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# Applying Deep-Learning-Based Computer Vision to Wireless Communications:
Methodologies, Opportunities, and Challenges ††thanks: Manuscript received on
August 3, 2020; revised on November 1, 2020 and November 30; accepted on
December 1, 2020 by IEEE Open Journal of the Communications Society. The
associate editor coordinating the review of this paper and approving it for
publication was Dr. Gayan Aruma Baduge. (Corresponding author: Gaofeng Pan.)
††thanks: Authors are with Computer, Electrical and Mathematical Sciences and
Engineering Division, King Abdullah University of Science and Technology
(KAUST), Thuwal 23955-6900, Saudi Arabia.

Yu Tian, Gaofeng Pan, , and Mohamed-Slim Alouini

###### Abstract

Deep learning (DL) has seen great success in the computer vision (CV) field,
and related techniques have been used in security, healthcare, remote sensing,
and many other areas. As a parallel development, visual data has become
universal in daily life, easily generated by ubiquitous low-cost cameras.
Therefore, exploring DL-based CV may yield useful information about objects,
such as their number, locations, distribution, motion, etc. Intuitively, DL-
based CV can also facilitate and improve the designs of wireless
communications, especially in dynamic network scenarios. However, so far, such
work is rare in the literature. The primary purpose of this article, then, is
to introduce ideas about applying DL-based CV in wireless communications to
bring some novel degrees of freedom to both theoretical research and
engineering applications. To illustrate how DL-based CV can be applied in
wireless communications, an example of using a DL-based CV with a millimeter-
wave (mmWave) system is given to realize optimal mmWave multiple-input and
multiple-output (MIMO) beamforming in mobile scenarios. In this example, we
propose a framework to predict future beam indices from previously observed
beam indices and images of street views using ResNet, 3-dimensional ResNext,
and a long short-term memory network. The experimental results show that our
frameworks achieve much higher accuracy than the baseline method, and that
visual data can significantly improve the performance of the MIMO beamforming
system. Finally, we discuss the opportunities and challenges of applying DL-
based CV in wireless communications.

###### Index Terms:

Computer vision, deep learning, multiple-input and multiple-output,
beamforming, beam tracking, long short-term memory, wireless communications

##  I Introduction

Recently, deep learning (DL) has seen great success in the computer vision
(CV) field. DL networks comprise networks such as deep neural networks, deep
belief networks, recurrent neural networks (RNNs), and convolutional neural
networks (CNNs). Many DL networks with various structures have emerged with
the availability of large image and video datasets and high-speed graphic
processing units (GPUs) [1]. DL networks can achieve success in CV because
they discover and integrate low-/middle-/high-level features in images and
leverage them to accomplish specific tasks [2]. DL can easily fulfill CV
applications with remarkably high performance, such as semantic segmentation,
image classification, and object detection/recognition [1]. DL-based CV has
therefore been widely utilized in public security, healthcare, and remote
sensing, as such fields generate much visual data [3]. However, DL-based CV is
rarely seen in the design and optimization of wireless communication systems
in which the researchers mainly focus on the transmission quality of the
information bits/packets, e.g., transmission rate, bit/packet error,
traffic/user fairness, etc. via purely exploiting the information on the
transmission behaviors of radio frequency signals (e.g., the power, direction,
phase, transmission duration, etc.), rather than making use of the geometry
information of the surrounding space. Thus, such presented design and
optimization of wireless communications cannot achieve the optimal performance
with no doubts.

Nowadays, high-definition cameras are installed almost everywhere because of
their low cost and small size. In some public areas, cameras have long existed
for monitoring purposes. Therefore, visual data can easily be obtained in
wireless communication systems in real life [4]. As useful information about
static system topology (including terminals’ numbers, positions, distances
among themselves, etc.) and dynamic system information (including moving
speed, direction, and changes in the number of the terminals) can be
recognized, estimated, and extracted from these multi-medium data via DL-based
CV techniques, new potential benefits can be exploited for wireless
communications to aid system design/optimization, such as resource scheduling
and allocations, algorithm design, and more.

Figure 1: Framework of applying DL-based CV to wireless communications (PHY:
Physical layer; MAC: MAC layer; Net.: Network layer; AI: artificial
intelligence)

Fig. 1 presents the framework of applying DL-based CV to wireless
communications, the core idea of which is to explore the useful information
obtained/forecasted by DL-based CV techniques to facilitate the design of
wireless communications via DL-based/traditional optimization methods. In the
following, we introduce some applications of DL-based CV in wireless systems
in three aspects: the physical layer, medium access control (MAC) layer, and
network layer.

1) In the physical layer of wireless communication systems, traditional
methods usually first estimate the channel state by sending pilot signals from
the transmitter to receivers [5]. Then according to the achieved channel state
information (CSI), specific modulation, source encoding, channel encoding, and
power control strategies can be selected to realize the optimal utilization of
system resources (e.g., bandwidth and energy budgets). However, the CSI only
contains amplitudes and phases information of the channel fading rather than
the locations, number, and environmental information of the users which can be
easily obtained from visual data by object detection and segmentation
techniques in CV, leading to the fact that the real optimal system performance
cannot be realized. However, with the aid of the more comprehensive users’
information, dynamic modulation, encoding, and power control can be easily and
optimally formulated and implemented. For example, in multiple-input and
multiple-output (MIMO) beamforming communication systems, the direction and
power of beams can be scheduled using the knowledge of users’ locations and
blocking cases in the visual data, which cannot be obtained via traditional
methods.

2) In the MAC layer, like in cellular wireless networks, receiver-to-
transmitter feedback information and cell-to-cell CSI is very important
information to be utilized to allocate resources and to guarantee the quality
of service in the traditional methods [6]. Thus, a long time delay may always
exist when analyzing the feedback and CSI in crowded scenarios in which a huge
number of users are served by the network. By jointly using these information
and the density or distribution of users obtained from the visual data in the
serving area of the BS, channel resources (including frequency bands, time
slots, etc.) can be efficiently reserved and allocated to achieve optimal
overall performance. For example, smart homes have various kinds of terminals
such as smartphones, televisions, laptops, and other intelligent home
appliances. As such, channel resources can be dynamically scheduled by
considering the information obtained from the visual data, such as the number
and locations of the users. Unlike traditional handover algorithms that adopt
the measured fluctuation of received signal power to estimate the distance
between the terminal and BS, the moving information including velocity and its
variations can be fully estimated from visual data to accurately facilitate
channel resource allocation in the handover process. This will be quite useful
in fifth-generation wireless networks due to the shrinking sizes of the
serving zones.

3) For the network layer, taking multi-hop transmission scenarios as an
example, traditional routing algorithms are mostly running based on the length
of the routing path estimated by the pilot and feedback signals which cannot
reflect the timely location changes in mobile scenarios [7]. By exploiting
system topology information from the visual data, novel routing algorithms can
be designed to efficiently improve transmission performance, such as the end-
to-end delivery delay, packet loss rate, jam rate, and system throughput. For
another instance, wireless sensor networks have numerous sensors that can be
deployed in target areas to monitor, gather, and transmit information about
their surrounding environments. Then, the system topology information from
visual data can be used to design multi-hop transmissions, which are required
due to the inherent resource limitations and hardware constraints of the
sensors.

In general, traditional algorithms adopted in wireless communication systems
depend on traditional channel/network state estimation methods to grab the CSI
and network state information which unavoidably suffer time delay and/or feed
errors, resulting in low efficiency or even wrong decisions. Especially, it is
hard or impossible to get accurate CSI or network state information in high
dynamic network scenarios through traditional methods. Thanks to the inherent
merits of DL-based CV techniques, the static and dynamic system information
can be accurately and efficiently extracted from visual data, bringing vital
benefits to the design and optimization of wireless communication systems.

In this context, this article introduces the methodologies, opportunities, and
challenges of applying DL-based CV in wireless communications as an essential
reference/guide for theoretical research and engineering applications.

The rest of this article is organized as follows. Section II overviews related
work from two perspectives: datasets and applications. Section III presents an
example of applying a DL-based CV to mmWave MIMO beamforming and elaborates on
the problem definition, framework architecture, pipeline, the results of the
example, and practical application. Section IV introduces and discusses some
challenges and open problems of applying DL-based CV to wireless
communications. Finally, Section V concludes the article.

##  II An Overview of Related Work

Applying DL-based CV to wireless communications has two essential dimensions:
datasets and applications. In the following, we give a brief overview of
recent work in these two aspects.

##### Datasets

Building datasets is an essential step as DL is data-hungry. In [8], the
authors proposed a parametric, systematic, scalable dataset framework called
Vision-Wireless (ViWi). They utilized this framework to build the first-
version dataset containing four scenarios with different camera distributions
(co-located and distributed) and views (blocked and direct). These scenarios
were based on a millimeter wave (mmWave) MIMO wireless communication system.
Each scenario contained a set of images captured by the cameras and raw
wireless data (signal departure/arrival angles, path gains, and channel
impulse responses). Using the provided MATLAB script, they could view the
user’s location and channel information in each image from the raw wireless
data. Later, the same authors built the second-version dataset called ViWi
Vision-Aided Millimeter-Wave Beam Tracking (ViWi-BT) [9] and posted it for the
machine learning competition at the IEEE International Conference on
Communications (ICC) 2020. This dataset contains images captured by the co-
located cameras and mmWave MIMO beam indices under a predefined codebook.
Section III-D1 covers the details of this dataset. The authors of [10]
introduced another dataset called Raymobtime which contains ray-tracing,
LIDAR, matrix channel, GPS, and image data in mmWave MIMO vehicle-to-
infrastructure wireless communication systems. Notably, ray-tracing data
provides path parameters such as received power, time of arrival, angle of
departure, angle of arrival, line of a sight ray status, and ray phase while
GPS user info data has line-of-sight (LOS) status, channel valid or not
information, number of the TX in the vehicle, and the 3D coordinates. Beam
selection and channel estimation challenges are held based on this dataset in
International Telecommunication Union (ITU) Artificial Intelligence/Machine
Learning in 5G Challenge. As both ViWi and Raymobtime datasets are versatile,
except the challenges in ICC and ITU, many other interesting applications
(e.g., blockage prediction, power prediction, angle estimation, LOS decision,
etc.) can be explored according to their abundant data. In [11], a dataset
consisting of depth image frames from recorded videos was built and can be
applied in channel estimation tasks.

##### Applications

There are plenty of interesting applications of DL-based CV techniques
designed to tackle problems in wireless communications. A framework to
implement beam selection in mmWave communication systems by leveraging
environmental information was presented by [4]. The authors used the images
with different perspectives captured by one camera to construct a three-
dimensional (3D) scene and generate corresponding point cloud data. They built
a model based on 3D CNN to learn the wireless channel from the point cloud
data and predict the optimal beam. Based on the first-version ViWi dataset,
[12] proposed a modified ResNet18 model to conduct beam and blockage
prediction from the images and channel information. Based on the second-
version ViWi-BT dataset, the authors of [9] provided a baseline method based
on Gated Recurrent Units (GRUs) without the images, only the beam indices.
They believe they can achieve better performance if they leverage both kinds
of data. Based on the Raymobtime dataset, CNN and deep reinforcement learning
(DRL) were utilized to select proper pair of beams for vehicles with images
generated from GPS locations data in vehicle-to-infrastructure scenarios in
[10]. The authors also compared DL-based methods with other traditional
machine learning methods such as SVM, AdaBoost, decision tree, and random
forest. The results showed that DL-based method has the best performance. In
[13], two CNNs were proposed to conduct line-of-sight decision and beam
selection by using LIDAR point cloud data in the Raymobtime dataset. In [14],
authors proposed a neural network containing CNNs and an RNN-based recurrent
prediction network to predict the dynamic link blockages using red, green,
blue (RGB) images and beamforming vectors provided by the extended ViWi-BT
dataset. In [11], authors developed a CNN-based framework called VVD to
estimate the wireless communication channels only from only depth images in
mmWave systems. In [15], a framework consisting of CNN and convolutional LSTM
(convLSTM) network was presented to proactively predict the received power
through depth images in mmWave networks and exhibited the highest accuracy
compared with the random forest algorithm and a CNN-based method. In [16], a
proactive handover management framework was proposed to make handover
decisions by using camera images and DRL. In [17], a multimodal split learning
method based on convLSTM networks was presented to predict mmWave received
power through camera images and radio frequency signals while considering
communication efficiency and privacy protection. All aforementioned papers are
summarized in Table I for comparison purposes.

