Contents. 1 Introduction 3

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Preface

This document is written in fulfilment of my honour degree in pure mathematics at the University of Sydney. Our initial intention was to give a complete treatment of the Kardar–Parisi–Zhang equation, which interested me because of its popularity after the year 2014, during which a fields medallist level work was associated with it. The equation also had significant relations with the notion of universality classes in probability, which I always found to be quiet fascinating. In order to restrict ourselves to topics in analysis, we decided to focus on an alternative of the original theory, nowadays known as the method of paracontrolled distributions, which relies on nothing more than well-known stochastic and Fourier analytic techniques. However, gradually we realised that the size of this project was too ambitious for a thesis which was limited to only 60 pages, and so in the final product we were forced to remove a number of interesting probabilistic topics, but only presented the functional analytic pillars of the arguments. As a consequence, the reading of this thesis requires almost no pre-requisites in stochastic analysis. What remains here is the result of that process.

We made efforts to introduce the theory with as much motivations as possible. In the introductory chapter we briefly look at some big pictures on singular stochastic partial differential equations, beginning from the very first theorem in the field. We paid particular attentions to how and why various ideas emerged by trying to understand the motivations of authors who wrote evolutionary papers during different times. In Chapter One we introduce the Fourier analytic tools which will be made use of. These will be the most technical aspects of this thesis. Starting from Chapter Two we will look at ordinary differential equations where the situation is very clear. In Chapter Three we will move on to partial differential equations, where the theory there is historically built on those of the ordinary cases.

I would like to thank first and foremost Professor Beniamin Goldys, who patiently supervised me in writing this thesis, while being the mentor of my academic life as well. From time to time I found myself reliant on his presences in order to move on from difficulties, some of them I suspect were beyond his responsibilities, yet he guided me through them nonetheless, and for that I am truly grateful. I would also like to thank those who lectured me through my undergraduate and honour curriculum, including but not limited to F.C. Cirstea, A. Fish, L. Tzou, Z. Zhang, H. Wu, U. Keich, M. Wechselberger and R. Marangell. I would like to thank in particular Professor Daniel Daners, who single handedly constructed the curriculum in mathematical analysis for the School of Mathematics and Statistics at the University of Sydney. The elegance in his proofs have been the main factor for my choice to study mathematical analysis in depths; and Professor Laurentiu Paunescu, who acted as the honour coordinator and assisted me with everything I ever needed as a honour student. Finally, I would like to thank my family and friends who supported me financially throughout the years. I would not have not survived without them. I would also like to thank all those baristas who sold me plenty of coffees as well.

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3

Chapter 1

Introduction

The area of stochastic partial differential equations (SPDEs) is an active field of re-search where problems are often heavily related to the physical sciences. Since exper-iments are often probabilistic in nature, mathematical models which capture random-ness are sometimes looked in favour over the deterministic ones. However, whereas mathematicians take considerations of well-posedness, scientists working in the gen-eral discipline pose their equations caring primarily the capacity of their models in explaining experimental results. As a consequence, stochastic equations explaining fundamental phenomenons in nature which are very well-established in physics turn out to be complete nonsense in the language of mathematics.

In the late nineteenth century there had been many such equations, and it was apparent that the existing tools in stochastic analysis were insufficient to give them meanings. Such problems arise mostly because the graph of sufficiently well-behaved random signals most also be quiet rough. For an example, consider the one-dimensional

Kardar–Parisi–Zhang (KPZ) equation:

∂th(x, t, ω)−ν∂_x2h(x, t, ω) = λ 2(∂xh)

2₍_{x, t, ω}_{) +}√_Dη₍_{x, t, ω}₎_,

whereη:R_×₍₀_,_∞₎_×_Ω _→R_{is the stochastic}_{Gaussian space-time white noise}_{for some}

suitable probability space (Ω, P), the constants (ν, λ, D)∈R2_×₍₀_,_∞_{) are parameters}

and h is continuous in both space and time. Named after its creators M. Kardar, G. Parisi and Y. Cheng, the equation was first applied to model random interface growth and has gained enough popularity to become the default setting for the topic.

Mathematically, for every possible out comeω _∈Ω, the smoothness of the solution

h(ω) is inevitably dependent on the smoothness of η(ω), which is known to be very rough, one could show that h(ω) is at most H¨older continuous for indices s <1/2. It is then a technical matter to describe what it is meant of the derivative∂xh(ω), which turns out to require the introduction of an entirely new class of elements known as tempered distributions. In particular, the product (∂xh)2₍_ω_{) cannot even be interpreted}

as the product of functions, which means the the KPZ equation is a priori ill-defined. Then centre of the problem now concerns the establishment of a working framework in which one could discuss the KPZ equation meaningfully.

