# Transferring Dexterous Manipulation from GPU Simulation to a Remote Real-
World TriFinger

Arthur Allshire1,3, Mayank Mittal2,3, Varun Lodaya1, Viktor Makoviychuk3,
Denys Makoviichuk4,
Felix Widmaier5, Manuel Wüthrich5, Stefan Bauer5, Ankur Handa3, Animesh
Garg1,3
1University of Toronto, Vector Institute, 2ETH Zurich, 3Nvidia, 4Snap, 5MPI
Tubingen

###### Abstract

We present a system for learning a challenging dexterous manipulation task
involving moving a cube to an arbitrary 6-DoF pose with only 3-fingers trained
with NVIDIA’s IsaacGym simulator. We show empirical benefits, both in
simulation and sim-to-real transfer, of using keypoints as opposed to
position+quaternion representations for the object pose in 6-DoF for policy
observations and in reward calculation to train a model-free reinforcement
learning agent. By utilizing domain randomization strategies along with the
keypoint representation of the pose of the manipulated object, we achieve a
high success rate of 83% on a remote TriFinger system maintained by the
organizers of the Real Robot Challenge. With the aim of assisting further
research in learning in-hand manipulation, we make the codebase of our system,
along with trained checkpoints that come with billions of steps of experience
available, at https://s2r2-ig.github.io

##  1 Introduction

Through cloud-based robotic systems, Robotics as-a-Service (RaaS) promises to
alleviate many of the upfront requirements to install and maintain robot
hardware [1]. The advantages of this model include easier management and
scalability, greater flexibility and savings in terms of compute power and
utilities. While RaaS has been proposed for parallel-grid computing on demand,
collective robot learning, and crowd-sourcing access to remote human
expertise, a great need for such a model has emerged in the robotics community
during the current pandemic. In this work, we show how large-scale simulation
done on a desktop grade GPU and cloud-based robotics can enable roboticists to
perform research in robotic learning with modest resources. We focus on 6-DoF
object manipulation by using a dexterous multi-finger manipulator as a case
study.

Dexterous manipulation requires dealing with high-dimensionality of the
system, hybrid dynamics, and uncertainties about the environment [2]. Prior
work has trained a control policy for in-hand manipulation of a block with a
Shadow Dexterous Hand [3]. This was achieved through scientific compute
clusters for distributed training in simulation and access to specialized
hardware (the “cage”) to receive reliable state information during object
interaction in the real-world. While impressive, the requirement of such an
exorbitant infrastructure makes this kind of study typically non-reproducible
bordering impractical, hence the paucity of results building upon this work
for further research into learning manipulation.

Contrarily, Wüthrich et al. [4] designed an open-sourced low-cost robotic
platform for dexterous manipulation called _TriFinger_. They showed that the
robot is suitable for deploying learned policies due to its hardware
robustness and software safety checks. Building upon this work, the authors
organized the ‘Real Robot Challenge (RRC)’ [5], for which they developed a
farm of TriFinger systems. For the challenge the authors provided access to
PyBullet simulation of the robot [6]. This challenge reinforced the
inaccessibility of applying learning-based systems for such tasks: most
competing teams elected to use structured policies with manually tuned high-
level controllers or residual learning on top [7, 8, 9].

One of the reasons for the above is the ubiquity of CPU-based simulators [10,
11]. Due to its low sample generating rate, it is highly time-consuming to
tune and train a successful policy on PyBullet (a CPU based simulator) for
such a complex task. The TriFinger platform is based on low-cost open-source
hardware [4]. The TriFinger system uses a vision-based tracker to triangulate
the object pose. The tracker functions at a low frequency and provides noisy
estimates of the cube pose, making reliable policy inference difficult.
Lastly, working on the cloud-based platform Trans-Atlantic with limited access
to the hardware slows down the iteration cycles. While the overall cost of the
system is reduced when compared to a high-end setup like the Shadow Hand used
by OpenAI, more noise and delays are present due to the commodity nature of
the hardware. This makes sim-to-real transfer non-trivial.

This effort aims to overcome these limitations through a systems approach to
robotics with infrastructure around a GPU-accelerated simulator coupled with a
remote TriFinger system for successful sim-to-real transfer of a learned
policy, as shown in Figure 1. Using NVIDIA’s IsaacGym [12], we train a policy
to learn 6-DoF object manipulation successfully in under a day on a single
desktop-grade GPU. This number is in contrast to previous efforts, for
example, OpenAI’s distributed training infrastructure, which took several days
to learn a robust policy for cube rotation on a large distributed server
involving hordes of CPU and GPU clusters. Additionally, we investigate
different object pose representations for observations and rewards
formulation. To allow successful sim-to-real transfer, we perform domain
randomization on various physics properties in the simulator along with
different noise models for observations and actions.

This paper primarily makes robot systems contributions as follows:

  1. 1.

We provide a framework for learning similar in-hand manipulation tasks with
sim-to-real transfer using far fewer computational resources (1 GPU & CPU)
than prior work but that can also benefit from large scale training.

  2. 2.

We show the benefits of using keypoints as representations of object pose and
in reward computation with RL algorithms for in-hand manipulation, especially
when reposing in $SE(3)$.

  3. 3.

We demonstrate the ability to learn a challenging 6-DoF manipulation task, re-
posing a cube, with simulation data alone and deploying on a third-party
remote physical robot setup.

  4. 4.

We open-source the software platform to run training in simulation and
inference of the resulting policies for other researchers to build on top of.

Figure 1: Top: Our system learns to manipulate a cube to desired 6-DoF goal
poses (position+orientation) entirely in simulation. Training is done on
desktop-scale compute using a GPU-based physics engine. Bottom: We then
transfer to a real robot located thousands of kilometers away from where
development work is done.

##  2 Related Work

Advances in reinforcement learning (RL) algorithms and computational hardware
have enabled rapid progress in using these algorithms for tasks on real
robots. Techniques such as domain randomization and large-scale training have
enabled results across a variety of tasks with sim2real, including in-hand
manipulation [3, 13], as well as in legged locomotion [14, 15]. Active
identification of system parameters has also been shown to be helpful in the
context of learning manipulation tasks [16]. However, few results have focused
on learning-based control over a full 6-DoF pose in-hand. Furthermore, none of
the existing systems for training in-hand manipulation at a large scale have
been provided in an accessible manner for further work in robot learning to
build on.

The most widely used simulators for robot learning research are MuJoCo [11]
and Pybullet [10]. While both have proven to be successful for various
robotics locomotion and manipulation tasks, they are often slow for
complicated environments and require CPU clusters putting some limits on their
scalability. Brax [17], on the other hand, supports GPU / TPU acceleration,
but it comes at the cost of simplified physics simulation assumptions and
simple environments. IsaacGym [12] offers high-fidelity physics modelling and
GPU acceleration support. It also supports directly sharing observations and
actions through GPU memory between policy network and the physics engine,
massively increasing throughput. Part of our contribution is to demonstrate
the viability of the approach of GPU-based simulation to sim2real for in-hand
manipulation.

Figure 2: Our system trains using the IsaacGym simulator222Available for
download at: https://developer.nvidia.com/isaac-gym[12], we train on 16,384
environments in parallel on a single NVIDIA Tesla V100 or RTX 3090 GPU.
Inference is then conducted remotely on a TriFinger robot located across the
Atlantic in Germany using the uploaded actor weights. The infrastructure on
which we perform Sim2Real transfer is provided courtesy of the organisers of
the Real Robot Challenge [5].

##  3 Method

###  3.1 Simulation Environment

We train on the IsaacGym simulator [12], a simulation environment tailored
towards allowing policy learning with a high sampling rate (>50K samples/sec
in policy inference on Tesla V100 and around 100K samples/sec on RTX 3090) on
a single GPU. This is crucial in our goal of providing an accessible yet
generalisable framework for sim2real with in-hand manipulation. We simulate
the physical system with $\Delta t=0.02s$, which we found gave a good balance
between simulation fidelity, speed of learning, and ability to learn high
frequency motions required in such a manipulation task.

