# Leveraging Cloud Computing to Make Autonomous Vehicles Safer

Peter Schafhalter1 Sukrit Kalra1 Le Xu2 Joseph E. Gonzalez1 Ion Stoica1 1 UC
Berkeley 2 UT Austin

###### Abstract

The safety of autonomous vehicles (AVs) depends on their ability to perform
complex computations on high-volume sensor data in a timely manner. Their
ability to run these computations with state-of-the-art models is limited by
the processing power and slow update cycles of their onboard hardware. In
contrast, cloud computing offers the ability to burst computation to vast
amounts of the latest generation of hardware. However, accessing these cloud
resources requires traversing wireless networks that are often considered to
be too unreliable for real-time AV driving applications.

Our work seeks to harness this unreliable cloud to enhance the accuracy of an
AV’s decisions, while ensuring that it can always fall back to its on-board
computational capabilities. We identify three mechanisms that can be used by
AVs to safely leverage the cloud for accuracy enhancements, and elaborate why
current execution systems fail to enable these mechanisms. To address these
limitations, we provide a system design based on the speculative execution of
an AV’s pipeline in the cloud, and show the efficacy of this approach in
simulations of complex real-world scenarios that apply these mechanisms.

##  I Introduction

Autonomous Vehicles (AVs) are predicted to make our roads significantly safer
by eliminating the vast majority of traffic accidents that contributed to
$38,824$ fatalities in $2020$ in the U.S. alone [1]. However, to fulfill their
potential, AVs must surpass the safety levels of human drivers and handle a
diverse set of challenging driving scenarios [2, 3]. Prior works [4, 5, 6]
have demonstrated the inextricability of driving safety with the accuracy of
the AV’s computational pipeline, which is governed by the following three
characteristics:

High-fidelity data. Integrating data from a wide array of high-fidelity
sensors can significantly enhance computational accuracy [7]. As a result,
newer generations of AVs seek to expand the number and quality of onboard
sensors, such as cameras, LiDARs, and radars. For example, Waymo’s $5$th-
generation AVs feature $29$ cameras as opposed to $19$ in the $4$th-
generation. These additional sensors form a new peripheral vision system that
bolsters safety by providing the AV with higher quality data at higher
frequencies [8, 9].

State-of-the-art computation. The increasing amounts of available data need to
be processed by state-of-the-art algorithms and models to ensure highly-
accurate results. However, more accurate algorithms and models often come at
the cost of increased parameter sizes and more floating point operations
(FLOPs) [10, 11, 12]. Notably, the increased scale of modern deep neural
networks (DNNs) forms the “primary ingredient“ in achieving state-of-the-art
accuracy [13]. Innovative solutions to reduce the computational requirements
of DNNs [14, 15] have usually resulted in reduced accuracy, which hampers the
AV’s safety in complex environments.

Timely results. To surpass the safety levels of human drivers, an AV must
respond more quickly than humans [16], whose reaction times vary from $390$ ms
[17] to $1.2$ s [18] depending on factors such as road conditions, driver
attentiveness, age, etc. To achieve these reaction times and ensure safety in
challenging driving scenarios, the AV’s computational pipeline must operate in
“real-time” to meet latency goals when executing state-of-the-art computation
on high-fidelity data.

Thus, to drive safely, AVs must produce highly-accurate and timely results
using state-of-the-art algorithms and models that consume high-fidelity sensor
data. The combination of these characteristics requires AVs to exploit the
compute capabilities of cutting-edge hardware. However, the deployment of such
hardware in an AV is constrained by its cooling, power, and stability
requirements [16]. For example, the DRIVE platform [19], NVIDIA’s flagship
hardware for AVs, is updated every 3 years. Its most recent iteration, the
DRIVE Orin [20], was put into production vehicles in 2023 and its successor,
the DRIVE Atlan [21], is slated for release in 2026 [22]. Moreover, upgrading
the hardware of previously-deployed AVs is often infeasible due to the cost
and complexity involved with a recall [23]. As a result, modern AVs are forced
to trade-off accuracy, and hence, safety, for computational resources and
timely results, by either reducing the amount of data fused from multiple
sensors, or deploying algorithms and models that require a lower number of
parameters and FLOPs [2].

In light of this fundamental mismatch between the pace of development of
compute technologies with the update cycles of vehicles [24], we propose to
augment the computational resources in an AV with the compute capabilities of
the cloud. Cloud computing platforms provide the illusion of infinite
computing resources [25], and enable low-cost access to state-of-the-art
hardware [26, 27, 28]. In contrast to the $3$-year update cycle in AVs, the
hardware and software in the cloud is frequently updated [29, 30]. For AVs,
the cloud enables the deployment of compute-intensive, rapidly-evolving
algorithms and models which can exploit state-of-the-art hardware without
requiring complex recalls for hardware and software updates [31, 32]. As a
result, AVs can enhance safety by executing highly-accurate algorithms and
models on powerful cloud resources to provide timely results.

Despite the potential benefits of the cloud, its practical use in AVs has
remained largely unexplored due to concerns regarding the limited bandwidth of
the network available on AVs and the high, often unpredictable latency of
cloud systems. In this paper, we evaluate the efficacy of augmenting the
computational resources on-board AVs with the unreliable resource pool of the
cloud. Specifically, we seek to enhance the accuracy of an AV’s decisions by
harnessing the capabilities of the cloud, while ensuring that the AV can
always fall back to its on-board computational capabilities in order to
strictly exceed its current safety standards. To this effect, the paper makes
the following key contributions:

  * (1)

We identify three mechanisms that can be used by AVs to leverage the cloud for
safety enhancements (§ III).

  * (2)

We identify limitations in the ability of current execution systems to achieve
these safety-enhancement mechanisms (§ IV), and propose a system design based
on speculative execution of computation in the cloud that enables their
efficient execution and maximizes accuracy (§ V).

  * (3)

We evaluate the efficacy of our design under current cloud and network
capabilities using the three mechanisms on complex scenarios from the NHTSA
crash scenarios (§ VI).

We argue that the network and compute trends are in favor of the feasibility
of our approach, and discuss specific challenges that must be resolved by the
community to enable AVs to reap the complete safety benefits of the cloud (§
VII).

Figure 1: A modular AV pipeline with multiple sensors feeding data to the
perception, prediction, planning and control modules.

##  II Background & Motivation

The prevalent design of an AV’s computation is a modular multi-stage pipeline
(Fig. 1) [33, 34, 4, 16, 35, 36] where the inputs from the vast array of
sensors (e.g., cameras, LiDARs, radars, etc.) are processed by the perception
and localization modules. The perception module detects and tracks nearby
objects of interest, such as pedestrians, vehicles, traffic lights, etc. By
fusing the perception results with the location from the localization module,
the AV pipeline constructs an internal representation of the environment. The
prediction module forecasts the behavior of non-static objects (i.e.,
pedestrians, vehicles) which enables the planning module to compute a safe and
comfortable trajectory for the AV to follow. Finally, the control module
converts the trajectory plan into steering and acceleration commands which are
applied to the AV.

TABLE I: runtime disparity between av hardware and cloud hardware Model | Runtime [ms] | Speedup
---|---|---
| Orin | A100 |
DETR-ResNet-50 | $301.7$ | $102.2$ | $2.95{\times}$
DETR-ResNet-101 | $407.7$ | $118.2$ | $3.45{\times}$
DETR-ResNet-101-DC | $859.2$ | $146.6$ | $5.86{\times}$
DINO-SWIN-Tiny | $722.1$ | $90.1$ | $8.01{\times}$
DINO-SWIN-Small | $903.5$ | $107.1$ | $8.43{\times}$
DINO-SWIN-Large | $1529.9$ | $180.8$ | $8.46{\times}$

To compute safe control commands, AVs depend on high-fidelity data from the
sensors and timely results from its algorithms and models (see § I). However,
due to limitations in the power of the hardware relative to the amount of
sensor data generated, AV developers must make decisions about about _which_
data to process using _what_ algorithms and _when_ to return results to ensure
the safety of the vehicle. For example, Baidu’s Apollo AV [36] only utilizes
specialized cameras for detecting traffic lights when the AV is notified of
the presence of a nearby traffic light via a pre-computed map of the city
[37]. While this strategy makes efficient use of the hardware resources on-
board, the strategy fails to detect temporary traffic signals installed by
construction or emergency vehicles, which may lead to unsafe driving
scenarios. Augmenting AV compute resources with the cloud will enable more
comprehensive data processing which avoids discarding data due to resource
limitations, thus improving safety without impacting the critical-path
computation.

After deciding _which_ data to process, AVs must decide _what_ algorithms and
models to execute. As discussed in § I, AVs must often compromise the accuracy
of its algorithms and models due to resource constraints stemming from the
hardware available on-board. Even after accounting for the slow upgrade cycle
which often puts the hardware in AVs several revisions behind the state-of-
the-art in the cloud, the hardware available in the cloud is much more
powerful than the hardware available in AVs, even if the architecture and the
revision are the same. For example, the NVIDIA DRIVE Orin platform [20], which
is slated to be available in production vehicles in 2023 and is the latest
revision supported by Baidu’s Apollo AV [38], uses NVIDIA’s Ampere
microarchitecture to deliver a performance of $5.2$ FP$32$ TFLOPs [39]. The
equivalent Ampere cloud GPU is the NVIDIA A$100$, which was released in 2020
and delivers a performance of $19.5$ FP$32$ TFLOPs [40], a $3.75{\times}$
increase over the DRIVE Orin. Table I measures the effects of this disparity
by executing open-source implementations of DETR [41, 42] and SWIN [10, 43],
two state-of-the-art vision transformers which have been adapted for object
detection, atop both the NVIDIA Orin and the A$100$. We observe a significant
speedup of up to $8.4{\times}$ by executing the same model on the same input
on a cloud GPU. Similar trends have been shown by prior work for other safety-
critical algorithms applicable to AVs, such as motion planning [44].

Finally, an AV must decide by _when_ to compute the control commands at the
end of the pipeline, thus mandating a deadline on the computation being
executed. For example, to surpass human driving in safety, the pipeline must
execute faster than human reaction times. In addition, prior works [45, 46, 2]
indicate this deadline varies widely according to the environment around the
AV. Intuitively, driving slowly on a crowded urban street may tolerate a lax
deadline for the computation to finish, whereas swerving to prevent an
accident on the freeway requires the AV to respond quickly [47, 48]. This
variability in the required response time of the computation further
complicates the deployment of more accurate, but slower algorithms and models
since they cannot meet tighter deadlines [2].

##  III Enhancing Safety Using The Cloud

Conventional wisdom suggests that the latency and availability of cellular
network connections makes using the cloud on the critical-path of the
computation infeasible [49]. However, Table I presents an arbitrage
opportunity whereby an AV could potentially return more accurate results
faster by exploiting the computational power of the hardware in the cloud (see
Fig. 2). We argue that AVs should instead take a best-effort speculative
approach to leverage the cloud when it is available and utilize reliable
fallback mechanisms that use on-board computation when the cloud is not
immediately accessible. We discuss three such mechanisms that enable a best-
effort augmentation of an AV’s safety below:

Higher-accuracy models. AVs can selectively offload data from their input
sensors to the cloud, allowing higher accuracy models to be executed in the
cloud. For example, in Table I, an AV can choose to execute DETR-
ResNet-$101$-DC in the cloud on its camera data, which provides a
${\sim}3$-point increase in average precision over DETR-ResNet-$50$ [41] and
executes faster. Thus, while an AV with an end-to-end deadline of $500$ ms can
only execute DETR-ResNet-$50$ on its local hardware, it can optimistically
exploit the accuracy of all models in Table I when the latency to the cloud is
low.