The Theory of Rough Path

It was in the 1990s when British mathematician T.J. Lyons first considered a deter-ministic approach to solving stochastic differential equations

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for the Brownian motion B and au:R_×_Ω _{is continuous in time. In the probabilistic}

literature, such an equation is usually interpreted as the Stratonovich or Ito’s integral equation:

u(t, ω) =u0(ω) +

Z t

0 f(u)(s, ω)dB(s, ω), (1.1)

where the pathwise regularity of B(ω) satisfies the condition

kB(ω)kVp₍_I₎ def= sup P⊂I X tj∈P |B(tj, ω)−B(tj+1, ω)|p 1/p <∞, p >2,

and the surepmum is taken over all partitions _P = (tj)j ofI. In other words,B(ω) has

finite p variation for all p > 2. Such a condition deems that the integral (1.1) cannot be finite for every fixed ω∈ Ω, and Lyons wanted to know whether it was possible to make sense of this integral and to what extend the corresponding differential equation is well-posed, as at his time this integrability condition was already pushed to the boundary case by L.C.Young, who proved that

Z t 0 u(s)dX(s) def = lim |P|→0 X tj∈P u(tj)(X(tj+1)−X(tj)) exists, provided

kukVq₍_I₎,kXk_Vp₍_I₎ <∞, for all q, p such that 1/q+ 1/p >1.

Here|P|is the mesh ofP. It turns out this is doable provided we restrict out attention to a suitable space ofgeometric rough paths, for which we denote by_X. This construc-tion by Lyons is algebraic in nature and somewhat very technical. More precisely, let

∆T be the simplex

∆T def= {(s, t)∈[0, T] | 0≤s≤t≤T}, T >0.

Lyons observed that for every continuous pathx: [0, T]→Rn_and_P _≥_{1, it is sufficient}

to study information of x stored in its step-_⌊p_⌋ signature:

X⌊p⌋:∆T →T⌊p⌋(Rn), T⌊p⌋(Rn) def= R⊕

M

1≤j≤⌊p⌋

(Rn₎⊗j_,

X⌊p⌋(s, t) def= (1, X(1)(s, t), ..., X(⌊p⌋)(s, t))∈R⊗Rn⊗(Rn)⊗2⊗...⊗(Rn)⊗⌊p⌋,

where the elements in the tensor products are of the form

X(k)₍_{s, t}₎ def₌ X 1≤i1,...,ik≤n Z s<u1<...<uk<t dxi1 u1...dx ik uk (ei1 ⊗...⊗eik)

A map X :∆T → T⌊p⌋(Rn) is then in X if there exists a sequence of such continuous

maps (x(m))m≥1 with finite 1-variation such that limm→∞X⌊p⌋(m) = X in the ⌊p⌋-th

variation metric: dp(X, X⌊p⌋(m)) def= max 1≤k≤⌊p⌋_P⊂sup_[0_,T_] X tj∈P |X(k)(tj−1, tj)−X⌊(pk⌋)(m)(tj−1, tj)|p/k 1/p .

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Chapter 1. Introduction 5 Occasionally a geometric rough path is also denoted byX_{. For every p-geometric rough}

path Lyons defined what is meant by (1.1). Moreover, the pathwise Brownian motion

B(ω) is a p-geometric rough path for all 2 < p < 3. To this end, Lyons also gave a meaning to the pathwise solution of the stochastic differential equation (1.1), which he calls the Universal Limit Theorem:

Theorem 1. Let X be a p-geometric path and (x(m))m∈N be a sequence of continuous

paths with finite 1-variation with_⌊p_⌋-step signaturesX⌊p⌋(m). Let f :Rn→ L(Rd;Rn)

be a vector field with at least_⌊p_⌋bounded derivatives and is H¨older continuous of order s > p. Let u(m) be the solution to the ordinary differential equation

du(m)(t) =f((u))(m)(t)dX(m)(t), u(m)|t=0 =u0(m)

with limm→∞u0(m) = u0. Then the sequence of step-⌊p⌋ signature u⌊p⌋(m) converges

to to a p-geometric rough path u in the p-variation metric. Moreover, u solves the differential equation

du(t) = f(u)(t)dX(t), u|t=0 =u0

driven by the p-geometric rough path X.

We will prove the theory of Lyons under different frameworks in Chapter Two. For a detailed treatment of the theory of rough path, see [5].

A Fourier Analytic Formulation

The theory of rough path gave a first example of special structures to solutions of equations with stochastic noise. The weakness in the theory being that its formulation in terms of path is specialised to work only for ordinary differential equation. In the meanwhile, it hinted the fact that when dealing with equations with rough signals, one cannot always expect to make sense of the equation, and additional constraints must postulated.

Following Lyon’s footsteps, Italian mathematician Massimiliano Gubinelli and his collaborators were successful in reformulateing Lyons’ idea in the language of Fourier analysis and moreover, generalised it to frameworks which were applicable to SPDEs. Taking advantages of the theory of distributions and the fact that the topology induced by the p-variation norm correspond exactly to the 1/p-H¨older topology. The theorem of Lyons is equivalent to finding the correct structure of the solution u to the rough differential equation:

d

dtu(t) =f(u(t))dX(t), u|t=0 =u0 for rough signalsX which is H¨older continuous

kXkC_b0,s def = kukL∞ + sup t6=t′ |X(t)−X(t′₎_| |t₋t′_|s <∞, ∀s <1/2,

and satisfying some approximation condition similar to geometric rough paths, such that the non-linearityf(u)dX is a priori well-defined and the solution udepends Lips-chitz continuously on the rough signalX and initial condition u0. One natural Fourier

