# Leasing the Cloud-Edge Continuum, à la Carte††thanks: Research partly funded
by projects: Energy-aware management of software applications in Cloud-IoT
ecosystems (RIC2021PON_A18), funded with ESF REACT-EU resources by the Italian
Ministry of University and Research through the PON Ricerca e Innovazione
2014–20 and hOlistic Sustainable Management of distributed softWARE systems
(OSMWARE), UNIPI PRA_2022_64, funded by the University of Pisa, Italy.

Stefano Forti University of Pisa
Pisa, Italy<EMAIL_ADDRESS>University of Birmingham
Birmingham, United KingdomUniversity of Pisa
Pisa, Italy Ahmad Ibrahim University of Birmingham
Birmingham, United Kingdom<EMAIL_ADDRESS>University of Pisa
Pisa, Italy Antonio Brogi University of Pisa
Pisa, Italy<EMAIL_ADDRESS>
###### Abstract

Next-gen computing paradigms foresee deploying applications to virtualised
resources along a continuum of Cloud-Edge nodes. Much literature has focused
on how to place applications onto such resources so as to meet their
requirements. To lease resources to application operators, infrastructure
providers need to identify a portion of their Cloud-Edge assets to meet
application operators’ requirements. This article proposes a novel declarative
resource selection strategy, prototyped in Prolog, to determine a suitable
infrastructure portion that satisfies all requirements. The proposal is
showcased over a lifelike scenario.

##  1 Introduction

Recent utility computing paradigms (e.g. Fog, Edge, Mist, Cloud-IoT computing)
foresee deploying applications to virtualised resources along a continuum of
Cloud-Edge computing, storage and networking resources. Much literature
(surveyed, for instance, by [14] and [18]) has focussed on how to place
(multi-service) applications onto such resources so as to meet their stringent
functional and non-functional requirements (e.g. in terms of latency,
bandwidth, security, deployment location, Quality of Service (QoS), energy
sources, costs). Application placement has been proven an NP-hard problem
([2]), featuring worst-case exp-time complexity of $O(N^{S})$ for placing $S$
application services onto an infrastructure of $N$ nodes. Most of the proposed
solutions (i) take the perspective of application operators that need to
deploy their application services, and (ii) determine eligible placements by
trying to match each service with any node in the infrastructure.

We tackle here a different, yet complementary, problem by (a) taking the
perspective of infrastructure providers leasing Infrastructure-as-a-Service
(IaaS) resources in the Cloud-Edge continuum to paying application operators
that need to deploy their services, and by (b) identifying a portion of the
available Cloud-Edge infrastructure that meets the requirements set by
application operators and maximizes the expected profit that the
infrastructure provider will get from leasing such infrastructure portion.

Said otherwise, we consider a setting in which:

  1. (1)

Application operators request to lease a Cloud-Edge continuum portion,
featuring specific (hardware, software, security, network) resources and
suitably guaranteeing performance metrics (e.g. sustainability, availability),

  2. (2)

Infrastructure providers identify a portion that meets all the requirements of
applications operators and that maximises their profit, by exploiting an
automated (virtual) resource selection mechanism,

  3. (3)

Application operators can seamlessly deploy applications onto the leased
infrastructure portion by employing any appplication placement strategy of
their choice.

It is worth noting that such a change of perspective enables taming the exp-
time complexity of the application placement problem by considering only a
small portion of $n<N$ nodes in the available infrastructure. For instance,
our experiments in [6] and [8] show how placement execution times grow
exponentially at increasing sizes of the considered infrastructure and,
therefore, how reducing decision-making to smaller portions of such an
infrastructure brings considerable speed-ups, viz. from 2$\times$ up to a
$1000\times$ in realistic use cases.

This article presents a declarative strategy to automate the aforementioned
step (2) by solving the following resource selection problem:

Let $N$ be a set of heterogeneous nodes of a Cloud-Edge infrastructure managed
by an infrastructure provider. Let $p:N\rightarrow\mathbb{R}$ be the function
that defines the profit of the infrastructure provider for leasing a node. Let
$R$ be a request of resources of an application operator. A solution to the
considered resource selection problem is a portion $C=\\{n_{1},n_{2},\dots
n_{m}\\}\subseteq N$ that guarantees all requirements in $R$ while maximising
the infrastructure provider profit. Formally, given $N$, $p$, and $R$, we aim
at determining $C$ such that

$\operatorname*{arg\\!\max}_{\begin{subarray}{c}C\subseteq N\\\ C\models
R\end{subarray}}\sum_{n\in N}p(n).$

We present our novel open-source prototype111Available at
https://github.com/di-unipi-socc/fogCutter fogCutter, entirely written in
Prolog. fogCutter allows infrastructure providers to describe the capabilities
of their nodes and application operators to input their requests with a
declarative syntax – implying better readability, maintainability and
extensibility. fogCutter will benefit both infrastructure providers and
application operators. Infrastructure providers will identify suitable
infrastructure portions to lease to maximise their profit. Application
operators will reduce decision-making times for managing their distributed
software by considering an infrastructure portion built à la carte, specially
based on their deployment needs.

The rest of this article is organised as follows. We describe a motivating
scenario (Sect. 2) and illustrate the fogCutter Prolog prototype (Sect. 3),
which we then use over the motivating scenario (Sect. 4). Last, after
discussing some related work (Sect. 5), we conclude pointing out to directions
for future work (Sect. 6).

##  2 Motivating Scenario

Consider the Cloud-Edge continuum of Fig. 1, provided by a single
infrastructure operator across different geographical regions (viz. US, EU,
China). The infrastructure is made of 20 nodes: 10 access points (APs)
ap1–ap10, 7 more edge gateways n1–n7 and 3 Cloud datacentres c1–c3. Assume the
application operator connects to the network through ap3.

Figure 1: Available Cloud-Edge infrastructure.

Each node in the available Cloud-Edge infrastructure of Fig. 1 is
characterised by its geographical location (represented as a flag), security
capabilities (represented as icons), share of renewable energy it exploits to
run (indicated by the colour of the node), and availability (reported in the
bottom right side of each node). Finally, each node features a different
amount of hardware and is associated to a profit value established by the
infrastructure provider. Hardware and profit for each node are listed in Table
1.

As an example, the Cloud node c2 is located in the EU, features encrypted
storage, anti-malware and security audit capabilities, is powered by 50% of
renewable energy, has an availability of 95%, features 16 free hardware units,
and – when leased – brings a profit of 7 €/hour to the infrastructure
provider.

Node Id | Hardware Units | Profit |  | Node Id | Hardware Units | Profit
---|---|---|---|---|---|---
ap1 | 2 | 2.5 |  | n1 | 8 | 4
ap2 | 4 | 1 |  | n2 | 12 | 5
ap3 | 3 | 1.75 |  | n3 | 4 | 3
ap4 | 4 | 3 |  | n4 | 12 | 4
ap5 | 2 | 2.5 |  | n5 | 2 | 2.5
ap6 | 2 | 1.5 |  | n6 | 8 | 4
ap7 | 4 | 3 |  | n7 | 8 | 4
ap8 | 1 | 2.25 |  | c1 | 24 | 9
ap9 | 3 | 1.75 |  | c2 | 16 | 7
ap10 | 8 | 4 |  | c3 | 12 | 6

Table 1: Node hardware and profit.

As they are deployed at different geographical locations and interconnected
with different (wired and wireless) technologies, end-to-end latency and
bandwidth between nodes vary depending on the type of node and its location.
For instance, on average, APs in the same location reach out to each other
with latency of 10 ms and average symmetric bandwidth of 100 Mbps; those at
different locations incur in a latency of 150 ms with a bandwidth of 25 Mbps.
Besides, APs reach out edge nodes in their location with a latency of 60 ms,
an upload bandwidth of 30 Mbps and a download bandwidth of 200 Mbps (viz.
200/30 Mbps). Last, APs connect to Cloud nodes at their location experiencing
130 ms latency and a 90/15 Mbps connection; those at a different location are
reached out to with a latency of 200 ms and a 25/10 bandwidth. Data about the
QoS of all other end-to-end links can be found online222Available at:
https://github.com/di-unipi-
socc/fogCutter/blob/main/scenarios/smarttraffic.pl.

An application operator aims at deploying a Smart Traffic application onto a
portion of the available infrastructure. Smart traffic management is a
classical use case for Cloud-Edge applications ([3, 16]), having to collect
and process data from various sensors (i.e. live video feed from traffic
cameras, speed radar, air quality monitor sensor) and services (weather, date-
time, location) to detect traffic congestion, issue speeding fines,
dynamically adjust the price of tollgates and speed limits, control traffic
signal duration, and send alerts and notifications.

Assume that the application operator accesses the network from node ap3. To
deploy her software artefacts, the application operator estimates she will
need at least 20 free hardware units, reachable within 250 ms from node ap3
with a 10 Mbps bandwidth. Not to incur in too high management costs (i.e. time
to decide where to place/migrate her application services at deployment and
runtime, respectively), she would like her Cloud-Edge portion to include at
most 4 nodes (ap3 included).

Receiving the above request, the infrastructure provider might wonder the
following:

> Q1. Is there a portion of the available Cloud-Edge infrastructure that
> features at least 20 hardware units, and is made of at most four nodes with
> end-to-end connections featuring at least 10 Mbps bandwidth and at most 250
> ms latency?

Before she can deploy her Smart Traffic application, the application operator
realises she also would need to meet GDPR constraints and deploy the software
only to resources in the EU, provided with an antimalware and encrypted
storage capabilities. She consequently updates her request to the
infrastructure provider, who must answer the following question:

> Q2. Is there a portion of the available Cloud-Edge infrastructure that
> satisfies all requirements at Q1, is fully located in the EU, and features
> antimalware and encrypted storage on all nodes?

Lastly, the application operator also determines that she wants the resources
onto which to deploy her application to feature an overall availability of 85%
and to be likely to be powered by renewable energy at least 30% of the time.
Therefore, considering this new request, the infrastructure provider will look
for an answer to the following question:

> Q3. Is there a portion of the available Cloud-Edge infrastructure that
> satisfies all requirements at Q1 and Q2 , and features 85% availability and
> 30% sustainability score?

##  3 Our Methodology

In this section, we first illustrate how fogCutter enables describing
infrastructure capabilities and user requests with a simple and extensible
model (Sect. 3.1). Then, we illustrate the logic program that, by processing
such a model, determines solutions to the considered problem (Sect. 3.2).

###  3.1 Infrastructure and request model

Nodes are declared with a valid N (e.g. their IP address, a symbolic name
within the DNS) by means of facts like

node(N).

Each node is associated to the profit that its lease brings to the
infrastructure provider, as in

profit(N, Profit).

After declaring a node, it is possible to declare its capabilities333Node
capabilities can be obtained by tools targeting Cloud-Edge infrastructure
monitoring, e.g. [7]. through facts like

nodeCap(N, CapabilityType, FeaturedCapabilities).

that specifies the CapabilityType and the currently FeaturedCapabilities for
such a type at node N. For instance, node c1 of the motivating scenario of
Sect. 2 is declared as in:

node(c1). nodeCap(c1, hardware, 24). nodeCap(c1, location, us). nodeCap(c1,
security, [antimalware, encryptedStorage, audit]). nodeCap(c1, availability,
0.99). nodeCap(c1, sustainability, 0.75).

Similarly, the capabilities of end-to-end links between nodes N and M can be
declared as facts like

linkCap(N, M, Capability, FeaturedCapabilities).