TABLE I: Paper review

Paper | Task | Network | Data type |  Dataset public available?
---|---|---|---|---
[4] | Beam selection | 3D CNN | Point clouds generated from camera images |  No
[12] | Beam and blockage prediction | Modified ResNet18 | Camera images |  Yes
[9] | Beam prediction | GRU | Previous beams |  Yes
[10] | Beam selection | CNN and DRL | Images generated from vehicle locations |  Yes
[13] | Beam selection | CNN | LIDAR point clouds |  Yes
[14] | Blockage prediction | CNN and RNN | Camera images and previous beams |  Yes
[11] | Channel estimation | CNN | Depth images |  Yes
[15] | Received power prediction | CNN and convLSTM network | Depth images |  No
[16] | Handover decision | DRL | Camera images |  No
[17] | Received power prediction | Multimodal split learning and convLSTM networks | Camera images and radio frequency signals |  No

Figure 2: Scenario of applying DL-based CV to mmWave MIMO beamforming

##  III An Example of Applying DL-based CV to Beamforming

Figure 3: Architecture of our proposed framework

###  III-A Problem Definition

MmWave communication is a promising technique in the fifth-generation
communication system, thanks to its broad available bandwidth and ultra-high
data-transmitting rate [12, 8, 9]. MIMO and beamforming are widely used in
mmWave communication systems and should be implemented in a large antenna
array to achieve the required high power gain and direction. The classic
beamforming and beam tracking algorithms suffer a common disadvantage:
complexity increases dramatically with the number of antennas, resulting in
substantial computational overhead. DL-based CV is a promising candidate to
address this overhead issue.

In this section, we give an example of applying the DL-based CV to mmWave MIMO
beamforming. As defined in [9], the considered scenario contains 2 BSs located
on the opposite sides of a street with a distance of 60 meters. As shown in
Fig. 2, each BS forms a MIMO beam to serve a target user moving along the
street. Therefore, the beam direction must be dynamically adjusted to catch
the target mobile user. The target user may be blocked at some moments, such
as $t_{8}$ in Fig. 2, and then the beam cannot directly reach the target user,
while proper reflection from other objects, such as buildings and vehicles,
must be designed. Meanwhile, three cameras installed at the BS capture RGB
images of the whole street view to assist the beamforming process. The problem
here is how to utilize the eight pairs of previously-observed consecutive
beams and corresponding images to predict the future one, three, and five
beams through DL model. Notably, these beams are represented as beam indices
under the same predefined codebook.

A sequence containing the eight pairs of previously-observed images and
corresponding beam indices for the $u$th user at the time instance $t$ is
given as

$\displaystyle S_{u}[t]=\\{(X_{u}[i],b_{u}[i])\\}_{i=t-7}^{t},$ (1)

where $X_{u}[i]$ is the RGB image taken at the $i$th time instance and
$b_{u}[i]$ is the corresponding beam index.

Let $f_{\Theta}(S_{u}[t])$ be a prediction function of a DL model and
$\hat{b}_{u}[t+n]$ ($n=1,...,5$) be the predicted beam index at the time
instance $t+n$. $f_{\Theta}(S_{u}[t])$ takes in the sequence $S_{u}[t]$ and
outputs a predicted sequence $\\{\hat{b}_{u}[t+n]\\}_{n=1}^{5}$. $\Theta$ is a
set of parameters of the DL model which is obtained by training the model with
the training set. The training set consists of labelled sequences, i.e.,
$\mathcal{D}=\\{(S_{u}[t],\\{g_{u}[t+n]\\}_{n=1}^{5})\\}_{u=1}^{U}$ where each
pair consists of an observed sequence and five groundtruth future beam
indices.

Equivalent to the defined problem, our goal is to get the prediction function
which can maximize the joint success probability of all data samples in
$\mathcal{D}$. The object function is expressed as

$\displaystyle\max_{f_{\Theta}(S_{u}[t])}\quad\prod_{u=1}^{U}\prod_{n=1}^{5}{\bf{Pr}}\left\\{\left.\hat{b}_{u}[t+n]=g_{u}[t+n]\right|S_{u}[t]\right\\},$
(2)

where each success probability only relies on its observed sequence
$S_{u}[t]$.

###  III-B Framework Architecture and Methods

We propose a DL network framework shown in Fig. 3 composed of ResNet[2], 3D
ResNext[18], a feature-fusion module (FFM)[19], and a predictive network.

####  III-B1 ResNet, ResNext and 3D ResNext

Figure 4: Structure of residual block, ResNet, and ResNext block

ResNet consists of several residual blocks, as presented in Fig. 4. Each block
contains two or more convolutional layers and superimposes its input to its
output through identity mapping. It can efficiently address the vanishing
gradient issue caused by the rising number of convolutional layers. If a
specific number of such blocks are concatenated, as depicted in Fig. 4, ResNet
is available to achieve as many as 152 layers.

Fig. 4 also presents the structure of the ResNext block[20], an improved
version of the residual block, that adds a ‘next’ dimension, also called
‘Cardinality’. It sums the outputs of $K$ parallel convolutional layer paths
that share the same topology and inherits the residual structure of the
combination. As $K$ diversities are achieved by $K$ paths, this block can
focus on more than one specific feature representations of the images.

In 3D ResNext, a similar structure can be observed but with 3D convolutional
layers instead of two-dimensional (2D) ones. The 3D convolutional layer is
designed to capture spatiotemporal 3D features from raw videos.

ResNet and 3D ResNext have been widely used as feature extractors for their
powerful feature-representation abilities. If they are used in a DL network
directly, however, the training time will become extremely long, and many
computational resources will be occupied due to the large number of layers.
Therefore, researchers commonly apply a pre-trained ResNet on the ImageNet
dataset to extract visual features from images and a 3D ResNext on the
Kinetics dataset to extract spatiotemporal features from videos [21]. These
features are then fed to the DL network as inputs.

####  III-B2  Long Short-Term Memory (LSTM) Network

Figure 5:  Structure of LSTM cell, method with 1D LSTM network, and method
with 2D LSTM network

The LSTM[22] network is designed for the tasks that contain time-series data,
such as prediction, speech recognition, text generation, etc. Hence, it is a
suitable candidate for our predictive network. The network comprises several
LSTM cells, as depicted in Fig. 5. Event (current state), previous long-term
memory (cell state), and previous short-term memory (hidden state) are the
inputs of an LSTM cell, in which learn, forget, remember, and output gates are
employed to explore the information from the inputs. The LSTM cell outputs new
long-term memory and short-term memory in which the latter is also regarded as
a prediction.

When an LSTM cell is recursively utilized in a 1D array form, a 1D LSTM
network is obtained, as presented in Fig. 5. At each moment, the cell and
hidden states of the previous moment are used to generates the outputs of the
current moment.

As shown in Fig. 5, a 2D LSTM network can be realized when the LSTM cell is
recursively in a 2D mesh form [23]. Each LSTM cell utilizes the hidden and
cell states from the two neighboring cells in the left and below positions in
the mesh, and its states are delivered to its neighboring cells in the right
and top positions. Obviously, the number of predictions is equal to the number
of rows.

####  III-B3 Feature-Fusion Module (FFM)

Fig. 3 shows the structure of the FFM which comprises two LSTM networks and a
cross-gating block. The features from ResNet and 3D ResNext are aggregated by
the LSTM networks and then high-level features are obtained. The cross-gating
block can make full use of the related semantic information between these two
kinds of features by multiplication and summation operations. Thus, the merged
features can be obtained through a linear transformation.

###  III-C Pipeline of our Framework

In the pipeline of the considered DL network, eight consecutive images are
inputted and utilized. As each is equivalent to a video clip, they contain
motion information, which is helpful for the beam prediction. Combined with
the visual information from each image, location, motion, and blockage
information can be extracted from these RGB images. The pre-trained 3D ResNext
with 101 layers (3D ResNext101) is adopted to extract motion features and the
pre-trained ResNet with 152 layers (ResNet152) is used to extract visual
features. These features are then merged as a vector through FFM and sent to
the predictive network.

As depicted in Fig. 5, there are three kinds of inputs in the predictive
network, namely, initial state, embedded beam vectors, and merged feature
vector. The initial state is set as a vector of all zeros. Embedding is
mapping a constant (beam index) to a vector and can represent the relation
between constants well. So the embedded vector is utilized to represent the
beam index. In each LSTM cell, the embedded beam vector and the merged feature
vector are firstly transformed to the same shape and then sum them up as the
‘event’. According to the event, short and long term memories are obtained
from the previous LSTM cell, and each cell predicts future 1 output vector
whose index of the maximum element is the beam index. Notably, all the LSTM
cells share the same merged features.

Based on the 1D and 2D LSTM networks introduced in Section III-B2, three
methods are proposed and explained below.

TABLE II: Performance of Top-1 Accuracy and Exponential Decay Scores

Top-1 Accuracy Exponential Decay Score Running Time (s) for ‘1 future beam’
prediction 1 future beam 3 future beams 5 future beams 1 future beam 3 future
beams 5 future beams Method with 1D LSTM Network 0.9170 0.7719 0.6448 0.9238
0.8206 0.7356 0.42+0.016 Method with Modified 1D LSTM Network 0.8887 0.6260
0.4800 0.8974 0.7137 0.6129 0.42+0.024 Method with 2D LSTM Network 0.8704
0.5893 0.4503 0.8803 0.6857 0.5877 0.42+0.064 Baseline Method in [9] 0.85 0.60
0.50 0.86 0.68 0.60 0.0016

Figure 6: Training procedure of method with 1D LSTM netowrk

####  III-C1 Method with 1D LSTM Network

When the predictive network in Fig. 3 is a 1D LSTM network, the first method
is obtained, as presented in Fig. 5. The LSTM cell is recursive 12 times. The
cell at the $k$th moment is denoted as the ‘$k$th LSTM cell’.

As shown in Fig. 6, during the training process, the pipeline of our first
method is the following:

  1. Step 1:

Eight consecutive images are fed to the pre-trained ResNet152 and 3D
ResNext101 and then visual features and motion features are obtained;

  2. Step 2:

These features from step 1 are merged through the FFM;

  3. Step 3:

The output vector from the FFM is fed to each LSTM cell as an input;

  4. Step 4:

The embedded vectors of the first 12 beam indices go through the first to the
last LSTM cells to update the hidden states and generate 12 output vectors;

  5. Step 5:

The 12 output vectors are used to calculate the training loss with the ground
truth and train the network.

Figure 7: Testing procedure of method with 1D LSTM netowrk

During the testing process, as we only have the first eight beam indices and
images, the fourth step above is not applicable and is separated into two sub-
steps as depicted in Fig. 7:

  1. Substep 4.1:

The embedded vectors of the first eight beam indices go through the first to
seventh LSTM cells and update the hidden states;

  2. Substep 4.2:

The eighth to twelfth LSTM cells are used to predict the future beam indices
which are obtained by acquiring the indices of the maximum element in these
output vectors. Each cell is fed with the hidden state and the embedded beam
index from the prediction of the previous LSTM.

The fifth step is skipped during testing.

####  III-C2 Method with Modified 1D LSTM Network

In our first method, the training and testing procedures are different.
Actually, the first method essentially aims to predict the next beam index as
we utilize all the first 12 beam indices as inputs during the training. During
the testing process, among the eighth to twelfth predicting beam indices, the
previous one’s correct prediction is important for the next prediction. To
make training and testing processes consistent, we designed a modified version
of the first method, in which the output vector of each of the last five LSTM
cells undergoes a linear transformation module and is fed to the next cell as
the embedded input. In this way, only the first eight beam indices are used as
inputs, and the training and testing can be the same.

####  III-C3 Method with 2D LSTM Network

When we apply the 2D LSTM network to the predictive network, the third method
can be realized as shown in Fig. 5. In this method, we need to input the
initial state, the merged feature vector and embedded vectors of the first
eight beam indices into the LSTM network, and then get five outputs vectors
directly. The training process is the same as the testing one.

###  III-D Experiment

In this section, we evaluate our three proposed methods on the ViWi-BT
dataset. All the experiments are conducted in the framework of PyTorch on one
NVIDIA V100 GPU.

####  III-D1 Dataset

The VIWI-BT dataset contains a training set with 281,100 samples, a validation
set with 120,468 samples, and a test set with 10,000 samples. There are 13
pairs of consecutive beam indices and corresponding images of street views in
each sample of the training and validation sets. Furthermore, the first eight
pairs are the observed beams for the target user and the sequence of the
images where the target user appears. The last five pairs are groundtruth data
containing the future beams and images of the same user. In this experiment,
the first eight pairs serve as the inputs of the designed DL network to
generate the predicted future five beam indices which are compared with the
groundtruth ones.

####  III-D2 Implementation Details

We first use the pre-trained ResNet152 and 3D ResNext101 to extract
2048-dimensional visual and 8192-dimensional motion features from the first
eight images of each sample. The merged features are embedded as a
463-dimensional vector and fed to the predictive LSTM network. There are a
512-dimensional hidden size and a 129-dimensional output vector in each LSTM
cell. The training pipeline mentioned in Section III-C is then implemented to
train the proposed network.

During the training, the designed DL network is optimized by the Adam
optimizer. The learning rate is set as $4\times 10^{-4}$ at first and reduced
by half every eight epochs. The batch size is set as 256. The cross-entropy
loss is utilized for the loss function.

####  III-D3 Performance

Following the evaluation in [9], the performances of our proposed methods are
evaluated on the validation set with the same metrics, which are the top-1
accuracy and exponential decay score.

