# An efficient Deep Spatio-Temporal Context Aware decision Network (DST-CAN)
for Predictive Manoeuvre Planning

Jayabrata Chowdhury1, Suresh Sundaram2, Nishant Rao3, Narasimhan Sundararajan4
1Jayabrata Chowdhury is with Robert Bosch Centre for Cyber-Physical Systems &
Artificial Intelligence and Robotics Lab, Dept. of Aerospace Engineering,
Indian Institute of Science, Bangalore, India. 2Suresh Sundaram and 3Nishant
Rao are with the Artificial Intelligence and Robotics Lab, Dept. of Aerospace
Engineering, Indian Institute of Science, Bangalore, India. 4Narasimhan
Sundararajan is a Technical Consultant, WIRIN project, Dept. of Aerospace
Engineering, Indian Institute of Science, Bangalore, India. Email:
<EMAIL_ADDRESS>and<EMAIL_ADDRESS>received April 5, 2022

###### Abstract

To ensure the safety and efficiency of its maneuvers, an Autonomous Vehicle
(AV) should anticipate the future intentions of surrounding vehicles using its
sensor information. If an AV can predict its surrounding vehicles’ future
trajectories, it can make safe and efficient manoeuvre decisions. In this
paper, we present such a Deep Spatio-Temporal Context-Aware decision Network
(DST-CAN) model for predictive manoeuvre planning of AVs. A memory neuron
network is used to predict future trajectories of its surrounding vehicles.
The driving environment’s spatio-temporal information (past, present, and
predicted future trajectories) are embedded into a context-aware grid. The
proposed DST-CAN model employs these context-aware grids as inputs to a
convolutional neural network to understand the spatial relationships between
the vehicles and determine a safe and efficient manoeuvre decision. The DST-
CAN model also uses information of human driving behavior on a highway.
Performance evaluation of DST-CAN has been carried out using two publicly
available NGSIM US-101 and I-80 datasets. Also, rule-based ground truth
decisions have been compared with those generated by DST-CAN. The results
clearly show that DST-CAN can make much better decisions with 3-sec of
predicted trajectories of neighboring vehicles compared to currently existing
methods that do not use this prediction.

###### Index Terms:

Predictive planning, Autonomous Vehicle (AV), spatio-temporal context, safe
decision.

##  I Introduction

Autonomous Vehicle (AV) developments have resulted in many technological
advancements in recent years. Further, advancements in deep learning have led
to some significant improvements in perception using cameras [1, 2], and
LiDARs [3], [4]. Deep learning-based perception schemes effectively detect
objects and their poses in different environmental conditions. Driving on
public roads requires safe decision planning. Many accidents occur during the
lane changes and lane merging maneuvers due to bad driving decisions [5, 6].
In this context, for a safe and efficient decision-making, the AV should
understand the intentions of surrounding vehicles from their past and present
contextual information provided by its perception system. These intentions can
be estimated by predicting the future trajectories of its surrounding
vehicles. Based on these inferences, safety margins can be defined for
planning future decisions. However, there are always some uncertainties in
accurately predicting these trajectories, resulting in wrong decision-making.
Hence, Predictive Manoeuvre Planning (PMP), which can incorporate contextual
information and uncertainties in the neighbouring vehicle’s intention
predictions is crucial for safe decision planning for an AV.

Recent works have addressed some aspects of inferring the future intentions of
surrounding vehicles by predicting their trajectories. Since recurrent neural
networks are especially good for temporal information processing and it has
been widely used for predicting vehicle trajectories [7], [8], [9], [10]. In
[7], each of the surrounding vehicles’ states and ego vehicle’s states were
passed through an Long Short Term Memory (LSTM) network for encoding. Then
another LSTM network was utilized to generate the future predicted
trajectories. A beam search technique was employed to predict multi-modal
trajectories on the occupancy grid map. In [8], one vehicle in the same lane
as the target vehicle and two nearest vehicles in two adjacent lanes were
chosen. An LSTM based encoder-decoder architecture was used to predict the
target vehicle’s future trajectory. The encoder network captured interactions
among vehicles, and the decoder network generated future trajectories. In
[10], the trajectory prediction was formulated as a sequence generation
problem, and a recurrent network called the Memory Neuron Network (MNN) [11]
was utilized to find a functional relationship between past trajectory samples
and the future trajectory samples. On the other hand, some works used inverse
reinforcement learning for trajectory prediction [12], [13]. However, these
works mainly focused on trajectory prediction and did not address the
decision-making process.

Other works that focused on decision-making capability using reinforcement
learning include [14], [15]. In [14], a Deep-Q-Network (DQN) was used with
continuous action and states space for automated lane change maneuvers in a
simulated environment. In [15], an asynchronous DQN was used for high-level
policy making for maneuver decisions. However, the DQN was trained with
handcrafted traffic scenarios. Also, these works did not try to infer the
intentions of other vehicles during their decision-making. Some works have
used Convolutional Neural Networks (CNN) to preserve the spatial construction
of the context features, as they can capture spatial relationships in an
efficient manner. In [16], a CNN-based lane change intention prediction
framework was used in a model predictive controller. It used a sequence of
simplified bird’s eye view images to indicate other vehicles present in this
context and then determined their lane change intention probabilities.
However, it did not incorporate the temporal relationships between vehicles to
infer their manoeuvre intentions. Recently in [17], the spatial and temporal
relationships among the surrounding vehicles is combined for decision-making.
Here, LSTM network is used to encode the past trajectory information of the
ego vehicle and its surrounding vehicles. LSTM states corresponding to the
spatial positions of the surrounding vehicles w.r.t (the distance from) the
ego vehicle are used to construct a social tensor. Since this social tensor
has spatio-temporal relationships, a CNN was used to pool out (from [18]) the
spatial interaction features of the vehicles to identify the lateral and
longitudinal manoeuvre decisions. In [19] a CNN was used for finding the
correlations between the spatial and temporal information of surrounding
vehicles. Aforementioned works uses only the past information for manoeuvre
prediction and did not consider the future intentions of neighbour vehicles.
Hence, there is a need to develop a PMP approach that considers both past and
future spatio-temporal context information to predict the manoeuvre decision
that is safe, fuel-efficient, and follows traffic rules.

In this paper, a novel approach called the Deep Spatio-Temporal Context-Aware
decision Network (DST-CAN) is proposed to overcome the above-mentioned
shortcomings in the predictive planning approach. The proposed approach has
three major blocks: Perception block, spatio-temporal context generator block,
and decision engine block. It is assumed that a state-of-the-art perception
block will provide positions of surrounding vehicles for previous time steps.
Hence, the publicly available real-world highway traffic data sets are used to
construct the spatio-temporal context information by context generator block.
From the dataset, context information is extracted as an occupancy grid map
around the ego vehicle. To understand the surrounding vehicles’ intentions,
the future trajectories of those vehicles are predicted using a memory neural
network [11]. MNN is used to predict look-ahead trajectories of vehicles in
the neighbourhood. A probabilistic occupancy map (POM) is then created from
these predicted trajectories. The past occupancy grid maps and the predicted
POMs are then combined to form a spatio-temporal context-aware grid. This
spatio-temporal context-aware grid map has both spatial and temporal
information. The decision engine then processes the spatio-temporal grid map
to decide on the manoeuvre to be taken by the ego vehicle at every instant. It
has been found that there are many scenarios where the human drivers’
decisions are not safe and efficient. Those actions also do not satisfy common
traffic rules for driving. Hence, ground truth decisions based on traffic
rules are generated for comparative analysis. The publicly available NGSIM
traffic dataset has been used for the performance evaluation of DST-CAN. Two
models of the DST-CAN are trained based on human driving decisions and ground
truth decisions. From the study on the different prediction horizons, it has
been found that a $3$ seconds prediction horizon gives better results than $1$
or $5$ seconds prediction horizon. The performance DST-CAN is compared with
the state-of-the-art interaction aware model (CS-LSTM) [17] for both human
decision data and ground truth decision data. The results clearly show that
DST-CAN has an improvement of $21.15\%$ for lateral decisions and $12.12\%$
for longitudinal decisions compared to CS-LSTM. The results clearly show that
using a future trajectory-based context grid helps DST-CAN determine an
efficient predictive manoeuvre prediction models for AV.

The paper is organized as follows: Section II describes the procedure for
creating a spatio-temporal context-aware grid map and processing it for safe
decision-making. Section III describes the simulation setup and results using
the public traffic datasets. Finally, Section IV offers the conclusions from
this study.

##  II Deep Spatio-Temporal Context-Aware decision Network (DST-CAN) for
Predictive Manoeuvre Planning

This section describes the problem formulation for Predictive Maneuver
Planning (PMP) for an autonomous vehicle. The proposed decision-making
approach for an autonomous vehicle operating in a highway or urban environment
is presented in this section. The approach uses a deep spatio-temporal
convolutional neural network for solving the PMP problem. Before getting into
the details of the above approach, the benchmark datasets used in this study
are described first.

###  II-A PMP Problem Formulation

Recent advancements in both sensing and computing technologies have increased
the feasibility of an Autonomous Vehicle (AV) operation in an urban
environment. One of the crucial challenges for operating a vehicle in a
natural environment is to develop its capability to understand the intentions
of its surrounding vehicles and then plan its path based on its own goal. The
planned path should be such that the maneuvers are safe, energy-efficient, and
also should follow the normal traffic rules. To understand the importance of
the prediction of the neighbouring vehicle’s intentions, a typical traffic
scenario with an AV in operation with other non-ego vehicles (at any given
time instant $\it{t}$) is shown in Figure 1. The dotted circular lines
indicate the sensing region of the AV.

Figure 1: A typical traffic situation with an autonomous vehicle in operation

The figure shows that the AV cannot sense the static obstacles present in the
top row, which influences the traffic on the left lane. The AV can detect the
current and past poses of static/dynamic objects in the scene using its sensor
information. Based on these observations, the AV needs to anticipate the
neighbour’s intentions (contextual information) and plan its manoeuvre
accordingly. This planned manoeuvre should be safe (without colliding with
other vehicles), follow the standard traffic rules, and also be energy
efficient. The problem of manoeuvre planning (i.e., action taken by the AV
without colliding with other objects) by understanding the intentions of its
neighbours is referred to as the predictive manoeuvre planning problem. It is
assumed that the AV is equipped with an algorithm that can detect the relative
poses of both the static and dynamic objects. Thus, the AV has complete
histories of paths traversed by the dynamic objects present at any given time
and is also aware of the local map. The PMP problem is formulated based on the
histories of the paths traversed by the dynamic objects present in the scene
and also the local map.

Figure 2: A context region with an AV at the centre

With the above aim, a rectangular window of length $L$ and width $W$ with the
AV at the centre is considered, as shown in Figure 2. The past trajectories of
the vehicles are represented with dark lines, and the predicted look-ahead
trajectories of the dynamic objects (non-ego motion vehicles) are represented
using dashed lines. The figure shows the path traversed by AV for the past $n$
frames (${P_{a}}$) and the estimated paths traversed by the other dynamic
objects for the past $n$ frames seen at time $t$ (${P_{o}}$). In PMP, it is
important to predict the future look-ahead trajectories of the other dynamic
objects ($\hat{{P}_{o}}$) to understand the occupancy map into which the AV
will move in the near future.

$\displaystyle\\{\textbf{P}_{a}\\}=\\{(x^{-n}_{a},y^{-n}_{a}),(x^{-n+1}_{a},y^{-n+1}_{a}),...,(x^{0}_{a},y^{0}_{a})\\}$
(1)
$\displaystyle\\{\textbf{P}_{o}\\}=\\{(x^{-n}_{o_{i}},y^{-n}_{o_{i}}),(x^{-n+1}_{o_{i}},y^{-n+1}_{o_{i}}),...,(x^{0}_{o_{i}},y^{0}_{o_{i}})\\}$
$\displaystyle\\{\hat{\textbf{P}}_{o}\\}=\\{(\hat{x}^{1}_{o_{i}},\hat{y}^{1}_{o_{i}}),(\hat{x}^{2}_{o_{i}},\hat{y}^{2}_{o_{i}}),...,(\hat{x}^{m}_{o_{i}},\hat{y}^{m}_{o_{i}})\\}$
$\displaystyle\text{where }i=1,2,...,D;\text{D is number of dynamic objects}$

The ego vehicle’s action space ($A$) analysis is essentially divided into
analysis of longitudinal and lateral maneuver actions. The two decisions for
the longitudinal maneuver can be either ’cruise (C)’ or ’brake (B).’ The
lateral maneuver action has three possibilities, and it can be either ’keep
the same lane’ (S) or ’change to the left lane’ (L), or ’change to the right
lane’ (R). The required actions for the AV should also be such that the motion
planned by the AV is safe, energy-efficient, and follow the traffic rules.