###  3.2 Representation of the Cube Pose

Our task involves manipulating an object in 6 degrees of freedom. As such, we
must represent this pose in multiple stages of our training pipeline. Prior
work has shown the benefits of alternate representations of spatial rotation
when using neural networks [18]. We choose to represent the pose of the object
using 8 keypoints sampled in the object’s local space at each vertex of the
cube, $\mathsf{k_{local,i}}$ $i=1,\dots,8$. These locations of the keypoints
in the object’s local frame are constant. We denote the keypoints in the world
frame for the current pose of the object, $\mathsf{k_{curr,i}}$ and the goal
pose of the object $\mathsf{k_{target,i}}$ $i=1,\dots,8$. These are obtained
by a straightforward transformation of the $\mathsf{k_{local,i}}$ back into
the world frame using the current / goal poses of the object. This results in
a set of 8 keypoints in 3-D Euclidean space. When concatenated for policy
inference, the set of 8 keypoints yields a 24-D vector representing an object
pose. In Sec. 4.2, we contrast this representation to a position+quaterinon
formulation used in [3, 12], finding that it improves the policy’s success
rate. We note as long as we are able to get the pose of the object (eg. via
[19]), we are able to obtain the 8 keypoints of the bounding box, and that
this does not depend on the object morphology (as shown in Appendix A.4).

###  3.3 Observation and Action Spaces

The observations of the policy (actor) and value-function (critic) are
described in Table 1(a) and 1(b). We use the representation of cube pose
described in the previous Sec 3.2. The action space of our policy is torque on
each of the 9 joints of the robot. The torque on each joint is limited such
that it does not damage the equipment while in operation. A safety damping is
applied to the output torques in simulation to mimic those on the real-world
robot.

Observation space | Degrees of freedom
---|---
Finger joints | position | $3\text{ fingers}\cdot 3\text{ joints}\cdot 1[\mathbb{R}^{1}]=9$
velocity | $3\text{ fingers}\cdot 3\text{ joints}\cdot 1[\mathbb{R}^{1}]=9$
Cube pose | keypoints | $8\text{ keypoints}\cdot 3\text{ }[\mathbb{R}^{3}]=24$
Goal pose | keypoints | $8\text{ keypoints}\cdot 3\text{ }[\mathbb{R}^{3}]=24$
Last action | torque | $3\text{ fingers}\cdot 3\text{ joints}\cdot 1[\mathbb{R}^{1}]=9$
Total | 75

(a) Actor Observations

Observation space | Degrees of freedom
---|---
Actor Observations (w/o DR) | 75
Cube | velocity | $6\text{ }[\mathbb{R}^{6}]$
Fingertips state | pose | $3\text{ fingers}\cdot 7\text{ }[\mathbb{R}^{3}\times SO(3)]=21$
velocity | $3\text{ fingers}\cdot 6\text{ }[\mathbb{R}^{6}]=18$
wrench | $3\text{ fingers}\cdot 6\text{ }[\mathbb{R}^{6}]=18$
Finger joints | torque | $3\text{ fingers}\cdot 3\text{ joints}\cdot 1[\mathbb{R}^{1}]=9$
Total | 147

(b) Critic Observations

Table 1: Asymmetric actor-critic to learn dexterous manipulation. While the
actor receives noisy observations, which are added as a part of DR (Section
3.5), the critic receives the same information without any noise and also has
access to certain privileged information from simulator.

###  3.4 Reward Formulation & Curriculum

Our reward has three components, each of which we found to be helpful in
achieving good training and sim-to-real performance. Following [14], we use a
logistic kernel to convert tracking error in euclidean space into a bounded
reward function. We slightly generalise the given formulation to account for a
range of distance scales, defining,
$\mathcal{K}(x)=\left(e^{ax}+b+e^{-ax}\right)^{-1}$, where $a$ is a scaling
factor and $b$ controls the sensitivity to the kernel at low values of
distance. We use $a=30$ and $b=2$.

As noted in Sec 3.2, we use keypoints in order to calculate the reward. The
component of the reward corresponding to the distance between the cube’s
current pose and the desired target pose is given by:
$\texttt{object\\_goal\\_reward}=\sum_{i=1}^{N}\mathcal{K}(||\mathsf{k_{curr,i}}-\mathsf{k_{target,i}}||_{2})$,
where $\mathsf{k_{curr,i}}$ and $\mathsf{k_{target,i}}$ are each of the $N=8$
keypoints at the corners of the current and target cubes, respectively.

In order to encourage the fingers to reach the cube during initial
exploration, we give a reward for moving the fingers towards the cube, which
was also found to be helpful in [20]. This term is defined by sum of the
movement of each fingertip towards the goal per timestep:
$\texttt{fingertip\\_to\\_object}=\sum_{i=1}^{3}\Delta^{t}_{i}$, where
$\Delta$ denotes the change across the timestep of the fingertip distance to
the centroid of the object,
$\Delta^{t}_{i}=||\mathsf{ft_{i,t}}-\mathsf{p_{curr,centroid,t}}||_{2}-||\mathsf{ft_{i,t-1}}-\mathsf{p_{curr,centroid,t-1}}||_{2}$,
and $\mathsf{ft_{i}}$ denotes the position of the $i$-th fingertip.

Finally, we define a penalty on the movement of each finger, preventing sudden
motions that may be difficult to execute reliably on the real robot:
$\texttt{fingertip\\_velocity\\_penalty}=\sum_{i=1}^{3}||\dot{\mathsf{ft_{i}}}||_{2}^{2}$

Our total reward is defined as:

$\begin{split}R(s,a)=\,&w_{fo}\times\texttt{fingertip\\_to\\_object}\times\mathbbm{1}(\texttt{timesteps}\leq
5e7)\\\
&+w_{fv}\times\texttt{fingertip\\_velocity\\_penalty}+w_{og}\times\texttt{object\\_goal\\_reward}\end{split}$
(1)

where $w_{fo}=-750$, $w_{fv}=-0.5$ and $w_{og}=40$ are the weights of each
reward component. We also found in initial experimentation that the curriculum
reducing the weight of the fingertip_to_object reward to $0$ after $50$
million steps was needed in order to allow the robot to perform nonprehensile
manipulation of the cube (releasing it in order to reorient). However, having
the reward term during the initial phases of training dramatically sped up
learning by encouraging the robot to interact with the cube.

###  3.5 Domain Randomization

Domain Randomization (DR) is a straightforward yet practical technique for
improving the robustness of policies in sim2real transfer [21, 22, 23]. We
choose our Domain Randomization parameters to account for modelling errors in
the environment as well as noise in sensor measurement noise. These parameters
are listed in Table 2. In addition to these randomizations, we apply random
forces to the cube in the same manner as described in [3] in order to improve
the stability of grasps and represent unmodelled dynamics. We mimic the
dynamics of the camera on the real system, described in Sec 3.7, by repeating
the observation of the keypoints for 5 frames.

Parameter | Range | $\sigma$ | $\sigma_{corr}$
---|---|---|---
Observation Noise
Cube Position333The noise to keypoints is not applied directly. Rather the noise is added to the cube pose in the world frame before computing the keypoints through it. | [-0.30, 0.30] | 0.002 | 0.000
Cube Orientation††footnotemark:  | [-1.00, 1.00] | 0.020 | 0.000
Finger Joint Position | [-2.70, 1.57] | 0.003 | 0.004
Finger Joint Velocity | [-10.00, 10.0] | 0.003 | 0.004
Action Noise
Applied Joint Torque | $[-0.36,0.36]$ | 0.02 | 0.01

Parameter | Scaling Distribution
---|---
Environment Parameters
Object Scale | $\textrm{uniform}(0.97,1.03)$
Object Mass | $\textrm{uniform}(0.70,1.30)$
Object Friction | $\textrm{uniform}(0.70,1.30)$
Table Friction | $\textrm{uniform}(0.50,1.50)$
External Forces | Refer to [3, pp. 9]

Table 2: For observations and actions, $\sigma$ and $\sigma_{corr}$ are the
standard deviation of additive gaussian noise sampled every timestep and at
the start of each episode, respectively. For environment, the parameters
represent scaling factor applied to the nominal values in the real robot
model.

###  3.6 Policy Architecture & Learning

We train using the Proximal Policy Optimization algorithm [24], using the
implementation from [25], which vectorizes observations and actions on GPU
allowing us to take advantage of the parallelization provided by the simulator
(see Sec 3.1). We use the following hyper-parameters: discount factor
$\gamma=0.99$, clipping $\epsilon=0.2$. The learning rate is annealed linearly
over the course of training from $5\mathrm{e}{-3}$ to $1\mathrm{e}{-6}$;
detailed hyperparameters are described in Appendix A.1 The inputs to the
policy are described in Table 1(b). We use an asymmetric actor critic approach
[26] with an actor that has 4 hidden layers, 2 of size 256 followed by 2 of
size 128, and 9 outputs which are scaled to the torque ranges of the real
robot and a critic that has 2 layers of size 512, followed by 2 layers of size
256 and 128 each and produces a scalar value function as output.