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analytic instrument for this is known as the Paley-Littlewood theory, which gives a natural meaning of products of distributions in terms of a decomposition

f(u(t))dX(t) = f1(u(t), X(t)) +f2(u(t), X(t), R(X(t), dX(t))),

where f1(u(t), X(t)) is well-defined with the natural regularity condition, and that

f2(u(t), X(t), R(X(t), dX(t))) makes sense if and only if X(t) and R(X(t), dX(t))

can be approximated via smooth functions (Xǫ₍_t₎_{, R}₍_Xǫ₍_t₎_{, dX}ǫ₍_t₎₎₎

ǫ>0 in the

prod-uct H¨older topology for some function R. In that case, Gubinelli proved that the map (X, dX, u0)→u satisfies the required Lipschitz conditions.

This approach is advantageous in the sense that it is no longer restrictive to ordinary differential equations. It turns out for SPDEs as complicated as theKPZ equation, it is still possible to find the right structure

h=hfinite+hsingular, and S(η(ω)) = (S1(η(ω)), ..., Sm(η(ω)))

such that (∂xh)2_{becomes well-defined provided}_h_{satisfies the given structure, and that}

the pathwise stochastic signalsS1(η(ω), ..., Sm(η(ω))) admit smooth approximating

se-quences (Sǫ

1(η(ω)), ..., Smǫ (η(ω)))ǫ>0 which converge toS(η(ω)) in a suitable topology as

ǫ goes to zero. The idea is that we isolate the problematic term hsingular, which causes

the non-linearity to be ill-defined but can be fixed under the influence of S(η(ω)). The trade-off being that by postulating these conditions, one also needs to prove the exis-tence and uniqueness of such ah, which eventually must solve theKPZ equation. The required approximation ofS(η(ω)) must also be explicitly construable using stochastic analysis as well.

For general singular stochastic equations with some ill-defined non-linearities, one thus needs to consider:

- which elements of the stochastic signal need to be approximated by smooth functions; - what structure of the solution can isolate the singularity and whether such singularity is fixable by postulating the above stochastic approximations;

- whether these restrictions are appropriate such that a fixed point argument can be closed, or that the stochastic approximations can be concretely constructed via stochastic analysis.

Since these structures are usually found by applying paradifferential calculus, such an approach is also known to be the theory of paracontrolled distributions. Chapter One of this text will be concerned with introducing the necessary analytic tools and we will demonstrate this method on the KPZ equation in Chapter Three.

Other Generalisations: The Theory of Regularity Structures

Finally, it is worth noting that there exists more than one generalisation to the theory of rough path. Most significantly, Swiss mathematician M. Hairer independently de-veloped what is now known as thetheory of regularity structures. Prior to the method of paracontrolled distributions, Hairer gave the first formulation of a robust solution theory to the KPZ equation via regularity structures and was subsequently awarded the fields medal in 2014. See [9] and the references thereafter for more details. Many correspondences between the method of paracontrolled distributions and the theory of

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Chapter 1. Introduction 7 regularity structures have been shown, see [2]. To this date, researches are still consis-tently being conducted in the area.

Before we begin our formal discussion, we make the remark that since we will be deal-ing with a great number of estimates, throughout the text we will generally use C to denote any strictly positive constants, with subscripts indicating dependencies.

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9

Chapter 2

Analysis in Besov Spaces

In this chapter we introduce the appropriate function spaces where our analysis of partial differential equations will take place in. Although the theory of Besov space is quiet overreaching, we will only be concerned with the special case which generalises the spaces of H¨older continuous function. We will start with the basic tool that will be used throughout this text, that is the Fourier transform.

2.1 Fourier Transform on Tempered Distributions

In this section we will recall some elementary facts of Fourier transform on the space of

Schwartz functions. As an extension we will define the space of tempered distributions

and the corresponding Fourier transform that follows. We define _D(Rn_{) to be the}

space of all compactly supported complex valued smooth functionsφ with continuous derivatives of all orders. IfΩ is a subset ofRn_{, then we also let} _D₍_Ω_{) denote the space}

of all functions in_D(Rn_{) which are compactly supported in the closure of}_Ω_{. The space}

of Schwartz functions S(Rn_{) consists of all smooth functions} _φ_{∈ D}₍Rn_{) such that}

kφ_kk,S def= sup |α|≤kk

(1 +_{| · |}2)k∂αu_kL∞ <∞, k∈N.

It is straightforward to check that, the topology generated by the family of seminorms (k · kk,S)k∈N is equivalent to the topology generated by a Fr´echet space. Thus,

lin-ear operators defined on _S(Rn_{) are continuous if and only if they are bounded. In}

particular:

1. If L:_S(Rn₎_→C _{is a linear functional, then} _L _{is continuous if and only if there}

exists integer k such that

|hL, φi| ≤Ckφkk,S ∀φ∈ S(Rn),

where we follow the notation that hL, φi denotes the application of L onφ. 2. If L:_S(Rn₎_{→ S}₍Rn_{) is a linear operator, then} _L _{is continuous if and only if for}

each integer k, there exists a integerN such that

kLφkk,S ≤CkφkN,S.

