# Explainable Artificial Intelligence in Retinal Imaging
for the detection of Systemic Diseases

Ayushi Raj Bhatt
Rajkumar Vaghashiya  Meghna Kulkarni  Dr Prakash Kamaraj

###### Abstract

Explainable Artificial Intelligence (AI) in the form of an interpretable and
semiautomatic approach to stage grading ocular pathologies such as Diabetic
retinopathy, Hypertensive retinopathy, and other retinopathies on the backdrop
of major systemic diseases. The experimental study aims to evaluate an
explainable staged grading process without using deep Convolutional Neural
Networks (CNNs) directly. Many current CNN-based deep neural networks used for
diagnosing retinal disorders might have appreciable performance but fail to
pinpoint the basis driving their decisions. To improve these decisions’
transparency, we have proposed a clinician-in-the-loop assisted intelligent
workflow that performs a retinal vascular assessment on the fundus images to
derive quantifiable and descriptive parameters. The retinal vessel parameters
meta-data serve as hyper-parameters for better interpretation and
explainability of decisions. The semiautomatic methodology aims to have a
federated approach to AI in healthcare applications with more inputs and
interpretations from clinicians. The baseline process involved in the machine
learning pipeline through image processing techniques for optic disc
detection, vessel segmentation, and arteriole/venule identification.

##  1 Introduction

Retinal health is essential in monitoring of individual health since it is a
direct extension of the vasculature of the central nervous system. Changes in
the retinal vasculature, such as thickening/widening of venules and narrowing
of arteries, as well as tortuosity of the vessels and branching pattern of the
daughter vessels, have proven to be significant biomarkers for the detection
of pathologies that serve as a backdrop to systemic disorders [31, 33, 36,
28].

###  1.1 Impact of Deep Learning (DL) based models

The most common pathology observed in retinal imaging is Diabetic retinopathy
(DR), being one of the main causes of blindness at the global level,
especially in the working-age population [7]. Experiencing severe grades of DR
in the working age group can cause severe loss in productivity. Deep Learning
has become a key tool in terms of AI-assisted diagnosis of DR [5, 18]. There
are also instances of Deep learning modules, such as CNNs that play a role in
the classification of hypertensive retinopathy (HTR) [43]. Features such as
the narrowing of vessels in the retina, vascular bleeding, and cotton wool
spots play a role in the detection of retinopathy diseases of varied etiology.
Deep learning has also helped improve the staging of diabetic retinopathy
conditions [40].

###  1.2 Automated retinal AI for detection of diseases

The diagnosis of the majority of retinopathies from the fundus images by
clinicians is based on visual clues such as abnormal retinal vasculature
morphology, bleeding, exudates, hemorrhages, and so on. Thus, computer vision
greatly underpins the automatic detection of retinal conditions via image
processing techniques such as blood vessel segmentation, classification [4],
optic disc detection, and exudate localization. When combined, these
intelligent algorithms can help detect ocular-cum-systemic conditions such as
hypertension, diabetic retinopathy, and atherosclerosis. There have been
extensive CNN-based researches such as on the automatic grading of retinal
vessels [30] and detection of lesions, microaneurysms, cotton wool spots,
exudates, and retinal hemorrhages [3, 16, 21]. These techniques have attained
appreciable performance comparable to that of trained ophthalmologists.
Further, various systems have been developed for specific disease detection
and management, such as for hypertensive retinopathy [1, 24].

###  1.3 The explainable AI

Explainable AI (XAI) are a modern-day example of having more accountable
artificial intelligence systems where the results are open to interpretation,
paving for better understanding by humans. XAI has been a conscious
exploration in the realm of retinal imaging analysis as well [25]. However,
there is always uncertainty when it comes to deep learning-based models of
retinal diseases [39]. There is a constant need for an analysis system better
understood by an ophthalmologist to avoid erroneous diagnoses. The difficulty
is also experienced in advanced imaging systems such as retinal OCT diagnosis
[6]. There is a need to explore more on the domain with attributable risk
factors with more qualitative and quantitative analysis [38].

###  1.4 Diseases in focus

Analysis of retinal vasculature could be extrapolated not just in ophthalmic
conditions but in systemic diseases as well. The major systemic diseases that
draw inference from the retinal vascular analysis are retinopathy, associated
with the following conditions.

  * Diabetes [13]

  * Diabetic Nephropathy [20]

  * Chronic kidney disease [45]

  * Hypertension [9]

  * Cardiovascular diseases [14]

  * Neurocognitive diseases [10]

  * Infectious diseases due to immunodeficiency [41, 42]

  * Inflammatory diseases [17]

###  1.5 Retinal vascular parameters and retinopathies

Many of the above-mentioned research utilize black-box models, i.e., the
specific decisions taken by the model to conclude are not explainable. Thus,
such systems resist adoption by clinicians despite their great performance.

Apart from image processing techniques and deep learning methods, there is a
need to explore more meaningful parameters that contribute to specific
conditions. There have been significant findings that relate changes in the
retinal vasculature to certain systemic diseases and retinopathies. Specific
findings were noticed towards prediabetic and hypertensive conditions [31].
Measurement of arterio-venous ratio has been especially useful in the
detection of hypertensive conditions [33, 36]. Alterations in the retinal
vascular caliber (RVC) are observed during routine retina examination during
pre-pathological and pathological states [20]. RVC has also indicated
relevance in systemic disease conditions such as diabetic nephropathy and
diabetic neuropathy.

Retinal vasculature parameters such as vascular tortuosity also have an impact
on cardiovascular diseases. Retinal vascular tortuosity is associated with
higher levels of blood pressure and HDL levels [12]. Retinal image-specific
signs such as arterio-venous nicking have linear correlations with
cardiovascular disorders, and hypertension [37]. Automated methods have been
developed to quantify such parameters from color fundus images [32, 44].

Fractal dimension (FD) is an indicative parameter that analyzes retinal
vasculature as a fractal; the geometric parameterization helps to analyze the
branching pattern of retinal vessels [28]. Fractal dimension has been
associated with significance in conditions like stroke [11, 27]. The vascular
parameters-based approach would serve as an effective modus operandi for
explainable AI in healthcare.

##  2 Method

###  2.1 Diseases and corresponding datasets used in the study

This study is focused on the detection and stage-grading of three
retinopathies, namely, Diabetic Retinopathy (DR), Hypertensive Retinopathy
(HTR) and Diabetic Macular Edema (ME). Quite a few retinal datasets are
available for research purpose and additional analysis. Publicly available
datasets are especially important to analyze further, given the additional
confounding factors of varying datasets by ethnicity, age, sex, and other
etiological factors. Modern imaging analysis methods through powerful
computation methods such as machine learning help in drawing additional
inferences [23].

The dataset used in the study for DR and Macular Edema:

  * IDRID [35]

The datasets used in the study for HTR:

  * IDRID [35]

  * AVRDB [2]

  * JSIEC Images (Severe Hypertensive retinopathy category) [22]

##  3 Fundus Image Gradability Analysis

###  3.1 Grability analysis criteria

Fundus photographs acquired from the respective datasets were filtered based
on the criteria of the visibility of four major arterioles and venules. The
images were additionally analyzed to rule out the ones with prevailing co-
morbidities.

###  3.2 Diabetic Retinopathy Dataset Analysis

A total of 516 images filtered out from the IDRID dataset were considered for
DR analysis. The data analysis and filtering process for DR dataset is shown
in Fig. 1.

The grade wise interpretation for multi-class classification is based on the
disease severity levels as specified:

Figure 1: Filtering workflow for DR and ME datasets Severity Grade | Train Data | Test Data
---|---|---
Grade 0 | 52 | 33
Grade 1 | 20 | 5
Grade 2 | 51 | 32
Grade 3 | 40 | 19
Grade 4 | 32 | 6
Table 1: Sample division in Diabetic Retinopathy Dataset

  * Grade 0: Absence of DR

  * Grade 1: Mild DR

  * Grade 2: Moderate DR

  * Grade 3: Severe DR

  * Grade 4: Signs of proliferative DR

The final data sample distribution for DR is specified in Table 1. The binary
grade classification was considered on the presence or absence of the disease
i.e. the Grade 0 signifies the absence and the remaining higher grades signify
the presence of the disease.

###  3.3 Diabetic Macular Edema Dataset Analysis

516 images from the IDRID dataset for Diabetic Macular Edema (ME) were
filtered out and randomly sampled for the image analysis. The data analysis
and filtering process for ME dataset is shown in Fig. 1, and the final data
sample distribution for ME is specified in Table 2. The grade wise
interpretation is based on the disease severity levels as follows:

  * Grade 0: No visible exudates

  * Grade 1: Shortest distance between macula and exudates $>$ one optic disc diameter

  * Grade 2: Shortest distance between macula and exudates $\leq$ one optic disc diameter

Severity Grade | Train Data | Test Data
---|---|---
Grade 0 | 82 | 41
Grade 1 | 14 | 9
Grade 2 | 94 | 40
Table 2: Sample division in Macular Edema Dataset

###  3.4 Hypertensive Retinopathy Dataset Analysis

A total of 66 hypertensive retinopathy images, combined from the AVRDB and the
JSIEC dataset, were used for the analysis of higher grades of HTR. The control
images (Grade 0) were taken from the IDRID dataset after ensuring the absence
of comorbidities. The data collection, filtering and selection procedure is
specified as follows is specified in Fig. 2. The final data sample
distribution for HTR is specified in Table 2.

Due to the lack of test data, 80-20 ratio split on the filtered data was used
for training and testing the model respectively. The grade wise interpretation
is based on the following disease severity levels:

  * Grade 0: No visible abnormalities

  * Grade 1: Diffuse arteriolar narrowing

  * Grade 2: Grade 1 with focal arteriolar constriction

  * Grade 3: Grade 2 with retinal hemorrhage

  * Grade 4: Grade 3 with hard exudates, retinal edema, and optic disc swelling

Figure 2: Filtering workflow for HTR dataset Severity Grade | Train Data
---|---
Grade 0 | 85
Grade 1 | 11
Grade 2 | 15
Grade 3 | 20
Grade 4 | 16
Table 3: Sample division in Hypertensive Retinopathy Dataset

##  4 Calculation of Retinal Vascular Parameters

The filtered data was graded and retinal vessel characteristics were extracted
using the Singapore “I” Vessel Assessment (SIVA) software [26]. The semi-
automated retinal vascular network evaluation was based on the analysis of
vessels from the center of the optic disc and then to three successive zones
corresponding to 0.5 (zone A), 1 (zone B) and 2 (zone C) optic disc diameters,
as shown in Fig. 3. The parameters analyzed, per vessels, arterioles, and
venules, were as follows:

  * Central retinal arteriole equivalent (CRAE)

  * Central retinal venule equivalent (CRVE)

  * Arteriovenous ratio (AVR)

  * Fractal dimension (FD)

  * Mean vessel width (MW)

  * Standard deviation of vessel width (STDW)

  * Tortuosity (TORT)

  * Length to diameter ratio (LDR)

  * Branching coefficient (BC)

  * Asymmetry Factor (AF)

  * Branching angle (BA)

  * Asymmetry angle (AA)

  * Junction exponent (JE)

  * Number of branches & number of first branches

  * Number of arterioles and venules

An $a$ beside the parameter, for e.g. FD$a$, denotes arteriolar measurement
while a $v$, for e.g. FD$v$ denotes venular measurement. The derivation for
the mentioned parameters is specified in [26]. The workflow for the parametric
calculations is specified in Fig. 4.

Figure 3: Vessel classification with retinal zones in SIVA Figure 4: SIVA-
based parametric quantification workflow

###  4.1 Stepwise Linear Regression Analysis

A stepwise multi-linear regression analysis was performed using IBM SPSS v23
to analyze the study parameters and to check for multicollinearity in the
input retinal parameters.

####  4.1.1 DR data analysis

The input variables statistically significantly predicted the DR severity
stage, F(51, 238) = 3.118, p$<$.0005, R2 = .401. No variables were found that
added statistically significantly to the prediction, p$<$.05. As per the
analysis, the parameters Baa, AD2a, BAv, AAv, Bat, and AD2t were excluded from
the study. The details of the tests are provided in the supplementary.

####  4.1.2 HTR data analysis

The input variables statistically significantly predicted the HTR severity
stage, F(51, 95) = 6.850, p$<$.0005, R2 = .786. The variables AVR (of both
zone B and C), arteriolar MW and BMW, and arteriolar NumLDR added
statistically significantly to the prediction, p$<$.05. As per the analysis,
the parameters BAa, AD2a, BAv, AAv, BAt, and AAt were excluded from the study.
The details of the tests are provided in the supplementary.