As defined in [9], the top-1 accuracy of $n$ future beams is expressed as

$\displaystyle
Acc^{(n)}_{\rm{top\text{-}1}}=\frac{1}{M}\sum\limits_{i=1}^{M}\mathbbm{1}\\{\mathbf{\hat{g}}^{(n)}_{i}=\mathbf{g}^{(n)}_{i}\\},$
(3)

where $M$ is the number of the samples in the validation set,
$\mathbbm{1}\\{\cdot\\}$ denotes the indicator function, and
$\mathbf{\hat{g}}^{(n)}_{i}$ and $\mathbf{g}^{(n)}_{i}$ represent the
predictive and groundtruth beam indices vectors of the $i$th sample with the
length $n$.

The exponential decay score of $n$ future beams is given as

$\displaystyle{\rm{score}}^{(n)}=\frac{1}{M}\sum\limits_{i=1}^{M}\exp{\left(-\frac{||\mathbf{\hat{g}}^{(n)}_{i}-\mathbf{g}^{(n)}_{i}||_{1}}{n\sigma}\right)},$
(4)

where $\sigma=0.5$ is a penalization parameter.

Table II lists our results, in which the baseline method in [9] is considered
for comparison purposes. In the baseline method, the authors simply leveraged
the beam-index data and ignored image data.

From the top-1 accuracy, we can see that our proposed method with the 1D LSTM
network outperforms the baseline method in [9]. The method with a modified 1D
LSTM network is better than the baseline method on ’1 future beam’ and ’3
future beams’. The method with only the 2D LSTM network performs better than
the baseline method on ’1 future beam’.

For the exponential decay scores, the designed methods with the 1D LSTM
network and modified 1D LSTM network absolutely outperformed the baseline
method. The method with the 2D LSTM network is better than the baseline on ‘1
future beam’ and ‘3 future beams’ but a little worse on ‘5 future beams’.

Our proposed methods outperform the baseline method on predicting ‘1 future
beam’. Because the location, blockage, and speed information of the target
user is extracted from the images and represented as motion and visual
features to assist the prediction, and advanced LSTM networks are leveraged as
the predictive networks. Among the three proposed methods, the method with the
1D LSTM network shows the best beam prediction performance in the target
mobile scenarios. Compared with this method, the other two exhibit extra
linear transformation modules or more LSTM cells in their predictive networks
and need to be trained by more data. Therefore, performance degradation
occurring on the predictions of ‘3 future beams’ and ‘5 future beams’ are
caused by the small size of the training dataset.

The computational complexity here is measured by the running time of ‘1 future
beam’ prediction which exhibits the best performance and is more potential to
be implemented in the practical wireless communication systems. The running
time of our methods consists of the execution time of feature extraction, FFM,
and predictive network. It takes 0.42 seconds for the pre-trained 3D
ResNext101 and ResNet152 to extract features from each set of eight images.
Our method with 2D LSTM network exhibits the longest average running time due
to its most complex structure shown in Fig. 5. The baseline method runs for
the shortest time as it utilizes simple GRUs as the predictive network and
abandons the image data. The method with 1D LSTM network shows the best
predictive performance but a moderate prediction time, 0.016 seconds.

The feature extraction takes a little long time which will cause latency
issues but can be mitigated by employing more efficient CNNs in future work.
From Table 1 of [24], we can see that the TSN [25] takes 15.5 ms and the ECO
[26] takes 17.4 ms on NVIDIA Tesla P100 GPU to extract 3D spatiotemporal
features from eight images. In practical application, only one new image is
captured at each time instance, and the previously extracted spatiotemporal
features can be merged with the new image to reduce the latency. Similarly,
for the 2D feature extraction, only the new image needs to be processed at
each time instance. Table 2 of [27] shows that the Darknet-19 can achieve a
speed of 171 frames per second (5.85 ms per image) to extract 2D features on
NVIDIA Titan X GPU. 2D and 3D features extractions can be conducted
simultaneously which means that the longer latter determines the whole feature
extraction time. By jointly using TSN and Darknet-19, the running time of the
method with 1D LSTM network can be reduced to less than 15.5+16=31.5 ms on
V100 GPU which is more powerful than P100 and Titan X. Besides, in our
experiments, we built the LSTM cell by ourselves. If we use the basic building
block of LSTM in PyTorch, the comparable running time of the predictive
network can be obtained with the baseline method. In engineering practice,
even shorter latency can be achieved by utilizing GPU computing in the whole
process and designing a more efficient framework (platform).

####  III-D4 Practical Application

In practical scenarios, RGB images and their corresponding beam indices can be
obtained by cameras installed on the BS and the classic beamforming algorithm.
After obtaining sufficient data for the training set, our proposed network
will be pre-trained on these data and then run in the processors of the BS for
beamforming. At the beginning of the serving time, the first eight beam
indices can be estimated by the classic beamforming algorithm. Then the eight
pairs of images and beam indices are sent to the processors for future beam
predictions. Notably, after the first eight beams, the following beams will be
predicted by using previously-obtained images and beam indices. These
predicted beam indices and their corresponding images can be added to the
training set to enlarge the dataset and improve the performance.

##  IV Challenges and Open Problems

Although the previous sections elaborated on leveraging CV to tackle the
mmWave beamforming problem, some challenges and open problems still exist in
the front way of applying DL-based CV technologies in wireless communications.

###  IV-A Building Datasets

As DL is immensely data-hungry, a large dataset can guarantee the successful
application of DL-based CV techniques to wireless communications. A qualified
dataset in CV usually includes more than 10,000s of samples. For example,
there are more than 14 million images in ImageNet, 60,000 images in Cifar-10,
and roughly 650,000 video clips in Kinetics. It takes much time, money, and
labor to generate such a huge amount of visual data. However, building a
qualified dataset, which should be comprehensive and exhibit a balanced
diversity of data, is still far from accomplished. These data should be able
to represent all possibilities in the corresponding problem, and the amounts
of different kinds of data can not make such a difference. Usually, a training
set, a validation set, and a test set comprise a dataset. These three sets
should be homogeneous and not overlapping, so randomly sampling them from a
shuffled data pool is a better way to obtain the three sets. These data should
be well-organized and easily manipulated. Therefore, the hardest work in DL is
to build a satisfactory dataset.

###  IV-B Selecting CV Techniques

Many state-of-the-art DL techniques have been proved efficient and powerful in
CV, such as reinforcement learning, encoder-decoder architecture, generative
adversarial networks (GANs), Transformer, graph convolutional networks (GCNs),
etc. Reinforcement learning has been widely applied in tackling optimization
problems in wireless communications[28]. GCNs can be leveraged to address
network-related issues[29], and encoder-decoder architecture is widely used in
semantic segmentation and sequence-to-sequence tasks. The GAN is an immensely
powerful CNN to learn the statistics of training data and has been widely used
to improve the performance of other DL networks in CV[1]. Transformer built on
the attention mechanisms is a kind of the encoder-decoder architecture that
can handle unordered sequences of data[30]. Much CV research has shown that if
these techniques are jointly applied to make full use of the visual data,
better results can be obtained[19, 21, 31].

Thus, a single proper CV technique or an adequate combination of several CV
techniques are required to handle a specific problem in wireless systems. In
the example given in Section III-B, we combined ResNet, 3D ResNext, and an
LSTM network to achieve the required performance. Finding proper, efficient CV
techniques thus remains an open problem.

###  IV-C Open Problems in Vision-Aided Wireless Communications

Figure 8: Pipeline of applying DL-based CV to cellular networks Figure 9: An
example of applying DL-based CV to cellular networks Figure 10: An example of
applying DL-based CV to vehicle-to-everything communications

As many kinds of cameras and LIDARs operate in real life, an enormous amount
of visual data can be obtained through them, for more accurate motion and
position information in the terminals that can be recognized, analyzed, and
extracted from these multimedia data, which can also be explored to design and
optimize wireless communications. Thus, some open problems in wireless
communication scenarios are introduced and discussed as follows:

(1) Cellular networks: As shown in Fig. 8, visual data obtained at the BS in a
cellular network may contain the locations, number, and motion information of
the terminals in the open area. This information can be used for the BS to
adjust its transmitting power and beam direction to save power consumption and
reduce interference. Fig. 9 presents a real-life example: the motion
information of the users in the coverage area of a BS can be utilized to
forecast the future positions of these terminals and judge whether/when a
terminal goes out or comes into its serving area. Then, transmit power and
beam can be accurately assigned for these users who still stay in the coverage
area, and channel resource allocation can be set up for the handover process
to improve the utilization efficiency of the system resource.

(2) Vehicle-to-everything communications: Visual data captured by one vehicle
can reveal its environments, such as traffic conditions, which can be used to
set up links with neighboring terminals, access points, and vehicles.
Therefore, traffic schedules and jam/accident alarms can be conducted for
improved road safety, traffic efficiency, and energy savings. As depicted in
Fig 10, the BS located on the side of the road utilizes the visual information
to estimate the distance of a vehicle terminal from it and adjust its transmit
power accordingly for power saving and interference deduction purposes.
Moreover, the images or videos captured by the cameras assembled on the
vehicles can detect the traffic jam or accident, and then forwards the
observed traffic information to the traffic control center for alarm and
future traffic schedule.

Figure 11: An example of applying DL-based CV to UAV-ground communications

(3) Unmanned aerial vehicle (UAV)-ground communications: When a UAV serves as
an aerial BS, visual data captured by the UAV can be used to identify the
locations and distribution of ground terminals, which can be utilized in power
allocation, route/trajectory planning, etc. Moreover, when a ground BS
communicates with several UAVs, visual data captured by the ground terminal
can be used to define the serving range, allocate the channels/power, and so
forth. Fig. 11 illustrates an example in which a group of UAVs communicates
with a set of ground terminals. The header UAV first takes an image of the
whole area and detects all the terminals. Then the serving area is divided
into several subareas. Each UAV serves one specific subarea and designs a
route schedule according to the location information of these ground terminals
obtained from the images.

(4) Smart cities: Visual data captured by satellites or airborne crafts can be
applied to recognize and analyze the user’s distribution and schedule power
budget/serving ranges to achieve optimal energy efficiency.

Figure 12: An example of applying DL-based CV to IRS system

(5) Intelligent reflecting surfaces (IRSs): Usually, implementing channel
estimation and achieving network state information at an IRS is impossible
because there is no comparable calculation capacity and no radio frequency
(RF) signal transmitting or receiving capabilities at the IRS. Fortunately,
DL-based CV can offer useful information to compensate for this gap. Thus, a
proper control matrix can be optimally designed to accurately reflect the
incident signals to the target destination by utilizing the visual data
captured by the camera installed on the IRS, which includes the locations,
distances, and the number of terminals shown in Fig. 12.

##  V Conclusion

This article mainly presented the methodologies, opportunities, and challenges
of applying DL-based CV to wireless communications. First, we discussed the
feasibility of applying a DL-based CV in physical, MAC, and network layers in
wireless communication systems. Second, we overviewed related datasets and
work. Third, we gave an example of applying a DL-based CV to a mmWave MIMO
beamforming system. In this example, previously observed images and beam
indices were leveraged to predict future beam indices using ResNet, 3D
ResNext, and an LSTM network. The experimental results show that visual data
can significantly improve the accuracy of beam prediction. Finally, challenges
and possible research directions were discussed. We hope this work stimulates
future research innovations and fruitful results.