Accurate environment representation. While the earlier mechanism significantly
enhances an AV’s ability to process sensor data and understand its
surroundings, this mechanism does not apply to obstacles obscured by the AV’s
blind spots. To improve safety in these scenarios, AVs can share their
locations computed by the localization module to the cloud and subsequently
retrieve the locations of other nearby vehicles. Integrating the locations of
nearby vehicles via the cloud allows the planning module to generate safer
trajectories which avoid collisions with vehicles located in blind spots.

Figure 2: An arbitrage opportunity afforded by exploiting state-of-the-art
hardware to execute higher-accuracy models at a lower latency.

Contingency planning. During the course of computation, AVs must make
probabilistic decisions which affect the outputs of the modules. For example,
object detectors are configured with a confidence threshold to filter out
misdetections from the model’s outputs [42]. Similarly, the prediction module
generates several possible trajectories for nearby obstacles, and ranks them
based on their probability of occurring [50, 51]. The planning module then
uses the most likely trajectory of the obstacles to plan a trajectory for an
AV to follow. However, AVs can offload the computation of plans that handle
unlikely object trajectories to the cloud. When an object takes an unlikely
trajectory, AVs can access the corresponding cloud-computed plan, enabling
quick reactions to any sudden changes in the environment.

§ VI evaluates the efficacy of these mechanisms to ensure the safety of an AV
under real-world complex scenarios.

##  IV Related Work

Our principle design objective, as per § III, is to use the cloud as a best-
effort pool of resources in order to improve the accuracy of AV decision-
making while retaining the ability to fall back to locally-computed results to
handle delays and variability. We now discuss the feasibility of implementing
such an approach using existing execution systems.

The Robot Operating System (ROS) [52] is the current execution system-of-
choice for AVs and has been deployed by vendors including Autoware [35],
Cruise [53], BMW [54], and others [55, 56]. ROS’ highly-modular design enables
various initiatives to provide additional feature enhancements, such as real-
time support [57]). One such initiative which aims to add cloud support to ROS
is FogROS [44, 58]. FogROS allows users to designate ROS nodes that must
execute in the cloud and specify their resource requirements. FogROS then
provisions suitable cloud instances, configures the network connection between
the cloud and the local machine, and registers the cloud nodes with the local
coordinator, enabling seamless communication between the cloud and local
nodes.

While FogROS greatly simplifies the deployment of specific ROS nodes to the
cloud, it lacks out-of-the-box support for fall-back mechanisms that switch to
local computation when the connection to the cloud becomes too unreliable.
Moreover, implementing such an approach atop ROS’ callback-execution model is
a complex, tedious, and error-prone process because developers must implement
and manage varying deadlines (see § II) using fine-grained wall-clock timers
on every node [59]. Because these fine-grained timers may cause priority
inversions when they rapidly produce events which overfill a
SingleThreadedExecutor’s queue, developers must implement ROS nodes using
MultiThreadedExecutors which requires manually managing shared state using
locks in order to ensure the safe and correct fusion of results from the cloud
and from local computation. This complexity further increases if nodes attempt
to maximize the accuracy of computation under a deadline by concurrently
executing a mixture of algorithm and model implementations.

In addition to ROS, several related works have proposed designs that allow AVs
to utilize the cloud, however, these approaches focus on specific tasks such
as merging sensor data and intermediate representations across a fleet of
vehicles [60, 61, 62, 63, 64]. For example, both Carcel [60] and EMP [64]
enable a fleet of vehicles to create an accurate representation of their
environment by selectively sharing their data to an edge server while
remaining resilient to network fluctuations. Other work (e.g., Neurosurgeon
[65]) has examined _how_ to split computation between the cloud and the AV
[66, 67, 68]. However, none of these works propose a general system design for
AVs that falls back to results computed by on-board hardware in order to
handle faults in the cloud or the network. Therefore, we believe that these
works are complementary, and that AVs can integrate their methods with our
design in order to gain further accuracy improvements while maintaining a
reliable baseline accuracy through the ability to fall back to results from
local computation.

##  V Design

Given the limitations of the current systems, we seek to design an execution
system that enables AV developers to easily achieve our design goal of
augmenting the computation’s accuracy with the vast pool of resources in the
cloud. To ensure that the programming model remains familiar to the current AV
systems (e.g., ROS), our system models an AV pipeline as a directed graph in
which vertices, referred to as operators are akin to ROS nodes, and are
connected to other vertices via streams. The streams are statically-typed and
enforce an interface between the sender (akin to a ROS publisher) and a
receiver (akin to a ROS sender).

However, to safely exploit the cloud resources in the presence of network
delays and unavailability of cloud resources, our system allows operators to
define deadlines that bound the time which can elapse between the transmission
of a particular request to the cloud and the retrieval of its result. To do
so, the system must enable each operator to: (i) specify the computation that
executes locally and the computation that can execute in the cloud, (ii)
control what input data is transmitted to the cloud and what deadline is
assigned to its completion, and (iii) fuse the inputs from the cloud and the
on-board hardware to ensure that the maximum accuracy results available by the
deadline are used. We emphasize that our system design does not automatically
decide which models and algorithms are suitable for cloud execution, but
rather provides the abstractions and mechanisms to enable augmenting local
computation with the cloud in order to benefit overall accuracy. Fig. 3
provides an overview of our approach, and we now elaborate on how our system
achieves each of the aforementioned aspects below:

1

Operator configuration. Each operator must implement a setup method in which
the operator informs the system if its computation is to be assisted by a
cloud implementation. To do so, the operator initiates a connection to the
cloud for dispatching remote procedure calls (RPCs) and invokes the use_cloud
method in our system with the following API:

`use_cloud(handle, type, msg_handler, priority)`

The operator registers the handle to the RPC connection with the system along
with the type of the message to be relayed. Moreover, the system allows an
operator to invoke use_cloud multiple times in order to take advantage of
multiple cloud implementations which may span different cloud providers and
exploit vendor-specific hardware to provide the highest-accuracy results
within the given deadline [69]. By setting the priority, the system ensures a
total order over the results of the invocations in order to select which of
the potential results to output. For example, an object detection operator may
invoke several models from Table I in the cloud with the hope that one of them
returns a result by the given deadline, and may set their priorities according
to each detector’s accuracy on a static dataset in order to favor results from
more accurate detectors. Invoking this API informs the underlying execution
system that it needs to set up the mechanisms detailed below, which ensure the
correct collation of results from the cloud and the local execution.

Figure 3: Our speculative cloud execution design allows each operator to make
fine-grained decisions about when to contact the cloud and how much time to
allocate for a response. The results from the cloud are automatically
incorporated to ensure the highest-possible accuracy.

2

Deadline enforcement. While the execution of non-cloud enabled operators
continues as normal, cloud-enabled operators must be able to specify a
deadline on the time that they are able to wait for a higher-accuracy result
from one of the cloud implementations registered with use_cloud. To do so,
each invocation of use_cloud requires a msg_handler function that conforms to
the following API: `(input_message, timestamp) ->`
` Optional[output_message, deadline]`

For each message that the operator receives, the system invokes the registered
msg_handlers which may use the message content and the timestamp to decide
whether to send the message to the handler’s cloud implementation and what
deadline to assign to this request. Unlike ROS timestamps that correspond to
physical time, this timestamp is an internal logical representation of time
that corresponds to a counter of the messages received by the operator. In
addition to deciding whether to offload the computation of the messages to the
cloud based on its contents [46, 70], the handler functions can use the
logical timestamps to enforce simple policies such as sending non-consecutive
inputs to the cloud to reduce bandwidth usage, or alternating between
different implementations to avoid bottlenecks in the cloud service and ensure
fault-tolerance for the operator.

The operator makes the system aware of its intent to transmit a message to the
cloud service by returning an output_message from the handler. The
output_message must match the message definition in the RPC handle registered
with use_cloud, but can be different from the type of the input_message. This
type differentiation allows the operators to bundle extra state to the cloud
platform that is not communicated to other operators. For example, the object
tracking operator in the perception module assigns unique identifiers to
detected obstacles based on their past history. As a result, selectively
executing inputs to this operator in the cloud also requires transmitting the
state of the identifiers generated by the local object tracker to allow the
cloud-based object tracker to correctly assign identifiers.

In addition to the message to be conveyed to the cloud service, the handler
decides the relative deadline $d$ to be assigned to the arrival of its result.
The ability to return a new deadline $d$ for each incoming message enables the
system to adjust to varying deadlines required by the AV to respond to
dynamicity in the environment [45, 46, 2]. Given the deadline and the message,
the system decides if the request should be sent based on the length of its
outgoing queues, which fill up when network’s quality of service degrades. If
the system elects to send the request, the system indexes the request by its
timestamp and priority and installs a timer that triggers $d$ milliseconds
from the current time.

3

High-accuracy results. In parallel to executing the request in the cloud, the
system invokes the callbacks registered for the incoming message and executes
them on the local hardware with the lowest priority $p$. Upon completion, the
local callbacks invoke send with the output message to convey the results to
downstream operators. However, the system intercepts the invocation of the
send to check for pending cloud requests with the same timestamp and a higher
priority $p^{\prime}>p$. If a pending cloud request is found, the system
caches the local result and waits for the request’s timer to interrupt as
detailed below. However, if no pending cloud request is found and the system
already forwarded a message from a cloud execution to the downstream operators
(i.e., the result from the cloud arrives before the local computation
finishes), the local result is dropped. Otherwise, the message containing the
local result is sent to the downstream operators.

Upon arrival of a message from the cloud with priority $p$ and timestamp $t$,
the system checks if the message missed its deadline by querying the status of
the timer installed for ($t$, $p$). If the timer was triggered, the message
missed its deadline and is dropped. Similarly, if the system already sent a
message for $t$ (i.e., a higher priority request returned more quickly or
another request’s timer triggered earlier), the message is dropped and the
installed timer is deactivated. Otherwise, if no higher priority cloud request
is pending and the system has not sent a message for $t$, the message is sent
to the downstream operators and the installed timer is deactivated. However,
if there is a pending cloud request with a higher priority, the current
results are cached.

Finally, if an active timer triggers, the system receives an interrupt in
which it checks its cache for the highest priority message presently available
for the timestamp and forwards the message to the downstream operators.

##  VI Evaluation

We now evaluate the efficacy of our approach in augmenting the safety of
cloud-assisted AVs under complex, real-world scenarios while maintaining the
accuracy of an AV using only on-board computation. We evaluate our design to
demonstrate that our approach to augmenting AVs Specifically, we seek to
answer the following questions:

  1. 1.

Do current technologies support a low-latency and reliable connection to the
cloud? (§ VI-A)

  2. 2.