####  4.1.3 ME Data Analysis

The input variables statistically significantly predicted the ME severity
stage, F(51, 228) = 1.567, p$<$.0005, R2 = .260. The variables CRVE (of both
zone B and C) added statistically significantly to the prediction, p$<$.05. As
per the analysis, the parameters BAa, AD2a, BAv, AAv, AD1t, and AD2t were
excluded from the study. The details of the tests are provided in the
supplementary.

Most of the retinal parameters under consideration in this study convey
certain unique and causal information about the retinal vasculature morphology
in relation to the disease being diagnosed. There is a possibility of spurious
correlation among the input parameters that might negatively interfere and
affect the multi-linear regression analysis.

##  5 Machine Learning Study Design

The experiment was conducted on a cloud-based compute platform, Google Colab
[15]. Standard machine learning algorithms for classification were evaluated
on the data set, mostly using scikit-learn [34], namely:

  * K-Neighbors Classifier (KNC)

  * Extreme Gradient Boosting (XGB) [8]

  * Random Forest Classifier (RFC)

  * Multi-Layer Perceptron Classifier (MLP)

  * Decision Tree Classifier (DTC)

  * Gaussian Naive Bayes Classifier (GNB)

  * Gaussian Process Classifier (GPC)

  * AdaBoost Classifier (ABC)

  * Quadratic Discriminant Analysis (QDA)

  * Logistic Regression (LR)

  * Support Vector Classifier (SVC)

Figure 5: ML model evaluation workflow

For feature scaling, the data of each continuous parameter is transformed
using Min-Max scaling technique to normalize the variance to a standard range
i.e. between 0 and 1. This is done to balance the effect of parameters, e.g.
CRVE and CRAE, typically having higher values compared to the rest, e.g. AVR
and FD. Following this, tSNE was utilized to view the class representations
and check for inter-class separability.

Cross-validation has been utilized to ensure consistent performance with
minimal tolerance in the accuracy. Nested cross-validation on the training
dataset was carried out using 4 inner folds for determination of the optimal
model hyperparameters, and 6 outer folds for determination of the model
performance and stability on the dataset. The top 4 performing models were
then evaluated on the test set to determine the final model performance. The
workflow of model hyperparameter selection and evaluation is summarized in
Fig. 5.

A weighted one-vs-rest Receiver Operating Characteristics-Area Under the Curve
(ROC-AUC) [19] the scheme was used as the scoring criteria. ROCAUC is a
parameter-agnostic graphical performance metric for binary classification
problems used to depict how well a discriminator model minimizes the overlap
between the two class distributions, the diseased and the non-diseased. ROC is
the probability curve derived from the evaluation of the test samples while
AUC denotes the area under that curve. AUC$\leq$0.5 implies model performs no
better than a random guess, while AUC$>$0.5 implies the model is able to
discriminate well between the classes, with AUC of 1 being the maximum
possible score. It is adapted for multi-class problems by pitting one class
against the rest (one-vs-rest).

Range of Hyper-parameters

KNC: weights : [uniform, distance]

algorithm: [auto, ball tree, kd tree]

XGB: learning_rate: [$10^{-3}$, $10^{-2}$, $10^{-1}$]

n_estimators: [100, 120, 140]

RFC: max_depth: [1, 2, 4, 8, 16, 32, 64, None]

n_estimators: [8,9,10,11,12,13, 100, 120, 150, 200]

min_samples_split: [1,2,3]

MLP: hidden_layer_sizes: [10, 20, 40, 60, 80, 100, 120, 150, 200]

DTC: max_depth: [1, 2, 4, 8, 16, 32, None]

min_samples_split: [1,2,3]

criterion: [gini, entropy]

GNB: var_smoothing: [1e-9]

GPC: max_iter_predict: [25, 50, 75, 100, 120, 150]

ABC: n_estimators: [10, 20, 40, 50, 70, 100, 120]

QDA: tol: [1e-4]

LR : max_iter: [50, 100, 500, 100, 5000, 10000]

C: [$10^{-4}$, $10^{-3}$, $10^{-2}$, $10^{1}$, $10^{0}$, $10^{1}$, $10^{2}$]

SVC: C: [$10^{-4}$, $10^{-3}$, $10^{-2}$, $10^{1}$, $10^{0}$, $10^{1}$,
$10^{2}$]

kernel: [poly, rbf, sigmoid]

Figure 6: (a) DR Detection Results (b) DR Detection Feature Importance

##  6 Classification Results

###  6.1 DR Detection Results

ROC-AUC of 0.602 has been achieved for DR detection as depicted in Fig. 6. The
best model is XGB with hyperparameters learning_rate: 0.01 and n_estimators:
140.

For the explanation of the results, a game-theoretic approach named SHAP [29]
is used to analyze the individual impact of the input retinal features in the
decision-making process. The feature analysis for DR detection reveals that
the vessel width deviations (STDWa and STDWv) and the central venular
equivalent (CRVE) are the leading parameters driving the decision on the
detection of the disease.

###  6.2 DR Stage Grading Results

For DR stage grading, the performance of $\sim$0.67 has been attained for
stages 1 and 2, while ROCAUC of 0.812 and 0.863 has been attained for the most
severe stages i.e. Grade 3 and 4 respectively as depicted in Fig. 7. The best
model is Random Forest Classifier (RFC) with hyperparameters max_depth: 8,
min_samples split: 3, and n_estimators: 100.

Figure 7: (a) DR Stage grading Results (b) DR Stage grading Feature Importance

Consistent with the DR detection, the SHAP feature analysis reveals that the
vessel width deviations (STDWa and STDWv) and the central venular equivalent
(CRVE) are the leading parameters driving the decision on the DR stagegrading
as well.

###  6.3 HTR Detection Results

For HTR detection, high performance with ROCAUC of 0.937 has been achieved as
depicted in Fig. 8. The best model is XGB with hyperparameters learning_rate:
0.01 and n_estimators: 120. The SHAP analysis reveals that the tortuosity
(curvature tortuosity in this case) is the primary deciding factor in HTR
detection, and this fact has been corroborated by past findings from retinal
experts.

Figure 8: (a) HTR Detection Results (b) HTR Detection Feature Importance

###  6.4 HTR Stage Grading

For HTR stage grading, a high-performance ROC-AUC $\sim$ 0.95 has been
obtained for stages 0,2 and 3, while stage 1 has ROCAUC 0.877 and stage 4 has
ROCAUC 0.721 respectively, as depicted in Fig. 9. The best model is Random
Forest Classifier (RFC) with hyperparameters max_depth: 2, min_samples_split:
2, and n_estimators: 100. The SHAP feature analysis reveals that the central
venular and arteriolar equivalents as well as parameters like fractal
dimension jointly contribute towards the stage-grading of the disease.

Figure 9: (a) HTR Stage grading Results (b) HTR Stage grading Feature
Importance

###  6.5 ME Detection Results

The performance in ME is similar to that achieved in DR detection. For ME
detection, ROC-AUC of 0.602 has been achieved for DR detection as depicted in
Fig. 10. The best model is XGB with hyperparameters learning_rate: 0.001 and
n_estimators: 140. The SHAP analysis reveals that the detection depends on a
lot of factors such as central vessel equivalents(Zone B CRAE and CRVE),
arteriolar vessel tortuosity (cTORT), and mean arteriolar width (Zone B MWa).

Figure 10: (a) ME Detection Results (b) ME Detection Feature Importance

###  6.6 ME Stage Grading

For ME stage-grading, the performance of ROC-AUC $\sim$0.61 for Grades 0 and
1, while a ROC-AUC of 0.468 for Grade 2 is achieved as depicted in Fig. 11.
The best model is AdaBoost Classifier (ABC) with hyperparameters n_estimators:
40.

Similar to the ME detection, the SHAP feature analysis reveals that the ME
stage grading jointly depends on factors such as central vessel equivalents
(CRAE and CRVE), vessel tortuosity (cTORT), and fractal dimension (Zone C
venular FD) as observed above. The final results of the study are shown in
Table 4. This the study explores how the changes in vascular morphology, when
quantified and employed, could serve as the much-needed explanatory biomarkers
that help in the understanding of the decision-making process as well result
interpretation of the intelligent models.

Figure 11: (a) ME Stage grading Results (b) ME Stage grading Feature Importance Task | Best | Train Perf. | Test Perf. | Train Time
---|---|---|---|---
| Model | ROCAUC | ROCAUC | in min
Disease Detection
DR | XGB | .776$\pm$.095 | .602 | 8.077
HTR | XGB | .986$\pm$.027 | .937 | .048
ME | XGB | .650$\pm$.115 | .620 | 0.986
Disease Grading
DR | RFC | .799$\pm$.049 | .721 | 2.675
HTR | RFC | .976$\pm$.009 | .889 | 1.877
ME | ABC | .486$\pm$.077 | .572 | 2.735
Table 4: Experiment Results

##  7 Conclusion and Future work

Despite having a relatively smaller dataset to work with, a multivariate
approach of specific retinal vascular parameters have shown a reasonable
association with retinal signs caused due to systemic pathologies. Deep
learning methods need a larger dataset to work, needing at least 2000 images
to have reasonable classifier results. The retinal vascular parameters
accounting for reliable parameters showing specific associations to systemic
diseases, we could have better results on larger datasets with dedicated
preprocessing and filtering for other comorbidities that could affect the co-
variate results. Better literature review and clinical studies for the
specific parameters in tandem with other secondary clinical parameters will
lead to better results and thus paving the way for explainable AI of retinal
vascular analysis.