Recall that the Fourier transform on_S(Rn_{) is defined by}

Fφ(ξ) def=

Z

Rnφ(x)e

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It is also customary to denote such a operation by φb(ξ). Moreover, by the classical results in Fourier analysis, the mappingF :S(Rn₎_{→ S}₍Rn_{) is a linear homeomorphism}

with inverse F−1_φ₍_x_{) =} 1 (2π)n Z Rnφ(x)e −ξ·x_dξ.

Using these information, we can describe tempered distributions.

Definition 2.1.1. A tempered distribution on Rn _{is any continuous linear functional}

u on _S(Rn_{). The space of tempered distributions is denoted by} _S′₍Rn_).

We equip S′₍_Rn_{) with the weak-}_∗ _{topology with respect to} _S₍_Rn_{). Thus, a sequence}

(un)n∈N of tempered distributions converges to u inS′(Rn) if and only if

∀φ_{∈ S}(Rn₎_, _lim

n→∞hun, φi=hu, φi.

It is most convenient to define linear operators on tempered distributions via duality.

Proposition 2.1.2. Let L be a linear continuous map from _S into _S. The formula htLu, φ_i def= _hu, Lφ_i

then defines a tempered distribution. Moreover, tL is sequential continuous.

Proof. By assumption, the linear map L is continuous, hence there exists a constant

C and an integer N such that

∀φ∈ S, kLφkk,S ≤CkφkN,S.

Thus if u is a tempered distribution, then an integer k and a constant C′ _{exists such}

that

∀φ∈ S, |htLu, φi|=|hu, Lφi| ≤C′kLφkk,S ≤CkφkN,S.

The sequential continuity now follows from

htLun, φi=hun, Lφi −→ hu, Lφi=htLu, φi.

The proposition is thus proved.

Let us now provide two important examples of tempered distributions.

Example 2.1.3. The space L1

loc of locally integrable functions f on Rn such that (1 + |x_|2₎−N_f₍_x₎_∈_L1 _{for some} _N _∈_N _{can be identified with a subset of}_S′ _{by the formula}

L1_loc _{→ S}′ :f _7−→Tf, with _hTf, φ_i def=

Z

Rnf(x)φ(x)dx.

Example 2.1.4. Any finite Borel measureµ_{∈ M}B can be identified with an element of

S′ _{by the formula}

MB → S′ :µ7−→Tµ, with hTµ, φi def=

Z

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2.1. Fourier Transform on Tempered Distributions 11 We now list a few situations where Proposition2.1.2 applies.

Example 2.1.5. The differential and multiplication operators₋∂α _or_M

xα withα∈Nn

are continuous since

∀φ ∈ S, k(−∂)αφkk,S ≤ kφkk+|α|,S and kMxαφk_k_S =kxαφk_k,_S ≤ kφk_k₊_|_α_|_,_S.

Thus the corresponding dual operators t(₋∂α₎_,t

Mxα are continuous linear maps on

tempered distributions. Similarly, iff is a smooth function, then an integer N exists such that sup x∈Rn(1 +|x| k₎−N _sup |α|≤k| ∂α_f₍_x₎_|_<_∞_.

Thus, the multiplications t_Mf : S′ → S′ defines linear operators on tempered

distri-bution as well.

Example 2.1.6. Let θ be a Schwartz function, the convolution between θ and φ _{∈ S} is

θ_∗φ def=

Z

Rnθ(x−y)φ(y)dy.

Because of the identity

∀φ_{∈ S}, _kθˇ_∗ϕ_kk,S ≤Ckkθkk+n+1,Skϕkk,S, θˇdef= θ(−·)

the operationφ _7→θˇ_∗φ is continuous. Thus the formula

∀φ_{∈ S}, _hθ_∗u, φ_i def= _hu,θˇ_∗φ_i

defines a linear operator on tempered distribution. Hence convolution is defined on_S′_.

Example 2.1.7. In view of the continuity of F on S, the formula

∀φ _{∈ S}, _ht_Fu, φ_i def= _hu,_Fφ_i

defines a linear opeartor on tempered distribution.

Example 2.1.8. Let A be a linear automorphism ofRn _{and define}

LAφ def= 1

det(A)φ◦A

−1_.

ThenLA also satisfies the hypothesis of Proposition 2.1.2.

In order to extend classical operations to tempered distributions, in what follows we will make the abuse of notation and write∂α_,_M

θ,LAandF as linear mappingS′ → S′

respectively in places of t(−∂α_), t_M

θ, tLA and tF.

The Fourier transform on tempered distributions allows for many classical proper-ties defined on S to be transferred to the case ofS′_.