Does our system enable the three techniques (§ III) to exploit cloud resources
to enhance AV safety? (§ VI-B)

Figure 4: Cellular network latency of a 5G connection while driving through a
route in San Francisco frequented by Waymo and Cruise AVs.

###  VI-A Feasibility of Cloud Access

We investigate whether modern cellular networks are able to provide the speed
and bandwidth necessary to execute data-intensive AV operators. To measure
network performance in realistic setting, we conduct a field test by following
a route in San Francisco where Cruise and Waymo already provide fully
autonomous rides [71, 72]. The route contains both urban and highway driving
and is visualized in Fig. 4. We discuss our experiment setup and the findings
below.

Experiment Setup. We model the transmission of HD camera footage from an AV to
the cloud. In our vehicle, we connect a Lenovo ThinkPad P1 Gen 2 laptop to an
Inseego MiFi X PRO 5G hotspot on the Verizon network via USB-C. On a Google
Pixel 5, we collect timestamped GPS coordinates at $1$ Hz. The laptop executes
a multithreaded gRPC [73] client which sends $33.3$ KB messages at $30$ Hz to
a server to match the bitrate of 30 FPS HD camera footage [74]. We establish a
connection to a Google Cloud Platform n1-highmem-8 instance in the us-west2-a
zone which executes a gRPC server that responds to messages from the client
with $1$ KB acknowledgments. The client measures the round-trip latency of
sending a message to receiving an reply. Because network delays may cause the
client to queue messages and overstate the network latency, the client waits
for all messages to process if it detects more than 30 queued messages.

Findings. The 5G network in San Francisco frequently provides latencies that
enable cloud execution. We measure a median round-trip-latency of $68$ ms
(Fig. 4) which demonstrates an opportunity to take advantage of the hundreds
of milliseconds in runtime disparity between cloud and AV hardware (Table I).
In addition, the long tail of network latencies from $336$ ms at the $90$th
percentile to $3027$ ms at the $99$th percentile substantiates the need to
manage network delays. In § VI-B, we demonstrate that our design takes
advantage of the fast common-case latencies to enable critical safety benefits
while mitigating the impact of long tail-end latencies by falling back to
results from local computation.

###  VI-B Study of Scenarios

We study of the safety benefits of our design (§ V) and how it enables the
three mechanisms which use the cloud to improve AV safety (§ III). For each
mechanism, we demonstrate its efficacy under a complex, real-world scenario
executed using the CARLA simulator [75]. We use the pseudo-asynchronous mode
of execution from the Pylot AV platform [4] to simulate the delay of
calculating a demand for different end-to-end deadlines, retrieved from
different end-to-end deadlines, retrieved from Fig. 4111Videos of scenarios
are available at https://tinyurl.com/26fzrabu.

Figure 5: Traffic Jam Scenario leverages the cloud’s ability to run higher-
accuracy models at a reduced latency to reduce the AV’s response time, thus
minimizing its reaction time and avoiding a collision with the motorcycle.

Figure 6: Running a Red Light Scenario leverages the cloud’s ability to build
accurate environment representations to detect the occluded vehicle running
the red light in time to avoid a collision.

Traffic Jam. This scenario from [2] simulates merging into a traffic jam,
visualized in Fig. 5. The AV drives at a high speed on a two lane undivided
road and must come to a halt behind the motorcycle stopped in the distance.
The motorcycle complicates this scenario because the AV must perceive the
stopped obstacle from afar to prevent a collision. Moreover, the AV cannot
swerve to avoid a collision due the vehicles in the opposite lane. Since this
scenario requires both far-away detections and rapid responses due to the high
driving speed, the technique of exploiting the cloud to run higher-accuracy
models quickly ensures maximum safety.

To evaluate safety, we use a simple planner that brakes once the AV’s object
detection operator identifies the obstacle on three consecutive camera frames.
We then investigate the following three operator configurations (Table II):

  * (1)

Local which executes DETR-ResNet-$50$ on a local NVIDIA Orin GPU. We choose
DETR-ResNet-50 since it is the only model that provides response times
required to ensure human-level safety on the local GPU (Table I and § I).

  * (2)

Cloud which executes DETR-ResNet-$101$ on an NVIDIA A$100$ GPU running on a
Google Cloud a2-highgpu-1g instance. We choose DETR-ResNet-$101$ to ensure
that the local and cloud models belong to the same architecture.

  * (3)

Ours which enables operators to specify deadlines on response time from the
cloud and fall back to local results when the cloud is unable to meet the
deadline.

We sweep the entire range of driving speeds in California (i.e., $25$ mph to
$65$ mph), and simulate cloud response times up to the p$99$ latency collected
from our drive through San Francisco (§ VI-A). We note that the higher cloud
response times do not apply to the Local approach. We find that at higher
speeds (e.g., $22$ m/s), the lower-latency access to higher-accuracy models
afforded by the cloud is critical in ensuring the safety of the AV. However,
without appropriate mechanisms to fall back to local computation using
deadlines as proposed in our design, the Cloud approach incurs more safety
violations than the Local approach when network latency is high (e.g., a $3$
second cloud response time when the AV drives at $18$ m/s).

Fig. 5 investigates how executing a higher-accuracy object detector locally
affects collision-avoidance in high-speed scenarios. We compare a Local
execution of DETR-ResNet-$101$ to a Cloud execution with median network
latency. We find that the local execution fails to break in time due to its
$222$ ms longer response time, resulting in a collision.

Running a Red Light. This scenario from the NHTSA pre-crash typology [76, 77]
simulates a vehicle running a red light which forces the AV to perform a
collision avoidance maneuver. The AVs can only perceive the other vehicle just
before a potential collision, challenging its ability to respond in time. We
evaluate three variations of this scenario with different intersections, and
find that neither local nor cloud executions of object detectors are able to
avoid a collision.

However, using the cloud’s ability to build an accurate environment
representation (e.g., by sharing location data with nearby AVs), allows the AV
to plan its trajectory with other vehicles in its blind spots. This enables
the AV to brake early and avoid a collision in this scenario.

TABLE II: Configurations that avoid a collision are marked in green, while configurations that collide are marked in red. Speed |  Approach | Cloud Response Time [s]
---|---|---
[m/s] |  |  0.5 |  0.75 |  1.0 |  1.25 |  1.5 |  3.0
11 |  Local |  |  |  |  |  |
|  Cloud |  |  |  |  |  |
|  Ours |  |  |  |  |  |
18 |  Local |  |  |  |  |  |
|  Cloud  |  |  |  |  |  |
|  Ours |  |  |  |  |  |
20 |  Local |  |  |  |  |  |
|  Cloud  |  |  |  |  |  |
|  Ours |  |  |  |  |  |
22 |  Local  |  |  |  |  |  |
|  Cloud |  |  |  |  |  |
|  Ours |  |  |  |  |  |
24 |  Local  |  |  |  |  |  |
|  Cloud |  |  |  |  |  |
|  Ours |  |  |  |  |  |

Person Jaywalking. This scenario simulates a person unexpectedly entering the
street, requiring the AV to quickly respond in order to avoid a collision.
Fig. 7 evaluates the scenario in which an AV executes its planning module
locally, which has a $500$ ms end-to-end response time. Because the pedestrian
enters the street when the AV is only $10$ m away, the AV cannot generate an
emergency swerving maneuver or stop in time, resulting in a collision with the
pedestrian.

However, we perform contingency planning using the cloud which generates a
plan for the low-likelihood case that the pedestrian enters the street,
downloads the plan to the AV, and caches the plan in the AV’s planner. When
the cloud-assisted AV detects the pedestrian entering the street, the AV
enacts the cached contingency plan and bypasses the local planner, lowering
the response time to $400$ ms. We observe that the cloud-computed contingency
plan enables the AV to swerve in time and successfully avoid a collision.

Figure 7: Person Jaywalking Scenario leverages the cloud’s ability to do
contingency planning for the unlikely case that the pedestrian enters the
street, allowing the AV to use the cached plan quickly and avoid a collision.

##  VII Discussion & Conclusion

Our experiments show that AVs can leverage the cloud with current technology
to exploit safety benefits. We present a design which ensures that cloud
execution strictly improves safety by executing models and algorithms
speculatively in the cloud. When the cloud fails to produce results by a
deadline, our design falls back to results from local execution which are
computed concurrently. Although our design focuses on the enforcement of
provided deadlines, the calculation of deadlines for cloud execution presents
an exciting opportunity for further exploration.

Moreover, trends in technology demonstrate that AVs of the future will harness
the benefits of the cloud. AVs increasingly rely on higher-fidelity data
collected from more sensors which they process using larger and more compute-
intensive algorithms and models. Meanwhile, cloud and network technologies
continue to improve, and will present more arbitrage opportunities in which
the cloud can benefit safety. We hope that this research will bring attention
to the potential advantages of augmenting AVs with the cloud and inspire
further exploration and development in this area.

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# Non-Volterra property of some class of compact operators

B.N. Biyarov

Key words: Perturbations, Schatten-von Neumann class, Laplace operator,
maximal (minimal) operator, Volterra operator, Volterra well-defined
restrictions and extensions of operators, elliptic operator.

AMS 2010 Mathematics Subject Classification: 47A05, 47A10.

Abstract. The authors Matsaev and Mogulskii singled out a wide class of weak
perturbation of a positive compact operator $H$, of the form $H(I+S)$, where
$S$ is such a compact operator that $I+S$ is continuously invertible, which
does not have a nonzero eigenvalue, i.e., is Volterra. On the other hand, such
weak perturbations have a complete system of root vectors if the self-adjoint
operator $H$ is from the Schatten-von Neumann class. In this paper, we
consider the compact operators $A$ representable as a sum of two compact
operators $A=C+T$, i.e., $A$ is not necessarily a weak perturbation, where $C$
is a non-negative operator. We will prove existence theorems of nonzero
eigenvalues for such operators. The study of the spectral properties of
operators generated by differential equations with Cauchy initial data
involve, as a rule, Volterra boundary-value problems that are well posed. But
Hadamard’s example shows that the Cauchy problem for the Laplace equation is
ill posed. At present, not a single Volterra well-defined restriction or
extension for elliptic-type equations is known. Thus, the following question
arises: Does there exist a Volterra well-defined restriction of a maximal
operator $\widehat{L}$ or a Volterra well-defined extension of a minimal
operator $L_{0}$ generated by the Laplace operator? The obtained existence
theorems for eigenvalues give that a wide class of well-defined restrictions
of the maximal operator $\widehat{L}$ and a wide class of well-defined
extensions of the minimal operator $L_{0}$ generated by the Laplace operator
cannot be Volterra. Moreover, in the two-dimensional case it is proven that
there are no Volterra well-defined restrictions or extensions for Laplace
operator at all.

##  1 Introduction

Let us present some definitions, notation, and terminology.

In a Hilbert space $H$, we consider a linear operator $L$ with domain $D(L)$
and range $R(L)$. By the kernel of the operator $L$ we mean the set

$\mbox{Ker}\,L=\bigl{\\{}f\in D(L):\;Lf=0\bigl{\\}}.$

######  Definition 1.1.

An operator $L$ is called a restriction of an operator $L_{1}$, and $L_{1}$ is
called an extension of $L$ if

1) $D(L)\subset D(L_{1})$;

2) $Lf=L_{1}f$ for all $f$ from $D(L)$.
Usually, one writes $L\subset L_{1}$.

