# ViViD++ : Vision for Visibility Dataset

Alex Junho Lee1, Younggun Cho2, Young-sik Shin3, Ayoung Kim4 and Hyun Myung5∗
1Alex Junho Lee is with the Department of Civil and Environmental Engineering,
KAIST, Daejeon, S. Korea<EMAIL_ADDRESS>Cho is with the Department
of Electrical Engineering, Inha University, Incheon, S. Korea.
<EMAIL_ADDRESS>Shin is with Korea Institute of Machinery and Materials,
Daejeon, S. Korea<EMAIL_ADDRESS>Kim is with the Department of
Mechanical Engineering, SNU, Seoul, S. Korea<EMAIL_ADDRESS>Myung is
with the School of Electrical Engineering, KAIST, Daejeon, S. Korea.
<EMAIL_ADDRESS>work was supported in part by Institute of Information
& Communications Technology Planning & Evaluation (IITP) grant funded by Korea
government (MSIT) (No.2020-0-00440, Development of Artificial Intelligence
Technology that Continuously Improves Itself as the Situation Changes in the
Real World), and in part by “Indoor Robot Spatial AI Technology Development”
project funded by KT (KT award B210000715). The students are supported by the
BK21 FOUR from the Ministry of Education (Republic of Korea).

###### Abstract

In this paper, we present a dataset capturing diverse visual data formats that
target varying luminance conditions. While RGB cameras provide nourishing and
intuitive information, changes in lighting conditions potentially result in
catastrophic failure for robotic applications based on vision sensors.
Approaches overcoming illumination problems have included developing more
robust algorithms or other types of visual sensors, such as thermal and event
cameras. Despite the alternative sensors’ potential, there still are few
datasets with alternative vision sensors. Thus, we provided a dataset recorded
from alternative vision sensors, by handheld or mounted on a car, repeatedly
in the same space but in different conditions. We aim to acquire visible
information from co-aligned alternative vision sensors. Our sensor system
collects data more independently from visible light intensity by measuring the
amount of infrared dissipation, depth by structured reflection, and
instantaneous temporal changes in luminance. We provide these measurements
along with inertial sensors and ground-truth for developing robust visual SLAM
under poor illumination. The full dataset is available at:
https://visibilitydataset.github.io/

##  I Introduction

With recent interests in autonomous navigation, a robot’s ability to achieve
localization and recognize the surroundings has been a critical feature for
mobile applications. To solve the autonomous navigation problem in the real
world, cameras have been widely used for their cost-effectiveness and
intuitiveness. Numerous studies based on images have focused on developing
robust visual SLAM (SLAM) algorithms to cope with real-world disturbances,
such as lighting and motion variances. The scene’s visual deviation arises
from the natural and artificial illumination changes. To overcome the visual
disturbances and develop robust visual SLAM, numbers of datasets covering
variations were introduced.

Figure 1: An overview of the sensor system and the dataset. We set up
different sensor configurations for handheld (upper left) and driving (upper
right). Sensor systems include RGB, thermal, events, depth, and interial
measurements. Each sensor is indicated with the letter box. Each sensor’s
visibility differences are displayed in the lower row. Relative to RGB,
thermal and event show less variance to the illumination condition.

Public datasets provide large environmental variations from cameras. Some of
them provide measurements from indoor [1, 2, 3] and the others arrange outdoor
scenes [4, 5, 6] providing a large scale of sensor measurements. Synthetic
datasets [7, 8, 9] were also provided for diversity.

However, achieving robust visual SLAM under low visibility still remains
challenging. For typical cameras that collect information by integrating
photons during the exposure time, the captured image is concentrated on the
lighting more than the object. Thus, the ideas of collecting information from
visual domains other than visible light intensity have been introduced. The
alternative vision sensors have advantages over typical cameras. For instance,
the thermal cameras could capture infrared radiation, and the event cameras
[10] could detect temporal changes. These special abilities make the sensor
measurement to be more independent from external lighting and motion
conditions. For datasets including alternative vision sensors, event datasets
[11, 12, 13, 14, 15, 16] and thermal datasets [17, 18] were publicly released.

In this paper, we present a dataset for developing robust visual SLAM in the
real world by providing:

  * •

the first dataset to enclose information from multiple types of aligned
alternative vision sensors;

  * •

multi-sensory measurements with ground-truth from external positioning system
and generated from SLAM;

  * •

wide range of environments in indoor and outdoor, recorded from multiple
platforms.

##  II Related Works

###  II-A Related Datasets

Vision sensors produce a wide range of information, but they are easily
influenced by external factors such as illumination and motion. As a result, a
visual SLAM must be evaluated in a wide range of scenarios. Existing datasets
cover a wide range of scenarios and include a large number of sensors, so as
to develop robust SLAM algorithms. Various datasets, encompassing a wide range
of environments, were released, including a wide range of visual and
structural scenes. These datasets served as benchmark baselines and enabled
the advancements for research.

In TUM RGB-D [1], image and depth data were collected, with ground-truths from
the motion capture system. To gradually test the robustness of visual sensors
over movement, sequences were separated by the motion speed. Similarly, in
EuRoC [2], images from stereo cameras were measured along with IMU (IMU) and
ground-truth. They separated the sequences with the difficulty of camera
motion to test the robustness over kinematic variances. Further in SceneNet
[8], photo-realistic videos, including motion blur and nonlinear camera
response were provided with ground-truth. In InteriorNet [9], a large scale of
synthetic environment with various customizable parameters, i.e., texture,
illumination, artificial lights, motion blur, etc., were presented with a wide
choice of sensors such as depth, fisheye, and event cameras. In NCLT [19] and
TUM monoVO [20], a large scale of indoor and outdoor data with light and
weather changes, was provided for long-term visual SLAM. Including all the
variances mentioned above in urban environment, datasets such as KITTI [4],
Cityscapes [5], and Complex urban dataset [6] were suggested.

Although many studies already covered a wide range of real-world noises, the
problem of classical cameras, which are highly dependent on external light,
remains. To overcome the appearance change and motion problem of conventional
cameras, datasets using alternative vision sensors, such as thermal and event
cameras, were also introduced.

###  II-B Alternative Vision Datasets

Unlike conventional cameras that capture intensity from the visible light
spectrum, thermal cameras capture infrared intensity from object surfaces. The
infrared measurements are generally more dependent on the object’s temperature
than external light; thermal cameras can capture the scene information
independently from external light conditions. Further, the event cameras [10]
are visual sensors that capture relative luminance changes over time, not the
absolute intensity. To code the instant light changes while maintaining low
latency, the event camera uses a particular type of data representation:
streams of tuples, not the sequence of images. Thus, the sensors’ output
characteristics are distinctive from typical cameras. The event cameras
require unique methods to process data, and several datasets have been
released.

TABLE I: Comparison with previous alternative vision datasets Dataset |  |  | Sensor |  |
---|---|---|---|---|---
Intensity | Depth | Thermal | Event | GT
Mueggler et al. [12] | ✓ |  |  | ✓ | ✓
MVSEC [13] | ✓ | ✓ |  | ✓ | ✓
UZH-FPV [14] | ✓ |  |  | ✓ | ✓
DSEC [16] | ✓ | ✓ |  | ✓ | ✓
Fischer et al. [15] | ✓ |  |  | ✓ | ✓
Maddern et al. [17] | ✓ |  | ✓ |  | ✓
Choi et al. [18] | ✓ | ✓ | ✓ |  | ✓
ViViD++ (proposed) | ✓ | ✓ | ✓ | ✓ | ✓

In early stages, [11] combined eDVS sensor with depth sensor and released data
in indoor sequences. Later the Event-Camera Dataset and Simulator [12] was
released, enabling the simulated events and widening the selection of test
environments.

MVSEC [13] is a dataset with various lighting and motion measured with stereo
event camera and inertial sensors. The dataset contains a large variety of
condition changes, suggesting the utilization of the event camera for high-
speed motion estimation. However, the dataset does not include repeated
measurements in outdoor environments. In UZH-FPV [14], aggressively moving
drones were used to provide event camera data for extreme motion cases. Also,
a large-scale event camera dataset was presented in [16], covering most of
applicable environments.

To investigate the appearance change problem with the event camera, in the
Brisbane-Event-VPR dataset [15], the event camera was mounted on the car and
acquired data along the same trajectory at different times. The dataset
provides a large appearance change due to sun elevation changes. Especially in
midnight sequences, street lights largely affect the cameras, and VPR becomes
nearly impossible, opening a problem to robustly run event-based algorithms
under both motion and appearance changes.

Also for thermal cameras, datasets covering illumination changes have been
released. In [17], the handheld rig of RGB, thermal, and GPS (GPS) receiver
was constructed, and collected data at different times. For object detection
in thermal images, Then, KAIST Multi-Spectral Day/Night Dataset [18] have
presented a sensor system including stereo RGB, LiDAR, and thermal camera for
SLAM purposes. Their data were collected with GPS at a large scale and along
with daytime temperature changes.

As listed above, several datasets have provided environmental variances with
different types of sensors in various environments. However, as in Table I,
none of the datasets include multi-sensor configurations for solving motion
disturbances and illumination conditions together, which are two significant
variances in the real world. We insist that both sensors should be utilized
for complementing each other because thermal cameras are more robust by less
dependency on external light sources, and event cameras are better for extreme
motion cases. Therefore we compose a dataset with multiple types of visual
information, at different wavelengths and temporal intensity difference, along
with depth measurements and ground-truth for SLAM.

TABLE II: Sensor specifications and data types Sensors | Specifications | Topic name | Description | Message type
---|---|---|---|---
Thermal |  | FLIR A65
---
640×512 pixel, 30Hz / 20Hz (handheld)
FOV : 90 °vert., 69 °horiz.
Spectral Range : 7.5-13 $\mu$m
| /thermal/camera_info
---
/thermal/image_raw
| Header
---
Image (16bit, 1ch)
| sensor_msgs/CameraInfo
---
sensor_msgs/Image
RGB-D |  | Asus Xtion Pro Live
---
640×480 pixel, 30Hz
FOV : 45 °vert., 58 °horiz.
| /depth/camera_info
---
/depth/image_raw
/rgb/camera_info
/rgb/image
| Header
---
Image (16bit, 1ch)
Header
Image (8bit, 3ch)
| sensor_msgs/CameraInfo
---
sensor_msgs/Image
sensor_msgs/CameraInfo
sensor_msgs/Image
RGB |  | Flea®3 FL3-U3-13E4C-C
---
1280×1024 pixel, 60Hz
FOV : 89 °vert., 73.8 °horiz.
| /camera/camera_info
---
/camera/image_color
| Header
---
Image (8bit, 3ch)
| sensor_msgs/CameraInfo
---
sensor_msgs/Image
Event |  | Handheld : DAVIS 240C
---
240×180 pixel, upto 12 MEPS
Driving : DVXplorer
640×480 pixel, upto 165 MEPS
/dvs/events | Event | dvs_msgs/EventArray
GPS |  | NovAtel Pwrpak7 E-1
---
VEXXIS GNSS-500 Dual Antenna
| /gps
---
/bestpos
GPS |  | gps_common/GPSFix
---
novatel_gps_msgs/NovatelPosition
Inertial |  | LORD Microstrain 3DM-GX5-25
---
On-board IMU (MPU 6150)
On-board IMU (MPU 9250)
On-board IMU (InvenSense ICM-20948)
| /imu/data
---
/dvs/imu
/dvs/imu
/os1_cloud_node/imu
IMU | sensor_msgs/Imu
LiDAR |  | Handheld : Velodyne VLP-16
---
Driving : Ouster OS1-64
| /velodyne_point_cloud
---
/os1_cloud_node/points
Pointcloud | sensor_msgs/PointCloud2

##  III Environment

###  III-A Sensors and Data types

We setup different types of vision sensors: RGB, thermal, and event. As
mentioned in Fig. 1, we configured different sensor systems for handheld and
driving sequences. In both scenarios, the three types of visual sensors were
included, but details differ. IMU was installed for handheld, and the depth
sensor was changed from depth camera to LiDAR for outdoor because we observed
that the depth measurements of a structured light depth camera were unreliable
on the outside. The dataset is provided with the binary format in rosbag. The
rostopic names, sensor model, and specifications are listed in Table II. The
timestamp from the clock running on each sensor frame, was initialized to
rostime at startup and recorded in the header of each message. For the thermal
camera, FLIR A65 was used with the raw format of 16 bits instead of the
typical 8 bits. Note that the actual bits used for image demonstration are 14
bits due to two empty bits in the front. The non-uniformity-correction (NUC)
was set to manual and done at the beginning of each sequence but not during
the experiment. For the event camera, different sensors were used for each
setup. DAVIS240C, which produces events and intensity images, was used for
handheld sequences. And DVXplorer, with higher spatial resolution but without
intensity images, was used for driving sequences. The LiDARs used for each
system also differs. Velodyne LiDAR with 16 channels and 100m range was used
for handheld sequences, and Ouster LiDAR with 64 channels and 120m range was
used for driving sequences.

###  III-B Ground-Truth Generation

####  III-B1 Handheld indoor

A cortex motion capture system KARPE (KAIST Arena with Real-time Positioning
Environment) [21] was used to track the sensor system’s 6 Degree of Freedom
(DoF) pose. The system uses multiple infrared strobes to track the reflected
light from the markers, assigning marker IDs to each track. The ID given to
each marker did not change during the experiment, and the minimum number of
markers to compose a 6DoF tracking was three. To ensure a robust track, we
installed additional markers over the minimum number. As the motion capture
system only provides the position of each marker, not the rotation, we defined
the 6DoF marker frames from the tracked position of markers and provided them
as the ground-truth.

####  III-B2 Handheld outdoor

We have generated poses for handheld outdoor sequences because the motion
capture system was only available with preinstalled indoor equipment, and the
GPS receiver was not able to solve the precise location in the experiment
site. Thus as an evaluation baseline, we used the pointcloud from scans of
VLP-16 attached to the top of the sensor jig and generated pose with LOAM
variants [22][23] with scan context [24] for loop closure.