Proposition 2.1.9. For any (u, θ) ∈ S′ _{× S}_, _λ _∈ _R_\{₀_} _and ₍_{a, ω}₎ _∈ _Rn_×_Rn_{, we} have1 ∂α_u_ˆ₌_F₍_ix₎α_u_, ₍_iξ₎α_u_ˆ₌_F₍_∂α_u₎_{, e}−ia·ξ_u_ˆ₌_F₍_τ au), τωuˆ=F(eix·ωu), λ−nuˆ(λ−1ξ) = F u(λ_·), and _F(θ_∗u) = ˆθu.ˆ 1

Here we letτa denotes translation by−afor arbitrarya∈R n

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Proof. The first five properties follow obviously from their counterpart on S. To show the convolution theorem on S′_{, notice that we have}

hF(θ∗u), φi=hu,θˇ∗ Fφi.

By the simple calculation ˇ θ∗ Fφ= Z Rnθ(η−ξ) Z Rnφ(x)e −ix·η_dx_dη = Z Rne −ix·ξ Z Rne −i(x·η−ξ)_θ₍_η₋_ξ₎_dη_φ₍_x₎_dx₌_F_(ˆ_θφ₎

we thus infer that

hF(θ_∗u), φ_i=_hu,_Fθφˆ _i=_hu,ˆ θφˆ _i=_hθˆu, φˆ _i which proves the result

2.2 Dyadic Partition of Unity

We now define a special kind of partition of unity, which will become immensely useful for our theory throughout the rest of this text.

Proposition 2.2.1. Let _C be the annulus _{ξ _∈ Rn _| ₃_/₄ _{≤ |}_ξ_{| ≤} ₈_/₃_}_{. There exists}

radial functions χ and ϕ, valued in [0,1], belonging respectively to _D(B(0,4/3)) and D(C), and such that

∀ξ_∈Rn _{, χ}₍_ξ_{) +}X

j≥0

ϕ(2−jξ) = 1, (2.1)

|j₋j′_{| ≥}2_⇒Supp ϕ(2−j_·)_∩Supp ϕ(2−j′_·) =_∅, (2.2)

j _≥1_⇒Supp χ_∩Supp ϕ(2−j_·) = _∅. (2.3)

Proof. Take α in (1,4/3) and denote by ˜_C the annulus with small radius α−1 _{and big}

radius 2α. Use the existence of radial smooth cut-off function to choose θ valued in [0,1], with value 1 in a neighbourhood of ˜C. Without loss of generality assume j′ _≥_j_,

then 2j′C ∩2jC 6=∅ ⇒2j′ ×3/4≤4×2j+1/3 and j′ −j ≤1. Therefore |j−j′| ≥2⇒2j′C ∩2jC =∅. (2.4) Let S(ξ) def= X j∈Z

θ(2jξ) and ϕ(ξ) def= θ(ξ)/S(ξ), χ(ξ) def= 1₋X

j≥0

ϕ(2−jξ).

With these definitions, Supp ϕ(2j_·₎_⊆₂j_C _{and Supp} _χ_⊆ _B₍₀_,₄_._{3). (}_2.4_{) then ensures}

that S is locally finite and so smooth, and (2.2), (2.3) are also satisfied. Finally

χ(ξ) +X j≥0 ϕ(2−jξ) = 1−X j≥0 ϕ(2−jξ) +X j≥0 ϕ(2−jξ) = 1,

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2.2. Dyadic Partition of Unity 13 TheFourier multiplier with symbolθ(aξ) is defined for every real number andθ∈ D

the unique function which satisfies

Fθ(aD) =θ(aξ)ˆu.

We now let u_{∈ S}(Rn_{) and set}

∆ju def= 0 if j ≤ −2, ∆−1u def= χ(D)u=F−1χ∗u,

and ∆ju def= ϕ(2−jD)u=F−1_ϕ₍₂−j_·₎_∗_u _if _j _≥₀_.

The mappings (∆j)j≥−1 areLittlewood-Paley blocksor simply dyadic blocks. Moreover,

the nonhomogenous low-frequency cut-off operator is the mapping

Sju def= X

j′_≤_j₋₁

∆j′u=χ(2−jD)u, j ≥0

By letting hj =F−1_ϕ₍₂−j_·_{) and ˜}_h₌_F−1_χ_{, we also have}

∆ju= 2jn Z Rnh(2 j_y₎_u₍_x₋_y₎_dy _and _∆ −1u= 2jn Z Rn ˜ h(2jy)u(x−y)dy. Similarly, we let Sju= 2jn Z Rn ˜ h2jyu(x−y)dy.

Remark2.2.2. The dyadic partition of unity constructed in Proposition (2.2.1) is clearly not unique. In general, any (χ, ϕ) in _D(B′₎_{× D}₍_C′_{) which satisfies (}_2.1_{) and almost}

disjointness in the sense of (2.2) are dyadic partitions of unity. Since for every u _∈ L∞₍_Rn_{) and} _j _≥_{0 we have} sup x∈Rn 2jn Z Rnh 2j(x₋y)u(y)dy≤ ku_kL∞khk_L1

and likewise for the case of ˜h. The operators ∆ju and Sj map L∞₍_Rn_{) to} _L∞₍_Rn₎

continuously for all of j.