######  Definition 1.2.

A linear closed operator $L_{0}$ in a Hilbert space $H$ is called minimal if
there exists a bounded inverse operator $L_{0}^{-1}$ on $R(L_{0})$ and
${R(L_{0})}\not=H$; a linear closed operator $\widehat{L}$ in a Hilbert space
$H$ is called maximal if $R(\widehat{L})=H$ and
$\mbox{Ker}\,\widehat{L}\not=\\{0\\}$; a linear closed operator $L$ in a
Hilbert space $H$ is called well defined if there exists a bounded inverse
operator $L^{-1}$ defined on all of $H$.

We say that a well defined operator $L$ in a Hilbert space $H$ is a well-
defined extension of a minimal operator $L_{0}$ (well-defined restriction of a
maximal operator $\widehat{L}$) if $L_{0}\subset L$ ($L\subset\widehat{L}$). A
well defined operator $L$ in a Hilbert space $H$ is a boundary well-defined
extension of a minimal operator $L_{0}$ with respect to a maximal operator
$\widehat{L}$ if $L$ is both a well-defined restriction of the maximal
operator $\widehat{L}$ and a well-defined extension of the minimal operator
$L_{0}$, i.e., $L_{0}\subset L\subset\widehat{L}$.

At the beginning of the 1950s, Vishik [1] extended the theory of self-adjoint
extensions of von Neumann–Krein symmetric operators to nonsymmetric operators
in Hilbert space.

Later, Bitsadze and Samarskii [2] discovered a correct problem not contained
among the problems described by Vishik. For ordinary differential equations,
problems of such type were studied by Dezin [3].

At the beginning of the 1980s, Otelbaev and his disciples proved abstract
theorems whose application makes it possible to describe all well-defined
extensions of some minimal operator using any single known well-defined
extension in terms of an inverse operator. Here such extensions need not be
restrictions of a maximal operator. Similarly, all possible well-defined
restrictions of some maximal operator that need not be extensions of a minimal
operator were described (see [4]). This description covers problems of
Bitsadze-Samarskii type.

Suppose that $\widehat{L}$ is a maximal linear operator in a Hilbert space
$H$, $L$ is any known well-defined restriction of $\widehat{L}$, and $K$ is an
arbitrary linear bounded (in $H$) operator satisfying the following condition:

$R(K)\subset\mbox{Ker}\,\widehat{L}.$ (1.1)

Then the operator $L_{K}^{-1}$ defined by the formula

$L_{K}^{-1}f=L^{-1}f+Kf,$ (1.2)

describes the inverse operators to all possible well-defined restrictions
$L_{K}$ of $\widehat{L}$, i.e., $L_{K}\subset\widehat{L}$.

Suppose that $L_{0}$ is a minimal operator in a Hilbert space $H$, $L$ is any
known well-defined extension of $L_{0}$, and $K$ is a linear bounded operator
in $H$ satisfying the conditions

a) $R(L_{0})\subset\mbox{Ker}\,K$,

b) $\mbox{Ker}\,(I+KL)=\\{0\\}$,
then the operator $L_{K}^{-1}$ defined by formula (1.2) describes the inverse
operators to all possible well-defined extensions $L_{K}$ of $L_{0}$, i.e.,
$L_{0}\subset L_{K}$.

Let $L$ be any known boundary well-defined extension of $L_{0}$, i.e.,
$L_{0}\subset L\subset\widehat{L}$. The existence of at least one boundary
well-defined extension $L$ was proved by Vishik in [1]. Let $K$ be a linear
bounded (in $H$) operator satisfying the conditions

a) $R(L_{0})\subset\mbox{Ker}\,K$,

b) $R(K)\subset\mbox{Ker}\,\widehat{L}$,
then the operator $L_{K}^{-1}$ defined by formula (1.2) describes the inverse
operators to all possible boundary well-defined extensions $L_{K}$ of $L_{0}$,
i.e., $L_{0}\subset L_{K}\subset\widehat{L}$.

######  Definition 1.3.

A bounded operator $A$ in a Hilbert space $H$ is called quasinilpotent if its
spectral radius is zero, that is, the spectrum consists of the single point
zero.

######  Definition 1.4.

An operator $A$ in a Hilbert space $H$ is called a Volterra operator if $A$ is
compact and quasinilpotent.

######  Definition 1.5.

A well-defined operator $L$ is said to be Volterra if the inverse operator
$L^{-1}$ is a Volterra operator.

Denote by $\mathfrak{S}_{\infty}\left(H,H_{1}\right)$ the set of all linear
compact operators from a Hilbert space $H$ to a Hilbert space $H_{1}$. If
$T\in\mathfrak{S}_{\infty}\left(H,H_{1}\right)$, then $T^{*}T$ is a
nonnegative self-adjoint operator from the set
$\mathfrak{S}_{\infty}\left(H\right)\equiv\mathfrak{S}_{\infty}\left(H,H\right)$
and, moreover, there is a nonnegative unique self-adjoint root
$\left|T\right|=\left(T^{*}T\right)^{1/2}$ in
$\mathfrak{S}_{\infty}\left(H\right)$. The eigenvalues
$\lambda_{n}\left(\left|T\right|\right)$ constitute a monotone sequence of
nonnegative numbers converging to zero. These numbers are usually called
$s$-numbers (see [5]) of the operator $T$ and denoted by
$s_{n}\left(T\right)$, $n\in\mathbb{N}$. By
$\mathfrak{S}_{p}\left(H,H_{1}\right)$ we denote the set of all compact
operators $T\in\mathfrak{S}_{\infty}\left(H,H_{1}\right)$ satisfying the
condition

$\sum_{j=1}^{\infty}s_{j}^{p}\left(T\right)<\infty,\quad 0<p<\infty.$

Obviously, if rank $|T|=r<\infty$, then $s_{n}(T)=0$, for $n=r+1,r+2,...$.
Operators of finite rank certainly belong to the classes
$\mathfrak{S}_{p}\left(H,H_{1}\right)$ for all $p>0$.

##  2 Main Result

In the work [6], a wide class of weak perturbation of a positive compact
operator $H$, of the form $H(I+S)$, was singled out, where $S$ is such a
compact operator that $I+S$ is continuously invertible, which does not have a
nonzero eigenvalue, i.e., is Volterra. On the other hand, such weak
perturbations have a complete system of root vectors if the self-adjoint
operator $H$ is from the Schatten-von Neumann class (see Theorem 8. 1 [5,
p.257]). In this paper, we consider compact operators $A$ representable as a
sum of two compact operators $A=C+T$, where $C$ is a non-negative operator. In
this section 2, we will prove existence theorems for nonzero eigenvalues for
such operators.

######  Theorem 2.1.

If the operator $A$ in a Hilbert space $H$ can be represented in the form

$A=C+T,$

where $C$ is a non-negative operator such that

$\lim_{n\rightarrow\infty}n^{2}s_{n}(C)=\infty,\quad\lim_{n\rightarrow\infty}n^{\alpha}s_{n}(C)=a,\quad\mbox{for}\quad
0<\alpha<2,\;\;\;0<a<\infty,$

and the operator $T\in S_{q}(H)$ for all $q\leq\frac{1}{\alpha}$, then the
operator $A$ cannot be Volterra.

###### Proof.

It is clear from the condition of Theorem 2.1 that $s_{n}(C)$ and $s_{n}(T)$
tend to zero as $n$ tends to infinity, i.e., they are compact operators. It is
known (see [5, p.95]) that

$\lim_{n\rightarrow\infty}n^{\frac{1}{q}}s_{n}(T)=0.$

Notice that

$\lim_{n\rightarrow\infty}n^{\alpha}s_{n}(T)=\lim_{n\rightarrow\infty}\bigg{(}\frac{n^{\alpha}}{n^{\frac{1}{q}}}\bigg{)}n^{\frac{1}{q}}s_{n}(T)=0,$

since $\alpha\leq\frac{1}{q}$. Then by Theorem 2.3 [5, p.32]

$\lim_{n\rightarrow\infty}n^{\alpha}s_{n}(A)=a.$

To prove Theorem 2.1 assume the contrary, i.e., suppose that the operator $A$
is Volterra.

Case 1. Let the given parameter $\alpha$ be in the interval $(0,1)$. Then
$1<\frac{1}{\alpha}<\infty$ and $0<q\leq\frac{1}{\alpha}<\infty$. It is clear
that

$T_{J}=\frac{T-T^{*}}{2i}\neq 0.$

Otherwise, $A$ is self-adjoint and contradicts the Volterra property of $A$.

a) If $0<q\leq 1/2$ then by Theorem 11.5 [5, p.220] we get that

$\lim_{n\rightarrow\infty}n^{2}s_{n}(C)$

exists and is finite. This contradicts the condition of Theorem 2.1.

b) Consider the case $1/2<q\leq 1$. Then by Theorem 11.3 [5, p.217] we have
that $C\in S_{q}$ for all $1/2<q\leq 1$, i.e., $\rho(C)=1/2$. But we consider
the case $0<\alpha<1$, i.e., $C\in S_{p}$, for all $p>1/\alpha>1$,i.e,
$\rho(C)=1/\alpha>1,$. This is a contradiction to the fact that $\rho(C)=1/2$.

c) Now we consider the case $1<q\leq 1/\alpha<\infty$. Despite the fact that
$T\in S_{q},\;q>1$ it may turn out that $T_{J}\in S_{q},\;q\leq 1$. Then by
Theorem 2.1 [7, p.83] we have that $A\in S_{\Omega}$ which is of order
$\rho(A)=\inf\\{p:\;A\in S_{\Omega}\\}\leq 1$. But we have $A_{R}=C+T_{R}$. By
the condition of Theorem 2.1,

$\rho(C)=\inf\\{p:\;C\in S_{p}\\}=1/\alpha>1.$

Then

$\rho(A_{R})=\rho(C)=1/\alpha>1,$

since $T_{R}\in S_{q},\;1<q\leq 1/\alpha$, i.e., $\rho(T_{R})\leq\rho(T)=1$
(see [7, p.118]), the orders of the Hermitian components of operators $T$ do
not exceed the order of $T$. Then we have $\rho(A)=1/\alpha>1$. This
contradicts the fact that $\rho(A)\leq 1$.

Now we consider the case $T_{J}\in S_{q},\;1<q\leq 1/\alpha$. Then by Theorem
11.1 [5, p.215] we obtain

$A_{R}=C+T_{R}\in S_{q},\;1<q\leq 1/\alpha,$

i.e., $\rho(A_{R})=1$. Then $\rho(A)=1$. But by the condition of Theorem 2.1
we have $\rho(A)=\rho(C)=1/\alpha>1$. These contradictions prove that A is not
Volterra.