Figure 3:  The actor and critic networks are parameterized using fully-
connected layers with ELU activation functions [27].

###  3.7 Policy Inference on Remote Real Robot

We evaluate our policy remotely on the TriFinger system [4] provided by the
organisers of the real robot challenge [7]. The cube is tracked on the system
using 3 cameras, described in [28]. We convert the position+quaternion
representation output by this system into the keypoints representation
described in 3.2 and use it as input to the policy. Observations of the cube
pose from the camera system are provided at 10Hz. Proprioceptive measurements
are available at up to 1KHz. Our policy is evaluated at 50Hz, matching the
simulation timestep. We repeat the camera-based cube-pose observations for
subsequent rounds of policy evaluation to allow the policy to take advantage
of the higher-frequency proprioceptive data available to the robot. The
resulting observations are identical to what we use in simulation (Table
1(a)).

##  4 Experiments

In our experiments, we aim to answer the following four questions pertaining
to learning a robust policy for this task, as well as evaluating how well it
transfers to the real world:

  1. 1.

How well does our system using large-scale simulation train on this task with
a reward function similar to what has been previously proposed?

  2. 2.

How does training performance change when we use a different representation -
keypoints - for reward computation and policy input?

  3. 3.

Is our policy robust to sensor noise and varying environment parameters, and
is robustness improved by use of Domain Randomization during training?

  4. 4.

How well do our policies, trained entirely in simulation, transfer to the real
TriFinger system?

###  4.1 Experiment 1: Training

Figure 4:  Training curves on a reward function similar to prior work [7, 8]
for the setting with DR. We take the average of 5 seeds; the shaded areas show
standard deviation, noting that curves for Orientation and
Position+Orientation overlap during training. It is worth noting that the
nature of the reward makes it very difficult for the policy to optimize,
particularly achieving an orientation goal.

The aim in our 6-DoF manipulation task is to get the position and orientation
of the cube to a specified goal position and orientation. We define our metric
for ‘success’ in this task as getting the position within 2 cm, and
orientation within 22°of the target goal pose as used in [3]; comparable to
mean results obtained in [7]. Following previous works dealing with similar
tasks [3, 13, 20], we attempted applying a reward based on the position and
orientation components of error individually.

We spent considerable effort experimenting with a variety of kernels and
tuning the parameters of the translation / rotation based reward. The best
candidate reward of this format was:
$\texttt{object\\_goal\\_reward}=\mathcal{K}(||\mathsf{t_{curr}}-\mathsf{t_{target}}||_{2})+\frac{1}{3\times\mathsf{|rot\\_dist|}+0.01}$,
where
$\texttt{rot\\_dist}=2\times\arcsin(\min(1.0,||\mathsf{q_{diff}}||_{2})),\mathsf{q_{diff}}=\mathsf{q_{curr}}\mathsf{q_{target}^{*}}$.
The argument of $\mathcal{K}$ is the logistic kernel that takes L2 norm
between the current and target cube position as input, and
$\mathsf{rot\\_dist}$ is the distance in radians between the current and
target cube orientation. We use the alternative scaling parameter $a=50$ in
$\mathcal{K}$, which we found to work better in this reward formulation (see
Sec. 3.4). We use the same weightings for each of the 3 components of the
reward as in Sec 3.4.

The results are shown in Figure 4. We found that while this formulation of the
reward was good at allowing PPO to learn a policy to get the cube to the goal,
even after 1 Billion steps in an environment with no Domain Randomization it
was learning very slowly to achieve the orientation goal.

(a) No DR.  (b) With DR.

Figure 5: Success Rate over the course of training without and with domain
randomization. Each curve is the average of 5 seeds; the shaded areas show
standard deviation.

###  4.2 Experiment 2: Representation of Pose

The poor results in Experiment 1 (Sec 4.1) lead us to search for alternative
representations of cube pose in the calculation of the reward and policy
observations; these are described in Sec 3.2 & 3.4. We compared our method of
using keypoints to represent the object pose and using positions and
quaternions along two axes. Firstly, using it as the policy input as compared
to a position and quaternion representation, and secondly, using it to
calculate the reward as compared to a reward based on the linear and angular
rotational distances individually.

For the observations, in order to provide a fair comparison between
position/quaternion and keypoints as policy input, we ensured that we applied
observation noise and delays in the same manner (by applying them in the
position and quaternion space before transforming to keypoints, as noted in
Sec. 3.5). Also note that both representations only rely on the spatial pose
information of the cube to compute. We represent the pose of the cube with a
7-dim vector involving translation and quaternion ($\mathsf{t}$,
$\mathsf{q}$). We provide the position and quaternion of the goal pose as
input to the actor and critic, replacing the keypoints in Tables 1(a) & 1(b).

For the reward, in order to provide a fair comparison to the keypoints reward,
as mentioned previously we spent many hours tuning the kernels and parameters
used in the translation based reward, described in Experiment 1. In
comparison, little effort was spent tuning the keypoints function, with only
one tweak to the weightings in the logistic kernel, showing the relative
simplicity of working with this formulation.

Figure 5 shows the results of training, with both timesteps and wall-clock
time. In the curve without any Domain Randomization, we trained for 1 Billion
steps over the course of 6 hours on a single GPU. Using Keypoints in
observations and the reward function performs the best of the four policies,
also exhibiting a low variance among seeds.

When Domain Randomization is applied, the two curves with a keypoints-based
reward are far better in terms of success rate at the end of training and in
terms of convergence rate, however in this case having observations be
keypoints seems to matter somewhat less. This is perhaps due to the longer
training (4B steps & 24 hours on a single GPU) overwhelming the inductive bias
introduced by using keypoints as representations. However, using keypoints to
compute the reward provided a large benefit in both cases, showing the
improvement caused by calculating the reward in Euclidean space rather than
mixing linear and angular displacements through addition.

Figure 6: We show the robustness to varying the object parameters outside the
DR range it was trained for. In the evaluations, all the other DR is turned
off to ensure a controlled setting. Each success is evaluated over 1024 runs
with random goal and object initialization.

###  4.3 Experiment 3: Robustness of Policies in Simulation

In order to investigate the impact that Domain Randomization (DR, see Sec 3.5)
has on the robustness of policies of a hand in this configuration, we ran
experiments by varying parameters outside of the normal domain randomization
ranges in simulation. Figure 6 shows the results. We find that, despite only
being randomised initially within a range of 0.97-1.03x nominal size, our
policies with Domain Randomization achieve over an 80% success rate even with
a scale of 0.6 and 1.2x nominal size, while those without DR have a success
rate that drops off much more quickly outside the normal range. We find
similar results when scaling the object mass relative to the nominal range,
however in this case we find that the policies using keypoints-based reward
even without DR is much more robust at masses 3x nominal.

Figure 7:  Success Rate on the real robot plotted for different trained
agents. O-PQ and O-KP stand for position+quaternion and keypoints observations
respectively, and R-PQ and R-KP stand for linear+angular and keypoints based
displacements respectively, as discussed in Sec 4.2. Each mean made of N=40
trials and error bars calculated based on an 80% confidence interval.

###  4.4 Experiment 4: Simulation to Remote Real Robot Transfer

We ran experiments on the real robot to determine the success rate of the
policies trained with Domain Randomization under the metric defined in Sec 4.
We performed $N=40$ trials for each policy; the results for each of the four
ablations on keypoints which we tested are shown in Figure 7.

Out of the four models discussed in 4.2, the best policy achieved a success
rate of 82.5%. This was achieved with the use of keypoints used in
observations of the policy as well as the reward function during training
(O-KP+R-KP). The policy using position+quaternion representations but with a
reward calculated with keypoints (O-PQ+R-KP) achieved a 77.5% success rate.
These first two policies were well within each others’ confidence intervals.
This is likely due to the impact of the better representation of keypoints
being mitigated somewhat after 4 Billion steps of training, as discussed in
4.2. In contrast, neither of the policies trained using the position &
quaternion based reward achieved good success rates, with the policy using
keypoints-based observations (O-KP+R-PQ) achieving only a 60% success rate
while the one with position and quaternion observations (O-PQ+R-PQ) only
achieved a 55% success rate. These results show the importance of having a
reward function which effectively balances learning to achieve the goal in
$R^{3}$ and $SO(3)$ in order to have policies with a high success rate in
simulation and thus a high corresponding success rate after real robot
transfer.