####  III-B3 Driving

With the dual antenna RTK-GPS installed on the car roof, we obtained a precise
RTK-GPS position as the ground-truth. To ensure the validity of Global
Navigation Satellite System (GNSS), the status of the calculated position was
also recorded with the $/bestpos$ topic.

###  III-C Calibration

For calibration results, we offered a chain of relative poses between sensors.
The sensor chains were connected to the RGB(-D) camera for both handheld and
driving sensor systems, except for the on-board IMU sensors. More details of
sensor layout, sample results, sequence statistics, and the validity of
calibration can be found on the dataset homepage:
https://visibilitydataset.github.io/.

####  III-C1 Calibration between cameras and external IMU

The relative poses between the cameras and an external IMU were estimated by
moving the sensor system in front of the checkerboard and the grid of AprilTag
[25], and calibrated with Kalibr [26]. Also the temporal offset between the
RGB camera and the IMU was estimated by correlating the angular velocity norms
of the sensors. The results of extrinsic calibration are shown in Fig. 2.

(a) RGB-D pointcloud projected
to event frames in the handheld seq.

(b) LiDAR pointcloud projected
to RGB frames in the driving seq.

Figure 2: Pointclouds transformed into the other cameras, by the calibration
results. In (b), LiDAR pointclouds are psudocolored with the depth values.

(a) RGB image of
PCB checkerboard.

(b) Thermal image of
heated PCB.

(c) Image of AprilTag
recovered from events.

Figure 3: Checkerboard pattern used for extrinsic calibration between
alternative vision sensors. In (b), the difference in the heat dissipation
rate of each material results in a temperature pattern. In (c), an intensity
image is reconstructed from events using E2VID [27] and used for calibration.

####  III-C2 Calibration of thermal camera

Because the thermal camera only captures the temperature but not the intensity
difference, a special type of checkerboard was required to make the pattern
detectable for both intensity and thermal cameras. We used a calibration board
made of aluminum coating on the printed circuit board (PCB) as in Fig. 3(a).
To produce detectable edges in the thermal camera, the board was heated before
calibration. Because the heat dissipation of aluminum is higher than plastic,
the aluminum pattern is cooled faster and showed a lower temperature. Then the
checkerboard pattern became visible from both RGB and thermal cameras as in
Fig. 3(b), enabling the thermal camera to be calibrated as the same procedure
of the RGB camera.

####  III-C3 Calibration of event cameras

The DAVIS240C event camera in the handheld sequences produces an intensity
image, enabling image-based calibration. However, in the driving sequences,
the DVXplorer event camera did not have images. Thus we chose to use
reconstructed intensity images from events, using E2VID [27]. With the
reconstructed AprilTag images as in Fig. 3(c), the event camera could be
calibrated as the same way RGB camera was calibrated.

####  III-C4 Calibration between LiDAR and camera

The RGB camera in the handheld sequences contains depth measurements from
structured light. Thus the transformation between a LiDAR and a camera could
be directly computed by running ICP (ICP) [28] on the pointclouds obtained
from both sensors. The transformation is obtained by matching planes and edges
from each sensor. For best results, we have collected measurements at the
intersection of three orthogonal planes for calibration. The relative poses of
sensors were calculated by aligning orthogonal plane points.

In the driving sequences, the extrinsic parameter between a LiDAR and a RGB
camera was found by minimizing reprojection errors of the marker board plane
extracted from a LiDAR and a camera. The extrinsic transformation is $4\times
4$ homogenous transform matrix $\mathbf{T}^{rgb}_{lidar}$, that transforms all
points from the LiDAR to the RGB frame. We captured $i$ numbers of snapshots
with fully visible checkerboard on both the camera and the LiDAR. Then, the
extrinsic calibration was calculated with both point-wise Euclidean and plane
vector distance. The distances are defined as
$\mathbf{D}_{point}=\left\lVert{\mathbf{\hat{n}}_{rgb}\cdot(\mathbf{p}_{rgb}-\mathbf{T}^{rgb}_{lidar}\mathbf{p}_{lidar})}\right\rVert$
and
$\mathbf{D}_{vector}=d(\mathbf{\hat{n}}_{rgb},\mathbf{T}^{rgb}_{lidar}\mathbf{\hat{n}}_{lidar})$,
where $\mathbf{p}_{rgb}$, $\mathbf{\hat{n}}_{rgb}$ are plane points and
normals from the RGB camera, and $\mathbf{p}_{lidar}$,
$\mathbf{\hat{n}}_{lidar}$ are plane points and normals from the LiDAR, and
$d$ is a quaternion distance. Then the extrinsic could be found by solving the
following minimization problem:
$\operatorname*{argmin}_{\mathbf{T}^{rgb}_{lidar}}\sum_{i}(\mathbf{D}_{i,point}+\mathbf{D}_{i,vector})$.
The temporal offset between a LiDAR and sensor system was calculated by
comparing the trajectory obtained from LiDAR and visual odometry. Given the
extrinsic calibration between LiDAR and camera from the method mentioned
above, the temporal offset was estimated by minimizing the Average Trajectory
Error (ATE) [29] upon different temporal offset values.

####  III-C5 Calibration between marker ground-truth and sensor system

The calibration between the motion capture system and the camera was achieved
by comparing the marker’s 6DoF path of the sensor measurements. We defined a
virtual marker frame using four markers, and compared the marker frame
trajectory with the camera poses estimated by recording a checkerboard. The
relative transformation between the marker frame and the camera was obtained
by following Hand-Eye Calibration [30]. We defined the camera frame and the
marker frame at time $t$ as $\mathbf{T}_{Ct}$ and $\mathbf{T}_{Mt}$, and the
transformation of the marker frame from the camera as $\mathbf{T}_{CM}$. Then
we solve the following equation:
$\mathbf{T}^{~{}}_{Ci}\mathbf{T}^{~{}}_{CM}\mathbf{T}^{-1}_{Mi}=\mathbf{T}^{~{}}_{Cj}\mathbf{T}^{~{}}_{CM}\mathbf{T}^{-1}_{Mj}$.
Also, then using relative poses
$\mathbf{T}_{Lij}=\mathbf{T}^{-1}_{Lj}\mathbf{T}^{~{}}_{Li}$ and
$\mathbf{T}_{Mij}=\mathbf{T}^{-1}_{Mj}\mathbf{T}^{~{}}_{Mi}$, the equation is
reformulated into
$\mathbf{T}_{Cij}\mathbf{T}_{CM}=\mathbf{T}_{CM}\mathbf{T}_{Mij}$. Then we
obtain the form of solving $A=\mathbf{T}B\mathbf{T}^{-1}$ for $n$ pairs, and
find error-minimizing transformation $\mathbf{T}_{CM}$. The temporal offset
between ROS and the motion capture system was calculated for each sequence, by
correlating the angular velocity norms of the IMU and the motion capture poses
[31].

(a) PR curves of campus-day1 seq.
matched to campus-day2 DB. (b) PR curves of campus-night seq.
matched to campus-day2 DB.

Figure 4: VPR performances of event(blue) and RGB(yellow) cameras, trained
with images from all of other “day” sequences and tested on the Campus
sequences. By consisting a DB only with Campus-Day2, we tested the robustness
of VPR module to match Campus-Day1/Night images to DB. The event-based VPR has
shown more robustness over appearance changes.

##  IV Dataset

We recorded multiple repetitions along similar trajectories under distinct
illumination conditions and motion constraints. To show the robustness of
alternative vision sensors over illumination conditions, we have shown the
performance of event-based VPR [32] in Fig. 4. To overview the sequences, 3D
reconstructions and samples are in the figures: Figs. 5 and 6 for handheld and
Figs. 7 and 8 for driving.

###  IV-A Handheld sequences

TABLE III: Environment setting for handheld sequences Location | Illumination | Motion | Duration | GT
---|---|---|---|---
Indoor | Global | Slow | 52.5s | Vicon
Unstable | 23.9s
Aggressive | 16.9s
Dark | Slow | 35.0s
Unstable | 20.5s
Aggressive | 16.9s
Local | Slow | 35.0s
Unstable | 17.1s
Aggressive | 13.4s
Varying | Slow | 35.8s
Outdoor | Day1 | Slow | 117s |  LOAM (baseline)
Day2 | Slow | 108s
Night1 | Slow | 101s
Night2 | Slow | 104s

For handheld sequences, we recorded multiple sessions in indoor and outdoor
environments, respectively. In Fig. 6, the trajectory and sample scene images
are displayed. The full sequence list is detailed in Table III.

For handheld indoor, sequences were divided by the level of motion deviation
and lighting conditions. We have repeated trajectories at three different
motion levels (42.7 to 120.7∘/s), which are slow, unstable, and aggressive.
Further, for each motion, we changed the illumination condition, from light
turned on (global), turned off (dark), and flashlight on (local). We also
included a bonus sequence (varying) to gradually change the flashlight
intensity from dark to local. The sensor system’s global poses were captured
via a motion capture system with 12 motion capture cameras mounted on the
wall. The scale of the room was $12.3\text{\,}\mathrm{m}$ $\times$
$8.9\text{\,}\mathrm{m}$.

For handheld outdoor, we trailed the same trajectory at different times
clockwise and counterclockwise. The approximate size of the space was about
$60\text{\,}\mathrm{m}$ $\times$ $40\text{\,}\mathrm{m}$. In the environment,
GPS was not sufficient because multi-story buildings enclosed the experiment
site and yielded a multipath reception problem [33]. However, we were able to
obtain LiDAR odometry from the plenty of planar and edge structures at the
building.

###  IV-B Driving sequences

TABLE IV: Environment setting for driving sequences Type | Location | Time | Duration | GT
---|---|---|---|---
Vision +LiDAR |  Campus (3.6km) | Day1 | 445s |  GPS, LOAM
Day2 | 463s
Evening | 438s
Night | 445s
City (9.3km) | Day1 | 914s
Day2 | 1218s
Evening | 832s
Night | 840s
Vision |  Campus (3.6km) | Morning | 495s | GPS
Day1 | 667s
Day2 | 484s
Evening | 487s
Night | 443s
City (9.3km) | Morning | 858s
Day1 | 1037s
Day2 | 890s
Evening | 1127s
Night | 855s
Urban (3.8km) | Morning | 652s
Day | 785s
Evening | 593s
Night | 598s

For driving sequences, we repeated measurements along the predefined
trajectories at different times. The repetitions were done throughout the
entire day and night to enclose sun elevation changes. The appearances of the
same place broadly change at different times, as seen in Fig. 8. The path was
carefully selected to include visual loop closure, and the recording was done
repeatedly over identical paths.

We set up three different trajectories, around the campus, in the urban
canyon, and along the city’s arterial road. Each of the sequences was named as
Campus, City, and Urban in Table IV. The campus sequence consisted of
pedestrians, bicycles, and medium-rise buildings. In the city sequences, the
vehicle drives along the river, under and over the bridges, and underpasses.
In the urban sequences, the sensor observed high-rise buildings and tall trees
(urban canyon). To provide extra data for unsupervised or self-supervised
learning, we prepared “Vision” sequences recorded only with cameras without a
LiDAR. To ensure reproducibility, we recommend to use the Campus sequences for
validation, as in the setup of Fig. 4. Each traverse was from 3.6 to 9.3km
long, with a maximum linear and angular velocity of 95 km/h and 39 ∘/s. As the
sun elevation angle changes, shadows differ, and large color shifts occurred
in the evening scenes. Further, at night, artificial lights, including street
lamps, were turned on. These illumination changes made the urban city very
different from itself along the time of the day. Although the vehicle followed
the same trajectory, the average speed dropped to 65% by the traffic signals
and traffic congestion.

##  V Issues

The spectral response of the DAVIS240C used for our experiment overlaps with
the infrared wavelength used from Cortex motion capture system. Thus, directly
using DAVIS240C in the motion capture room resulted in an event camera’s high
malfunction, capturing all the flashes of infrared strobes and ruining the
event measurement. Therefore, we attached a C-Mount IR cut filter in front of
the event camera to remove the effects of motion capture strobes.

The event cameras possess very high sensitivity and provide a high dynamic
range from their relative form of luminance illustration. However, due to the
sensitivity, we identified that the event camera produced erroneous output and
decreased data quality when directly observing the light source. As is shown
in the last column of Fig. 8, the event camera shows artifacts from the street
lamps and loses visual information near the lamp. Further studies are required
to neglect artifacts and recover the dynamic range of events.

Figure 5: The samples of appearance changes and corresponding ground-truth or
baseline trajectories (motion capture for indoor, SLAM for outdoor) from
handheld sequences. For indoor trajectories, the measurements were done in the
same room, but with different illumination and motion speed. Outdoor sequences
were recorded by walking around the environment in a clockwise and counter-
clockwise direction at different times.

Figure 6: Handheld scene overview with 3D reconstruction of the environment.
From left to right, sampled images from RGB, depth, event and thermal camera
are shown. The text label at each row represents illumination condition. The
indoor environment was reconstructed with known poses from a motion capture
and a RGB-D camera, and the outdoor was reconstructed by LeGO-LOAM run from
VLP-16 LiDAR attached at the top of the sensor system.

Figure 7: The samples of appearance changes and corresponding GPS trajectories
from Vision for Visibility Dataset driving sequences. Outdoor sequences were
recorded by following predefined path, at different times. Due to the
illumination changes, visible appearance largely shifts as in the figure.

Figure 8: Vision+LiDAR sequences of City and Campus, with reconstruction from
A-LOAM and the current viewpoint indicated with a white arrow. In the city,
road infrastructures such as overpass and underpass / high-rise and large
buildings / riverside are included. In the campus, parked cars / vegetation /
medium-sized buildings / construction sites are observed. We could observe the
light artifacts from the street lamps, in the last row of the event outputs.

##  VI Conclusion

This paper provides a vision for visibility dataset to overcome poor lighting
conditions in robotics applications: beyond the visible light spectrum and
temporal luminance differences. Besides providing a reference environment to
test the performance of the thermal and event cameras, our dataset enables
researchers to determine the vision sensors’ required abilities for robotic
applications: fully identifying the potential of passive cameras in the real
world. We hope our work will help solve robust robot vision regardless of
motion or environmental disturbances by encouraging visual SLAM based on
alternative vision sensors, proposing a test-bed to test optimal camera
characteristics in the real world.