For solving the problem of PMP, at first, the intentions of surrounding
vehicles are inferred from the predicted trajectories of surrounding vehicles.
Then it is converted into a classification problem of deciding which decisions
to take, and the classification problem is then solved. For every step, both
the lateral and longitudinal manoeuvre class’s probabilities have to be
predicted as Lat pred = [P(S), P(L), P(R)] and Lon pred = [P(C), P(B)]. Then,
the actions with the highest probabilities are chosen from these
classification probabilities. In a nutshell, one can say that the PMP problem
essentially consists of a trajectory prediction problem followed by a decision
classification problem. For carrying out this, as a first step, one needs to
encode the context information up to the current instant for the AV and then
develop a model to predict the look-ahead occupancy map of the neighbouring
dynamic obstacles. Next, one needs to approximate the functional relationships
between the context information of the surrounding vehicles and the possible
action class for the AV meeting the desired objectives. Since imitation
learning [20] tries to mimic the behaviour of humans in different conditions,
for PMP, imitation learning from real-world traffic data has been utilized to
derive this functional relationship between the context information and the
action class. Before getting into the details of the method, a brief
description of the traffic datasets used in this study is given next.

###  II-B Traffic Dataset Description

In this work, the traffic data sets NGSIM I-80 and US-101 [21], where the
vehicle positions are given for different time frames, have been used. The two
datasets consist of 45 minutes of traffic data at two different locations.
Each dataset has three segments of a low, medium, and highly congested traffic
data. Each segment has $15$ minutes of traffic data. The datasets are sampled
at $10Hz$ implying ten frames per second. The raw dataset gives the positions
of all the vehicles and vehicle ids at each time frame. As a first step, to
make this raw dataset suitable for prediction purposes, the dataset has to be
pre-processed. Each vehicle in the frame is represented as an ego vehicle. If
a vehicle’s lane id changed from the previous $4$ seconds to the next $4$
seconds, it was assumed that it changed its lane. From this change in the
lane, the lateral manoeuvre done is recorded. Suppose the lane number
decreases within this $\pm$ $4$ seconds, then the vehicle changes to the left
lane. Otherwise, if the lane number increases in this interval, the vehicle
changes to the right lane. Also, if a vehicle’s average speed over the next
$5$ seconds is lower than $0.8$ times current speed, it is assumed that the
vehicle is braking.

It has also been found that there are some scenarios where the actual
decisions taken by a human driver do not satisfy common traffic rules. Hence,
if only human decisions are used for learning the possible driving decisions,
then the AV can also make unsafe decisions. Hence, there is a need for
generating ground truth decisions for each scenario to provide for safe
actions. For this reason, ground truth decisions have also been developed in
this study for both datasets. More details about ground truth can be found in
supplementary document.

###  II-C Deep Spatio-Temporal Context-Aware decision Network

This paper presents a Deep Spatio-Temporal Context-Aware decision Network
referred to as DST-CAN to predict the motion planning actions required for an
autonomous vehicle. The schematic diagram of the proposed DST-CAN using
context-aware decision making using the PMP model is given in Figure.3. Fig.3
shows that the overall scheme has three major blocks: viz., a perception
block, a context generator block, and a CNN decision engine block. The
perception engine block of AV detects and tracks other objects. The context
generator block generates the current context and future occupancy maps of the
neighbouring vehicles. As the final step, based on the above information, the
CNN decision engine predicts the manoeuvre decisions which follows the traffic
rule with minimum energy. The details of the workings of the different blocks
are described next.

AVPerceptionEngine$\begin{pmatrix}\boldsymbol{P_{a}}\\\
\boldsymbol{P_{o}}\end{pmatrix}$MNNLook-
aheadPredictor$\begin{pmatrix}\boldsymbol{P_{a}}\\\
\boldsymbol{P_{o}}\end{pmatrix}$past$\left(\boldsymbol{\hat{P_{o}}}\right)$predictedfutureContextbased
POMEncoderSpatio - Temporal context based POM GeneratorContext &
POMframeConvolutionalNeuralNetworkCNN Decision Engine$\begin{Bmatrix}S\\\ L\\\
R\end{Bmatrix}$$\begin{Bmatrix}C\\\ B\end{Bmatrix}$ Figure 3: A Schematic
diagram for PMP using Deep Spatio Temporal Context-Aware decision Network
(DST-CAN)

The perception engine in AV identifies the dynamic objects at the current
instant and tracks the paths traversed by these objects, as shown in the first
block of Figure.3. After the perception engine block, DST-CAN essentially
consists of two modules: a context-aware POM generation module using a look
ahead predictor and a CNN-based action engine to relate the context frames and
action classes. The context-aware POM generator uses the paths traversed by
the AV and dynamic objects up to time $t$ and generates their look-ahead
trajectories (predictions) using a memory neuron network [11], [10]. A context
encoder then encodes these predicted trajectories into a probability occupancy
density map. The resultant context and occupancy map frames are then used as
inputs to a convolutional neural network to obtain the final motion planning
decision (classification).

####  II-C1 Spatio-temporal Context and probabilistic occupancy map (POM)
Generator

The objective of the context and POM encoder is to generate a context and
occupancy map in which the autonomous vehicle was present in the past. It also
predicts the potential look-ahead trajectories of the other dynamic objects in
the scene to understand the future intentions of these objects. The AV
perception engine provides the path traversed by the AV (Pa) for n-frames and
the paths traversed by the dynamic objects up to the current time frame (Po).
Data-driven look-ahead prediction approaches have been widely used for
prediction, such as recurrent neural networks [10], long-short term memory
networks [22], [23], and fuzzy neural networks [24]. Recently, it was shown
that the MNN based approach [10] is effective for look-ahead trajectory
prediction than other state-of-the-art approaches.

Hence, in this paper, an MNN has been used and its training has been done with
backpropagation in time similar to [10] for look-ahead trajectory predictions.
A series-parallel model is used for training, and a parallel-parallel model is
used during testing to predict the look-ahead trajectories [25]. From the
paths traversed by the dynamic objects ($\\{\bf{P}_{o}\\}$), changes in
displacements in both $x$ and $y$ directions are computed. The inputs to the
MNNs are $\Delta x(t-1)$ and $\Delta y(t-1)$, and the outputs of the network
are $\Delta x(t)$ and $\Delta y(t)$. The MNN architecture used in the paper is
shown in Figure. 4. Each layer has a network neuron (blank circle), and each
network neuron has one memory neuron (dark circle). The net output by
$\phi_{j}^{h}(t)$ of the $j^{th}$ network neuron in the hidden layer $h$ is:

$\phi^{h}_{j}(t)=g^{h}\left(\sum_{k=1}^{2}w_{kj}^{i}\psi_{k}^{i}(t)+\sum_{k=1}^{2}f_{kj}^{i}v_{k}^{i}(t)\right)$
(2)

where,

  * •

$\psi_{k}^{i}(t)$ is the output of the $k^{th}$ network neuron in the input
layer $i$. In our case, $\psi_{1}^{i}(t)=\Delta{\hat{x}_{t-1}}$ and
$\psi_{2}^{i}(t)=\Delta{\hat{y}_{t-1}}$. $(x,y)$ is position of a vehicle.

  * •

$g^{h}(.)=tanh(.)$ is the activation function of the network neurons present
in the hidden layer.

  * •

H is number of network neurons in hidden layer. In this work H=6.

The output of the memory neuron corresponding to the $j^{th}$ network neuron
in the layer $l$ is given by:

$v_{j}^{j}(t)=\alpha_{j}^{l}\psi^{l}_{j}(t-1)+(1-\alpha^{l}_{j})v^{l}_{j}(t-1)$
(3)

where the $\alpha$ is weight between each network neuron and memory neuron.
The net output $n^{L}_{j}(t)$ of the $j^{th}$ network neuron in the last layer
$L$ is calculated as:

$\phi^{L}_{j}(t)=g^{L}\left(\sum_{k=1}^{H}w^{h}_{kj}\psi^{h}_{kj}(t)+\sum_{k=1}^{H}f^{h}_{kj}v^{h}_{k}(t)+\beta^{L}_{j}v^{L}_{j}(t)\right),$
(4)

where $1\leq j\leq 2$. The inherent feedback connections in the memory neurons
help in accurately capturing the dynamics. The network neurons in the hidden
layer employ a bipolar sigmoidal activation function, and the output layer
employs a linear activation function.

$z^{\shortminus 1}$$z^{\shortminus
1}$$\alpha_{1}^{i}$$1\shortminus\alpha_{1}^{i}$Network neuronMemory
neuron$\Delta\hat{x}_{t\shortminus 1}$$\Delta\hat{y}_{t\shortminus
1}$$\vdots$$\Delta\hat{x}_{t}$$\Delta\hat{y}_{t}$$\psi_{k}^{i}$$v_{k}^{i}$$w_{kj}^{i}$$f_{kj}^{i}$$\beta_{2}^{L}$
Figure 4: MNN architecture for look-ahead trajectory prediction

The root mean square error function for predicting future positions during
training the MNN is given as:

$RMSE=\frac{\sum_{n=1}^{N}\sqrt{\frac{\sum_{t=1}^{T_{H}}{\lVert\mathbf{x}_{t}^{(n)}-\hat{\mathbf{x}}_{t}^{(n)}\rVert}^{2}}{T_{H}}}}{N}$
(5)

where $\mathbf{x}_{t}=\left[x(t),y(t)\right]$ and
$\hat{\mathbf{x}}_{t}=\left[\hat{x}(t),\hat{y}(t)\right]$ are ground truth and
predicted positions respectively. N is the number of vehicles at time t. The
network is trained using backpropagation through time algorithm as in [11].
One can find more details on the parallel-parallel implementation used to
predict the future $m$-frames, and a comparison with other state-of-the-art
method in [10].

Next, the workings of the spatio-temporal context and POM encoder are
described. This encoder converts the trajectories of AV and the other dynamic
objects in the scene into a context-aware grid. For this purpose, the encoder
divides the road segment of length $L$ with AV at the centre into three tracks
with equal widths. Further, it divides the road segment into $p$ equal parts.
A spatial context map is developed for every past time frame. Since the
perception engine gives the distances to the other vehicles very accurately
(using sensors like LiDAR), it can be assumed that uncertainty in distance
estimation is much smaller compared to each grid length. The grid value on the
road segment is set to one if a vehicle occupies the grid. Otherwise, it is
set to zero.

(a) context grid map

(b) POM

(c) POM

Figure 5: Context and Probabilistic Occupancy Map (POM) embedding for
different time steps

As a typical example, the spatial context map for the current instant is shown
in Figure.5(a). The ego vehicle is shown in grey colour. Dynamic objects are
shown in red colours. For red colour occupancy, the darker the colour, the
higher is the probability of occupancy. There is uncertainty in the predicted
positions of surrounding vehicles. The Root Mean Square Error (RMSE) for
predictions is given in [10]. To develop a function that can capture the
uncertainty well, RMSE for every second has been normalized based on the max
error observed. Using this normalized error, a discrete graph showing how
certainty reduces with time is shown in Figure.6. A decaying function of time
has been fitted to closely relate the uncertainty of prediction to create the
POM. This decaying function shows how the certainty of prediction decreases
with future time. This decaying function of time is calculated as given below:

$\textbf{P}(t)=0.47+\sqrt{0.236-0.04\left(\frac{t}{10.0}\right)}$ (6)

where $0\leq t\leq 50$ is the time index.

$0$$1$$2$$3$$4$$5$$6$$0$$0.2$$0.4$$0.6$$0.8$$1$prediction time horizon
(sec)$Certainty,\textbf{C}(t)$ Figure 6: Predicted position certainty over
different prediction time horizon occupancy mapat
$t=\tau$$t$$\tau-3$$\tau$$\tau+1$$\tau+T$Generation of $13\times 3\times 60$
context-aware grid$11\times 1\times 60$CNN decision engine$3\times 3$
conv$9\times 12$$3\times 1$ conv$5\times 12$$2\times
1$maxpoolfullyconnectedLateraldecisionnetwork$\left[S,L,R\right]$Longitudinaldecisionnetwork$\left[C,B\right]$
Figure 7: Context-aware grid formation and CNN decision engine for AV decision
making

1:Inputs: The set of past track history
$\\{\mathbf{X}_{h}\\}_{i=1}^{\mathcal{N}}$ where $\mathcal{N}$ is the number
of surrounding vehicles detected by the sensors, pre-trained weights for
Memory Neuron Network (MNN), current time step $t$, prediction horizon $T$

2:Output: predicted maneuver $\mathbf{M}$, encoded as a one-hot vector

3:Initialize: $\mathcal{G}$ is a $13\times 3\times 60$ array of zeros

4:Step 1: Obtain the context-aware grid $\mathcal{G}$:

5:for $i\leftarrow 1$ to $\mathcal{N}$ do

6: for $\tau\leftarrow t-30$ to $t+T$ do

7: if $\tau\leq$ t then

8: $\mathcal{G}(i,j,\tau)=1.0$, where $(i,j)$ is the grid
a a aa indices obtained using the relative position $\mathbf{X}_{h}$
a a a . of vehicle $i$ w.r.t ego vehicle’s position at time $\tau$.