Figure 8: (a) Achieving a challenging goal at the edge of the workspace. (b)
Recovery from dropping due to a bad grasp.

We noticed a variety of emergent behaviours used to achieve sub-goals within
the overall cube-reposing task. We display some of these in the panel in
Figures 1 and 8. The most prominent of these is "dropping and regrasping". In
this maneuver, the robot learns to drop the cube when it is close to the
correct position, re-grasp, and pick it back up. This enables the robot to get
a stable grasp on the cube in the right position. The robot also learns to use
the motion of the cube to the correct location in the arena as an opportunity
to simultaneously rotate it on the ground to make achieving the correct grasp
in challenging target locations far from the center of the fingers’ workspace.
Our policy is also robust towards dropping - it is able to recover from a cube
falling out of the hand and retrieve it from the ground.

##  5 Summary

This paper emphasizes the empirical value of a systems approach to robot
learning through a case study in dexterous manipulation. We introduced a
framework for learning in-hand manipulation tasks and transferring the
resulting policies to the real world. Using GPU-based simulation, we showed
how this can be done with order of magnitude fewer computational resources
than prior work. Furthermore, we show how RL algorithms for in-hand
manipulation can benefit from using keypoints as opposed to the more ordinary
angular and linear displacement-based reward and observation computation. This
paper shows a path for democratization of robot learning and a viable solution
through large scale simulation and robotics-as-a-service.

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##  Appendix A Appendix

###  A.1 Training

####  A.1.1 Learning Algorithm Details

Hyperparameter | Value
---|---
Discount Factor | 0.99
GAE Discount Factor ($\tau$) | 0.95
Learning Rate (start of training) | 5e-4
Learning Rate (end of training, linear decay) | 1e-6
Batch Size | 65356
Mini-batch Size | 16384
Number of Epochs | 8
Clip Range ($\epsilon$) | 0.2
Entropy Coefficient | 0
Table 3: PPO Hyperparameters.

We used the open-source version of PPO from [25] which provides the ability to
work with highly vectorised environments. The hyperparameters used are listed
in Table 3.

###  A.2 Success for rotation and position

We break out position and rotation success rates individually in Figure 9.
They show that the keypoint-based reward formulation fixes the issues
identified in Experiment 1 from the paper, namely that summing position and
orientation components of reward leads to poor orientation success rate. Using
keypoints improves orientation performance without sacrificing achieving the
position goal. It is still apparent that progress can be made with reducing
this gap as the orientation reward still continues improving until 4 Billion
steps of experience, and this is a direction of ongoing work.

(a) Position Goal at End of Episode  (a) Orientation Goal at End of Episode

Figure 9: End of episode success for keypoints and no keypoints.

####  A.2.1 Reproducing results on a consumer GPU

The reward curves and times stated in the main text were produced on a single
NVIDIA V-100 GPU. We were able to reproduce these results on a desktop machine
with a consumer-grade NVIDIA RTX3090 GPU. This produced the same reward curves
but actually reduced the training times from around 24 hours to 20 hours,
showing the ability of our system to train on a desktop.

###  A.3 Details of Sim2Real Transfer

####  A.3.1 Success Thresholds

Figure 10: We investigated the impact of varying the thresholds on success
rate. The results are shown in the heatmap.

The success rates on the real robot for different thresholds of position and
orientation are shown in Figure 10. We see a graceful degradation as the
success thresholds are tightened. We note that these are necessarily based on
noisy camera observations due to the remote nature of the setup; at 0.01m of
position and 0.1rad of orientation error this becomes a particular problem.
Note also that ’success’ for us is based off a different metric than some
other works (eg. [3]): we define ’success’ as being within the goal at the end
of an episode instead of achiving it at any point during it. This is because
part of the challenge of the Trifinger orientation task is being able to grasp
and hold the cube in position, as the upside-down orientation of Trifinger
making this challenging.

####  A.3.2 Hardware setup

As mentioned in the main text, we perform inference on the Trifinger platform
remotely. The interface is described in the corresponding whitepaper [4].

Inference, including camera tracking and running the network, is performed on
CPU on the same computer that hand-written solutions to last year’s real robot
challenge [7, 8, 9] were written on. An entire setup to run our system,
including training, inference and physical robot hardware, could be purchased
for less than US$10,000.

####  A.3.3 Software details

Inference is done in the Python; the time from getting the observations to
sending the actions to the hardware platform is on the order of 5-8ms, a delay
consisting of generating keypoints observations and running the policy.
Reducing this delay by moving our inference code to C++ is a direction for
future improvements to our system.

####  A.3.4 Pose Filtering

Unlike some previous works using visual information to perform in-hand
manipulation, our system uses the pose estimator provided in [28]. This
performs iterative optimization without reference to the history, and thus can
provide temporally inconsistent quaternion inputs to the policy, with the
quaternion value flipping between $+q$ and $-q$. We found that this
destabilised the policies which were provided position and quaternion inputs
during inference, and so implemented a simple filter over the input: if the
quaternion from the last camera measurement $q_{last}$ was within 0.2 of the
negated quaternion from a new camera measurement $-q_{new}$, we used
$-q_{new}$ in the policy input. While this had no impact on the keypoints
model (it performs an analytic transformation prior to policy inference which
is invariant to this issue) we found it important to perform this
transformation to allow stable grasps in policies which took raw quaternions
as input and thus to provide a fair comparison.

We tried using an Extended Kalman Filter using the formulation from [29] in
order to account for the noise in camera observations. However, we did not
find that the performance of our policies on the real-robot was noticeably
improved as compared with policies, likely due to the high variance in the
unknown acceleration in in-hand manipulation.

###  A.4 Other objects

Figure 11: We experimented with seeing how our learned policy generalized to
other objects.

We experimented with our system to see what the 0-shot transfer performance to
different object morphologies was. In order to do this, we swapped the objects
in the simulator, ran inference the O-KP+R-KP policy (see Section 4) that
produced the best sim2real transfer results, and measured the success rate. We
do not change the keypoints representation, but rather keep the 8 keypoints as
if they lie on the original 6.5cm3 cube despite the changing object
morphology. We were only able to perform these experiments in simulation, as
in the remote Trifinger setup we did not have the capacity to swap out
objects. However, using an off the shelf pose detector (eg. [19]) we are
confident that the same system would produce good sim2real transfer results.

We tested a sphere of diameter similar to the cube side length, cuboids of
different sizes, and a few objects from the YCB dataset [30]. The results are
listed in Table 4. Our our policy generalises surprisingly well on different
object morphologies, for example by achieving nearly 70% accuracy on a mug
(depicted in figure 3 in Figure 11). However, it struggles with long and
skinny objects. This is unsurprising given the difficulty in grasping the cube
at less than 0.5x the original scale (or 3cm).

Object | Success Rate
---|---
Cube 6.5cm3 [Training Object] | 92.1%
Ball radius=3.75cm | 96.7%
Cuboid 2x8x2cm | 2.0%
Cuboid 2x8x4cm | 42.9%
Cuboid 4x8x4cm | 92.4%
Cuboid 2x6.5x2cm | 2.44%
Cuboid 2x6.5x4cm | 48.9%
Cuboid 4x6.5x4cm | 94.6 %
YCB Mug (025_mug) | 68.8 %
YCB Banana (011_banana) | 28.0 %
YCB Potted Meat Can (010_potted_meat_can) | 81.1 %
YCB Foam Brick (061_foam_brick) | 91.7 %
Table 4: 0-shot object transfer performance. Numbers calculated from N=1024
trials in simulation. For these, we disabled Environment and Observation &
Action Randomizations (though still only gave 10Hz observations to simualte
camera tracking). Objects prefixed with YCB were used from the YCB dataset.
[30]


# Commutator Theorems for Fractional Integral Operators on Weighted Morrey
Spaces

Zengyan Si  Zengyan Si
School of Mathematics and Information Science
Henan Polytechnic University
Jiaozuo 454000
P. R. China<EMAIL_ADDRESS>
###### Abstract.

