smooth increasing functions such that $\varphi(0)=\psi(0)=0$, $\varphi(r)\geq r$ and $\psi(r)\leq r$ for all $r>0$. For technical reasons, we assume that the function $t\mapsto \frac t {\varphi^{-1}(t)^2}$ is decreasing and that there exist constants $C_1,C_2$ so that $C_1\varphi(t)\le \varphi(2t)\le C_2\varphi(t)$ and $C_1\psi(t)\le \psi(2t)\le C_2\psi(t)$ for all $t>0.$ If $\varphi^{-1}=\psi$ above, as in the introduction, $\Omega$ will simply be called $\psi$-LC. One could relax joinability by an arc above to joinability by a continuum, but this leads to the same concept; see [@hy88 Theorem 3-17]. Notice that if $\Omega$ is simply connected and bounded, then $\varphi$-LC-1 guarantees that $\Omega$ is a Jordan domain.

Finally we define the central tool for us – the modulus of a path family. A Borel function $\rho\colon{\ensuremath{\mathbb{R}}}^2\to[0,\infty\mathclose]$ is said to be admissible for a path family $\Gamma$ if $\int_\gamma\rho\,ds\geq1$ for each locally rectifiable $\gamma\in\Gamma$. The modulus of the path family $\Gamma$ is then $$\operatorname{mod}(\Gamma):=
\inf\Big\{\int_{\Omega}\rho^2(x)\,dx :  \rho
\text{ is admissible for }  \Gamma\}.$$ For subsets $E$ and $F$ of ${\overline{\Omega}}$ we write $\Gamma(E,F,\Omega)$ for the path family consisting of all locally rectifiable paths joining $E$ to $F$ in $\Omega$ and abbreviate $\operatorname{mod}(\Gamma(E,F,\Omega))$ to $\operatorname{mod}(E,F,
\Omega).$ In what follows, $\gamma(x,y)$ refers to a curve or an arc from $x$ to $y$.

When we write $f(x)\lesssim g(x)$, we mean that $f(x)\leq Cg(x)$ is satisfied for all $x$ with some fixed constant $C\geq 1$. Similarly, the expression $f(x)\gtrsim g(x)$ means that $f(x)\geq C^{-1}g(x)$ is satisfied for all $x$ with some fixed constant $C\geq 1$. We write $f(x)\approx g(x)$ whenever $f(x)\lesssim g(x)$ and $f(x)\gtrsim g(x)$.

Auxiliary results
=================

We begin this section by showing the equivalence of the generalized three point property and generalized local connectivity mentioned in the introduction.

\[lemma:equivalence of three point property and LLC\] Let $\Omega\subset{\ensuremath{\mathbb{R}}}^2$ be a Jordan domain. Then $\Omega$ has the three point property with the control function $\psi$ if and only if $\Omega$ is $\psi^{-1}$-locally connected.

First, suppose that $\Omega$ has the three point property with the control function $\psi$. We want to show that $\Omega$ is $\psi$-LC-1. To this end, let $x,y\in B(z,r)\cap\Omega$. We may assume that there exist $x',y'\in B(z,r)\cap {\partial \Omega}$ such that $$d(x,x')=d(x,{\partial \Omega}), d(y,y')=d(y,{\partial \Omega})$$ and that $x$ can be connected to $x'$ by an arc $\beta_1$ in ${\overline{\Omega}}\cap B(z,r)$ and $y$ can be connected to $y'$ by an arc $\beta_2$ in ${\overline{\Omega}}\cap B(z,r)$. Let $\alpha_1$ and $\alpha_2$ be the components of ${\partial \Omega}\backslash \{x',y'\}$. We may assume that $\alpha_1\leq \alpha_2$. Then $$\operatorname{diam}(\alpha_1)\leq \psi(|x'-y'|)\leq \psi(2r).$$ Hence, $\gamma=\beta_1\cup\alpha_1\cup\beta_2$ is a curve that connects $x$ and $y$ in ${\overline{\Omega}}$ with diameter less than $2\psi(2r)$. Then the Jordan assumption for $\Omega$ implies that we may connect $x$ to $y$ in $\Omega$ by a curve with diameter no more than $3\psi(2r)$. This together with Lemma \[lemma:consequence of technical assumption\] below implies that $\Omega$ is $\psi$-LC-1. Similarly, one can prove that ${\ensuremath{\mathbb{R}}}^2\backslash {\overline{\Omega}}$ is $\psi$-LC-1. Then the duality result in [@gk12] implies that $\Omega$ is $\psi$-LC.