## References

  * [1] S. Akbar, M. U. Akram, M. Sharif, A. Tariq, and S. A. Khan.  Decision support system for detection of hypertensive retinopathy using arteriovenous ratio.  Artificial intelligence in medicine, 90:15–24, 2018.  Publisher: Elsevier.
  * [2] S. Akbar, T. Hassan, M. U. Akram, U. U. Yasin, and I. Basit.  AVRDB: annotated dataset for vessel segmentation and calculation of arteriovenous ratio.  pages 129–134. The Steering Committee of The World Congress in Computer Science, Computer …, 2017.
  * [3] M. U. Akram, S. Akbar, T. Hassan, S. G. Khawaja, U. Yasin, and I. Basit.  Data on fundus images for vessels segmentation, detection of hypertensive retinopathy, diabetic retinopathy and papilledema.  Data in brief, 29:105282, 2020.  Publisher: Elsevier.
  * [4] M. Arsalan, M. Owais, T. Mahmood, S. W. Cho, and K. R. Park.  Aiding the diagnosis of diabetic and hypertensive retinopathy using artificial intelligence-based semantic segmentation.  Journal of clinical medicine, 8(9):1446, 2019.  Publisher: MDPI.
  * [5] N. Asiri, M. Hussain, F. Al Adel, and N. Alzaidi.  Deep learning based computer-aided diagnosis systems for diabetic retinopathy: A survey.  Artificial intelligence in medicine, 99:101701, 2019.  Publisher: Elsevier.
  * [6] A. S. Brar.  Explainable AI for retinal OCT diagnosis.  2021\.  Publisher: University of Waterloo.
  * [7] R. Cheloni, S. A. Gandolfi, C. Signorelli, and A. Odone.  Global prevalence of diabetic retinopathy: protocol for a systematic review and meta-analysis.  BMJ open, 9(3):e022188, 2019.  Publisher: British Medical Journal Publishing Group.
  * [8] T. Chen and C. Guestrin.  Xgboost: A scalable tree boosting system.  pages 785–794, 2016.
  * [9] C. Y.-l. Cheung, M. K. Ikram, C. Sabanayagam, and T. Y. Wong.  Retinal microvasculature as a model to study the manifestations of hypertension.  Hypertension, 60(5):1094–1103, 2012.  Publisher: Am Heart Assoc.
  * [10] C. Y.-l. Cheung, S. Ong, M. K. Ikram, Y. T. Ong, C. P. Chen, N. Venketasubramanian, and T. Y. Wong.  Retinal vascular fractal dimension is associated with cognitive dysfunction.  Journal of Stroke and Cerebrovascular Diseases, 23(1):43–50, 2014\.  Publisher: Elsevier.
  * [11] C. Y.-l. Cheung, W. T. Tay, M. K. Ikram, Y. T. Ong, D. A. De Silva, K. Y. Chow, and T. Y. Wong.  Retinal microvascular changes and risk of stroke: the Singapore Malay Eye Study.  Stroke, 44(9):2402–2408, 2013.  Publisher: Am Heart Assoc.
  * [12] C. Y.-l. Cheung, Y. Zheng, W. Hsu, M. L. Lee, Q. P. Lau, P. Mitchell, J. J. Wang, R. Klein, and T. Y. Wong.  Retinal vascular tortuosity, blood pressure, and cardiovascular risk factors.  Ophthalmology, 118(5):812–818, 2011.  Publisher: Elsevier.
  * [13] R. Crosby-Nwaobi, L. Z. Heng, and S. Sivaprasad.  Retinal vascular calibre, geometry and progression of diabetic retinopathy in type 2 diabetes mellitus.  Ophthalmologica, 228(2):84–92, 2012.  Publisher: Karger Publishers.
  * [14] A. Deiseroth, T. Marcin, C. Berger, D. Infanger, J. Schäfer, B. Bannert, A. Schmidt-Trucksäss, R. E. Voll, D. Kyburz, and H. Hanssen.  Retinal vessel diameters and physical activity in patients with mild to moderate rheumatic disease without cardiovascular comorbidities.  Frontiers in physiology, 9:176, 2018.  Publisher: Frontiers Media SA.
  * [15] B. Ekaba.  Google colaboratory.  Building Machine Learning and Deep Learning Models on Google Cloud Platform; A Comprehensive Guide for Beginners, 2019.
  * [16] K. N. Fatima, M. U. Akram, and S. A. Bazaz.  Papilledema detection in fundus images using hybrid feature set.  pages 1–4. IEEE, 2015.
  * [17] A. J. Green, S. McQuaid, S. L. Hauser, I. V. Allen, and R. Lyness.  Ocular pathology in multiple sclerosis: retinal atrophy and inflammation irrespective of disease duration.  Brain, 133(6):1591–1601, 2010.  Publisher: Oxford University Press.
  * [18] V. Gulshan, L. Peng, M. Coram, M. C. Stumpe, D. Wu, A. Narayanaswamy, S. Venugopalan, K. Widner, T. Madams, and J. Cuadros.  Development and validation of a deep learning algorithm for detection of diabetic retinopathy in retinal fundus photographs.  Jama, 316(22):2402–2410, 2016.  Publisher: American Medical Association.
  * [19] K. Hajian-Tilaki.  Receiver operating characteristic (ROC) curve analysis for medical diagnostic test evaluation.  Caspian journal of internal medicine, 4(2):627, 2013.  Publisher: Babol University of Medical Sciences.
  * [20] M. K. Ikram, C. Y. Cheung, M. Lorenzi, R. Klein, T. L. Jones, T. Y. Wong, and NIH/JDRF Workshop on Retinal Biomarker for Diabetes Group.  Retinal vascular caliber as a biomarker for diabetes microvascular complications.  Diabetes care, 36(3):750–759, 2013.  Publisher: Am Diabetes Assoc.
  * [21] S. Irshad, M. Salman, M. U. Akram, and U. Yasin.  Automated detection of cotton wool spots for the diagnosis of hypertensive retinopathy.  pages 121–124. IEEE, 2014.
  * [22] Joint Shantou International Eye Centre (JSIEC).  1000 fundus images with 39 categories., 2019.
  * [23] S. M. Khan, X. Liu, S. Nath, E. Korot, L. Faes, S. K. Wagner, P. A. Keane, N. J. Sebire, M. J. Burton, and A. K. Denniston.  A global review of publicly available datasets for ophthalmological imaging: barriers to access, usability, and generalisability.  The Lancet Digital Health, 3(1):e51–e66, 2021.  Publisher: Elsevier.
  * [24] S. Khitran, M. U. Akram, A. Usman, and U. Yasin.  Automated system for the detection of hypertensive retinopathy.  pages 1–6. IEEE, 2014.
  * [25] A. Kind and G. Azzopardi.  An explainable AI-based computer aided detection system for diabetic retinopathy using retinal fundus images.  pages 457–468. Springer, 2019.
  * [26] Q. P. Lau, M. L. Lee, W. Hsu, T. Y. Wong, E. Ng, U. Acharya, A. Campillo, and J. Suri.  The singapore eye vessel assessment system.  Image analysis and modeling in ophthalmology, pages 143–160, 2014\.  Publisher: CRC Press Boca Raton, FL.
  * [27] S. Lemmens, A. Devulder, K. Van Keer, J. Bierkens, P. De Boever, and I. Stalmans.  Systematic review on fractal dimension of the retinal vasculature in neurodegeneration and stroke: assessment of a potential biomarker.  Frontiers in neuroscience, 14:16, 2020.  Publisher: Frontiers Media SA.
  * [28] G. Liew, J. J. Wang, N. Cheung, Y. P. Zhang, W. Hsu, M. L. Lee, P. Mitchell, G. Tikellis, B. Taylor, and T. Y. Wong.  The retinal vasculature as a fractal: methodology, reliability, and relationship to blood pressure.  Ophthalmology, 115(11):1951–1956, 2008.  Publisher: Elsevier.
  * [29] S. M. Lundberg and S.-I. Lee.  A unified approach to interpreting model predictions.  Advances in neural information processing systems, 30, 2017.
  * [30] D. Maji and A. A. Sekh.  Automatic grading of retinal blood vessel in deep retinal image diagnosis.  Journal of Medical Systems, 44(10):1–14, 2020.  Publisher: Springer.
  * [31] T. T. Nguyen, J. J. Wang, and T. Y. Wong.  Retinal vascular changes in pre-diabetes and prehypertension: new findings and their research and clinical implications.  Diabetes care, 30(10):2708–2715, 2007.  Publisher: Am Diabetes Assoc.
  * [32] U. T. Nguyen, A. Bhuiyan, L. A. Park, R. Kawasaki, T. Y. Wong, J. J. Wang, P. Mitchell, and K. Ramamohanarao.  An automated method for retinal arteriovenous nicking quantification from color fundus images.  IEEE Transactions on Biomedical Engineering, 60(11):3194–3203, 2013\.  Publisher: IEEE.
  * [33] H. M. Pakter, E. Ferlin, S. C. Fuchs, M. K. Maestri, R. S. Moraes, G. Nunes, L. B. Moreira, M. Gus, and F. D. Fuchs.  Measuring arteriolar-to-venous ratio in retinal photography of patients with hypertension: development and application of a new semi-automated method.  American journal of hypertension, 18(3):417–421, 2005.  Publisher: Oxford University Press.
  * [34] F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, and V. Dubourg.  Scikit-learn: Machine learning in Python.  the Journal of machine Learning research, 12:2825–2830, 2011.  Publisher: JMLR. org.
  * [35] P. Prasanna, P. Samiksha, K. Ravi, K. Manesh, D. Girish, S. Vivek, and M. Fabrice.  Indian diabetic retinopathy image dataset (idrid).  IEEE Dataport, 2018.
  * [36] A. Rani and D. Mittal.  Measurement of arterio-venous ratio for detection of hypertensive retinopathy through digital color fundus images.  Journal of Biomedical Engineering and Medical Imaging, 2(5):35, 2015\.
  * [37] S. SHELBURNE, J. L. HAWLEY, and A. McGee.  Retinal arteriovenous nicking: relation to enlargement of the heart in ambulatory patients with hypertension.  Archives of Internal Medicine, 69(2):213–221, 1942.  Publisher: American Medical Association.
  * [38] A. Singh, S. Sengupta, A. R. Mohammed, I. Faruq, V. Jayakumar, J. Zelek, and V. Lakshminarayanan.  What is the optimal attribution method for explainable ophthalmic disease classification?  pages 21–31. Springer, 2020.
  * [39] A. Singh, S. Sengupta, M. A. Rasheed, V. Jayakumar, and V. Lakshminarayanan.  Uncertainty aware and explainable diagnosis of retinal disease.  volume 11601, pages 116–125. SPIE, 2021.
  * [40] H. Takahashi, H. Tampo, Y. Arai, Y. Inoue, and H. Kawashima.  Applying artificial intelligence to disease staging: Deep learning for improved staging of diabetic retinopathy.  PloS one, 12(6):e0179790, 2017.  Publisher: Public Library of Science San Francisco, CA USA.
  * [41] P. Tan, D. C. Lye, T. K. Yeo, C. Y. Cheung, T.-L. Thein, J. G. Wong, R. Agrawal, L.-J. Li, T.-Y. Wong, and V. C. Gan.  A prospective case-control study to investigate retinal microvascular changes in acute dengue infection.  Scientific Reports, 5(1):1–8, 2015.  Publisher: Nature Publishing Group.
  * [42] P. B. Tan, O. K. Hee, C. Cheung, T. K. Yeo, R. Agrawal, J. Ng, T. H. Lim, T. Y. Wong, and S. C. Teoh.  Retinal vascular parameter variations in patients with human immunodeficiency virus.  Investigative ophthalmology & visual science, 54(13):7962–7967, 2013.  Publisher: The Association for Research in Vision and Ophthalmology.
  * [43] B. K. Triwijoyo, W. Budiharto, and E. Abdurachman.  The classification of hypertensive retinopathy using convolutional neural network.  Procedia Computer Science, 116:166–173, 2017.  Publisher: Elsevier.
  * [44] J. Wigdahl, P. Guimarães, G. Leontidis, A. Triantafyllou, and A. Ruggeri.  Automatic Gunn and Salus sign quantification in retinal images.  pages 5251–5254. IEEE, 2015.
  * [45] T. Y. Wong, D. Xu, D. Ting, S. Nusinovici, C. Cheung, T. E. Shyong, C.-Y. Cheng, M. L. Lee, W. Hsu, and C. Sabanayagam.  Artificial intelligence deep learning system for predicting chronic kidney disease from retinal images.  Investigative Ophthalmology & Visual Science, 60(9):1468–1468, 2019.  Publisher: The Association for Research in Vision and Ophthalmology.


# Sobolev extension on $L^{p}$-quasidisks

Zheng Zhu  Department of Mathematics and Statistics, P.O.Box 35 (MaD) FI-40014
University of Jyväskylä, Finland<EMAIL_ADDRESS>
###### Abstract.

In this paper, we study the Sobolev extension property of $L^{p}$-quasidisks
which are the generalizations of classical quasidisks. After that, we also
find some applications of this property.

###### Key words and phrases:

Homeomorphism of finite distortion, Sobolev extension domains,
$L^{p}$-quasidisks.

The author was supported by the Academy of Finland (project No. 323960). The
author thanks Prof. P. Koskela and Prof. J. Onninen for some interesting
discussion and Prof. T. Kilpeläinen for improving the writing of the paper.

##  1\. Introduction

Let ${\Omega}\subset{\mathbb{R}}^{2}$ be a domain. A homeomorphism
$h:{\Omega}\to{\mathbb{R}}^{2}$ is said to be quasiconformal, if $h\in
W^{1,2}_{\rm loc}({\Omega},{\mathbb{R}}^{2})$ and the inequality

(1.1) $|Dh(z)|^{2}\leq KJ_{h}(z),$

holds for almost every $z\in{\Omega}$ with a constant $K\in[1,\infty)$
independent of $x$. A bounded simply connected domain
${\Omega}\subset{\mathbb{R}}^{2}$ is called a quasidisk, if there exists a
global quasiconformal mapping from
$h\colon{\mathbb{R}}^{2}\xrightarrow[]{{}_{\\!\\!\textnormal{onto\,\,}\\!\\!}}{\mathbb{R}}^{2}$
with $h({\Omega})={\mathbb{D}}$. Quasidisks have lots of nice geometrical and
potential properties, see the textbook by Gehring and Hag [16] and references
therein. For example, a bounded simply connected planar domain is a quasidisk
if and only if it is a uniform domain. Uniform domains were first introduced
by Martio and Sarvas [40]. A domain ${\Omega}\subset{\mathbb{R}}^{2}$ is
called uniform, if there exists a constant $C<\infty$ such that for every
$z_{1},z_{2}\in{\Omega}$ there is a curve
$\gamma_{z_{1},z_{2}}\subset{\Omega}$ with endpoints $z_{1}$ and $z_{2}$ and

$l(\gamma_{z_{1},z_{2}})\leq Cd(z_{1},z_{2})$

and for all $z\in\gamma_{z_{1},z_{2}}$ it holds

$\min\left\\{l(\gamma_{z_{1},z}),l(\gamma_{z,z_{2}})\right\\}\leq
Cd(z,{\mathbb{R}}^{2}\setminus{\Omega}),$

where $\gamma_{z_{1},z}$ and $\gamma_{z,z_{2}}$ mean the subcurves of
$\gamma_{z_{1},z_{2}}$ from $z_{1}$ to $z$ and from $z$ to $z_{2}$
respectively. By Jone’s result in [30], uniform domains hence quasidisks are
Sobolev $(p,p)$-extension domains for arbitrary $1\leq p\leq\infty$. We say
${\Omega}\subset{\mathbb{R}}^{2}$ is a Sobolev $(p,q)$-extension domain, if
for every $u\in W^{1,p}({\Omega})$, there exists a function $E(u)\in
W^{1,q}({\mathbb{R}}^{2})$ with $E(u)\big{|}_{\Omega}\equiv u$ and

$\|E(u)\|_{W^{1,q}({\mathbb{R}}^{2})}\leq C\|u\|_{W^{1,p}({\Omega})}$

with a positive constant $C$ independent of $u$.