## References

  * [1] I. Goodfellow, Y. Bengio, and A. Courville, _Deep Learning_. MIT press, 2016.
  * [2] K. He, X. Zhang, S. Ren, and J. Sun, “Deep residual learning for image recognition,” in _Pro. of the IEEE Conf. on Comput. Vision and Pattern Recognit._ , Las Vegas, NV, USA, Jun. 27-30, 2016, pp. 770–778.
  * [3] V. Bharti, B. Biswas, and K. K. Shukla, “Recent trends in nature inspired computation with applications to deep learning,” in _2020 10th Int. Conf. on Cloud Comput., Data Sci. & Eng. (Confluence)_. Noida, Uttar Pradesh, India: IEEE, Jan. 29-31, 2020, pp. 294–299.
  * [4] W. Xu, F. Gao, S. Jin, and A. Alkhateeb, “3D scene based beam selection for mmWave communications,” _arXiv preprint arXiv:1911.08409_ , 2019.
  * [5] S. Zhou and G. B. Giannakis, “Adaptive modulation for multiantenna transmissions with channel mean feedback,” _IEEE Trans. on Wireless Commun._ , vol. 3, no. 5, pp. 1626–1636, 2004.
  * [6] D. Gesbert, S. G. Kiani, A. Gjendemsjo, and G. E. Oien, “Adaptation, coordination, and distributed resource allocation in interference-limited wireless networks,” _Proc. of the IEEE_ , vol. 95, no. 12, pp. 2393–2409, 2007.
  * [7] C. Jung, K. Kim, J. Seo, B. N. Silva, and K. Han, “Topology configuration and multihop routing protocol for bluetooth low energy networks,” _IEEE Access_ , vol. 5, pp. 9587–9598, 2017.
  * [8] M. Alrabeiah, A. Hredzak, Z. Liu, and A. Alkhateeb, “ViWi: A deep learning dataset framework for vision-aided wireless communications,” _arXiv preprint arXiv:1911.06257_ , 2019.
  * [9] M. Alrabeiah, J. Booth, A. Hredzak, and A. Alkhateeb, “ViWi vision-aided mmWave beam tracking: Dataset, task, and baseline solutions,” _arXiv preprint arXiv:2002.02445_ , 2020.
  * [10] A. Klautau, P. Batista, N. González-Prelcic, Y. Wang, and R. W. Heath, “5G MIMO data for machine learning: Application to beam-selection using deep learning,” in _2018 Inf. Theory and Applications Workshop (ITA)_ , 2018, pp. 1–9.
  * [11] S. Ayvaşık, H. M. Gürsu, and W. Kellerer, “Veni Vidi Dixi: Reliable wireless communication with depth images,” in _Proc. of the 15th Int. Conf. on Emerg. Netw. Exp. and Technol._ , 2019, pp. 172–185.
  * [12] M. Alrabeiah, A. Hredzak, and A. Alkhateeb, “Millimeter wave base stations with cameras: Vision aided beam and blockage prediction,” _arXiv preprint arXiv:1911.06255_ , 2019.
  * [13] A. Klautau, N. González-Prelcic, and R. W. Heath, “LIDAR data for deep learning-based mmWave beam-selection,” _IEEE Wireless Commun. Lett._ , vol. 8, no. 3, pp. 909–912, 2019.
  * [14] G. Charan, M. Alrabeiah, and A. Alkhateeb, “Vision-aided dynamic blockage prediction for 6G wireless communication networks,” _arXiv preprint arXiv:2006.09902_ , 2020.
  * [15] T. Nishio, H. Okamoto, K. Nakashima, Y. Koda, K. Yamamoto, M. Morikura, Y. Asai, and R. Miyatake, “Proactive received power prediction using machine learning and depth images for mmWave networks,” _IEEE J. on Sel. Areas in Commun._ , vol. 37, no. 11, pp. 2413–2427, 2019\.
  * [16] Y. Koda, K. Nakashima, K. Yamamoto, T. Nishio, and M. Morikura, “Handover management for mmWave networks with proactive performance prediction using camera images and deep reinforcement learning,” _IEEE Trans. on Cogn. Commun. and Netw._ , vol. 6, no. 2, pp. 802–816, 2020.
  * [17] Y. Koda, J. Park, M. Bennis, K. Yamamoto, T. Nishio, M. Morikura, and K. Nakashima, “Communication-efficient multimodal split learning for mmWave received power prediction,” _IEEE Commun. Lett._ , vol. 24, no. 6, pp. 1284–1288, 2020.
  * [18] K. Hara, H. Kataoka, and Y. Satoh, “Can spatiotemporal 3D CNNs retrace the history of 2D CNNs and ImageNet?” in _Proc. of the IEEE Conf. on Comput. Vision and Pattern Recognit._ , Salt Lake City, Utah, USA, Jun. 18-23, 2018, pp. 6546–6555.
  * [19] B. Wang, L. Ma, W. Zhang, W. Jiang, J. Wang, and W. Liu, “Controllable video captioning with POS sequence guidance based on gated fusion network,” in _Proc. of the IEEE Int. Conf. on Comput. Vision_ , Seoul, South Korea, Oct. 27-Nov. 2, 2019, pp. 2641–2650.
  * [20] S. Xie, R. Girshick, P. Dollár, Z. Tu, and K. He, “Aggregated residual transformations for deep neural networks,” in _Proc. of the IEEE Conf. on Comput. Vision and Pattern Recognit._ , Venice, Italy, Oct. 22-29, 2017, pp. 1492–1500.
  * [21] J. S. Park, M. Rohrbach, T. Darrell, and A. Rohrbach, “Adversarial inference for multi-sentence video description,” in _Proc. of the IEEE Conf. on Comput. Vision and Pattern Recognit._ , Long Beach, CA, USA, Jun. 16-20, 2019, pp. 6598–6608.
  * [22] S. Hochreiter and J. Schmidhuber, “Long short-term memory,” _Neural Comput._ , vol. 9, no. 8, pp. 1735–1780, 1997.
  * [23] P. Bahar, C. Brix, and H. Ney, “Towards two-dimensional sequence to sequence model in neural machine translation,” _arXiv preprint arXiv:1810.03975_ , 2018.
  * [24] Z. Liu, D. Luo, Y. Wang, L. Wang, Y. Tai, C. Wang, J. Li, F. Huang, and T. Lu, “Teinet: Towards an efficient architecture for video recognition.” in _AAAI_ , 2020, pp. 11 669–11 676.
  * [25] L. Wang, Y. Xiong, Z. Wang, Y. Qiao, D. Lin, X. Tang, and L. Van Gool, “Temporal segment networks: Towards good practices for deep action recognition,” in _Pro. of the European Conf. on Comput. Vision (ECCV)_. Springer, 2016, pp. 20–36.
  * [26] M. Zolfaghari, K. Singh, and T. Brox, “Eco: Efficient convolutional network for online video understanding,” in _Pro. of the European Conf. on Comput. Vision (ECCV)_ , 2018, pp. 695–712.
  * [27] A. Farhadi and J. Redmon, “Yolov3: An incremental improvement,” _Pro. of the IEEE Conf. on Comput. Vision and Pattern Recognit._ , 2018.
  * [28] N. C. Luong, D. T. Hoang, S. Gong, D. Niyato, P. Wang, Y.-C. Liang, and D. I. Kim, “Applications of deep reinforcement learning in communications and networking: A survey,” _IEEE Commun. Surv. & Tut._, vol. 21, no. 4, pp. 3133–3174, 2019.
  * [29] K. Rusek, J. Suárez-Varela, A. Mestres, P. Barlet-Ros, and A. Cabellos-Aparicio, “Unveiling the potential of graph neural networks for network modeling and optimization in SDN,” in _Proc. of the 2019 ACM Symp. on SDN Research_ , San Jose, CA, USA, Apr. 3-4, 2019, pp. 140–151.
  * [30] A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, Ł. Kaiser, and I. Polosukhin, “Attention is all you need,” in _Adv. in Neural Inf. Process. Syst._ , 2017, pp. 5998–6008.
  * [31] Y. Hua, R. Li, Z. Zhao, X. Chen, and H. Zhang, “GAN-powered deep distributional reinforcement learning for resource management in network slicing,” _IEEE J. on Sel. Areas in Commun._ , vol. 38, no. 2, pp. 334–349, 2020.


# Bounded Weak Solutions of Degenerate $p$-Poisson Equations

Sullivan F. MacDonald and Scott Rodney  Scott Rodney
Dept. of Mathematics, Physics and Geology
Cape Breton University
Sydney, NS B1Y3V3, CA<EMAIL_ADDRESS>Sullivan F. MacDonald, MSc student
in the Dept. of Mathematics & Statistics
McMaster University
Hamilton, ON L8S4L8, CA<EMAIL_ADDRESS>
(Date: July 2023)

###### Abstract.

In this work we study global boundedness and exponential integrability of weak
solutions to degenerate $p$-Poisson equations using an iterative method of De
Giorgi type. Given a symmetric, non-negative definite matrix valued function
$Q$ defined on a bounded domain $\Omega\Subset\mathbb{R}^{n}$, a weight
function $v\in L^{1}_{\textrm{loc}}(\Omega,dx)$, and a suitable non-negative
function $\tau$, we give sufficient conditions for any weak solution to the
Dirichlet problem

$\displaystyle\left\\{\begin{array}[]{rccl}-\displaystyle\frac{1}{v}\mathrm{{div}}\left(\left|\sqrt{Q}\nabla
u\right|^{p-2}Q\nabla u\right)+\tau\left|u\right|^{p-2}u&=&f&\textrm{in
}\Omega,\\\ u&=&0&\textrm{on }\partial\Omega\end{array}\right.$

to be bounded and exponentially integrable when the data function $f$ belongs
to an appropriate Orlicz space.

###### Key words and phrases:

Orlicz Spaces, Degenerate Elliptic PDE, Global Regularity

###### 1991 Mathematics Subject Classification:

28A25, 35B45, 35D30, 35J25, 46E30, 47H99.

S. Rodney was supported by the NSERC Discovery Grant Program. S.F. MacDonald
was supported by the NSERC USRA Program.

##  1\. Introduction

In a bounded domain $\Omega\Subset\mathbb{R}^{n}$, the authors of [1]
investigated _a priori_ boundedness of weak solutions to a class of Dirichlet
problems for linear second order operators that includes

$\displaystyle\left\\{\begin{array}[]{rcl}-\Delta u(x)&=&f(x)\textrm{ for
}x\in\Omega\\\ u(x)&=&0~{}~{}~{}\textrm{ for
}x\in\partial\Omega,\end{array}\right.$

where the data function $f$ is permitted to be in the Orlicz class
$L^{A}(\Omega,dx)$ with $A(t)=t^{n/2}\log(e-1+t)^{q}$ for $q>0$. The main
result of that paper shows that a weak solution to such a problem is bounded
on $\Omega$ whenever $q>n/2$. This paper continues the study of such problems.
Indeed, we investigate _a priori_ boundedness and exponential integrability of
weak solutions to Dirichlet problems for operators of $p$-Laplace type with
rough coefficients. As in [1], the aim of this note is to explore optimal
conditions on data functions that ensure regularity of the corresponding weak
solution. Our approach uses De Giorgi type iterative techniques to prove our
results, and they and are modelled on the techniques used in [1]. To
effectively state the problems we study, we first describe the relevant class
of operators.

Letting $S_{n}$ denote the collection of $n\times n$ symmetric, non-negative
definite matrices, we let $Q:\Omega\rightarrow S_{n}$ be a matrix-valued
function with real measurable entries. The pointwise operator norm of $Q$ is
henceforth denoted by $\omega:\Omega\rightarrow\mathbb{R}$. That is, for
$x\in\Omega$ we write
$\omega(x)=\|Q(x)\|_{\textrm{op}}=\sup_{|\xi|=1}\left|Q(x)\xi\right|$. For
$p>1$, let $v$ be a weight function in $\Omega$ that satisfies

(1.1) $\displaystyle v(x)\geq C\omega^{p/2}(x),\textrm{ with}~{}\omega\in
L^{p/2}_{\textrm{loc}}(\Omega,dx).$

More, we will always assume that the matrix function $Q$ satisfies a Sobolev
inequality with gain $\sigma>1$. In particular, we assume there is are
constants $C>0$ and $\sigma>1$ such that

(1.2)
$\left(\int_{\Omega}\left|\varphi\right|^{p\sigma}~{}vdx\right)^{1/p\sigma}\leq
C\left(\int_{\Omega}|\sqrt{Q}\nabla\varphi|^{p}~{}dx\right)^{1/p}$

holds for any $\varphi\in\mathrm{Lip}_{0}(\Omega)$. The dual exponent of
$\sigma$ plays a significant role in our results, and we denote it by
$\sigma^{\prime}$ at all times.

Given $\tau:\Omega\rightarrow[0,\infty]$, we consider Dirichlet problems of
the form

(1.5)
$\displaystyle\left\\{\begin{array}[]{rccl}-\displaystyle\frac{1}{v}\mathrm{{div}}(|\sqrt{Q}\nabla
u|^{p-2}Q\nabla u)+\tau\left|u\right|^{p-2}u&=&f&\textrm{in }\Omega\\\
u&=&0&\textrm{on }\partial\Omega,\end{array}\right.$

where the data function $f$ is in a $v$-weighted Orlicz class. Our main
results are the following two theorems.

######  Theorem 1.6.

Let $1<p<\infty$ and $\sigma>1$ be as in (1.2), and assume that
$v(\Omega)<\infty$. Let $\tau\in L^{p\sigma^{\prime}}(\Omega,v)$ be a non-
negative, real valued function on $\Omega$, and let $f\in
L^{\Gamma}(\Omega,v)$ for a Young function $\Gamma(t)$ for which
$t^{-\sigma^{\prime}}\Gamma(t)$ is increasing on $[0,\infty)$ and

(1.7)
$\displaystyle\int_{1}^{\infty}\frac{\Gamma^{\prime}(t)}{\Gamma(t)}\left(\frac{t}{\Gamma(t)^{\frac{1}{\sigma^{\prime}}}}\right)^{\frac{1}{p-1}}~{}dt<\infty.$

Then, any weak sub-solution $(u,\nabla u)\in QH^{1,p}_{0}(\Omega,v)$ of (1.5)
is bounded satisfying

$\|u\|_{L^{\infty}(\Omega,v)}\leq
C\|f\|_{L^{\Gamma}(\Omega,v)}^{\frac{1}{p-1}}$

for a constant $C$ independent of both $(u,\nabla u)$ and $f$.

######  Theorem 1.8.

Let $f\in L^{\sigma^{\prime}}(\Omega,v)$, and let $u\in
QH_{0}^{1,p}(\Omega,v)$ be a bounded weak solution to (1.5) with $\tau=0$.
There is a constant $\gamma>0$ and a constant $C>0$ such that

$\int_{\Omega}e^{\gamma u}vdx\leq Cv(\Omega).$

We note that both $\gamma$ and $C$ are independent of the weak solution
$(u,\nabla u)$.