Case 2. We consider the case $1\leq\alpha<2,\;1/2<1/\alpha\leq 1$, then
$0<q\leq 1/\alpha\leq 1$.

a) When $0<q\leq 1/2$ we get the contradiction with the condition of Theorem
2.1 by virtue of Theorem 11.5 [5, p.220], similarly as in case 1, a).

b) In the case of $1/2<q\leq 1/\alpha\leq 1$, by the condition of Theorem 2.1
we have that $\rho(C)=1/\alpha>1/2$. But, by virtue of Theorem 11.3 [5,
p.217], which states that $C\in S_{q},\;1/2<q\leq 1/\alpha\leq 1$, we have
that $\rho(C)=1/2$. Then we get the contradiction. Thus we proved A is not
Volterra. ∎

######  Definition 2.2.

The function $L(x)\,(b\leq x<\infty;\,b>0)$ will be called the slowly varying
if it is positive and continuously differentiable, and if [5, p.144]

$\lim_{x\rightarrow\infty}\frac{xL^{\prime}(x)}{L(x)}=0.$

Further, we will need the following properties of slowly varying functions:

  1. 1)

for each $\lambda>0$ we have

$\lim_{x\rightarrow\infty}\frac{L(\lambda x)}{L(x)}=1;$ (2.1)

  2. 2)

for any $\epsilon>0$, one can always find constants $C^{\prime}_{\epsilon}$,
$C^{\prime\prime}_{\epsilon}$ for a given slowly varying function $L(x)$ such
that

$C^{\prime}_{\epsilon}x^{-\epsilon}<L(x)<C^{\prime\prime}_{\epsilon}x^{\epsilon}\;(1\leq
x<\infty);$

  3. 3)

for any $l>0$ one has

$\lim_{x\rightarrow\infty}\frac{L(x+l)}{L(x)}=1.$ (2.2)

As an example of a slowly varying function we can take any function

$L(x)=(\ln x)^{\beta}$

where $\beta$ is an arbitrary real number.

######  Theorem 2.3.

If in the representation $A=C+T$ of the compact operator $A$ in a Hilbert
space $H$, the $C$ is a non-negative compact operator such that

$\lim_{n\rightarrow\infty}\frac{1}{L(n)}s_{n}(C)=a,\quad\mbox{for}\;\;0<a<\infty,\;\;{and}\quad
T\in S_{q},\;\;q>0,,$

then the operator $A$ cannot be Volterra.

###### Proof.

By Corollary 2.2 [5, p.29], if $C,T\in S_{\infty}$ are compact then

$s_{n+m-1}(C+T)\leq s_{n}(C)+s_{m}(T),\quad n,m=1,2,\dots.$

Then

$\begin{split}s_{2n+1}(A)&=s_{2n+1}(C+T)\leq s_{2n}(A)=s_{2n}(C+T)\leq
s_{2n-1}(C+T)\\\ &\leq s_{n}(C)+s_{n}(T),\quad\;\;n=1,2,\dots.\end{split}$

Thus

$\begin{split}\frac{1}{L(n)}s_{2n+1}(C+T)&\leq\frac{1}{L(n)}s_{2n}(C+T)\leq\frac{1}{L(n)}s_{n}(C)+\frac{1}{L(n)}s_{n}(T)\\\
&=\frac{1}{L(n)}s_{n}(C)+\frac{1}{L(n)\cdot n^{\frac{1}{q}}}\cdot
n^{\frac{1}{q}}s_{n}(T),\end{split}$
$\begin{split}\Rightarrow\quad\frac{L(2n+1)}{L(n)L(2n+1)}s_{2n+1}(C+T)&\leq\frac{L(2n)}{L(n)L(2n)}s_{2n}(C+T)\\\
&\leq\frac{1}{L(n)}s_{n}(C)+\frac{1}{L(n)\cdot n^{\frac{1}{q}}}\cdot
n^{\frac{1}{q}}s_{n}(T).\end{split}$

Then by (2.1), (2.2) we have

$\lim_{n\rightarrow\infty}\frac{s_{n}(C+T)}{L(n)}\leq a.$

On the other hand $C=A-T$ and $s_{2n}(C)\leq s_{n}(A)+s_{n}(T)$. Furthermore

$\begin{split}s_{2n+1}(C)\leq s_{2n}(C)\leq
s_{n}(A)+s_{n}(T),\quad\;\;n=1,2,\dots.\end{split}$

Thus

$\begin{split}\frac{1}{L(n)}s_{2n+1}(C)&\leq\frac{1}{L(n)}s_{2n}(C)\leq\frac{1}{L(n)}s_{n}(A)+\frac{1}{L(n)}s_{n}(T)\\\
&=\frac{1}{L(n)}s_{n}(A)+\frac{1}{L(n)\cdot n^{\frac{1}{q}}}\cdot
n^{\frac{1}{q}}s_{n}(T),\end{split}$
$\begin{split}\Rightarrow\quad\frac{L(2n+1)}{L(n)L(2n+1)}s_{2n+1}(C)&\leq\frac{L(2n)}{L(n)L(2n)}s_{2n}(C)\\\
&\leq\frac{1}{L(n)}s_{n}(A)+\frac{1}{L(n)\cdot n^{\frac{1}{q}}}\cdot
n^{\frac{1}{q}}s_{n}(T).\end{split}$

and we have

$a\leq\lim_{n\rightarrow\infty}\frac{1}{L(n)}s_{n}(A),$

then consequently

$\lim_{n\rightarrow\infty}\frac{1}{L(n)}s_{n}(A)=\lim_{n\rightarrow\infty}\frac{1}{L(n)}s_{n}(C+T)=a.$

Thus we have $A=C+T\notin S_{p}$, for all $p>0$.

To prove Theorem 2.3 assume the contrary, i.e., suppose that the operator $A$
is Volterra.

Consider the following cases

1) If $0<q\leq 1/2$ then by Theorem 11.5 [5, p.220] we get that

$\lim_{n\rightarrow\infty}n^{2}s_{n}(C)$

exists and is finite. This contradicts the condition of Theorem 2.3.

2) If $1/2<q\leq 1$ then by Theorem 11.3 [5, p.217] we get $C\in
S_{q},\;\;1/2<q\leq 1$, but this contradicts with the condition of Theorem
2.3, i.e., with $C\notin S_{p}$, for all $p>0$.

3) If $1<q<\infty$ then we consider following cases:

a) If $T_{J}\in S_{q},\;\;1<q<\infty$ then by Theorem 11.1 [5, p.215]
$C+T_{R}=A_{R}\in S_{q},\;\;1<q<\infty\quad\Rightarrow\quad A\in
S_{q},\;\;1<q<\infty.$ But this contradicts with the condition of Theorem 2.3,
i.e., with the $A=C+T\notin S_{p}$, for all $p>0$.

b) If $T_{J}\in S_{q},\;\;0<q\leq 1$ and $T_{R}\in S_{q},\;\;1<q<\infty$, then
by the Theorem 2.1 [7, p.83] we have $A\in S_{\Omega}$ which is of order
$\rho(A)\leq 1$. But this contradicts with the condition of Theorem 2.3, i.e.,
with the $A=C+T\notin S_{p}$, for all $p>0$.

∎

######  Corollary 2.4.

The results of Theorem 2.1 and Theorem 2.3 are also true for the operator
$A^{*}$.

##  3 An application of Theorem 2.1 and Theorem 2.3 to the Laplace operator

As a rule, the study of the spectral properties of operators generated by
differential equations of hyperbolic or parabolic type with Cauchy initial
data involves Volterra boundary well-defined problems. But Hadamard’s example
shows that the Cauchy problem for the Laplace equation is ill posed. At
present, not a single Volterra well-defined restriction or extension for
elliptic-type equations is known. Thus, the following question arises: Does
there exist a Volterra well-defined restriction of the maximal operator
$\widehat{L}$ or a Volterra well-defined extension of the minimal operator
$L_{0}$ generated by the Laplace operator (3.1)? This section is devoted to
the study of this question.

In the Hilbert space $L_{2}(\Omega)$, where $\Omega$ is a bounded domain in
$\mathbb{R}^{m}$ with infinitely smooth boundary $\partial\Omega$, let us
consider the minimal $L_{0}$ and maximal $\widehat{L}$ operators generated by
the Laplace operator

$-\Delta
u=-\biggl{(}\frac{\partial^{2}u}{\partial{x_{1}^{2}}}+\frac{\partial^{2}u}{\partial{x_{2}^{2}}}+\cdots+\frac{\partial^{2}u}{\partial{x_{m}^{2}}}\biggr{)}.$
(3.1)

The closure $L_{0}$ in the space $L_{2}(\Omega)$ of the Laplace operator (3.1)
with domain $C_{0}^{\infty}(\Omega)$ is called the minimal operator
corresponding to the Laplace operator.

The operator $\widehat{L}$, adjoint to the minimal operator $L_{0}$
corresponding to the Laplace operator is called the maximal operator
corresponding to the Laplace operator (see [8]). Note that

$D(\widehat{L})=\\{u\in L_{2}(\Omega):\;\widehat{L}u=-\Delta u\in
L_{2}(\Omega)\\}.$

Denote by $L_{D}$ the operator corresponding to the Dirichlet problem with the
domain

$D(L_{D})=\\{u\in W_{2}^{2}(\Omega):\;u|_{\partial\Omega}=0\\}.$

Then, in view of (1.2), the inverse operators $L^{-1}_{K}$ to all possible
well-defined restrictions of the maximal operator $\widehat{L}$ corresponding
to the Laplace operator (3.1) have the following form:

$u\equiv L^{-1}_{K}f=L_{D}^{-1}f+Kf,$ (3.2)

where, in view of (1.1), $K$ is an arbitrary bounded (in $L_{2}(\Omega)$)
operator with

$R(K)\subset\mbox{Ker}\,\widehat{L}=\\{u\in L_{2}(\Omega):\>-\Delta u=0\\}.$

Then the direct operator $L_{K}$ is determined from the following problem:

$\widehat{L}u=f,\quad f\in L_{2}(\Omega),$ (3.3) $D(L_{K})=\\{u\in
D(\widehat{L}):\;(I-K\widehat{L})u|_{\partial\Omega}=0\\},$ (3.4)

where $I$ is the unit operator in $L_{2}(\Omega)$. There are no other linear
well-defined restrictions of the operator $\widehat{L}$ (see [9]).

The operators $(L^{*}_{K})^{-1}$, corresponding to the operators $L^{*}_{K}$,

$v\equiv(L^{*}_{K})^{-1}g=L_{D}^{-1}g+K^{*}g,$

describe the inverse operators to all possible well-defined extensions of the
minimal operator $L_{0}$ if and only if $K$ satisfies the condition (see [9])

$\mbox{Ker}\,(I+K^{*}L_{D})=\\{0\\}.$

Note that the last condition is equivalent to the following one:
$\overline{D(L_{K})}=L_{2}(\Omega)$. If the additional condition

$KR(L_{0})=\\{0\\}$

is imposed on the operator $K$ from (3.2), then the operator $L_{K}$
corresponding to problem (3.3), (3.4), will turn out to be boundary well-
defined.