There are lots of planar simply connected domains are not quasidisks, for
example inward and outward cuspidal domains, see [21, 23, 25, 28, 33, 34, 35].
Hence, it is natural to study generalizations of quasiconformal mappings. For
instance, we concentrate on homeomorphisms of finite distortion here. A
homeomorphism $h:{\Omega}\to{\mathbb{R}}^{2}$ is said to be a homeomorphism of
finite distortion, if $h\in W^{1,1}_{\rm loc}({\Omega},{\mathbb{R}}^{2})$ and
the inequality

(1.2) $|Dh(z)|^{2}\leq K(z)J_{h}(z),$

holds for almost every $z\in{\Omega}$ with a measurable function
$K(z)\in[1,\infty)$. For a homeomorphism of finite distortion $h$, we denote
$K_{h}$ to be the optimal distortion function for (1.2) which will be defined
below. If $K_{h}\in L^{\infty}({\Omega})$, then $h$ is quasiconformal. The
inverse of a quasiconformal mapping is still quasiconformal. However, if we
relax the regularity of the distortion function from essentially boundedness
to some other weaker one, we cannot hope the distortion of the inverse can
attain the same regularity, see [22, 23]. Motivated by the Sobolev extension
property of quasidisks, we can arise two following interesting problems.

  1. (1)

What is the best Sobolev extension property of those domains which can be
mapped onto the unit disk by homeomorphisms of finite distortion whose
distortion functions satisfy some regularity?

  2. (2)

What is the best Sobolev extension property of domains which are images of the
unit disk under homeomorphisms of finite distortion whose distortion functions
satisfy some regularity?

In this paper, we mainly concentrate on the first problem with the condition
that distortion functions are locally $L^{p}$-integrable for $1\leq p<\infty$.
Under this condition, the corresponding bounded domains are called
$L^{p}$-quasidisks. The terminology $L^{p}$-quasidisk was firstly introduced
in the paper [28] by author with Iwaniec and Onninen, where we gave
characterizations to polynomial cuspidal domains which are $L^{p}$-quasidisks
simultaneously. In [28], we showed that every bounded domain with a
rectifiable boundary is a $L^{1}$-quasidisk. Hence, distortion function is
locally $L^{1}$-integrable is not enough to permit Sobolev extension.

######  Theorem 1.1.

There exists a $L^{1}$-quasidisk which is not a Sobolev $(p,q)$-extension
domain for any $1\leq q\leq p<\infty$.

The main result in [28] tells us that for a fixed $1<p<\infty$, polynomial
cuspidal domains with a relatively not very sharp singularities are
$L^{p}$-quasidisks. Hence, for $1<p<\infty$, we can not hope every
$L^{p}$-quasidisk is a Sobolev $(k,k)$-extension domain for $1\leq\ k<\infty$,
see [43, 44, 45, 46]. The following theorem will imply the Sobolev extension
property of $L^{p}$-quasidisks which is actually a special case of the result
in the theorem.

######  Theorem 1.2.

Let ${\Omega}\subset{\mathbb{R}}^{2}$ be a bounded simply connected domain and
$0<R<\infty$ be a large enough constant with $\overline{\Omega}\subset
B(0,R)$. If there exists a homeomorphism of finite distortion
$h:{\mathbb{R}}^{2}\xrightarrow[]{{}_{\\!\\!\textnormal{onto\,\,}\\!\\!}}{\mathbb{R}}^{2}$
with $h({\Omega})={\mathbb{D}}$ and $K_{h}\in L^{p}({\Omega})\cap
L^{q}(B(0,R)\setminus\overline{\Omega})$ for $1<p,q\leq\infty$. Then
${\Omega}$ is a Sobolev $\left(\frac{2p}{p-1},\frac{2q}{q+1}\right)$-extension
domain and ${\mathbb{R}}^{2}\setminus\overline{\Omega}$ is a Sobolev
$\left(\frac{2q}{q-1},\frac{2p}{p+1}\right)$-extension domain.

Inward cuspidal domains with polynomial-type singularity will shows us the
sharpness of this result. Also, the combination of this result with the
Sobolev extension property of polynomial inward cuspidal domains due to Maz’ya
and Poborchi [43, 44, 45, 46] will give a new and simpler proof to the
necessary part of the main result in [28]. By a result due to Hencl and
Koskela in [22], for a homeomorphism of finite distortion
$h:{\mathbb{R}}^{2}\xrightarrow[]{{}_{\\!\\!\textnormal{onto\,\,}\\!\\!}}{\mathbb{R}}^{2}$
with $\exp\left(\lambda K_{h}\right)\in L^{1}_{\rm loc}({\mathbb{R}}^{2})$ for
some large enough positive constant $\lambda$, we have $K_{h^{-1}}\in
L^{p}_{\rm loc}({\mathbb{R}}^{2})$ for $1<p<c_{2}\lambda$ with a constant
$c_{2}<2$ independent of $p$. Hence, as an application of Theorem 1.2, we have
the following result towards to the second problem above.

######  Theorem 1.3.

Let
$h:{\mathbb{R}}^{2}\xrightarrow[]{{}_{\\!\\!\textnormal{onto\,\,}\\!\\!}}{\mathbb{R}}^{2}$
be a homeomorphism of finite distortion with $\exp\left(\lambda
K_{h}\right)\in L^{1}_{\rm loc}({\mathbb{R}}^{2})$ for some $\lambda>0$ large
enough. Then ${\Omega}:=h({\mathbb{D}})$ is a Sobolev
$\left(\frac{2p_{\lambda}}{p_{\lambda}-1},\frac{2p_{\lambda}}{p_{\lambda}+1}\right)$-extension
domain for some $1<p_{\lambda}<\infty$.

However, we have not obtained the best Sobolev extension property for all
bounded simply connected domains which are images of the unit disk under
global homeomorphisms of finite distortion with locally exponentially
integrable distortion function.

##  2\. Preliminarily

The notation ${\Omega}$ always means a domain in the Euclidean plane
${\mathbb{R}}^{2}$.
$\widehat{{\mathbb{R}}^{2}}:={\mathbb{R}}^{2}\cup\\{\infty\\}$ is the one-
point compactification of the plane ${\mathbb{R}}^{2}$. $B(z,r)$ is a disk
with the center $z\in{\mathbb{R}}^{2}$ and radius $0<r<\infty$.
${\mathbb{D}}:=B(0,1)$ means the unit disk in ${\mathbb{R}}^{2}$. For
$0<r<R<\infty$, we denote $A(r,R):=B(0,R)\setminus\overline{B(0,r)}$ to be an
annulus. Typically, $C$ will be a constant that depend on various parameters
and may differ even on the same line of inequalities. For a measurable subset
$A\subset{\Omega}$ with $0<|A|<\infty$ and a function $u\in L^{1}_{\rm
loc}({\Omega})$, $u_{A}$ is the integral average defined by setting

$u_{A}:={\hbox to0.0pt{\bf\kern
4.49997pt--\hss}\int}\ignorespaces_{A}u(z)dz=\frac{1}{|A|}\int_{A}u(z)dz.$

Let us give the definition of Sobolev spaces first.

######  Definition 2.1.

Let $1\leq p\leq\infty$ and ${\Omega}\subset{\mathbb{R}}^{2}$ be a domain. we
say a function $u\in L^{1}_{\rm loc}({\Omega})$ belongs to the homogeneous
Sobolev space $\dot{W}^{1,p}({\Omega})$ if it is weakly differentiable and its
weak derivative satisfies $|\nabla u|\in L^{p}({\Omega})$. The homogeneous
Sobolev space $\dot{W}^{1,p}({\Omega})$ is equipped with the semi-norm

$\|u\|_{\dot{W}^{1,p}({\Omega})}:=\left(\int_{\Omega}|\nabla
u(z)|^{p}dz\right)^{\frac{1}{p}}.$

If $u\in L^{p}({\Omega})$ at the same time, we say $u$ is contained in the
Sobolev space $W^{1,p}({\Omega})$. The Sobolev space $W^{1,p}({\Omega})$ is
equipped with the norm

$\|u\|_{W^{1,p}({\Omega})}:=\left(\int_{\Omega}|u(z)|^{p}+|\nabla
u(z)|^{p}dz\right)^{\frac{1}{p}}.$

Next, we define Sobolev extension domains and homogeneous Sobolev extension
domains.

######  Definition 2.1.

Let $1\leq q\leq p\leq\infty$. A bounded domain
${\Omega}\subset{\mathbb{R}}^{2}$ is said to be a Sobolev $(p,q)$-extension
domain, if for every $u\in W^{1,p}({\Omega})$ there exists an extension
function $E(u)\in W^{1,q}({\mathbb{R}}^{2})$ with $E(u)\big{|}_{\Omega}\equiv
u$ and

$\|E(u)\|_{W^{1,q}({\mathbb{R}}^{2})}\leq C\|u\|_{W^{1,p}({\Omega})}$

for a constant $C$ independent of $u$. Simply replace $W^{1,p}({\Omega})$ by
$\dot{W}^{1,p}({\Omega})$ and replace $W^{1,q}({\mathbb{R}}^{2})$ by
$\dot{W}^{1,q}({\mathbb{R}}^{2})$ in above, we get the definition of
homogeneous Sobolev $(p,q)$-extension domains.

In [24], authors proved that a bounded domain is homogeneous Sobolev
$(p,p)$-extension if and only of it is a Sobolev $(p,p)$-extension for all
$1\leq p<\infty$. Their argument at least also implies that a bounded
homogeneous Sobolev $(p,q)$-extension domain is also a Sobolev
$(p,q)$-extension domain for $1\leq q\leq p<\infty$. For the convenience of
readers, we represent this observation here.

######  Lemma 2.1.

Let $1\leq q\leq p<\infty$. A bounded homogeneous Sobolev $(p,q)$-extension
domain is also a Sobolev $(p,q)$-extension domain.

###### Proof.

Let ${\Omega}\subset{\mathbb{R}}^{2}$ be a bounded homogeneous Sobolev
$(p,q)$-extension domain for $1\leq q\leq p<\infty$. And let
$B\subset{\mathbb{R}}^{2}$ be a large enough disk with ${\Omega}\subset\subset
B$. Fix $u\in W^{1,p}({\Omega})\subset\dot{W}^{1,p}({\Omega})$. Since
${\Omega}$ is a homogeneous Sobolev $(p,q)$-extension domain, we have an
extension function $E(u)\in\dot{W}^{1,q}({\mathbb{R}}^{2})$ with
$E(u)\big{|}_{\Omega}\equiv u$ and

$\|\nabla E(u)\|_{L^{q}({\mathbb{R}}^{2})}\leq C\|\nabla
u\|_{L^{P}({\Omega})}$

for a constant $C$ independent of $u$. By [24, Lemma 4.2], we have
$E(u)\big{|}_{B}\in W^{1,q}(B)$. Then, Lemma $4.1$ in [24] implies

(2.1) $\left(\int_{B}|E(u)(z)-u_{\Omega}|^{q}dz\right)^{\frac{1}{q}}\\\ \leq
2^{n}{\mathop{\mathrm{missing}}{\,diam\,}}(B)\left(\frac{|B|}{|{\Omega}|}\right)^{\frac{1}{q}}\left(\int_{B}|\nabla
E(u)(z)|^{q}dz\right)^{\frac{1}{q}}\\\ \leq C\left(\int_{\Omega}|\nabla
u(z)|^{p}dz\right)^{\frac{1}{p}}.$

By (2.1), the triangle inequality and the Hölder inequality, we obtain the
desired inequality that

(2.2) $\left(\int_{B}\lvert E(u)(z)\rvert^{q}+\lvert\nabla
E(u)(z)\rvert^{q}dz\right)^{\frac{1}{q}}\\\ \leq
C\left(\int_{\Omega}\lvert\nabla
u(z)\rvert^{p}\right)^{\frac{1}{p}}+C\left(\frac{|B|}{|{\Omega}|}\right)^{\frac{1}{q}}\left(\int_{\Omega}\lvert
u(z)\rvert^{q}dz\right)^{\frac{1}{q}}\\\ \leq C\left(\int_{\Omega}\lvert
u(z)\rvert^{p}dz+\lvert\nabla u(z)\rvert^{p}dz\right)^{\frac{1}{p}}.$

∎

Let us define homeomorphisms of finite distortion and $L^{p}$-quasidisks.