Let $L$ be the infinitesimal generator of an analytic semigroup on
$L^{2}(R^{n})$ with Gaussican kernel bounds, and let $L^{-\alpha/2}$ be the
fractional integrals of $L$ for $0<\alpha<n.$ For any locally integrable
function $b$, The commutators associated with $L^{-\alpha/2}$ are defined by
$[b,L^{-\alpha/2}](f)(x)=b(x)L^{-\alpha/2}(f)(x)-L^{-\alpha/2}(bf)(x)$. When
$b\in BMO(\omega)$(weighted $BMO$ space) or $b\in BMO$, the author obtain the
necessary and sufficient conditions for the boundedness of $[b,L^{-\alpha/2}]$
on weighted Morrey spaces respectively.

###### Key words and phrases:

weighted $BMO$ spaces, weighted Morrey spaces, fractional integrals.

###### 2000 Mathematics Subject Classification:

Primary 42B20, 42B25. Secondary 46B70, 47B38.

##  1\. INTRODUCTION AND MAIN RESULTS

Morrey [morrey] introduced the classical Morrey spaces to investigate the
local behavior of solutions to second order elliptic partial differential
equations. Chiarenza and Frasca [chiarenzafrasca] established the boundedness
of the Hardy-Littlewood maximal operator, the fractional operator and a
singular integral operator on the Morrey spaces. On the other hand, Coifman
and Fefferman [coifman], Muckenhoupt [muckenhoupt] studied the boundedness of
these operator on weighted $L^{p}$ spaces. Motivated by these work, Komori and
Shirai [ks] introduced the following weighted Morrey space and investigated
the boundedness of classical operators in harmonic analysis, that is, the
Hardy-Littlewood maximal operator, a Calderón-Zygmund operator, the fractional
integral operator, etc.

Let $1\leq p<\infty$ and $0\leq k<1$. Then for two weights $\mu$ and $\nu$,
the weighted Morrey space is defined by

$L^{p,k}(\mu,\nu)=\\{f\in
L_{loc}^{p}(\mu):||f||_{L^{p,k}(\mu,\nu)}<\infty\\},$

where

$||f||_{L^{p,k}(\mu,\nu)}=\sup_{Q}\left(\frac{1}{\nu(Q)^{k}}\int_{Q}|f(x)|^{p}\mu(x)dx\right)^{\frac{1}{p}}.$

and the supremum is taken over all cubes $Q$ in $R^{n}$.

If $\mu=\nu,$ then we have the classical Morrey space $L^{p,k}(\mu)$ with
measure $\mu$. When $k=0,$ then $L^{p,k}(\mu,\nu)=L^{p}(\mu)$ is the Lebesgue
space with measure $\mu$.

Suppose that $L$ is a linear operator on $L^{2}(R^{n})$ which generates an
analytic semigroup $e^{-tL}$ with a kernel $p_{t}(x,y)$ satisfying a Gaussian
upper bound, that is,

(1.1) $|p_{t}(x,y)|\leq\frac{C}{t^{\frac{n}{2}}}e^{-c\frac{|x-y|^{2}}{t}}$

for $x,y\in R^{n}$ and all $t>0.$

For $0<\alpha<n,$ the fractional integral $L^{-\alpha/2}$ of the operator $L$
is defined by

$L^{-\alpha/2}f(x)=\frac{1}{\Gamma(\frac{\alpha}{2})}\int_{0}^{\infty}e^{-tL}(f)\frac{dt}{t^{-\alpha/2+1}}(x).$

Note that if $L=-\Delta$ is the Laplacian on $R^{n}$, then $L^{-\alpha/2}$ is
the classical fractional integral $I_{\alpha}$ which plays important roles in
many fields. It is well known that $I_{\alpha}$ is bounded from $L^{p}(R^{n})$
to $L^{q}(R^{n})$ for all $p>1,1/q=1/p-\alpha/n>0$ and is also of weak type
$(1,n/(n-\alpha)).$

Let $1\leq p<\infty$ and $\omega$ be a weight function. A locally integral
function $b$ is said to be in $BMO_{p}(\omega)$ if

$||b||_{BMO_{p}(\omega)}=\sup_{Q}\left(\frac{1}{\omega(Q)}\int_{Q}|b(x)-b_{Q}|^{p}\omega(x)^{1-p}dx\right)^{\frac{1}{p}}\leq
C<\infty,$

where $b_{Q}=\frac{1}{|Q|}\int_{Q}b(y)dy$ and the supremum is taken over all
cube $Q\in R^{n}.$

Let $\omega\in A_{1}$, García-Cuerva [garcia] proved that the spaces
$BMO_{p}(\omega)$ coincide, and the norm of $||\cdot||_{BMO_{p}(\omega)}$ are
equivalent with respect to different values of provided that $1\leq p<\infty.$

Let $b$ be a locally integrable function on $R^{n}$, we consider the
commutator $[b,L^{-\alpha/2}]$ defined by

$[b,L^{-\alpha/2}](f)(x)=b(x)L^{-\alpha/2}(f)(x)-L^{-\alpha/2}(bf)(x).$

Chanillo [chanillo] proved that the commutator $[b,I_{\alpha}]$ of the
multiplication operator by $b\in BMO$ is bounded on $L^{p}$ for $1<p<\infty.$

Duong and Yan [duongyan] proved $[b,L^{-\alpha/2}]$ is bounded from $L^{p}$ to
$L^{q},$ where $b\in BMO,1<p<n/\alpha,1/q=1/p-\alpha/n,0<\alpha<n.$

Mo and Lu [molu] proved the multilinear commutator generated by $\vec{b}$ and
$L^{-\alpha/2}$ is bounded from $L^{p}$ to $L^{q},$ where
$1<p<n/\alpha,1/q=1/p-\alpha/n,0<\alpha<1$,
$\vec{b}=(b_{1},\cdots,b_{m}),b_{i}\in BMO,$ for $i=1,\cdots,m.$

Lu, Ding and Yan [ludingyan] proved $[b,I_{\alpha}]$ is bounded from $L^{p}$
to $L^{q}$ if and only if $b\in BMO.$

Wang [wang] proved that $[b,I_{\alpha}]$ is bounded from $L^{p,k}(\omega)$ to
$L^{q,kq/p}(\omega^{1-(1-\alpha/n)q},\omega),$ where $b\in BMO(\omega)$,
$0<\alpha<n,1<p<n/\alpha,1/q=1/p-\alpha/n,0<k<p/q$ and $\omega^{q/p}\in
A_{1}.$

Inspired by the above results, we study the boundedness properties of the
commutator $[b,L^{-\alpha/2}]$ on weighted Morrey spaces in this work. The
main theorems are stated as follows.

######  Theorem 1.1.

Let $0<\alpha<n,1<p<n/\alpha,1/q=1/p-\alpha/n,$ $0\leq k<p/q,$
$\omega^{\frac{q}{p}}\in A_{1}$ and $r_{\omega}>\frac{1-k}{p/q-k},$ where
$r_{\omega}$ denotes the critical index of $\omega$ for the reverse Hölder
condition. Then the following conditions are equivalent:

$(a)$ $b\in BMO(\omega).$

$(b)$ $[b,L^{-\alpha/2}]$ is bounded from $L^{p,k}(\omega)$ to
$L^{q,kq/p}(\omega^{1-(1-\alpha/n)q},\omega)$.

Specially, when $k=0$ in Theorem 1.1, we get

######  Corollary 1.1.

Let $0<\alpha<n,1<p<n/\alpha,1/q=1/p-\alpha/n,$ $\omega^{\frac{q}{p}}\in
A_{1}$ and $r_{\omega}>\frac{q}{p},$ where $r_{\omega}$ denotes the critical
index of $\omega$ for the reverse Hölder condition. Then the following
conditions are equivalent:

$(a)$ $b\in BMO(\omega).$

$(b)$ $[b,L^{-\alpha/2}]$ is bounded from $L^{p}(\omega)$ to
$L^{q}(\omega^{1-(1-\alpha/n)q})$.

Further more, if $L=-\Delta$ is the Laplacian, Then the following conditions
are equivalent:

$(a^{{}^{\prime}})$ $b\in BMO(\omega).$

$(b^{{}^{\prime}})$ $[b,I_{\alpha}]$ is bounded from $L^{p}(\omega)$ to
$L^{q}(\omega^{1-(1-\alpha/n)q})$.