9: else

10: $\mathcal{G}(i,j,\tau)=\mathbf{P}(\tau)$, where $(i,j)$ is the grid
a a a a indices obtained using the predicted position of
a a a . vehicle $i$ (from MNN), and $\mathbf{P}(\tau)$ is calculated
a a a . using Eq. 6

11: end if

12: end for

13:end for

14:Step 2: Calculate the predicted maneuver $\mathbf{M}$ by passing the
context aware grid $\mathcal{G}$ to the convolutional social pooling algorithm

Algorithm 1 DST-CAN Context-aware grids generation pseudo code

Based on their past trajectories, POMs are created for the future predicted
positions of the surrounding vehicles present at the current instant in the
occupancy grid map. The probability that a vehicle will be at the predicted
location decreases exponentially with time. The difference in the position of
a non-ego vehicle from frame ($t-1$) to frame $t$ is given as an input to the
trained MNN. The MNN then outputs the next predicted change in the position.
This predicted change in the position is added with the past position to get
the next predicted position, as shown in equation 7.

$\hat{x}_{t+1}=x_{t}+\Delta\hat{x}_{t}\text{ and
}\hat{y}_{t+1}=y_{t}+\Delta\hat{y}_{t}$ (7)

The vehicle will be at that predicted position in the grid with a probability
$P(t)$ as given in equation 6 and as shown in 5(b). Since there is always
uncertainty in a vehicle’s future predicted position, it is assumed that the
vehicle could be at any one of the eight surrounding grids encircled by the
MNN predicted grid. It is anticipated that the vehicle could be at any one of
the surrounding grids with a probability $(1-P(t))/8$ as shown in Fig.5(b) for
the first future frame. Once the first predicted location is found for one
future instant, the model takes the change in location from the last time
frame and passes that change to the MNN to get the new predicted location
change for the second future frame. This new location calculated as a running
sum will then have the probability of occupancy calculated according to
equation 6 as given by equation 8 (shown in Fig.5(c)).

$\displaystyle\hat{x}_{t+(i+1)}=\hat{x}_{t+i}+\Delta\hat{x}_{t+i}\text{ and
}\hat{y}_{t+(i+1)}=\hat{y}_{t+i}+\Delta\hat{y}_{t+i}$ (8)

where $i=1,2,...,(m-1)$. A context-aware grid with POM is formed in this way,
as shown left part of in Fig.7 for the rest of the prediction horizon. These
spatio-temporal context-aware grid frames and POMs are given as input to a
convolutional neural network to generate the manoeuvre decisions.

####  II-C2 Convolutional Neural Network Action Engine

The spatio-temporal context and occupancy map information are embedded into
context-aware grid frames and are used to determine the motion planning
decisions. Here a convolutional neural network is used to find out the spatial
relationships of these vehicles in every frame. The architecture used is shown
in the right part of Figure.7. The convolutional neural network employs two
three-dimensional convolutional layers and one max-pooling layer. The leaky-
ReLU activation function is used for the convolutional layers. The extracted
features are flattened and passed through two fully connected networks. The
softmax function output gives the probability distribution of lateral and
longitudinal motion planning decisions. The mode with the highest probability
is chosen as the final decision taken by the ego vehicle.

1:Input: Context Aware Spatio Temporal Grids for every instant as tensors with
maneuver decisions (either Ground Truth (GT) decisions or human decisions) at
that moment

2:Initialize: Convolutional Neural Network (CNN) with final output dimension
of maneuver action classes and a blank list L for storing tensor indexes

3:Step 1: For first epoch, pass all tensors through CNN

4:for $i\leftarrow 1$ to Total number of social tensors do

5: if predicted decision = actual decision (either GTa decision or human
decision) then

6: Save the tensor index $i$ to the list L

7: end if

8:end for

9:Step 2: Remove the tensors in list L from training dataset and get new
training dataset with remaining data

10:Step 3: Train the CNN with new training dataset

11:for $i\leftarrow 1$ to E number of epochs do

12: for $i\leftarrow 1$ to batch size B do

13: predict the maneuver action classes as probabili a . a -ties and calculate
the Binary Cross Entropy (BCE) a a loss $\mathcal{L}$ as given in equation 9

14: end for

15:end for

Algorithm 2 DST-CAN training pseudo code

The training is done with respect to the binary cross-entropy loss given in
Eqn.9. The training loss convergence graph for a 3-sec prediction horizon with
and without regularization (as described in algorithm 2) is given in Fig.8.
The data samples that give good results for the first epoch have been removed
from the training dataset for further training for regularizing the training
process. It has been found that those samples overfit the DST-CAN model, as
shown by the loss curve in Fig.8.

$0$$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$$11$$12$$0.16$$0.17$$0.18$$0.19$$0.2$$0.21$$0.22$$0.23$$0.24$$0.25$EpochsBinary
cross entropy losswith regularizationwithout regularization Figure 8: Network
training error history

$\mathcal{L}=-\frac{1}{N}\sum_{i=1}^{N}y_{i}log(p(y_{i}))+(1-y_{i})log(1-p(y_{i}))$
(9)

where $y_{i}$ is the ground truth and $p(y_{i})$ is the predicted output. $N$
is the total number of samples. The model is implemented in PyTorch[26] and is
trained using an RMSProp optimizer.

##  III Performance Evaluation of DST-CAN

In this section, the performance evaluation results of DST-CAN using benchmark
traffic data sets NGSIM I-80, and US-101 datasets [21] are presented. Before
presenting the performance results, details of the simulation setup for these
studies are described, followed by the performance analysis of DST-CAN for
different prediction horizons. Finally, the performance of DST-CAN using a
3-sec prediction horizon is compared with the earlier state-of-the-art
approach of Convolutional Social Long-Short Term Memory (CS-LSTM) [17] in
terms of correct decision making.

###  III-A Simulation setup

Publicly available NGSIM US-101 and I-80 datasets [21] have been used for this
study. The low and highly congested traffic data from both the datasets have
been used for training and the medium congested data for testing. A summary of
the details of the datasets is given in supplementary material. For
constructing the spatio-temporal grids, the sensor range considered is 90
feet. Each processed data frame consists of vehicles in the range of $\pm$90
feet and two adjacent lanes of the ego vehicle. It is assumed that each lane
is 15 feet wide. For prediction, 3 sec of past data is used. Three prediction
time horizons consisting of 1 sec, 3 sec, and 5-sec were used to find the one
that gave a better result. Most of the data in the datasets correspond to
decision-making during cruising in the same lane. Since keeping the same lane
manoeuvre is a major portion of the dataset, only a small portion ($20\%$) of
those maneuvers are taken for imitative driving model training. By doing this
processing, the model can predict lane-changing and longitudinal maneuvers
well.

This work here attempts to predict safe and fuel-efficient maneuvers in real
scenarios. In this context, if one looks at some of the decisions taken by a
human driver, it can be seen that it is not optimal for some situations. Hence
ground truth (GT) decisions for every condition have been computed based on
simple rule base. For example, to estimate the relative velocity of the
vehicle in the front, the vehicle’s position should be tracked for some time.
If the front vehicle slows down and there is not enough space for safe
driving, it is more reasonable to look for a lane-changing manoeuvre. It
should also be checked that there is enough free space to change safely for
changing lanes. If at least two grids (i.e., $\sqrt{5}$ in L2 distance from
the ego vehicle) are unoccupied on the adjacent lanes on both the forward and
rear sides, it is safe to switch the lane. If there is enough space in front
of the ego vehicle and the relative velocity of the front vehicle is the same
or more than the ego vehicle, then it is safe to cruise on the same lane. If
there is not enough space for cruising on the same lane, the ego vehicle
should check for the availability of enough space for changing the lanes. If
none of the adjacent lanes are free to move safely, then the ego vehicle
should brake on the same lane. More details on ground truth decision
generation and rules can be found in supplementary document.

Two DST-CAN models, DST-CAN-Imitation and DST-CAN-GT, have been trained to
understand the importance of following standard traffic rules. DST-CAN-
Imitation is trained with the actual human imitation data and DST-CAN-GT with
the ground truth data.

###  III-B Ablation Study

The trained DST-CAN model has been evaluated using the moderate congestion
traffic data. The model finally outputs the probability distribution of the
control decisions. The output mode with the highest probability has been taken
as the suggested action. That suggested action is given a value of $1.0$, and
all other actions are given $0.0$ in the output vector for comparing with the
action taken by a human driver. The suggested actions are compared with GT
decisions as GT decisions follow the standard traffic rules. There are cases
where the decisions taken by humans and GT decisions are conflicting. Since GT
decisions are optimal for both safety and fuel efficiency, a comparison has
been made with the GT decisions. The percentage of cases when predicted
decisions by DST-CAN and GT decisions are the same are compared. Analytical
metrics such as root mean squared error, negative log-likelihood, etc., show
differences in the results compared to actual human decision data. However, in
many cases, human decisions are not good enough, and many of the proposed
network predicted decisions could be much better than the actual decisions
that human drivers took at that condition. Thus, comparing with the analytical
results makes a poor reflection of the model performance. Hence, the
percentage of cases where the predicted decisions are the same as ground truth
decisions are considered for comparison.

####  III-B1 Study on prediction horizon

Prediction horizon | Lateral Consensus cases | Lateral Conflict cases | Longitudinal Consensus cases | Longitudinal Conflict cases
---|---|---|---|---
accuracy (%) | accuracy (%) | accuracy (%) | accuracy (%)
1 sec | $99.62$ | $13.79$ | $95.14$ | $32.31$
3 sec | 99.67 | 13.85 | 95.35 | 32.81
5 sec | $99.64$ | $13.61$ | $95.24$ | $32.35$
TABLE I: Prediction accuracies for different time horizons

To find the prediction horizon that gives the best result, $1$, $3$, and
$5$-seconds horizons were chosen, and detailed studies were undertaken. The
comparison results for different horizons are shown in Table I. Consensus
cases are those where actual human decisions are the same as the ground truth
decisions. Conflict cases are those where the human decisions and ground truth
decisions are different. In Table I, the DST-CAN model is trained based only
on human decisions. The accuracies for both the consensus and conflict cases
are shown in Table I. The results show that the $3$ seconds prediction horizon
is better than $1$ second and $5$ seconds prediction horizons. Hence, a $3$
seconds prediction horizon is selected for comparison in this study.

####  III-B2 Analysis of conflicting cases

(a) $1^{st}$ scene

(b) $2^{nd}$ scene

(c) $3^{rd}$ scene

(d) $4^{th}$ scene

Figure 9: Heat maps of different prediction scenarios

In this section, the cases where the proposed model predicted a different
manoeuvre than the actual decision taken by a human driver are analysed. In
these cases, the proposed model predicted better control decisions (maneuvers)
many times from correctly understanding the behaviours of the surrounding
vehicles from their past and the predicted future trajectories. Some of these
conditions for mismatching maneuvers are shown in Fig.9. The ego vehicle is at
the centre of the heat map, as shown by the yellow vehicle. The heat maps of
Fig.9 show the probabilistic occupancy maps of the surrounding vehicles for
the immediate next frame. The red dots show the actual trajectories taken by
the surrounding cars for ten future frames projected on the heat map to
understand the best decision to be taken by the ego vehicle at that instant.

The maneuvers are encoded as one-hot vectors. In Fig.9(a), one probabilistic
occupancy grid map is shown. Here DST-CAN predicts an decision that requires
it to cruise on the same lane. However, the actual human action taken is
braking and changing to the left lane. It may be noted that there is not
enough space on the rear side to change to the left lane safely. There is a
high chance of collision with another vehicle in the left lane, as can be seen
from the red dots on Fig.9(a). Hence, in this case, the human decision is
unsafe and inefficient. In this condition, DST-CAN predicts a much better safe
and efficient decision. In fig.9(b), another prediction case is shown. Here
the model predicts an action to cruise on the same lane. The human driver
decided to slow down and move to the left lane. In this case, going to the
left lane is safe but not fuel-efficient as the lane in front is free to move.
Here DST-CAN’s predicted decision is both safe and fuel-efficient since there
is no need for braking. For the scenario shown in Fig.9(c), there will be some
space in the left lane to move after some time. Nevertheless, moving to the
left lane is not very safe as vehicles are in the rear and forward end of the
left lane. Here following the same lane is much safer and more fuel-efficient.
There are a few cases where predicted maneuvers based on the DST-CAN are not
that good. For example, in Fig.9(d), the human decision is to change to the
left lane and cruise. DST-CAN decided to slow down on the same lane. However,
one can see from the heat map that cruising on the same lane is the best fuel-
efficient manoeuvre.