######  Definition 2.2.

We say a homeomorphism
$h:{\Omega}\xrightarrow[]{{}_{\\!\\!\textnormal{onto\,\,}\\!\\!}}h({\Omega})$
has finite distortion if $h\in W^{1,1}_{\rm loc}({\Omega},{\mathbb{R}}^{2})$
and there is a measurable function $K:{\Omega}\to[1,\infty]$ with
$K(z)<\infty$ almost everywhere such that

$|Dh(z)|^{2}\leq K(x)J_{h}(z)\ {\rm for\ almost\ all}\ z\in{\Omega}.$

For a homeomorphism of finite distortion, we define the optimal distortion
function by setting

(2.3) $K_{h}(z):=\begin{cases}\frac{|Dh(z)|^{2}}{J_{h}(z)},&\ {\rm for}\
z\in\\{J_{h}>0\\},\\\ 1,&\ {\rm for}\ z\in\\{J_{h}=0\\}.\end{cases}$

######  Definition 2.3.

Let $1\leq p\leq\infty$. An bounded simply connected domain
${\Omega}\subset{\mathbb{R}}^{2}$ is said to be a $L^{p}$-quasidisk, if there
exists a global homeomorphism of finite distortion
$h:{\mathbb{R}}^{2}\xrightarrow[]{{}_{\\!\\!\textnormal{onto\,\,}\\!\\!}}{\mathbb{R}}^{2}$
with $h({\Omega})={\mathbb{D}}$ and $K_{h}\in L^{p}_{\rm
loc}({\mathbb{R}}^{2})$.

Let ${\Omega}\subset{\mathbb{R}}^{2}$ be a bounded simply connected domain. A
self homeomorphism
$h:{\mathbb{R}}^{2}\xrightarrow[]{{}_{\\!\\!\textnormal{onto\,\,}\\!\\!}}{\mathbb{R}}^{2}$
is called a reflection over the boundary $\partial{\Omega}$ if
$h({\Omega})=\widehat{{\mathbb{R}}^{2}}\setminus\overline{\Omega}$ and
$h(z)=z$ for every $z\in\partial{\Omega}$. Base on the homeomorphism of finite
distortion in Theorem 1.2, we can construct a reflection
$\mathcal{R}_{h}:{\mathbb{R}}^{2}\xrightarrow[]{{}_{\\!\\!\textnormal{onto\,\,}\\!\\!}}{\mathbb{R}}^{2}$
over $\partial{\Omega}$ which can induce a bounded linear extension operator
from $\dot{W}^{1,\frac{2p}{p+1}}({\Omega})$ to
$\dot{W}^{1,\frac{2q}{q-1}}({\mathbb{R}}^{2})$ and a bounded linear extension
operator from
$\dot{W}^{1,\frac{2q}{q+1}}({\mathbb{R}}^{2}\setminus\overline{\Omega})$ to
$\dot{W}^{1,\frac{2p}{p-1}}({\mathbb{R}}^{2})$. In general, we say a
reflection
$\mathcal{R}:\widehat{{\mathbb{R}}^{2}}\xrightarrow[]{{}_{\\!\\!\textnormal{onto\,\,}\\!\\!}}\widehat{{\mathbb{R}}^{2}}$
over $\partial{\Omega}$ can induce a bounded linear Sobolev $(p,q)$-extension
operator for ${\Omega}$ with $1\leq q\leq p<\infty$, if there exists a bounded
Lipschitz domain $U$ containing $\partial{\Omega}$ such that, for every
function $u\in\dot{W}^{1,p}({\Omega})$, the function $v$ defined by setting
$v=u$ on $U\cap{\Omega}$ and $v=u\circ\mathcal{R}$ on
$U\setminus\overline{{\Omega}}$ has a representative which belongs to the
Sobolev space $\dot{W}^{1,q}(U)$ and we have

(2.4) $\|v\|_{\dot{W}^{1,q}(U)}\leq C\|u\|_{\dot{W}^{1,p}({\Omega})}$

with a positive constant $C$ independent of $u$. Similarly, we say the
reflection $\mathcal{R}$ induces a bounded linear Sobolev $(p,q)$-extension
operator for ${\mathbb{R}}^{2}\setminus\overline{\Omega}$ with $1\leq q\leq
p<\infty$, if for every
$u\in\dot{W}^{1,p}({\mathbb{R}}^{2}\setminus\overline{\Omega})$, the function
$\tilde{v}$ defined by setting $\tilde{v}=u$ on $U\setminus\overline{\Omega}$
and $\tilde{v}=u\circ\mathcal{R}$ on $U\cap{\Omega}$ has a representative
which belongs to the Sobolev space $\dot{W}^{1,q}(U)$ and we have

$\|\tilde{v}\|_{\dot{W}^{1,q}(U)}\leq
C\|u\|_{\dot{W}^{1,p}({\mathbb{R}}^{2}\setminus\overline{\Omega})}$

with a positive constant $C$ independent of $u$. By using a suitable cut-off
function, it is easy to see ${\Omega}$ or
${\mathbb{R}}^{2}\setminus\overline{\Omega}$ is a homogeneous Sobolev
$(p,q)$-extension domain. Here the introduction of the bounded open set $U$ is
a convenient way to overcome the non-essential difficulty that functions in
$\dot{W}^{1,p}(G)$ do not necessarily belong to $\dot{W}^{1,q}(G)$ when $1\leq
q<p<\infty$ and $G$ has infinite volume. The following technical lemma
justifies our terminology.

######  Proposition 2.1.

Let ${\Omega}\subset{\mathbb{R}}^{2}$ be a bounded domain with
$|\partial{\Omega}|=0$ and
$\mathcal{R}:\widehat{{\mathbb{R}}^{2}}\to\widehat{{\mathbb{R}}^{2}}$ be a
reflection over $\partial{\Omega}$. If $\mathcal{R}$ induces a bounded linear
extension operator from $\dot{W}^{1,p}({\Omega})$ to
$\dot{W}^{1,q}({\mathbb{R}}^{2})$ in the sense of (2.4) $($from
$\dot{W}^{1,p}({\mathbb{R}}^{2}\setminus\overline{\Omega})$ to
$\dot{W}^{1,q}({\mathbb{R}}^{2})$, respectively$)$ for $1\leq q\leq p<\infty$,
then ${\Omega}$ (${\mathbb{R}}^{2}\setminus\overline{\Omega}$, respectively)
is a homogeneous Sobolev $(p,q)$-extension domain with a linear extension
operator.

###### Proof.

We only consider the case of ${\Omega}$, since the case of
${\mathbb{R}}^{2}\setminus\overline{\Omega}$ is analogous. Let
$U\subset{\mathbb{R}}^{2}$ be the corresponding Lipschitz domain which
contains $\partial{\Omega}$. For a given function
$u\in\dot{W}^{1,p}({\Omega})$, we define a function $E_{\mathcal{R}}(u)$ by
setting

(2.5) $E_{\mathcal{R}}(u)(z):=\left\\{\begin{array}[]{ll}u(\mathcal{R}(z)),&\
{\rm for}\ z\in U\setminus\overline{{\Omega}},\\\ 0,&\ {\rm for}\
z\in\partial{\Omega},\\\ u(z),&\ {\rm for}\ z\in\Omega.\end{array}\right.$

Then $E_{\mathcal{R}}(u)$ has a representative that belongs to
$\dot{W}^{1,q}(U)$ with

$\|E_{\mathcal{R}}(u)\|_{\dot{W}^{1,q}(U)}\leq
C\|u\|_{\dot{W}^{1,p}({\Omega})}.$

Since $U\subset{\mathbb{R}}^{2}$ is a bounded Lipschitz domain, by the result
due to Herron and Koskela [24, Theorem 4.5], for every $1\leq q<\infty$, there
exists a bounded linear extension operator
$L:\dot{W}^{1,q}(U)\to\dot{W}^{1,q}({\mathbb{R}}^{2})$. Although Herron and
Koskela only discussed the case for $1<q<\infty$, however their argument also
works for $q=1$. For every function $u\in\dot{W}^{1,p}({\Omega})$, we define
an extension function
$\tilde{E}_{\mathcal{R}}(u)\in\dot{W}^{1,q}({\mathbb{R}}^{2})$ by setting

(2.6) $\tilde{E}_{\mathcal{R}}(u):=L(E_{\mathcal{R}}(u)).$

Then, we have $\tilde{E}_{\mathcal{R}}(u)\big{|}_{\Omega}\equiv u$ with

$\|\tilde{E}_{\mathcal{R}}(u)\|_{\dot{W}^{1,q}({\mathbb{R}}^{2})}\leq
C\|u\|_{\dot{W}^{1,p}({\Omega})}$

for a constant $C$ independent of $u$. ∎

A strictly increasing function
$\rho:[0,\infty)\xrightarrow[]{{}_{\\!\\!\textnormal{onto\,\,}\\!\\!}}[0,\infty)$
of class $C^{1}(0,\infty)\cap C[0,\infty)$ is called a cuspidal function if
$\rho(1)=1$, $\rho^{\prime}$ is increasing in $(0,\infty)$ and

$\lim_{x\to 0^{+}}\rho^{\prime}(x)=0.$

The corresponding inward and outward cuspidal domains are defined by setting

(2.7)
${\Omega}^{i}_{\rho}:=B(0,1)\setminus\\{(x,y)\in[0,1]\times{\mathbb{R}}:|y|<\rho(x)\\}$

and

(2.8)
${\Omega}^{o}_{\rho}:=B((2,0),\sqrt{2})\cup\\{(x,y)\in(0,1]\times{\mathbb{R}}:|y|<\rho(x)\\}$

respectively. See the Figure below for the exemplary cuspidal function
$\rho(x)=x^{\frac{4}{3}}$.

Figure 1. Inward and outward cuspidal domains

If the cuspidal function is $\rho(x)=x^{s}$ for some $1<s<\infty$, the
corresponding inward and outward cuspidal domains are called polynomial inward
and outward cuspidal domains with the degree $s$. The following result about
Sobolev extension property of polynomial inward and outward cuspidal domains
dues to Maz’ya and Poborchi [43, 44, 45, 46].

######  Proposition 2.2.

The polynomial outward cuspidal domain ${\Omega}^{o}_{t^{s}}$ is a Sobolev
$(p,q)$-extension domain for $1\leq q<p<\infty$ if and only if
$s<\frac{2p}{q}-1$. The polynomial inward cuspidal domain
${\Omega}^{i}_{t^{s}}$ is a Sobolev $(p,q)$-extension domain for
$1<q<p\leq\infty$ if and only if $s<\frac{pq+p-2q}{pq-p}$. And for every
$1<s<\infty$, the polynomial inward cuspidal domain ${\Omega}^{i}_{t^{s}}$ is
a Sobolev $(1,1)$-extension domain.

By combining Theorem 1.2 and Proposition 2.2, we can give a new and simpler
proof to the necessary part of the following proposition which is the main
result in the paper [28] by author, Iwaniec and Onninen. The proof for the
sufficient part comes from the construction of desired homeomorphisms of
finite distortion in [28].

######  Proposition 2.3.

Let ${\Omega}^{i}_{t^{s}}\subset{\mathbb{R}}^{2}$ be a polynomial inward
cuspidal domain with the degree $1<s<\infty$. Then there exists a
homeomorphism of finite distortion
$h:{\mathbb{R}}^{2}\xrightarrow[]{{}_{\\!\\!\textnormal{onto\,\,}\\!\\!}}{\mathbb{R}}^{2}$
with $h({\Omega}^{i}_{t^{s}})={\mathbb{D}}$ and $K_{h}\in
L^{p}({\Omega}^{i}_{t^{s}})\cap
L^{q}(B(0,2)\setminus\overline{{\Omega}^{i}_{t^{s}}})$ for $1<p,q\leq\infty$
with $\min\\{p,q\\}<\infty$ if and only if $s<\frac{pq+p+2q}{pq-p}$.

Combine Proposition 2.2 and Proposition 2.3, we can also obtain the sharpness
of Theorem 1.2. The polynomial inward cuspidal domain ${\Omega}^{i}_{t^{s}}$
shows that there exists a bounded simply connected domain
${\Omega}\subset{\mathbb{R}}^{2}$ which satisfies assumptions of Theorem 1.2,
such that we can not hope ${\Omega}$ has a better Sobolev extension property
than it is a Sobolev $\left(\frac{2p}{p-1},\frac{2q}{q+1}\right)$-extension
domain and ${\mathbb{R}}^{2}\setminus\overline{\Omega}$ is a Sobolev
$\left(\frac{2q}{q-1},\frac{2p}{p+1}\right)$-extension domain.

##  3\. Proofs of Theorem 1.1 and Theorem 1.2

###  3.1. Proof of Theorem 1.1

By [28, Theorem 1.8], every simply connected Jordan domain with a rectifiable
boundary is a $L^{1}$-quasidisk. Hence, an inward cuspidal domain
${\Omega}^{i}_{\rho}$ with cuspidal function
$\rho(x)=e/\exp\left(\frac{1}{x}\right)$ is a $L^{1}$-quasidisk. By
Proposition 2.2, ${\Omega}^{i}_{\rho}$ cannot be a Sobolev $(p,q)$-extension
domain for any $1\leq q\leq p<\infty$.