Next, we assume that $\Omega$ is $\psi^{-1}$-LC. Then, again by the duality result in [@gk12], we know that both $\Omega$ and ${\ensuremath{\mathbb{R}}}^2\backslash{\overline{\Omega}}$ are $\psi$-LC-1. Let $x,y\in {\partial \Omega}$ and let $\alpha_1$, $\alpha_2$ be the components of ${\partial \Omega}\backslash \{x,y\}$. We may assume that $\operatorname{diam}(\alpha_1)\leq \operatorname{diam}(\alpha_2)$. Let $z=\frac{x+y}{2}$ and $r=|x-y|$. Then $x,y\in B(z,r)$. We may choose two points $x'$ and $y'$ in $\Omega\cap B(z,r)$ such that $x$ can be connected to $x'$ by an arc $\beta_1$ in ${\overline{\Omega}}\cap B(z,r)$ and $y$ can be connected to $y'$ by an arc $\beta_2$ in ${\overline{\Omega}}\cap B(z,r)$. Then we may connect $x'$ to $y'$ by an arc $\gamma$ in $\Omega\cap B(z,2\psi(r))$. Then the curve $\eta=\beta_1\cup\gamma\cup\beta_2$ forms a crosscut of $\Omega$ with diameter no more than $4\psi(r)$. Similarly, we may form a crosscut $\eta'$ of ${\ensuremath{\mathbb{R}}}^2\backslash{\overline{\Omega}}$ with diameter no more than $4\psi(r)$. Thus $\eta\cup\eta'$ is a Jordan curve which contains the Jordan arc $\alpha_1$. Therefore, the diameter of $\alpha_1$ is no more than $8\psi(r)$. This together with Lemma \[lemma:consequence of technical assumption\] below implies that $\Omega$ has the three point property with the control function $\psi$.

\[lemma:consequence of technical assumption\] Let $C_1\geq 1$, $C_2\geq 1$, and $C_3\geq 1$ be given. There exists a constant $C$, depending only on $C_0$, $C_1$, $C_2$ and $C_3$, such that $$\label{eq:consequence of technical assumption}
C_1\varphi(C_2t)+C_3t\leq \varphi(Ct)$$ for all $t>0$. Above, $C_0$ is the doubling constant of $\varphi^{-1}$.

The following two modulus estimates are standard, see e.g. [@v88].

\[lemma:modululower\] Let $E, F$ be disjoint nondegenerate continua in $B(x,R).$ Then $$C_0^{-1}\frac{1}{\log(1+t)}\geq \operatorname{mod}(E,F,B(x,R))\geq C_0\frac{1}{\log(1+t)},$$ where $t=\frac{\operatorname{dist}(E,F)}{\min\{\operatorname{diam}E, \operatorname{diam}F\}}$ and $C_0$ is an absolute constant.

\[lemma:modulusupper\] Let $\Gamma$ be a curve family such that for all $t\in (r,R)$, the circle $|z-z_1|=t$ contains a curve $\gamma\in \Gamma$. Then $$\operatorname{mod}(\Gamma)\geq \frac{1}{2\pi}\log\frac{R}{r}.$$

Next, we recall the following result on the modulus of continuity of a quasiconformal mapping. The proof can be found in [@kot01]; also see [@g13].

\[lemma:modulus of continuity\] Suppose $g\colon \Omega\to \mathbb D$ is a $K$-quasiconformal mapping from a simply connected domain $\Omega$ onto the unit disk. Then there exists a positive constant $C$, (depending on $f$), such that for any $\omega,\xi\in \Omega$, $$|g(\omega)-g(\xi)|\leq Cd_{I}(\omega,\xi)^{\frac{1}{2K}},$$ where $d_{I}(\omega,\xi)$ is defined as $\inf_{\gamma(\omega,\xi)\subset \Omega}\operatorname{diam}(\gamma(\omega,\xi))$. In particular, if $\Omega$ above is $\varphi$-LC-1, then $$|g(\omega)-g(\xi)|\leq C\varphi(|\omega-\xi|)^{\frac{1}{2K}}.$$

Finally, we need the following key estimate.