The point of the above construction is to justify the following convergence

Id = X

j≥−1

∆j in S′(Rn) (2.5)

and thus extend the definition for dyadic blocks to tempered distribution.

Proposition 2.2.3. Let u be in _S′₍_Rn₎_{. If} ₍_{χ, φ}₎ _{is a dyadic partition of unity, then} u= limj→∞Sju in S′(Rn).

Proof. Notice thathu−Sju, φi=hu, φ−Sjφi for all φ inS(Rn_{) and} _u _in_S′₍Rn_{). For}

every _|α_{| ≤}k, we can apply the Leibniz rule to see that

∂α_Fφ₋χ(2−jξ)_Fφ=1₋χ(2−jξ)∂α_Fφ+ X

0<β≤α

Cα,β2−j|β|(∂βχ)(2−jξ)∂α−β_Fφ.

Since_kχ_kL∞ ≤1 with χ= 1 near the origin, it follows that kF(φ₋Sjφ)_kk,S ≤CkkFφkk,S

for j large enough. Because the Fourier transform is an automorphism of _S(Rn_{) the}

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When defined on S′₍_Rn_{), we refer to (}_2.5_{) as the} _{Littlewood-Paley decomposition} _or

simply dyadic decomposition of the tempered distribution u.

We now consider two extremely useful Bernstein-type estimates.

Lemma 2.2.4. Let C be an annulus and B a ball. A constant C exists such that for any nonnegative integer k, any couple (p, q) in [1,_∞]2 _with _q _≥ _p_, _{λ >} ₀ _{and any}

function u of Lp_{, we have}

Supp ˆu_⊂λB=_{⇒ k}Dku_kLq def= sup

|α|=kk ∂αu_kLq ≤Ck+1λ(k+n)+n( 1 p−1q)kuk Lp Supp ˆu⊂λC =⇒C−k−1λkkukLp ≤ kDkuk_Lp ≤Ck+1λkkuk_Lp.

Proof. Let φ _{∈ D}(Rn_{) be such that} _φ _{= 1 near} _B _and _|_α_{| ≤} _k_{. Then we have}

ˆ

u(ξ) = ˆu(ξ)φ(λ−1_ξ_{) and therefore}

∂αu=λ|α|−nu_∗∂αg(λx) with ∂αg def= ∂α_F−1φ.

Applying Young’s convolution inequality we get

k∂αu_kLq ≤λ|α|−nk∂αg(λ·)k_Lrkuk_Lp with 1 r =− 1 p+ 1 q + 1. Since k∂α_g₍_λ_·₎_k Lr ≤λ(−1+ 1 p− 1 q)nk∂αgk Lr and k∂αgk_Lr ≤ k∂αgk_L∞+k∂αgk_L1 ≤Ck+1 we therefore conclude our first claim. The right hand side of the second claim follows readily from the first one by putting q = p. On the other hand, using the algebraic identity that for some constants Aα,

|ξ_|2k = X

1≤j1,...,jk≤n

ξ_j2₁ _{· · ·}ξ_j2_k = X

|α|=k

Aα(iξ)α(₋iξ)α,

we have for every φ_{∈ D}(Rn_\{₀_}_{) such that} _φ_{= 1 near} _C_{, that}

u= X

|α|=k

gα,λ∗∂αu with gα,λ def=Aα(−iξ)α|ξ|−2k_φ₍_λ−1_ξ₎_.

Thus, applying again Young’s inequality

kukLp ≤

X

|α|=k

kgα,λkL1k∂αuk_Lp,

and it suffices to note that

kgα,λ_kL1 =λ|α|kgαk_L1, with gα def= gα,₁ to conclude the proof.

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2.3. H¨older-Besov Spaces 15

2.3 H¨

older-Besov Spaces

We are now ready to discuss the function spaces in which our analysis will take place.

Definition 2.3.1. Let s _∈ R_{, the} _{H¨older-Besov spaces} _Cs₍Rn_;C_{) consists of all}

tem-pered distributionsu such that

ku_k_Cs def = (2js_k∆jukL∞)_j_∈Z _ℓ∞₍Zn₎<∞.

Clearly this defines a norm.

In the above definition we fixed a particular dyadic partition of unity. In what follows we will often assume, without further explanation that such a choice coincide with the one constructed in (2.2.1). However, it is important to know that the definition of Besov space is independent of such choices. For this we need the following important lemma.

Lemma 2.3.2. Let C′ _{be an annulus and B a ball. Let} ₍_uj₎

j≥−1 be a sequence of smooth

function on Rn _and

1. there exists constant C′ _{such that the functions satisfy}

Supp ˆuj _⊂2j_C′ and _kuj_kL∞ ≤C′2−js, j ≥ −1, s ∈R or;

2. there exists constant C such that the functions satisfy

Supp ˆuj ⊂2jB and kujkL∞ ≤C2−js, j ≥ −1, s >0 In either case, we then have

u def= X j≥−1 uj ∈ Cs _and _k_u_k Cs ≤Cs (2jskujkL∞)_j_≥−₁ _ℓ∞₍Z₎.