As an example, the end-to-end link connecting nodes ap1 and n1 in the previous
example can be expressed through the following facts:

linkCap(ap1,n1,latency,60). linkCap(ap1,n1,bandwidth,30).
linkCap(n1,ap1,latency,60). linkCap(n1,ap1,bandwidth,200).

We distinguish two kinds of capability types: local and global. Local
capabilities are checked on a single resource, either a node or a link. Such
capability types are declared through facts like

capType(Capability, Resource, ComparisonOperator).

that specify the Resource for which such capability can be declared (viz. node
s or link s), a Comparison- Operator to check whether the FeaturedCapabilities
at a node satisfy a request on such Capability. For instance, fogCutter
defines the following four local capabilities

capType(security, node, supset). capType(location, node, member).
capType(latency, link, smaller). capType(bandwidth, link, greater).

that enable checking444For the sake of conciseness, comparison operators are
not listed here. The interested reader can find them in the online project
repository. the set of security capabilities featured by a node against a set
of requested capabilities, the location of a node against a list of permitted
locations, and the latency and bandwidth featured by communication links
against an upper and a lower bound, respectively.

Global capabilities are instead checked against a portion of the available
infrastructure and also need to be aggregated. They can be declared through
facts like

capType(CapabilityType, Resource, ComparisonOperator, AggregationOperator).

that also specify a suitable AggregationOperator to compute them over a
selected portion of the whole infrastructure. Again, fogCutter features the
following default capabilities

capType(hardware, node, smaller, sum). capType(availability, node, smaller,
product). capType(sustainability, node, smaller, product).

that correspond to the total amount of hardware of an infrastructure portion,
and to the overall availability555We define availability as the probability
that a system is operational at a given time. It is obtained over a portion of
an infrastructure by multiplying the availability of the single nodes in the
portion. and sustainability666We define sustainability as the probability that
a system is fully powered by renewable energy. It is obtained over a portion
of an infrastructure by multiplying the sustainability of the single nodes in
the portion.  featured by such a portion. The model above is designed to be
easily extensible by taking into account further node/link capabilities of
arbitrary (local and global) types.

fogCutter receives requests for Cloud-Edge portions through facts like

request(RequestId, SourceNode, MaxNodes, Requirements).

where RequestId is a unique identifier for the request, SourceNode is the
identifier of the user node from which the request is issued, MaxNodes is the
maximum number of nodes the user wants in her portion, and requirements is a
list of pairs (CapabilityType, R) that specifies which global/local capability
is needed and the target amount or threshold R that is requested by the user.
For instance, the fact

request(req42, ap3, 3, [(hardware,20), (latency,250),(bandwidth,10),
(security,[antimalware, encryptedStorage]), (location,[eu]), (availability,
0.85), (sustainability, 0.3)]).

requires an infrastructure portion of at most 3 nodes in addition to ap3, with
at least 20 hardware units, reachable through end-to-end link featuring at
most 250 ms latency and at least 10 Mbps bandwidth, featuring antimalware and
encrypted storage capabilities, located in the EU, and with at least 85%
availability and 30% sustainability.

###  3.2 Resource Selection: a Logic Programming Strategy

Predicate portion/3 inputs a RequestId, determines a Portion of the available
Cloud-Edge continuum that satisfies all the request’s requirements and
associates the Portion with an estimate of the Profit that the infrastructure
provider will get from leasing such Portion.

portion(RequestId, (Portion, Profit)) :- request(RequestId, N, MaxNodes,
Reqs), splitRequirements(Reqs, NodeReqs, LinkReqs, GlobalReqs), portion(N,
MaxNodes, NodeReqs, LinkReqs, GlobalReqs, [N], Portion),
portionProfit(Portion, Profit). Predicate portion/3 first fetches the data (N,
MaxNodes, Reqs) of the input RequestId. Then predicate splitRequirements/4
simply partitions the list of request’s requirements Reqs into three lists,
depending on whether they refer to local capabilities of nodes (NodeReqs) or
of links (LinkReqs), or on whether they are global requirements on the overall
infrastructure portion (GlobalReqs).

Predicate portion/7 determines a Portion of the infrastructure that contains
node N and at most other MaxNodes nodes and satisfies all the NodeReqs,
LinkReqs and GlobalReqs requirements.

portion(N, MaxNodes, NodeReqs, LinkReqs, GlobalReqs, I, NewI) :- MaxNodes ¿ 0,
“+ satisfiesGlobalReqs(GlobalReqs, I), node(M), “+ member(M,[N—I]),
satisfiesNodeReqs(M,NodeReqs), satisfiesLinkReqs(M,N,LinkReqs), NewMaxNodes is
MaxNodes-1, portion(N, NewMaxNodes, NodeReqs, LinkReqs, GlobalReqs, [M—I],
NewI). portion(˙, ˙, ˙, ˙, GlobalReqs, I, I) :-
satisfiesGlobalReqs(GlobalReqs, I). Until all the global requirements
GlobalReqs are satisfied, predicate portion/7 recursively adds to the
infrastructure portion a new node M that satisfies all the NodeReqs and
LinkReqs local requirements.

Predicate satisfiesGlobalReqs/2 verifies whether the (non empty)
infrastructure portion determined so far satisfies the list GlobalReqs of
global requirements.

satisfiesGlobalReqs(Rs, [I—Is]) :- satisfies(Rs, [I—Is]). satisfies([], ˙).
satisfies([(P,V,Op,Ag)—Rs], I) :- findall(Vm, (member(M,I),nodeCap(M,P,Vm)),
Vs), aggregatedListValue(Vs,Ag,AV), compareValue(V,Op,AV), satisfies(Rs, I).
aggregatedListValue([V—Vs],Ag,AV):- aggregatedListValue(Vs,Ag,AV2),
aggregatedValue(V,Ag,AV2,AV). Predicate satisfies/2 verifies that each global
requirement (P,V,Op,Ag) is satisfied by the infrastructure portion I
determined so far. After fetching into a list Vs all the values of property P
in the nodes M in I, satisfies/2 aggregates all the values in Vs according to
the indicated aggregator operator Ag and compares the obtained aggregated
value AV with the value P of the considered global requirement according to
the indicated comparison operator Op. For instance, the aggregator operators
sum and product and the comparison operators smaller and greater are defined
as follows:

aggregatedListValue([],sum,0). aggregatedListValue([],product,1).
aggregatedValue(X,sum,Y,Z) :- Z is X+Y. aggregatedValue(X,product,Y,Z) :- Z is
X*Y. compareValue(X,smaller,Y) :- X ¡ Y. compareValue(X,greater,Y) :- X ¿ Y.
Predicates satisfiesNodeReqs/2 and satisfiesLinkReqs/2 verify that each local
node and link requirement is satisfied if the new node M is added to the
infrastructure portion I determined so far.

satisfiesNodeReqs(M,[(P,V,Op)—Rs]) :- nodeCap(M,P,Vm), compareValue(Vm,Op,V),
satisfiesNodeReqs(M,Rs). satisfiesNodeReqs(˙,[]).

satisfiesLinkReqs(M,N,[(P,V,Op)—Rs]) :- linkCap(M,N,P,Vmn),
compareValue(Vmn,Op,V), satisfiesLinkReqs(M,N,Rs). satisfiesLinkReqs(˙,˙,[]).
Predicate portionProfit/2 associates an infrastructure Portion with an
estimate of the Profit that the infrastructure provider will get from leasing
such Portion. The estimate of the overall Profit can be simply obtained by
summing the profit associated with each node in the determined Portion.

portionProfit([N—Ns], Profit) :- profit(N, N˙Profit), portionProfit(Ns,
Ns˙Profit), Profit is N˙Profit + Ns˙Profit. portionProfit([], 0). Finally,
predicate fogCutter/2 inputs a RequestId and determines the Portion of the
available Cloud-Edge continuum that satisfies all the request’s requirements
and that corresponds to the highest estimated Profit for the infrastructure
provider.

fogCutter(RequestId, (Portion,Profit)) :- setof(X, portion(RequestId, X),
CandidatePortions), bestPortion(CandidatePortions, (Portion,Profit)).
bestPortion(CandidatePortions, (Portion,Profit)) :- member((Portion,Profit),
CandidatePortions), “+ (member((˙,HigherProfit), CandidatePortions), Profit ¡
HigherProfit).

##  4 fogCutter in Action

In this section, we exploit the fogCutter prototype to answer questions Q1, Q2
and Q3 of Sect. 2.

We start by retaking question Q1:

> Q1. Is there a portion of the available Cloud-Edge infrastructure that
> features at least 20 hardware units, and is made of at most four nodes with
> end-to-end connections featuring at least 10 Mbps bandwidth and at most 250
> ms latency?

The request of the application operator is denoted by a fact like:

request(req42, ap3, 3, [(hardware,20), (latency,250), (bandwidth,10)]).

The infrastructure provider can suitably query predicate fogCutter/2,
obtaining the following as a result:

?- fogCutter(req42,P). P = ([ap3, ap8, c1, c2], 20.0) ; false.

Therefore, there exist only one best candidate portion to lease to the
application operator, associated to a profit of 20 €/h, and spanning node ap3,
ap8, c1 and c2. Note that there exist other 412 alternative solutions777We
determined the number of alternative solutions by querying predicate
setof(X,portion(RequestId, X),CandidatePortions) and counting the found
results. that satisfy all the requirements of the application operator but
feature a lower profit.

We now consider question Q2:

> Q2. Is there a portion of the available Cloud-Edge infrastructure that
> satisfies all requirements at Q1, is fully located in the EU, and features
> antimalware and encrypted storage on all nodes?

In this case, the request of the application operator is extended with
security and location requirements, and can be expressed through the fact:

request(req42, ap3, 3, [(hardware,20), (latency,250), (bandwidth,10),
(security,[antimalware, encryptedStorage]), (location,[eu])]).

Querying again predicate fogCutter/2, the infrastructure provider obtains:

?- fogCutter(req42,P). P = ([ap3, ap4, c2, n2], 16.75) ; P = ([ap3, ap7, c2,
n2], 16.75) ; false.

Therefore, this request can be satisfied by two different candidate portions
of the available infrastructure that bring a maximum profit of 16.75 €/h, and
span nodes ap3, ap4, c2 and n2, or ap3, ap7, c2 and n2, respectively. Also in
this case, other 35 alternative portions exist that do not maximise the profit
of the infrastructure provider.

Last, we focus on answering question Q3:

> Q3. Is there a portion of the available Cloud-Edge infrastructure that
> satisfies all requirements at Q1 and Q2 , and features 85% availability and
> 30% sustainability score?

Here, the request of the application operator becomes:

request(req42, ap3, 3, [(hardware,20), (latency,250),(bandwidth,10),
(security,[antimalware, encryptedStorage]), (location,[eu]), (availability,
0.85), (sustainability, 0.3)]).

The query to fogCutter/2 from the infrastructure provider returns:

P = ([ap3, ap4, c2], 11.75).

Hence, the request can be satisfied by providing the application operator with
the portion made from nodes ap3, ap4 and c2, with an associated profit of
11.75 €/h. For this last question, such a portion also represents the only
eligible solution to the considered resource selection problem instance.

Table 2 recapitulates on the performance of the experiments in our motivating
scenario, showing the number of Prolog inferences and the CPU time
needed888Prolog inferences and CPU time are obtained by relying on SWI-
Prolog’s time/1 predicate. Experiments are obtained by running fogCutter in
SWI-Prolog v. 8.4.3 ([20]) on a machine equipped with Apple Silicon M1 and
16GB of RAM. to answer the queries. It also lists the number of optimal
solutions (i.e. those that maximise profit and satisfy all requirements set by
the application operator) and the number of eligible solutions (i.e. those
that satisfy all requirements set by the application operator). It is worth
noting that fogCutter determines an optimal solution almost instantaneously
for all three questions.