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# Boundedness of fractional integral operators on non-homogeneous metric
measure spaces

Rulong Xie and Lisheng Shu  Rulong Xie, 1. School of Mathematical Sciences,
University of Science and Technology of China, Hefei 230026, China 2.
Department of Mathematics, Chaohu University, Chaohu 238000, China
<EMAIL_ADDRESS>Lisheng Shu, Department of Mathematics, Anhui Normal
University, Wuhu 241000, China<EMAIL_ADDRESS>
###### Abstract.

In this paper, the fractional integral operator on non homogeneous metric
measure spaces is introduced, which contains the classical fractional integral
operator, fractional integral with non-doubling measures and fractional
integral with fractional kernel of order $\alpha$ and regularity $\epsilon$
introduced by García-Cuerva and Gatto as special cases. And the
$(L^{p}(\mu),L^{q}(\mu))$-boundedness for fractional integral operators on
non-homogeneous metric measure spaces is established. From this, the
$(L^{p}(\mu),L^{q}(\mu))$-boundedness for commutators and multilinear
commutators generated by fractional integral operators with $RBMO(\mu)$
function are further obtained. These results in this paper include the
corresponding results on both the homogeneous spaces and non-doubling measure
spaces.

###### 1991 Mathematics Subject Classification:

42B20, 42B25

Supported by National Natural Science Foundation of China (No.10371087) and
Natural Science Foundation of Education Committee of Anhui Province
(No.KJ2011A138, No.KJ2012B116).

##  1\. Introduction

As we know, the theory on spaces of homogeneous type need to assume that
measure $\mu$ of metric spaces $(X,d,\mu)$ satisfies the doubling measure
condition, i.e. there exists a constant $C>0$ such that $\mu(B(x,2r))\leq
C\mu(B(x,r))$ for all $x\in\rm{supp}\mu$ and $r>0$. Recently, many classical
theory have been proved still valid without the assumption of doubling measure
condition, see [2,4-6,8-10,18,19,21-24] and their references. In case of non-
doubling measures, a Radon measure $\mu$ on $R^{n}$ only need to satisfy the
polynomial growth condition, i.e., for all $x\in R^{n}$ and $r>0$, there
exists a constants $C>0$ and $k\in(0,n]$ such that,

(1.1) $\ \mu(B(x,r))\leq C_{0}r^{k},$

where $B(x,r)=\\{y\in R^{n}:|y-x|<r\\}$. The analysis on non-doubling measures
has important applications in solving the long-standing open
Painlev$\acute{e}$’s problem (see [21]).

However, as pointed out by Hytönen in [12], the measure satisfying (1.1) do
not include the doubling measure as special cases. For this reason, a kind of
metric measure spaces $(X,d,\mu)$ satisfying geometrically doubling and the
upper doubling measure condition (see Definition 1.1 and 1.2) is introduced by
Hytönen in [12], which is called non-homogeneous metric measure spaces. The
advantage of this kind of spaces is that it includes both the homogeneous
spaces and metric spaces with polynomial growth measures as special cases.
From then on, some results paralled to homogeneous spaces and non-doubling
measure spaces are obtained (see [1,3,11-17] and the references therein). For
example, Hytönen et al. in [14] and Bui and Duong in [1] independently
introduced the atomic Hardy space $H^{1}(\mu)$ and obtained that the dual
space of $H^{1}(\mu)$ is $RBMO(\mu)$. Hytönen and Martikainen [13] established
the $T_{b}$ theorem in this surroundings. In [1], Bui and Duong also obtained
that Calderón-Zygmund operator and commutators of Calderón-Zygmund operators
with $RBMO(\mu)$ function are bounded in $L^{p}(\mu)$ for $1<p<\infty$. Later,
Lin and Yang [16] introduced the space $RBLO(\mu)$ and proved the maximal
Calderón-Zygmund operators is bounded from $L^{\infty}(\mu)$ into $RBLO(\mu)$.
Recently, some equivalent characterizations are established by Liu, Yang Da.
and Yang Do. in [17] for the boundedness of Carder on-Zygmund operators on
$L^{p}(\mu)$ for $1<p<\infty$. The boundedness and weak type endpoint estimate
of multilinear commutators of Calderón-Zygmund operators on non-homogeneous
metric spaces is established by Fu, Yang and Yuan in [3]. And weighted
estimate for multilinear Calderón-Zygmund operators on non-homogeneous metric
spaces is also obtained by Hu, Meng and Yang in [11].

The purpose of this paper is to establish the theory of fractional integral
operators on non-homogeneous metric measure spaces. At first, fractional
integral operators on non-homogeneous metric measure spaces is introduced.
This kind of fractional integral operators contains the classical fractional
integral operator, fractional integral with non-doubling measures and
fractional integral with fractional kernel of order $\alpha$ and regularity
$\epsilon$ introduced by García-Cuerva and Gatto in [4] as special cases. The
$(L^{p}(\mu),L^{q}(\mu))$-boundedness for fractional integral operators on
non-homogeneous metric measure spaces is obtained. From this result, the
boundedness of commutators and multilinear commutators generated by fractional
integral operators with $RBMO(\mu)$ function are also established. The results
in this paper include the corresponding results on both the homogeneous spaces
and non-doubling measure spaces.

For the sake of the reader’s convenience, let us give some references about
previous results of fractional integral operators which are closely related to
results in this paper. Classical fractional integral theory can be founded in
[20]. In the circumstance of non-doubling measures,
$(L^{p}(\mu),L^{q}(\mu))$-boundedness for fractional integral operators is
established in [4,5]. The boundedness of commutators and multilinear
commutators generated by fractional integral operators with $RBMO(\mu)$
function were established in [2] and [8] respectively. In addition, it’s worth
mentioning that the Besicovitch covering lemma is only applicable to $R^{n}$,
but is not applicable to non-homogeneous metric measure spaces. Therefore, in
the process of proving Lemma 2.2 in this paper, we will adopt a covering lemma
introduced in [7].

Before stating the main results, we firstly recall some notations and
definitions.

Definition 1.1.[12] A metric spaces $(X,d)$ is called geometrically doubling
if there exist some $N_{0}\in\mathbb{N}$ such that, for any ball
$B(x,r)\subset X$, there exists a finite ball covering
$\\{B(x_{i},r/2)\\}_{i}$ of $B(x,r)$ such that the cardinality of this
covering is at most $N_{0}$.

Definition 1.2.[12] A metric measure space $(X,d,\mu)$ is said to be upper
doubling if $\mu$ is a Borel measure on $X$ and there exists a dominating
function $\lambda:X\times(0,+\infty)\rightarrow(0,+\infty)$ and a constant
$C_{\lambda}>0$ such that for each $x\in X,r\longmapsto(x,r)$ is non-
decreasing, and for all $x\in X,r>0$,

(1.2) $\mu(B(x,r))\leq\lambda(x,r)\leq C_{\lambda}\lambda(x,r/2)$

Remark 1.1.(i) A space of homogeneous type is a special case of upper doubling
spaces, where one can take the dominating function
$\lambda(x,r)\equiv\mu(B(x,r))$. On the other hand, a metric space $(X,d,\mu)$
satisfying the polynomial growth condition (1.1) (in particular,
$(X,d,\mu)\equiv(R^{n},|\cdot|,\mu)$ with $\mu$ satisfying (1.1) for some
$k\in(0,n])$) is also an upper doubling measure space if we take
$\lambda(x,r)\equiv Cr^{k}$.

(ii) Let$(X,d,\mu)$ be an upper doubling space and $\lambda$ be a dominating
function on $X\times(0,+\infty)$ as in Definition 1.2. In [15], it was showed
that there exists another dominating function $\widetilde{\lambda}$ such that
for all $x,y\in X$ with $d(x,y)\leq r$,

(1.3) $\widetilde{\lambda}(x,r)\leq\widetilde{C}\widetilde{\lambda}(y,r).$

Thus, in this paper, we always suppose that $\lambda$ satisfies (1.3).

Definition 1.3. Let $\alpha,\beta\in(1,+\infty)$. A ball $B\subset X$ is
called $(\alpha,\beta)$-doubling if $\mu(\alpha B)\leq\beta\mu(B)$.

As stated in Lemma 2.3 of [1], there exist plenty of doubling balls with small
radii and with large radii. In the rest of the paper, unless $\alpha$ and
$\beta$ are specified otherwise, by an $(\alpha,\beta)$ doubling ball we mean
a $(6,\beta_{0})$-doubling with a fixed number
$\beta_{0}>\max\\{C_{\lambda}^{3log_{2}6},6^{n}\\}$, where $n=log_{2}N_{0}$ be
viewed as a geometric dimension of the spaces.

Definition 1.4. Let $0\leq\beta<n$ and $N_{B,Q}$ be the smallest integer
satisfying $6^{N_{B,Q}}r_{B}\geq r_{Q}$, then we set

(1.4)
$K^{(\beta)}_{B,Q}=1+\sum_{k=1}^{N_{B,Q}}\biggl{[}\frac{\mu(6^{k}B)}{\lambda(x_{B},6^{k}r_{B})}\biggr{]}^{1-\beta/n}.$

When $\beta=0$, then we denote $K^{(0)}_{B,Q}$ by $K_{B,Q}$, which is firstly
defined in [22].

Definition 1.5.[1] Let $\rho>1$ be some fixed constant. A function $b\in
L_{loc}^{1}(\mu)$ is said to belong to $RBMO(\mu)$ if there exists a constant
$C>0$ such that for any ball $B$

(1.5) $\frac{1}{\mu(\rho B)}\int_{B}|b(x)-m_{\widetilde{B}}b|d\mu(x)\leq C,$

and for any two doubling balls $B\subset Q$,

(1.6) $|m_{B}(b)-m_{Q}(b)|\leq CK_{B,Q}.$

$\widetilde{B}$ is the smallest $(\alpha,\beta)$-doubling ball of the form
$6^{k}B$ with $k\in{\mathbb{N}}\bigcup\\{0\\}$, and $m_{\widetilde{B}}(b)$ is
the mean value of $b$ on $\widetilde{B}$, namely,

$m_{\widetilde{B}}(b)=\frac{1}{\mu(\widetilde{B})}\int_{\widetilde{B}}b(x)d\mu(x).$

The minimal constant $C$ appearing in (1.5) and (1.6) is defined to be the
$RBMO(\mu)$ norm of $b$ and denoted by $||b||_{\ast}$. The norm $||b||_{\ast}$
is independent of $\rho>1$.

Next, let us introduce fracional integral operator on nonhomogeneous metric
measure spaces.

Definition 1.6. Let $0<\alpha<n$ and $0<\epsilon\leq 1$. A function
$K_{\alpha}(\cdot,\cdot)\in L_{loc}^{1}(X\times X\backslash\\{(x,y):x=y\\})$
is said to be a fractional kernel of order $\alpha$ and regularity $\epsilon$
if it satisfies the following two conditions:

(i)

(1.7)
$\quad|K_{\alpha}(x,y)|\leq\frac{C}{\biggl{[}\lambda(x,d(x,y))\biggr{]}^{1-\alpha/n}}$

for all $x\neq y$.

(ii) There exists $0<\epsilon\leq 1$ such that

(1.8)
$|K_{\alpha}(x,y)-K_{\alpha}(x^{\prime},y)|+|K_{\alpha}(y,x)-K_{\alpha}(y,x^{\prime})|\leq\frac{Cd(x,x^{\prime})^{\epsilon}}{d(x,y)^{\epsilon}\biggl{[}\lambda(x,d(x,y))\biggr{]}^{1-\alpha/n}},$

proved that $Cd(x,x^{\prime})\leq d(x,y)$.

A operator $I_{\alpha}$ is called a fractional integral operator on non-
homogeneous metric measure spaces with the above fractional kernel
$K_{\alpha}$ satisfying (1.7) and (1.8) if, for $f\in L^{\infty}$ functions
with compact support and $x\notin suppf$,

(1.9) $I_{\alpha}f(x)=\int_{X}K_{\alpha}(x,y)f(y)d\mu(y).$

Remark 1.2. By taking $\lambda(x,d(x,y))=Cd(x,y)^{n}$, it is easy to see that
Definition 1.6 in this paper contains Definition 3.1 and Definition 4.1
introduced by García-Cuerva and Gatto in [4]. Obviously, it also contains the
classical fractional integral operator

$I_{\alpha}f(x)=\int_{R^{d}}\frac{f(y)}{|x-y|^{n-\alpha}}d\mu(y)$

as special case.