Proof. If the Fourier transform of uj is supported in 2j_C′_{. then there exists positive}

integer N such that

|j′₋j_|> N =_⇒∆j′u_j = 0. Thus we have k∆jukL∞ ≤ X |j′₋_j_|≤_N k∆juj′k_L∞ ≤2−jsCs (2jskujkL∞)_j_≥−₁ _ℓ∞₍Z₎. (2.6)

If the Fourier transform of uj is supported in 2jB. Then there exists positive integer

N′ _{such that}

j′ > j+N =⇒∆j′uj = 0,

and the same bound (2.6) applies with a geometric series. The claim follows. This gives rises to the aforementioned corollary.

Corollary 2.3.3. The spaceCs _{does not depend on the choice of dyadic decomposition.} Proof. Let u be in Cs _{and suppose (}_χ′_{, ϕ}′_{) is another dyadic partition of unity with}

the corresponding dyadic blocks∆′

j. Proposition (2.2.2), (2.2.3) together with Lemma

(2.3.2) then implies there exists constant Cs such that

u= X j≥−1 ∆′_ju and _ku_kCs ≤C_s (2js_k∆′_ju_kL∞)_j_∈Z ℓ∞₍Z₎.

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2.3.1 Completeness and Embeddings

Let us gather some topological properties of H¨older-Besov spaces. A natural first observation is that they are complete.

Proposition 2.3.4. For every s _∈ R _{the space} _Cs _{is continuously embedded into} _S′_.

Moreover, (_Cs_, _{k · k}

Cs) is a Banach space.

Proof. Let us apply the idea used in the proof of Lemma (2.2.4). Since the Fourier transform of ∆ju is supported in 2jC, we can choose φ ∈ D(C\{0}) such that φ = 1

near _C, then for all j _≥0,

∆ju= 2−jk X

|α|=k

2jngα(2j·)∗∂α∆ju.

Hence, for any Schwartz function φ_{∈ S}, we have

h∆ju, φi= (−1)k2−js

X

|α|=k

h∆ju,gαˇ ∗∂αφi,

where we recall that

gα =Aα(−iξ)α|ξ|−2k_φ₍_ξ₎

as in the proof of the Bernstein estimates Lemma 2.2.4. Hence we can find constant

Mk such that |h∆ju, φi| ≤2−j(2j(1−k)k∆jukL∞)kφk_M k,S X |α|=k kgˇαk_L1 2. Since we can choose k toe be as large as we want, it follows that

|h∆ju, φi| ≤Ck2−jkukCskφk_M k,S.

Now take summation over all j _{≥ −}1 to see that

|hu, φi| ≤CkkukCskφk_M_k_,_S.

This concludes the continuity of the embedding _Cs _֒_{→ S}′_{. If (}_u(k)₎

k∈N is a Cauchy

sequence in Cs_{, then (}_h_u(n)_{, φ}_i₎

k∈N must be a Cauchy sequence in R. Thus by the

Banach-Steinhaus theorem in Fr´echet space, the limit limk→∞hu(k), φidefine a tempered

distribution. Moreover, (∆ju(k))k∈N needs to be a Cauchy sequence in L∞ for every j.

Since L∞ _֒_{→ S}′ _{is a continuous embedding and that the actions of dyadic blocks are}

continuous in _S′ _{we also have lim}

k→∞∆(jk)u(k) =∆ju. Therefore, for everyǫ >0 there

exists a positive integer N such the

2jsku(k)−u(k′)kCs ≤ ku(k)−u(k ′₎

kCs ≤ǫ

for all k, k′ _≥_N_{. By taking the limit as} _k′ _{goes to infinity and supremum over all} _j _we

obtain the desired inequality.

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2.3. H¨older-Besov Spaces 17

Proposition 2.3.5. Let s be a real number, then the H¨older-Besov spaces Cs _satisfies the following embedding properties:

1. _Cs _֒_→_L∞ _if _{s >}₀_;

2. L∞ _֒_{→ C}s _if _s_≤₀_; 3. _Cs _֒_{→ C}r _if _r_≤_s_;

4. Let (s, r)_∈[1,_∞]2 _and_Bs

p,r to be the space of all tempered distributions such that kukBs p,r def = (2jsk∆jukLp)_j_∈Z ℓr₍Z₎.

Let 1≤p1 ≤p2 ≤ ∞ and 1≤r1 ≤r2 ≤ ∞, then we have Bps1,r2 ֒→B

s−n(_p1

1− 1

p2)

p2,r2 .

Proof. The first embedding is a strightforward consequences of

ku_kL∞ ≤ X j≥−1 k∆jukL∞ ≤ X j≥−1 2−js_ku_kCs

and the fact that s >0. For the second claim we have

ku_kCs = (2−j|s|_k∆jukL∞)_j_∈Z ℓ∞₍Z₎ ≤ (_k∆jukL∞)_j_∈Z ℓ∞₍Z₎≤CkukL∞

whenever j _≥ 0 and s _≤ 0. The third claim is a trivial consequence of the first two. Using the embeddings ℓr1₍_Z₎ _֒→ _ℓr2₍_Z_{) and an application of the Bernstein lemma} which yields

k∆jukLp2 ≤C2

jn(1

p₁−p1₂)_k_∆_ju_k Lp1, we arrive at the final claim as well. This concludes the proof.