###  3.2. Proof of Theorem 1.2

Let ${\Omega}\subset{\mathbb{R}}^{2}$ be a bounded simply connected domain
with $\overline{\Omega}\subset\subset B(0,R)$ for a large enough $R>0$.
Suppose there exists a homeomorphism of finite distortion
$h:{\mathbb{R}}^{2}\xrightarrow[]{{}_{\\!\\!\textnormal{onto\,\,}\\!\\!}}{\mathbb{R}}^{2}$
with $h({\Omega})={\mathbb{D}}$ and

(3.1)
$\int_{\Omega}K_{h}^{p}(z)dz+\int_{B(0,R)\setminus\overline{\Omega}}K_{h}^{q}(z)dz<\infty.$

The circle inversion map
$\mathcal{R}:\widehat{{\mathbb{R}}^{2}}\xrightarrow[]{{}_{\\!\\!\textnormal{onto\,\,}\\!\\!}}\widehat{{\mathbb{R}}^{2}}$,

$\mathcal{R}(z):=\begin{cases}\frac{z}{|z|^{2}},&\ {\rm if}\ z\neq 0,\\\
\infty,&\ {\rm if}\ z=0.\end{cases}$

is an anticonformal reflection over the unit circle $\partial\mathbb{D}$,
which means that at every point it preserves angles and reverses orientation.
Then a self-homeomorphism
$\widetilde{\mathcal{R}}:\widehat{{\mathbb{R}}^{2}}\xrightarrow[]{{}_{\\!\\!\textnormal{onto\,\,}\\!\\!}}\widehat{{\mathbb{R}}^{2}}$
defined on every $z\in\widehat{{\mathbb{R}}^{2}}$ by setting

$\widetilde{\mathcal{R}}(z):=h^{-1}\circ\mathcal{R}\circ h(z)$

is a reflection over the boundary $\partial{\Omega}$.

With respect to different domains ${\Omega}$ and
${\mathbb{R}}^{2}\setminus\overline{{\Omega}}$, we divide the full proof of
Theorem 1.2 into two parts. First, let us prove the Sobolev extension property
of the bounded domain ${\Omega}$.

######  Theorem 3.1.

Under the assumption of Theorem 1.2, ${\Omega}$ is a Sobolev
$\left(\frac{2p}{p-1},\frac{2q}{q+1}\right)$-extension domain.

###### Proof.

By Lemma 2.1, it is sufficient to show that ${\Omega}$ is a homogeneous
Sobolev $\left(\frac{2p}{p-1},\frac{2q}{q+1}\right)$-extension domain. Let
$u\in\dot{W}^{1,\frac{2p}{p-1}}({\Omega})$ be arbitrary. We define an
extension function $E_{\widetilde{\mathcal{R}}}(u)$ on $B(0,R)$ by setting

(3.2)
$E_{\widetilde{\mathcal{R}}}(u)(z):=\begin{cases}u(\widetilde{\mathcal{R}}(z)),&\
{\rm if}\ z\in B(0,R)\setminus\overline{\Omega},\\\ 0,&\ {\rm if}\
z\in\partial{\Omega},\\\ u(z),&\ {\rm if}\ z\in{\Omega}.\end{cases}$

By Proposition 2.1, it suffices to prove
$E_{\widetilde{\mathcal{R}}}(u)\in\dot{W}^{1,\frac{2q}{q+1}}(B(0,R))$ with the
following inequality that

(3.3)
$\left(\int_{B(0,R)}|DE_{\widetilde{\mathcal{R}}}(u)(z)|^{\frac{2q}{q+1}}dz\right)^{\frac{q+1}{2q}}\leq
C\left(\int_{{\Omega}}|Du(z)|^{\frac{2p}{p-1}}dz\right)^{\frac{p-1}{2p}}$

for a positive constant $C$ independent of $u$. Since $1<\frac{2q}{q+1}\leq
2\leq\frac{2p}{p-1}<\infty$, the Hölder inequality implies

$\left(\int_{{\Omega}}|DE_{\widetilde{\mathcal{R}}}(u)(z)|^{\frac{2q}{q+1}}dz\right)^{\frac{q+1}{2q}}\leq
C\left(\int_{{\Omega}}|Du(z)|^{\frac{2p}{p-1}}dz\right)^{\frac{p-1}{2p}}.$

By [23, Theorem 4.13], $\partial{\Omega}$ must be of measure zero. Hence, we
only need to prove

(3.4)
$\left(\int_{B(0,R)\setminus\overline{\Omega}}|DE_{\widetilde{\mathcal{R}}}(u)(z)|^{\frac{2q}{q+1}}dz\right)^{\frac{q+1}{2q}}\leq
C\left(\int_{{\Omega}}|Du(z)|^{\frac{2p}{p-1}}dz\right)^{\frac{p-1}{2p}}.$

We define a function $v$ on ${\mathbb{D}}$ by setting $v(z):=u\circ h^{-1}(z)$
for every $z\in{\mathbb{D}}$ and define an extension function
$E_{\mathcal{R}}(v)$ on $h(B(0,R))$ with $z=(x,y)$ by setting

(3.5) $E_{\mathcal{R}}(v)(z):=\begin{cases}v(\mathcal{R}(z)),&\ {\rm if}\ z\in
h(B(0,R))\setminus\overline{\mathbb{D}},\\\ 0,&\ {\rm if}\
z\in\partial{\mathbb{D}},\\\ v(z),&\ {\rm if}\ z\in{\mathbb{D}}.\end{cases}$

By the definitions of functions and reflections, for every $z\in B(0,R)$, we
have

$E_{\widetilde{\mathcal{R}}}(u)(z)=E_{\mathcal{R}}(v)(h(z)).$

We divide the following argument into three steps.

Step $1$: We would like to show $v\in\dot{W}^{1,2}({\mathbb{D}})$ with

(3.6) $\left(\int_{{\mathbb{D}}}|Dv(z)|^{2}dz\right)^{\frac{1}{2}}\leq
C\left(\int_{{\Omega}}|Du(z)|^{\frac{2p}{p-1}}dz\right)^{\frac{p-1}{2p}}$

for a positive constant $C$ independent of $u$. By [23, Theorem 1.6 and
Theorem 2.24], $h^{-1}\in W^{1,1}({\mathbb{D}},{\mathbb{R}}^{2})$ and it is
differentiable almost everywhere on ${\mathbb{D}}$. Hence, there exists a
subset $G\subset{\mathbb{D}}$ with $|G|=|{\mathbb{D}}|$ and $h^{-1}$ is
differentiable on every point $z\in G$. By the chain rule, for every $z\in G$,
we have

(3.7) $|Dv(z)|\leq|Du(h^{-1}(z))|\cdot|Dh^{-1}(z)|.$

Set $A\subset G$ to be the subset such that $J_{h^{-1}}(z)>0$ for every $z\in
A$. Then by [23, Theorem 1.6], $|Dh^{-1}(z)|=0$ for almost every $z\in
G\setminus A$. Hence, we have

$\displaystyle\int_{{\mathbb{D}}}|Dv(z)|^{2}dz$ $\displaystyle=$
$\displaystyle\int_{G}|Dv(z)|^{2}dz$ $\displaystyle\leq$
$\displaystyle\int_{A}|Du(h^{-1}(z))|^{2}\cdot|Dh^{-1}(z)|^{2}dz.$

If $p=\infty$, then $h\big{|}_{\Omega}$ is quasiconformal. By the fact that
the inverse of a quasiconformal mapping is also quasiconformal, we have

(3.9) $\int_{A}|Du(h^{-1}(z))|^{2}\cdot|Dh^{-1}(z)|^{2}dz\\\
\leq\int_{A}|Du(h^{-1}(z))|^{2}J_{h^{-1}}(z)\frac{|Dh^{-1}(z)|^{2}}{J_{h^{-1}}(z)}dz\\\
\leq C\int_{A}|Du(h^{-1}(z))|^{2}J_{h^{-1}}(z)dz.$

If $1<p<\infty$, the Hölder inequality implies

(3.10) $\int_{A}|Du(h^{-1}(z))|^{2}\cdot|Dh^{-1}(z)|^{2}dz\\\
\leq\int_{A}|Du(h^{-1}(z))|^{2}J_{h^{-1}}^{\frac{p-1}{p}}(z)\frac{|Dh^{-1}(z)|^{2}}{J^{\frac{p-1}{p}}_{h^{-1}}(z)}dz\\\
\leq\left(\int_{A}|Du(h^{-1}(z))|^{\frac{2p}{p-1}}J_{h^{-1}}(z)dz\right)^{\frac{p-1}{p}}\\\
\times\left(\int_{A}\frac{|Dh^{-1}(z)|^{2p}}{J^{p}_{h^{-1}}(z)}J_{h^{-1}}(z)dz\right)^{\frac{1}{p}}.$

For every $1<p\leq\infty$, the change of variables formula implies

(3.11)
$\int_{A}|Du(h^{-1}(z))|^{\frac{2p}{p-1}}J_{h^{-1}}(z)dz\leq\int_{h^{-1}(A)}|Du(z)|^{\frac{2p}{p-1}}dz.$

By [23, Lemma A.29], $h$ is differentiable on every $w\in h^{-1}(A)$ with
$Dh(w)=(Dh^{-1}(z))^{-1}$ for $w=h^{-1}(z)$. Hence, for every $z\in A$ with
$w=h^{-1}(z)$, we have

$K_{h^{-1}}(z)=\frac{|Dh^{-1}(z)|^{2}}{J_{h^{-1}}(z)}=\frac{|Dh(w)|^{2}}{J_{h}(w)}=K_{h}(w).$

Hence, the change of variables formula implies

(3.12)
$\int_{A}\frac{|Dh^{-1}(z)|^{2p}}{J^{p}_{h^{-1}}(z)}J_{h^{-1}}(z)dz\leq\int_{{\Omega}}K_{h}^{p}(z)dz.$

By combing inequalities (3.2), (3.9), (3.10), (3.11) and (3.12), we obtain the
desired inequality (3.6).

Step $2$: We prove $E_{\mathcal{R}}(v)\in\dot{W}^{1,2}(h(B(0,R)))$ with

(3.13) $\int_{h(B(0,R))}|DE_{\mathcal{R}}(v)(z)|^{2}dz\leq
C\int_{{\mathbb{D}}}|Dv(z)|^{2}dz$

for a positive constant $C$ independent of $v$. By [24, Lemma 4.2],
$v\in\dot{W}^{1,2}({\mathbb{D}})=W^{1,2}({\mathbb{D}})$. Since
$C^{\infty}({\mathbb{R}}^{2})\cap W^{1,2}({\mathbb{D}})$ is dense in
$W^{1,2}({\mathbb{D}})$, we can find a sequence of functions $v_{k}\in
C^{\infty}({\mathbb{R}}^{2})\cap W^{1,2}({\mathbb{D}})$ with

$\lim_{k\to\infty}v_{k}(z)=v(z)$

for almost every $z\in{\mathbb{D}}$, and

(3.14) $\lim_{k\to\infty}\|v_{k}-v\|_{\dot{W}^{1,2}({\mathbb{D}})}=0.$

With an extra diagonal argument if necessary, we can also assume the sequence
of weak gradients $\\{Dv_{k}\\}$ converges to $Dv$ almost everywhere on
${\mathbb{D}}$. For every $v_{k}\in C^{\infty}({\mathbb{R}}^{2})\cap
W^{1,2}({\mathbb{D}})$, we define the extension function
$E_{\mathcal{R}}(v_{k})$ as in (3.5). Since $v_{k}\in
C^{\infty}({\mathbb{R}}^{2})\cap W^{1,2}({\mathbb{D}})$,
$E_{\mathcal{R}}(v_{k})$ is $ACL$ on $h(B(0,R))$. By definition (3.5) and the
fact that $\mathcal{R}$ is anticonformal, it is easy to see that we have

(3.15) $\int_{h(B(0,R))}|DE_{\mathcal{R}}(v_{k})(z)|^{2}dz\leq
C\int_{{\mathbb{D}}}|Dv_{k}(z)|^{2}dz,$

for a positive constant $C$ independent of $k$. Since $v_{k}$ converges to $v$
almost everywhere on ${\mathbb{D}}$, by (3.5), $E_{\mathcal{R}}(v_{k})$
converges to $E_{\mathcal{R}}(v)$ almost everywhere on $h(B(0,R))$. Since
$\\{Dv_{k}\\}$ converges to $Dv$ almost everywhere on ${\mathbb{D}}$, by the
definitions of $E_{\mathcal{R}}(v_{k})$ and $E_{\mathcal{R}}(v)$ in (3.5),
$\\{DE_{\mathcal{R}}(v_{k})\\}$ converges to $DE_{\mathcal{R}}(v)$ almost
everywhere on ${\mathbb{R}}^{2}$. By (3.14) and (3.15), we have

(3.16)
$\lim_{k\to\infty}\int_{h(B(0,R))}|DE_{\mathcal{R}}(v_{k})(z)-DE_{\mathcal{R}}(v)(z)|^{2}dz=0.$

Then by Theorem $1$ in [41, Section 1.1.13],
$E_{\mathcal{R}}(v)\in\dot{W}^{1,2}(h(B(0,R)))$ with the weak gradient
$DE_{\mathcal{R}}(v)$. Combine (3.14), (3.15) and (3.16), we obtain

(3.17) $\int_{h(B(0,R))}|DE_{\mathcal{R}}(v)(z)|^{2}dz\leq
C\int_{{\mathbb{D}}}|Dv(z)|^{2}dz,$

for a positive constant independent of $u$.