Definition 1.7. The commutators $[b,I_{\alpha}]$ generated by fractional
integral operator $I_{\alpha}$ with $RBMO(\mu)$ function $b$ is defined by

(1.10) $[b,I_{\alpha}](f)(x)=b(x)I_{\alpha}(f)(x)-I_{\alpha}(bf)(x).$

Definition 1.8. The multilinear commutators $I_{\alpha,\vec{b}}$ is formally
defined by

(1.11)
$I_{\alpha,\vec{b}}f(x)=[b_{k},[b_{k-1},\cdot\cdot\cdot,[b_{1},I_{\alpha}]]]f(x)$

where $\vec{b}=(b_{1},b_{2},\cdot\cdot\cdot,b_{k})$, and

$[b_{1},I_{\alpha}]f(x)=b_{1}(x)I_{\alpha}f(x)-I_{\alpha}(b_{1}f)(x).$

For $1\leq i\leq k$, we denote by $C_{i}^{k}$ the family of all finite subsets
$\sigma=\\{\sigma(1),\sigma(2),\cdot\cdot\cdot,\sigma(i)\\}$ of
$\\{1,2,\cdot\cdot\cdot,k\\}$ with $i$ different elements. For any $\sigma\in
C_{i}^{k}$, the complementary sequences $\sigma^{\prime}$ is given by
$\sigma^{\prime}=\\{1,2,\cdot\cdot\cdot,k\\}\backslash\sigma$. Let
$\vec{b}=(b_{1},b_{2},\cdot\cdot\cdot,b_{k})$ be a finite family of locally
integrable functions. For all $1\leq i\leq k$ and
$\sigma=\\{\sigma(1),\cdot\cdot\cdot,\sigma(i)\\}\in C_{i}^{k}$, we set
$\vec{b}_{\sigma}=(b_{\sigma(1)},\cdot\cdot\cdot,b_{\sigma(i)})$ and the
product $b_{\sigma}=b_{\sigma(1)}\cdot\cdot\cdot b_{\sigma(i)}$. With this
notation, we write

$||\vec{b}_{\sigma}||_{\ast}=||b_{\sigma(1)}||_{\ast}\cdot\cdot\cdot||b_{\sigma(i)}||_{\ast}.$

In particular, for $i\in\\{1,\cdot\cdot\cdot,k\\}$ and
$\sigma=\\{\sigma(1),\cdot\cdot\cdot,\sigma(i)\\}\in C_{i}^{k}$,

$[b(y)-b(z)]_{\sigma}=[b_{\sigma(1)}(y)-b_{\sigma(1)}(z)]\cdot\cdot\cdot[b_{\sigma(i)}(y)-b_{\sigma(i)}(z)],$

and

$[b(y)-m_{\widetilde{B}}(b)]_{\sigma}=[b_{\sigma(1)}(y)-m_{\widetilde{B}}(b_{\sigma(1)})]\cdot\cdot\cdot[b_{\sigma(i)}(y)-m_{\widetilde{B}}(b_{\sigma(i)})],$

where $B$ is any ball in $X$ and $y,z\in X$. For the product of all the
functions, we simply write

$||\vec{b}||_{\ast}=||b_{1}||_{\ast}\cdot\cdot\cdot||b_{k}||_{\ast}.$

With any $\sigma\in C_{i}^{k}$, we set

$I_{\alpha,\vec{b_{\sigma}}}f(x)=[b_{\sigma(i)},[b_{\sigma(i-1)},\cdot\cdot\cdot,[b_{\sigma(1)},I_{\alpha}]]]f(x).$

In particular, when $\sigma=\\{1,\cdot\cdot\cdot,k\\}$, we denote
$I_{\alpha,\vec{b_{\sigma}}}$ simply by $I_{\alpha,\vec{b}}$.

The main results of this paper are stated as follows.

######  Theorem 1.1.

Let $0<\alpha<n$, $1<p<n/\alpha$ and $1/q=1/p-\alpha/n$. $I_{\alpha}$ is
defined by (1.9). Then there exists a constant $C>0$ such that for all $f\in
L^{p}(\mu)$,

(1.12) $||I_{\alpha}f||_{L^{q}(\mu)}\leq C||f||_{L^{p}(\mu)}.$

######  Theorem 1.2.

Let $0<\alpha<n$, $1<p<n/\alpha$ and $1/q=1/p-\alpha/n$. Suppose that $b\in
RBMO(\mu)$ and $[b,I_{\alpha}]$ is defined by (1.10). Then there exists a
constant $C>0$ such that for all $f\in L^{p}(\mu)$,

(1.13) $||[b,I_{\alpha}](f)||_{L^{q}(\mu)}\leq
C||b||_{\ast}||f||_{L^{p}(\mu)}.$

######  Theorem 1.3.

Let $0<\alpha<n$, $1<p<n/\alpha$ and $1/q=1/p-\alpha/n$. Suppose that
$b_{i}\in RBMO(\mu),i=1,2,\cdots,k$ and $I_{\alpha,\vec{b}}$ is defined by
(1.11). Then there exists a constant $C>0$ such that for all $f\in
L^{p}(\mu)$,

(1.14) $||I_{\alpha,\vec{b}}(f)||_{L^{q}(\mu)}\leq
C||\vec{b}||_{\ast}||f||_{L^{p}(\mu)}.$

Throughout this paper, $\chi_{E}$ denotes the characteristic function of set
$E$. $C$ always denotes a positive constant independent of the main parameters
involved, but it may be different from line to line. And $p^{\prime}$ is the
conjugate index of $p$, namely, $\dfrac{1}{p}+\dfrac{1}{p^{\prime}}=1.$

##  2\. Proof of Theorem 1.1

To prove Theorem 1.1, we need to give the following notations and lemmas.

Let $0\leq\beta<n$ and $f\in L_{loc}^{1}(\mu)$, the sharp maximal operator is
defined by

(2.1) $\begin{split}M^{\sharp,(\beta)}f(x)&=\sup_{B\ni
x}\frac{1}{\mu(6B)}\int_{B}|f(y)-m_{\widetilde{B}}(f)|d\mu(y)\\\
&\quad+\sup_{(B,Q)\in\Delta_{x}}\frac{|m_{B}(f)-m_{Q}(f)|}{K^{(\beta)}_{B,Q}},\end{split}$

where $\Delta_{x}:=\\{(B,Q):x\in B\subset Q\ \text{and $B,Q$ are two doubling
balls}\\}$ and the non centered doubling maximal operator is denoted by

$Nf(x)=\sup_{B\ni x,\ B\ doubling}\frac{1}{\mu(B)}\int_{B}|f(y)|d\mu(y).$

By the Lebesgue differentiation theorem, it is easy to see that for any $f\in
L_{loc}^{1}(\mu)$ and almost every $x\in X$,

(2.2) $|f(x)|\leq Nf(x).$

Moreover, $N$ is of weak type (1,1) and bounded on
$L^{p}(\mu),1<p\leq+\infty$.

From Theorem 4.2 in [1] or Lemma 4 in [2], it is easy to obtain that

######  Lemma 2.1.

Let $0\leq\beta<n$ and $f\in L^{1}_{loc}(\mu)$ with $\int_{X}f(x)d\mu(x)=0$ if
$||\mu||<\infty$. For $1<p<\infty$, if $\inf(1,Nf)\in L^{p}(\mu)$, then there
exists a constant $C>0$ such that

(2.3) $||N(f)||_{L^{p}(\mu)}\leq C||M^{\sharp,(\beta)}(f)||_{L^{p}(\mu)}.$

######  Lemma 2.2.

Let $\eta\geq 5$, $0\leq\beta<n/p$, $r<p<n/\beta$ and $1/q=1/p-\beta/n$. Then

(2.4) $||M_{r,(\eta)}^{(\beta)}f||_{L^{q}(\mu)}\leq C||f||_{L^{p}(\mu)},$

where the non-centered maximal operator $M_{r,(\eta)}^{(\beta)}$ is defined by

(2.5) $M_{r,(\eta)}^{(\beta)}f(x)=\sup_{B\ni x}\biggl{\\{}\frac{1}{\mu(\eta
B)^{1-\beta r/n}}\int_{B}|f(y)|^{r}d\mu(y)\biggr{\\}}^{1/r}.$

Remark 2.1. If $\beta=0$, we denote $M_{r,(\eta)}^{(\beta)}$ simply by
$M_{r,(\eta)}$. From [1], we obtain that if $\eta\geq 5$ and $q>r$, then

(2.6) $||M_{r,(\eta)}f||_{L^{q}(\mu)}\leq C||f||_{L^{q}(\mu)}.$

###### Proof.

(of Lemma 2.2) Let us firstly prove the following result.

(2.7)
$\mu(\\{x:M_{r,(\eta)}^{(\beta)}f(x)>\lambda\\})\leq(\frac{C}{\lambda}||f||_{L^{r}(\mu)})^{nr/(n-\beta
r)}.$

Let $E_{\lambda}=\\{x:M_{r,(\eta)}^{(\beta)}f(x)>\lambda\\}$. By the
definition of $M_{r,(\eta)}^{(\beta)}$, for any $x\in E_{\lambda}$, there
exists a ball $B_{x}$ containing $x$ such that

(2.8) $\frac{1}{\mu(\eta B_{x})^{1-\beta
r/n}}\int_{B_{x}}|f(x)|^{r}d\mu(x)\geq\lambda^{r}.$

Note that for $\eta\geq 5$. By Theorem 1.2 in [7], we can pick a disjoint
collection $\\{B_{x_{i}}\\}$ with $x_{i}\in E_{\lambda}$ and
$E_{\lambda}\subset\bigcup_{x\in
E_{\lambda}}B_{x}\subset\bigcup_{i}5B_{x_{i}}\subset\bigcup_{i}\eta
B_{x_{i}}$. Let $q=\frac{nr}{n-\beta r}$ and $r/q<1$, then

(2.9)
$\mu(E_{\lambda})^{r/q}\leq\mu(\bigcup_{i}5B_{x_{i}})^{r/q}\leq\mu(\bigcup_{i}\eta
B_{x_{i}})^{r/q}\leq\sum_{i}\mu(\eta B_{x_{i}})^{r/q}.$

Since $r/q=1-\beta r/n$, then

(2.10) $\sum_{i}\mu(\eta
B_{x_{i}})^{r/q}\leq\frac{1}{\lambda^{r}}\int_{X}|f|^{r}(\sum_{i}\chi_{B_{x_{i}}})d\mu.$

Therefore,

(2.11) $\mu(E_{\lambda})\leq\frac{C}{\lambda^{q}}||f||_{L^{r}(\mu)}^{q}.$

If $r<s<n/\beta$, using the Hölder’s inequality, we deduce

(2.12) $M_{r,(\eta)}^{(\beta)}f(x)\leq{M}_{s,(\eta)}^{(\beta)}f(x).$

By the preceding arguments, then we obtain

(2.13)
$\mu(E_{\lambda})\leq\mu(\\{x:{M}_{s,(\eta)}^{(\beta)}f(x)>\lambda\\})\leq(\frac{C}{\lambda}||f||_{L^{s}(\mu)})^{ns/(n-\beta
s)}.$

By the Marcinkiewicz interpolation theorem, the proof of Lemma 2.2 is
completed. ∎

With the similar method to proof of Lemma 3 in [2] or Lemma 2.1 in [22], it is
easy to obtain the following Lemma 2.3. Here we omit the details.

######  Lemma 2.3.

For $0\leq\beta<n$, we have the following properties:

(1) If $B\subset Q\subset R$ are balls in $X$, then $K^{(\beta)}_{B,Q}\leq
K^{(\beta)}_{B,R}$, $K^{(\beta)}_{Q,R}\leq K^{(\beta)}_{B,R}$ and
$K^{(\beta)}_{B,R}\leq C(K^{(\beta)}_{B,Q}+K^{(\beta)}_{Q,R})$

(2) If $B\subset Q$ have comparable sizes, then $K^{(\beta)}_{B,Q}\leq C$.

(3) If N is a positive integer and the balls $6B,6^{2}B,\cdots,6^{N-1}B$ are
non doubling balls, then $K^{(\beta)}_{B,6^{N}B}\leq C$.

Lemma 2.4 and Lemma 2.5 can be obtained by analogue to Lemma 3.11 and Lemma
3.12 in [3] or Lemma 5 and Lemma 6 in [2] respectively. Here we omit the
details of proof.

######  Lemma 2.4.

For $0\leq\beta<n$, there exists a positive constant $P_{\beta}$ (big enough),
depending on $\beta$, $n$ and $C_{\lambda}$, such that if $B_{1}\subset
B_{2}\subset\cdots\subset B_{m}$ are concentric balls with
$K^{(\beta)}_{B_{i},B_{i+1}}>P_{\beta}$ for $i=1,2,\cdots,m-1$, then there
exists a positive constant $C$, depending on $\beta$, $n$ and $C_{\lambda}$,
such that $\sum_{i=1}^{m-1}K^{(\beta)}_{B_{i},B_{i+1}}\leq
CK^{(\beta)}_{B_{1},B_{m}}$.

######  Lemma 2.5.

For $0\leq\beta<n$, there exists a positive constant $P^{\prime}_{\beta}$ (big
enough), depending on $\beta$, $n$ and $C_{\lambda}$, such that if $x\in X$ is
some fixed point and $\\{f_{B}\\}_{B\ni x}$ is a set of numbers such that
$|f_{B}-f_{Q}|\leq C_{x}$ for all doubling balls $B\subset Q$ with $x\in B$
such that $K^{(\beta)}_{B,Q}\leq P^{\prime}_{\beta}$, then there exists a
positive constant $C$, depending on $\beta$, $n$ and $C_{\lambda}$, such that
for all foubling balls $B\subset Q$ with $x\in B$, $|f_{B}-f_{Q}|\leq
CK^{(\beta)}_{B,Q}C_{x}$.

###### Proof.

(of Theorem1.1) For all $1<p<n/\alpha$, we firstly establish the following
sharp maximal function estimate

(2.14) $M^{\sharp,(\alpha)}(I_{\alpha}f)(x)\leq CM^{(\alpha)}_{r,(5)}f(x).$

Suppose (2.14) is valid for a moment. Choosing $r$ such that $1<r<p<n/\alpha$
and $1/q=1/p-\alpha/n$. By Lemma 2.1 and Lemma 2.2, we have

(2.15)
$\begin{split}&||I_{\alpha}f||_{L^{q}(\mu)}\leq||N(I_{\alpha}f)||_{L^{q}(\mu)}\leq
C||M^{\sharp,(\alpha)}(I_{\alpha}f)||_{L^{q}(\mu)}\\\
\leq&C||M^{(\alpha)}_{r,(5)}f||_{L^{q}(\mu)}\leq
C||f||_{L^{p}(\mu)}.\end{split}$

As in the proof of Theorem 9.1 in [22], to obtain (2.14), by Lemma 2.4 and
Lemma 2.5, it suffices to show that

(2.16)
$\frac{1}{\mu(6B)}\int_{B}|I_{\alpha}f(y)-m_{B}(I_{\alpha}(f\chi_{X\backslash\frac{6}{5}B}))|d\mu(y)\leq
CM^{(\alpha)}_{r,(5)}f(x).$

holds for any $x$ and ball $B$ with $x\in B$, and

(2.17)
$|m_{B}(I_{\alpha}(f\chi_{X\backslash\frac{6}{5}B}))-m_{Q}(I_{\alpha}(f\chi_{X\backslash\frac{6}{5}Q}))|\leq
CK^{(\alpha)}_{B,Q}M^{(\alpha)}_{r,(5)}f(x)$

for all balls $B\subset Q$ with $x\in B$, where $B$ is an arbitrary ball and
$Q$ is a doubling ball.