\[lemma:key modulus estimate\] Let $\Omega\subset{\ensuremath{\mathbb{R}}}^2$ be a Jordan domain that has the three point property with the control function $\psi$. Let $\alpha_1$ and $\alpha_2$ be two disjoint arcs in ${\partial \Omega}$ and let $\Gamma$ and $\Gamma'$ be the family of curves which join $\alpha_1$ and $\alpha_2$ in $\Omega$ and ${\ensuremath{\mathbb{R}}}^2\backslash{\overline{\Omega}}$, respectively. If $\operatorname{mod}(\Gamma)\leq C$, then $$\label{eq:diam bounds}
\min\{\operatorname{diam}(\alpha_1),\operatorname{diam}(\alpha_2)\}\leq \psi\circ\psi(d(\alpha_1,\alpha_2))$$ and hence $$\label{eq:modulus bounds}
\operatorname{mod}(\Gamma')\leq C_0^{-1}\log^{-1}\Big(1+\frac{\psi^{-1}\circ\psi^{-1}(\min\{\operatorname{diam}(\alpha_1),\operatorname{diam}(\alpha_2)\})}{\min\{\operatorname{diam}(\alpha_1),\operatorname{diam}(\alpha_2)\}} \Big).$$

The idea of the proof is similar to that of the proof of Theorem 5.1 in [@gkt13]. Let $\alpha_1$ and $\alpha_2$ be two disjoint arcs in ${\partial \Omega}$. Choose $z_1\in \alpha_1$, $z_2\in\alpha_2$ so that $$|z_1-z_2|=d(\alpha_1,\alpha_2):=d.$$ Without loss of generality, we may assume that $$r:=\operatorname{diam}(\alpha_1)\leq \operatorname{diam}(\alpha_2).$$

Our aim is to show that $r\leq 2\psi\circ\psi(d)$. Thus we may clearly assume that $r>2\psi\circ\psi(d)$. Note that our assumption on $\psi$ implies that $r>\psi(d)$. Since $\Omega$ has the three point property with the control function $\psi$, $$\min_{i=1,2}\operatorname{diam}(\gamma_i)\leq \psi(d),$$ where $\gamma_1$, $\gamma_2$ are the components of ${\partial \Omega}\backslash \{z_1,z_2\}$. Again, we may assume that $$\operatorname{diam}(\gamma_1)\leq \psi(d).$$ Let $\beta_1$, $\beta_2$ be the components of ${\partial \Omega}\backslash (\alpha_1\cup\alpha_2)$, labeled so that $\beta_i\subset\gamma_i$. Then $\beta_1\subset\gamma_1\subset{\overline{B}}(z_1,\psi(d))$. Choose $z_0\in\beta_2$ and let $\delta_1$, $\delta_2$ denote the components of ${\partial \Omega}\backslash \{z_0,z_1\}$ labeled so that $\alpha_2\subset\delta_2$. Then the fact $\Omega$ has the three point property with the control function $\psi$ implies that $$\min_{i=1,2}\operatorname{diam}(\delta_i)\leq \psi(|z_1-z_0|).$$ Choose $\omega_1,\omega_2\in\alpha_1$ so that $$r=|\omega_1-\omega_2|=\operatorname{diam}(\alpha_1).$$ Then $\omega_i\in \gamma_1\cup\delta_1$, and the fact that $\operatorname{diam}(\gamma_1)\leq \psi(d)<r$ implies that not both of these points can lie in $\gamma_1$. If $\omega_1\in \gamma_1$, then $$\begin{aligned}
\operatorname{diam}(\delta_1)&\geq |\omega_2-z_1|\geq |\omega_1-\omega_2|-|z_1-\omega_1|\\
&\geq r-\operatorname{diam}(\gamma_1)\geq r-\psi(d)\geq \frac{r}{2}.\end{aligned}$$ If both lie in $\delta_1$, then $$\operatorname{diam}(\delta_1)\geq |\omega_1-\omega_2|=r.$$ Thus $$\frac{r}{2}\leq \min_{i=1,2}\operatorname{diam}(\delta_i)\leq \psi(|z_1-z_0|).$$ It follows that $$\beta_2\cap B(z_1,\psi^{-1}(\frac{r}{2}))=\emptyset.$$ In particular, the circle $|z-z_1|=t$ separates $\beta_1$ and $\beta_2$ for $t\in (\psi(d),\psi^{-1}(\frac{r}{2}))$ and hence must contain an arc $\gamma$ joining $\alpha_1$ and $\alpha_2$ in $\Omega$. Thus Lemma \[lemma:modulusupper\] implies that $$\frac{1}{2\pi}\log\frac{\psi^{-1}(r/2)}{\psi(d)}\leq \operatorname{mod}(\Gamma)\leq C$$ from which the claim follows. The desired inequality  follows from Lemma \[lemma:modululower\] directly.