Step $3$: We prove
$E_{\widetilde{\mathcal{R}}}(u)\in\dot{W}^{1,\frac{2q}{q+1}}(B(0,R))$ and we
have

(3.18)
$\left(\int_{B(0,R)\setminus\overline{\Omega}}|DE_{\widetilde{\mathcal{R}}}(u)(z)|^{\frac{2q}{q+1}}dz\right)^{\frac{q+1}{2q}}\\\
\leq
C\left(\int_{h(B(0,R))}|DE_{\mathcal{R}}(v)(z)|^{2}dz\right)^{\frac{1}{2}}$

for a positive constant $C$ independent of $u$. As we know,
$E_{\widetilde{\mathcal{R}}}(u)(z)=E_{\mathcal{R}}(v)(h(z))$ for every $z\in
B(0,R)$. By [23, Theorem 1.7 and Theorem 2.24], $h$ is differentiable almost
everywhere on $B(0,R)\setminus\overline{\Omega}$ with positive determinant
value. Hence, there exists a subset $\widehat{G}\subset
B(0,R)\setminus\overline{\Omega}$ with
$|\widehat{G}|=|B(0,R)\setminus\overline{\Omega}|$ and $h$ is differentiable
on every point in $z\in\widehat{G}$ with $J_{h}(z)>0$. By the chain rule, for
every $z\in\widehat{G}$, we have

(3.19)
$|DE_{\widetilde{\mathcal{R}}}(u)(z)|\leq|DE_{\mathcal{R}}(v(h(z)))|\cdot|Dh(z)|.$

If $q=\infty$, then $h$ restricts to $B(0,R)\setminus\overline{\Omega}$ is
quasiconformal. Then we have

(3.20)
$\int_{B(0,R)\setminus\overline{\Omega}}|DE_{\widetilde{\mathcal{R}}}(u)(z)|^{2}dz\\\
\leq\int_{B(0,R)\setminus\overline{\Omega}}|DE_{\mathcal{R}}(v)(h(z))|^{2}J_{h}(z)\cdot\frac{|Dh(z)|^{2}}{J_{h}(z)}dz\\\
\leq
C\int_{B(0,R)\setminus\overline{\Omega}}|DE_{\mathcal{R}}(v)(h(z))|^{2}J_{h}(z)dz.$

If $1<q<\infty$, then we have

(3.21)
$\int_{B(0,R)\setminus\overline{\Omega}}|DE_{\widetilde{\mathcal{R}}}(u)(z)|^{\frac{2q}{q+1}}dz=\int_{\widehat{G}}|DE_{\widetilde{\mathcal{R}}}(u)(z)|^{\frac{2q}{q+1}}dz\\\
\leq\int_{\widehat{G}}|DE_{\mathcal{R}}(v)(h(z))|^{\frac{2q}{q+1}}\cdot|Dh(z)|^{\frac{2q}{q+1}}dz.$

The Hölder inequality implies

(3.22)
$\int_{\widehat{G}}|DE_{\mathcal{R}}(v)(h(z))|^{\frac{2q}{q+1}}\cdot|Dh(z)|^{\frac{2q}{q+1}}dz\\\
\leq\int_{\widehat{G}}|DE_{\mathcal{R}}(v)(h(z))|^{\frac{2q}{q+1}}J_{h}^{\frac{q}{q+1}}(z)\frac{|Dh(z)|^{\frac{2q}{q+1}}}{J_{h}^{\frac{q}{q+1}}(z)}dz\\\
\leq\left(\int_{\widehat{G}}|DE_{\mathcal{R}}(v)(h(z))|^{2}J_{h}(z)dz\right)^{\frac{q}{q+1}}\\\
\times\left(\int_{\widehat{G}}\left(\frac{|Dh(z)|^{2}}{J_{h}(z)}\right)^{q}dz\right)^{\frac{1}{q}}.$

The change of variables formula implies

(3.23)
$\int_{\widehat{G}}|DE_{\mathcal{R}}(v)(h(z))|^{2}J_{h}(z)dz\leq\int_{h(B(0,R))}|DE_{\mathcal{R}}(v)(z)|^{2}dz.$

By the definition of distortion function and the integral condition (3.1), we
have

(3.24)
$\int_{\widehat{G}}\left(\frac{|Dh(z)|^{2}}{J_{h}(z)}\right)^{q}dz\leq\int_{B(0,R)\setminus\overline{\Omega}}K_{h}^{q}(z)dz<\infty.$

Combine inequalities (3.20)-(3.24), we obtain the desired inequality (3.18).

Finally, combine inequalities (3.6), (3.13) and (3.18), we obtain the desired
inequality (3.4) and finish the proof of Theorem 1.2. ∎

Next, we consider the Sobolev extension property of the complementary domain
${\mathbb{R}}^{2}\setminus\overline{\Omega}$.

######  Theorem 3.2.

Under the assumption of Theorem 1.2,
${\mathbb{R}}^{2}\setminus\overline{\Omega}$ is a Sobolev
$\left(\frac{2q}{q-1},\frac{2p}{p+1}\right)$-extension domain.

The idea of the proof is very similar with the proof of last theorem. The main
difference is that ${\mathbb{R}}^{2}\setminus\overline{\Omega}$ is unbounded
here. Hence, we give a sketch proof here.

###### Proof.

From the geometrical viewpoint, it is easy to see
${\mathbb{R}}^{2}\setminus\overline{\Omega}$ has a same Sobolev extension
property with the bounded domain $B(0,R)\setminus\overline{\Omega}$. Hence, by
Lemma 2.1, it suffices to prove $B(0,R)\setminus\overline{\Omega}$ is a
homogeneous Sobolev $\left(\frac{2q}{q-1},\frac{2p}{p+1}\right)$-extension
domain. Since
$h:{\mathbb{R}}^{2}\xrightarrow[]{{}_{\\!\\!\textnormal{onto\,\,}\\!\\!}}{\mathbb{R}}^{2}$
is a homeomorphism with $h({\Omega})={\mathbb{D}}$, there exist two positive
constants $\epsilon_{1}>0$ and $\epsilon_{2}>0$ such that
$B(0,1+\epsilon_{1})\subset h(B(0,R))$ and

$\mathcal{R}(A(1-\epsilon_{2},1+\epsilon_{1}))=A(1-\epsilon_{2},1+\epsilon_{1}).$

We define $\mathcal{A}:=h^{-1}(A(1-\epsilon_{2},1))$,
$\mathbf{U}:=\left(B(0,R)\setminus{\Omega}\right)\cup\mathcal{A}$ and
$\widetilde{\mathbf{U}}:=h(\mathbf{U})$. Let
$u\in\dot{W}^{1,\frac{2q}{q-1}}({\mathbb{R}}^{2}\setminus\overline{\Omega})$
be arbitrary. Then

$u\big{|}_{B(0,R)\setminus\overline{\Omega}}\in\dot{W}^{1,\frac{2q}{q-1}}(B(0,R)\setminus\overline{\Omega}).$

To simplify the notation, we still denote
$u\big{|}_{B(0,R)\setminus\overline{{\Omega}}}$ by $u$. Then we define a
function $E_{\widetilde{\mathcal{R}}}(u)$ on $\mathbf{U}$ by setting

(3.25)
$E_{\widetilde{\mathcal{R}}}(u)(z):=\begin{cases}u(\widetilde{\mathcal{R}}(z)),&\
{\rm if}\ z\in\mathcal{A},\\\ 0,&\ {\rm if}\ z\in\partial{\Omega},\\\ u(z),&\
{\rm if}\ z\in B(0,R)\setminus\overline{{\Omega}}.\end{cases}$

It suffices to prove
$E_{\widetilde{\mathcal{R}}}(u)\in\dot{W}^{1,\frac{2p}{p+1}}(\mathbf{U})$ with

(3.26)
$\left(\int_{\mathbf{U}}|DE_{\widetilde{\mathcal{R}}}(u)(z)|^{\frac{2p}{p+1}}dz\right)^{\frac{p+1}{2p}}\\\
\leq
C\left(\int_{B(0,R)\setminus\overline{\Omega}}|Du(z)|^{\frac{2q}{q-1}}dz\right)^{\frac{q-1}{2q}}.$

for a positive constant $C$ independent of $u$. By the definition of
$E_{\widetilde{\mathcal{R}}}(u)$, the Hölder inequality implies

(3.27)
$\left(\int_{B(0,R)\setminus\overline{\Omega}}|DE_{\widetilde{\mathcal{R}}}(u)(z)|^{\frac{2p}{p+1}}\right)^{\frac{p+1}{2p}}\\\
\leq
C\left(\int_{B(0,R)\setminus\overline{\Omega}}|Du(z)|^{\frac{2q}{q-1}}dz\right)^{\frac{q-1}{2q}}$

for a positive constant $C$ independent of $u$. By [23, Theorem 4.13],
$\partial{\Omega}$ must be of measure zero. Hence, it suffices to prove the
inequality

(3.28)
$\left(\int_{\mathcal{A}}|DE_{\widetilde{\mathcal{R}}}(u)(z)|^{\frac{2p}{p+1}}dz\right)^{\frac{p+1}{2p}}\\\
\leq
C\left(\int_{B(0,R)\setminus\overline{\Omega}}|Du(z)|^{\frac{2q}{q-1}}dz\right)^{\frac{q-1}{2q}}$

for a positive constant $C$ independent of $u$. We define a function $v\in
h(B(0,R)\setminus\overline{{\Omega}})$ by setting $v(z):=u(h^{-1}(z))$ for
every $z\in h(B(0,R)\setminus\overline{\Omega})$. By a similar argument to the
inequality (3.6), we obtain
$v\in\dot{W}^{1,2}(h(B(0,R)\setminus\overline{\Omega}))$ with

(3.29)
$\left(\int_{h(B(0,R)\setminus\overline{\Omega})}|Dv(z)|^{2}dz\right)^{\frac{1}{2}}\leqq
C\left(\int_{B(0,R)\setminus\overline{\Omega}}|Du(z)|^{\frac{2q}{q-1}}\right)^{\frac{q-1}{2q}}$

for a positive constant $C$ independent of $u$. Define an extension function
$E_{\mathcal{R}}(v)$ on $\widetilde{\mathbf{U}}$ by setting

(3.30) $E_{\mathcal{R}}(v)(z):=\begin{cases}v(\mathcal{R}(z)),&\ {\rm if}\
z\in A(1-\epsilon_{2},1),\\\ 0,&\ {\rm if}\ z\in\partial{\mathbb{D}},\\\
v(z),&\ {\rm if}\
z\in\widetilde{\mathbf{U}}\setminus\overline{{\mathbb{D}}}.\end{cases}$

It is easy to see for every $z\in\mathbf{U}$, we have

$E_{\widetilde{\mathcal{R}}}(u)(z)=E_{\mathcal{R}}(v)(h(z)).$

By (3.29), $v\big{|}_{A(1,1+\epsilon_{1})}$ belongs to the class
$\dot{W}^{1,2}(A(1,1+\epsilon_{1}))$. To simplify the notation, we still
denote $v\big{|}_{A(1,1+\epsilon_{1})}$ by $v$. By facts that
$\mathcal{R}(A(1-\epsilon_{2},1+\epsilon_{1}))=A(1-\epsilon_{2},1+\epsilon_{1})$,
$\dot{W}^{1,2}(A(1-\epsilon_{2},1+\epsilon_{1}))=W^{1,2}(A(1-\epsilon_{2},1+\epsilon_{1}))$
and $C^{\infty}({\mathbb{R}}^{2})\cap
W^{1,2}(A(1-\epsilon_{2},1+\epsilon_{1}))$ is dense in
$W^{1,2}(A(1-\epsilon_{2},1+\epsilon_{1}))$. With a similar argument to the
inequality (3.17), we obtain
$E_{\mathcal{R}}(v)\in\dot{W}^{1,2}(\widetilde{\mathbf{U}})$ with

(3.31)
$\int_{A(1-\epsilon_{2},1+\epsilon_{1})}|DE_{\mathcal{R}}(v)(z)|^{2}dz\\\ \leq
C\int_{A(1,1+\epsilon_{1})}|Dv(z)|^{2}dz\\\ \leq
C\int_{h(B(0,R)\setminus\overline{\Omega})}|Dv(z)|^{2}dz$

for a positive constant $C$ independent of $u$. Then, with a similar argument
to the inequality (3.18), we obtain the inequality

(3.32)
$\left(\int_{\mathcal{A}}|DE_{\mathcal{R}}(u)(z)|^{\frac{2p}{p+1}}dz\right)^{\frac{p+1}{2p}}\\\
\leq
C\left(\int_{A(1-\epsilon_{2},1)}|DE_{\mathcal{R}}(v)(z)|^{2}dz\right)^{\frac{1}{2}}$

for a positive constant $C$ independent of $u$.