######  Theorem 1.2.

Let $0<\alpha<n,0\leq k<p/q,1/q=1/p-\alpha/n,$ and $1<r,s<\infty$ such that
$1<rs<p<n/\alpha$, $\omega^{rs}\in A_{p/rs,q/rs}.$ Then the following
conditions are equivalent:

$(a)$ $b\in BMO.$

$(b)$ $[b,L^{-\alpha/2}]$ is bounded from $L^{p,k}(\omega^{p},\omega^{q})$ to
$L^{q,kq/p}(\omega^{q})$.

Specially, when $k=0$ in Theorem 1.2, we obtain

######  Corollary 1.2.

Let $0<\alpha<n,1/q=1/p-\alpha/n,$ and $1<r,s<\infty$ such that
$1<rs<p<n/\alpha$, $\omega^{rs}\in A_{p/rs,q/rs}.$ Then the following
conditions are equivalent:

$(a)$ $b\in BMO.$

$(b)$ $[b,L^{-\alpha/2}]$ is bounded from $L^{p}(\omega^{p})$ to
$L^{q}(\omega^{q})$.

Further more, if $L=-\Delta$ is the Laplacian, Then the following conditions
are equivalent:

$(a^{{}^{\prime}})$ $b\in BMO.$

$(b^{{}^{\prime}})$ $[b,I_{\alpha}]$ is bounded from $L^{p}(\omega^{p})$ to
$L^{q}(\omega^{q})$.

######  Remark 1.1.

It is easy to see that our results extend the results in
[chanillo],[duongyan],[ludingyan],[wang] significantly.

##  2\. PREREQUISITE MATERIAL

Let us first recall some definitions.

######  Definition 2.1.

The Hardy-Littlewood maximal operator $M$ is defined by

$M(f)(x)=\sup_{x\in Q}\frac{1}{|Q|}\int_{Q}|f(y)|dy.$

Let $\omega$ be a weight. The weighted maximal operator $M_{\omega}$ is
defined by

$M_{\omega}(f)(x)=\sup_{x\in Q}\frac{1}{\omega(Q)}\int_{Q}|f(y)|\omega(y)dy.$

For $0<\alpha<n,r\geq 1,$ the fractional maximal operator $M_{\alpha,r}$ is
defined by

$M_{\alpha,r}(f)(x)=\sup_{x\in Q}\left(\frac{1}{|Q|^{1-\frac{\alpha
r}{n}}}\int_{Q}|f(y)|^{r}dy\right)^{\frac{1}{r}};$

and the fractional weighted maximal operator $M_{\alpha,r,\omega}$ is defined
by

$M_{\alpha,r,\omega}(f)(x)=\sup_{x\in
Q}\left(\frac{1}{\omega(Q)^{1-\frac{\alpha
r}{n}}}\int_{Q}|f(y)|^{r}\omega(y)dy\right)^{\frac{1}{r}}.$

For any $f\in L^{p}(R^{n}),p\geq 1,$ the sharp maximal function
$M^{\sharp}_{L}f$ associated the generalized approximations to the identity
$\\{e^{-tL},t>0\\}$ is given by

$M^{\sharp}_{L}f(x)=\sup_{x\in
Q}\frac{1}{|Q|}\int_{Q}|f(y)-e^{-t_{Q}L}f(y)|dy$

where $t_{Q}=r^{2}_{Q}$ and $r_{Q}$ is the radius of the ball $Q$.

In the above definitions, the supremum is taken over all cubes $Q$ containing
$x.$

######  Definition 2.2.

A weight function $\omega$ is in the Muckenhoupt class $A_{p}$ with
$1<p<\infty$ if for every cube $Q$ in $R^{n}$, there exists a positive
constant $C$ which is independent of $Q$ such that

$\left(\frac{1}{|Q|}\int_{Q}\omega(x)dx\right)\left(\frac{1}{|Q|}\int_{Q}\omega(x)^{-\frac{1}{p-1}}dx\right)^{p-1}\leq
C.$

When $p=1,\omega\in A_{1},$ if

$\left(\frac{1}{|Q|}\int_{Q}\omega(x)dx\right)\leq C{\texttt{ess}\,\inf}_{x\in
Q}\,\omega(x).$

When $p=\infty,\omega\in A_{\infty},$ if there exist positive constants
$\delta$ and $C$ such that given a cube $Q$ and $E$ is a measurable subset of
$Q$, then

$\frac{\omega(E)}{\omega(Q)}\leq C\left(\frac{|E|}{|Q|}\right)^{\delta}.$

######  Definition 2.3.

A weight function $\omega$ belongs to $A_{p,q}$ for $1<p<q<\infty$ if for
every cube $Q$ in $R^{n}$, there exists a positive constant $C$ which is
independent of $Q$ such that

$\left(\frac{1}{|Q|}\int_{Q}\omega(x)^{q}dx\right)^{\frac{1}{q}}\left(\frac{1}{|Q|}\int_{Q}\omega(x)^{-p^{{}^{\prime}}}dx\right)^{\frac{1}{p^{{}^{\prime}}}}\leq
C.$

where $p^{{}^{\prime}}$ denotes the conjugate exponent of $p>1,$ that is
$1/p+1/p^{{}^{\prime}}=1.$

######  Definition 2.4.

A weight function $\omega$ belongs to the reverse Hölder class $RH_{r}$ if
there exist two constants $r>1$ and $C>0$ such that the following reverse
Hölder inequality

$\left(\frac{1}{|Q|}\int_{Q}\omega(x)^{r}dx\right)^{\frac{1}{r}}\leq
C\left(\frac{1}{|Q|}\int_{Q}\omega(x)dx\right)$

holds for every cube $Q$ in $R^{n}$.

It is well known that if $\omega\in A_{p}$ with $1\leq p<\infty$, then there
exists $r>1$ such that $\omega\in RH_{r}.$ It follows from Hölder s inequality
that $\omega\in RH_{r}$ implies $\omega\in RH_{s}$ for all $1<s<r.$ Moreover,
if $\omega\in RH_{r},r>1,$ then we have $\omega\in RH_{r+\epsilon}$ for some
$\epsilon>0.$ We thus write $r_{w}=\sup\\{r>1:\omega\in RH_{r}\\}$ to denote
the critical index of $\omega$ for the reverse Hölder condition.

We will make use of the following lemmas. We first provide a weighted version
of the local good $\lambda$ inequality for $M^{\sharp}_{L}$ which allow us to
obtain an analog of the classical Fefferman-Stein(see [fefferman, coifman])
estimate on weighted Morrey spaces.

######  Lemma 2.1.

([martell]) Assume that the semigroup $e^{-tL}$ has a kernel $p_{t}(x,y)$
which satisfies the upper bound (1.1). Take $\lambda>0,f\in L^{1}_{0}(R^{n})$
and a ball $Q_{0}$ such that there exists $x_{0}\in Q_{0}$ with
$Mf(x_{0})\leq\lambda.$ Then, for every $\omega\in A_{\infty},0<\eta<1,$ we
can find $\gamma>0$(independent of $\lambda,Q_{0},f,x_{0}$) and constant
$C_{\omega,}r>0$(which only depend on $\omega$) .

$\omega\\{x\in Q_{0}:Mf>A\lambda,M^{\sharp}_{L}f(x)\leq\gamma\lambda\\}\leq
C_{\omega}\eta^{r}\omega(Q_{0}).$

where $A>1$ is a fixed constant which depends only on $n.$

As a consequence, by using the standard arguments, we have the following
estimates:

For every $f\in L^{p,k}(\mu,\nu)$, with $1<p<\infty.$ If $\mu,\nu\in
A_{\infty},1<p<\infty,0\leq k<1.$

$||f||_{L^{p,k}(\mu,\nu)}\leq||Mf||_{L^{p,k}(\mu,\nu)}\leq
C||M^{\sharp}_{L}f||_{L^{p,k}(\mu,\nu)}$

In particular, when $\mu=\nu=\omega$ and $\omega\in A_{\infty},$ we have

$||f||_{L^{p,k}(\omega)}\leq||Mf||_{L^{p,k}(\omega)}\leq
C||M^{\sharp}_{L}f||_{L^{p,k}(\omega)}$

######  Lemma 2.2.

([wang]) Let
$0<\alpha<n,1<p<\frac{n}{\alpha},\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n}$ and
$\omega^{q/p}\in A_{1}.$ Then if $0<k<p/q$ and $r_{\omega}>\frac{1-k}{p/q-k}$,
we have

$||M_{\alpha,1}f||_{L^{q,kq/p}(\omega^{q/p},\omega)}\leq
C||f||_{L^{p,k}(\omega)}.$

The same conclusion still hold for $I_{\alpha}$.