The reason behind the poor prediction in a few cases is due to that the DST-
CAN model proposes a local predictive path planning for an autonomous vehicle.
Here global path planning is not of primary concern because the destinations
of surrounding vehicles are not known. Sometimes, some surrounding vehicles
may maneuver like changing lanes to leave the highway and go to a particular
place. The model cannot figure out these conditions very well as it gets
trained on such inaccurate data and tries to predict a different decision from
the ground truth. These inaccurate data samples cannot be easily removed from
the training dataset. For e.g., the vehicle may change lanes to exit a
specific highway section without the presence of any surrounding vehicles. The
presence of such training samples may affect the generalization ability of the
network.

###  III-C Performance Comparison

Training data used | Lateral Consensus cases | Lateral Conflict cases | Longitudinal Consensus cases | Longitudinal Conflict cases
---|---|---|---|---
DST-CAN(our) | CS-LSTM | DST-CAN(our) | CS-LSTM | DST-CAN(our) | CS-LSTM | DST-CAN(our) | CS-LSTM
Human imitation data | 99.67 % |  $99.45$% | 13.85 % |  $8.79$% | 95.35 % |  $94.72$% | 32.81 % |  $26.44$%
Ground truth data | 99.69 % |  $98.11$% | 99.82 % |  $78.67$% | 99.92 % |  $97.95$% | 99.95 % |  $87.83$%
TABLE II: Percentage accuracy comparison for Deep Spatio Temporal Context-
Aware decision Network vs. CS-LSTM

In this section, we compare the performance of DST-CAN trained using human
imitation decisions and ground truth decision with state-of-the-art approach
in the literature [17]. The past $3$ second of trajectory history in the
sensing region is used by both DST-CAN and CS-LSTM for predicting the
manoeuvre decision. For the implementation of [17] (here), only the model up
to lateral and longitudinal manoeuvre prediction has been taken. For training,
the model in [17], only low and highly congested traffic data is used. The
medium congested data is applied for evaluation. For a fair comparison, two
models of CS-LSTM [17] are trained. One CS-LSTM model is trained with human
decisions, and the other model with ground truth decisions.

| CS-LSTM prediction |  |  | DST-CAN prediction
---|---|---|---|---
GT decision | Same lane |  | Take
---
left
| Take
---
right
Cruise | Brake |  | GT decision | Same lane |  | Take
---
left
| Take
---
right
Cruise | Brake
Same lane | 86679 | 616 | 3145 | 0 | 0 |  | Same lane | 89604 | 582 | 254 | 0 | 0
| Take
---
left
46952 | 104501 | 3733 | 0 | 0 |  |  | Take
---
left
1 | 155185 | 0 | 0 | 0
| Take
---
right
47648 | 858 | 188516 | 0 | 0 |  |  | Take
---
right
3 | 1 | 237018 | 0 | 0
Cruise | 0 | 0 | 0 | 303517 | 7287 |  | Cruise | 0 | 0 | 0 | 310698 | 106
Brake | 0 | 0 | 0 | 74841 | 289404 |  | Brake | 0 | 0 | 0 | 664 | 363581
| a) Confusion matrix for CS-LSTM |  |  | b) Confusion matrix for DST-CAN
TABLE III: Performance comparison based on confusion matrix for conflict scenarios for DST-CAN and CS-LSTM | CS-LSTM prediction |  |  | DST-CAN prediction
---|---|---|---|---
GT decision | Same lane |  | Take
---
left
| Take
---
right
Cruise | Brake |  | GT decision | Same lane |  | Take
---
left
| Take
---
right
Cruise | Brake
Same lane | 2405634 | 13781 | 30158 | 0 | 0 |  | Same lane | 2443422 | 5059 | 1092 | 0 | 0
| Take
---
left
1149 | 2851 | 93 | 0 | 0 |  |  | Take
---
left
0 | 4093 | 0 | 0 | 0
| Take
---
right
1419 | 7 | 2838 | 0 | 0 |  |  | Take
---
right
1 | 0 | 4263 | 0 | 0
Cruise | 0 | 0 | 0 | 2038301 | 14878 |  | Cruise | 0 | 0 | 0 | 2052489 | 690
Brake | 0 | 0 | 0 | 31570 | 180780 |  | Brake | 0 | 0 | 0 | 271 | 212079
| a) Confusion matrix for CS-LSTM |  |  | b) Confusion matrix for DST-CAN
TABLE IV: Performance comparison based on confusion matrix for consensus
scenarios for DST-CAN and CS-LSTM

From the ground truth decisions at every time step, the cases where human
decisions and ground truth decisions are different have been found. Since
ground truth decisions are optimal for both safety and fuel efficiency, these
are used for comparison. It has been found that the DST-CAN can predict better
decisions than CS-LSTM [17]. Table II shows a comparison of the results. From
the table, it is clear that for the consensus cases, the performance of DST-
CAN is slightly better than the CS-LSTM. It has been found that conflicting
cases are those where safety is of significant importance. DST-CAN performs
much better than CS-LSTM for conflicting cases. For the model trained with
ground truth decisions, there is a $21.15$ $\%$ and 12.12$\%$ improvement in
making correct decisions for conflicting cases for lateral and longitudinal
cases, respectively. There are $5.06\%$ and $6.37\%$ improvements for lateral
and longitudinal decisions for a model trained with human decisions. Class-
wise maneuver predictions are also shown to understand the significance of the
proposed model. Table III show the different class-wise predictions for
conflicting cases for DST-CAN and CS-LSTM models trained with ground truth
decisions. Those tables show the number of samples that are classified in the
corresponding classes. From the lane-changing maneuvers, it is clear that DST-
CAN performed much better than CS-LSTM. CS-LSTM predicts the decision of
following the same lane for many scenes where it should actually predict
changing lanes, as it can be seen from Table III. The tables clearly show that
DST-CAN predicts fewer misclassifications than the CS-LSTM model. Similarly,
Table IV show misclassifications for consensus cases. From tables for
consensus cases, it is clear that even with an imbalance in the number of
samples for different decisions classes, the self-regulation in learning of
DST-CAN helps in better performance than CS-LSTM. From the study, one can say
that the proposed DST-CAN accurately predicts safe and efficient lateral and
longitudinal decisions based on real traffic conditions.

##  IV CONCLUSIONS

This paper has presented a novel Deep Spatio Temporal Context-Aware decision
Network (DST-CAN), for predictive manoeuvre planning for autonomous vehicles.
The surrounding vehicles’ past, present, and accurately predicted future
trajectories have been encoded into a context grid in a probabilistic manner.
From these encoding, safe and fuel-efficient maneuvers for an AV are
predicted. Based on the studies, it was found that human decisions are not
safe and fuel-efficient for many scenarios. Hence, to ensure the above ground
truth decisions have been developed and are employed for comparison with the
earlier proposed CS-LSTM method. From the ablation study of prediction
horizons, it has been found that a $3sec$ prediction horizon gives the best
result. The study results also show that DST-CAN shows a $21.15\%$ improvement
the number of correct lateral decisions and $12.12\%$ improvement for correct
longitudinal decisions compared to CS-LSTM. The results clearly indicate the
importance of predicting other vehicles’ intentions for good motion planning
decisions.

## Acknowledgments

This work has been funded by the Wipro-IISc Research Innovation Network
(WIRIN), India.

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##  V Supplementary material

A summary of the details of the datasets is given in Table V. The ground truth
decision generation details are given in Algorithm 3.

| US101 | US101 | US101
---|---|---|---
| 07:50-08:05 | 08:05:08:20 | 08:20-08:35
Follow same lane | 94.19 % | 95.78 % | 95.93 %
Take left lane | 3.70 % | 2.67 % | 2.91 %
Take right lane | 2.11 % | 1.55 % | 1.16 %
Cruise on same lane | 90.49 % | 85.96 % | 83.19 %
Brake | 9.51 % | 14.04 % | 16.81 %
Total samples | 1180598 | 1403095 | 1515240
|  |  |
| I80 | I80 | I80
| 16:00-16:15 | 17:00-17:15 | 17:15-17:30
Follow same lane | 94.37 % | 95.87 % | 96.17 %
Take left lane | 4.62 % | 3.60 % | 3.19 %
Take right lane | 1.01 % | 0.53 % | 0.64 %
Cruise on same lane | 84.58 % | 78.77 % | 78.17 %
Brake | 15.42 % | 21.23 % | 21.83 %
Total samples | 1262678 | 1549918 | 1753791
TABLE V: Details of NGSIM traffic dataset

1:Inputs: Social occupancy grid map at current time. $D_{S}$: number of
unoccupied grids on the same lane as ego vehicle at current instant;
$D_{pre}$: number of unoccupied grids on the same lane as ego vehicle at 2
seconds ago (for relative velocity of vehicle in front of ego vehicle);
$D_{LB}$: L2 distance to nearest occupied grid in left-backward side;
$D_{LF}$: L2 distance to nearest occupied grid in left-forward side; $D_{RB}$:
L2 distance to nearest occupied grid in right-backward side; $D_{RF}$: L2
distance to nearest occupied grid in right-forward side; $I_{r}$: Occupancy of
immediate right grid of ego vehicle(1 if occupied, 0 otherwise); $I_{l}$:
Occupancy of immediate left grid of ego vehicle(1 if occupied, 0 otherwise)

2:Output: Ground truth decision at current time

3:$D_{diff}=D_{s}-D_{pre}$

4:if $D_{S}>2$ then

5: if $D_{diff}\geq 0$ then

6: lateral decision $\leftarrow$ Follow same lane;
a . . .. . . longitudinal decision $\leftarrow$ Cruise;

7: else

8: if [$D_{LB}>\sqrt{5}$ & $D_{LF}>\sqrt{5}$ & ($D_{RB}<=\sqrt{5}$ a . . . . .
. . or $D_{RF}<=\sqrt{5}$ or $I_{r}=1$) & $I_{l}=0$] then

9: lateral decision $\leftarrow$ Take left lane;
a . . .. . a . longitudinal decision $\leftarrow$ Cruise;

10: else if [$D_{RB}>\sqrt{5}$ & $D_{RF}>\sqrt{5}$ &
($D_{LB}<=\phantom{a}\phantom{a}\phantom{a}\phantom{a}\phantom{a}\phantom{a}\sqrt{5}$
or $D_{LF}<=\sqrt{5}$ or $I_{l}=1$) & $I_{r}=0$] then

11: lateral decision $\leftarrow$ Take right lane;
a . . .. . a . longitudinal decision $\leftarrow$ Cruise;

12: else if [($D_{RB}>\sqrt{5}$ & $D_{RF}>\sqrt{5}$) &
($D_{LB}>\phantom{a}\phantom{a}\phantom{a}\phantom{a}\phantom{a}\phantom{a}\sqrt{5}$
& $D_{LF}>\sqrt{5}$) & $I_{r}=0$ & $I_{l}=0$] then

13: lateral decision $\leftarrow$ Take right lane;
a . . .. . a . longitudinal decision $\leftarrow$ Cruise;

14: else

15: lateral decision $\leftarrow$ Follow same lane;
a . . .. . a . longitudinal decision $\leftarrow$ Brake;

16: end if

17: end if

18:else

19: if [$D_{LB}>\sqrt{5}$ & $D_{LF}>\sqrt{5}$ & ($D_{RB}<=\sqrt{5}$ a . . . .
. . . or $D_{RF}<=\sqrt{5}$ or $I_{r}=1$) & $I_{l}=0$] then

20: lateral decision $\leftarrow$ Take left lane;
a . . .. . a longitudinal decision $\leftarrow$ Cruise;

21: else if [$D_{RB}>\sqrt{5}$ & $D_{RF}>\sqrt{5}$ &
($D_{LB}<=\phantom{a}\phantom{a}\phantom{a}\phantom{a}\phantom{a}\phantom{a}\sqrt{5}$
or $D_{LF}<=\sqrt{5}$ or $I_{l}=1$) & $I_{r}=0$] then

22: lateral decision $\leftarrow$ Take right lane;
a . . .. . a longitudinal decision $\leftarrow$ Cruise;

23: else if [($D_{RB}>\sqrt{5}$ & $D_{RF}>\sqrt{5}$) &
($D_{LB}>\phantom{a}\phantom{a}\phantom{a}\phantom{a}\phantom{a}\phantom{a}\sqrt{5}$
& $D_{LF}>\sqrt{5}$) & $I_{r}=0$ & $I_{l}=0$] then

24: lateral decision $\leftarrow$ Take right lane;
a . . .. . a longitudinal decision $\leftarrow$ Cruise;

25: else

26: lateral decision $\leftarrow$ Follow same lane;
a . . .. . . longitudinal decision $\leftarrow$ Brake;

27: end if

28:end if

Algorithm 3 Ground truth generation pseudo-code


# On solid cores and hulls of weighted Bergman spaces $A_{\mu}^{1}$

José Bonet

Instituto Universitario de Matemática Pura y Aplicada IUMPA, Universitat
Politècnica de València, E-46071 Valencia, Spain

Wolfgang Lusky

Institut für Mathematik, Universität Paderborn, D-33098 Paderborn, Germany

Jari Taskinen (corresponding author)

Department of Mathematics and Statistics, P.O. Box 68, University of Helsinki,
00014 Helsinki, Finland

email<EMAIL_ADDRESS>

Keywords: Bergman space, weighted $L^{1}$-norm, unit disc, solid hull, solid
core

To appear in Journal of Mathematical Sciences

Abstract. We consider weighted Bergman spaces $A_{\mu}^{1}$ on the unit disc
as well as the corresponding spaces of entire functions, defined using non-
atomic Borel measures with radial symmetry. By extending the techniques from
the case of reflexive Bergman spaces we characterize the solid core of
$A_{\mu}^{1}$. Also, as a consequence of a characterization of solid
$A_{\mu}^{1}$-spaces we show that, in the case of entire functions, there
indeed exist solid $A_{\mu}^{1}$-spaces. The second part of the paper is
restricted to the case of the unit disc and it contains a characterization of
the solid hull of $A_{\mu}^{1}$, when $\mu$ equals the weighted Lebesgue
measure with weight $v$. The results are based on a duality relation of
weighted $A^{1}$\- and $H^{\infty}$-spaces, the validity of which requires the
assumption that $-\log v$ belongs to the class ${\mathcal{W}}_{0}$, studied in
a number of publications; moreover, $v$ has to satisfy condition $(b)$,
introduced by the authors. The exponentially decreasing weight
$v(z)=\exp(-1/(1-|z|)$ provides an example satisfying both assumptions.