For any ball $B$, it is easy to see that

(2.18)
$\begin{split}&\frac{1}{\mu(6B)}\int_{B}|I_{\alpha}f(y)-m_{B}(I_{\alpha}(f\chi_{X\backslash\frac{6}{5}B}))|d\mu(y)\\\
\leq&\frac{1}{\mu(6B)}\int_{B}|I_{\alpha}(f\chi_{\frac{6}{5}B})(y)|d\mu(y)\\\
&\ \ \
+\frac{1}{\mu(6B)}\int_{B}|I_{\alpha}(f\chi_{X\backslash\frac{6}{5}B})(y)-m_{B}(I_{\alpha}(f\chi_{X\backslash\frac{6}{5}B}))|d\mu(y)\\\
=:&I_{1}+I_{2}.\end{split}$

To estimate $I_{1}$, by (i) in Definition 1.6, the properties of function
$\lambda$ and the Hölder’s inequality, we get

(2.19)
$\begin{split}I_{1}&\leq\frac{C}{\mu(6B)}\int_{B}\int_{\frac{6}{5}B}\frac{|f(z)|}{\lambda(y,d(y,z))^{1-\alpha/n}}d\mu(z)d\mu(y)\\\
&\leq\frac{C}{\mu(6B)}\int_{B}\int_{\frac{6}{5}B}\frac{|f(z)|}{\lambda(y,r_{B})^{1-\alpha/n}}d\mu(z)d\mu(y)\\\
&\leq\frac{C}{\mu(6B)}\int_{B}\int_{\frac{6}{5}B}\frac{|f(z)|}{\lambda(y,r_{6B})^{1-\alpha/n}}d\mu(z)d\mu(y)\\\
&\leq\frac{C}{\mu(6B)}\int_{B}\int_{\frac{6}{5}B}\frac{|f(z)|}{\mu(6B)^{1-\alpha/n}}d\mu(z)d\mu(y)\\\
&\leq\frac{C}{\mu(6B)}\int_{B}\biggl{[}\frac{1}{\mu(6B)^{1-\frac{\alpha
r}{n}}}\int_{\frac{6}{5}B}|f(z)|^{r}d\mu(z)\biggr{]}^{\frac{1}{r}}\biggl{[}\frac{\mu(\frac{6}{5}B)}{\mu(6B)}\biggr{]}^{1-\frac{1}{r}}d\mu(y)\\\
&\leq CM^{(\alpha)}_{r,(5)}f(x).\\\ \end{split}$

For $y,y_{0}\in B$, by (ii) in Definition 1.6 and the properties of function
$\lambda$ and the Hölder’s inequality, we obtain

(2.20)
$\begin{split}&|I_{\alpha}(f\chi_{X\backslash\frac{6}{5}B})(y)-I_{\alpha}(f\chi_{X\backslash\frac{6}{5}B})(y_{0})|\\\
\leq&C\int_{X\backslash\frac{6}{5}B}\frac{d(y,y_{0})^{\epsilon}}{d(y,z)^{\epsilon}\lambda(y,d(y,z))^{1-\alpha/n}}|f(z)|d\mu(z)\\\
\leq&C\sum_{k=1}^{\infty}\int_{6^{k}\frac{6}{5}B\backslash
6^{k-1}\frac{6}{5}B}6^{-k\epsilon}\frac{|f(z)|d\mu(z)}{\lambda(y,r_{5\times
6^{k}\frac{6}{5}B})^{1-\alpha/n}}\\\
\leq&C\sum_{k=1}^{\infty}6^{-k\epsilon}\biggl{[}\frac{1}{\mu(5\times
6^{k}\frac{6}{5}B)^{1-\frac{\alpha
r}{n}}}\int_{6^{k}\frac{6}{5}B}|f(z)|^{r}d\mu(z)\biggr{]}^{\frac{1}{r}}\biggl{[}\frac{\mu(6^{k}\frac{6}{5}B)}{\mu(5\times
6^{k}\frac{6}{5}B)}\biggr{]}^{1-\frac{1}{r}}\\\
\leq&CM^{(\alpha)}_{r,(5)}f(x).\\\ \end{split}$

Taking the mean over $y_{0}\in B$, then we have

(2.21) $I_{2}\leq CM^{(\alpha)}_{r,(5)}f(x).$

So (2.16) holds from (2.18) to (2.21).

Now we prove (2.17). Consider two balls $B\subset Q$ with $x\in B$, where $B$
is an arbitrary ball and $Q$ is a doubling ball. Let $N=N_{B,Q}+1$, then we
have

(2.22)
$\begin{split}&\biggl{|}m_{B}\biggr{[}I_{\alpha}(f\chi_{X\backslash\frac{6}{5}B})\biggr{]}-m_{Q}\biggl{[}I_{\alpha}(f\chi_{X\backslash\frac{6}{5}Q})\biggr{]}\biggr{|}\\\
\leq&\biggl{|}m_{B}\biggl{[}I_{\alpha}(f\chi_{X\backslash
6^{N}B})\biggr{]}-m_{Q}\biggl{[}I_{\alpha}(f\chi_{X\backslash
6^{N}B})\biggr{]}\biggr{|}\\\ &\ \
+\biggl{|}m_{B}\biggl{[}I_{\alpha}(f\chi_{6^{N}B\backslash\frac{6}{5}B})\biggr{]}\biggr{|}+\biggl{|}m_{Q}\biggl{[}I_{\alpha}(f\chi_{6^{N}B\backslash\frac{6}{5}Q})\biggr{]}\biggr{|}\\\
=&:J_{1}+J_{2}+J_{3}.\end{split}$

With the same method to estimate $I_{2}$, we immediately get

(2.23) $J_{1}\leq CM^{(\alpha)}_{r,(5)}f(x).$

To estimate $J_{2}$, for $z\in B$, it is easy to see that

(2.24)
$|I_{\alpha}(f\chi_{6^{N}B\backslash\frac{6}{5}B})(z)|\leq|I_{\alpha}(f\chi_{6^{N}B\backslash
6B})(z)|+|I_{\alpha}(f\chi_{6B\backslash\frac{6}{5}B})(z)|.$

By (i) in Definition 1.6 and the Hölder’s inequality, we deduce

(2.25) $\begin{split}&|I_{\alpha}(f\chi_{6^{N}B\backslash 6B})(z)|\\\
\leq&C\int_{6^{N}B\backslash
6B}\frac{|f(y)|}{\lambda(z,d(z,y))^{1-\alpha/n}}d\mu(y)\\\
\leq&C\sum_{k=1}^{N-1}\int_{6^{k+1}B\backslash
6^{k}B}\frac{|f(y)|}{\lambda(z,r_{6^{k+1}B})^{1-\frac{\alpha}{n}}}d\mu(y)\\\
\leq&C\sum_{k=1}^{N-1}[\frac{\mu(5\times 6^{k+1}B)}{\lambda(z,r_{5\times
6^{k+1}B})}]^{1-\frac{\alpha}{n}}\frac{1}{\mu(5\times
6^{k+1}B)^{1-\frac{\alpha}{n}}}\int_{6^{k+1}B}|f(y)|d\mu(y)\\\
\leq&C\sum_{k=1}^{N-1}[\frac{\mu(5\times 6^{k+1}B)}{\lambda(z,r_{5\times
6^{k+1}B})}]^{1-\frac{\alpha}{n}}\biggl{[}\frac{1}{\mu(5\times
6^{k+1}\frac{6}{5}B)^{1-\frac{\alpha
r}{n}}}\int_{6^{k+1}\frac{6}{5}B}|f(y)|^{r}d\mu(y)\biggr{]}^{\frac{1}{r}}\\\
&\ \times\biggl{[}\frac{\mu(6^{k+1}\frac{6}{5}B)}{\mu(5\times
6^{k+1}\frac{6}{5}B)}\biggr{]}^{1-\frac{1}{r}}\\\
\leq&CK^{(\alpha)}_{B,Q}M^{(\alpha)}_{r,(5)}f(x).\\\ \end{split}$

Also, we obtain that

(2.26) $\begin{split}&|I_{\alpha}(f\chi_{6B\backslash\frac{6}{5}B})(z)|\\\
\leq&C\int_{6B\backslash\frac{6}{5}B}\frac{|f(y)|}{\lambda(z,r_{B})^{1-\alpha/n}}d\mu(y)\\\
\leq&\frac{C}{\mu(5\times 6B)^{1-\alpha/n}}\int_{6B}|f(y)|d\mu(y)\\\
\leq&C\biggl{[}\frac{1}{\mu(5\times 6B)^{1-\frac{\alpha
r}{n}}}\int_{6B}|f(y)|^{r}d\mu(y)\biggr{]}^{\frac{1}{r}}\biggl{[}\frac{\mu(6B)}{\mu(5\times
6B)}\biggr{]}^{1-\frac{1}{r}}\\\ \leq&CM^{(\alpha)}_{r,(5)}f(x).\\\
\end{split}$

Then

(2.27) $|I_{\alpha}(f\chi_{6^{N}B\backslash\frac{6}{5}B})(z)|\leq
CK^{(\alpha)}_{B,Q}M^{(\alpha)}_{r,(5)}f(x).$

Taking mean for $z$ over $B$, we have

(2.28) $J_{2}\leq CK^{(\alpha)}_{B,Q}M^{(\alpha)}_{r,(5)}f(x).$

Similarly, we also obtain that

(2.29) $J_{3}\leq CK^{(\alpha)}_{B,Q}M^{(\alpha)}_{r,(5)}f(x).$

From (2.22) to (2.29), we yield (2.17) holds. Hence the proof of Theorem 1.1
is completed. ∎

##  3\. Proof of Theorem 1.2

Let us firstly give the equivalent definition of $RBMO(\mu)$, which is useful
in proving Theorem 1.2.

Definition 3.1.[8] Let $\rho>1$ be some fixed constant. A function $b\in
L_{loc}^{1}(\mu)$ is said to belong to $RBMO(\mu)$ if there exists a constant
$C>0$ such that for any ball $B$, a number $b_{B}$ such that

(3.1) $\frac{1}{\mu(\rho B)}\int_{B}|b(x)-b_{B}|d\mu(x)\leq C,$

and for any two balls $B\subset Q$,

(3.2) $|b_{B}-b_{Q}|\leq CK_{B,Q}.$

The minimal constant $C$ appearing in (3.1) and (3.2) is defined to be the
$RBMO(\mu)$ norm of $f$ and denoted by $||b||_{\ast}$. The norm $||b||_{\ast}$
is independent of $\rho>1$.

######  Lemma 3.1.

[22]For any ball $B$, we have

(3.3) $|b_{B}-b_{6^{k}\frac{6}{5}B}|\leq Ck||b||_{\ast}.$

###### Proof.

(of Theorem1.2) For all $1<p<n/\alpha$, we firstly establish the following
sharp maximal function estimate

(3.4) $M^{\sharp,(\alpha)}([b,I_{\alpha}]f)(x)\leq
C||b||_{\ast}[M^{(\alpha)}_{r,(5)}f(x)+M_{r,(6)}(I_{\alpha}f)(x)+I_{\alpha}(|f|)(x)].$

Suppose (3.4) holds for a moment. By Lemma 3.3 in [22], we can assume that
$b\in L^{\infty}(\mu)$. Choosing $r$ such that $1<r<p<n/\alpha$ and
$1/q=1/p-\alpha/n$. By Lemma 2.1, Lemma 2.2 and Theorem 1.1 in this paper, we
obtain

(3.5)
$\begin{split}&||[b,I_{\alpha}]f||_{L^{q}(\mu)}\leq||N([b,I_{\alpha}]f)||_{L^{q}(\mu)}\leq
C||M^{\sharp,(\alpha)}([b,I_{\alpha}]f)||_{L^{q}(\mu)}\\\ \leq
C&\\{||M^{(\alpha)}_{r,(5)}f||_{L^{q}(\mu)}+||M_{r,(6)}(I_{\alpha}f)||_{L^{q}(\mu)}+||I_{\alpha}(|f|)||_{L^{q}(\mu)}\\}\\\
\leq C&\\{||f||_{L^{p}(\mu)}+||I_{\alpha}f||_{L^{q}(\mu)}\\}\leq
C||f||_{L^{p}(\mu)}.\end{split}$

Let $\\{b_{B}\\}$ be a collection of numbers satisfying for ball $B$,

(3.6) $\int_{B}|b-b_{B}|d\mu\leq 2\mu(6B)||b||_{\ast},$

and for two balls $B\subset Q$,

(3.7) $|b_{B}-b_{Q}|\leq 2K_{B,Q}||b||_{\ast}.$

As in the proof of Theorem 1 in [2], to obtain (3.4), by Lemma 2.4 and Lemma
2.5 in Section 2 of this paper, it suffices to deduce that

(3.8) $\frac{1}{\mu(6B)}\int_{B}|[b,I_{\alpha}](f)(y)-h_{B}|d\mu(y)\leq
C||b||_{\ast}(M^{(\alpha)}_{r,(5)}f(x)+M_{r,(6)}(I_{\alpha}f)(x)).$

holds for any $x$ and ball $B$ with $x\in B$, and

(3.9) $|h_{B}-h_{Q}|\leq
C||b||_{\ast}K_{B,Q}K^{(\alpha)}_{B,Q}(M^{(\alpha)}_{r,(5)}f(x)+I_{\alpha}(|f|)(x))$

for all balls $B\subset Q$ with $x\in B$, where $B$ is an arbitrary ball, $Q$
is a doubling ball and for any ball $B$, we denote

$h_{B}:=m_{B}(I_{\alpha}[(b-b_{B})f\chi_{X\backslash\frac{6}{5}B}]).$

To obtain (3.8), we write $[b,I_{\alpha}]f$ as follows.