######  Remark 1.9.

  * (1)

The existence of weak solutions to (1.5) has been studied extensively; see for
example, [2, 3, 6, 8, 10]. Weak solutions to problems similar to (1.5) exist
in the degenerate matrix weighted Sobolev spaces defined as closures of
Lipschitz functions with respect to the Sobolev norm as described in §2.2.
Notation differs across these works; note for example that
$QH^{1,p}_{0}(\Omega,v)$ as in Theorem 1.6 is described in Chapter 4 of [3],
where the authors use the notation $\hat{H}^{1,p}_{Q,0}(\Omega)$.

  * (2)

Elements of the matrix weighted Sobolev space $QH^{1,p}_{0}(\Omega,v)$ are
pairs of the form $(u,\vec{g})\in
L^{p}(\Omega,v)\times\mathcal{L}^{p}(\Omega,Q)$, where
$\mathcal{L}^{p}(\Omega,Q)$ is the Banach space of $\mathbb{R}^{n}$-valued
measurable functions as described in [2]. The function $\vec{g}$ is a
generalized derivative of $u$ in the sense that there exists a sequence
$\\{u_{j}\\}$ of Lipschitz functions with compact support in $\Omega$ such
that

$\lim_{j\rightarrow\infty}\|u-u_{j}\|_{L^{p}(\Omega,v)}=0\quad\textrm{and}\quad\lim_{j\rightarrow\infty}\|\sqrt{Q}(\vec{g}-\nabla
u_{j})\|_{\mathcal{L}^{p}(\Omega,Q)}=0.$

We note that completeness of the matrix weighted space
$\mathcal{L}^{p}(\Omega,Q)$ was first established for the case $p=2$ in [12],
and for general $p\in[1,\infty)$ by the authors of [3].

  * (3)

A classical result of Gilbarg and Trudinger [5, Thm. 8.16] is recovered in the
case $p=2$ by setting $Q=I$, taking $\sigma=\frac{n}{n-2}$, and using
$\Gamma=t^{q}$ for $q>n/2$. We note that (1.7) holds exactly when the well-
known condition $q>\frac{n}{2}$ is satisfied.

Theorem 1.6 generalizes [1, Thm. 1.3] in several ways. First, [1, Thm. 1.3] is
a special case of Theorem 1.6 obtained by taking $p=2$, $\tau=0$, with Young
function $\Gamma(t)=t^{\sigma^{\prime}}\log(e-1+t)^{q}$ for
$q>\sigma^{\prime}$ with $\sigma>1$. Condition (1.7) is satisfied since

$\int_{1}^{\infty}\frac{t\Gamma^{\prime}(t)}{\Gamma(t)^{1+\frac{1}{\sigma^{\prime}}}}\leq(q+\sigma^{\prime})\int_{1}^{\infty}\frac{1}{t\log(e-1+t)^{\frac{q}{\sigma^{\prime}}}}~{}dt,$

and the integral on the right-hand side converges precisely when
$q>\sigma^{\prime}$. Second, our method can accommodate the presence of the
first-order drift term $\tau|u|^{p-2}u$ in (1.5). This feature is useful and
related to the existence of Sobolev and Poincaré inequalities, see [2, 3].
Lastly, we do not assume a special form of the Young function $\Gamma$ in
replacing $\Gamma(t)=t^{\sigma^{\prime}}\log(e-1+t)^{q}$ with any Young
function satisfying (1.7). For example, our results are easily shown to apply
when $\Gamma(t)$ is an iterated log bump Young function of the form

(1.10)
$\Gamma(t)=\bigg{(}t\prod_{j=1}^{k-1}\underbrace{\log\cdots\log}_{j\;\mathrm{times}}(c_{j}+t)\bigg{)}^{\sigma^{\prime}}\underbrace{\log\cdots\log}_{k\;\mathrm{times}}(c_{k}+t)^{q},$

where $q>\sigma^{\prime}$ and $c_{1},\dots,c_{k}$ are chosen independent of
$q$ and $\sigma$ so that $\Gamma(1)=1$.

The plan for the rest of this paper is as follows. In Section 2 we recall the
relevant notions of Orlicz spaces and weak solutions, and we discuss special
properties of a family of test functions specially related to our iterative
technique. Section 3 is devoted to the proof of Theorem 1.6. Following this,
Section 4 discusses exponential integrability of weak solutions to (1.5) and
contains the proof of Theorem 1.8.

##  2\. Preliminaries

This section provides a brief overview of the theory of Orlicz spaces needed
for our results. Comprehensive treatments of this theory can be found in many
of the works cited in this section.

###  2.1. Orlicz Spaces

First we introduce the Young functions which induce Orlicz spaces. Such
functions are commonly defined with respect to a density, which is any non-
decreasing and right-continuous function
$\psi:[0,\infty)\rightarrow[0,\infty)$ which satisfies $\psi(t)=0$ if and only
if $t=0$, and $\psi(t)\rightarrow\infty$ as $t\rightarrow\infty$. Given a
density, the associated Young function $\Psi$ is defined for $t\geq 0$ by
setting

$\Psi(t)=\int_{0}^{t}\psi(s)ds.$

From this formulation, one sees that $\Psi$ is continuous, strictly
increasing, and convex on $(0,\infty)$. Moreover, it is clear that $\Psi(0)=0$
and $\Psi(t)\rightarrow\infty$ as $t\rightarrow\infty$.

Equipped with a Young function $\Psi$ and a non-negative function $v\in
L^{1}_{\textrm{loc}}(\Omega,dx)$ which we call a weight, one defines the
Orlicz space $L^{\Psi}(\Omega,v)$ associated with a domain
$\Omega\subset\mathbb{R}^{n}$ to be the collection of real-valued functions on
$\Omega$ for which the norm

(2.1)
$\|f\|_{L^{\Psi}(\Omega,v)}=\inf\left\\{\lambda>0~{}:~{}\int_{\Omega}\Psi\left(\frac{|f(t)|}{\lambda}\right)~{}v(t)dt\leq
1\right\\}$

is well-defined and finite. It is well-known that $L^{\Psi}(\Omega,v)$ is a
Banach space; see [11]. Orlicz spaces generalize the classical Lebesgue
spaces, and they are useful since they provide a finer scale of norms in the
following sense: if $\Psi(t)=t^{p}\log(e-1+t)^{q}$ for $p\geq 1$ and $q>0$,
and if $v$ is any non-vanishing weight, then for any $\varepsilon>0$ we have

$L^{p+\varepsilon}(\Omega,v)\subsetneq L^{\Psi}(\Omega,v)\subsetneq
L^{p}(\Omega,v).$

The strictness of these inclusions follows from the examples in [7]. One can
thus think of the ‘log-bump’ space $L^{\Psi}(\Omega,v)$ as an intermediate
between Lebesgue spaces.

Just as the space $L^{p}(\Omega,v)$ has a topological dual which is isomorphic
to $L^{\frac{p}{p-1}}(\Omega,v)$, one can identify a dual Orlicz space to
$L^{\Psi}(\Omega,v)$. Given a Young function $\Psi$, the conjugate Young
function is defined via the Legendre transform

$\overline{\Psi}(t)=\sup_{s\geq 0}\\{st-\Psi(s)\\},$

and the Orlicz space $L^{\overline{\Psi}}(\Omega,v)$ is (isomorphic to) the
dual of $L^{\Psi}(\Omega,v)$. This fact can be verified using the properties
summarized in the following lemma. For proof of these properties, the reader
is referred to [9].

######  Lemma 2.2.

If $\Psi$ and $\overline{\Psi}$ are conjugate Young functions, then the
following hold:

  * (1)

Young’s inequality: If $s\geq 0$ and $t\geq 0$ then
$st\leq\Psi(s)+\overline{\Psi}(t)$,

  * (2)

Conjugate equivalence: If $t\geq 0$ then
$t\leq\Psi^{-1}(t)\overline{\Psi}^{-1}(t)\leq 2t$,

  * (3)

Hölder’s Inequality: If $f\in L^{\Psi}(\Omega,v)$ and $g\in
L^{\overline{\Psi}}(\Omega,v)$ then

$\bigg{|}\int_{\Omega}fg\;vdx\bigg{|}\leq
2\left\|f\right\|_{L^{\Psi}(\Omega,v)}\left\|g\right\|_{L^{\overline{\Psi}}(\Omega,v)}.$

It is useful to note that like Hölder’s inequality, many other well-known
properties of the Lebesgue spaces extend to Orlicz spaces in a natural way.
One of these is Chebyshev’s inequality, which generalizes in the following way
when we replace the usual $L^{p}$ norm with an Orlicz norm.

######  Theorem 2.3 (Chebyshev’s Inequality).

If $f\in L^{\Psi}(\Omega,v)$ then for any $\alpha>0$,

(2.4)
$\Psi^{-1}(v(\\{x\in\Omega:|f(x)|\geq\alpha\\})^{-1})^{-1}\leq\frac{1}{\alpha}\|f\|_{L^{\Psi}(\Omega,v)}.$

###### Proof.

Verification of this inequality in $L^{1}(\Omega,v)$ is an exercise common to
most graduate measure theory courses, and we take for granted that for any
$f\in L^{1}(\Omega,v)$,

$v(\\{x\in\Omega:|f(x)|\geq\alpha\\})\leq\frac{1}{\alpha}\int_{\Omega}|f|vdx.$

To establish (2.4) using the inequality above, we observe that it suffices to
prove the estimate when $\|f\|_{L^{\Psi}(\Omega,v)}=1$. Since $\Psi$ is
increasing and $\Psi(|f|)\in L^{1}(\Omega,v)$, we have

$v(\\{x\in\Omega:|f(x)|\geq\alpha\\})=v(\\{x\in\Omega:\Psi(|f(x)|)\geq\Psi(\alpha)\\})\leq\frac{1}{\Psi(\alpha)}\int_{\Omega}\Psi(|f|)vdx.$

Owing to our assumption that $\|f\|_{L^{\Psi}(\Omega,v)}=1$, we find that the
integral above is at most one. Consequently we have
$v(\\{x\in\Omega:|f(x)|\geq\alpha\\})\leq\Psi(\alpha)^{-1}$, and using the
fact that $\Psi^{-1}$ is increasing it is straightforward to rearrange and
obtain (2.4). ∎

Using the properties of Orlicz norms established above, one can show that if
$\Psi$ is a Young function and if $\chi_{S}$ denotes the indicator of a
measurable set $S\subseteq\Omega$, then

(2.5) $\|\chi_{S}\|_{L^{\Psi}(\Omega,v)}=\Psi^{-1}((v(S)^{-1})^{-1}.$

The final property of Orlicz norms which we require in the sections which
follow is a generalized version of Hölder’s inequality.

######  Lemma 2.6.

Let $\Phi$, $\Psi_{1}$, and $\Psi_{2}$ be Young functions and assume for each
$t\geq 0$ that $t\geq\Phi(\Psi_{1}^{-1}(t)\Psi_{2}^{-1}(t))$. If $f\in
L^{\Psi_{1}}(\Omega,v)$ and $g\in L^{\Psi_{2}}(\Omega,v)$, then

$\|fg\|_{L^{\Phi}(\Omega,v)}\leq
2\|f\|_{L^{\Psi_{1}}(\Omega,v)}\|g\|_{L^{\Psi_{2}}(\Omega,v)}.$

###### Proof.

Once again, we observe that it suffices to prove the claimed estimate when

$\|f\|_{L^{\Psi_{1}}(\Omega,v)}=\|g\|_{L^{\Psi_{2}}(\Omega,v)}=1.$

Define $\Omega_{1}=\\{x\in\Omega:\Psi_{1}(|f(x)|)\geq\Psi_{2}(|g(x)|)\\}$ and
set $\Omega_{2}=\Omega\setminus\Omega_{1}$. If $x\in\Omega_{1}$ then

$\Phi(|fg|)\leq\Phi(\Psi_{1}^{-1}(\Psi_{1}(|f|))\Psi_{2}^{-1}(\Psi_{1}(|f|)))\leq\Psi_{1}(|f|),$

while if $x\in\Omega_{2}$ then $\Phi(|fg|)\leq\Psi_{2}(|g|)$. Since
$\Omega\subseteq\Omega_{1}\cup\Omega_{2}$ and
$\|f\|_{\Psi_{1}}=\|g\|_{\Psi_{2}}=1$,

$\int_{\Omega}\Phi(|fg|)vdx\leq\int_{\Omega_{1}}\Psi_{1}(|f|)vdx+\int_{\Omega_{2}}\Psi_{2}(|g|)vdx\leq
2.$

Finally, as $\Phi$ is a Young function it is convex, meaning that
$\|fg\|_{\Phi}\leq 2$ by (2.1) since

$\int_{\Omega}\Phi\bigg{(}\frac{|fg|}{2}\bigg{)}vdx\leq\frac{1}{2}\int_{\Omega}\Phi(|fg|)vdx\leq
1.$

The claimed norm inequality then follows by dehomogenizing. ∎

Generalizing the proof above, one can show that if
$\Psi_{1}^{-1}\cdots\Psi_{m}^{-1}\leq\Phi^{-1}$ pointwise, and if $f_{j}\in
L^{\Psi_{j}}(\Omega,v)$ for $j=1,\dots,m$, then

$\|f_{1}\cdots f_{m}\|_{L^{\Phi}(\Omega,v)}\leq
m\|f_{1}\|_{L^{\Psi_{1}}(\Omega,v)}\cdots\|f_{m}\|_{L^{\Psi_{m}}(\Omega,v)}.$

Orlicz spaces are well-studied, and they enjoy many more properties analogous
to the Lebesgue spaces. For our purposes though, the brief outline given above
suffices.