######  Lemma 2.3.

([wang]) Let
$0<\alpha<n,1<p<\frac{n}{\alpha},\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n}$ and
$\omega^{q/p}\in A_{1}.$ Then if $0<k<p/q,1<r<p$ and
$r_{\omega}>\frac{1-k}{p/q-k}$, we have

$||M_{r,\omega}f||_{L^{q,kq/p}(\omega^{q/p},\omega)}\leq
C||f||_{L^{q,kq/p}(\omega^{q/p},\omega)}.$

######  Lemma 2.4.

([wang])
$0<\alpha<n,1<p<\frac{n}{\alpha},\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n}$,
$0<k<p/q$, $\omega\in A_{\infty}.$ For any $1<r<p,$ we have

$||M_{\alpha,r,\omega}f||_{L^{q,kq/p}(\omega)}\leq C||f||_{L^{p,k}(\omega)}.$


in Example 5. Let $X$ denote the character space of
$L^{\infty}({\mathbb{T}},\lambda)$. Then, via the Gelfand transform, we know
that $L^{\infty}({\mathbb{T}},\lambda)\cong\mathrm{C}(X)$. Let
$A\subset\mathrm{C}(X)$ denote the unital norm-closed subalgebra corresponding
to $H^{\infty}({\mathbb{D}})$ under this identification. Let $\tau$ denote the
state on $\mathrm{C}(X)$ corresponding to integration against $\lambda$ on
$L^{\infty}({\mathbb{T}},\lambda)$. Let $\Delta=\\{\tau\\}$. In this case, $A$
has the Amar–Lederer property in $B$ with respect to $\Delta$ by a classical
result of Amar and Lederer [AL1971]. Of course understanding this requires
also the dictionary between closed sets and closed projections, and peak sets
and closed projections in $A^{\perp\perp}$, that we have discussed earlier. On
the other hand, properties (iv) and (v) in Theorem 6.5 fail. To see this note
that the latter properties, in the specific case of this example, imply that
closed ‘Lebesgue-null’ sets are generalized peak sets. Yet [pelczynski1977,
Proposition 6.3] gives a closed ‘Lebesgue-null’ $G_{\delta}$ set in this
context which is not a peak set. Hence, it is not a generalized peak set,
since $G_{\delta}$ generalized peak sets are peak sets [gamelin1969, Lemma
II.12.1].

For $H^{\infty}({\mathbb{D}})\subset L^{\infty}({\mathbb{T}},\lambda)$ as
above, the Amar–Lederer property was established in [AL1971] as a tool in
characterizing the exposed points of the unit ball of
$H^{\infty}({\mathbb{D}})$. Subsequently, it also played a key role in Ando’s
proof of uniqueness for the predual of $H^{\infty}({\mathbb{D}})$ [ando1978],
and in establishing a Lebesgue decomposition for $H^{\infty}({\mathbb{D}})$.
In light of these developments, it stands to reason that the Amar–Lederer
property should also be meaningful in the context of certain noncommutative
analogues of the inclusion $H^{\infty}({\mathbb{D}})\subset
L^{\infty}({\mathbb{T}},\lambda)$. Indeed the Amar–Lederer property has
inspired several deep papers in noncommutative function theory, some of them
concerned with the topic of uniqueness of predual. Ueda and the first author
with Labuschagne have proved in [ueda2009, BL2018] a partial Amar–Lederer
theorem for maximal subdiagonal algebras in $\sigma$-finite von Neumann
algebras, and used this to give several powerful applications such as a
Lebesgue decomposition for such algebras. We say ‘partial’ because of a severe
restriction imposed in those papers on the projections in the theorem.
Accordingly, we set out to remove this restriction and establish the full
noncommutative Amar–Lederer theorem for the class of maximal subdiagonal
algebras. The first part of argument has little to do with those algebras, but
instead is purely von Neumann algebraic in nature.

###  9.1. Von Neumann algebraic preliminaries

Let $B$ be a $\mathrm{C}^{*}$-algebra and let $q\in B^{**}$ be a projection.
Let $\varphi$ be a state on $B$. We say that $\varphi$ is _regular at $q$_ if
for all projections $p\in B$ dominating $q$ and for all $\varepsilon>0$, there
exists an open projection $u\in B^{**}$ such that $p\geq u\geq q$ and
$\varphi(u)<\varphi(q)+\varepsilon$.

For example, let $X$ be a compact Hausdorff space and let $B=\mathrm{C}(X)$.
Let $\mu$ be regular Borel measure on $X$, and let $\varphi$ be the state on
$B$ of integration against $\mu$. Let $q=\chi_{E}\in B^{**}$ for some Borel
subset $E\subset X$. Then, $\varphi$ is regular at $q$. Indeed, given
$\varepsilon>0$, by the regularity property of the measure $\mu$, we may
choose an open projection $u\in B^{**}$ such that $u_{0}\geq q$ and
$\varphi(u_{0})<\varphi(q)+\varepsilon$. If we are also given a projection
$p\in B$ such that $p\geq q$, then we may simply take $u=u_{0}p$, which is
still open and has the required property.

######  Proposition 9.1.

Let $B$ be a unital $\mathrm{C}^{*}$-algebra and let $q\in B^{**}$ be a
projection. Let $\varphi$ be a state on $B$. Two sufficient conditions for
regularity of $\varphi$ at $q$ are the following:

  1. (i)

$q$ is open or closed.

  2. (ii)

$q$ is of type $F_{\sigma}$ and $\varphi$ is tracial with $\varphi(q)=0$.

###### Proof.

(i) The ‘open’ case is trivial, so assume $q$ closed. We adapt [akemann1969,
Proposition II.3]. Write $q^{\perp}=1-q$. Let $\varepsilon>0$ and let $p$ be a
projection in $B$ with $p\leq q^{\perp}$. The desired conclusion is equivalent
to showing that there is a closed projection $r\in B^{**}$ such that $p\leq
r\leq q^{\perp}$ and $\varphi(r)>\varphi(q^{\perp})-\varepsilon$. Because $q$
is closed and is orthogonal to $p$, $p+q$ is closed [akemann1969, Theorem
II.7], and hence $q^{\perp}-p=1-(q+p)$ is open. Thus, there is an increasing
net of contractions $(b_{i})$ in $B$ converging to $q^{\perp}-p$ in the
weak-$*$ topology of $B^{**}$. Consider the positive element $a_{i}=b_{i}+p\in
B$. The increasing net $(a_{i})$ converges to $q^{\perp}$ in the weak-$*$
topology of $B^{**}$. Choose an index $i$ large enough so that
$\varphi(q^{\perp}-a_{i})<\varepsilon/2$. Note that $a_{i}p=p$ since
$pb_{i}p\leq p(q^{\perp}-p)p=0$. Thus for any polynomial $f$ we have
$f(a_{i})p=f(1)p$, and by uniform approximation the same is true for any
continuous function on $[0,1]$. The proof of [akemann1969, Proposition II.3]
constructs a sequence $(f_{n})$ of continuous functions on $[0,1]$ with
$f_{n}(1)=1$ such that $(f_{n}(a_{i}))$ increases in the weak-$*$ topology to
the closed projection $r=\chi_{[\varepsilon/2,1]}(a_{i})\in B^{**}$. Because
$f_{n}(a_{i})p=p$ for each $n$, we find $rp=p$ or $r\geq p$. Finally, arguing
as in [akemann1969, Proposition II.3], we see that
$\varphi(q^{\perp}-r)<\varepsilon$ as desired.

(ii) Since $q$ is of type $F_{\sigma}$, there is a countable collection
$\\{q_{k}:k\geq 1\\}$ of closed projections in $B^{**}$ such that
$q=\bigvee_{k}q_{k}$. Let $\varepsilon>0$, and let $p$ be a projection in $B$
dominating $q$. For each $k\geq 1$, we apply (i) to find an open projection
$u_{k}\in B^{**}$ with $\varphi(u_{k})<\varepsilon/2^{k}$ and $q_{k}\leq
u_{k}\leq p$. The projection $u=\bigvee_{k}u_{k}$ is open and clearly
dominates $q$ and is dominated by $p$. Furthermore, since $\varphi$ is tracial
on $B$, it is also tracial on $B^{**}$ so we see that
$\varphi(u)\leq\sum_{k}\varphi(u_{k})$ by [takesaki2002, Proposition V.1.6].
Therefore $\varphi(u)<\varepsilon$. ∎

Remark. Proposition 9.1 is true for nonunital $\mathrm{C}^{*}$-algebras with
‘closed’ in (i) replaced by ‘compact’. This follows almost immediately by
applying the unital case in the unitization $B^{1}$, since $q$ is compact if
it is closed in $B^{1}$.