Extension of a conformal welding
================================

Before stating the main result of this section, let us describe the standard way of extending a conformal map $f\colon \mathbb D\to\Omega,$ where $\Omega$ is a Jordan domain, to a mapping of the entire plane. First of all, $f$ can be extended to a homeomorphism between ${\overline{\mathbb D}}$ and ${\overline{\Omega}}$. For simplicity, we denote this extended homeomorphism also by $f$. It follows from the Riemann Mapping Theorem that there exists a conformal mapping $g\colon {\ensuremath{\mathbb{R}}}^2\setminus{{\overline{\mathbb D}}}\to{\ensuremath{\mathbb{R}}}^2\setminus{{\overline{\Omega}}}$ such that the complement of the closed unit disk gets mapped to the complement of $\overline \Omega$. In this correspondence the boundary curve $\Gamma={\partial \Omega}$ is mapped homeomorphically onto the boundary circle $\partial \mathbb D$ and hence the composed mapping $G=g^{-1}\circ f$ is a well-defined circle homeomorphism, called conformal welding. Suppose we are able to extend $G$ to the exterior of the unit disk, with the extension still denoted by $G$. Then the mapping $G'=g\circ G$ will be well-defined outside the unit disk and it coincides with $f$ on the boundary circle $\partial \mathbb D$. Finally, if we define $$F(x)=
  \begin{cases}
    G'(x) & \text{if } |x| \geq 1 \\
    f(x) & \text{if } |x| \leq 1 ,
  \end{cases}$$ then we obtain an extension of $f$ to the entire plane. In the case of a quasidisk, that is when $\Omega$ is linearly locally connected (LLC), the extension $G$ can be chosen to be quasiconformal and hence the obtained map $F$ is also quasiconformal.

On the other hand, the extendability of a conformal mapping $f:\mathbb D\to \Omega$ to a homeomorphism $\hat f:{\ensuremath{\mathbb{R}}}^2
\to {\ensuremath{\mathbb{R}}}^2$ of locally integrable distortion is essentially equivalent to being able to extend the conformal welding $G'$ above to this class. Indeed, if $\hat f$ extends $f$, then $g^{-1}\circ \hat f$ extends $G$ to the exterior of $\mathbb D$ and has the same distortion as $\hat f.$ Reflecting (twice) with respect to the unit circle one then further obtains an extension to $\mathbb D\setminus \{0\}.$ Hence, one obtains an extension $\hat G'$ of $G'$ to ${\ensuremath{\mathbb{R}}}^2\setminus \{0\}$ with distortion that has the same local integrability degree as the distortion of $\hat f.$ If the latter distortion is sufficiently nice in a neighborhood of infinity (e.g. bounded), then this holds in all of ${\ensuremath{\mathbb{R}}}^2$ as well.

Given a sense-preserving homeomorphism $f\colon \partial \mathbb D\to \partial \mathbb D$ and $0< t< \frac{\pi}{2}$, set $$\delta_f(\theta,t)=\max \Big\{\frac{|f(e^{i(\theta+t)})-f(e^{i\theta})|}{|f(e^{i\theta})-f(e^{i(\theta-t)})|}, \frac{|f(e^{i(\theta-t)})-f(e^{i\theta})|}{|f(e^{i\theta})-f(e^{i(\theta+t)})|}\Big\}.$$ Clearly $\delta_f$ is continuous in both variables, $\delta_f\geq 1$ and $\delta_f(\theta+2\pi,t)=\delta_f(\theta,t)$. The scalewise distortion of $f$ is defined as $\rho_f(t)=\sup_{\theta}\delta_f(\theta,t)$.

In the following, we discuss a standard way of extending a conformal welding $G:{\partial \mathbb}{D}:\to{\partial \mathbb}{D}$ to a global homeomorphism of the whole plane with controlled distortion. More precisely, we want to present the following result, which is implicitly contained in [@z08 Section 2 and Section 3].

\[prop:results for conformal welding\] Given a conformal welding $G:{\partial \mathbb}{D}:\to{\partial \mathbb}{D}$, there exists a homeomorphism $\hat{G}:{\ensuremath{\mathbb{R}}}^2\to {\ensuremath{\mathbb{R}}}^2$ with the following property:

-   For some $\delta\in (0,\frac{1}{2})$, $\hat{G}(z)=z$ if $|z|<\delta$ or $|z|>\frac{1}{\delta}$.