Question | Prolog Inferences | CPU Time | Optimal Solution(s) | Eligible Solution(s)
---|---|---|---|---
Q1 | 1,712,813 | 0.186 s | 1 | 413
Q2 | 208,776 | 0.033 s | 2 | 37
Q3 | 343,393 | 0.048 s | 1 | 1

Table 2: Performance of fogCutter over the motivating scenario.

More in general, given the maximum number MaxNodes of nodes requested for a
portion, the worst-case time complexity for running fogCutter is
$O(N^{\texttt{\small MaxNodes}})$. As mentioned before, determining valid
placements of $S$ services onto $N$ nodes also features an exp-time complexity
of $O(N^{S})$. Setting, however, $\texttt{\small MaxNodes}<S$ permits to
quickly identify a portion of the infrastructure where to deploy applications
and can dramatically reduce decision-making times related to placements.
Indeed, as shown for instance in [6] and [8], considering smaller problem
instances can reduce execution times from minutes to milliseconds, with speed-
ups from 2$\times$ up to a $1000\times$ in realistic use cases. Also, note
that portions of the Cloud-Edge continuum can be pre-computed by the
infrastructure provider over pre-defined request templates, and be immediately
ready to be returned to application operators, when needed.

##  5 Related Work

Resource allocation is an important challenge in the Cloud-edge continuum.
Lahmar and Boukadi [10] conducted a comprehensive survey of resource
allocation challenges that should be considered while deploying an application
over a Cloud-Edge continuum. A similar survey by Shakarami et al. [17]
classified existing resource allocation algorithms into five categories i.e.,
heuristic, framework, model, machine learning and game theory based, discussed
their pros and cons and common performance metrics that are relevant for the
application operators. Faticanti et al. [4] tackle a different problem which
is to maximise the profit of the infrastructure provider by maximising the
number of Fog applications in a batch that can be deployed over the Cloud-Edge
continuum. To accomplish that their algorithm partition the application’s
components into two groups i.e., Cloud-only and Fog-only. Using throughput as
QoS metrics, their algorithm tried to recommend appropriate regions for these
components.

The role of the broker in the Cloud is also well investigated. Mei et al. [12]
discuss the role and necessity of brokers in a Cloud-Edge continuum. According
to them, the presence of a Cloud broker can lead to better pricing for the end
user and at the same time, high utilisation of providers’ resources. Cloud
brokers can also earn profit in these transactions. Wang et al. [19] discussed
the idea that a Cloud broker can maximise resource utilisation by
differentiating between latency sensitive tasks (such as Fog applications)
versus non-latency sensitive applications. They highlighted different
strategies to deal with fairly pricing the two categories of applications for
the broker that could make the offer attractive for a wide variety of users
and providers.

The concept of IoT brokers was described in [13, 15]. Sciancalepore et al.
[15] discussed a strategy that is aimed towards IoT brokers that can service
different customers using IoT slices. In their approach, resources can be
increased for a given slice as per demand in real-time. A subscription based
model for IoT providers on the Cloud-Edge continuum was described by Niyato et
al. [13]. In their scheme, IoT owners and IoT data users have to subscribe to
a service that enables data exchange data based on an agreed pricing model.
They also described different strategies for IoT owners to bundle their
offerings and make them attractive to the Cloud-Edge continuum users.

Slicing in a network is also thoroughly investigated although it is a slightly
different problem for a different setting (i.e., telecom) [21]. IoT slicing is
another related topic that deals with creating slices (partitions) of IoT
resources in a Cloud-Edge continuum to satisfy different types of user
requirements. Fernandez et al. [5] proposed a concept of an IoT orchestration
service that could satisfy a customer request, especially on a resource
constraint platform using IoT slices. Similarly, Hwang et al. [9] described a
central service to manage incoming requests (tasks), partition the
infrastructure into different slices of the IoT sensors and distribute the
tasks to different IoT slices to answer the query. Experimental results
conducted by Luthra et al. [11] show that different operator placement
algorithms are needed to suit the dynamic environment and conflicting QoS
demands.

Our approach is different from the existing approaches as we try to solve a
resource selection problem marginally addressed before. Our approach allows a
declarative and extensible definition of parameters which can be adopted as
per application operator and infrastructure operator requirements.

##  6 Concluding Remarks

The problem of selecting an infrastructure portion matching the requirements
set by an application operator is challenging in the Cloud-Edge continuum due
to the large number of assets with heterogeneous capabilities. The problem is
important both for application operators, who need to suitably deploy their
applications, and for infrastructure providers, who wish to maximize their
profit in leasing infrastructure resources to application operators.

In this article, we have presented a declarative resource selection strategy
for the Cloud-Edge continuum, implemented in the open source prototype
fogCutter. Using a motivating example, we have shown how fogCutter can be
successfully exploited to reply to different requests of customers (viz.
application operators). The main advantages of the declarative solution
implemented by fogCutter are its readability and ease of extensibility to
include new types of requirements. Indeed, it is straightforward to include
further local (e.g. availability of IoT devices or specialised hardware, link
jitter) and global requirements (e.g. QoS-assurance, nodes’ trust) by simply
programming new capType predicates and the associated comparison and
aggregation operators. Directions for future work include:

  * •

returning partial solutions (when not all requirements can be met),
considering partial allocations of Cloud-Edge resources, and allowing
customers to declare hard and soft constraints in requests,

  * •

performing a thorough scalability assessment at varying sizes of the input
Cloud-Edge infrastructure and number of considered requirements for the
portion to be determined, also extending fogCutter with heuristic strategies,
and

  * •

assessing the tool with real data (possibly against alternative solutions,
e.g. constraint-based) in simulated, laboratory or testbed settings, also in
pipeline with placement strategies to better estimate the obtainable speed-
ups.

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# Smoothing estimates for weakly nonlinear internal waves in a rotating ocean

M. B. Erdoğan and N. Tzirakis  Department of Mathematics
University of Illinois
Urbana, IL 61801, U.S.A<EMAIL_ADDRESS>Department of Mathematics
University of Illinois
Urbana, IL 61801, U.S.A<EMAIL_ADDRESS>
###### Abstract.

In this paper we study the effect of rotation on nonlinear wave phenomena in
weakly dispersive media modeled by the Korteweg-de Vries equation on the real
line. It is well known that smoothing in the case of the KdV equation with
periodic boundary conditions is a result of the presence of high frequency
waves that weaken the nonlinearity through time averaging, [1], [12]. High
frequency interactions through normal form transformations can be thought of
as adding a fast rotating term in certain toy PDE models that arrest the
development of singularities, [1]. It is crucial for this phenomena that the
zero Fourier mode can be removed due to the conservation of the mean. On the
real line this mechanism breaks down as the resonance sets close to zero
frequency are sizable and normal form transformations are not useful, [20],
and hence smoothing fails. The model we study is a perturbation of the KdV
equation on a rotating frame of reference. A combination of a suitable normal
form transformation along with the restricted norm method of Bourgain [2]
implies smoothing on the real line.

The first author is partially supported by NSF grant DMS–2154031.

##  1\. Introduction

On this paper we study a perturbation of the Korteweg-de Vries (KdV) equation
that models weakly nonlinear waves in the ocean taking into account the
rotation of the earth, [22], [23]. Unlike KdV, the equation is not known to be
completely integrable, see Section 3.2 in [17]. The main purpose of our
exposition is to prove that, when the data are in classical Sobolev spaces,
the nonlinear part of the equation on the real line is smoother than the
initial data. Nonlinear smoothing in dispersive equations has recently played
an important role in analyzing the local and global behavior of low regularity
solutions [3], [24], [16]. There exist a variety of applications of nonlinear
smoothing in both linear and nonlinear set-ups for dispersive partial
differential equations (PDEs) with various boundary conditions. The most
common situations are the open boundary case of the real line and bounded
intervals with periodic boundary conditions. Also of interest are the
Euclidean spaces in general, semi-infinite intervals, and bounded intervals
where boundary conditions for the values of the data or their derivatives are
present. Some 30 years ago through the seminal work of Bourgain, it was
realized that nonlinear smoothing plays an important role in obtaining global-
in-time solutions for dispersive equations and systems with infinite energy,
[5]. Smoothing was also used to obtain higher order Sobolev norms estimates
for globally defined solutions of dispersive equations that do not scatter,
[4].

For periodic problems smoothing has been used to study the problem of
dispersive quantization (or Talbot effect) in dispersive PDEs, see [14], [6],
[11] and the references therein. Moreover, for both periodic and non-periodic
PDEs smoothing leads to the asymptotic compactness of the global evolution,
which in turn implies the existence and uniqueness of global attractors for
forced and dissipative dispersive PDE, [15]. Finally another important
application is the almost everywhere convergence of solutions of dispersive
equations to the initial data in Sobolev spaces. For similar developments the
interested reader can consult the book [16].

In the current manuscript we prove nonlinear smoothing for a variant of the
KdV equation after transforming it using normal form transformations. More
precisely we study the Cauchy problem on ${\mathbb{R}}$ associated with the
equation

(1) $\left\\{\begin{array}[]{l}u_{t}+\beta
u_{xxx}-\gamma\partial_{x}^{-1}u+\partial_{x}(u^{2})=0,\,\,\,\,x\in{\mathbb{R}},t\in{\mathbb{R}},\\\
u(x,0)=f(x),\end{array}\right.$

which describes weakly nonlinear wave processes in the ocean taking into
account the earth rotation, [22], [23]. In the literature, this equation is
often called the Ostrovsky equation. The parameter $\gamma$ measures the
effect of rotation and the parameter $\beta$ the strength of the dispersion.
Note that the operator $\partial_{x}^{-1}$ is defined on the Fourier side by

$\widehat{\partial_{x}^{-1}f}(\xi)=\frac{\widehat{f(\xi)}}{i\xi}.$

In our paper, we consider the case where $\beta\cdot\gamma>0$. In particular,
it is enough to consider $\beta=\gamma=1$. Later in an Appendix we show that
the nonlinear solution is not smoother than the initial data when
$\beta\cdot\gamma<0$. For any $\beta\cdot\gamma\neq 0$ the equation is
optimally well–posed at the $H^{-\frac{3}{4}}$ level, [18], [21], [26]. A
result of Tsugawa, [25] slightly improved the result in [18] by proving
well–posedness in anisotropic Sobolev spaces that contains the $-\frac{3}{4}$
threshold as a special case. This last result also proves that the solution of
the equation (1) converges to the solution of the KdV equation when the
rotation parameter $\gamma$ goes to $0$ and the common initial data is in the
$L^{2}$ space. To prove such a result the author needed to obtain a new,
uniform in $\gamma$, bilinear estimate.

Smooth solutions of (1) satisfy $L^{2}$ conservation, namely
$\|u(t)\|_{L_{x}^{2}({\mathbb{R}})}=\|f\|_{L^{2}({\mathbb{R}})}$. The above
results then establish that the solutions are in fact global at least for
$L^{2}$ data. This result was extended in [19] to include global-in-time
solutions for any $s>-\frac{3}{10}$.