##  1\. Introduction and preliminaries

The solid hulls and cores of spaces of analytic functions on the unit disc
${\mathbb{D}}=\\{z\in\mathbb{C}:|z|<1\\}$ or the entire plane ${\mathbb{C}}$
have been investigated by many authors. We refer the reader to the recent
books [12] and [16] and the many references therein. In the series of papers
[2]–[5] the authors have presented the solid hulls and cores of the weighted
$H^{\infty}$-spaces $H^{\infty}_{v}$ on ${\mathbb{D}}$ or ${\mathbb{C}}$ for a
large class of radial weights $v$ as well as their Bergman space analogues
$A_{\mu}^{p}$ for $1<p<\infty$. Earlier, the cases of standard weights and
$d\mu(r)=(1-r)^{\alpha}dr$, $\alpha>0$, were considered in [1] and [12].

In this note we want to extend the results of [5] to weighted Bergman spaces
$A_{\mu}^{p}$ for $p=1$. The spaces are defined on the unit disc
${\mathbb{D}}$ or on the entire plane. (Fock spaces are usually considered as
the Bergman space analogues of spaces of entire functions, but these are
defined with Gaussian weight functions, which is not required here. Thus, we
keep here the term Bergman space also for entire functions.) Consider $R=1$ or
$R=\infty$. We study holomorphic functions
$f:R\cdot\mathbb{D}\rightarrow\mathbb{C}$ where $R\cdot\mathbb{D}=\mathbb{D}$
if $R=1$ and $R\cdot\mathbb{D}=\mathbb{C}$ if $R=\infty$. Let $\hat{f}(k)$ be
the Taylor coeffients of $f$, i.e. $f(z)=\sum_{k=0}^{\infty}\hat{f}(k)z^{k}$.
We take a non-atomic positive bounded Borel measure $\mu$ on $[0,R[$ such that
$\mu([r,R[)>0$ for every $r>0$ and $\int_{0}^{R}r^{n}d\mu(r)<\infty$ for all
$n>0$. Put, for $1\leq p<\infty$,

$\|f\|_{p}=\left(\frac{1}{2\pi}\int_{0}^{R}\int_{0}^{2\pi}|f(re^{i\varphi})|^{p}d\varphi
d\mu(r)\right)^{1/p}$

and let

$A_{\mu}^{p}=\\{f:R\cdot\mathbb{D}\rightarrow\mathbb{C}:f\mbox{ holomorphic
with }\|f\|_{p}<\infty\\}.$

We will also consider the weighted spaces

$H_{v}^{\infty}=\\{f:{\mathbb{D}}\to{\mathbb{C}}\mbox{
holomorphic}:\|f\|_{v}=\sup\limits_{z\in{\mathbb{D}}}v(z)|f(z)|<\infty\\},$

where the weight $v:{\mathbb{D}}\to(0,\infty)$ is a continuous and radial
($v(z)=v(|z|)$) function which is decreasing with respect to $r=|z|$, and
$\lim_{r\to 1^{-}}v(r)=0$.

Let $A$ be a vector space of holomorphic functions on $R\cdot\mathbb{D}$
containing the polynomials. The solid core is defined as

$s(A)=\\{f\in A:g\in A\mbox{ for all holomorphic }g\mbox{ with
}|\hat{g}(k)|\leq|\hat{f}(k)|\mbox{ for all }k\\}$

and the solid hull as

$S(A)=\\{g:\mathbb{D}\rightarrow\mathbb{C}\mbox{ holomorphic : there is }f\in
A\mbox{ with }|\hat{g}(k)|\leq|\hat{f}(k)|\mbox{ for all }k\\}.$

The space $A$ is called solid if $A=S(A)$. The concept of a solid hull will
also be discussed in the beginning of Section 3.

Here, in Theorem 2.4 we will transfer Theorem 4.1. of [5] to the case $p=1$.
This result concerns the characterization of solid Bergman spaces
$A_{\mu}^{1}$, and it is motivated by the fact that such spaces indeed exist
in the case $R=\infty$ (only), as will be shown in Example 2.7 and Corollary
2.6. In Theorem 2.8 we determine the solid cores for all Bergman spaces
$A_{\mu}^{1}$.

We also present in Section 3 how duality theory can be used for new results on
certain solid hulls; see the beginning of Section 3 for detailed definitions.
In particular, we construct the solid hull $S_{BK}(A_{\mu}^{1})$ of
$A_{\mu}^{1}$ for $R=1$ by using the known solid core of the space
$H^{\infty}_{v}$ in [4]. This result is more special than the above one for
solid cores, since we need to restrict to the case $\mu$ is the weighted
Lebesgue measure $d\mu=vdA=v\pi^{-1}rdrd\varphi$, where the weight $v$ needs
to satisfy some special assumptions in addition to those mentioned above.
Examples of such weights include important cases like exponentially decreasing
weights.

For a holomorphic $g$ and $r>0$ we define

$M_{p}(g,r)=\left(\frac{1}{2\pi}\int_{0}^{2\pi}|g(re^{i\varphi})|^{p}d\varphi\right)^{1/p}$

and denote the Dirichlet projections by
$P_{n}g(z)=\sum_{k=0}^{n}\hat{g}(k)z^{k}$, $n\in{\mathbb{N}}$. It is well-
known that, for $1<p<\infty$, there are constants $c_{p}>0$, not depending on
$g$, $n$ or $r$, such that $M_{p}(P_{n}g,r)\leq c_{p}M_{p}(g,r)$. Moreover we
have $\lim_{n\rightarrow\infty}M_{p}(g-P_{n}g,r)=0$. Hence we obtain

$\|P_{n}f\|_{p}\leq c_{p}\|f\|_{p}\mbox{ for all }f\in A_{\mu}^{p}\mbox{ and
all }n\mbox{ and }\lim_{n\rightarrow\infty}\|f-P_{n}f\|_{p}=0.$

In particular we see that the monomials $z\mapsto z^{n}$,
$n\in{\mathbb{N}}_{0}=\\{0,1,2,\ldots\\}={\mathbb{N}}\cup\\{0\\}$, form a
(Schauder) basis of $A_{\mu}^{p}$ if $1<p<\infty$. On the other hand, denoting
by $H^{1}$ the Hardy space of all holomorphic functions on $\mathbb{D}$ which
are bounded under $\sup_{0\leq r<1}M_{1}(\cdot,r)$, it is well known that the
operator norm of $P_{n}:H^{1}\to H^{1}$ tends to infinity as
$n\rightarrow\infty$. Details can be seen in [9] and [17]. For the terminology
and definitions on bases in Banach spaces, see also [13].

In the rest of the article $[r]$ denotes the largest integer smaller or equal
than $r>0$. By $c,c_{1},c_{2},C,C^{\prime}$ etc. we denote generic positive
constants, the actual value of which may vary from place to place.

##  2\. Solid core and examples of solid $A_{\mu}^{1}$-spaces.

In this section we extend Theorem 4.1. of [5] concerning the characterization
of solid Bergman spaces to the case $p=1$ and also determine the solid cores
for all spaces $A_{\mu}^{1}$. We consider both cases $R=1$ or $R=\infty$
unless otherwise specified. At first we recall a fundamental result from [10],
which concerns equivalent representations of the norm of the space
$A_{\mu}^{1}$.

######  Theorem 2.1.

There are sequences $0<s_{1}<s_{2}<\ldots<R$ and $0=m_{0}<m_{1}<m_{2}<\ldots$,
non-negative numbers $d_{n}$, $t_{n,k}$ (with $n\in{\mathbb{N}}$ and
$[m_{n-1}]<k\leq[m_{n+1}]$ ) and constants $c_{1}$, $c_{2}>0$ such that for
all $g(z)=\sum_{k=0}^{\infty}\alpha_{k}z^{k}$ we have

(2.1) $\displaystyle
c_{1}\|g\|_{1}\leq\sum_{n=0}^{\infty}M_{1}(T_{n}g,s_{n})d_{n}\leq
c_{2}\|g\|_{1},$

where

(2.2) $\displaystyle
T_{n}g=\sum_{[m_{n-1}]+1}^{[m_{n+1}]}t_{n,k}\alpha_{k}z^{k}.$

We will need the following consequence of this result.

######  Corollary 2.2.

Let $(n_{j})_{j=1}^{\infty}$ be an increasing sequence of indices such that
$n_{j+1}-n_{j}\geq 2$ for all $j$ and let
$h_{j}(z)=\sum_{k=[m_{n_{j}}]+1}^{[m_{n_{j}+1}]}\alpha_{k}z^{k}$ be a
polynomial. We have

(2.3) $\displaystyle\|h\|_{1}\leq\sum_{j=0}^{\infty}\|h_{j}\|_{1}\leq
C\|h\|_{1}\ \ \ \mbox{for all}\ h=\sum_{j=1}^{\infty}h_{j}\in A_{\mu}^{1}.$

Proof. Applying (2.1) to $h_{j}$ yields that $\|h_{j}\|_{1}$ and

$M_{1}(T_{n_{j}}h_{j},s_{n_{j}})d_{n_{j}}+M_{1}(T_{n_{j}+1}h_{j},s_{n_{j}+1})d_{n_{j}+1}$

are proportional quantities. Moreover, $T_{n}h=0$, if $n$ is not equal to
$n_{j}$ or $n_{j}+1$ for any $j$, and

(2.4) $\displaystyle T_{n_{j}}h=T_{n_{j}}h_{j},\ \
T_{n_{j}+1}h=T_{n_{j}+1}h_{j}\ \ \ \mbox{for all }j.$

Hence, by another application of (2.1),

$\displaystyle\|h\|_{1}\leq\sum_{j=0}^{\infty}\|h_{j}\|_{1}\leq
C\sum_{j=0}^{\infty}\big{(}M_{1}(T_{n_{j}}h_{j},s_{n_{j}})d_{n_{j}}+M_{1}(T_{n_{j}+1}h_{j},s_{n_{j}+1})d_{n_{j}+1}\big{)}$
$\displaystyle=$ $\displaystyle
C\sum_{n=0}^{\infty}M_{1}(T_{n}h,s_{n})d_{n}\leq C^{\prime}\|h\|_{1}.\ \hskip
142.26378pt\Box$

Let us make a remark concerning the numbers and constants in the above
results.

######  Remark 2.3.

$1^{\circ}.$ Theorem 2.1 is a reformulation of Theorem 1.3. of [10], where the
sequences $(s_{n})_{n=1}^{\infty}$ and $(m_{n})_{n=0}^{\infty}$ were chosen,
by using induction, such that, for all $n\in{\mathbb{N}}$,

$\int_{0}^{s_{n}}r^{m_{n}}d\mu=b\int_{s_{n}}^{R}r^{m_{n}}d\mu\ \ \ \mbox{ and
}\ \ \ \int_{0}^{s_{n}}r^{m_{n+1}}d\mu=\int_{s_{n}}^{R}r^{m_{n+1}}d\mu.$

where $b>5$ is some constant. Then, the numbers $d_{n}$ were set to be

(2.5) $\displaystyle
d_{n}=\left(\int_{0}^{s_{n}}\left(\frac{r}{s_{n}}\right)^{m_{n}}d\mu+\int_{s_{n}}^{R}\left(\frac{r}{s_{n}}\right)^{m_{n+1}}d\mu\right).$

As proven in Section 5 of [10], it is always possible to find these sequences,
although calculating them exactly for given concrete weights seems in general
to be difficult.