-   The distortion of $\hat{G}$ is bounded above by the scalewise distortion of $G$, $$\label{eq:estimate of scalewise distortion}
    K_{\hat{G}}(z)\leq C\rho_{G}(\log |z|)=C\sup_{\theta\in [0,2\pi]}\delta_{G}(\theta,\log |z|),$$ if $\delta\leq |z|\leq \frac{1}{\delta}$ for some absolute constant $C>0$.

Let us describe below the argument leading to Proposition \[prop:results for conformal welding\]. Given a conformal welding $G:{\partial \mathbb}{D}\to{\partial \mathbb}{D}$, we first want to extend $G$ to a homeomorphism $\tilde{G}:{\overline{\mathbb{D}}}\to{\overline{\mathbb{D}}}$. We may represent $G$ in the form $$G(e^{2\pi ix})=e^{2\pi ih(x)},$$ where $h:{\ensuremath{\mathbb{R}}}\to {\ensuremath{\mathbb{R}}}$ is a homeomorphism of the real line which commutes with the unit translation $x\mapsto x+1$. For simplicity, we may assume that $G(1)=1$ and hence $h(0)=0$.

Next, we extend our mapping $h$ to a homeomorphism $H:\mathbb{H}\to \mathbb{H}$. This can be done via the well-known Beurling-Ahlfors extension. To be more precise, for $0<y<1$, set $$\label{eq:B-A extension}
H(x+iy)=\frac{1}{2}\int_0^1 (h(x+ty)+h(x-ty))dt+i\int_0^1 (h(x+ty)-h(x-ty))dt.$$ Then $H=h$ on the real axis and $H$ is a $C^1$-smooth homeomorphism of $\mathbb{H}$. Since $h(x+1)=h(x)+1$, for $y=1$ $$H(x+i)=x+i+C_0,$$ where $C_0=\int_0^1 h(t)dt-\frac{1}{2}\in [-\frac{1}{2},\frac{1}{2}]$. For $1\leq y\leq 2$ we extend $H$ linearly by setting $$H(z)=z+(2-y)C_0, \quad z=x+iy.$$ Finally, we set $H(z)=z$ if $y=Im(z)\geq 2$. It is easy to check that $H(z+k)=H(z)+k$ for $k\in \mathbb{Z}$. We set $$\tilde{G}=\textbf{e}\circ H\circ L,$$ where $\textbf{e}:z\mapsto e^{2\pi iz}$ is the lifting mapping and $L:z\mapsto \frac{\log z}{2\pi i}$ is the logarithmic mapping. We claim that $\tilde{G}:\mathbb{D}\backslash \{0\}\to \mathbb{D}\backslash \{0\}$ is a well-defined homeomorphism. To see thisu\], let $z=re^{0i}=re^{2\pi i}$ be as in Figure \[fig:homeomorphism\]. We need to show that $L$ is well-defined on the segment $P:=\{z:r\leq |z|\leq 1\}$. Note that in Figure \[fig:homeomorphism\], the vertical line $[0,L(z)]$ corresponds to the image of $P$ with argument $0$ and the vertical line $[1,L(z)+1]$ corresponds to the image of $P$ with argument $2\pi$ under the mapping $L$. Note also that $L(re^{0i})=\frac{\log r}{2\pi i}$ and $L(re^{2\pi i})=\frac{\log r+2\pi i}{2\pi i}$. Since $H$ satisfies that $H(z+1)=H(z)+1$ and $\textbf{e}$ is $1$-periodic, the mapping $\tilde{G}$ is a homeomorphism in the annulus $\mathbb{D}\backslash B(0,r)$.

![The homeomorphism $\tilde{G}$[]{data-label="fig:homeomorphism"}](Fig1.eps "fig:"){width="12cm"}\

Moreover, $\tilde{G}=G$ on ${\partial \mathbb}{D}$ and $\tilde{G}(z)=z$ for $0<|z|\leq \delta:= e^{-4\pi}$. Thus $\tilde{G}$ is well-defined homeomorphism of the unit disk if we additionally set $\tilde{G}(0)=0$.