In [18], the authors prove that the initial value problem (1) is locally
well–posed in $H^{s}$, $s>-\frac{3}{4}$, by constructing a unique solution
$u\in X^{s,b}$ for any $s>-\frac{3}{4}$ and $b>\frac{1}{2}$ valid in $[-T,T]$
where $T=T(\|f\|_{H^{s}})$. In effect, the local solution can be obtained by
means of Picard iterations of the Duhamel operator

$u=e^{-tL}f(x)-e^{-tL}\int_{0}^{t}e^{sL}(uu_{x})ds.$

Here we prove that the difference of the nonlinear solution with the linear
evolution (the integral term above) is in a smoother Sobolev space than the
initial data as long as the solution is locally well–posed by the result in
[18]. For the original KdV equation smoothing is not possible on the real
line, [20]. To accomplish this we apply the differentiation by parts method of
[1]. It may be possible to obtain a smoothing estimate by using the sharp
result of [18] within the framework of the $X^{s,b}$ method, but the smoothing
gained would be rather marginal, much less than the values we outline here. We
now state the main result of this paper:

######  Theorem 1.1.

Fix $s>-\frac{3}{4}$ and $a<\min(s+\frac{3}{4},\frac{1}{2})$. Consider the
real valued solution of the equation (1) (with $\beta=\gamma=1$) on
$\mathbb{R}\times[0,T_{LWP}]$ with initial data $u(x,0)=f(x)\in
H^{s}(\mathbb{R})$. Then $u(t)-e^{tL}f\in C^{0}_{t}H^{s+a}_{x}$ and

$\|u(t)-e^{-tL}f\|_{H^{s+a}}\leq C(s,\|f\|_{H^{s}}),$

on $[0,T_{LWP}]$ where $L=\partial_{xxx}-\partial_{x}^{-1}$.

As in [12], one can concatenate the smoothing statement and obtain a global
smoothing estimate in the full range where global wellposedness is known,
$s>-\frac{3}{10}$.

To prove Theorem 1.1 we introduce a normal form transformation on (1) which is
similar to the one we applied to the periodic KdV problem in [12]. We should
mention here that although Sobolev space smoothing fails for the KdV equation
on ${\mathbb{R}}$, Colliander et al. in [9] proved that the nonlinear part of
the KdV equation on the real line is smoother than the initial data for
solutions that are restricted on high frequencies. Thus in general low–low and
high–high interactions are not problematic in KdV type equations. Therefore,
we split the nonlinearity into two parts and estimate them separately. The
first part includes low frequencies and high-high interactions for which the
smoothing can be established without a normal form transformation. These terms
provide an upper bound on the smoothing estimate $a=s+\frac{3}{4}$. As it is
usually the case, the smoothing fails at the $-\frac{3}{4}$ threshold. The
harder to estimate terms are in the second part, which includes the low-high
interactions (corresponding to the resonant terms for periodic KdV). We handle
these terms with the help of a normal form tranformation.

The main idea in our result rests on the fact that for the perturbation of KdV
that we study the dispersive symbol can be lower bounded away from zero (see
(4) below), and thus the smoothing effect takes hold even for the low–high
interactions on the real line. For problems with periodic boundary conditions,
because of the conservation of the mean, one can remove the zero Fourier mode
and hence the low–high interactions do not cause a problem. As such there are
now many smoothing results for certain variants of the KdV equation on
$\mathbb{T}$. We should also note that, in [7, 8], the authors obtained many
smoothing estimates for dispersive PDEs on ${\mathbb{R}}^{d}$ with
nonlinearities of order 3 and higher. However, their method fails for the
quadratic nonlinearities as it is expected from the failure of smoothing in
the KdV case.

For our model the improvement in the estimates come from the phase function
$\phi(\xi)=\xi^{3}-\frac{1}{\xi}$. After the normal form transformation and
because of the nonlinear interactions, the following function is introduced in
the denominator of the multilinear estimates:

(2)
$\phi(\xi)-\phi(\xi_{1})-\phi(\xi-\xi_{1})=\xi^{3}-\xi_{1}^{3}-(\xi-\xi_{1})^{3}-\frac{1}{\xi}+\frac{1}{\xi_{1}}+\frac{1}{\xi-\xi_{1}}\\\
=\frac{3\xi^{2}\xi_{1}^{2}(\xi-\xi_{1})^{2}+\xi^{2}+\xi_{1}^{2}-\xi\xi_{1}}{\xi\xi_{1}(\xi-\xi_{1})},$

which implies

(3)
$\frac{1}{|\phi(\xi)-\phi(\xi_{1})-\phi(\xi-\xi_{1})|}\lesssim\frac{|\xi\xi_{1}(\xi-\xi_{1})|}{\xi^{2}\xi_{1}^{2}(\xi-\xi_{1})^{2}+\xi^{2}+\xi_{1}^{2}}$

Notice that for the low–high interactions ($|\xi_{1}|\ll|\xi|$), this term can
be further bounded by

(4)
$\frac{1}{|\phi(\xi)-\phi(\xi_{1})-\phi(\xi-\xi_{1})|}\lesssim\frac{|\xi_{1}|}{\xi^{2}\xi_{1}^{2}+1}=K(\xi,\xi_{1}),$

and that the maximum of the $K$ function is of the order $\frac{1}{\xi}$. This
recovers the derivative of the nonlinearity. For the details, see Section 3.

The paper is organized as follows. In Section 2, we introduce our notation and
define the spaces that we use. In addition, we state an elementary lemma that
we use throughout the paper in order to prove the multilinear smoothing
estimates. Section 3 introduces the normal form method for equation (1) and
contains the heart of our arguments where different interactions are
estimated. Section 4 contains an Appendix where we show that smoothing fails
for equation (1) for any $\gamma\leq 0$ when $\beta=1$. As we have already
mentioned, the case $\gamma=0$ corresponds to the KdV result of [20].

##  2\. Notation

Recall that for $s\in\mathbb{R}$, $H^{s}(\mathbb{R})$ is defined as a subspace
of $L^{2}$ via the norm

$\|f\|_{H^{s}(\mathbb{R})}:=\sqrt{\int_{\mathbb{R}}\langle\xi\rangle^{2s}|\widehat{f}(\xi)|^{2}\
d\xi},$

where $\langle\xi\rangle:=(1+{\xi}^{2})^{1/2}$ and
$\widehat{f}(\xi)=\int_{\mathbb{R}}f(x)e^{-2\pi ix\xi}dx$ are the Fourier
coefficients of $f$. Plancherel’s theorem takes the form

$\sum_{\xi\in{\mathbb{R}}}|\widehat{f}(\xi)|^{2}=\int_{{\mathbb{R}}}|f(x)|^{2}dx.$

We denote the linear propagator of the equation as $e^{tL}$, where it is
defined on the Fourier side as
$\widehat{(e^{tL}f)}(\xi)=e^{-it\phi(\xi)}\widehat{f}(\xi)$ and
$\phi(\xi)=\xi^{3}-\frac{1}{\xi}$. We also use $(\cdot)^{+}$ to denote
$(\cdot)^{\epsilon}$ for all $\epsilon>0$ with implicit constants depending on
$\epsilon$.

The Bourgain spaces, $X^{s,b}$, will be defined as the closure of compactly
supported smooth functions under the norm

$\|u\|_{X^{s,b}}:=\|e^{-tL}u\|_{H^{b}_{t}(\mathbb{R})H^{s}_{x}(\mathbb{R})}=\|\langle\tau-\phi(\xi)\rangle^{b}\langle\xi\rangle^{s}\widehat{u}(\xi,\tau)\|_{L_{\tau}^{2}L^{2}_{\xi}}.$

We close this section by presenting an elementary lemma that will be used
repeatedly. For the proof see [13].

######  Lemma 2.1.

If $\beta\geq\gamma\geq 0$ and $\beta+\gamma>1$, then

$\int\frac{1}{\langle x-a_{1}\rangle^{\beta}\langle
x-a_{2}\rangle^{\gamma}}dx\lesssim\langle
a_{1}-a_{2}\rangle^{-\gamma}\phi_{\beta}(a_{1}-a_{2}),$

where

$\phi_{\beta}(a)\sim\begin{cases}1&\beta>1\\\ \log(1+\langle
a\rangle)&\beta=1\\\ \langle a\rangle^{1-\beta}&\beta<1.\end{cases}$

##  3\. Normal form transform

In this section we apply the normal form transformation to the equation (1)
following the differentiation by parts method of [1]. See Proposition 3.1 for
the transformed equation. A similar method was used in our work in [12] for
the periodic KdV (also see [10] for an application on ${\mathbb{R}}$,
especially Proposition 6.1). Notice that for more regular data we gain half a
derivative in our smoothing estimate, which is sharp (see Remark (3.3)), while
in the case of periodic KdV we gained a full derivative. In both cases the
formulas contain boundary terms, resonant terms and non-resonant terms. The
non-resonant terms can be further transformed if needed. As noted above, the
upper bound in the smoothing estimate for more regular solutions comes from
the boundary terms, $B(u)$ in (7), which indicates that further applications
of the normal form transformation that exist in the literature may not improve
the result.

To perform differentiation by parts we define
$L=\partial_{xxx}-\partial_{x}^{-1}$ and $N(u)=\partial_{x}(u^{2})$, and we
write (1) for $\beta=\gamma=1$ as

(5) $u_{t}+Lu+N(u)=0.$

We observe that smoothing for the low frequency part and the high-high
interactions can be obtained without normal-forms. Therefore, we further write

$N(u)=R(u)+\widetilde{N}(u),$

where

$R(u):=P_{|\xi|\lesssim 1}N(u)+P_{|\xi|\gg 1}N(P_{>|\xi|/100}u),$
${\mathcal{F}}(\widetilde{N}(u))(\xi)=i\xi\int_{\mathbb{R}}m(\xi,\xi_{1})u(\widehat{\xi}_{1},t)u(\widehat{\xi-\xi_{1}},t)d\xi_{1}$

where

$m(\xi,\xi_{1})=\chi_{|\xi|\gg
1}\big{(}\chi_{|\xi_{1}|\leq|\xi|/100}+\chi_{|\xi-\xi_{1}|\leq|\xi|/100}\big{)}.$

Letting $\phi(\xi)=\xi^{3}-\tfrac{1}{\xi}$, we have

$e^{Lt}u={\mathcal{F}}^{-1}\big{[}e^{-i\phi(\xi)t}u(\widehat{\xi},t)\big{]}.$

The following proposition is the main tool for proving Theorem 1.1.