In representation (3.2) we can always take any other positive well-defined
restriction $L$ of the maximal operator $\widehat{L}$ instead of $L_{D}$, such
that:

$lim_{n\rightarrow\infty}n^{2}s_{n}(L^{-1})=\infty$

and

$\lim_{n\rightarrow\infty}n^{\alpha}s_{n}(L^{-1})=a>0,\quad\mbox{for}\quad
0<\alpha<2,\quad\mbox{or}\quad\lim_{n\rightarrow\infty}\frac{1}{L(n)}s_{n}(L^{-1})=a>0.$

Note that the existence of such operator $L$ is achieved using the operator
$K$ in (3.2). Then the well-defined restriction of the maximal operator
$\widehat{L}$ generated by (3.1) have the inverse operators of the following
form

$u\equiv L^{-1}_{K}f=L^{-1}f+Kf,$ (3.5)

where $K$ is an arbitrary bounded operator (in $L^{2}(\Omega))$ with

$R(K)\subset\mbox{Ker}\,\widehat{L},\quad D(L_{K})=\big{\\{}u\in
D(\widehat{L}):\,(I-K\widehat{L})u\in D(L)\big{\\}}.$

An application of Theorem 2.1 and Theorem 2.3 to the Laplace operator gives
following theorems.

######  Theorem 3.1.

If the operator $L_{K}^{-1}$ in a Hilbert space $L_{2}(\Omega)$ can be
represented in the form (3.5), where $L^{-1}$ is a non-negative operator such
that

$\lim_{n\rightarrow\infty}n^{2}s_{n}(L^{-1})=\infty,\quad\lim_{n\rightarrow\infty}n^{\alpha}s_{n}(L^{-1})=a,\quad\mbox{for}\quad
0<\alpha<2,\;\;\;0<a<\infty,$

and the operator $K\in S_{q}(L_{2}(\Omega))$ for all $q\leq\frac{1}{\alpha}$,
then the operator $L_{K}^{-1}$ cannot be Volterra, i.e., $L_{K}$ is the non
Volterra well-defined restriction of the maximal operator $\widehat{L}$.

######  Remark 3.2.

In the special case when $L=L_{D}$, i.e., $\alpha=2/m$, this result was
obtained by the author in [10].

######  Theorem 3.3.

If in the representation (3.5) of the operator $L_{K}^{-1}$ in a Hilbert space
$L_{2}(\Omega)$, the $L^{-1}$ is a non-negative operator such that

$\lim_{n\rightarrow\infty}\frac{1}{L(n)}s_{n}(L^{-1})=a,\;\;\quad\mbox{for}\;\;0<a<\infty,\quad\mbox{and}\quad
K\in S_{q},\;\;q>0,$

then the operator $L_{K}^{-1}$ cannot be Volterra, i.e., $L_{K}$ is the non
Volterra well-defined restriction of the maximal operator $\widehat{L}$.

######  Remark 3.4.

The results obtained in Theorem 3.1 and Theorem 3.3 are also valid for
elliptic operators of order $2l$ ($l\in\mathbb{N}$):

$(\mathcal{L}u)(x)=\sum_{|\alpha|,|\beta|\leq
l}(-1)^{|\alpha|+|\beta|}D^{\alpha}(a_{\alpha\beta}(x)D^{\beta}u),\quad
x\in\Omega,$

where $a_{\alpha\beta}\in C^{l}(\Omega)$ are the real-valued functions for all
multi-indices $\alpha,\beta$, and $\Omega$ is a bounded domain in
$\mathbb{R}^{m}$ with infinitely smooth boundary $\partial\Omega$. We assume
the uniform ellipticity condition is satisfied, i.e., for some $\nu>0$

$\sum_{|\alpha|=|\beta|=l}a_{\alpha\beta}(x)\xi^{\alpha}\xi^{\beta}\geq\nu|\xi|^{2l},\quad\mbox{for
all}\;\;x,\xi\in\Omega.$

######  Corollary 3.5.

The results of Theorem 3.1 and Theorem 3.3 are also true for the operator
$L_{K}^{*}$.

##  4 On the absence of Volterra well-defined restrictions and extensions of
the Laplace operator in the case $m=2$

In Sections 2 and 3, we found a sufficient conditions for non-Volterra
operators of one perturbation class of positive operators, i.e., we proved the
existence of at least one nonzero eigenvalue. In this section, we prove that
for the Laplace operator in the two-dimensional case there are no Volterra
well-defined restrictions or extensions at all.

We pass to the polar coordinate system:

$x=r\cos\varphi,\>y=r\sin\varphi.$

Then the Laplace operator (3.1) has the form

$\widehat{L}u\equiv-\Delta
u=-\frac{\partial^{2}u}{\partial{x^{2}}}-\frac{\partial^{2}u}{\partial{y^{2}}}=-u_{rr}-\frac{1}{r}u_{r}-\frac{1}{r^{2}}u_{\varphi\varphi}=f(r,\varphi),$
(4.1)

on

$D(\widehat{L})=\\{u\in L_{2}(\Omega):\;\Delta u\in L_{2}(\Omega)\\},$

is the maximal operator (see [10]), where $\Omega$ is unit circle. In this
case denote by $L_{D}$ the operator corresponding to the Dirichlet problem
with the domain

$D(L_{D})=\\{u\in W_{2}^{2}(\Omega):\;u|_{r=1}=0\\}.$

Then any well-defined restriction $L_{K}$ from (3.2) acts as the maximal
operator $\widehat{L}$ on the domain

$D(L_{K})=\\{u\in D(\widehat{L}):\;[(I-K\widehat{L})u]|_{r=1}=0\\},$ (4.2)

where $K$ is any bounded linear operator in $L_{2}(\Omega)$ such that
$R(K)\subset\mbox{Ker}\,\widehat{L}$. Compactness of $L^{-1}_{K}$ is necessary
condition for $L_{K}$ to be Volterra. From (3.2) note that $L^{-1}_{K}$ is
compact if and only if $K$ is a compact operator, since $L^{-1}_{D}$ is
compact. Then for $K$ the Schmidt expansion takes place (see [5, p. 47(28)])

$K=\sum_{j=1}^{\infty}s_{j}(\>\cdot\,,\,Q_{j})F_{j}$ (4.3)

where $\\{Q_{j}\\}_{1}^{\infty}$ is orthonormal system in $L_{2}(\Omega)$,
$\,\\{F_{j}\\}_{1}^{\infty}$ is orthonormal system in
$\mbox{Ker}\,\widehat{L}\,$, and $\,\\{s_{j}\\}_{1}^{\infty}$ is a monotone
sequence of non-negative numbers converging to zero. The series on the right
side of (4.3) converges in the uniform operator norm. We now state the main
result of this paper.

######  Theorem 4.1.

Let $\widehat{L}$ be a maximal operator generated by the Laplace (3.1) in
$L_{2}(\Omega)$ when $m=2$. Then any well-defined restriction $L_{K}$ of the
maximal operator $\widehat{L}$, with the compact operator $K$ in
representation (3.2), cannot be Volterra, i.e., the problem (4.1) and (4.2)
cannot be Volterra.

###### Proof.

Let us prove by contradiction. Suppose that there exists a Volterra well-
defined restriction $L_{K}$. This is equivalent to the existence of a such
compact operator $K$ that the operator $L_{K}$ has no nonzero eigenvalue. The
general solution of the equation

$\widehat{L}u=-\Delta
u=-u_{rr}-\frac{1}{r}u_{r}-\frac{1}{r^{2}}u_{\varphi\varphi}=\lambda^{2}u,$

from the space $L_{2}(\Omega)$ has the form (see [11])

$u(r,\varphi)=u_{0}(r,\varphi)-\int_{0}^{r}u_{0}(\rho,\varphi)\frac{\partial}{\partial\rho}J_{0}(\lambda\sqrt{r(r-\rho)})\,d\rho,$

where $\lambda$ is any complex number, $u_{0}(r,\varphi)$ is the solution of
the equation

$\widehat{L}u_{0}\equiv-\Delta
u_{0}=-u_{0_{rr}}-\frac{1}{r}u_{0_{r}}-\frac{1}{r^{2}}u_{0_{\varphi\varphi}}=0,$

which is a harmonic function from the space $L_{2}(\Omega)$ and

$J_{0}(z)=\sum_{n=0}^{\infty}\dfrac{(-1)^{n}}{(n!)^{2}}\Bigl{(}\dfrac{z}{2}\Bigr{)}^{2n}$

is the Bessel function. Then, by virtue of (4.2) we obtain the equation

$\begin{split}&u_{0}(1,\varphi)-\int_{0}^{1}u_{0}(\rho,\varphi)\frac{\partial}{\partial\rho}J_{0}(\lambda\sqrt{1-\rho})\,d\rho\\\\[5.0pt]
&\,-\lambda^{2}\sum_{j=1}^{\infty}s_{j}F_{j}(1,\varphi)\cdot\frac{1}{2\pi}\int_{0}^{2\pi}\int_{0}^{1}u_{0}(\rho,\theta)\cdot\overline{Q_{j}(\rho,\theta)}\rho\,d\rho\,d\theta\\\\[5.0pt]
&\,+\lambda^{2}\sum_{j=1}^{\infty}s_{j}F_{j}(1,\varphi)\frac{1}{2\pi}\int_{0}^{2\pi}\int_{0}^{1}\overline{Q_{j}(\rho,\theta)}\cdot\int_{0}^{\rho}u_{0}(\tau,\theta)\frac{\partial}{\partial\tau}J_{0}(\lambda\sqrt{\rho(\rho-\tau)}\,d\tau\rho\,d\rho\,d\theta=0.\end{split}$
(4.4)

The considered problem on the spectrum for the Laplace operator has no
eigenvalues if and only if the equation (4.4) has no zeros as a function of
$\lambda$. The harmonic function $u_{0}(\rho,\varphi)$ does not depend on
$\lambda$. It is easy to notice that the left side of the equation is an
entire function of exponential type not higher than the first order. Then, by
virtue of Picard’s theorem (see [12, p. 264, 266]), this function has the form
$Ce^{d\lambda}$, where $C(\varphi)$ and $d(\varphi)$ are a functions which are
independent of $\lambda$. If we notice that the left side of the equation
(4.4) is even with respect to the sign of $\lambda$, then $d=0$. By equating
these functions when $\lambda=0$ we have $C=u_{0}(1,\varphi)$. Then we get the
following

$\begin{split}&-\int_{0}^{1}u_{0}(\rho,\varphi)\frac{\partial}{\partial\rho}J_{0}(\lambda\sqrt{1-\rho})\,d\rho\\\\[5.0pt]
&\,-\lambda^{2}\sum_{j=1}^{\infty}s_{j}F_{j}(1,\varphi)\cdot\frac{1}{2\pi}\int_{0}^{2\pi}\int_{0}^{1}u_{0}(\rho,\theta)\cdot\overline{Q_{j}(\rho,\theta)}\rho\,d\rho\,d\theta\\\\[5.0pt]
&\;\;+\lambda^{2}\sum_{j=1}^{\infty}s_{j}F_{j}(1,\varphi)\cdot\frac{1}{2\pi}\int_{0}^{2\pi}\int_{0}^{1}\overline{Q_{j}(\rho,\theta)}\int_{0}^{\rho}u_{0}(\tau,\theta)\frac{\partial}{\partial\tau}J_{0}(\lambda\sqrt{\rho(\rho-\tau)}\,d\tau\,\rho\,d\rho\,d\theta=0.\end{split}$
(4.5)