Finally, by combining inequalities (3.29), (3.31) and (3.32), we obtain the
desired inequality (3.28). ∎

##  4\. Sharpness and applications of Theorem 1.2

###  4.1. Sharpness of Theorem 1.2

In this section, we discuss the sharpness of Theorem 1.2. Let
${\Omega}={\Omega}^{i}_{t^{s}}\subset{\mathbb{R}}^{2}$ be a polynomial inward
cuspidal domain with the degree $1<s<\infty$ such that there exists a
homeomorphism of finite distortion
$h:{\mathbb{R}}^{2}\xrightarrow[]{{}_{\\!\\!\textnormal{onto\,\,}\\!\\!}}{\mathbb{R}}^{2}$
with $h({\Omega})={\mathbb{D}}$, and the distortion function satisfies the
integral condition (3.1). It is easy to see
${\mathbb{R}}^{2}\setminus\overline{{\Omega}^{i}_{t^{s}}}$ has a same Sobolev
extension property with the polynomial outward cuspidal domain
${\Omega}^{o}_{t^{s}}$. By combining Proposition 2.2 and Proposition 2.3, we
obtain that we cannot obtain a better Sobolev extension result than ${\Omega}$
is a Sobolev $\left(\frac{2p}{p-1},\frac{2q}{q+1}\right)$-extension domain and
${\mathbb{R}}^{2}\setminus\overline{{\Omega}^{i}_{t^{s}}}$ is a Sobolev
$\left(\frac{2q}{q-1},\frac{2p}{p+1}\right)$-extension domain. It shows the
sharpness of Theorem 1.2.

###  4.2. A new proof of the necessity of Proposition 2.3

Actually, by combing Theorem 1.2 and Proposition 2.2, we give a new proof to
the necessary part of Proposition 2.3, which is entirely different with the
proof in paper [28] where we gave an argument based on some geometrical facts.

###  4.3. An application of Theorem 1.2

In this section, we prove Theorem 1.3, which can be regarded as an application
of Theorem 1.2.

###### Proof of Theorem 1.3.

By [22, Theorem 1.6], there exists a positive constant $0<\beta<2$ such that
for every homeomorphism of finite distortion
$h:{\mathbb{R}}^{2}\xrightarrow[]{{}_{\\!\\!\textnormal{onto\,\,}\\!\\!}}{\mathbb{R}}^{2}$
with $\exp\left(\lambda K_{h}\right)\in L^{1}_{\rm loc}({\mathbb{R}}^{2})$
with $\lambda>0$, we have $K_{h^{-1}}\in L^{p_{\lambda}}_{\rm
loc}({\mathbb{R}}^{2})$ with $p_{\lambda}=\beta\lambda$. Let
$h:{\mathbb{R}}^{2}\xrightarrow[]{{}_{\\!\\!\textnormal{onto\,\,}\\!\\!}}{\mathbb{R}}^{2}$
be a homeomorphism of finite distortion with $\exp\left(\lambda
K_{h}\right)\in L^{1}_{\rm loc}({\mathbb{R}}^{2})$ for some large enough
$\lambda>0$ with $p_{\lambda}=\beta\lambda>1$. Then by Theorem 1.2,
${\Omega}:=h^{-1}({\mathbb{D}})$ is a $L^{p_{\lambda}}$-quasidisk and is a
Sobolev
$\left(\frac{2p_{\lambda}}{p_{\lambda}-1},\frac{2p_{\lambda}}{p_{\lambda}+1}\right)$-extension
domain. ∎

## References

  * [1] L. V. Ahlfors, Quasiconformal reflections, Acta Math. 109 (1963) 291–301.
  * [2] K. Astala, T. Iwaniec, G. Martin and J. Onninen, Extremal mappings of finite distortion, Proc. London Math. Soc. (3) 91 (2005), no. 3, 655-702.
  * [3] K. Astala, T. Iwaniec, and G. Martin, Elliptic partial differential equations and quasiconformal mappings in the plane, Princeton University Press, 2009.
  * [4] K. Borsuk, Theory of retracts, Monografie Matematyczne, Tom 44Państwowe Wydawnictwo Naukowe, Warsaw ,1967.
  * [5] D. Burago, Y. Burago and S. Ivanov, A course in metric geometry, Graduate Studies in Mathematics, 33. American Mathematical Society, Providence, RI, 2001. xiv+415 pp.
  * [6] V. I. Burenkov, On a method of extension of differentiable functions, Trudy Mat. Inst. Steklov 140 (1976), 27-67 (Russian); English transl. in Proc. Steklov Inst. Math., American Mathematical Society, Providence, Rhode Island 140 (1979, issue 1).
  * [7] V. I. Burenkov, Extension of functions preserving certain smoothness and compactness of embeddings for spaces of differentiable functions. Trudy Math. Inst. Steklov, 248 (2005), 74-85 (Russian). English transl. in Proc. Steklov Inst. Math., American Mathematical Society, Providence, Rhode Island 248 (2005).
  * [8] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics. CRC Press, Boca Raton (1992).
  * [9] B. L. Fain, Extension of functions from Sobolev spaces for irregular domains preserving the index of smoothness. (Russian) Dokl. Akad. Nauk SSSR 285 (1985), no. 2, 296–301.
  * [10] H. Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, 2nd edn, Band 153. Springer, New York (1996).
  * [11] F. W. Gehring, Univalent functions and the Schwarzian derivative. Comment. Math. Helv. 52 (1977) 561-572.
  * [12] F. W. Gehring, Characteristic properties of quasidisks, Séminaire de Mathématiques Supérieures [Seminar on Higher Mathematics], 84. Presses de l’Université de Montréal, Montreal, Que., (1982).
  * [13] F. W. Gehring, Uniform domains and the ubiquitous quasidisk. Jber. Deutsche Math. Verein. 89 (1987) 88-103.
  * [14] F. W. Gehring and K. Hag, Remorks on uniform and quasiconformal extension domains. Complex Var. Elliptic Equ. 9 (1987) 175-188.
  * [15] F. W. Gehring and K. Hag, Reflections on reflections in quasidisks, Papers on analysis, 81-90, Rep. Univ. Jyväskylä Dep. Math. Stat., 83, Univ. Jyväskylä, Jyväskylä, 2001.
  * [16] F. W. Gehring and K. Hag, The ubiquitous quasidisk. With contributions by Ole Jacob Broch, Mathematical Surveys and Monographs, 184. American Mathematical Society, Providence, RI, 2012.
  * [17] F. W. Gehring and O. Lehto, On the total differentiability of functions of a complex variable, Ann. Acad. Sci. Fenn. Ser. A I No. 272 1959 9.
  * [18] L. Greco, T. Iwaniec and G. Moscariello, Limits of the Improved Integrability of the Volume Forms, Indiana Univ. Math. J. 44 (1995), no. 2, 305–339.
  * [19] V. M. Gol’dshtein and L. G. Gurov, Applications of change of variables operators for exact embedding theorems, Integral Equations Operator Theory 19 (1994), no. 1, 1-24.
  * [20] V. M. Gol’dshtein and V. N. Sitnikov, Continuation of functions of the class $W^{1}_{p}$ across Hölder boundaries. (Russian) Imbedding theorems and their applications, 31-43, Trudy Sem. S. L. Soboleva, No. 1, 1982, Akad. Nauk SSSR Sibirsk. Otdel., Inst. Mat., Novosibirsk, 1982.
  * [21] C. Guo, P. Koakela and J. Takkinen, Generalized quasidisks and conformality. Publ. Mat. 58 (2014), no. 1, 193–212.
  * [22] S. Hencl and P. Koskela,Regularity of the inverse of a planar Sobolev homeomorphism, Arch. Ration. Mech. Anal. 180 (2006), no. 1, 75–95.
  * [23] S. Hencl and P. Koskela, Lectures on mappings of finite distortion, Lecture Notes in Mathematics, 2096. Springer, Cham, (2014).
  * [24] D. A. Herron and P. Koskela, Uniform, Sobolev extension and quasiconformal circle domains. J. Anal. Math. 57 (1991), 172–202.
  * [25] T. Iwaniec and G. Martin, Geometric Function Theory and Non-linear Analysis, Oxford Mathematical Monographs, Oxford University Press, 2001.
  * [26] T. Iwaniec and J. Onninen, Deformations of finite conformal energy: Boundary behavior and limit theorems, Trans. Amer. Math. Soc. 363 (2011), no. 11, 5605–5648.
  * [27] T. Iwaniec and J. Onninen, Deformations of finite conformal energy: existence and removability of singularities, Proc. Lond. Math. Soc. (3) 100 (2010), no. 1, 1–23.
  * [28] T. Iwaniec, J. Onninen and Z. Zhu, Singularities in $\mathscr{L}^{p}$-quasidisks, arXiv:1909.01573.
  * [29] T. Iwaniec, J. Onninen and Z. Zhu, Bi-Sobolev boundary singularities, submitted.
  * [30] P. Jones, Quasiconformal mappings and extendability of functions in Sobolev spaces, Acta Math. 147 (1981), no. 1-2, 71–88.
  * [31] N. D. Kazarinoff, Geometric inequalities, New Mathematical Library, 4 Random House, New York-Toronto 1961.
  * [32] P. Koskela, P. Pankka and Y. Zhang, Ahlfors reflection theorem for $p$-morphisms. arXiv:1912.09200.
  * [33] P. Koskela and J. Takkinen, Mappings of finite distortion: formation of cusps, Publ. Mat. 51 (2007), no. 1, 223–242.
  * [34] P. Koskela and J. Takkinen, Mappings of finite distortion: formation of cusps. III., Acta Math. Sin. (Engl. Ser.) 26 (2010), no. 5, 817–824.
  * [35] P. Koskela and J. Takkinen, A note to ”Mappings of finite distortion: formation of cusps II”, Conform. Geom. Dyn. 14 (2010), 184–189.
  * [36] P. Koskela, A. Ukhlov and Z. Zhu, The volume of the boundary of Sobolev $(p,q)$-extension domains. Preprint.
  * [37] P. Koskela and Z. Zhu, Sobolev extension via reflections, arXiv:1812.09037.
  * [38] P. Koskela and Z. Zhu, Sobolev extension via reflections over Cantor-cusp graph. Preprint.
  * [39] A. Koski and J. Onninen, Sobolev homeomorphic extensions, arXiv:1812.02085.
  * [40] O. Martio and J. Sarvas, Injectivity theorems in plane and space. Ann. Acad. Sci. Fenn. 4 (1978-1979) 383-401.
  * [41] V. G. Maz’ya, Sobolev spaces. Translated from the Russian by T. O. Shaposhnikova. Springer Series in Soviet Mathematics. Springer-Verlag, Berlin, 1985.
  * [42] V. G. Maz’ya, Sobolev spaces with applications to elliptic partial differential equations. Second, revised and augmented edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 342. Springer, Heidelberg, 2011.
  * [43] V. G. Maz’ya and S. V. Poborchi, On extension of functions in Sobolev classes to the exterior of a domain with the vertex of a peak on the boundary, Dokl. Akad. Nauk SSSR 275:5 (1984) 1066-1069 (Russian). English translation: Soviet Math. 29 (1984) 361-364.
  * [44] V. G. Maz’ya and S. V. Poborchi, Extension of functions in Sobolev classes to the exterior of a domain with the vertex of a peak on the boundary I, Czech. Math. Journ. 36:111 (1986) 634-661 (Russian).
  * [45] V. G. Maz’ya and S. V. Poborchi, Extension of functions in Sobolev classes to the exterior of a domain with the vertex of a peak on the boundary II, Czech. Math. Journ. 37:112 (1987) 128-150 (Russian).
  * [46] V. G. Maz’ya and S. V. Poborchi, Differentiable functions on bad domains, World Scientific Publishing Co., Inc., River Edge, NJ, 1997.
  * [47] J. Takkinen, Mappings of finite distortion: formation of cusps. II. Conform. Geom. Dyn. 11 (2007), 207-218.
  * [48] A. D. Ukhlov, On mappings that generate embeddings of Sobolev spaces, (Russian) Sibirsk. Mat. Zh. 34(1993), no.1, 185-192; translation in Siberian Math. J. 34 (1993), no.1, 165-171.
  * [49] W. P. Ziemer,Weakly differentiable functions. Sobolev spaces and functions of bounded variation, Graduate Texts in Mathematics, 120. Springer-Verlag, New York, 1989.