(3.10)
$\begin{split}&[b,I_{\alpha}]f(y)=(b(y)-b_{B})I_{\alpha}f(y)-I_{\alpha}((b-b_{B})f)(y)\\\
=&(b(y)-b_{B})I_{\alpha}f(y)-I_{\alpha}((b-b_{B})f_{1})(y)-I_{\alpha}((b-b_{B})f_{2})(y),\end{split}$

where $f_{1}=f\chi_{\frac{6}{5}B}$ and $f_{2}=f-f_{1}$. Now, by the Hölder’s
inequality, we have

(3.11)
$\begin{split}&\frac{1}{\mu(6B)}\int_{B}|(b(y)-b_{B})I_{\alpha}f(y)|d\mu(y)\\\
\leq&(\frac{1}{\mu(6B)}\int_{B}|b(y)-b_{B}|^{r^{\prime}}d\mu(y))^{1/r^{\prime}}(\frac{1}{\mu(6B)}\int_{B}|I_{\alpha}f(y)|^{r}d\mu(y))^{1/r}\\\
\leq&C||b||_{\ast}M_{r,(6)}(I_{\alpha}f)(x).\end{split}$

We take $s=\sqrt{r}$ and let $1/t=1/s-\alpha/n$. Using the Hölder’s
inequality, the result of Theorem 1.1 and Definition 3.1, we obtain

(3.12)
$\begin{split}&\frac{1}{\mu(6B)}\int_{B}|I_{\alpha}((b-b_{B})f_{1})(y)|d\mu(y)\\\
\leq&\frac{\mu(B)^{1-1/t}}{\mu(6B)}||I_{\alpha}((b-b_{B})f_{1})||_{L^{t}(\mu)}\\\
\leq&\frac{\mu(B)^{1-1/t}}{\mu(6B)}||((b-b_{B})f_{1})||_{L^{s}(\mu)}\\\
\leq&\biggl{(}\frac{1}{\mu(6B)}\int_{\frac{6}{5}B}|b-b_{B}|^{ss^{\prime}}d\mu(y)\biggr{)}^{\frac{1}{ss^{\prime}}}\\\
&\ \times\biggl{(}\frac{1}{\mu(6B)^{1-\alpha
r/n}}\int_{\frac{6}{5}B}|f(y)|^{r}d\mu(y)\biggr{)}^{\frac{1}{r}}\\\
\leq&C||b||_{\ast}M^{(\alpha)}_{r,(5)}f(x).\end{split}$

Next, to prove (3.8), we only need to compute
$|I_{\alpha}((b-b_{B})f_{2})(y)-h_{B}|$. For $y,\ y_{0}\in B$, by Lemma 3.1,
we have

(3.13)
$\begin{split}&|I_{\alpha}((b-b_{B})f_{2})(y)-I_{\alpha}((b-b_{B})f_{2})(y_{0})|\\\
\leq&C\int_{X\backslash\frac{6}{5}B}\frac{d(y,y_{0})^{\epsilon}}{d(y,z)^{\epsilon}\lambda(y,d(y,z))^{1-\frac{\alpha}{n}}}|b(z)-b_{B}||f(z)|d\mu(z)\\\
\leq&C\sum_{k=1}^{\infty}\int_{6^{k}\frac{6}{5}B\backslash
6^{k-1}\frac{6}{5}B}\frac{6^{-k\epsilon}}{\lambda(y,r_{6^{k}\frac{6}{5}B})^{1-\frac{\alpha}{n}}}\\\
&\ \
\times(|b(z)-b_{6^{k}\frac{6}{5}B}|+|b_{B}-b_{6^{k}\frac{6}{5}B}|)|f(z)|d\mu(z)\\\
\leq&C\sum_{k=1}^{\infty}6^{-k\epsilon}\frac{1}{\mu(5\times
6^{k}\frac{6}{5}B)^{1-\frac{\alpha}{n}}}\int_{6^{k}\frac{6}{5}B}|b(z)-b_{6^{k}\frac{6}{5}B}||f(z)|d\mu(z)\\\
&+C\sum_{k=1}^{\infty}k6^{-k\epsilon}||b||_{\ast}\frac{1}{\mu(5\times
6^{k}\frac{6}{5}B)^{1-\frac{\alpha}{n}}}\int_{6^{k}\frac{6}{5}B}|f(z)|d\mu(z)\\\
\leq&C\sum_{k=1}^{\infty}6^{-k\epsilon}\biggl{[}\frac{1}{\mu(5\times
6^{k}\frac{6}{5}B)^{1-\frac{\alpha
r}{n}}}\int_{6^{k}\frac{6}{5}B}|f(z)|^{r}d\mu(z)\biggr{]}^{1/r}\\\
&\times\biggl{[}\frac{1}{\mu(5\times
6^{k}\frac{6}{5}B)}\int_{6^{k}\frac{6}{5}B}|b(z)-b_{6^{k}\frac{6}{5}B}|^{r^{\prime}}d\mu(z)\biggr{]}^{1/r^{\prime}}\\\
&+C\sum_{k=1}^{\infty}k6^{-k\epsilon}||b||_{\ast}\biggl{[}\frac{1}{\mu(5\times
6^{k}\frac{6}{5}B)^{1-\frac{\alpha
r}{n}}}\int_{6^{k}\frac{6}{5}B}|f(z)|^{r}d\mu(z)\biggr{]}^{\frac{1}{r}}\\\ &\
\times\biggl{[}\frac{\mu(6^{k}\frac{6}{5}B)}{\mu(5\times
6^{k}\frac{6}{5}B)}\biggr{]}^{\frac{1}{r^{\prime}}}\\\
\leq&C||b||_{\ast}M^{(\alpha)}_{r,(5)}f(x).\end{split}$

Taking the mean over $y_{0}\in B$, we obtain

(3.14) $|I_{\alpha}((b-b_{B})f_{2})(y)-h_{B}|\leq
C||b||_{\ast}M^{(\alpha)}_{r,(5)}f(x).$

So (3.8) is proved.

To prove (3.9), we consider two balls $B\subset Q$ with $x\in B$, where $B$ is
an arbitrary ball, $Q$ is a doubling ball. Denote $N=N_{B,Q}+1$, we write

(3.15)
$\begin{split}&|m_{B}(I_{\alpha}((b-b_{B})f\chi_{X\backslash\frac{6}{5}B}))-m_{Q}(I_{\alpha}((b-b_{Q})f\chi_{X\backslash\frac{6}{5}Q}))|\\\
\leq&|m_{B}(I_{\alpha}((b-b_{B})f\chi_{6B\backslash\frac{6}{5}B}))|\\\ &\
+|m_{B}(I_{\alpha}((b_{B}-b_{Q})f\chi_{X\backslash 6B}))|\\\ &\
+|m_{B}(I_{\alpha}((b-b_{Q})f\chi_{6^{N}B\backslash 6B}))|\\\ &\
+|m_{B}(I_{\alpha}((b-b_{Q})f\chi_{X\backslash
6^{N}B}))-m_{Q}(I_{\alpha}((b-b_{Q})f\chi_{X\backslash 6^{N}B}))|\\\ &\
+|m_{Q}(I_{\alpha}((b-b_{Q})f\chi_{6^{N}B\backslash\frac{6}{5}Q}))|\\\
=:&L_{1}+L_{2}+L_{3}+L_{4}+L_{5}.\end{split}$

For $y\in B$, by the Hölder’s inequality, we get

(3.16)
$\begin{split}&|I_{\alpha}((b-b_{B})f\chi_{6B\backslash\frac{6}{5}B})(y)|\\\
\leq&C\int_{6B}\frac{|b(z)-b_{B}||f(z)|}{\lambda(y,d(y,z))^{1-\alpha/n}}d\mu(z)\\\
\leq&C\biggl{[}\frac{1}{\mu(5\times 6B)^{1-\frac{\alpha
r}{n}}}\int_{6B}|f(z)|^{r}d\mu(z)\biggr{]}^{\frac{1}{r}}\\\ &\
\times\biggl{[}\frac{1}{\mu(5\times
6B)}\int_{6B}|b(z)-b_{B}|^{r^{\prime}}d\mu(z)\biggr{]}^{\frac{1}{r^{\prime}}}\\\
\leq&C||b||_{\ast}M^{(\alpha)}_{r,(5)}f(x).\end{split}$

Then we get $L_{1}\leq C||b||_{\ast}M^{(\alpha)}_{r,(5)}f(x).$

For $x,\ y\in B$, it is easy to see that

(3.17) $|I_{\alpha}(f\chi_{X\backslash 6B})(y)|\leq
I_{\alpha}(|f|)(x)+CM^{(\alpha)}_{r,(5)}f(x).$

Then, by Definition 3.1, we get

(3.18) $L_{2}\leq
CK_{B,Q}||b||_{\ast}(I_{\alpha}(|f|)(x)+M^{(\alpha)}_{r,(5)}f(x)).$

Let us estimate $L_{3}$. For $y\in B$, we have

(3.19) $\begin{split}&|I_{\alpha}((b-b_{Q})f\chi_{6^{N}B\backslash 6B})(y)|\\\
\leq&C\sum_{k=1}^{N-1}\int_{6^{k+1}B\backslash
6^{k}B}\frac{|b(z)-b_{Q}||f(z)|}{\lambda(y,d(y,z))^{1-\alpha/n}}d\mu(z)\\\
\leq&C\sum_{k=1}^{N-1}[\frac{\mu(5\times 6^{k+1}B)}{\lambda(y,r_{5\times
6^{k+1}B})}]^{1-\alpha/n}\\\ &\times\frac{1}{\mu(5\times
6^{k+1}B)^{1-\alpha/n}}\int_{6^{k+1}B}|b(z)-b_{Q}||f(z)|d\mu(z)\\\
\leq&CK^{(\alpha)}_{B,Q}\biggl{[}\frac{1}{\mu(5\times
6^{k+1}B)^{1-\frac{\alpha
r}{n}}}\int_{6^{k+1}B}|f(z)|^{r}d\mu(z)\biggr{]}^{\frac{1}{r}}\\\
&\times\biggl{[}\frac{1}{\mu(5\times
6^{k+1}B)}\int_{6^{k+1}B}|b(z)-b_{Q}|^{r^{\prime}}d\mu(z)\biggr{]}^{\frac{1}{r^{\prime}}}\\\
\leq&CK^{(\alpha)}_{B,Q}M^{\alpha}_{r,(5)}f(x)\biggl{[}\frac{1}{\mu(5\times
6^{k+1}B)}\int_{6^{k+1}B}|b(z)-b_{6^{k+1}B}\\\
&+b_{6^{k+1}B}-b_{Q}|^{r^{\prime}}d\mu(z)\biggr{]}^{\frac{1}{r^{\prime}}}\\\
\leq&CK_{B,Q}K^{(\alpha)}_{B,Q}||b||_{\ast}M^{(\alpha)}_{r,(5)}f(x).\end{split}$

Here we have used the fact that $|b_{6^{k+1}B}-b_{Q}|\leq
CK_{B,Q}||b||_{\ast}$.

Taking the mean over $B$, we have

$L_{3}\leq CK_{B,Q}K^{(\alpha)}_{B,Q}||b||_{\ast}M^{(\alpha)}_{r,(5)}f(x).$

For $L_{4}$. Operating as in (3.13), for any $y\in B$ and $z\in Q$, we obtain

$|T((b-b_{Q})f\chi_{X\backslash 6^{N}B})(y)-T((b-b_{Q})f\chi_{X\backslash
6^{N}B})(z)|\leq C||b||_{\ast}M^{(\alpha)}_{r,(5)}f(x).$

Taking the mean over $B$ for $y$ and over $Q$ for $z$, we have

$L_{4}\leq C||b||_{\ast}M^{(\alpha)}_{r,(5)}f(x).$

For $L_{5}$, similar to $L_{1}$, we can deduce $L_{5}\leq
C||b||_{\ast}M^{(\alpha)}_{r,(5)}f(x).$ Thus (3.9) is valid and the proof of
Theorem 1.2 is finished . ∎

##  4\. Proof of Theorem 1.3

To prove Theorem 1.3, we need the following some lemmas.

######  Lemma 4.1.

[8,22]Let $1\leq p<\infty$ and $1<\rho<\infty$. If $b\in RBMO(\mu)$, then for
any ball $B\in X$,

(4.1) $\biggl{\\{}\frac{1}{\mu(\rho
B)}\int_{B}|b(x)-m_{\widetilde{B}}(b)|^{p}d\mu(x)\biggr{\\}}^{1/p}\leq
C||b||_{\ast}.$

######  Lemma 4.2.

[9]For any ball $B$, we have

(4.2) $|m_{\widetilde{6^{j}\frac{6}{5}B}}(b)-m_{\widetilde{B}}(b)|\leq
Cj||b||_{\ast}.$

###### Proof.

(of Theorem 1.3) We prove the theorem by induction on $k$. If $k=1$, the
result of Theorem 1.2 asserts that $[b,I_{\alpha}]$ is bounded from
$L^{p}(\mu)$ to $L^{q}(\mu)$ for any $1<p<n/\alpha$, $0<\alpha<n$ and
$1/q=1/p-\alpha/n$. Now we assume that $k\geq 2$ is an integer and that for
any $1\leq i\leq k-1$ and any subset
$\sigma=\\{\sigma(1),\cdot\cdot\cdot,\sigma(i)\\}$ of
$\\{1,\cdot\cdot\cdot,k\\}$, $I_{\alpha,\vec{b}_{\sigma}}$ is bounded from
$L^{p}(\mu)$ to $L^{q}(\mu)$ for any for any $1<p<n/\alpha$, $0<\alpha<n$ and
$1/q=1/p-\alpha/n$. We next claim that for any $1<r<\infty$,
$I_{\alpha,\vec{b}}$ satisfies the following sharp maximal function estimate

(4.3) $\begin{split}M^{\sharp,(\alpha)}(I_{\alpha,\vec{b}}f)(x)&\leq
C||\vec{b}||_{\ast}\\{M_{r,(6)}(I_{\alpha}f)(x)+M^{(\alpha)}_{r,(5)}f(x)\\}\\\
&\ +C\sum_{i=1}^{k-1}\sum_{\sigma\in
C_{i}^{k}}||\vec{b}_{\sigma}||_{\ast}M_{r,(6)}(I_{\alpha,\vec{b}_{\sigma^{\prime}}}f)(x).\end{split}$

Suppose (4.3) holds for a moment. Now we prove $T_{\vec{b}}$ satisfies (1.14).
By Lemma 3.3 in [22], we can assume that $b_{i}\in L^{\infty}(\mu)$ for $1\leq
i\leq k$. Choosing $r$ such that $1<r<p<n/\alpha$ and $1/q=1/p-\alpha/n$. By
Lemma 2.1, Lemma2.2, Theorem 1.1 and Theorem 1.2 in this paper, we deduce