Divide both sides of (4.5) by $\lambda^{2}$ and let $\lambda$ tend to zero.
Then

$\sum_{j=1}^{\infty}s_{j}F_{j}(1,\varphi)\cdot\frac{1}{2\pi}\int_{0}^{2\pi}\int_{0}^{1}u_{0}(\rho,\theta)\overline{Q_{j}(\rho,\theta)}\rho\,d\rho\,d\theta=-\frac{1}{4}\int_{0}^{1}u_{0}(\rho,\varphi)\,d\rho.$
(4.6)

Under the condition that (4.6) is fulfilled we obtain

$\begin{split}&-\int_{0}^{1}u_{0}(\rho,\varphi)\biggl{[}\frac{\partial
J_{0}}{\partial\rho}(\lambda\sqrt{1-\rho})+\frac{1}{4}\biggr{]}d\rho\\\\[5.0pt]
&\,+\sum_{j=1}^{\infty}s_{j}F_{j}(1,\varphi)\cdot\frac{1}{2\pi}\int_{0}^{2\pi}\int_{0}^{1}\overline{Q_{j}(\rho,\theta)}\int_{0}^{\rho}u_{0}(\tau,\theta)\frac{\partial}{\partial\tau}J_{0}(\lambda\sqrt{\rho(\rho-\tau)}\,d\tau\,\rho\,d\rho\,d\theta=0.\end{split}$
(4.7)

On the left side of the equation (4.7) we make the change of variables: in the
first summand $t=\sqrt{1-\rho}$, in the second summand
$t=\sqrt{\rho(\rho-\tau)}$. Then we have

$\begin{split}&\int_{0}^{1}u_{0}(1-t^{2},\varphi)\biggl{[}\frac{J_{0}^{\prime}(\lambda
t)}{2\lambda t}+\frac{1}{4}\biggr{]}2t\,dt\\\\[5.0pt]
&\,-\sum_{j=1}^{\infty}s_{j}F_{j}(1,\varphi)\cdot\frac{1}{2\pi}\int_{0}^{2\pi}\int_{0}^{1}J_{0}^{\prime}(\lambda
t)\int_{t}^{1}u_{0}\Bigl{(}\frac{\rho^{2}-t^{2}}{\rho},\theta\Bigr{)}\overline{Q_{j}(\rho,\theta)}\rho\,d\rho\,dt\,d\theta=0.\end{split}$
(4.8)

For the Bessel function the following equalities hold

$\dfrac{J^{\prime}_{0}(\lambda t)}{2\lambda
t}+\dfrac{1}{4}=\dfrac{1}{4}\sum_{n=1}^{\infty}\dfrac{(-1)^{n+1}}{((n+1)!)^{2}}\Bigl{(}\dfrac{\lambda
t}{2}\Bigr{)}^{2n}\cdot(n+1),$

and

$\lambda J^{\prime}_{0}(\lambda
t)=\sum_{n=1}^{\infty}\dfrac{(-1)^{n}}{(n!)^{2}}\Bigl{(}\dfrac{\lambda
t}{2}\Bigr{)}^{2n}\cdot n\cdot\dfrac{2}{t}.$

Substitute them into (4.8) and equate the coefficients of $\lambda^{2n}$ to
zero

$\begin{split}&\int_{0}^{1}u_{0}(1-t^{2},\varphi)\frac{-1}{4(n+1)}\cdot
t^{2n}\cdot 2t\,dt\\\\[5.0pt]
&\,-\sum_{j=1}^{\infty}s_{j}F_{j}(1,\varphi)\cdot\frac{1}{2\pi}\int_{0}^{2\pi}\int_{0}^{1}nt^{2n}\cdot\frac{2}{t}\int_{t}^{1}u_{0}\Bigl{(}\frac{\rho^{2}-t^{2}}{\rho},\theta\Bigr{)}\overline{Q_{j}(\rho,\theta)}\rho\,d\rho\,dt\,d\theta=0.\end{split}$

We make the transformation of the following type

$\frac{1}{n+1}\cdot t^{2n}=\frac{2}{t^{2}}\int_{0}^{t}\tau^{2n+1}d\tau,\qquad
n\cdot t^{2n}=\frac{t}{2}\cdot\frac{\partial}{\partial
t}\Bigl{(}t^{2n}\Bigr{)}.$

Then

$\begin{split}&\int_{0}^{1}t^{2n}\cdot\biggl{\\{}t\int_{t}^{1}u_{0}(1-\tau^{2},\varphi)\frac{d\tau}{\tau}\,dt\\\\[5.0pt]
&\quad-\frac{1}{2\pi}\int_{0}^{2\pi}\frac{\partial}{\partial
t}\sum_{j=1}^{\infty}s_{j}F_{j}(1,\varphi)\cdot\int_{t}^{1}u_{0}\Bigl{(}\frac{\rho^{2}-t^{2}}{\rho},\theta\Bigr{)}\overline{Q_{j}(\rho,\theta)}\rho\,d\rho\,d\theta\biggr{\\}}\,dt=0.\end{split}$

In view of the completeness of the system of functions
$\\{t^{2n}\\}_{1}^{\infty}$ in $L_{2}(0,1)$ we obtain (see [13, p. 107])

$t\int_{t}^{1}u_{0}(1-\tau^{2},\varphi)\frac{d\tau}{\tau}-\frac{1}{2\pi}\int_{0}^{2\pi}\frac{\partial}{\partial
t}\sum_{j=1}^{\infty}s_{j}F_{j}(1,\varphi)\cdot\int_{t}^{1}u_{0}\Bigl{(}\frac{\rho^{2}-t^{2}}{\rho},\theta\Bigr{)}\overline{Q_{j}(\rho,\theta)}\rho\,d\rho\,d\theta=0.$

Integrating this equation from $t$ to $1$, we get

$\begin{split}&\,\int_{t}^{1}u_{0}(1-\tau^{2},\varphi)\frac{\tau^{2}-t^{2}}{2\tau}\,d\tau\\\\[5.0pt]
&\quad+\frac{1}{2\pi}\int_{0}^{2\pi}\int_{t}^{1}u_{0}\Bigl{(}\frac{\rho^{2}-t^{2}}{\rho},\theta\Bigr{)}\sum_{j=1}^{\infty}s_{j}F_{j}(1,\varphi)\overline{Q_{j}(\rho,\theta)}\rho\,d\rho\,d\theta=0.\end{split}$
(4.9)

where $\,0\leq t\leq 1,\;\,0\leq\varphi<2\pi$. Note that the condition (4.9)
contains the condition (4.6) as a particular case when $t=0$. Condition (4.9)
will turn out to be the Volterra criterion of the well-defined restriction
$L_{K}$, if it holds for any harmonic function $u_{0}(r,\varphi)$ from
$L_{2}(\Omega)$.

By Poisson’s formula

$u_{0}(r,\varphi)=\frac{1}{2\pi}\int_{0}^{2\pi}\dfrac{1-r^{2}}{1-2r\cos(\varphi-\gamma)+r^{2}}u_{0}(1,\gamma)\,d\gamma,$

the equality (4.9) is transformed to

$\begin{split}&\frac{1}{2\pi}\int_{0}^{2\pi}u_{0}(1,\gamma)\biggl{\\{}\int_{t}^{1}\dfrac{1-(1-\tau^{2})^{2}}{1-2(1-\tau^{2})\cos(\varphi-\gamma)+(1-\tau^{2})^{2}}\cdot\dfrac{\tau^{2}-t^{2}}{2\tau}\,d\tau\\\\[5.0pt]
&\,+\frac{1}{2\pi}\int_{0}^{2\pi}\int_{t}^{1}\dfrac{1-\bigl{(}\frac{\rho^{2}-t^{2}}{\rho}\bigr{)}^{2}}{1-2\bigl{(}\frac{\rho^{2}-t^{2}}{\rho}\bigr{)}\cos(\theta-\gamma)+\bigl{(}\frac{\rho^{2}-t^{2}}{\rho}\bigr{)}^{2}}\sum_{j=1}^{\infty}s_{j}F_{j}(1,\varphi)\overline{Q_{j}(\rho,\theta)}\rho\,d\rho\,d\theta\biggr{\\}}\,d\gamma=0.\end{split}$

Taking into account the density of the set of functions $u_{0}(1,\varphi)$ in
$L_{2}(0,2\pi)$ for almost all values of $t\,(0\leq t\leq
1),\;\varphi\;(0\leq\varphi<2\pi)$ we obtain the equality

$\begin{split}&\int_{t}^{1}\dfrac{1-(1-\tau^{2})^{2}}{1-2(1-\tau^{2})\cos(\varphi-\gamma)+(1-\tau^{2})^{2}}\cdot\dfrac{\tau^{2}-t^{2}}{2\tau}\,d\tau\\\\[5.0pt]
&\,+\frac{1}{2\pi}\int_{0}^{2\pi}\int_{t}^{1}\dfrac{1-\bigl{(}\frac{\rho^{2}-t^{2}}{\rho}\bigr{)}^{2}}{1-2\bigl{(}\frac{\rho^{2}-t^{2}}{\rho}\bigr{)}\cos(\theta-\gamma)+\bigl{(}\frac{\rho^{2}-t^{2}}{\rho}\bigr{)}^{2}}\sum_{j=1}^{\infty}s_{j}F_{j}(1,\varphi)\overline{Q_{j}(\rho,\theta)}\rho\,d\rho\,d\theta=0.\end{split}$
(4.10)

Now the equation (4.10) is the Volterra criterion of the well-defined
restriction $L_{K}$ of the maximal operator $\widehat{L}$ generated by the
Laplace operator (3.1) in $L_{2}(\Omega)$, where $\Omega$ is the unit disk.

Further, we apply to the equation (4.10) the Poisson operator of the variables
$r$ and $\varphi$. The first summand we transform with the formula of the
superposition of two Poisson integrals (see [14, p. 140]), and in the second
summand the harmonic function $F_{j}(r,\varphi)$ is reproduced by Poisson’s
formula. We have

$\begin{split}&\int_{t}^{1}\dfrac{1-r^{2}(1-\tau^{2})^{2}}{1-2r(1-\tau^{2})\cos(\varphi-\gamma)+r^{2}(1-\tau^{2})^{2}}\cdot\dfrac{\tau^{2}-t^{2}}{2\tau}\,d\tau\\\\[5.0pt]
&\,+\frac{1}{2\pi}\int_{0}^{2\pi}\int_{t}^{1}\dfrac{1-\bigl{(}\frac{\rho^{2}-t^{2}}{\rho}\bigr{)}^{2}}{1-2\bigl{(}\frac{\rho^{2}-t^{2}}{\rho}\bigr{)}\cos(\theta-\gamma)+\bigl{(}\frac{\rho^{2}-t^{2}}{\rho}\bigr{)}^{2}}\sum_{j=1}^{\infty}s_{j}F_{j}(r,\varphi)\overline{Q_{j}(\rho,\theta)}\rho\,d\rho\,d\theta=0.\end{split}$

From this equality by using the orthonormality of the system
$\\{F_{j}(r,\varphi)\\}_{1}^{\infty}$ we obtain the relation between the
orthonormal systems $\\{F_{j}\\}_{1}^{\infty}$ and $\\{Q_{j}\\}_{1}^{\infty}$
of the following form