Finally, we may define our mapping $\hat{G}:{\ensuremath{\mathbb{R}}}^2\to{\ensuremath{\mathbb{R}}}^2$ by setting $$\hat{G}(z)=
  \begin{cases}
    \tilde{G}(z) & \text{if } |z| \geq 1 \\
    R\circ\tilde{G}\circ R(z) & \text{if } |z| \leq 1 ,
  \end{cases}$$ where $R(z)=\frac{1}{\bar{z}}$ is the inversion with respect to the unit circle. To complete the proof of Proposition \[prop:results for conformal welding\], we need to estimate the distortion of $\hat{G}$.

It is clear that we only need to estimate the distortion of $\tilde{G}$. Since $\textbf{e}$ and $L$ are conformal mappings, it follows that $$\begin{aligned}
K_{\tilde{G}}(z)=K_H(\omega),\quad z=e^{2\pi i\omega},\ \omega\in \mathbb{H}.\end{aligned}$$ So we reduce all distortion estimates for $\tilde{G}$ to the corresponding ones for $H$. Since $H$ is conformal for $y>2$ and linear for $y\in [1,2]$, it suffices to estimate $K_H$ in the strip $S={\ensuremath{\mathbb{R}}}\times [0,1]$. The desired estimate $$K_H(x+iy)\leq C_0\rho_h(y),\quad x+iy\in S$$ follows from the calculation in [@cch96], where $$\begin{aligned}
\rho_h(t)&=\sup_{\theta\in {\ensuremath{\mathbb{R}}}}\delta_h(\theta,t)\\
&:=\sup_{\theta\in {\ensuremath{\mathbb{R}}}}\max \Big\{\frac{|h(\theta+t)-h(\theta)|}{|h(\theta)-h(\theta-t)|},\frac{|h(\theta-t)-h(\theta)|}{|h(\theta)-h(\theta+t)|}\Big\}.\end{aligned}$$ Note that if $t\in [0,1]$, then $$\delta_{G}(\theta,t)\approx \delta_h(\theta,t)$$ and hence $$\rho_G(t)\approx\rho_h(t).$$ The proof of Proposition \[prop:results for conformal welding\] is complete.

As an application of Proposition \[prop:results for conformal welding\], we easily obtain the following corollary.

Let $\delta$ be as in Proposition \[prop:results for conformal welding\] and let $\varepsilon<\delta$ is sufficiently small such that $$\log |z|\leq 2|r-1|$$ for $z=re^{i\theta}\in A:={\overline{B(0,1+\varepsilon)}}\backslash B(0,1-\varepsilon)$. Proposition \[prop:results for conformal welding\] implies that $$\begin{aligned}
  K_{\hat{G}}(z)\leq C\rho_G(\log |z|)\leq C\rho_G(2|r-1|)\end{aligned}$$ for $z\in A$. If $\rho_G(t)\leq C't^{-\alpha}$ as $t\to 0$, then $$\begin{aligned}
K_{\hat{G}}(z)\leq C |r-1|^{-\alpha},\end{aligned}$$ for $z\in A$. Integrating in polar coordinates, we immediately obtain the following corollary.

\[coro:sufficient for extending\] Let $G:{\partial \mathbb}{D}:\to{\partial \mathbb}{D}$ be a conformal welding. If $$\rho_G(t)=O(\log\frac{1}{t}) \quad \text{as}\quad t\to 0,$$ then $G$ extends to a homeomorphism of the entire plane of locally exponentially integrable distortion. Moreover, if $$\rho_G(t)=O(t^{-\alpha}) \quad \text{as}\quad t\to 0$$ for some $\alpha>0$, then $G$ extends to a homeomorphism of the entire plane of locally $p$-integrable distortion with any $p\in (0,\frac{1}{\alpha})$.

Main proofs
===========

Theorem \[thm:thma\] follows from the following more general result.

\[thm: general thmc\] If $\Omega\subset{\ensuremath{\mathbb{R}}}^2$ is a Jordan domain that has the three point property with a control function $\psi$ such that $$\label{eq:sufficient condition}
    \limsup_{r\to 0}\frac{r}{\psi^{-1}\circ\psi^{-1}(r)\log\frac{1}{r}}\leq C'\\$$ for some constant $C'$, then $\Omega$ is a generalized quasidisk.