(4.4) $\begin{split}&||I_{\alpha,\vec{b}}f||_{L^{q}(\mu)}\leq
C||N(I_{\alpha,\vec{b}}f)||_{L^{q}(\mu)}\leq
C||M^{\sharp,(\alpha)}(I_{\alpha,\vec{b}}f)||_{L^{q}(\mu)}\\\
\leq&C||\vec{b}||_{\ast}\\{||M_{r,(6)}(I_{\alpha}f)||_{L^{q}(\mu)}+||M^{(\alpha)}_{r,(5)}f||_{L^{q}(\mu)}\\}\\\
&\ +C\sum_{i=1}^{k-1}\sum_{\sigma\in
C_{i}^{k}}||\vec{b}_{\sigma}||_{\ast}||M_{r,(6)}(I_{\alpha,\vec{b}_{\sigma^{\prime}}}f)||_{L^{q}(\mu)}\\\
\leq&C||\vec{b}||_{\ast}(||I_{\alpha}f||_{L^{q}(\mu)}+||f||_{L^{p}(\mu)})+C\sum_{i=1}^{k-1}\sum_{\sigma\in
C_{i}^{k}}||\vec{b}_{\sigma}||_{\ast}||I_{\alpha,\vec{b}_{\sigma^{\prime}}}f||_{L^{q}(\mu)}\\\
\leq&C||\vec{b}||_{\ast}||f||_{L^{p}(\mu)}\end{split}$

As in the proof of Theorem 2 in [9], to obtain (4.3), by Lemma 2.4 and Lemma
2.5 in this paper, it only need to show that

(4.5)
$\begin{split}&\frac{1}{\mu(6B)}\int_{B}|I_{\alpha,\vec{b}}f(y)-h_{B}|d\mu(y)\\\
\leq&C||\vec{b}||_{\ast}\\{M_{r,(6)}(I_{\alpha}f)(x)+M^{(\alpha)}_{r,(5)}f(x)\\}+C\sum_{i=1}^{k-1}\sum_{\sigma\in
C_{i}^{k}}||\vec{b}_{\sigma}||_{\ast}M_{r,(6)}(I_{\alpha,\vec{b}_{\sigma^{\prime}}}f)(x)\end{split}$

hold for all $x$ and $B$ with $x\in B$, and

(4.6) $\begin{split}|h_{B}-h_{Q}|&\leq
C(K_{B,Q})^{k}K^{(\alpha)}_{B,Q}||\vec{b}||_{\ast}\\{M_{r,(6)}(I_{\alpha}f)(x)+M^{(\alpha)}_{r,(5)}f(x)\\}\\\
&\ +C(K_{B,Q})^{k}\sum_{i=1}^{k-1}\sum_{\sigma\in
C_{i}^{k}}||\vec{b}_{\sigma}||_{\ast}M_{r,(6)}(I_{\alpha,\vec{b}_{\sigma^{\prime}}}f)(x)\end{split}$

holds for any cube $B\subset Q$ with $x\in B$, where $B$ is an arbitrary cube
and $Q$ is a doubling cube. We denote

$h_{B}=m_{B}\left(I_{\alpha}\left[\left(b_{1}-m_{\widetilde{B}}(b_{1})\right)\cdot\cdot\cdot\left(b_{k}-m_{\widetilde{B}}(b_{k})\right)f\chi_{X\backslash\frac{6}{5}B}\right]\right),$

and

$h_{R}=m_{R}\left(I_{\alpha}\left[\left(b_{1}-m_{R}(b_{1})\right)\cdot\cdot\cdot\left(b_{k}-m_{R}(b_{k})\right)f\chi_{X\backslash\frac{6}{5}R}\right]\right).$

Let us firstly estimate (4.5). It is easy to see that

(4.7)
$\prod_{i=1}^{k}\left[b_{i}(z)-m_{\widetilde{B}}(b_{i})\right]=\sum_{i=0}^{k}\sum_{\sigma\in
C_{i}^{k}}[b(z)-b(y)]_{\sigma^{\prime}}[b(y)-m_{\widetilde{B}}(b)]_{\sigma}$

for $y,z\in X$, where if $i=0$, then $\sigma^{\prime}=\\{1,2\cdot\cdot\cdot
k\\}$, $\sigma=\emptyset$ and $[b(y)-m_{\widetilde{B}}(b)]_{\emptyset}=1$.
Hence

$I_{\alpha,\vec{b}}f(y)=I_{\alpha}\left(\prod_{i=1}^{k}\left[b_{i}-m_{\widetilde{B}}(b_{i})\right]f\right)(y)-\sum_{i=1}^{k}\sum_{\sigma\in
C_{i}^{k}}\left[b(y)-m_{\widetilde{B}}(b)\right]_{\sigma}I_{\alpha,\vec{b}_{\sigma^{\prime}}}f(y),$

and if $i=k$, we denote $I_{\alpha}f(y)$ by
$I_{\alpha,\vec{b}_{\sigma^{\prime}}}f(y)$. Thus,

(4.8)
$\begin{split}&\frac{1}{\mu(6B)}\int_{B}|I_{\alpha,\vec{b}}f(y)-h_{B}|d\mu(y)\\\
\leq&\frac{1}{\mu(6B)}\int_{B}\left|I_{\alpha}\left(\prod_{i=1}^{k}\left[b_{i}-m_{\widetilde{B}}(b_{i})\right]f\chi_{\frac{6}{5}B}\right)(y)\right|d\mu(y)\\\
\ &+\sum_{i=1}^{k}\sum_{\sigma\in
C_{i}^{k}}\frac{1}{\mu(6B)}\int_{B}\left|\left[b(y)-m_{\widetilde{B}}(b)\right]_{\sigma}\right|\left|I_{\alpha,\vec{b}_{\sigma^{\prime}}}f(y)\right|d\mu(y)\\\
\
&+\frac{1}{\mu(6B)}\int_{B}\left|I_{\alpha}\left(\prod_{i=1}^{k}\left[b_{i}-m_{\widetilde{B}}(b_{i})\right]f\chi_{X\backslash\frac{6}{5}B}\right)(y)-h_{B}\right|d\mu(y)\\\
=:&II_{1}+II_{2}+II_{3}.\end{split}$

Write

$b_{i}(y)-m_{\widetilde{Q}}(b_{i})=b_{i}(y)-m_{\widetilde{\frac{4}{3}Q}}(b_{i})+m_{\widetilde{\frac{4}{3}Q}}(b_{i})-m_{\widetilde{Q}}(b_{i})$

for $i=1,\cdot\cdot\cdot,k$. By Lemma 4.1, we obtain

(4.9)
$\int_{B}\prod_{i=1}^{k}|b_{i}(y)-m_{\widetilde{B}}(b_{i})|^{ss^{\prime}}d\mu(y)\leq
C||\vec{b}||_{\ast}^{ss^{\prime}}\mu(6B).$

Take $s=\sqrt{r}$ and let $1/t=1/s-\alpha/n$. By Theorem 1.1, the Hölder’s
inequality and (4.9), we deduce

(4.10)
$\begin{split}II_{1}&\leq\frac{\mu(B)^{1-1/t}}{\mu(6B)}\left|\left|I_{\alpha}\left(\prod_{i=1}^{k}\left[b_{i}-m_{\tilde{B}}(b_{i})\right]f\chi_{\frac{6}{5}B}\right)\right|\right|_{L^{t}(\mu)}\\\
&\leq
C\frac{\mu(B)^{1-1/t}}{\mu(6B)}\left|\left|\prod_{i=1}^{k}\left[b_{i}-m_{\tilde{B}}(b_{i})\right]f\chi_{\frac{6}{5}B}\right|\right|_{L^{s}(\mu)}\\\
&\leq
C\left(\frac{1}{\mu(6B)}\int_{\frac{6}{5}B}\prod_{i=1}^{k}\left|b_{i}(y)-m_{\tilde{B}}(b_{i})\right|^{ss^{\prime}}d\mu(y)\right)^{\frac{1}{ss^{\prime}}}\\\
&\ \ \times\left(\frac{1}{\mu(6B)^{1-\frac{\alpha
r}{n}}}\int_{\frac{6}{5}B}|f(y)|^{r}d\mu(y)\right)^{\frac{1}{r}}\\\ &\leq
C||\vec{b}||_{\ast}M^{(\alpha)}_{r,(5)}f(x).\end{split}$

From the Hölder’s inequality and Lemma 4.1, it follows that

(4.11) $\begin{split}II_{2}&\leq\sum_{i=1}^{k}\sum_{\sigma\in
C_{i}^{k}}\left(\frac{1}{\mu(6B)}\int_{B}\left|\left[b(y)-m_{\tilde{B}}(b)\right]_{\sigma}\right|^{r^{\prime}}d\mu(y)\right)^{\frac{1}{r^{\prime}}}\\\
&\ \
\times\left(\frac{1}{\mu(6B)}\int_{B}|I_{\alpha,\vec{b}_{\sigma^{\prime}}}f(y)|^{r}d\mu(y)\right)^{\frac{1}{r}}\\\
&\leq C\sum_{i=1}^{k}\sum_{\sigma\in
C_{i}^{k}}||\vec{b}_{\sigma}||_{\ast}M_{r,(6)}(I_{\alpha,\vec{b}_{\sigma^{\prime}}}f)(x)\\\
&=C\sum_{i=1}^{k-1}\sum_{\sigma\in
C_{i}^{k}}||\vec{b}_{\sigma}||_{\ast}M_{r,(6)}(I_{\alpha,\vec{b}_{\sigma^{\prime}}}f)(x)+C||b||_{\ast}M_{r,(6)}(I_{\alpha}f)(x).\end{split}$

Let us estimate $II_{3}$. For $y,\ y_{0}\in B$, by the condition (ii) in
Definition 1.6 and the Hölder’s inequality, we have

(4.12)
$\begin{split}&\biggl{|}I_{\alpha}\left(\prod_{i=1}^{k}\left[b_{i}-m_{\tilde{B}}(b_{i})\right]f\chi_{X\backslash\frac{6}{5}B}\right)(y)\\\
&-I_{\alpha}\left(\prod_{i=1}^{k}\left[b_{i}-m_{\tilde{B}}(b_{i})\right]f\chi_{X\backslash\frac{6}{5}B}\right)(y_{0})\biggr{|}\\\
\leq&C\int_{X\backslash\frac{6}{5}B}\frac{d(y,y_{0})^{\epsilon}}{d(y,z)^{\epsilon}\lambda(y,d(y,z))^{1-\frac{\alpha}{n}}}\prod_{i=1}^{k}\left|b_{i}(z)-m_{\widetilde{B}}(b_{i})\right||f(z)|d\mu(z)\\\
\leq&C\sum_{j=1}^{\infty}\int_{6^{k}\frac{6}{5}B}6^{-j\epsilon}\frac{1}{\lambda(y,r_{6^{j}\frac{6}{5}B})^{1-\frac{\alpha}{n}}}\prod_{i=1}^{k}\biggl{(}\left|b_{i}(z)-m_{\widetilde{6^{j}\frac{6}{5}B}}(b_{i})\right|\\\
&\ \
+\biggl{|}m_{\widetilde{6^{j}\frac{6}{5}B}}(b_{i})-m_{\tilde{B}}(b_{i})\biggr{|}\biggr{)}|f(z)|d\mu(z)\\\
\leq&C\sum_{j=1}^{\infty}\sum_{i=0}^{k}\sum_{\sigma\in
C_{i}^{k}}6^{-j\epsilon}j^{k-i}||\vec{b}_{\sigma^{\prime}}||_{\ast}\\\ &\ \
\times\frac{1}{\mu(5\times
6^{j}\frac{6}{5}B)^{1-\alpha/n}}\int_{6^{j}\frac{6}{5}B}\left|\left[b(z)-m_{\widetilde{6^{j}\frac{6}{5}B}}(b_{i})\right]_{\sigma}\right||f(z)|d\mu(z)\\\
\leq&C\sum_{i=0}^{k}\sum_{\sigma\in
C_{i}^{k}}\sum_{j=1}^{\infty}6^{-j\epsilon}j^{k-i}||\vec{b}_{\sigma}||_{\ast}||\vec{b}_{\sigma^{\prime}}||_{\ast}M^{(\alpha)}_{r,(5)}f(x)\\\
\leq&C||\vec{b}||_{\ast}M^{(\alpha)}_{r,(5)}f(x).\\\ \end{split}$

From the above estimate and the definition of $h_{B}$, we have

(4.13)
$\begin{split}&\left|I_{\alpha}\left(\prod_{i=1}^{k}\left[b_{i}-m_{\tilde{B}}(b_{i})\right]f\chi_{X\backslash\frac{6}{5}B}\right)(y)-h_{B}\right|\\\
=&\biggl{|}I_{\alpha}\left(\prod_{i=1}^{k}\left[b_{i}-m_{\tilde{Q}}(b_{i})\right]f\chi_{X\backslash\frac{6}{5}B}\right)(y)\\\
&-m_{B}\left[I_{\alpha}\left(\prod_{i=1}^{k}\left[b_{i}-m_{\tilde{Q}}(b_{i})\right]f\chi_{X\backslash\frac{6}{5}B}\right)\right]\biggr{|}\\\
\leq&C||\vec{b}||_{\ast}M^{(\alpha)}_{r,(5)}f(x).\\\ \end{split}$

Then

$II_{3}\leq C||\vec{b}||_{\ast}M^{(\alpha)}_{r,(5)}f(x).$

The estimate for $II_{1}$, $II_{2}$ and $II_{3}$ yields (4.5).

Now we turn to the estimate for (4.6). For any balls $B\subset Q$ with $x\in
B$ and $Q$ is a doubling ball, we denote $N_{B,Q}+1$ simply by $N$.