######  Proposition 3.1.

The solution $u$ of equation (1) satisfies

(6)
$\partial_{t}\left(e^{Lt}u-e^{Lt}B(u)\right)=-e^{Lt}\big{(}R(u)+NR(u)\big{)},$

where

(7) $\widehat{B(u)}(\xi)=\xi\int
m(\xi,\xi_{1})\frac{u_{\xi_{1}}u_{\xi-\xi_{1}}}{\phi(\xi)-\phi(\xi_{1})-\phi(\xi-\xi_{1})}d\xi_{1},$
(8) $\widehat{NR(u)}(\xi)=\xi\int
m(\xi,\xi_{1})\frac{u_{\xi_{1}}w_{\xi-\xi_{1}}+w_{\xi_{1}}u_{\xi-\xi_{1}}}{\phi(\xi)-\phi(\xi_{1})-\phi(\xi-\xi_{1})}d\xi_{1}.$

Here $u_{\xi}(t):=u(\widehat{\xi},t)$ and $w_{\xi}(t):=w(\widehat{\xi},t)$
with

$w=e^{-Lt}\partial_{t}(e^{Lt}u)=-N(u).$

###### Proof.

The following calculations can be justified by smooth approximations. First
observe that by (5)

$\partial_{t}(e^{Lt}u)=-e^{Lt}N(u)=-e^{Lt}(R(u)+\widetilde{N}(u)).$

On the Fourier side, we have

${\mathcal{F}}(e^{Lt}\widetilde{N}(u))(\xi)=i\xi\int
e^{-i\phi(\xi)t}m(\xi,\xi_{1})u_{\xi_{1}}u_{\xi-\xi_{1}}d\xi_{1}.$

We rewrite the integral above as

$i\xi\int
e^{-i[\phi(\xi)-\phi(\xi_{1})-\phi(\xi-\xi_{1})]t}m(\xi,\xi_{1})[e^{Lt}u]_{\xi_{1}}[e^{Lt}u]_{\xi-\xi_{1}}d\xi_{1}\\\
=-\xi\partial_{t}\int\frac{e^{-i[\phi(\xi)-\phi(\xi_{1})-\phi(\xi-\xi_{1})]t}}{[\phi(\xi)-\phi(\xi_{1})-\phi(\xi-\xi_{1})]}m(\xi,\xi_{1})[e^{Lt}u]_{\xi_{1}}[e^{Lt}u]_{\xi-\xi_{1}}d\xi_{1}\\\
+\xi\int\frac{e^{-i[\phi(\xi)-\phi(\xi_{1})-\phi(\xi-\xi_{1})]t}}{[\phi(\xi)-\phi(\xi_{1})-\phi(\xi-\xi_{1})]}m(\xi,\xi_{1})\partial_{t}\big{(}[e^{Lt}u]_{\xi_{1}}[e^{Lt}u]_{\xi-\xi_{1}}\big{)}d\xi_{1}\\\
=-\partial_{t}{\mathcal{F}}(e^{Lt}B(u))+{\mathcal{F}}(e^{Lt}NR(u)).$

∎

The following propositions estimate the terms that appear in (6).

######  Proposition 3.2.

For fixed $s>-\frac{3}{4}$, we have

$\displaystyle\|B(u)\|_{H^{s+a}}\lesssim\|u\|_{H^{s}}^{2},\,\,\,\,a\leq\frac{1}{2}$
$\displaystyle\|R(u)\|_{X^{s+a,-\frac{1}{2}+}}\lesssim\|u\|_{X^{s,\frac{1}{2}+}}^{2},\,\,\,\,a<\frac{3}{4}+s.$

###### Proof.

Recalling that $\phi(\xi)=\xi^{3}-\frac{1}{\xi}$, we have

(9)
$\phi(\xi)-\phi(\xi_{1})-\phi(\xi-\xi_{1})=\xi^{3}-\xi_{1}^{3}-(\xi-\xi_{1})^{3}-\frac{1}{\xi}+\frac{1}{\xi_{1}}+\frac{1}{\xi-\xi_{1}}\\\
=\frac{3\xi^{2}\xi_{1}^{2}(\xi-\xi_{1})^{2}+\xi^{2}+\xi_{1}^{2}-\xi\xi_{1}}{\xi\xi_{1}(\xi-\xi_{1})}.$

Therefore,

(10)
$\frac{1}{|\phi(\xi)-\phi(\xi_{1})-\phi(\xi-\xi_{1})|}\lesssim\frac{|\xi\xi_{1}(\xi-\xi_{1})|}{\xi^{2}\xi_{1}^{2}(\xi-\xi_{1})^{2}+\xi^{2}+\xi_{1}^{2}}$

We start with the claim for $B$. By symmetry and noting the support of
$m(\xi,\xi_{1})$, we can assume that $|\xi_{1}|\leq|\xi|/100<|\xi-\xi_{1}|$.
Therefore,

(11)
$\frac{1}{|\phi(\xi)-\phi(\xi_{1})-\phi(\xi-\xi_{1})|}\lesssim\frac{|\xi_{1}|}{\xi^{2}\xi_{1}^{2}+1}.$

Using this in (7), we have

$\|B(u)\|_{H^{s+a}}\lesssim\Big{\|}\xi^{1+s+a}\int
m(\xi,\xi_{1})\frac{|\xi_{1}|\langle\xi_{1}\rangle^{-s}\langle\xi-\xi_{1}\rangle^{-s}f(\xi_{1})f(\xi-\xi_{1})}{\xi^{2}\xi_{1}^{2}+1}\
d\xi_{1}\Big{\|}_{L^{2}},$

where $f(\xi)=\langle\xi\rangle^{s}|u_{\xi}|$; $\|f\|_{L^{2}}=\|u\|_{H^{s}}$.

By duality and Cauchy-Schwarz inequality

$\|B(u)\|_{H^{s+a}}^{2}\lesssim\|u\|_{H^{s}}^{4}\sup_{|\xi|\gg
1}\xi^{2+2s+2a}\int_{|\xi_{1}|\ll|\xi|}\frac{|\xi_{1}|^{2}\langle\xi_{1}\rangle^{-2s}\langle\xi-\xi_{1}\rangle^{-2s}}{(\xi^{2}\xi_{1}^{2}+1)^{2}}\
d\xi_{1}.$

Note that the supremum is bounded by

$\sup_{|\xi|\gg
1}|\xi|^{2a-1}\int_{|\eta|\ll\xi^{2}}\frac{|\eta|^{2}\langle\eta/\xi\rangle^{-2s}}{(\eta^{2}+1)^{2}}\
d\eta\lesssim 1,$

provided that $a\leq\frac{1}{2}$, $s>-\frac{3}{4}$.

We now consider $R(u)$:

$R(u):=P_{|\xi|\lesssim 1}N(u)+P_{|\xi|\gg 1}N(P_{>|\xi|/100}u).$

The first summand satisfies the claim by the local wellposedness theory, for
any $a$. The contribution of the second summand to the $X^{s+a,-\frac{1}{2}+}$
norm is bounded by

$\Big{\|}\chi_{|\xi|\gg
1}\int_{|\xi_{1}|,|\xi-\xi_{1}|\gtrsim|\xi|}\frac{|\xi|^{1+s+a}|\xi_{1}|^{-s}|\xi-\xi_{1}|^{-s}f(\xi_{1},\tau_{1})f(\xi-\xi_{1},\tau-\tau_{1})}{\langle\sigma_{1}\rangle^{\frac{1}{2}+}\langle\sigma_{2}\rangle^{\frac{1}{2}+}\langle\sigma\rangle^{\frac{1}{2}-}}d\xi_{1}d\tau_{1}\Big{\|}_{L^{2}_{\xi,\tau}},$

where $\|f\|_{L^{2}_{\xi,\tau}}\lesssim\|u\|_{X^{s,\frac{1}{2}+}}$. Here
$\sigma_{1}:=\tau_{1}-\phi(\xi_{1})$, and
$\sigma_{2}:=\tau-\tau_{1}-\phi(\xi-\xi_{1})$, and $\sigma:=\tau-\phi(\xi)$.
We consider only the case when $\rho:=-s\in[0,\frac{3}{4})$; the case $s>0$ is
easier. Also note that in the domain of the integral, we always have
$|\xi-\xi_{1}|\approx|\xi_{1}|\gtrsim|\xi|$.

By symmetry, it suffices to consider the cases when $\langle\sigma_{1}\rangle$
is the largest and when $\langle\sigma\rangle$ is the largest. When
$\langle\sigma_{1}\rangle$ is the largest, using duality and Cauchy-Schwarz
inequality, it suffices to bound

$\sup_{\xi_{1},\tau_{1}}\int_{1\ll|\xi|\lesssim|\xi_{1}|\approx|\xi-\xi_{1}|}\frac{|\xi|^{2+2a-2\rho}|\xi_{1}|^{2\rho}|\xi-\xi_{1}|^{2\rho}d\xi
d\tau}{\langle\sigma_{1}\rangle^{1+}\langle\sigma_{2}\rangle^{1+}\langle\sigma\rangle^{1-}}.$

Integrating in $\tau$ and using the fact that

$\langle\sigma_{1}\rangle=\max(\langle\sigma_{1}\rangle,\langle\sigma_{2}\rangle,\langle\sigma\rangle)\gtrsim|\phi(\xi)-\phi(\xi_{1})-\phi(\xi-\xi_{1})|\gtrsim|\xi\xi_{1}(\xi-\xi_{1})|\approx|\xi|\xi_{1}^{2}$

(the second inequality follows from (9)), we have the bound

$\sup_{\xi_{1},\tau_{1}}\int_{1\ll|\xi|\lesssim|\xi_{1}|\approx|\xi-\xi_{1}|}\frac{|\xi|^{2+2a-2\rho}|\xi_{1}|^{4\rho}}{|\xi|^{1+}|\xi_{1}|^{2+}\langle\tau_{1}+\phi(\xi-\xi_{1})-\phi(\xi)\rangle^{1-}}\
d\xi$
$\lesssim\sup_{\xi_{1},\tau_{1}}\int_{1\ll|\xi|\lesssim|\xi_{1}|\approx|\xi-\xi_{1}|}\frac{|\xi|^{1+2a-2\rho-}|\xi_{1}|^{4\rho-2-}}{\langle\tau_{1}+(\xi-\xi_{1})^{3}-\xi^{3}\rangle^{1-}}\
d\xi.$

Letting $\eta=(\xi-\xi_{1})^{3}-\xi^{3}$, we see that

$d\eta\approx|\xi_{1}|^{1/2}|\eta+\xi_{1}^{3}/4|^{1/2}d\xi.$

Also note that when $|\xi_{1}|\gg|\xi|$

$d\eta\approx|\xi_{1}|^{2-}|\eta+\xi_{1}^{3}/4|^{0+}d\xi.$

In the case $|\xi|\approx|\xi_{1}|$, we bound the integral by

$\int\frac{|\xi_{1}|^{2a+2\rho-\frac{3}{2}}d\eta}{|\eta+\xi_{1}^{3}/4|^{1/2}\langle\tau_{1}+\eta\rangle^{1-}}\lesssim|\xi_{1}|^{2a+2\rho-\frac{3}{2}}\lesssim
1$

provided that $a\leq\frac{3}{4}-\rho$. In the case $1\ll|\xi|\ll|\xi_{1}|$, we
have the bound

$\int\frac{|\xi(\eta)|^{1+2a-2\rho}|\xi_{1}|^{4\rho-4+}d\eta}{|\eta+\xi_{1}^{3}/4|^{0+}\langle\tau_{1}+\eta\rangle^{1-}}\lesssim\int\frac{|\xi(\eta)|^{-3+2a+2\rho+}d\eta}{|\eta+\xi_{1}^{3}/4|^{0+}\langle\tau_{1}+\eta\rangle^{1+}}\lesssim
1$

provided that $a<\frac{3}{2}-\rho$.

The case $\langle\sigma\rangle$ is the largest is analogous by integrating in
$\xi_{1},\tau_{1}$ instead of $\xi,\tau$. ∎

######  Remark 3.3.

We note that the regularity of the $B$ term in Proposition 3.2 cannot be
improved for any $s$ by considering the example
$f(\xi_{1})=N^{\frac{1}{2}}\chi_{[\frac{1}{N},\frac{2}{N}]}(\xi_{1})+\chi_{[N,N+1]}(\xi_{1})$,
$N\gg 1$.