$2^{\circ}$. If $R=1$ and $d\mu=rv(r)drd\theta$ with
$v(r)=\exp\big{(}-\alpha(1-r^{\ell})^{-\beta}\big{)}$ for some constants
$\alpha,\beta,\ell>0$, then the numbers $m_{n}$ and $s_{n}$,
$n\in{\mathbb{N}}$, ($m_{0}=0$), were calculated by a different method than in
$1^{\circ}$ in Propositions 3.1 and 3.3.$(ii)$ of [6]:

(2.6) $\displaystyle
m_{n}=\ell\beta^{2}\Big{(}\frac{\beta}{\alpha}\Big{)}^{1/\beta}n^{2+2/\beta}-\ell\beta^{2}n^{2}\
\ \mbox{and}\ \
s_{n}=\Big{(}1-\Big{(}\frac{\alpha}{\beta}\Big{)}^{1/\beta}n^{-2/\beta}\Big{)}^{1/\ell}.$

$3^{\circ}$. In the citations mentioned in $1^{\circ}$ and $2^{\circ}$, the
numbers $t_{n,k}$ were chosen as the coefficients of certain de la Valleé
Poussin operators, more precisely,

(2.7)
$t_{n,k}=\left\\{\begin{array}[]{ll}{\displaystyle\frac{k-[m_{n}]}{[m_{n}]-[m_{n-1}]}},&\
\mbox{if}\ m_{n-1}<|k|\leq m_{n},\\\ &\\\
{\displaystyle\frac{[m_{n+1}]-k}{[m_{n+1}]-[m_{n}]}},&\ \mbox{if}\
m_{n}<|k|\leq m_{n+1}.\end{array}\right.$

Let us next state our result on the characterization of solid $A_{\mu}^{1}$
spaces.

######  Theorem 2.4.

The following are equivalent:

(i) $A_{\mu}^{1}$ is solid,

(ii) $s(A_{\mu}^{1})=A_{\mu}^{1}$,

(iii) the monomials $(z^{n})_{n=0}^{\infty}$ are an unconditional basis of
$A_{\mu}^{1}$,

(iv) the normalized monomials $(z^{n}/\|z^{n}\|_{1})_{n=0}^{\infty}$ are
equivalent to the unit vector basis of $\ell^{1}$,

(v) $\sup\limits_{n\in{\mathbb{N}}}(m_{n+1}-m_{n})<\infty$ for the numbers
$m_{n}$ in Theorem 2.1.

In the following we retain the numbers $m_{n}$, $s_{n}$ of Theorem 2.1 and
consider the Dirichlet projections $P_{n}$.

######  Lemma 2.5.

Assume that $\limsup_{n\rightarrow\infty}(m_{n+1}-m_{n})=\infty$. Then, for
every $N>0$ there exist an arbitrarily large $n\in{\mathbb{N}}$, an index
$M<m_{n+1}$ and a polynomial
$f(z)=\sum_{k=[m_{n}]+1}^{[m_{n+1}]}\alpha_{k}z^{k}$ with $\|f\|_{1}\leq 1$
but $\|P_{M}f\|_{1}=\|(P_{M}-P_{[m_{n}]})f\|_{1}\geq N$.

Proof. Due to the unboundedness of the operator norms of $P_{n}$ on $H^{1}$,
see Section 1, we find an index $K$ and a polynomial
$g(z)=\sum_{j=0}^{L}\beta_{j}z^{j}$ with $M_{1}(g,1)=1$ but
$M_{1}(P_{K}g,1)>N$. By assumption we find $n\in{\mathbb{N}}$, as large as we
wish, such that $m_{n+1}-m_{n}>L+1$. Then put

$f(z)=\sum_{k=[m_{n}]+1}^{[m_{n}]+L+1}\beta_{k-[m_{n}]-1}\frac{1}{s_{n}^{k}}z^{k}.$

We obtain

$M_{1}(f,s_{n})=M_{1}(g,1)=1\ \ \ \mbox{ and }\ \ \
M_{1}(P_{m_{n}+K}f,s_{n})=M_{1}(P_{K}g,1)>N.$

Put $M=K+[m_{n}]+1$ and use Theorem 2.1 to complete the proof of the lemma. We
have $P_{M}f=(P_{M}-P_{[m_{n}]})f$ just by the choice of $f$. $\Box$

Proof of Theorem 2.4. $(i)\Leftrightarrow(ii)$: follows from the definition.
$(iv)\Rightarrow(iii)\Rightarrow(ii)$: these are obvious.
$(ii)\Rightarrow(v)$: Assume that
$\limsup_{n\rightarrow\infty}(m_{n+1}-m_{n})=\infty$. For every
$j\in{\mathbb{N}}$ we find, by Lemma 2.5, a polynomial $f_{j}\in$
span$\\{z^{[m_{n_{j}}]+1},\ldots,z^{[m_{n_{j+1}]}}\\}$ for some $m_{n_{j}}$
with

(2.8) $\displaystyle\|f_{j}\|_{1}=2^{-j}\ \ \mbox{and
$\big{\|}P_{k_{j}}f_{j}\big{\|}_{1}\geq 1$ for some
$k_{j}\in(m_{n_{j}},m_{n_{j}+1})$. }$

We may assume that $n_{j+1}-n_{j}\geq 2$. Put $f=\sum_{j}f_{j}$ and
$g=\sum_{j}P_{k_{j}}f_{j}=\sum_{j}(P_{k_{j}}-P_{[m_{n_{j}}]})f_{j}$. Then,
$f\in A_{\mu}^{1}$ but in view of (2.8), (2.3) we have $g\not\in A_{\mu}^{1}$.
Hence $f\not\in s(A_{\mu}^{1})$.
$(v)\Rightarrow(iv):$ Let
$g(z)=\sum_{k=[m_{n-1}]+1}^{[m_{n+1}]}\alpha_{k}z^{k}$. By $(v)$ we obtain a
constant independent of $n$, $r$ and $g$ with

$M_{1}(g,r)\leq\sum_{k=[m_{n-1}]+1}^{[m_{n+1}]}|\alpha_{k}|r^{k}\leq
cM_{1}(g,r).$

Then, (2.1) yields numbers $\delta_{k}=t_{n,k}s_{n}^{k}d_{n}$ such that for
all functions $f(z)=\sum_{k=0}^{\infty}\alpha_{k}z^{k}\in A_{\mu}^{1}$ we
have, with universal constants $c_{1},c_{2}$,

$c_{1}\|f\|_{1}\leq\sum_{k=0}^{\infty}\delta_{k}|\alpha_{k}|\leq
c_{2}\|f\|_{1}.$

This proves $(iv)$. $\Box$

######  Corollary 2.6.

If $R<\infty$ then $A_{\mu}^{1}$ is never solid.

Proof. It follows from Proposition 2.1. of [10] that in this case we always
have $\limsup_{n\rightarrow\infty}(m_{n+1}-m_{n})=\infty$. $\Box$

######  Example 2.7.

There are indeed examples where $A_{\mu}^{1}$ is solid. Let $R=\infty$ and
$d\mu(r)=\exp(-\log^{2}(r))dr$. It was shown in [10], Example 2a) that here
$\sup_{n}(m_{n+1}-m_{n})<\infty$.

######  Theorem 2.8.

Let $m_{n}$, $s_{n}$ and $d_{n}$ be the numbers of Theorem 2.1. The solid core
of $A_{\mu}^{1}$ equals

(2.9) $\displaystyle s(A_{\mu}^{1})=$ $\displaystyle\bigg{\\{}$ $\displaystyle
g:R\cdot\mathbb{D}\rightarrow\mathbb{C}:g(z)=\sum_{k=0}^{\infty}\hat{g}(k)z^{k}$
$\displaystyle\mbox{ with
}\sum_{n=1}^{\infty}d_{n}\Big{(}\sum_{k=[m_{n}]+1}^{[m_{n+1}]}|\hat{g}(k)|^{2}s_{n}^{2k}\Big{)}^{1/2}<\infty\bigg{\\}}.$

Proof. For a holomorphic function $g(z)=\sum_{k=0}^{\infty}\hat{g}(k)z^{k}$ we
write

$g_{n}(z)=\sum_{k=[m_{n}]+1}^{[m_{n+1}]}\hat{g}(k)z^{k}\ \ \mbox{ and }\ \
g_{I}(z)=\sum_{n=0}^{\infty}g_{2n}(z),\ \
g_{II}(z)=\sum_{n=0}^{\infty}g_{2n+1}(z).$

Let us denote by $V$ the function space on the right-hand side of (2.9).
Moreover, for all $n$, let $\Delta_{n}=\\{+1,-1\\}^{[m_{n+1}]-[m_{n}]}$, and
for $\Theta_{n}=(\theta_{[m_{n}]+1},\ldots,\theta_{[m_{n+1}]})\in\Delta_{n}$
put

$g_{\Theta_{n}}(z)=\sum_{k=[m_{n}]+1}^{[m_{n+1}]}\theta_{k}\hat{g}(k)z^{k}.$

At first assume that $g\in V$. Then $g_{I},g_{II}\in V$. Let $f$ be
holomorphic with $|\hat{f}(k)|\leq|\hat{g_{I}}(k)|$ for all $k$. By (2.3) and
Theorem 2.1

$\displaystyle\|f\|_{1}$ $\displaystyle\leq$
$\displaystyle\sum_{n=0}^{\infty}\|f_{2n}\|_{1}\leq
c\sum_{n=0}^{\infty}d_{2n}M_{1}(f_{2n},s_{2n})$ $\displaystyle\leq$
$\displaystyle c\sum_{n=0}^{\infty}d_{2n}M_{2}(f_{2n},s_{2n})\leq
c\sum_{n=0}^{\infty}d_{2n}M_{2}(g_{2n},s_{2n})<\infty$

where $c>0$ is a universal constant and we also used the definition of the
space $V$ in the last step. Hence $f\in A_{\mu}^{1}$, in particular $g_{I}\in
A_{\mu}^{1}$. We conclude $g_{I}\in s(A_{\mu}^{1})$. The same proof shows that
$g_{II}$ and hence $g\in s(A_{\mu}^{1})$.

Conversely, let $g\in s(A_{\mu}^{1})$. Then $g_{I},g_{II}\in s(A_{\mu}^{1})$.
Let ${\tilde{\Theta}}_{n}\in\Delta_{n}$ be such that

$a_{1}\left(\sum_{k=[m_{n}]+1}^{[m_{n+1}]}|\hat{g}(k)|^{2}\right)^{1/2}\leq\frac{1}{2^{[m_{n+1}]-[m_{n}]}}\sum_{\Theta_{n}\in\Delta_{n}}M_{1}(g_{\Theta_{n}},s_{n})\leq
M_{1}(g_{{\tilde{\Theta}}_{n}},s_{n}).$

Here we used the Khintchine inequality (see [18], Ch. V, Thm. 8.4) with the
Khintchine constant $a_{1}$. Put
$h_{I}=\sum_{n=0}^{\infty}g_{{\tilde{\Theta}}_{2n}}$. Then we obtain
$|\hat{h_{I}}(k)|=|\hat{g_{I}}(k)|$ for all $k$. Hence $h_{I}\in A_{\mu}^{1}$.
The choice of ${\tilde{\Theta}}_{n}$ and Theorem 2.1 applied to $h_{I}$ yield

$\displaystyle\sum_{n=0}^{\infty}d_{2n}\Big{(}\sum_{k=[m_{2n}]+1}^{[m_{2n+1}]}|\hat{g}(k)|^{2}s_{2n}^{2k}\Big{)}^{1/2}=\sum_{n=0}^{\infty}d_{2n}\Big{(}\sum_{k=[m_{2n}]+1}^{[m_{2n+1}]}|\hat{h_{I}}(k)|^{2}s_{2n}^{2k}\Big{)}^{1/2}$
$\displaystyle\leq$
$\displaystyle\frac{1}{a_{1}}\sum_{n=0}^{\infty}d_{2n}M_{1}(g_{{\tilde{\Theta}}_{2n}},s_{2n})\leq\frac{c_{2}}{a_{1}}\|h_{I}\|_{1}<\infty.$

Here, $c_{2}$ is the constant of Theorem 2.1. We conclude $g_{I}\in V$, and
similarly we see that $g_{II}\in V$. Hence $g\in V$, which implies
$V=s(A_{\mu}^{1})$. $\Box$

##  3\. On solid hulls

In this section we assume $R=1$. We start by the remark that in addition to
the definition of a solid hull as in Section 1, there exist two other a priori
different definitions in the literature: in [1], the solid hull $S_{{\rm
vect}}(X)$ of a space $X$ of analytic functions on ${\mathbb{D}}$ is defined
as the intersection of all solid vector spaces of analytic functions on
${\mathbb{D}}$. Obviously, $S(X)$ is a vector space if and only if for every
$f,g\in X$ there is $h\in X$ such that the Taylor coefficients satisfy
$|\hat{f}(k)|+|\hat{g}(k)|\leq|\hat{h}(k)|$ for all $k$.