###  2.2. Sobolev Spaces and Test Functions

Fix $1\leq p<\infty$ and let $Q$ be an $n\times n$ symmetric, non-negative
definite matrix-valued function with pointwise operator norm $\omega$.

The $Q$-weighted vector-valued Lebesgue space $\mathcal{L}^{p}(\Omega,Q)$ is
defined as the set of all functions $\vec{g}:\Omega\rightarrow\mathbb{R}^{n}$
for which the norm

$\|\vec{g}\|_{\mathcal{L}^{p}(\Omega,Q)}=\bigg{(}\int_{\Omega}|\sqrt{Q}~{}\vec{g}|^{p}dx\bigg{)}^{\frac{1}{p}}<\infty.$

Identifying any two measurable $\mathbb{R}^{n}$-valued functions $\vec{f}$ and
$\vec{g}$ if $\|\vec{f}-\vec{g}\|_{\mathcal{L}^{p}(\Omega,Q)}=0$, it is
established in [2] that $\mathcal{L}^{p}(\Omega,Q)$ is a Banach space. Using
this norm, the Sobolev space $QH^{1,p}_{0}(\Omega,v)$ is defined as the
completion of $\mathrm{Lip}_{0}(\Omega)$ with respect to the norm

(2.7)
$\displaystyle\|u\|_{QH^{1,p}(\Omega,v)}=\|u\|_{L^{p}(\Omega,v)}+\|\nabla
u\|_{\mathcal{L}^{p}(\Omega,Q)}.$

In other words, $QH^{1,p}(\Omega,v)$ is the collection of ordered pairs
$(u,\vec{g})\in L^{p}(\Omega,v)\times\mathcal{L}^{p}(\Omega,Q)$ for which
there exists a sequence $\\{u_{j}\\}$ of Lipschitz functions with compact
support in $\Omega$ that satisfies

(2.8)
$\lim_{j\rightarrow\infty}\|u-u_{j}\|_{L^{p}(\Omega,v)}=0\qquad\mathrm{and}\qquad\lim_{j\rightarrow\infty}\|\vec{g}-\nabla
u_{j}\|_{\mathcal{L}^{p}(\Omega,Q)}=0.$

The space $QH^{1,p}(\Omega,v)$ is defined analogously via completion of the
collection of $\mathrm{Lip}(\Omega)$ functions with respect to (2.7).

In some settings, owing to degeneracy of $Q$, the function $\vec{g}$ need not
be uniquely determined by $u$. An example of this behaviour is given in [4].
Nevertheless, we often abuse notation by writing $\nabla u$ in place of
$\vec{g}$ and referring to $\nabla u$ as a formal gradient of $u$, but we
caution the reader that this identification is made only in the sense of
(2.8). More, to simplify notation, we often write $u\in
QH^{1,p}_{0}(\Omega,v)$ in place of $(u,\nabla u)\in QH^{1,p}_{0}(\Omega,v)$.

####  2.2.1. Weak Solutions

Now we give a definition of weak solution of the Dirichlet problem (1.5). The
conditions required on the data function $f$ and non-negative function $\tau$
ensure that the integrals in the definition are finite.

######  Definition 2.9.

Let $Q$ and the weight $v$ satisfy (1.1) and (1.2). Given a function $f\in
L^{(p\sigma)^{\prime}}(\Omega,v)$ and non-negative $\tau\in
L^{\sigma^{\prime}}(\Omega,v)$, a pair $(u,\vec{g})\in QH^{1,p}_{0}(\Omega,v)$
is called a weak solution to (1.5) if the identity

(2.10) $\displaystyle\int_{\Omega}|\sqrt{Q}~{}\vec{g}|^{p-2}\nabla\varphi\cdot
Q~{}\vec{g}~{}dx+\int_{\Omega}\tau\left|u\right|^{p-2}u\varphi~{}vdx=\int_{\Omega}f\varphi~{}vdx.$

holds for every $\varphi\in\mathrm{Lip}_{0}(\Omega)$.

Our notion of weak solution is well-defined since the integrals in (2.10)
converge by Hölder’s inequality and our assumptions on $f$, $\tau$, $u$, and
$\vec{g}$. Using a simple density argument together with Fatou’s lemma, we can
expand our test functions from $\mathrm{Lip}_{0}(\Omega)$ to include all
member pairs of $QH^{1,p}_{0}(\Omega,v)$. That is, given any pair
$(w,\vec{h})\in QH^{1,p}_{0}(\Omega,v)$ one has that

(2.11)
$\displaystyle\int_{\Omega}\left|\sqrt{Q}~{}\vec{g}\right|^{p-2}\vec{h}\cdot
Q\vec{g}~{}dx+\int_{\Omega}\tau\left|u\right|^{p-2}uw~{}vdx=\int_{\Omega}fw~{}vdx.$

###  2.3. Technical Results

In Sections 3 and 4, given a weak solution $u\in QH^{1,p}_{0}(\Omega,v)$ we
will use test functions $\varphi_{\alpha}=(u-\alpha)_{+}$ and $w=e^{\alpha
u}-1$ for arbitrary $\alpha>0$. The following technical lemmas demonstrate
that these are valid test functions that can be used in (2.11).

######  Lemma 2.12.

Let $Q$ be as above, let inequality (1.1) hold, and fix $\alpha>0$. Given a
pair $(u,\nabla u)\in QH^{1,p}_{0}(\Omega,v)$, set
$\varphi_{\alpha}=(u-\alpha)_{+}$. Then
$(\varphi_{\alpha},\chi_{\\{u\geq\alpha\\}}\nabla u)\in
QH^{1,p}_{0}(\Omega,v)$.

###### Proof.

Recalling the definition of $QH^{1,p}_{0}(\Omega,v)$, we see that it is enough
to identify a sequence $\\{\psi_{j}\\}$ in $\mathrm{Lip}_{0}(\Omega)$ for
which

$\lim_{j\rightarrow\infty}\|\psi_{j}-\varphi_{\alpha}\|_{L^{p}(\Omega,v)}=0\qquad\textrm{and}\qquad\lim_{j\rightarrow\infty}\|\nabla\psi_{j}-\chi_{\\{u\geq\alpha\\}}\nabla
u\|_{\mathcal{L}^{p}(\Omega,Q)}=0.$

Since $u\in QH^{1,p}_{0}(\Omega,v)$, there is a sequence $\\{u_{j}\\}$ of
$\mathrm{Lip}_{0}(\Omega)$ functions that converge to $u$ in
$L^{p}(\Omega,v)$, and whose gradients converge to the formal gradient $\nabla
u$ in $\mathcal{L}^{p}(\Omega,Q)$. Passing to a subsequence if necessary, we
may assume that $u_{j}\rightarrow u$ $v$-a.e. and that $\sqrt{Q}\nabla
u_{j}\rightarrow\sqrt{Q}\nabla u$ a.e. in $\Omega$. For each $j\in\mathbb{N}$,
define $\psi_{j}=(u_{j}-\alpha)_{+}$ so that pointwise $v$-a.e. we have
$\psi_{j}\rightarrow\varphi_{\alpha}$. In addition,

$|\psi_{j}-\varphi_{\alpha}|^{p}\leq|u-u_{j}|^{p}\leq
2^{p-1}(|u|^{p}+|u_{j}|^{p})$

holds $v$-a.e., and as the right-hand side converges pointwise $v$-a.e. to
$2^{p}|u|^{p}\in L^{1}(\Omega,v)$ it follows from the dominated convergence
theorem that $\psi_{j}\rightarrow\varphi_{\alpha}$ in $L^{p}(\Omega,v)$ also.

To see the convergence of gradients, we note that
$\nabla\psi_{j}=\chi_{\\{u_{j}\geq\alpha\\}}\nabla u_{j}$ for each
$j\in\mathbb{N}$ by [5, Lemma 7.6], and that
$\chi_{\\{u_{j}\geq\alpha\\}}\rightarrow\chi_{\\{u\geq\alpha\\}}$ pointwise
$v$-a.e. in $\Omega$. Since

$|\nabla\psi_{i}-\chi_{\\{u\geq\alpha\\}}\nabla
u|\leq|(\chi_{\\{u_{j}\geq\alpha\\}}-\chi_{\\{u\geq\alpha\\}})\nabla
u|+|\chi_{\\{u_{j}\geq\alpha\\}}(\nabla u_{j}-\nabla u)|,$

another use of dominated convergence shows that
$\nabla\psi_{i}-\chi_{\\{u\geq\alpha\\}}\nabla u$ converges to zero in
$\mathcal{L}^{p}(\Omega,Q)$, meaning that $(\varphi_{\alpha},\nabla
u\chi_{S_{\alpha}})\in QH^{1,p}_{0}(\Omega,v)$ as claimed. ∎

######  Proposition 2.13.

Let $(u,\nabla u)\in QH^{1,p}_{0}(\Omega,v)$ with $u\geq 0$ $v$-a.e. in
$\Omega$, and assume that $u$ is essentially bounded. Then there exists a
sequence $\\{u_{j}\\}$ of $\mathrm{Lip}_{0}(\Omega)$ functions satisfying the
following criteria:

  1. (1)

$0\leq u_{j}(x)\leq\|u\|_{L^{\infty}(\Omega,v)}+1$ $v$-a.e. in $\Omega$.

  2. (2)

$u_{j}\rightarrow u$ both pointwise $v$-a.e. in $\Omega$ and in
$L^{p}(\Omega,v)$.

  3. (3)

$\nabla u_{j}\rightarrow\nabla u$ in $\mathcal{L}^{p}(\Omega,Q)$ and
$\sqrt{Q}\nabla u_{j}\rightarrow\sqrt{Q}\nabla u$ $v$-a.e. in $\Omega$.

  4. (4)

$\|\nabla u_{j}\|_{\mathcal{L}^{p}(\Omega,Q)}\leq\|\nabla
u\|_{\mathcal{L}^{p}(\Omega,Q)}+1$ for all $j\in\mathbb{N}$.

###### Proof.

Let $\\{\psi_{j}\\}$ be a sequence of $\mathrm{Lip}_{0}(\Omega)$ functions
that converge to $u$ in the norm of $QH^{1,p}_{0}(\Omega,v)$. Passing to
subsequences if necessary, we may assume that $\psi_{j}\rightarrow u$ holds
pointwise $v$-a.e. and in $L^{p}(\Omega,v)$, that
$\nabla\psi_{j}\rightarrow\nabla u$ in $\mathcal{L}^{p}(\Omega,Q)$ and
$\sqrt{Q}\nabla\psi_{j}\rightarrow\sqrt{Q}\nabla u$ pointwise a.e. in
$\Omega$, and finally that
$\|\nabla\psi_{j}\|_{\mathcal{L}^{p}(\Omega,Q)}\leq\|\nabla
u\|_{\mathcal{L}^{p}(\Omega,Q)}+1$ for each $j\in\mathbb{N}$.

Following [1], we set $w_{j}=|\psi_{j}|$ so that $|w_{j}-u|\leq|\psi_{j}-u|$
pointwise $v$-a.e. in $\Omega$, from which it follows that $w_{j}\rightarrow
u$ pointwise $v$-a.e. and in $L^{p}(\Omega,v)$. Since $w_{j}$ is Lipschitz, it
is differentiable $v$-a.e. by the Rademacher-Stepanov Theorem, and $v$-a.e. we
also have $\nabla w_{j}=\mathrm{sgn}(\psi_{j})\nabla\psi_{j}$. This gives
$|\sqrt{Q}\nabla w_{j}|=|\sqrt{Q}\nabla\psi_{j}|$ a.e., and we see that the
sequence $\\{w_{j}\\}$ inherits the properties of $\\{\psi_{j}\\}$ outlined
above.

We construct the sequence $\\{u_{j}\\}$ as in [1, Lemma 2.15] by setting
$M=\|u\|_{L^{\infty}(\Omega,v)}$ and letting $\phi\in C^{1}([0,\infty)$ be a
non-negative increasing function for which $\phi(t)=t$ when $t\leq
M-\frac{1}{2}$, $\phi(t)=M$ for $t\geq M+1$, and $0\leq\phi^{\prime}(t)\leq 1$
globally. Taking $u_{j}=\phi(w_{j})$, it is easy to verify that (1)-(4) hold,
as is done in [1, Lemma 2.15]. ∎

######  Lemma 2.14.

Let $u\in QH_{0}^{1,p}(\Omega,v)$ be non-negative and essentially bounded.
Given $\phi\in C^{1}(\mathbb{R})$ such that $\phi(0)=0$, we have
$(\phi(u),\phi^{\prime}(u)\nabla u)\in QH_{0}^{1,p}(\Omega)$.