The idea is similar to that used in [@gkt13 Theorem 5.1]. Since $\Omega$ is a Jordan domain, $f$ extends to a homeomorphism between $\mathbb D$ and ${\overline{\Omega}}$ and we denote also this extension by $f$. Let $e^{i(\theta-t)},\ e^{i\theta}$ and $e^{i(\theta+t)}$ be three points on $S$. Since $f$ is a sense-preserving homeomorphism, $f(e^{i(\theta-t)}),\ f(e^{i\theta})$ and $f(e^{i(\theta+t)})$ will be on the boundary of $\Omega$ in order. Let $g\colon {\ensuremath{\mathbb{R}}}^2\setminus {{\overline{\mathbb D}}}\to{\ensuremath{\mathbb{R}}}^2\setminus{{\overline{\Omega}}}$ be a conformal mapping from the Riemann Mapping Theorem. Then $g$ extends to a homemorphism between ${\ensuremath{\mathbb{R}}}^2\setminus{\mathbb D}$ and ${\ensuremath{\mathbb{R}}}^2\setminus{\Omega}$. As before, we still denote this extension by $g$. Based on the discussion in the previous section, we only need to estimate the scale-wise distortion of the conformal welding $G:=g^{-1}\circ f$.

Let $P=e^{i(\theta+\pi)}$ be the anti-polar point of $e^{i\theta}$ on ${\partial \mathbb}{D}$ and let $\gamma_f(P,\theta-t)$ denote the arc from $f(P)$ to $f(e^{i(\theta-t)})$ on ${\partial \Omega}$ . There exists a $t_0$ small enough such that $\operatorname{diam}(\gamma_f(-1,\theta-t))\geq \operatorname{diam}(\gamma_f(\theta,\theta+t))$ when $t\in [0,t_0)$. Let $\Gamma_1$ be the family of curves in $\mathbb{D}$ joining $\gamma(P,e^{i(\theta-t)})$ and $\gamma(e^{i\theta},e^{i(\theta+t)})$. Then Lemma \[lemma:modululower\] implies that $$\operatorname{mod}(\Gamma_1)\leq C_1$$ for some absolute constant $C_1>0$. The conformal invariance of modulus gives us that $$\operatorname{mod}(\Gamma):=\operatorname{mod}(f(\Gamma_1))\leq C_2.$$ Thus, we may use Lemma \[lemma:key modulus estimate\] for $\alpha_1=\gamma_f(\theta,\theta+t)$ and $\alpha_2=\gamma_f(P,\theta-t)$ and conclude that $$\operatorname{diam}(\gamma_f(\theta,\theta+t))\leq \psi\circ\psi(d),$$ where $d=d(\alpha_1,\alpha_2)$ is the distance between these two arcs. Moreover, $$\label{eq:modulus estimate for outer family}
\operatorname{mod}(\Gamma')\leq C\log^{-1}\Big(1+\frac{\psi^{-1}\circ\psi^{-1}(\operatorname{diam}(\alpha_1))}{\operatorname{diam}(\alpha_1)} \Big),$$ where $\Gamma'$ is the family of curves joining $\alpha_1$ and $\alpha_2$ in ${\ensuremath{\mathbb{R}}}^2\backslash{\overline{\Omega}}$. Again by conformal invariance of modulus, we obtain that $$\label{eq:modulus from conformal invar}
\log^{-1}(1+\frac{1}{\delta_{G}(\theta,t)})\leq C_0^{-1}\operatorname{mod}(\Gamma'),$$ where $C_0$ is the constant from Lemma \[lemma:modululower\]. Note that $$\frac{1}{\log(1+t)}\approx \frac{1}{t}\quad\text{as}\quad t\to 0.$$ Combining  with  gives us the estimate $$\delta_{G}(\theta,t)\leq \frac{C\operatorname{diam}(\alpha_1)}{\psi^{-1}\circ\psi^{-1}(\operatorname{diam}(\alpha_1))} .$$ Since $\frac{t}{\psi^{-1}\circ\psi^{-1}(t)}$ is non-increasing, Lemma \[lemma:modulus of continuity\] implies that $$\label{eq:important estimate for scalewise distortion}
\delta_{G}(\theta,t)\leq \frac{Ct^2}{\psi^{-1}\circ\psi^{-1}(t^2)} .$$ Theorem \[thm: general thmc\] follows immediately from ,  and Corollary \[coro:sufficient for extending\].

This is basically contained in the proof of Theorem \[thm: general thmc\]. In this case, the desired bound  reads as follows: $$\delta_{G}(\theta,t)\leq Ct^{2(1-\frac{1}{s^2})}.$$ The claim follows directly from Corollary \[coro:sufficient for extending\] with $\alpha=2(\frac{1}{s^2}-1)$.