One more variant appears in the theory of so called BK-spaces. By definition,
a BK-space is a vector space of complex sequences $f=(f_{k})_{k=0}^{\infty}$
endowed with a norm which makes it into a Banach space, such that the
coordinate functionals become bounded operators. In the theory of BK-spaces,
see [8], the solid hull $S_{BK}(X)$ of a BK-space $X$ is defined as the
intersection of all solid $BK$-spaces containing $X$. By using Taylor
coefficients we consider Banach spaces of analytic functions on ${\mathbb{D}}$
as BK-spaces, and, in particular, we will characterize in the sequel the solid
hull $S_{BK}(A_{\mu}^{1})$ although we will avoid using the terminology of BK-
spaces, except for the proof of Proposition 3.1. It is quite easy to see that

(3.1) $\displaystyle S(X)\subset S_{{\rm vect}}(X)\subset S_{BK}(X)$

for a BK-space $X$ as above. All results on solid hulls $S(X)$ in the
literature, which are known to the authors, happen to be vector spaces which
can be endowed with norms making them into solid BK-spaces. Thus, in all of
these cases one actually has $S(X)=S_{BK}(X)$.

Our aim is to use the known duality relations between weighted $A^{1}$ and
$H^{\infty}$-spaces and existing results of the solid core of $H_{v}^{\infty}$
in order to find the solid hull $S_{BK}(A_{\mu}^{1})$. We focus on the case
the measure $\mu$ is the weighted Lebesgue measure $vdA$ with a radial weight
$v$ making the Bergman space into a ”large” one: the admissible weights
include the exponentially decreasing weights, see Example 3.3, below.

We start by some general considerations.

Given a sequence $\theta=(\theta_{k})_{k=0}^{\infty}$ with $|\theta_{k}|\leq
1$ for all $k$, we denote by $M_{\theta}$ the operator
$M_{\theta}\sum_{k=0}^{\infty}\hat{f}(k)z^{k}=\sum_{k=0}^{\infty}\theta_{k}\hat{f}(k)z^{k}$.
We will need to consider analytic function spaces on ${\mathbb{D}}$ such that
the norm of the space satisfies

(3.2) $\displaystyle\|M_{\theta}f\|\leq\|f\|$

for all $f=\sum_{k=0}^{\infty}\hat{f}(k)z^{k}\in X$ and all sequences
$\theta=(\theta_{k})_{k=0}^{\infty}$ with $|\theta_{k}|\leq 1$ for all $k$.

The following result is essentially known.

######  Proposition 3.1.

If $(X,\|\cdot\|_{X})$ is a Banach space of analytic functions on the unit
disc ${\mathbb{D}}$ such that all coordinate functionals $f\mapsto\hat{f}(k)$
are bounded operators, then its solid hull $S_{BK}(X)$ can be endowed with a
norm $\|\cdot\|_{S}$ such that

$(i)$ the embedding $X\hookrightarrow S_{BK}(X)$ is continuous,

$(ii)$ the norm $\|\cdot\|_{S}$ satisfies (3.2),

$(iii)$ if $p:S_{BK}(X)\to{\mathbb{R}}_{0}^{+}$ is any norm with (3.2) such
that $p(f)\leq\|f\|_{X}$ for all $f\in X$, then $p(f)\leq C\|f\|_{S}$ for a
constant $C>0$ and all $f\in S_{BK}(X)$,

$(iv)$ the normed space $\big{(}S_{BK}(X),\|\cdot\|_{S}\big{)}$ is complete,
and

$(v)$ if the subspace of polynomials ${\mathcal{P}}$ is dense in $X$, then it
is dense in $\big{(}S_{BK}(X),\|\cdot\|_{S}\big{)}$, too.

Proof. Let us explain how the claims follow from the theory of BK-spaces, see
[7], [8]. For the sake of the simplicity of notation, let us consider $X$ as a
BK-sequence space in the following, which we can do by assumption. We denote
by $y\cdot f$ the coordinatewise product of two complex sequences $y$ and $f$.
The space $\ell^{\infty}\widehat{\otimes}X$ is defined in [7] to consist of
sequences $g=(g_{k})_{k=0}^{\infty}$ having a coordinatewise convergent
representation

(3.3) $\displaystyle g=\sum_{j=1}^{\infty}y^{(j)}\cdot f^{(j)}\ \ \ \mbox{with
$y^{(j)}=\big{(}y^{(j)}_{k}\big{)}_{k=0}^{\infty}\in\ell^{\infty}$,
$f^{(j)}=\big{(}f^{(j)}_{k}\big{)}_{k=0}^{\infty}\in X\ \forall\,j$}$

such that

(3.4)
$\displaystyle\sum_{j=1}^{\infty}\|y^{(j)}\|_{\ell^{\infty}}\|f^{(j)}\|_{X}<\infty.$

The norm $\|\cdot\|_{S}$ of $g\in\ell^{\infty}\widehat{\otimes}X$ is defined
by taking the infimum of the quantity (3.4) over all possible representations
(3.3) of $g$. Theorem 3 of [7] yields that the resulting space is complete,
and Theorem 8 of [8] says that $\ell^{\infty}\widehat{\otimes}X$ equals the
solid hull $S_{BK}(X)$. The completeness of the space is included in the same
reference, hence, property $(iv)$ holds.

If $f\in X$, then we have ${\rm e}\cdot f=f$, where ${\rm
e}=(1,1,1,\dots)\in\ell^{\infty}$, and in view of the above definition of the
norm $\|\cdot\|_{S}$, this implies that $\|f\|_{S}\leq\|f\|_{X}$ for all $f\in
X$ so that the embedding of $X$ into $\big{(}S_{BK}(X),\|\cdot\|_{S}\big{)}$
is continuous.

Also, if $g\in S_{BK}(X)$ has a representation (3.3) and $\theta$ is given as
in (3.2), then $M_{\theta}g$ has a coordinatewise convergent representation

(3.5) $\displaystyle M_{\theta}g=\sum_{j=1}^{\infty}(M_{\theta}y^{(j)})\cdot
f^{(j)},$

and property $(ii)$ follows from the definition of $\|\cdot\|_{BK}$.

In the proof of Theorem 3 of [7] it is shown if $p$ is the norm of any BK-
space containing $S_{BK}(X)$, then there exists $C>0$ such that

$\displaystyle p(y\cdot f)\leq C\|y\|_{\infty}\,\|f\|_{X}$

for all $y\in\ell^{\infty}$, $f\in X$. This implies

$\displaystyle p\Big{(}\sum_{j=1}^{\infty}y^{(j)}\cdot f^{(j)}\Big{)}\leq
C\sum_{j=1}^{\infty}\|y^{(j)}\|_{\infty}\,\|f^{(j)}\|_{X}\ \ \mbox{for all}\
g=\sum_{j=1}^{\infty}y^{(j)}\cdot f^{(j)}\in\ell^{\infty}\widehat{\otimes}X,$

and property $(iii)$ follows from the definition of $\|\cdot\|_{S}$. Finally,
as for property $(v)$, it follows from Theorem 2 of [7] that finite linear
combinations of functions $y\cdot f$, $y\in\ell^{\infty}$, $f\in X$, form a
dense subspace of $\ell^{\infty}\widehat{\otimes}X=S(X)$. If $y$, $f$ and
$\varepsilon>0$ are given, we use the assumption in $(v)$ to find a polynomial
$h$ such that $\|f-h\|_{X}<\varepsilon/(1+\|y\|_{\infty})$. Then, $y\cdot h$
is a polynomial, which satisfies

$\displaystyle p(y\cdot f-y\cdot
h)=p(y\cdot(f-h))\leq\|y\|_{\infty}\|f-h\|_{X}\leq\varepsilon.$

Property $(v)$ follows from these arguments. $\Box$

######  Lemma 3.2.

Let $X$ be a Banach space of analytic functions on the unit disc
${\mathbb{D}}$ such that the subspace ${\mathcal{P}}$ of polynomials is dense
in $X$, and let $w$ be a radial weight function on ${\mathbb{D}}$. Let $Y$ be
the space of all analytic functions $g$ on the disc such that

(3.6) $\displaystyle\sup\limits_{f\in B_{X}}|\langle f,g\rangle|<\infty,\ \ \
\mbox{where}\ \ \ \ \langle
f,g\rangle=\int\limits_{\mathbb{D}}f\overline{g}wdA$

and $B_{X}$ denotes the unit ball of $X$. If $X$ is solid and there exists a
constant $C>0$ such that

(3.7) $\displaystyle\|M_{\theta}f\|_{X}\leq C\|f\|_{X}$

for all numerical sequences $\theta=(\theta_{k})_{k=0}^{\infty}$ with
$|\theta_{k}|\leq 1$, then $Y$ is solid, too.

We point out given a Banach space $X$ as in the assumption, it is not in
general known whether its dual space has a representation as a space of
analytic functions with dual norm coming from (3.6).

Proof. If $g=\sum_{k=0}^{\infty}\hat{g}(k)z^{k}\in Y$ and $\theta$ is as
above, then for $M_{\theta}g$ we have by (3.7)

(3.8) $\displaystyle\sup\limits_{f\in B_{X}}|\langle
f,M_{\theta}g\rangle|=\sup\limits_{f\in
B_{X}}\sum_{k=0}^{\infty}\bar{\theta}_{k}\hat{f}(k)\overline{\hat{g}(k)}\int\limits_{0}^{1}r^{2k+1}w(r)dr$
$\displaystyle=$ $\displaystyle\sup\limits_{f\in B_{X}}|\langle
M_{\bar{\theta}}f,g\rangle|\leq\sup\limits_{\stackrel{{\scriptstyle\scriptstyle
f\in X}}{{\|f\|_{X}\leq C}}}|\langle f,g\rangle|<\infty.$

Thus, $M_{\theta}g\in Y$. $\Box$

We next recall an elementary fact concerning Banach sequence spaces. Assume
that the sequences $(\beta_{k})_{k=0}^{\infty}$ and
$(\gamma_{k})_{k=0}^{\infty}$ of positive numbers are given and
$\alpha_{k}=\gamma_{k}\beta_{k}^{-1}$ for all $k$. Let also
$(\mu_{n})_{n=0}^{\infty}$ be an increasing, unbounded sequence of non-
negative numbers; denote $\mu_{-1}=-1$ and let

(3.9) $\displaystyle A$ $\displaystyle=$
$\displaystyle\big{\\{}a=(a_{k})_{k=0}^{\infty}\,:\,\|a\|_{A}=\sum_{n\in{\mathbb{N}}}\max_{\mu_{n-1}<k\leq\mu_{n}}\alpha_{k}|a_{k}|<\infty\big{\\}},$
(3.10) $\displaystyle B$ $\displaystyle=$
$\displaystyle\big{\\{}b=(b_{k})_{k=0}^{\infty}\,:\,\|b\|_{B}=\sup\limits_{n\in{\mathbb{N}}}\sum_{\mu_{n-1}<k\leq\mu_{n}}\beta_{k}|b_{k}|<\infty\big{\\}}.$

Then, $B$ is the dual of $A$ with respect to the dual pairing

(3.11) $\displaystyle\langle
a,b\rangle=\sum_{k=0}^{\infty}\gamma_{k}b_{k}\overline{a_{k}},\ \ \
\mbox{where}\ a=(a_{k})_{k=0}^{\infty}\in A,\ b=(b_{k})_{k=0}^{\infty}\in B.$

From now on we consider radial weights $v:{\mathbb{D}}\to{\mathbb{R}}^{+}$
satisfying the following two assumptions.

$(I)$ We have

(3.12) $\displaystyle v(z)=\exp(-\varphi(z)),$

where $\varphi$ belongs to the class ${\mathcal{W}}_{0}$ of [11].

We will not need a detailed definition of ${\mathcal{W}}_{0}$, but recall that
$\varphi\in{\mathcal{W}}_{0}$, if it is a twice continuously differentiable
real valued function with $\Delta\varphi>0$ on ${\mathbb{D}}$ and there exists
a function $\rho:{\mathbb{D}}\to{\mathbb{R}}$ and a constant $C>0$ such that

(3.13)
$\displaystyle\frac{1}{C}\rho(z)\leq\frac{1}{\sqrt{\Delta\varphi(z)}}\leq
C\rho(z)\ \ \forall\,z\in{\mathbb{D}};$

the function $\rho$ must also satisfy the Hölder-property

(3.14)
$\displaystyle\sup\limits_{z,w\in{\mathbb{D}},z\not=w}\frac{|\rho(z)-\rho(w)|}{|z-w|}<\infty$

as well as the Lipschitz-property

(3.15)
$\displaystyle\forall\,\varepsilon>0\,\exists\,\mbox{compact}\,E\subset{\mathbb{D}}:\
|\rho(z)-\rho(w)|\leq\varepsilon|z-w|\ \forall\,z,w\in{\mathbb{D}}\setminus
E.$

For more details, see [11]. Note that the considerations in [11] are not
restricted to radial weights, contrary to our situation.