######  Proposition 3.4.

For fixed $s>-\frac{3}{4}$ and $a<\min(\frac{3}{4}+s,\frac{1}{2})$, we have

$\big{\|}NR(u)\big{\|}_{X^{s+a,-\frac{1}{2}+}}\lesssim\|u\|_{X^{s,\frac{1}{2}+}}^{3}$

###### Proof.

Recall that

$m(\xi,\xi_{1})=\chi_{|\xi|\gg
1}\big{(}\chi_{|\xi_{1}|\leq|\xi|/100}+\chi_{|\xi-\xi_{1}|\leq|\xi|/100}\big{)}.$

By $\xi_{1}\leftrightarrow\xi-\xi_{1}$ symmetry, it suffices to consider only
the contribution of the first summand of $m$ to $NR(u)$:

$\xi\int
m(\xi,\xi_{1})\frac{u_{\xi_{1}}w_{\xi-\xi_{1}}+w_{\xi_{1}}u_{\xi-\xi_{1}}}{\phi(\xi)-\phi(\xi_{1})-\phi(\xi-\xi_{1})}\
d\xi_{1}=\\\ \xi\chi_{|\xi|\gg
1}\int_{|\xi_{1}|\leq|\xi|/100}\frac{(\xi-\xi_{1})\
u_{\xi_{1}}u_{\xi_{2}}u_{\xi-\xi_{1}-\xi_{2}}+\xi_{1}u_{\xi-\xi_{1}}u_{\xi_{2}}u_{\xi_{1}-\xi_{2}}}{\phi(\xi)-\phi(\xi_{1})-\phi(\xi-\xi_{1})}\
d\xi_{1}\ d\xi_{2}.$

Below, we provide details for the estimate of the first summand in the
numerator. As $\xi_{1}$ is smaller compared to $\xi-\xi_{1}$, the analysis of
the second term is similar and easier.

Letting $\rho=-s\in[0,3/4)$ and by duality, it suffices to bound the integral
below by $\|f\|_{L^{2}_{\xi,\tau}}^{4}$

$\int_{|\xi|\gg
1,|\xi_{1}|\leq|\xi|/100}\frac{|\xi|^{2-\rho+a}\langle\xi_{1}\rangle^{\rho}\langle\xi_{2}\rangle^{\rho}\langle\xi-\xi_{1}-\xi_{2}\rangle^{\rho}f_{\xi_{1}}f_{\xi_{2}}f_{\xi-\xi_{1}-\xi_{2}}f_{\xi}}{|\phi(\xi)-\phi(\xi_{1})-\phi(\xi-\xi_{1})|\langle\sigma_{1}\rangle^{\frac{1}{2}+}\langle\sigma_{2}\rangle^{\frac{1}{2}+}\langle\sigma_{3}\rangle^{\frac{1}{2}+}\langle\sigma\rangle^{\frac{1}{2}-}}d\xi_{1}d\xi_{2}d\xi
d\tau_{1}d\tau_{2}d\tau.$

Here $\sigma$’s are defined as in the proof of Proposition 3.2. Also using
(11), we have the bound

(12) $\int_{|\xi|\gg
1,|\xi_{1}|\leq|\xi|/100}\frac{|\xi|^{2-\rho+a}|\xi_{1}|\langle\xi_{1}\rangle^{\rho}\langle\xi_{2}\rangle^{\rho}\langle\xi-\xi_{1}-\xi_{2}\rangle^{\rho}|f_{\xi_{1}}f_{\xi_{2}}f_{\xi-\xi_{1}-\xi_{2}}f_{\xi}|}{[\xi^{2}\xi_{1}^{2}+1]\
\langle\sigma_{1}\rangle^{\frac{1}{2}+}\langle\sigma_{2}\rangle^{\frac{1}{2}+}\langle\sigma_{3}\rangle^{\frac{1}{2}+}\langle\sigma\rangle^{\frac{1}{2}-}}d\xi_{1}d\xi_{2}d\xi
d\tau_{1}d\tau_{2}d\tau.$

Note that,

(13)
$\langle\sigma-\sigma_{1}-\sigma_{2}-\sigma_{3}\rangle=\langle\phi(\xi_{1})+\phi(\xi_{2})+\phi(\xi-\xi_{1}-\xi_{2})-\phi(\xi)\rangle\\\
\approx\big{\langle}3(\xi-\xi_{1})(\xi-\xi_{2})(\xi_{1}+\xi_{2})+\frac{1}{\xi_{1}}+\frac{1}{\xi_{2}}+\frac{1}{\xi-\xi_{1}-\xi_{2}}\big{\rangle}:=\langle
M\rangle.$

We will consider the following cases:

  * •

Case 1) $|\xi_{1}|\lesssim 1$, $|\xi_{2}|\lesssim 1$.

  * •

Case 2) $1\ll|\xi_{1}|\ll|\xi|$ and $|\xi_{2}|\lesssim 1$.

  * •

Case 3) $1\ll|\xi_{1}|\ll|\xi|$, $1\ll|\xi_{2}|\lesssim|\xi|$ and
$|\xi-\xi_{1}-\xi_{2}|\gg 1$.

  * •

Case 4) $|\xi_{1}|\lesssim\frac{1}{\langle\xi(\xi-\xi_{2})\xi_{2}\rangle}$ and
$|\xi_{2}|,|\xi-\xi_{1}-\xi_{2}|\gg 1$.

  * •

Case 5) $1\gtrsim|\xi_{1}|\gg\frac{1}{\langle\xi(\xi-\xi_{2})\xi_{2}\rangle}$
and $|\xi_{2}|,|\xi-\xi_{1}-\xi_{2}|\gg 1$.

  * •

Case 6) $1\ll|\xi_{1}|\ll|\xi|\ll|\xi_{2}|$.

Case 1) $|\xi_{1}|\lesssim 1$, $|\xi_{2}|\lesssim 1$. By the change of
variable $\xi_{2}\to\xi-\xi_{1}-\xi_{2}$, this also takes care of the case
$|\xi_{1}|,|\xi-\xi_{1}-\xi_{2}|\lesssim 1$. By Cauchy-Schwarz inequality, it
suffices to prove that

$\sup_{|\xi|\gg 1}\ \int_{|\xi_{1}|,|\xi_{2}|\lesssim
1}\frac{|\xi|^{4+2a}|\xi_{1}|^{2}}{[\xi^{2}\xi_{1}^{2}+1]^{2}\ \langle
M\rangle^{1-}}d\xi_{1}d\xi_{2}<\infty.$

Note that

$\langle M\rangle\approx\langle
3(\xi-\xi_{1})(\xi-\xi_{2})(\xi_{1}+\xi_{2})+\frac{1}{\xi_{1}}+\frac{1}{\xi_{2}}\big{\rangle}=\big{\langle}(\xi_{1}+\xi_{2})[3(\xi-\xi_{1})(\xi-\xi_{2})+\frac{1}{\xi_{1}\xi_{2}}]\big{\rangle}.$

Letting
$\eta=(\xi_{1}+\xi_{2})[3(\xi-\xi_{1})(\xi-\xi_{2})+\frac{1}{\xi_{1}\xi_{2}}]$
in the $\xi_{2}$ integral, we see that

$d\eta=[3(\xi-\xi_{1})(\xi-2\xi_{2}-\xi_{1})-\frac{1}{\xi_{2}^{2}}]d\xi_{2}.$

Note that the Jacobian is $\gtrsim\xi^{2}$ if
$|\xi_{2}|\not\approx\frac{1}{|\xi|}$. In this case we estimate the $\xi_{2}$
integral by

$\Big{(}\int_{|\xi_{2}|\lesssim 1}\langle
M\rangle^{-1-}d\xi_{2}\Big{)}^{1-}\lesssim|\xi|^{-2+}\Big{(}\frac{1}{\langle
s\rangle^{1+}}ds\Big{)}^{1-}\lesssim|\xi|^{-2+}.$

This leads to the bound

$\int_{|\xi_{1}|\lesssim
1}\frac{|\xi|^{2+2a+}|\xi_{1}|^{2}}{[\xi^{2}\xi_{1}^{2}+1]^{2}}d\xi_{1}\lesssim|\xi|^{-1+2a+},$

which is bounded on the set $|\xi|>1$ provided that $a<\frac{1}{2}$.

When $|\xi_{2}|\approx\frac{1}{|\xi|}$, consider the cases
$|\xi_{1}|\not\approx\frac{1}{|\xi|}$ and $|\xi_{1}|\approx\frac{1}{|\xi|}$
seperately. In the former case, $\langle
M\rangle\gtrsim\langle(\xi_{1}+\xi_{2})\xi^{2}\rangle$, which leads to (for
$a<\frac{1}{2}$)

$\int_{1\gtrsim|\xi_{1}|\not\approx\frac{1}{|\xi|},|\xi_{2}|\approx\frac{1}{|\xi|}}\frac{|\xi|^{2+2a+}|\xi_{1}|^{2}}{[\xi^{2}\xi_{1}^{2}+1]^{2}\
|\xi_{1}+\xi_{2}|^{1-}}d\xi_{1}d\xi_{2}\lesssim|\xi|^{-1+2a+}\lesssim 1.$

In the latter case, we have $\langle
M\rangle\gtrsim\langle(\xi_{1}+\xi_{2})(3\xi^{2}+\frac{1}{\xi_{1}\xi_{2}})\rangle\approx\langle(\xi_{1}+\xi_{2})\xi^{3}(\xi_{2}+\frac{1}{3\xi_{1}\xi^{2}})\rangle$,
which leads to

$\int_{|\xi_{1}|,|\xi_{2}|\approx\frac{1}{|\xi|}}\frac{|\xi|^{-1+2a+}}{|\xi_{1}+\xi_{2}|^{1-}|\xi_{2}+\frac{1}{3\xi_{1}\xi^{2}}|^{1-}}d\xi_{1}d\xi_{2}\lesssim\int_{|\xi_{1}|\approx\frac{1}{|\xi|}}\frac{|\xi|^{-1+2a+}}{|\xi_{1}-\frac{1}{3\xi_{1}\xi^{2}}|^{1-}}d\xi_{1}\\\
\lesssim\int_{|\xi_{1}|\approx\frac{1}{|\xi|}}\frac{|\xi|^{-2+2a+}}{|\xi_{1}^{2}-\frac{1}{3\xi^{2}}|^{1-}}d\xi_{1}\lesssim|\xi|^{-1+2a+}\lesssim
1$

provided that $a<\frac{1}{2}$.

Case 2) $1\ll|\xi_{1}|\ll|\xi|$ and $|\xi_{2}|\lesssim 1$. By the change of
variable $\xi_{2}\to\xi-\xi_{1}-\xi_{2}$, this also takes care of the case
$1\ll|\xi_{1}|\ll|\xi|$, $|\xi-\xi_{1}-\xi_{2}|\lesssim 1$. By Cauchy-Schwarz
inequality

$\sup_{|\xi|\gg 1}\ \int_{1\ll|\xi_{1}|\ll|\xi|,\ |\xi_{2}|\lesssim
1}\frac{|\xi|^{4+2a}|\xi_{1}|^{2+2\rho}}{[\xi^{2}\xi_{1}^{2}+1]^{2}\ \langle
M\rangle^{1-}}d\xi_{1}d\xi_{2}\lesssim\sup_{|\xi|\gg 1}\
\int_{1\ll|\xi_{1}|\ll|\xi|,\ |\xi_{2}|\lesssim
1}\frac{|\xi|^{2a}|\xi_{1}|^{2\rho-2}}{\langle
M\rangle^{1-}}d\xi_{1}d\xi_{2}.$

Note that

$\langle M\rangle\approx\langle
3(\xi-\xi_{1})(\xi-\xi_{2})(\xi_{1}+\xi_{2})+\frac{1}{\xi_{2}}\big{\rangle}$

Note that the contribution of the set
$|\xi_{2}|\lesssim\frac{1}{\xi^{2}|\xi_{1}|}$ is $\lesssim 1$ provided that
$a<1$. If $|\xi_{2}|\gg\frac{1}{\xi^{2}|\xi_{1}|}$, then $\langle
M\rangle\approx\xi^{2}|\xi_{1}|$, which leads to the bound

$\sup_{|\xi|\gg 1}\ \int_{1\ll|\xi_{1}|\ll|\xi|,\ |\xi_{2}|\lesssim
1}|\xi|^{2a-2+}|\xi_{1}|^{2\rho-3+}d\xi_{1}d\xi_{2}\lesssim 1,$

provided that $a<1$.