###### Proof.

By Proposition 2.13, we may select a sequence $\\{u_{k}\\}$ of non-negative
bounded $\mathrm{Lip}_{0}(\Omega)$ functions which converge to $u$ in the norm
of $QH^{1,p}_{0}(\Omega,v)$, and which satisfy properties (1) through (4). It
is clear that $\phi(u_{k})\in\mathrm{Lip}_{0}(\Omega)$ for each $k$ as
$\phi\in C^{1}(\mathbb{R})$. Moreover, the continuity of $\phi$ ensures that
$\phi(u_{k})$ converges to $\phi(u)$ $v$-a.e. in $\Omega$.

Let $M$ be such that $u_{j}\leq M$ for all $j$. Since $\phi^{\prime}$ is
continuous it is bounded on $[0,M]$, and as $\phi(0)=0$ there exists $C>0$
such that $|\phi(t)|=|\phi(t)-\phi(0)|\leq C|t|$ for $t\leq M$. Therefore

$|\phi(u_{k})-\phi(u)|^{p}\leq 2^{p-1}(|\phi(u_{k})|^{p}+|\phi(u)|^{p})\leq
2^{p-1}C^{p}(|u_{k}|^{p}+|u|^{p}),$

and since the right-hand side converges to $2^{p}C^{p}|u|^{p}$ in
$L^{1}(\Omega,v)$, the dominated convergence theorem implies that
$\phi(u_{k})\rightarrow\phi(u)$ in $L^{p}(\Omega,v)$. The gradient converges
in a similar manner; we have
$\sqrt{Q}\nabla\phi(u_{k})=\phi^{\prime}(u_{k})\sqrt{Q}\nabla u_{k}$ almost
everywhere, and continuity of $\phi^{\prime}$ thus ensures that
$\sqrt{Q}\nabla\phi(u_{k})\rightarrow\phi^{\prime}(u)\sqrt{Q}\nabla u$
pointwise a.e. in $\Omega$. Additionally we have

$|\sqrt{Q}\nabla\phi(u_{k})-\phi^{\prime}(u)\sqrt{Q}\nabla u|^{p}\leq
2^{p-1}C^{p}(|\sqrt{Q}\nabla u_{k}|^{p}+|\sqrt{Q}\nabla u|^{p}),$

and once again the right-hand side converges to $2^{p}C^{p}|\sqrt{Q}\nabla
u|^{p}$ in $\mathcal{L}^{1}(\Omega,dx)$. The dominated convergence theorem
thus lets us conclude that $\sqrt{Q}\nabla\phi(u)$ converges to
$\phi^{\prime}(u)\sqrt{Q}\nabla u$ in $L^{p}(\Omega,v)$, giving
$(\phi(u),\nabla\phi(u))\in QH_{0}^{1,p}(\Omega,v)$ as required. ∎

As a last note to this section, we show that under the hypotheses of Theorem
1.6, the weak formulation (2.10) of problem (1.5) is well-defined. Recall, we
assume that $f\in L^{\Gamma}(\Omega,v)$ where $\Gamma$ is a Young function for
which $t^{-\sigma^{\prime}}\Gamma(t)$ is non-decreasing on $[0,\infty)$ and
(1.7) holds. In contrast, (2.10) is well defined only under the assumption
$f\in L^{(p\sigma)^{\prime}}(\Omega,v)$. We now show that the former condition
implies the latter.

######  Lemma 2.15.

Suppose $v(\Omega)<\infty$, $1<p<\infty$, and let $\sigma,v$ be as in (1.1)
and (1.2). Let $\Gamma$ be a Young function satisfying (1.7). Then,
$L^{\Gamma}(\Omega,v)\subseteq L^{(p\sigma)^{\prime}}(\Omega,v)$.

###### Proof.

First we let $q$ be the non-negative function for which
$q(t)\Gamma(t)=t^{(p\sigma)^{\prime}}$, and we show that $q$ must be bounded
for $t\geq 1$. Assume to the contrary that $q(t)\rightarrow\infty$ as
$t\rightarrow\infty$, and observe that since $\Gamma(t)\leq
t\Gamma^{\prime}(t)$ we have

$\int_{1}^{\infty}\frac{\Gamma^{\prime}(t)}{\Gamma(t)}\left(\frac{t}{\Gamma(t)^{\frac{1}{\sigma^{\prime}}}}\right)^{\frac{1}{p-1}}dt\geq\int_{1}^{\infty}\frac{1}{t}\left(\frac{t}{\Gamma(t)^{\frac{1}{\sigma^{\prime}}}}\right)^{\frac{1}{p-1}}dt=\int_{1}^{\infty}t^{\frac{1}{p\sigma-1}-1}q(t)^{\frac{\sigma-1}{\sigma(p-1)}}dt.$

For any $k\geq 1$, it follows from the inequality $p\sigma>1$ and the estimate
above that

$\int_{1}^{\infty}\frac{\Gamma^{\prime}(t)}{\Gamma(t)}\left(\frac{t}{\Gamma(t)^{\frac{1}{\sigma^{\prime}}}}\right)^{\frac{1}{p-1}}dt\geq\int_{k}^{2k}t^{\frac{1}{p\sigma-1}-1}q(t)^{\frac{\sigma-1}{\sigma(p-1)}}dt\geq\log
2\bigg{(}\min_{t\geq k}q(t)^{\frac{\sigma-1}{\sigma(p-1)}}\bigg{)}.$

Taking a limit in $k$, we find that

$\int_{1}^{\infty}\frac{\Gamma^{\prime}(t)}{\Gamma(t)}\left(\frac{t}{\Gamma(t)^{\frac{1}{\sigma^{\prime}}}}\right)^{\frac{1}{p-1}}dt\geq\log
2\bigg{(}\liminf_{k\rightarrow\infty}q(k)^{\frac{\sigma-1}{\sigma(p-1)}}\bigg{)}=\infty,$

which contradicts our assumption that the integral on the left-hand side
converges. We conclude that $q(t)\leq C$ for $t\geq 1$, meaning that
$t^{(p\sigma)^{\prime}}\leq C\Gamma(t)$ for $t\geq 1$. For later convenience,
we replace $C$ with $\max\\{1,C\\}$ to assume that $C\geq 1$.

Taking
$\lambda=C\max\\{2,2v(\Omega)\\}^{\frac{1}{(p\sigma)^{\prime}}}\|f\|_{\Gamma}$
and making a straightforward estimate gives

$\int_{\Omega}\bigg{(}\frac{|f|}{\lambda}\bigg{)}^{(p\sigma)^{\prime}}vdx\leq\frac{v(\Omega\cap\\{|f|\leq
C\|f\|_{\Gamma}\\})}{2v(\Omega)}+\frac{1}{2}\int_{\Omega\cap\\{|f|>C\|f\|_{\Gamma}\\}}\bigg{(}\frac{|f|}{C\|f\|_{\Gamma}}\bigg{)}^{(p\sigma)^{\prime}}vdx.$

Thanks to the estimate $t^{(p\sigma)^{\prime}}\leq C\Gamma(t)$ for $t\geq 1$,
we can observe by convexity that

$\int_{\Omega\cap\\{|f|>C\|f\|_{\Gamma}\\}}\bigg{(}\frac{|f|}{C\|f\|_{\Gamma}}\bigg{)}^{(p\sigma)^{\prime}}vdx\leq
C\int_{\Omega}\Gamma\bigg{(}\frac{|f|}{C\|f\|_{\Gamma}}\bigg{)}vdx\leq\int_{\Omega}\Gamma\bigg{(}\frac{|f|}{\|f\|_{\Gamma}}\bigg{)}vdx\leq
1.$

For $\lambda$ as above then, we find that
$\|f\|_{(p\sigma)^{\prime}}\leq\lambda$ since

$\int_{\Omega}\bigg{(}\frac{|f|}{\lambda}\bigg{)}^{(p\sigma)^{\prime}}vdx\leq\frac{v(\Omega\cap\\{|f|\leq
C\|f\|_{\Gamma}\\})}{2v(\Omega)}+\frac{1}{2}\leq 1.$

If follows that if $f\in L^{\Gamma}(\Omega,v)$ and if $v(\Omega)<\infty$, then
$f\in L^{(p\sigma)^{\prime}}(\Omega,v)$ as claimed. ∎

If $\Gamma$ is as above and if $v$ is essentially constant on $\Omega$, the
preceding lemma and [3, Proposition 4.4] together imply that (1.5) has a weak
solution in $QH_{0}^{1,p}(\Omega,v)$. Existence of weak solutions to a similar
class of homogeneous Neumann problems is also discussed in [2, Chapter 4],
under more general hypotheses on the weight $v$.

##  3\. Proof of Theorem 1.6

Following standard convention, we use $C$ to denote a constant which may
change from line to line. Fix a Young function $\Gamma$ satisfying (1.7) for
which $t^{-\sigma^{\prime}}\Gamma(t)$ is non-decreasing, let $u\in
QH^{1,p}(\Omega,v)$ be a non-negative weak solution to (1.5), and assume
without loss of generality that $\Gamma(1)=1$ and $v(\Omega)=1$. For any
constant $\alpha>0$ we define $\varphi_{\alpha}=(u-\alpha)_{+}$, and we show
that $v_{\alpha}=v(\\{x\in\Omega:u(x)\geq\alpha\\})=0$ when $\alpha$ is
sufficiently large. From this identity it will follow that
$\|u\|_{L^{\infty}(\Omega,v)}\leq\alpha$.

For any $\beta>\alpha\geq 0$, we first estimate $v_{\beta}$ in terms of
$v_{\alpha}$ to obtain a recursion in the measures of level sets. Using
Theorem 2.3 and Hölder’s inequality, we find that

(3.1)
$(\beta-\alpha)v_{\beta}\leq\int_{\Omega}\varphi_{\alpha}vdx\leq\|\varphi_{\alpha}\|_{L^{p\sigma}(\Omega,v)}v_{\alpha}^{1-\frac{1}{p\sigma}}.$

Our aim is to estimate $\|\varphi_{\alpha}\|_{L^{p\sigma}(\Omega,v)}$ in terms
of $v_{\alpha}$. Since $\varphi_{\alpha}\in QH^{1,p}_{0}(\Omega,v)$ by Lemma
2.12, we can use it as a test function in (2.9) to get

$\int_{\Omega}|\sqrt{Q}\nabla u|^{p-2}\sqrt{Q}\nabla
u\cdot\sqrt{Q}\nabla\varphi_{\alpha}dx+\int_{\Omega}\tau|u|^{p-2}u\varphi_{\alpha}vdx=\int_{\Omega}f\varphi_{\alpha}vdx.$

The definition of $\varphi_{\alpha}$ gives $|\sqrt{Q}\nabla
u|^{p-2}\sqrt{Q}\nabla
u\cdot\sqrt{Q}\nabla\varphi_{\alpha}=|\sqrt{Q}\nabla\varphi_{\alpha}|^{p}$,
and moreover the second integral above is non-negative. It follows from (1.2)
that

$\|\varphi_{\alpha}\|_{L^{p\sigma}(\Omega,v)}^{p}\leq
C\|\sqrt{Q}\nabla\varphi_{\alpha}\|_{L^{p}(\Omega,dx)}^{p}\leq
C\int_{\Omega}f\varphi_{\alpha}vdx.$

To proceed, we estimate the integral on the right-hand side above by employing
Hölder’s inequality once again to get

$\int_{\Omega}f\varphi_{\alpha}vdx\leq
2\|f\|_{L^{\Gamma}(\Omega,v)}\|\varphi_{\alpha}\|_{L^{\overline{\Gamma}}(\Omega,v)}.$

If $\Psi$ is the Young function that satisfies
$\Psi^{-1}(t)=\overline{\Gamma}^{-1}(t)t^{-\frac{1}{p\sigma}}$ for $t\geq 0$,
then Lemma 2.6 gives us
$\|\varphi_{\alpha}\|_{L^{\overline{\Gamma}}(\Omega,v)}\leq
2\|\varphi_{\alpha}\|_{L^{p\sigma}(\Omega,v)}\|\chi_{\alpha}\|_{L^{\Psi}(\Omega,v)},$
where $\chi_{\alpha}$ is the indicator of the set
$\\{x\in\Omega:u(x)\geq\alpha\\}$. From (2.5) and item (2) of Lemma 2.2 we
also have

$\|\chi_{\alpha}\|_{L^{\Psi}(\Omega,v)}=\Psi^{-1}(v_{\alpha}^{-1})^{-1}\leq
v_{\alpha}^{1-\frac{1}{p\sigma}}\Gamma^{-1}(v_{\alpha}^{-1}),$

meaning that

$\int_{\Omega}f\varphi_{\alpha}vdx\leq
4\|f\|_{L^{\Gamma}(\Omega,v)}\|\varphi_{\alpha}\|_{L^{p\sigma}(\Omega,v)}v_{\alpha}^{1-\frac{1}{p\sigma}}\Gamma^{-1}(v_{\alpha}^{-1}).$

Combining our estimates involving $f$ and
$\|\varphi_{\alpha}\|_{L^{p\sigma}(\Omega,v)}$ and simplifying now gives

$\|\varphi_{\alpha}\|_{L^{p\sigma}(\Omega,v)}\leq
C\|f\|_{L^{\Gamma}(\Omega,v)}^{\frac{1}{p-1}}v_{\alpha}^{\frac{p\sigma-1}{p\sigma(p-1)}}\Gamma^{-1}(v_{\alpha}^{-1})^{\frac{1}{p-1}}.$

Using (3.1) and the estimate above, we obtain the desired recursion in
measure:

$(\beta-\alpha)v_{\beta}\leq
C\|f\|_{L^{\Gamma}(\Omega,v)}^{\frac{1}{p-1}}v_{\alpha}^{\frac{p\sigma-1}{p\sigma-\sigma}}\Gamma^{-1}(v_{\alpha}^{-1})^{\frac{1}{p-1}}.$

To simplify, we replace $u$ with
$\tilde{u}=u/C\rho\|f\|_{L^{\Gamma}(\Omega,v)}^{\frac{1}{p-1}}$ for $C$ as
above and $\rho>1$ to get

$(\beta-\alpha)v_{\beta}\leq\frac{1}{\rho}v_{\alpha}^{\frac{p\sigma-1}{p\sigma-\sigma}}\Gamma^{-1}(v_{\alpha}^{-1})^{\frac{1}{p-1}}.$

Exploiting this recursion, we show now that $\tilde{u}$ is essentially
bounded.