(4.14)
$\begin{split}&\biggl{|}m_{B}\biggl{[}I_{\alpha}\left(\prod_{i=1}^{k}\left[b_{i}-m_{\tilde{B}}(b_{i})\right]f\chi_{X\backslash\frac{6}{5}B}\right)\biggr{]}\\\
&-m_{Q}\left[I_{\alpha}\left(\prod_{i=1}^{k}\left[b_{i}-m_{Q}(b_{i})\right]f\chi_{X\backslash\frac{6}{5}Q}\right)\right]\biggr{|}\\\
\leq&\biggl{|}m_{Q}\left[I_{\alpha}\left(\prod_{i=1}^{k}\left[b_{i}-m_{\tilde{B}}(b_{i})\right]f\chi_{X\backslash
6^{N}B}\right)\right]\\\
&-m_{B}\left[I_{\alpha}\left(\prod_{i=1}^{k}\left[b_{i}-m_{\tilde{B}}(b_{i})\right]f\chi_{X\backslash
6^{N}B}\right)\right]\biggr{|}\\\
+&\biggl{|}m_{Q}\left[I_{\alpha}\left(\prod_{i=1}^{k}\left[b_{i}-m_{Q}(b_{i})\right]f\chi_{X\backslash
6^{N}B}\right)\right]\\\
&-m_{Q}\left[I_{\alpha}\left(\prod_{i=1}^{k}\left[b_{i}-m_{\tilde{B}}(b_{i})\right]f\chi_{X\backslash
6^{N}B}\right)\right]\biggr{|}\\\
+&\left|m_{B}\left[I_{\alpha}\left(\prod_{i=1}^{k}\left[b_{i}-m_{\tilde{B}}(b_{i})\right]f\chi_{{6^{N}B\backslash\frac{6}{5}B}}\right)\right]\right|\\\
+&\left|m_{Q}\left[I_{\alpha}\left(\prod_{i=1}^{k}\left[b_{i}-m_{Q}(b_{i})\right]f\chi_{{6^{N}B\backslash\frac{6}{5}Q}}\right)\right]\right|\\\
=:&JJ_{1}+JJ_{2}+JJ_{3}+JJ_{4}.\end{split}$

With the similar estimate for $II_{3}$, we easily get that

$JJ_{1}\leq C||\vec{b}||_{\ast}M^{(\alpha)}_{r,(5)}f(x).$

To estimate $JJ_{2}$, with the help of (4.7), we deduce that

(4.15)
$\begin{split}&\biggl{|}I_{\alpha}\left(\prod_{i=1}^{k}\left[b_{i}-m_{Q}(b_{i})\right]f\chi_{X\backslash
6^{N}B}\right)(y)\\\
&-I_{\alpha}\left(\prod_{i=1}^{k}\biggl{[}b_{i}-m_{\tilde{B}}(b_{i})\biggr{]}f\chi_{X\backslash
6^{N}B}\right)(y)\biggr{|}\\\
=&\biggl{|}I_{\alpha}\left(\prod_{i=1}^{k}\left[b_{i}-m_{Q}(b_{i})\right]f\chi_{X\backslash
6^{N}B}\right)(y)\\\ &\ -\sum_{i=0}^{k}\sum_{\sigma\in
C_{i}^{k}}\left[m_{Q}(b)-m_{\tilde{B}}(b)\right]_{\sigma^{\prime}}I_{\alpha}\left([b-m_{Q}(b)]_{\sigma}f\chi_{X\backslash
6^{N}B}\right)(y)\biggr{|}\\\ \leq&C\sum_{i=0}^{k-1}\sum_{\sigma\in
C_{i}^{k}}(K_{B,Q})^{k-i}||\vec{b}_{\sigma^{\prime}}||_{\ast}\left|I_{\alpha}\left([b-m_{Q}(b)]_{\sigma}f\chi_{X\backslash
6^{N}B}\right)(y)\right|\\\ \leq&C\sum_{i=0}^{k-1}\sum_{\sigma\in
C_{i}^{k}}(K_{B,Q})^{k-i}||\vec{b}_{\sigma^{\prime}}||_{\ast}\biggl{\\{}I_{\alpha}\left([b-m_{Q}(b)]_{\sigma}f\right)(y)\\\
&\ \
+I_{\alpha}\left([b-m_{Q}(b)]_{\sigma}f\chi_{6^{N}B}\right)(y)\biggr{\\}}\\\
\leq&C\sum_{i=0}^{k-1}\sum_{\sigma\in
C_{i}^{k}}(K_{B,Q})^{k-i}||\vec{b}_{\sigma^{\prime}}||_{\ast}\biggl{\\{}\sum_{j=0}^{i}\sum_{\eta\in
C_{j}^{i}}|[b(y)-m_{Q}(b)]_{\eta^{\prime}}|I_{\alpha,\vec{b}_{\eta}}f(y)|\\\
&+\biggl{|}I_{\alpha}\left([b-m_{Q}(b)]_{\sigma}f\chi_{6^{N}B\backslash\frac{6}{5}Q}\right)(y)\biggr{|}\\\
&+\biggl{|}I_{\alpha}\left([b-m_{Q}(b)]_{\sigma}f\chi_{\frac{6}{5}Q}\right)(y)\biggr{|}\biggr{\\}}.\\\
\end{split}$

Using the Hölder’s inequality and the fact that $Q$ is a doubling ball, it
follows that

(4.16)
$\begin{split}&\frac{1}{\mu(Q)}\int_{Q}|[b(y)-m_{Q}(b)]_{\eta^{\prime}}||I_{\alpha,\vec{b}_{\eta}}f(y)|d\mu(y)\\\
\leq&C\frac{\mu(6Q)}{\mu(Q)}\biggl{[}\frac{1}{\mu(6Q)}\int_{Q}|[b(y)-m_{Q}(b)]_{\eta^{\prime}}|^{r^{\prime}}d\mu(y)\biggr{]}^{1/r^{\prime}}\\\
&\
\times\biggl{[}\frac{1}{\mu(6Q)}\int_{Q}|I_{\alpha,\vec{b}_{\eta}}f(y)|^{r}d\mu(y)\biggr{]}^{1/r}\\\
\leq&C||\vec{b}_{\eta^{\prime}}||_{\ast}M_{r,(6)}(I_{\alpha,\vec{b}_{\eta}}f)(x).\end{split}$

By the Hölder’s inequality, Lemma 4.1 and the condition (i) in Definition 1.6,
it is easy to see that for $y\in Q$,

(4.17)
$\begin{split}&\left|I_{\alpha}\left([b-m_{Q}(b)]_{\sigma}f\chi_{6^{N}B\backslash\frac{6}{5}Q}\right)(y)\right|\\\
\leq&C\int_{6^{N}B\backslash\frac{6}{5}Q}\frac{1}{\lambda(y,d(y,z))^{1-\alpha/n}}|[b(z)-m_{Q}(b)]_{\sigma}||f(z)|d\mu(z)\\\
\leq&C\frac{1}{\mu(5\times
6^{N}B)^{1-\alpha/n}}\int_{6^{N}B}|[b(z)-m_{Q}(b)]_{\sigma}||f(z)|d\mu(z)\\\
\leq&C\left[\frac{1}{\mu(5\times
6^{N}B)}\int_{6^{N}B}|[b(z)-m_{Q}(b)]_{\sigma}|^{r^{\prime}}d\mu(z)\right]^{\frac{1}{r^{\prime}}}\\\
&\ \ \times\left[\frac{1}{\mu(5\times 6^{N}B)^{1-\frac{\alpha
r}{n}}}\int_{6^{N}B}|f(z)|^{r}d\mu(z)\right]^{\frac{1}{r}}\\\
\leq&C||\vec{b}_{\sigma}||_{\ast}M^{(\alpha)}_{r,(5)}f(x).\end{split}$

Taking the mean over $y\in Q$, it easily obtains that

(4.18)
$m_{Q}\left[\left|I_{\alpha}\left([b-m_{Q}(b)]_{\sigma}f\chi_{6^{N}B\backslash\frac{6}{5}Q}\right)\right|\right]\leq
C||\vec{b}_{\sigma}||_{\ast}M^{(\alpha)}_{r,(5)}f(x).$

With the similar method to estimate $II_{1}$, we deduce

(4.19)
$m_{Q}\left[\left|I_{\alpha}\left([b-m_{Q}(b)]_{\sigma}f\chi_{\frac{6}{5}Q}\right)\right|\right]\leq
C||\vec{b}_{\sigma}||_{\ast}M^{(\alpha)}_{r,(5)}f(x).$

Therefore,

(4.20)
$\begin{split}JJ_{2}\leq&C(K_{B,Q})^{k}\biggl{[}\sum_{i=0}^{k-1}\sum_{\sigma\in
C_{i}^{k}}||\vec{b}_{\sigma}||_{\ast}M_{r,(6)}(I_{\alpha,\vec{b}_{\sigma}}f)(x)+||\vec{b}||_{\ast}M^{(\alpha)}_{r,(5)}f(x)\biggr{]}\\\
=&C(K_{B,Q})^{k}\biggl{[}\sum_{i=1}^{k-1}\sum_{\sigma\in
C_{i}^{k}}||\vec{b}_{\sigma}||_{\ast}M_{r,(6)}(I_{\alpha,\vec{b}_{\sigma}}f)(x)\\\
&\ \
+||\vec{b}||_{\ast}\biggl{(}M_{r,(6)(I_{\alpha}f)(x)}+M^{(\alpha)}_{r,(5)}f(x)\biggr{)}\biggr{]}.\end{split}$

Now we turn to estimate $JJ_{3}$. By the Hölder’s inequality, Lemma 4.1 and
Lemma 4.2, for $y\in B$, we get

(4.21)
$\begin{split}&\left|I_{\alpha}(\prod^{k}_{i=1}[b_{i}-m_{\tilde{B}}(b_{i})]f\chi_{6^{N}B\backslash\frac{6}{5}B})(y)\right|\\\
\leq&C\sum_{j=1}^{N-1}\frac{1}{\lambda(y,r_{6^{j}B})^{1-\frac{\alpha}{n}}}\int_{6^{j+1}B}\prod_{i=1}^{k}|b_{i}(z)-m_{\tilde{B}}(b_{i})||f(z)|d\mu(z)\\\
&\
+\frac{C}{\lambda(y,r_{B})^{1-\frac{\alpha}{n}}}\int_{6B\backslash\frac{6}{5}B}\prod_{i=1}^{k}|b_{i}(z)-m_{\tilde{B}}(b_{i})||f(z)|d\mu(z)\\\
\leq&C\sum^{N-1}_{j=1}\biggl{[}\frac{\mu(5\times
6^{j+1}B)}{\lambda(y,r_{5\times
6^{j+1}B})}\biggr{]}^{1-\frac{\alpha}{n}}\biggl{[}\frac{1}{\mu(5\times
6^{j+1}B)^{1-\frac{\alpha
r}{n}}}\int_{6^{j+1}B}|f(z)|^{r}d\mu(z)\biggr{]}^{1/r}\\\ &\
\times\biggl{[}\frac{1}{\mu(5\times
6^{j+1}B)}\int_{6^{j+1}B}\prod_{i=1}^{k}|b_{i}(z)-m_{\widetilde{6^{j+1}B}}(b_{i})\\\
&\ \
+m_{\widetilde{6^{j+1}B}}(b_{i})-m_{\tilde{B}}(b_{i})|^{r^{\prime}}d\mu(z)\biggr{]}^{1/r^{\prime}}\\\
&\ +C\biggl{[}\frac{1}{\mu(5\times
6B)}\int_{6B}\prod_{i=1}^{k}|b_{i}(z)-m_{\tilde{B}}(b_{i})|^{r^{\prime}}d\mu(z)\biggr{]}^{1/r^{\prime}}\\\
&\ \times\biggl{[}\frac{1}{\mu(5\times 6B)^{1-\frac{\alpha
r}{n}}}\int_{6B}|f(z)|^{r}d\mu(z)\biggr{]}^{1/r}\\\
\leq&CK^{(\alpha)}_{B,Q}(K_{B,Q})^{k}||\vec{b}||_{\ast}M^{(\alpha)}_{r,(5)}f(x)+C||\vec{b}||_{\ast}M^{(\alpha)}_{r,(5)}f(x)\\\
\leq&CK^{(\alpha)}_{B,Q}(K_{B,Q})^{k}||\vec{b}||_{\ast}M^{(\alpha)}_{r,(5)}f(x).\end{split}$

Taking the mean over $y\in Q$, we obtain

$JJ_{3}\leq
CK^{(\alpha)}_{B,Q}(K_{B,Q})^{k}||\vec{b}||_{\ast}M^{(\alpha)}_{r,(5)}f(x).$

Finally, we estimate $JJ_{4}$. For $y\in Q$,

(4.22)
$\begin{split}&\biggl{|}I_{\alpha}(\prod_{i=1}^{k}[b_{i}-m_{Q}(b_{i})]f\chi_{6^{N}B\backslash\frac{6}{5}Q})(y)\biggr{|}\\\
\leq&C\int_{6^{N}B\backslash\frac{6}{5}Q}\frac{1}{\lambda(y,d(y,z))^{1-\frac{\alpha}{n}}}\prod_{i=1}^{k}|b_{i}(z)-m_{Q}(b_{i})||f(z)|d\mu(z)\\\
\leq&C\biggl{[}\frac{1}{\mu(5\times
6^{N}B)}\int_{6^{N}B}\prod_{i=1}^{k}|b_{i}(y)-m_{B}(b_{i})|^{r^{\prime}}d\mu(y)\biggr{]}^{1/r^{\prime}}\\\
&\ \ \times\biggl{[}\frac{1}{\mu(5\times 6^{N}B)^{1-\frac{\alpha
r}{n}}}\int_{6^{N}B}|f(y)|^{r}d\mu(y)\biggr{]}^{1/r}\\\
\leq&C||\vec{b}||_{\ast}M^{(\alpha)}_{r,(5)}f(x).\end{split}$

Taking the mean over $y\in Q$, it follows that

$JJ_{4}\leq C||\vec{b}||_{\ast}M^{(\alpha)}_{r,(5)}f(x).$

The estimate for $JJ_{1}$, $JJ_{2}$, $JJ_{3}$ and $JJ_{4}$ yields (4.6). Thus
the proof of Theorem 1.3 is completed. ∎

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