Case 3) $1\ll|\xi_{1}|\ll|\xi|$, $1\ll|\xi_{2}|\lesssim|\xi|$ and
$|\xi-\xi_{1}-\xi_{2}|\gg 1$. Once again we have

$\sup_{|\xi|\gg 1}\ \int_{1\ll|\xi_{1}|\ll|\xi|,\ 1\ll|\xi_{2}|\lesssim|\xi|,\
|\xi-\xi_{1}-\xi_{2}|\gg
1}\frac{|\xi|^{-2\rho+2a}|\xi_{1}|^{2\rho-2}|\xi_{2}|^{2\rho}|\xi-\xi_{1}-\xi_{2}|^{2\rho}}{\langle
M\rangle^{1-}}d\xi_{1}d\xi_{2}$ $\lesssim\sup_{|\xi|\gg 1}\
\int_{1\ll|\xi_{1}|\ll|\xi|}\frac{|\xi|^{2\rho+2a}|\xi_{1}|^{2\rho-2}}{|\xi|^{1-}|\xi-\xi_{2}|^{1-}|\xi_{1}+\xi_{2}|^{1-}}d\xi_{1}d\xi_{2}$
$\lesssim\sup_{|\xi|\gg 1}\
\int_{1\ll|\xi_{1}|\ll|\xi|}\frac{|\xi|^{2\rho+2a-1+}|\xi_{1}|^{2\rho-2}}{|\xi_{1}+\xi|^{1-}}d\xi_{1}\lesssim
1$

provided that $a<1-\rho$.

Case 4) $|\xi_{1}|\lesssim\frac{1}{\langle\xi(\xi-\xi_{2})\xi_{2}\rangle}$ and
$|\xi_{2}|,|\xi-\xi_{1}-\xi_{2}|\gg 1$.

By Cauchy-Schwarz inequality as above

$\sup_{|\xi|\gg 1}\
\int_{|\xi_{1}|\lesssim\frac{1}{|\xi(\xi-\xi_{2})\xi_{2}|}}|\xi|^{4-2\rho+2a}|\xi_{1}|^{2}\langle\xi_{2}\rangle^{2\rho}\langle\xi-\xi_{2}\rangle^{2\rho}d\xi_{1}d\xi_{2}\\\
\lesssim\sup_{|\xi|\gg
1}\int\frac{|\xi|^{4-2\rho+2a}\langle\xi_{2}\rangle^{2\rho}\langle\xi-\xi_{2}\rangle^{2\rho}}{|\xi(\xi-\xi_{2})\xi_{2}|^{3}}\
d\xi_{2}\lesssim\sup_{|\xi|\gg 1}|\xi|^{2a-2}<\infty,$

provided that $a\leq 1$.

Case 5) $1\gtrsim|\xi_{1}|\gg\frac{1}{\langle\xi(\xi-\xi_{2})\xi_{2}\rangle}$
and $|\xi_{2}|,|\xi-\xi_{1}-\xi_{2}|\gg 1$.
Note that in this case, we also have $|\xi-\xi_{2}|\gg 1$ and

$\langle M\rangle\approx\langle
3(\xi-\xi_{1})(\xi-\xi_{2})(\xi_{1}+\xi_{2})+\frac{1}{\xi_{1}}\big{\rangle}\approx|\xi(\xi-\xi_{2})\xi_{2}|.$

We bound (12) on this set by

$\int_{|\xi|\gg
1\gtrsim|\xi_{1}|\gg\frac{1}{\langle\xi(\xi-\xi_{2})\xi_{2}\rangle}}\frac{|\xi|^{2-\rho+a}|\xi_{1}||\xi_{2}|^{\rho}|\xi-\xi_{2}|^{\rho}|f_{\xi_{1}}f_{\xi_{2}}f_{\xi-\xi_{1}-\xi_{2}}f_{\xi}|}{[\xi^{2}\xi_{1}^{2}+1]\
\langle\sigma_{1}\rangle^{\frac{1}{2}+}\langle\sigma_{2}\rangle^{\frac{1}{2}+}\langle\sigma_{3}\rangle^{\frac{1}{2}+}\langle\sigma\rangle^{\frac{1}{2}-}}d\xi_{1}d\xi_{2}d\xi
d\tau_{1}d\tau_{2}d\tau.$

By symmetry, and since
$\max(\langle\sigma_{1}\rangle,\langle\sigma_{2}\rangle,\langle\sigma_{3}\rangle,\langle\sigma\rangle)\gtrsim\langle
M\rangle$, it suffices to consider the subcases
$\langle\sigma_{1}\rangle\gtrsim\langle M\rangle$,
$\langle\sigma_{2}\rangle\gtrsim\langle M\rangle$, and
$\langle\sigma\rangle\gtrsim\langle M\rangle$.

Subcase 1: $\langle\sigma_{1}\rangle\gtrsim\langle
M\rangle\approx|\xi(\xi-\xi_{2})\xi_{2}|$. By Cauchy-Schwarz inequality and
then integrating in $\tau,\tau_{2}$, and using
$\frac{|\xi_{1}|}{\xi^{2}\xi_{1}^{2}+1}\lesssim\frac{1}{|\xi|}$, it suffices
to consider

(14) $\sup_{|\xi_{1}|\lesssim 1,\tau_{1}}\int_{|\xi|\gg
1\gtrsim|\xi_{1}|\gg\frac{1}{\langle\xi(\xi-\xi_{2})\xi_{2}\rangle}}\frac{|\xi|^{1-2\rho+2a}|\xi_{2}|^{2\rho-1}|\xi-\xi_{2}|^{2\rho-1}}{\langle\tau_{1}+\phi(\xi_{2})+\phi(\xi-\xi_{1}-\xi_{2})-\phi(\xi)\rangle^{1+}}d\xi_{2}d\xi.$

Noting that the factor in the denominator is
$\approx\langle\tau_{1}+\xi_{2}^{3}+(\xi-\xi_{1}-\xi_{2})^{3}-\xi^{3}\rangle$,
we let $\eta=\xi_{2}^{3}+(\xi-\xi_{1}-\xi_{2})^{3}-\xi^{3}$. We have

$\Big{|}\frac{\partial\eta}{\partial\xi_{2}}\Big{|}\approx|\xi||\xi-\xi_{1}-2\xi_{2}|,\,\,\,\,\Big{|}\frac{\partial\eta}{\partial\xi}\Big{|}\approx|\xi_{2}||2\xi-\xi_{1}-\xi_{2}|.$

When $|\xi_{2}|\geq\frac{3}{2}|\xi|$, we use the change of variable $\eta$ in
the $\xi_{2}$ integral in (14), and obtain

$\lesssim\sup_{|\xi_{1}|\lesssim 1,\tau_{1}}\int_{|\xi|\gg
1}\frac{|\xi|^{-2\rho+2a}|\xi_{2}(\eta,\xi)|^{4\rho-3}}{\langle\tau_{1}+\eta\rangle^{1+}}d\eta
d\xi\lesssim\int_{|\xi|\gg 1}|\xi|^{2\rho+2a-3}d\xi\lesssim 1,$

provided that $\rho\leq\frac{3}{4}$ and $a\leq 1-\rho$.

When $|\xi|\geq\frac{3}{2}|\xi_{2}|$, we use the change of variable $\eta$ in
the $\xi$ integral:

$\lesssim\sup_{|\xi_{1}|\lesssim 1,\tau_{1}}\int_{|\xi_{2}|\gg
1}\frac{|\xi(\eta,\xi_{2})|^{2a-1}|\xi_{2}|^{2\rho-2}}{\langle\tau_{1}+\eta\rangle^{1+}}d\eta
d\xi_{2}\lesssim\int_{|\xi_{2}|\gg 1}|\xi_{2}|^{2\rho+2a-3}d\xi_{2}\lesssim
1,$

provided that $a\leq\min(\frac{1}{2},1-\rho)$.

When $\frac{2}{3}|\xi|\leq|\xi_{2}|\leq\frac{3}{2}|\xi|$, we use the change of
variable $\eta$ in the $\xi$ integral:

$\lesssim\sup_{|\xi_{1}|\lesssim 1,\tau_{1}}\int_{|\xi_{2}|\gg
1}\frac{|\xi_{2}|^{2a-2}|\xi(\eta,\xi_{2})-\xi_{2}|^{2\rho-1}}{\langle\tau_{1}+\eta\rangle^{1+}}d\eta
d\xi_{2}\lesssim\int_{|\xi_{2}|\gg
1}|\xi_{2}|^{2a-2+\max(2\rho-1,0)}d\xi_{2}\lesssim 1,$

provided that $a\leq\frac{1}{2}-\max(\rho-\frac{1}{2},0)$. In the second
inequality we used the bound $|\xi-\xi_{2}|\gg 1$, which is valid throughout
the Case 5.

Subcase 2: $\langle\sigma_{2}\rangle\gtrsim\langle
M\rangle\approx|\xi(\xi-\xi_{2})\xi_{2}|$. By Cauchy-Schwarz inequality and
then integrating in $\tau,\tau_{1}$, and using
$\frac{|\xi_{1}|^{2}}{[\xi^{2}\xi_{1}^{2}+1]^{2}}\lesssim\frac{1}{|\xi|^{3-}|\xi_{1}|^{1-}}$
it suffices to consider

$\sup_{|\xi_{2}|\gg 1,\tau_{2}}\int_{|\xi|\gg
1\gtrsim|\xi_{1}|\gg\frac{1}{\langle\xi(\xi-\xi_{2})\xi_{2}\rangle}}\frac{|\xi|^{-2\rho+2a+}|\xi_{2}|^{2\rho-1}|\xi-\xi_{2}|^{2\rho-1}}{|\xi_{1}|^{1-}\langle\tau_{2}+\phi(\xi_{1})+\phi(\xi-\xi_{1}-\xi_{2})-\phi(\xi)\rangle^{1+}}d\xi_{1}d\xi.$

Noting that the factor in the denominator is
$\approx\langle\tau_{2}+\phi(\xi_{1})+(\xi-\xi_{1}-\xi_{2})^{3}-\xi^{3}\rangle$,
we let $\eta=\tau_{2}+\phi(\xi_{1})+(\xi-\xi_{1}-\xi_{2})^{3}-\xi^{3}$. We
have

$\Big{|}\frac{\partial\eta}{\partial\xi}\Big{|}\approx|\xi_{2}||2\xi-\xi_{1}-\xi_{2}|.$

We use this change of variable when $|2\xi-\xi_{2}|\gg 1$ to obtain

$\sup_{|\xi_{2}|\gg 1,\tau_{2}}\int_{|\xi|\gg
1\gtrsim|\xi_{1}|}\frac{|\xi|^{-2\rho+2a+}|\xi_{2}|^{2\rho-2}|\xi-\xi_{2}|^{2\rho-1}}{|2\xi-\xi_{2}||\xi_{1}|^{1-}\langle\eta\rangle^{1+}}d\eta
d\xi_{1}\lesssim 1$

provided that

$\sup_{|\xi|,|\xi_{2}|,|\xi-\xi_{2}|,|2\xi-\xi_{2}|\gg
1}\frac{|\xi|^{-2\rho+2a+}|\xi_{2}|^{2\rho-2}|\xi-\xi_{2}|^{2\rho-1}}{|2\xi-\xi_{2}|}\lesssim
1.$

This inequality holds for $a<1-\rho$ by considering the cases
$|\xi|\approx|\xi_{2}|$ and $|\xi|\not\approx|\xi_{2}|$ seperately.