$\begin{split}&\int_{t}^{1}\biggl{\\{}\frac{1}{2\pi}\int_{0}^{2\pi}\int_{0}^{1}\dfrac{1-r^{2}(1-\tau^{2})^{2}}{1-2r(1-\tau^{2})\cos(\varphi-\gamma)+r^{2}(1-\tau^{2})^{2}}\overline{F_{j}(r,\varphi)}r\,dr\,d\varphi\biggr{\\}}\cdot\dfrac{\tau^{2}-t^{2}}{2\tau}\,d\tau\\\\[5.0pt]
&\,=-\frac{1}{2\pi}\int_{0}^{2\pi}\int_{t}^{1}\dfrac{1-\bigl{(}\frac{\rho^{2}-t^{2}}{\rho}\bigr{)}^{2}}{1-2\bigl{(}\frac{\rho^{2}-t^{2}}{\rho}\bigr{)}\cos(\theta-\gamma)+\bigl{(}\frac{\rho^{2}-t^{2}}{\rho}\bigr{)}^{2}}\cdot
s_{j}\overline{Q_{j}(\rho,\theta)}\rho\,d\rho\,d\theta,\quad
j=1,2,\ldots\end{split}$ (4.11)

In both parts of the equality (4.11) we use the expansion of the Poisson
kernel

$\frac{1-r^{2}}{1-2r\cos\varphi+r^{2}}=1+2\sum_{n=1}^{\infty}r^{n}\cos
n\varphi.$

We obtain the equality of the two Fourier series in the orthogonal system
$\\{1/2,\cos\gamma,\sin\gamma,\ldots,$ $\cos n\gamma,\sin n\gamma,\ldots\\}$
in $L_{2}(0,2\pi)$. Equating the coefficients, we get the following system of
equations

$\left\\{\begin{split}&\frac{1}{2\pi}\int_{0}^{2\pi}\int_{0}^{1}\overline{F_{j}(r,\varphi)}\cdot
r\,dr\,d\varphi\cdot\int_{t}^{1}\dfrac{\tau^{2}-t^{2}}{2\tau}\,d\tau=-\frac{1}{2\pi}\int_{0}^{2\pi}\int_{t}^{1}s_{j}\overline{Q_{j}(\rho,\theta)}\rho\,d\rho\,d\theta,\\\\[8.0pt]
&\frac{1}{\pi}\int_{0}^{2\pi}\int_{0}^{1}\overline{F_{j}(r,\varphi)}\cdot
r^{n+1}\cos
n\varphi\,dr\,d\varphi\int_{t}^{1}(1-\tau^{2})^{n}\dfrac{\tau^{2}-t^{2}}{2\tau}\,d\tau\\\\[4.0pt]
&\quad=-\int_{t}^{1}\frac{1}{\pi}\int_{0}^{2\pi}s_{j}\overline{Q_{j}(\rho,\theta)}\cdot\cos
n\theta\,d\theta\biggl{(}\dfrac{\rho^{2}-t^{2}}{\rho}\biggr{)}^{n}\rho\,d\rho,\\\\[8.0pt]
&\frac{1}{\pi}\int_{0}^{2\pi}\int_{0}^{1}\overline{F_{j}(r,\varphi)}\cdot
r^{n+1}\cdot\sin
n\varphi\,dr\,d\varphi\int_{t}^{1}(1-\tau^{2})^{n}\dfrac{\tau^{2}-t^{2}}{2\tau}\,d\tau\\\\[4.0pt]
&\quad=-\int_{t}^{1}\frac{1}{\pi}\int_{0}^{2\pi}s_{j}\overline{Q_{j}(\rho,\theta)}\cdot\sin
n\theta\,d\theta\biggl{(}\dfrac{\rho^{2}-t^{2}}{\rho}\biggr{)}^{n}\rho\,d\rho,\quad
j=1,2,\ldots,\quad n=1,2,\ldots.\end{split}\right.$ (4.12)

We denote

$A_{nj}=\frac{1}{\pi}\int_{0}^{2\pi}\int_{0}^{1}\overline{F_{j}(r,\varphi)}\cos
n\varphi\cdot r^{n}\cdot r\,dr\,d\varphi,\quad j=1,2,\ldots,\quad
n=0,1,2,\ldots,$
$B_{nj}=\frac{1}{\pi}\int_{0}^{2\pi}\int_{0}^{1}\overline{F_{j}(r,\varphi)}\sin
n\varphi\cdot r^{n}\cdot r\,dr\,d\varphi,\quad j=1,2,\ldots,\quad
n=1,2,\ldots.$

From the first equation of the system (4.12) it is easy to find that

$\frac{1}{2\pi}\int_{0}^{2\pi}s_{j}\overline{Q_{j}(t,\theta)}d\theta=\frac{1}{2}A_{0j}\cdot\ln
t.$

If we denote by

$\omega_{nj}(\rho)=\frac{1}{\pi}\int_{0}^{2\pi}s_{j}\overline{Q_{j}(\rho,\theta)}\cos
n\theta\,d\theta\dfrac{1}{A_{nj}\rho^{n}},$

then the second equation transforms to

$\int_{t}^{1}(1-\tau^{2})^{n}\dfrac{\tau^{2}-t^{2}}{2\tau}\,d\tau=-\int_{t}^{1}(\rho^{2}-t^{2})^{n}\omega_{nj}(\rho)\rho\,d\rho,\quad
n=1,2,\ldots,\quad j=1,2,\ldots$ (4.13)

The third equation is transformed into the same equation (4.13), if we denote
by

$\omega_{nj}(\rho)=\frac{1}{\pi}\int_{0}^{2\pi}s_{j}\overline{Q_{j}(\rho,\theta)}\sin
n\theta\,d\theta\,\dfrac{1}{B_{nj}\rho^{n}}.$

We solve the equation (4.13) with respect to $\omega_{nj}(\rho)$. Note that

$\omega_{1j}(t)=-\dfrac{1-t^{2}}{2t^{2}},\quad\omega_{2j}(t)=-\dfrac{1-t^{4}}{4t^{4}};$

Further, we get the recurrence relation

$\begin{split}&(1-t^{2})^{n-k}=-\int_{t}^{1}(\rho^{2}-t^{2})^{n-k-2}\cdot(n-k)(n-k-1)\cdot
4t^{2}\omega_{nj}(\rho)\rho\,d\rho\\\\[5.0pt]
&\,+k\int_{t}^{1}(\rho^{2}-t^{2})^{n-k-1}\cdot(n-k)\cdot
4\omega_{nj}(\rho)\rho\,d\rho,\quad n=2,3,4,\ldots,\quad
k=0,1,2,\ldots,n-2.\end{split}$

This relation is equivalent to the Cauchy problem

$\omega_{nj}^{\prime}(t)+\frac{2n}{t}\omega_{nj}(t)=\frac{1}{t},\quad\omega_{nj}(1)=0.$

By solving this problem we get

$\omega_{nj}(t)=\frac{1-t^{-2n}}{2n},\quad j=1,2,\ldots.$

Now we have the following relations between the orthonormal systems
$\\{Q_{j}\\}_{1}^{\infty}$ and $\\{F_{j}\\}_{1}^{\infty}$:

$\left\\{\begin{split}&\frac{1}{2\pi}\int_{0}^{2\pi}s_{j}\overline{Q_{j}(t,\theta)}\,d\theta=\frac{1}{2}A_{0j}\cdot\ln
t,\quad n=0,\,\,j=1,2,\ldots\\\\[5.0pt]
&\frac{1}{\pi}\int_{0}^{2\pi}s_{j}\overline{Q_{j}(t,\theta)}\cdot\cos
n\theta\,d\theta=-A_{nj}\cdot\dfrac{t^{n}-t^{-n}}{2n},\\\\[5.0pt]
&\frac{1}{\pi}\int_{0}^{2\pi}s_{j}\overline{Q_{j}(t,\theta)}\cdot\sin
n\theta\,d\theta=-B_{nj}\cdot\dfrac{t^{n}-t^{-n}}{2n},\quad n=1,2,\ldots,\quad
j=1,2,\ldots.\end{split}\right.$ (4.14)

Satisfying the Volterra criterion (4.10), we obtained the relation (4.14). By
assumption $Q_{j}(t,\theta)$ is in $L_{2}(\Omega)$. Then the integral with
respect to $t$ on the left-hand sides of the system of equations (4.14)
exists. However, for an arbitrary orthonormal system
$\\{F_{j}\\}_{1}^{\infty}$, for $n=1,2,\ldots$, the integral on the right-hand
sides of the system of equations (4.14) with respect to $t$ from $0$ to $1$
does not exist. This means that there are no orthonormal systems
$\\{F_{j}\\}_{1}^{\infty}$ and $\\{Q_{j}\\}_{1}^{\infty}$ satisfying the
equality (4.10). This contradicts our assumption that there exists a Volterra
well-defined restriction $L_{K}$. Thus, Theorem 4.1 is proved. ∎

######  Corollary 4.2.

There does not exist a Volterra well-defined extension $L_{K}$ of the minimal
operator $L_{0}$ generated by the Laplace operator (4.1) in a Hilbert space
$L_{2}(\Omega)$, where $\Omega$ is the unit disk.

###### Proof.

Suppose that there exists a Volterra well-defined extension $L_{K}$ of the
minimal operator $L_{0}$. From $L_{0}\subset L_{K}$ it follows that
$L^{*}_{K}\subset L_{0}^{*}=\widehat{L}$. The adjoint of a Volterra operator
is a Volterra operator. Then we get a contradiction to Theorem 4.1. This
completes the proof of Corollary 4.2. ∎

It was also noticed that in the case $m=1$ there exists many Volterra well-
defined restrictins and extensions. Indeed, if in (4.11) the function
$u(r,\varphi)$ does not depend on the angle $\varphi$, then we get one-
dimensional equation

$\widehat{L}u\equiv-\Delta u=-u_{rr}-\frac{1}{r}u_{r}-=f(r),$

in the weighted space $L_{2}(r;0,1)$ with weight $r$. Then the Volterra
criterion (4.10) and the equation (4.14) determine the operator $K$ of the
following form

$Kf=\int_{0}^{1}f(t)\ln t\cdot t\,dt.$

$K$ corresponds to the well-defined restriction $L$ with domain

$D(L_{K})=\\{u\in W_{2}^{1}(r;\,0,1):u(0)=0\\}.$

Then the well-defined restriction $L_{K}$ is Volterra, because its inverse
operator

$u(r)=L^{-1}_{K}f=\int_{0}^{r}\ln\frac{t}{r}f(t)t\,dt,$

is Volterra in space $L_{2}(r;\,0,1)$.

######  Remark 4.3.

Theorem 4.1 is true for every bounded simply connected domain in the plane,
for which the Dirichlet problem is well-defined and there exists a conformal
mapping onto the unit disk.

######  Remark 4.4.

The generalization of Theorem 4.1 to the $m$-dimensional ball (where
$m\geqslant\leavevmode\nobreak\ 3$) does not cause problems but it is rather
cumbersome to write down.

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Bazarkan Nuroldinovich Biyarov,

Institute of Mathematics and Mathematical Modeling,

National Academy of Science Republic of Kazakhstan,

28 Shevchenko str.,

050010 Almaty, Kazakhstan

E-mail<EMAIL_ADDRESS>