For a monotone increasing sequence $\\{s_{k}\\}$ to be chosen momentarily, we
define $v_{k}=\\{x\in\Omega:u(x)\geq s_{k}\\}$ so that the preceding
inequality yields

$(s_{k+1}-s_{k})v_{k+1}\leq\rho^{-1}v_{k}^{\frac{p\sigma-1}{p\sigma-\sigma}}\Gamma^{-1}(v_{k}^{-1})^{\frac{1}{p-1}}.$

To construct $\\{s_{k}\\}$ we set $s_{0}=0$ and note that $v_{0}\leq
v(\Omega)=1$. Given $s_{k}$ we let

$s_{k+1}=s_{k}+v_{k}^{\frac{1}{(p-1)\sigma^{\prime}}}\Gamma^{-1}(v_{k}^{-1})^{\frac{1}{p-1}}$

so that $v_{k+1}\leq\rho^{-1}v_{k}$ and $v_{k}\leq\rho^{-k}$. It follows that
$v_{k}\rightarrow 0$ as $k\rightarrow\infty$, and so

$\|\tilde{u}\|_{L^{\infty}(\Omega,v)}\leq\lim_{k\rightarrow\infty}s_{k}=\sum_{k=0}^{\infty}v_{k}^{\frac{1}{(p-1)\sigma^{\prime}}}\Gamma^{-1}(v_{k}^{-1})^{\frac{1}{p-1}}.$

Since $t^{-\sigma^{\prime}}\Gamma(t)$ is non-decreasing, it is easy to show
that $t^{\frac{1}{\sigma^{\prime}}}\Gamma^{-1}(t^{-1})$ is also non-
decreasing, whereby we conclude from the estimate $v_{k}\leq\rho^{-k}$ that

$\|\tilde{u}\|_{L^{\infty}(\Omega,v)}\leq\sum_{k=0}^{\infty}\rho^{-\frac{k}{(p-1)\sigma^{\prime}}}\Gamma^{-1}(\rho^{k})^{\frac{1}{p-1}}\leq
1+\int_{0}^{\infty}\rho^{-\frac{t}{(p-1)\sigma^{\prime}}}\Gamma^{-1}(\rho^{t})^{\frac{1}{p-1}}dt.$

Making the change of variables $\rho(t)=\Gamma(s)$ and using condition (1.7),
we see that

$\|\tilde{u}\|_{L^{\infty}(\Omega,v)}\leq
1+\frac{1}{\log(\rho)}\int_{1}^{\infty}\frac{\Gamma^{\prime}(s)}{\Gamma(s)}\bigg{(}\frac{s}{\Gamma(s)^{\frac{1}{\sigma^{\prime}}}}\bigg{)}^{\frac{1}{p-1}}ds\leq
C.$

It follows that $\|u\|_{L^{\infty}(\Omega,v)}\leq
C\|f\|_{L^{\Gamma}(\Omega,v)}^{\frac{1}{p-1}}$, as we wished to show. $\Box$

##  4\. Exponential Integrability

In this section we study exponential integrability of weak solutions to the
Dirichlet problem (1.5) when $\tau=0$. That is, if $u$ is a weak solution to
the degenerate Poisson-type problem

(4.1) $\left\\{\begin{aligned} -\frac{1}{v(x)}\mathrm{div}(|\sqrt{Q(x)}\nabla
u|^{p-2}Q(x)\nabla u(x))=\;&f(x)\quad&&x\in\Omega,\\\ \hfill
u(x)=&\;0\qquad&&x\in\partial\Omega,\end{aligned}\right.$

then $\|e^{\gamma u}\|_{L^{1}(\Omega,v)}\leq Cv(\Omega)$ when
$v(\Omega)<\infty$ and $\gamma$ is a small constant small enough. Our approach
is modelled on [1, Section 2], and we begin with the following norm estimate.

######  Lemma 4.2.

Let $u$ be a non-negative bounded weak solution to (4.1), and for constants
$\xi>0$ and $\alpha>0$ let $w=e^{\xi u}-1$ and $w_{\alpha}=(w-\alpha)_{+}$.
Then

(4.3) $\|\sqrt{Q}\nabla
w_{\alpha}\|_{L^{p}(\Omega,dx)}^{p}\leq\xi^{p-1}\int_{\Omega}|f|w_{\alpha}(w+1)^{p-1}vdx.$

###### Proof.

From Lemma 2.14 it follows that
$\psi_{\alpha}=((w+1)^{p}-(\alpha+1)^{p})_{+}\in QH_{0}^{1}(\Omega,v)$,
meaning that we can use $\psi_{\alpha}$ in the weak formulation of (4.1).
Further, observe that
$\sqrt{Q}\nabla\psi_{\alpha}=\chi_{\alpha}p\xi(w+1)^{p}\sqrt{Q}\nabla u$,
where $\chi_{\alpha}$ denotes the indicator function of the support of
$w_{\alpha}$. Consequently we have

$\int_{\Omega}|\sqrt{Q}\nabla u|^{p-2}\sqrt{Q}\nabla
u\cdot\sqrt{Q}\nabla\psi_{\alpha}dx=\frac{p}{\xi^{p-1}}\int_{\Omega}|\xi(w+1)\sqrt{Q}\nabla
u|^{p}\chi_{\alpha}dx,$

and the integral on the right-hand side is exactly $\|\sqrt{Q}\nabla
w_{\alpha}\|_{L^{p}(\Omega,dx)}^{p}$. Combining the identity above with the
weak formulation of (4.1) then, we get

$\|\sqrt{Q}\nabla
w_{\alpha}\|_{L^{p}(\Omega,dx)}^{p}=\frac{\xi^{p-1}}{p}\int_{\Omega}f\psi_{\alpha}vdx.$

Inequality (4.3) follows from the pointwise estimate $\psi_{\alpha}\leq
pw_{\alpha}(w+1)^{p-1}$ for $p\geq 1$. ∎

######  Theorem 4.4.

Let $f\in L^{\sigma^{\prime}}(\Omega,v)$, and let $u\in
QH_{0}^{1,p}(\Omega,v)$ be a bounded weak solution to (4.1). Then for $\xi$
sufficiently small we have

(4.5) $\|e^{\xi
u}\chi_{\alpha}\|_{L^{p\sigma}(\Omega,v)}\leq\frac{(1+\alpha)v(\\{e^{\xi
u}\geq
1+\alpha\\})^{\frac{1}{p\sigma}}}{1-C\xi\|f\|_{L^{\sigma^{\prime}}(\Omega,v)}^{\frac{1}{p-1}}}.$

###### Proof.

Fixing $\alpha\geq 0$ and applying Lemma 4.2 together with Hölder’s inequality
gives

$\|\sqrt{Q}\nabla
w_{\alpha}\|^{p}_{L^{p}(\Omega,dx)}\leq\xi^{p-1}\|w_{\alpha}\|_{L^{p\sigma}(\Omega,v)}\|f\|_{L^{\sigma^{\prime}}(\Omega,v)}\bigg{(}\int_{\Omega}(w+1)^{p\sigma}\chi_{\alpha}vdx\bigg{)}^{\frac{p-1}{p\sigma}},$

and from (1.2) it follows that $\|w_{\alpha}\|_{L^{p\sigma}(\Omega,v)}\leq
C\|\sqrt{Q}\nabla w_{\alpha}\|_{L^{p}(\Omega,dx)}$. Combining these
inequalities and rearranging, we see that

$\|w_{\alpha}\|_{L^{p\sigma}(\Omega,v)}\leq\xi
C\|f\chi_{\alpha}\|_{L^{\sigma^{\prime}}(\Omega,v)}^{\frac{1}{p-1}}\|e^{\xi
u}\chi_{\alpha}\|_{L^{p\sigma}(\Omega,v)}.$

Additionally, since $\|e^{\xi
u}\chi_{\alpha}\|_{L^{p\sigma}(\Omega,v)}=\|w_{\alpha}+(1+\alpha)\chi_{\alpha}\|_{L^{p\sigma}(\Omega,v)}$,
Minkowski’s inequality and the preceding estimate together give

$\|e^{\xi u}\chi_{\alpha}\|_{L^{p\sigma}(\Omega,v)}\leq\xi
C\|f\|_{L^{\sigma^{\prime}}(\Omega,v)}^{\frac{1}{p-1}}\|e^{\xi
u}\chi_{\alpha}\|_{L^{p\sigma}(\Omega,v)}+(1+\alpha)v(\\{w\geq\alpha\\})^{\frac{1}{p\sigma}}.$

For $\xi$ small (namely,
$\xi<1/C\|f\|_{L^{\sigma^{\prime}}(\Omega,v)}^{\frac{1}{p-1}}$), we can
rearrange to obtain (4.5). ∎

Since the condstants in the above are independent of $\alpha>0$, we may allow
$\alpha\rightarrow 0^{+}$ to obtain Theorem 1.8 as a corollary.

## References

  * [1] D. Cruz-Uribe and S. Rodney.  Bounded weak solutions to elliptic PDE with data in orlicz spaces.  Journal of Differential Equations, 297:409–432, 2021.
  * [2] David Cruz-Uribe, Scott Rodney, and Emily Rosta.  Poincaré inequalities and Neumann problems for the p-Laplacian.  Canadian Mathematical Bulletin, 61(4):738–753, 2018.
  * [3] David Cruz-Uribe, Scott Rodney, and Emily Rosta.  Global Sobolev inequalities and degenerate p-Laplacian equations.  Journal of Differential Equations, 268(10):6189–6210, 2020.
  * [4] E. B. Fabes, C. E. Kenig, and R. P. Serapioni.  The local regularity of solutions of degenerate elliptic equations.  Comm. Partial Diff. Equations, 7(1):77–116, 1982.
  * [5] David Gilbarg and Neil S. Trudinger.  Elliptic partial differential equations of second order.  Classics in Mathematics. Springer-Verlag, Berlin, 2001.  Reprint of the 1998 edition.
  * [6] Usman Hafeez, Théo Lavier, Lucas Williams, and Lyudmila Korobenko.  Orlicz-Sobolev inequalities and the Dirichlet problem for infinitely degenerate elliptic operators.  Electron. J. Differential Equations, pages Paper No. 82, 19, 2021\.
  * [7] Addison Mailhot.  On the limit of orlicz norms.  Cape Breton University Undergraduate Thesis, 2022.
  * [8] Dario D. Monticelli and Scott Rodney.  Existence and spectral theory for weak solutions of Neumann and Dirichlet problems for linear degenerate elliptic operators with rough coefficients.  J. Differential Equations, 259(8):4009–4044, 2015.
  * [9] M. M. Rao and Z. D. Ren.  Theory of Orlicz spaces, volume 146 of Monographs and Textbooks in Pure and Applied Mathematics.  Marcel Dekker Inc., New York, 1991.
  * [10] Scott Rodney.  Existence of weak solutions of linear subelliptic Dirichlet problems with rough coefficients.  Canad. J. Math., 64(6):1395–1414, 2012.
  * [11] W. Rudin.  Real and complex analysis.  McGraw-Hill Book Co., New York, third edition, 1987.
  * [12] Eric T. Sawyer and Richard L. Wheeden.  Degenerate sobolev spaces and regularity of subelliptic equations.  Transactions of the American Mathematical Society, 362(4):1869–1906, 2010.