When $|2\xi-\xi_{2}|\lesssim 1$, we have
$|\xi_{2}|,|\xi-\xi_{2}|\approx|\xi|$. Therefore, we have the bound

$\sup_{|\xi_{2}|\gg 1,\tau_{2}}\int_{|2\xi-\xi_{2}|,|\xi_{1}|\lesssim
1}\frac{|\xi|^{2\rho+2a-2+}}{|\xi_{1}|^{1-}}d\xi_{1}d\xi\lesssim 1$

provided that $a<1-\rho$ (since the $\xi$ integral is on an interval of length
$\lesssim 1$).

Subcase 3: $\langle\sigma\rangle\gtrsim\langle
M\rangle\approx|\xi(\xi-\xi_{2})\xi_{2}|$. By Cauchy-Schwarz inequality and
then integrating in $\tau_{1},\tau_{2}$, and using
$\frac{|\xi_{1}|^{2}}{[\xi^{2}\xi_{1}^{2}+1]^{2}}\lesssim\frac{1}{|\xi|^{3-}|\xi_{1}|^{1-}}$
it suffices to consider

$\sup_{|\xi|\gg
1,\tau}\int_{1\gtrsim|\xi_{1}|}\frac{|\xi|^{-2\rho+2a+}|\xi_{2}|^{2\rho-1+}|\xi-\xi_{2}|^{2\rho-1+}}{|\xi_{1}|^{1-}\langle\tau-\phi(\xi_{1})-\phi(\xi-\xi_{1}-\xi_{2})-\phi(\xi_{2})\rangle^{1+}}d\xi_{1}d\xi_{2}.$

This can be handled as in Subcase 2.

Case 6) $1\ll|\xi_{1}|\ll|\xi|\ll|\xi_{2}|$.

Note that in this case, we have

$\langle M\rangle\approx\langle
3(\xi-\xi_{1})(\xi-\xi_{2})(\xi_{1}+\xi_{2})\big{\rangle}\approx|\xi\xi_{2}^{2}|.$

We bound (12) on this set by

$\int_{1\ll|\xi_{1}|\ll|\xi|\ll|\xi_{2}|}\frac{|\xi|^{a-\rho}|\xi_{1}|^{\rho-1}|\xi_{2}|^{2\rho}|f_{\xi_{1}}f_{\xi_{2}}f_{\xi-\xi_{1}-\xi_{2}}f_{\xi}|}{\langle\sigma_{1}\rangle^{\frac{1}{2}+}\langle\sigma_{2}\rangle^{\frac{1}{2}+}\langle\sigma_{3}\rangle^{\frac{1}{2}+}\langle\sigma\rangle^{\frac{1}{2}-}}d\xi_{1}d\xi_{2}d\xi
d\tau_{1}d\tau_{2}d\tau.$

We consider the subcases as in Case 5. Since they can be handled similarly, we
give details only for subcase 1:

Subcase 1: $\langle\sigma_{1}\rangle\gtrsim\langle
M\rangle\approx|\xi\xi_{2}^{2}|$.

As above, it suffices to bound

$\sup_{|\xi_{1}|\gg
1,\tau_{1}}\int_{|\xi_{1}|\ll|\xi|\ll|\xi_{2}|}\frac{|\xi|^{2a-2\rho-1}|\xi_{1}|^{2\rho-2}|\xi_{2}|^{4\rho-2}}{\langle\tau_{1}+\phi(\xi_{2})+\phi(\xi-\xi_{1}-\xi_{2})-\phi(\xi)\rangle^{1+}}d\xi_{2}d\xi.$

Letting $\eta=\tau_{1}+\xi_{2}^{3}+(\xi-\xi_{1}-\xi_{2})^{3}-\xi^{3}$ we see
that

$\Big{|}\frac{\partial\eta}{\partial\xi}\Big{|}\approx|\xi_{2}|^{2}.$

This leads to the bound

$\int_{|\xi|\ll|\xi_{2}|}\frac{|\xi|^{2a-2\rho-1}|\xi_{2}|^{4\rho-4}}{\langle\eta\rangle^{1+}}d\xi_{2}d\eta\lesssim\sup_{|\xi|\gg
1}|\xi|^{2a+2\rho-4+}\int_{|\xi_{2}|\gg
1}\frac{|\xi_{2}|^{-1-}}{\langle\eta\rangle^{1+}}d\xi_{2}d\eta\lesssim 1$

provided that $\rho<\frac{3}{4}$ and $a<2-\rho$. ∎

We now outline the proof of Theorem 1.1. For more details and the global
version of the theorem, see [12].

###### Proof of Theorem 1.1.

Integrating (6) on $[0,t]$ we obtain

$u(t)-e^{-Lt}f=B(u(t))-e^{Lt}B(f)-\int_{0}^{t}e^{-L(t-s)}\big{(}R(u)+NR(u)\big{)}\
ds.$

The claim follows from this identity using the bounds in Proposition 3.2,
Proposition 3.4, standard $X^{s,b}$ inequalities, and the embedding
$X^{s,b}\subset C^{0}_{t}H^{s}_{x}$ for $b>\frac{1}{2}$. ∎

##  4\. Appendix

Consider the equation

(15)
$\left\\{\begin{array}[]{l}u_{t}+u_{xxx}-\gamma\partial_{x}^{-1}u+\partial_{x}(u^{2})=0,\,\,\,\,x\in{\mathbb{R}},t\in{\mathbb{R}},\\\
u(x,0)=f(x).\end{array}\right.$

The Duhamel formulation of the equation is

$u=e^{-Lt}f(x)-e^{-Lt}\int_{0}^{t}e^{Ls}N(u)ds,$

where $Lu=u_{xxx}-\gamma\partial_{x}^{-1}u$ and $N(u)=\partial_{x}(u^{2})$.
Now we consider the nonlinear part of the first Picard iterate, which is
obtained by replacing $u$ with $e^{-Lt}f(x)$. On the Fourier side, it is given
by

(16)
$\widehat{P_{t}(f)}:=-e^{i\phi(\xi)t}\xi\int_{{\mathbb{R}}}\int_{0}^{t}e^{-i[\phi(\xi)-\phi(\eta)-\phi(\xi-\eta)]s}\widehat{f}(\eta)\widehat{f}(\xi-\eta)dsd\eta\\\
=-ie^{i\phi(\xi)t}\xi\int_{{\mathbb{R}}}\frac{e^{-i[\phi(\xi)-\phi(\eta)-\phi(\xi-\eta)]t}-1}{\phi(\xi)-\phi(\eta)-\phi(\xi-\eta)}\widehat{f}(\eta)\widehat{f}(\xi-\eta)d\eta,$

where $\phi(\xi)=\xi^{3}-\frac{\gamma}{\xi}$.

We claim that there is no smoothing in the first Picard iterate when
$\gamma\leq 0$, namely $P_{t}(f)$ is not smoother than the initial data. This
is already known for $\gamma=0$, see [20]. We provide the proof for the cases
$\gamma=0$ and $\gamma=-1$ for completeness.

For $\gamma=0$, we rewrite (16) as

$\widehat{P_{t}(f)}=-ie^{i\xi^{3}t}\int\frac{\widehat{f}(\eta)\widehat{f}(\xi-\eta)}{\eta(\xi-\eta)}(e^{-it3\xi\eta(\xi-\eta)}-1)d\eta.$

Let $\widehat{f}_{N}(\eta)=i\ {\rm
sign}(\eta)\big{[}N\chi_{[\frac{\alpha}{N^{2}},\frac{\beta}{N^{2}}]}(|\eta|)+N^{-s}\chi_{[N,N+1]}(|\eta|)\big{]}$.
Note that $f_{N}$ is real-valued. Here $\alpha<\beta$ are chosen depending on
fixed $t$ so that

$[3t\alpha,3t\beta]\subset[\pi-\frac{1}{10},\pi+\frac{1}{10}].$

This guarantees that $\Re(1-e^{-it3\xi\eta(\xi-\eta)})\gtrsim 1$ in the
support of $\widehat{f}_{N}$. We also have $\|f_{N}\|_{H^{s}}\approx 1$.

Note that there are four hi-low contributions and they have the same sign
because of symmetry and because $\frac{\widehat{f}(\eta)}{\eta}$ is even.
Therefore, for large $N$ and $\xi\in[N,N+1]$, we have

$|\widehat{P_{t}(f_{N})}(\xi)|\approx\int_{[\frac{\alpha}{N^{2}},\frac{\beta}{N^{2}}]}N^{1-s}Nd\eta\approx
N^{-s}.$

We conclude that, for $a>0$, $\sup_{N}\|P_{t}(f_{N})\|_{H^{s+a}}=\infty$.

When $\gamma=-1$, we instead have

(17)
$\big{|}\widehat{P_{t}(f)}\big{|}=\Big{|}\int\frac{\xi\widehat{f}(\eta)\widehat{f}(\xi-\eta)}{\Phi(\xi,\eta)}(e^{-it\Phi(\xi,\eta)}-1)d\eta\Big{|},$

where

(18)
$\Phi(\xi,\eta)=\phi(\xi)-\phi(\eta)-\phi(\xi-\eta)=3\xi\eta(\xi-\eta)+\frac{1}{\xi}-\frac{1}{\eta}-\frac{1}{\xi-\eta}.$

Taking $N\gg 1$ and

$\widehat{f}_{N}(\eta)=i{\rm
sign}(\eta)\big{[}N\chi_{[\frac{1}{\sqrt{3}N-\beta},\frac{1}{\sqrt{3}N-\alpha}]}(|\eta|)+N^{-s}\chi_{[N,N+1]}(|\eta|)\big{]},$

we see that $f_{N}$ is real-valued and $\|f_{N}\|_{H^{s}}\approx 1$. This is
because

$\frac{1}{\sqrt{3}N-\alpha}-\frac{1}{\sqrt{3}N-\beta}\lesssim\frac{1}{N^{2}}.$

Fixing $\xi\in[N-c,N+c]$, by $\eta\leftrightarrow\xi-\eta$ symmetry in (17)
and (18), we can assume $|\eta|\lesssim\frac{1}{N}$. Therefore, we see that

$\Phi(\xi,\eta)=3\xi^{2}\eta-\frac{1}{\eta}+O(1/N).$

We now put additional restrictions on $c$ and on $0<\alpha<\beta$ depending
only on $t>0$ so that for $\xi\in[N-c,N+c]$ and
$|\eta|\in[\frac{1}{\sqrt{3}N-\beta},\frac{1}{\sqrt{3}N-\alpha}]$, we have

$\Phi(\xi,\eta)\ {\rm sign}(\eta)\approx 1$

and

$|t\Phi(\xi,\eta)|\in[k\pi-\frac{1}{10},k\pi+\frac{1}{10}],$

where $k=k(t)>0$ is odd.

A calculation as above yields the claim:

$|\widehat{P_{t}(f_{N})}(\xi)|\approx\int_{[\frac{1}{\sqrt{3}N-\beta},\frac{1}{\sqrt{3}N-\alpha}]}N^{1-s}Nd\eta\approx
N^{-s}.$

Therefore, we again conclude that, for $a>0$,
$\sup_{N}\|P_{t}(f_{N})\|_{H^{s+a}}=\infty$.

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