We also record a fact found in the proof of [akemann1969, Proposition II.14].

######  Lemma 9.2.

Let $M$ be a von Neumann algebra. Let $u\in M^{**}$ be an open projection and
$\varphi$ be a normal state on $M$. Then, there is a projection $p\in M$
dominating $u$ such that $\varphi(u)=\varphi(p)$.

We now establish the pivotal technical tool that we require.

######  Proposition 9.3.

Let $M$ be a von Neumann algebra, let $\varphi$ be a normal state on $M$ and
let $q\in M^{**}$ be a projection. Then $\varphi$ is regular at $q$ if and
only if there is a decreasing sequence of projections $(p_{n})$ in $M$
dominating $q$ such that $\lim_{n\to\infty}\varphi(p_{n})=\varphi(q)$.

###### Proof.

Suppose that there is a decreasing sequence of projections $(p_{n})$ in $M$
dominating $q$ such that $\lim_{n\to\infty}\varphi(p_{n})=\varphi(q)$, and
that $p$ is a projection in $M$ dominating $q$. For each $n$, we let
$u_{n}=p\wedge p_{n}\in M$, which is clopen. Since $q\leq u_{n}\leq p_{n}$ we
have that $\lim_{n\to\infty}\varphi(u_{n})=\varphi(q)$.

For the other direction, put $p_{0}=1$. Assume that we have constructed
projections $p_{0},\ldots,p_{n}\in M$ dominating $q$ such that $p_{j}\leq
p_{k}$ and $\varphi(p_{j})\leq\varphi(q)+1/(j+1)$ for every $0\leq k\leq j\leq
n$. Consider the von Neumann algebra $N=p_{n}Mp_{n}$. By regularity, there is
an open projection $u=p_{n}up_{n}\in M^{**}$ such that $u\geq q$ and
$\varphi(u)<\varphi(q)+1/(n+2)$. Also $u$ is open in $N^{**}$. Apply Lemma 9.2
to find a projection $p_{n+1}\in N$ such that $p_{n+1}\geq u\geq q$ and
$\varphi(p_{n+1})=\varphi(u)<\varphi(q)+1/(n+2)$. Since $p_{n+1}\in N$, we
must have that $p_{n+1}\leq p_{n}$. The existence of the desired sequence then
follows by induction. ∎

###  9.2. Subdiagonal algebras

Let us recall the setting of Arveson’s subdiagonal algebras, referring the
reader to [BL2007, ueda2009, BL2018] for additional detail and references.

Let $M$ be a $\sigma$-finite von Neumann algebra with a faithful normal state
$\varphi$. Let $A\subset M$ be a unital weak-$*$ closed subalgebra such that
$A+A^{*}$ is weak-$*$ dense in $M$. Put $D=A\cap A^{*}$ and let $E:M\to D$ be
a (normal faithful) conditional expectation. We say that $A$ is _subdiagonal_
in $M$ with respect to $E$ if $E$ is multiplicative on $A$. Such an algebra
$A$ is said to be a maximal subdiagonal algebra if $A$ is properly contained
in no larger subdiagonal algebra in $M$ with respect to $E$.

We now motivate the main result of this section. Let $\Delta\subset M^{*}$
denote the set of normal states on $M$. Recall that there is a central
projection $z\in M^{**}$ such that $M_{*}=M^{*}z$ [takesaki2002, Theorem
III.2.14]. Let $\psi$ be a state on $M$ and consider its support projection
$s_{\psi}\in M^{**}$, which is defined by

$\\{x\in M^{**}:\psi(x^{*}x)=0\\}=M^{**}(1-s_{\psi}).$

If we assume that $\psi\in\operatorname{SG}(\Delta)$, then by definition
$s_{\psi}r_{\Delta}=0$ or $s_{\psi}\in\operatorname{AC}(\Delta)^{\perp}$,
which by Lemma 7.3 is equivalent to $s_{\psi}\leq 1-z$. Under the additional
requirement that the given faithful normal state $\varphi$ be tracial, Ueda
showed in [ueda2009] that there always exists a closed projection
$p\in\operatorname{AC}(\Delta)^{\perp}\cap A^{\perp\perp}$ such that
$s_{\psi}\leq p$. This is a ‘partial Amar–Lederer theorem’. At the cost of
great technical difficulty, the tracial condition on $\varphi$ was removed in
[BL2018].

We now improve this result, where we replace the support projection of a
singular state by any $\varphi$-null regular projection. Fortunately, the very
lengthy and delicate argument found in [BL2018] can be adapted verbatim to our
situation, so we simply need to verify that the required conditions for its
application are met.

######  Theorem 9.4.

Let $M$ be a $\sigma$-finite von Neumann algebra with a faithful normal state
$\varphi$. Let $\Delta$ denote the set of normal states on $M$. Let $A\subset
M$ be a maximal subdiagonal algebra. Then, the following statements hold.

  1. (i)

$A$ has the Amar–Lederer property in $M$ with respect to $\Delta$. Indeed
every regular projection that is $\varphi$-null (or in
$\operatorname{AC}(\Delta)^{\perp}$) is dominated by a closed peak projection
in $\operatorname{AC}(\Delta)^{\perp}\cap A^{\perp\perp}$.

  2. (ii)

$A$ has the Forelli property in $M$ with respect to $\Delta$.

###### Proof.

(i) Note that $\operatorname{AC}(\Delta)=\operatorname{AC}(\varphi)$ by virtue
of Lemmas 7.3 and 7.2. In particular, it follows from [CT2023Henkin, Lemma
2.4] that a projection $p\in M^{**}$ lies in
$\operatorname{AC}(\Delta)^{\perp}$ if and only if $\varphi(p)=0$. Invoking
Proposition 9.1, it thus suffices to fix a given projection $q\in M^{**}$ such
that $\varphi(q)=0$ and with the property that $\varphi$ is regular at $q$,
and to show that there then exists a closed projection $p\in M^{**}$ lying in
$A^{\perp\perp}$ such that $\varphi(p)=0$ and $p\geq q$. By Proposition 9.3
there is a decreasing sequence of projections $(p_{n})$ in $M$ dominating $q$
such that $\lim_{n\to\infty}\varphi(p_{n})=0$. It follows that $p_{n}\searrow
0$ weak* in $M$ (since if $p_{n}\searrow r$ then $\varphi(r)=0$ and so $r=0$).
Let $p_{0}$ be the infimum (and weak* limit) of the $p_{n}$ in $M^{**}$, a
closed projection with $\varphi(p_{0})=0$.

We now have completed our generalization of the first (von Neumann algebraic)
part of the main theorem in [ueda2009] or [BL2018]. Fortunately the above is
all that is needed to be changed in the rest of the proofs of the latter. By
the very technical proof comprising pages 8223–8228 in [BL2018] we obtain a
contraction $b\in A$ with $bp_{0}=p_{0}=p_{0}b$. By the basic theory of peak
projections [hay2007, blecher2013] the real positive element $a=(1+b)/2$ peaks
at a projection $p\geq p_{0}$, with $a^{n}\to p$ weak* in $M^{**}$. As in
[ueda2009] we have $a^{n}\to 0$ weak* in $M$, so that
$p\in\operatorname{AC}(\Delta)^{\perp}\cap A^{\perp\perp}$. By construction
$p$ is a peak projection.

For the last assertion, simply note that (vi) in Theorem 6.5 implies the
Forelli property. ∎

Thus for example if we assume in addition that $\varphi$ is tracial then every
type $F_{\sigma}$ projection in $\operatorname{AC}(\Delta)^{\perp}$ is
dominated by a closed projection in $\operatorname{AC}(\Delta)^{\perp}\cap
A^{\perp\perp}$.

As noted in the proof we may replace $\Delta$ by $\\{\varphi\\}$ with no
change in the conclusions.

Finally, we remark that statement (i) above is almost certainly not provable
for general (even commutative) von Neumann algebras. Indeed there is a set
theoretic obstruction as is noted in the last section of [BL2018].

## References