If $\Omega$ is LLC-1, then Lemma \[lemma:modulus of continuity\] implies that $f^{-1}$ is uniformly Hölder continuous. On the other hand, the duality result implies that ${\ensuremath{\mathbb{R}}}^2\backslash{\overline{\Omega}}$ is LLC-2, which is further equivalent to ${\ensuremath{\mathbb{R}}}^2\backslash{\overline{\Omega}}$ being John by the results in [@nv91]. Then by the results in [@kot01], $g$ is also Hölder continuous. Hence $G^{-1}$ is uniformly Hölder continuous with some exponent $\alpha$. Therefore, for $t$ sufficiently small, we have $$\begin{aligned}
\delta_G(\theta,t)&\leq \max\Big\{\frac{|G(e^{i(\theta+t)})-G(e^{i\theta})|}{|G(e^{i\theta})-G(e^{i(\theta-t)})|}, \frac{|G(e^{i\theta})-G(e^{i(\theta-t)})|}{|G(e^{i(\theta+t)})-G(e^{i\theta})|}\Big\}\\
&\lesssim t^{-1/\alpha}.\end{aligned}$$ The claim follows from Corollary \[coro:sufficient for extending\].

Concluding remarks
==================

Definition of generalized quasidisks
------------------------------------

This was previously discussed briefly in the introduction. Recall that $\Omega\subset{\ensuremath{\mathbb{R}}}^2$ is a generalized quasidisk if it is the image of the unit disk $\mathbb D$ under a homeomorphism $f\colon{\ensuremath{\mathbb{R}}}^2\to{\ensuremath{\mathbb{R}}}^2$ of the entire plane with locally exponentially integrable distortion and $f$ is conformal in the unit disk $\mathbb{D}$. However, this is not natural from the technical point of view since our extended mapping $\hat{f}$ is the identity outside a compact disk.

On the other hand, from the point view of conformal welding, requiring that $f$ is identity at infinity is reasonable since it makes the two extension problems equivalent as discussed in Section 4.

From the point view of finding a geometric characterization of generalized quasidisks, this additional requirement is also natural. More precisely, the geometry of a generalized quasidisk $\Omega\subset{\ensuremath{\mathbb{R}}}^n$ should be determined by the geometry of its boundary (at least this is the case if $\Omega$ is a quasidisk). Intuitively the geometry of ${\partial \Omega}$ should have nothing to do with the behavior of the global homeomorphism $f$ at infinity.

These observations suggest that it is better to require the global homeomorphism $f$ to be identity at $\infty$ in the definition of a generalized quasidisk.

Inward pointing and outward pointing cusps
------------------------------------------

As we already observed, (polynomial) interior cusps are more dangerous than (polynomial) exterior cusps for our extension problems. This is not a big surprise from the technical point of view since our aim is to estimate the scalewise distortion of our conformal welding $G$. It is fairly easy to observe that this is closely related to the modulus of continuity of $G^{-1}$. On the other hand, combining the duality results in [@gk12] with the global Hölder continuity estimates of conformal mappings in [@g13; @kot01], one can immediately see how the role of $\Omega$ being $\varphi$-LC-1 or $\psi$-LC-2 is related the modulus of continuity of $G$ and $G^{-1}$. In fact, this is exactly the way we proved [@gkt13 Theorem 4.4].

Open problems
-------------

To end the article, we put forward some open problems, which are plausible to be true.

In Theorem \[thm:thma\], can we further relax the control function $\psi$ to be of the form $\psi(t)=Ct\log\frac{1}{t}$ ? By the result in [@gkt13], we know that the result fails for $\psi(t)=Ct\log^{1+\delta}\frac{1}{t}$ for any $\delta>0$.

In Theorem \[thm:thmc\], can we conclude that the extension has better integrability for the distortion, say locally exponentially integrable distortion, if we additionally assume that $\Omega$ is $\psi$-LC-2 for $\psi(t)=t^{s}$ with $s>1$?

If we require reasonable good moduli of continuity for both $f$, $g$ and their inverses, say both $f$ and $g$ are bi-Hölder continuous up to boundary, can we conclude that $\Omega$ is a generalized quasidisk ?

**Acknowledgements**

I wish to thank my supervisor Academy Professor Pekka Koskela for many useful suggestions.

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