According to [11], Theorem 4.3., if the weight $v$ satisfies condition $(I)$,
then the space $H_{v}^{\infty}$ is the dual of $A^{1}_{v}$ with respect to the
dual pairing

(3.16) $\displaystyle\langle
f,g\rangle=\int\limits_{\mathbb{D}}f\overline{g}v^{2}dA.$

The second requirement is the following:

$(II)$ The weight $v$ satisfies the condition $(b)$ of [3], [4].

Recall that the weight $v$ satisfies the condition $(b)$ if there exist
numbers $b>2$, $K>b$ and $0<\mu_{1}<\mu_{2}<\ldots$ with
$\lim_{n\rightarrow\infty}\mu_{n}=\infty$ such that

(3.17) $\displaystyle
b\leq\left(\frac{r_{\mu_{n}}}{r_{\mu_{n+1}}}\right)^{\mu_{n}}\frac{v(r_{\mu_{n}})}{v(r_{\mu_{n+1}})},\left(\frac{r_{\mu_{n+1}}}{r_{\mu_{n}}}\right)^{\mu_{n+1}}\frac{v(r_{\mu_{n+1}})}{v(r_{\mu_{n}})}\leq
K,$

where $r_{m}\in]0,1[$ denotes the global maximum point of the function
$r^{m}v(r)$ for any $m>0$. Theorem 2.4 of [4] states that the solid core of
the space $H_{v}^{\infty}$ equals

(3.18) $\displaystyle
s(H_{v}^{\infty})=\Big{\\{}(b_{k})_{k=0}^{\infty}\,:\,\|b\|_{v,s}=\sup\limits_{n\in{\mathbb{N}}}v(r_{\mu_{n}})\sum_{\mu_{n}<k\leq\mu_{n+1}}|b_{k}|\sigma_{k}<\infty\Big{\\}},$

where we denote $\sigma_{k}=r_{\mu_{n}}^{k}$. Let us define for every
$k\in{\mathbb{N}}_{0}$ the number

(3.19) $\displaystyle
S_{k}=\frac{\int\limits_{0}^{1}r^{2k+1}v(r)^{2}dr}{v(r_{\mu_{n}})\sigma_{k}}\
,$

where $n$ is the unique number such that $\mu_{n}<k\leq\mu_{n+1}$.

######  Example 3.3.

According to [4], all weights
$v(r)=\exp\big{(}-\alpha/(1-r^{2})^{\beta}\big{)}$ with $\alpha,\beta>0$,
satisfy condition $(b)$, and it is easy to see that they also satisfy
assumption $(I)$.

######  Theorem 3.4.

Let the weight $v$ satisfy the assumptions $(I)$ and $(II)$. Then, we have

(3.20) $\displaystyle
S_{BK}(A_{\mu}^{1})=\Big{\\{}b=(b_{k})_{k=0}^{\infty}\,:\,\|b\|_{\mu,S}=\sum_{n=0}^{\infty}\sup_{\mu_{n}<k\leq\mu_{n+1}}|b_{k}|S_{k}<\infty\Big{\\}},$

and the norm $\|\cdot\|_{S}$ given by Proposition 3.1 is equivalent with
$\|\cdot\|_{\mu,S}$.

Proof. Let the solid hull $S_{BK}(A_{\mu}^{1})$ be endowed with the norm
$\|\cdot\|_{S}$ of Proposition 3.1, and let us denote the Banach space on the
right-hand side of (3.20) by $Z$.

We note that by the duality relations explained above (see (3.18) for the
definition of $\|\cdot\|_{v,s}$), we have for all $f\in A_{\mu}^{1}$

(3.21)
$\displaystyle\|f\|_{1}=\sup\limits_{\stackrel{{\scriptstyle\scriptstyle g\in
H_{v}^{\infty}}}{{\scriptstyle\|g\|_{H_{v}^{\infty}}\leq 1}}}|\langle
f,g\rangle|\ \ \mbox{and}\ \
\|f\|_{\mu,S}=\sup\limits_{\stackrel{{\scriptstyle\scriptstyle g\in
s(H_{v}^{\infty})}}{{\scriptstyle\|g\|_{v,s}\leq 1}}}|\langle f,g\rangle|.$

It is proved in [4], equation (2.4) and the very end of the proof of Theorem
2.4, that $\|g\|_{H_{v}^{\infty}}\leq C\|g\|_{v,s}$ for $g\in
s(H_{v}^{\infty})$. Therefore $\|f\|_{1}\geq C\|f\|_{\mu,S}$ for all $f\in
A_{\mu}^{1}$. This implies in particular that $A_{\mu}^{1}\subset Z$. Clearly,
$Z$ is a solid Banach space and the coordinate functionals are continuous,
thus it contains the space $S_{BK}(A_{\mu}^{1})$. Moreover, we obtain
$\|f\|_{S}\geq C\|f\|_{\mu,S}$ for $f\in S_{BK}(A_{\mu}^{1})$ from Proposition
3.1.$(iii)$.

We show that the norms $\|\cdot\|_{\mu,S}$ and $\|\cdot\|_{S}$ are equivalent
in $S_{BK}(A_{\mu}^{1})$. To do this, we prove that
$C\|f\|_{\mu,S}\geq\|f\|_{S}$. Note that the space (3.18) is the dual space of
$Z$ in the dual pairing (3.16). Indeed, if $f=\sum_{k}\hat{f}(k)z^{k}$and
$g=\sum_{k}\hat{g}(k)z^{k}$ are polynomials, then, by a direct calculation,

(3.22) $\displaystyle\langle
f,g\rangle=\sum_{k=0}^{\infty}\hat{f}(k)\overline{\hat{g}(k)}\int\limits_{0}^{1}r^{2k+1}v(r)^{2}dr.$

The result follows from (3.9)–(3.10), in addition to the definitions
(3.16)–(3.20).

Suppose now by antithesis that $\|\cdot\|_{\mu,S}$ and $\|\cdot\|_{S}$ are
non-equivalent norms so that we can find a sequence
$(f_{n})_{n=1}^{\infty}\subset S_{BK}(A_{\mu}^{1})$ such that

(3.23) $\displaystyle\|f_{n}\|_{\mu,S}\leq 2^{-n}\|f_{n}\|_{S}\ \ \mbox{and}\
\ \|f_{n}\|_{S}=1\ \forall\,n\in{\mathbb{N}}.$

By property $(v)$ in Proposition 3.1 we can assume that $f_{n}$’s are
polynomials. We claim that it is possible to find polynomials $\tilde{f}_{n}$,
$n\in{\mathbb{N}}$, with property (3.23) such that they have distinct degrees,
more precisely

(3.24)
$\displaystyle\tilde{f}_{n}(z)=\sum_{k=K_{n}}^{K_{n+1}-1}\hat{f}(n,k)z^{k},\ \
n\in{\mathbb{N}},$

for some unbounded sequence $0=K_{0}<K_{1}<\ldots$ and some
$\hat{f}(n,k)\in{\mathbb{C}}$. Assume that $N\in{\mathbb{N}}$ and that such
polynomials $\tilde{f}_{n}$ have been found for $n\leq N$, and let
$M\in{\mathbb{N}}$ be the highest degree of these polynomials. Since
${\mathcal{P}}_{M}$ (the $M+1$-dimensional space of polynomials of degree at
most $M$) is finite dimensional, all norms are equivalent there and we thus
find a constant $K=K(M)>0$ such that

(3.25) $\displaystyle\|f\|_{S}\leq K\|f\|_{\mu,S}$

for all $f\in{\mathcal{P}}_{M}$. We pick up the polynomial $f_{L}$ as in
(3.23) with $L=M+K$ and write $f_{1}=P_{M}f_{L}$, $f_{2}=f_{L}-f_{1}$, where
$P_{M}$ is the $M$th Dirichlet projection from $S_{BK}(A_{\mu}^{1})$ onto
${\mathcal{P}}_{M}$, see Section 1. Then, we have
$\|f_{2}\|_{S}\geq\frac{1}{2}\|f_{L}\|_{S}$, since otherwise we get by (3.25)
and the triangle inequality

$\displaystyle\|f_{L}\|_{\mu,S}\geq\|f_{1}\|_{\mu,S}\geq\frac{1}{K}\|f_{1}\|_{S}\geq\frac{1}{2K}\|f_{L}\|_{S}>\frac{1}{2L}\|f_{L}\|_{S}$

which contradicts with (3.23). Now we get

(3.26) $\displaystyle\|f_{2}\|_{\mu,S}\leq\|f_{L}\|_{\mu,S}\leq
2^{-L}\|f_{L}\|_{S}\leq 2^{-L+1}\|f_{2}\|_{S}.$

Taking $f_{2}\|f_{2}\|_{S}^{-1}$ for $\tilde{f}_{N+1}$, the claim is proved.

Finally, for every $n$ we set

(3.27) $\displaystyle
T_{n}:=\Big{(}P^{(n)}\big{(}S_{BK}(A_{\mu}^{1})\big{)},\|\cdot\|_{S}\Big{)}\ \
\mbox{with}\ \ P^{(n)}=P_{K_{n+1}-1}-P_{K_{n}}$

and then, using the Hahn-Banach theorem, pick up a polynomial

$g_{n}=\sum_{k=K_{n}}^{K_{n+1}-1}\hat{g}(n,k)z^{k}$

which defines a bounded functional on $(T_{n},\|\cdot\|_{S})$ with respect to
the dual pairing (3.22), such that

(3.28) $\displaystyle\langle\tilde{f}_{n},g_{n}\rangle=1,\ \
\|g_{n}\|_{n,*}:=\sup\limits_{\stackrel{{\scriptstyle\scriptstyle f\in
T_{n}}}{{\scriptstyle\|f\|_{S}\leq 1}}}|\langle f,g_{n}\rangle|=1$

Then, we observe that $g_{n}$ extends via (3.22) to a functional on
$S_{BK}(A_{\mu}^{1})=:S$ such that

(3.29) $\displaystyle\sup\limits_{\stackrel{{\scriptstyle\scriptstyle f\in
S}}{{\scriptstyle\|f\|_{S}\leq 1}}}|\langle
f,g_{n}\rangle|=\sup\limits_{\stackrel{{\scriptstyle\scriptstyle f\in
S}}{{\scriptstyle\|f\|_{S}\leq 1}}}|\langle
P^{(n)}f,g_{n}\rangle|=\sup\limits_{\stackrel{{\scriptstyle\scriptstyle f\in
T_{n}}}{{\scriptstyle\|f\|_{S}\leq 1}}}|\langle f,g_{n}\rangle|=1,$

since the norm $\|\cdot\|_{S}$ of $S_{BK}(A_{\mu}^{1})$ satisfies $(ii)$ of
Proposition 3.1 and thus $\|P^{(n)}f\|_{S}\leq\|f\|_{S}$ for all $f\in
S_{BK}(A_{\mu}^{1})$. Consequently,

(3.30) $\displaystyle g=\sum_{n\in{\mathbb{N}}}\frac{1}{n^{2}}g_{n}$

is an analytic function which also is a bounded functional on
$\big{(}S_{BK}(A_{\mu}^{1}),\|\cdot\|_{S}\big{)}$ in the dual pairing (3.22).
However, $g$ is not a bounded functional on $Z$, since

(3.31) $\displaystyle\langle
2^{n}\tilde{f}_{n},g\rangle=\frac{2^{n}}{n^{2}}\langle\tilde{f}_{n},g_{n}\rangle=\frac{2^{n}}{n^{2}}$

and by (3.23), the $Z$-norm $\|2^{n}\tilde{f}_{n}\|_{\mu,S}$ is still at most
1.

The space $Y$ of all analytic functions on ${\mathbb{D}}$, which also are
bounded functionals on $\big{(}S_{BK}(A_{\mu}^{1}),\|\cdot\|_{S}\big{)}$ in
the dual pairing (3.22), equals the space $Y$ in Lemma 3.2, when
$X:=S_{BK}(A_{\mu}^{1})$. Hence, $Y$ is solid. Due to the characterization of
$H_{v}^{\infty}$ as the dual of $A_{v}^{1}$, see (3.16), we also have
$Y\subset H_{v}^{\infty}$. On the other hand, we observed in the beginning of
the proof that the solid core $s(H_{v}^{\infty})$, see (3.18), equals the dual
of $Z$ in the pairing (3.22). The properties of the function $g$, (3.30), show
that $s(H_{v}^{\infty})\subsetneq Y$, which contradicts the definition of a
solid core. We conclude that $C\|f\|_{\mu,S}\geq\|f\|_{S}$ for all $f\in
S_{BK}(A_{\mu}^{1})$.

We come to the conclusion that the norms $\|\cdot\|_{S}$ and
$\|\cdot\|_{\mu,S}$ are equivalent, hence, the spaces $S_{BK}(A_{\mu}^{1})$
and $Z$ coincide, since they both are complete. $\Box$

Statement on Compliance with Ethical Standards:

The research of Bonet was partially supported by the project MCIN
PID2020-119457GB-I00/AEI/10.13039/501100011033.

There are no conflicts of interest as regards to this article.

The research has been ethically conducted.

The research does not involve human participants and/or animals.

Data Availability Statements: n/a

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