2.3.2 The H¨

older Condition

In this subsection we will justify why the spaces _Cs _{are called H¨older-Besov spaces.}

Indeed, let s be in (0,1), recall that the space of H¨older continuous functions consists of real valued functionsusuch that_|u(x)₋u(y)_{| ≤}C_|x₋y_|s_{for some positive constant} C >0. If s is instead chosen to be in (0,_∞), then a natural generalisation of this idea is to consider the spaceC_bs,s−[s] of all bounded real valued functions such that

kuk_Cs,s−[s] b def = kuk_C[s] b + max|α|=[s]sup_x₆₌_y |∂α_u₍_x₎₋_∂α_u₍_y₎_| |x₋y_|s−[s] <∞.

It turns out that if s is not an integer, then C_bs−[s] could be characterised by _Cs_. Proposition 2.3.6. Let s be in R+_\N_{, then the spaces} _Cs _and _C[s],s−[s]

b coincide.

As a non-standard result, let us first prove a Taylor-type estimate for elements of

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Lemma 2.3.7. Let s be in R+_\N_{. Then for every} _u _∈ _C[s],s−[s]

b there exists positive constant Cs such that

u(x)− X |α|≤[s] (x₋y)α α! ∂ α_u₍_y₎ ≤ Cs [s]!|x−y| s_.

Proof. For functions f :R_→R_and _θ′ _∈₍₀_,_{1), it is easy to see the expansion}

f(1) = X k≤[s] θk k! dk dθkf(0) + θ[s] [s]! _d[s] dθ[s]f(θ ′₎ − d [s] dθ[s]f(0) .

Apply the above identity to f(θ) = uy+θ(x₋y), we obtain the expression

u(x) = X |α|≤[s] (x−y)α α! ∂ α_u₍_y_{) +} X |α|=[s] (x−y)α α! ∂αuy+θ′(x₋y)₋∂αu(y) , with X |α|=[s] (x−y)α α! ∂αuy+θ′(x₋y)₋∂αu(y) ≤ _[Cs_s_]!|x−y|s

which is the desired result.

From the classical estimate _k∂α_u_k

C_bs,s−[s] ≤ CskukC_bs+|α|,s−[s], it is also natural for the

differential operator defined on Cs _{spaces to be continuous.}

Lemma 2.3.8. Let (s, α)be in R_×Nn_{. Then the differential operator}_∂α _:_Cs_{→ C}s−|α|

is bounded.

Proof. Since for each j we have

∂α∆ju=∂αF−1

ϕ(2−jξ)ˆu(ξ)=_F−1(iξ)αϕ(2−jξ)ˆu(ξ)=_F−1ϕ(2−jξ)∂dα_u₍_ξ₎_.

By an application of the Bernstein Lemma (2.2.4), there exists annulus_C and constant

C such that

2j(s−|α|)_k_∂α_∆

jukL∞ ≤C22jsk∆_juk_L∞.

Summing over all j yields the required estimate. We are now ready to prove Proposition (2.3.6).

Proof of Proposition 2.3.6. Let us note first that, since

Z Rnx α_h₍_x₎_dx₌_i−αZ RF −1_∂α_ϕdx_{= 0}_, ∀α_∈Nn_,

owing to the binomial theorem, we can write

∆ju(x) = 2jn Z Rnh 2j(x−y)u(y)− [s] X |α|=1 (y₋x)α α! ∂ α_u₍_x₎_dy.

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2.4. Paradifferential Calculus 19 So by an application of the previously established Taylor expansion, we have

k∆jukL∞ ≤ (1 +Cs)

[s]! kukL∞2

−js

kh_{| · |}s_kL1,

together with the obvious bound _k∆−1ukL∞ ≤Ckuk_L∞ it is easy to see that kukCs ≤

(1 +Cs)

[s]! kukC_b[s],s−[s]kh| · | s_k

L1.

In virtue of the continuity of the differential operator, it suffices to prove the estimate

k∂α_u_k

C_b0,s−[s] ≤CskukCs. Suppose now that |x−y|>1, then by definition |∂αu(x)−∂αu(y)| ≤2k∂αukCs−[s]|x−y|s−[s],

we already have one part of the claim. If instead|x−y| ≤1, then choose j′ _{such that}

there exists c′_{, C}′ _{which satisfies} _c′₂−j′

≤ |x₋y_{| ≤}C′₂−j′ . We have |∂αu(x)−∂αu(y)| ≤ X j<j′ |∂α∆ju(x)−∂α∆ju(y)|+ X j≥j′ |∂α∆ju(x)−∂α∆ju(y)|.

Forj < j′_{, we make an application of the Bernstein lemma and the mean value theorem}

to obtain that

|∂α∆ju(x)₋∂α∆ju(y)| ≤C22j(1−s+[s])2j(s−[s])k∂α∆jukL∞|x−y| ≤C22j(1−s+[s])_k∂αu_k_Cs−[s]|x−y|.

Forj _≥j′_{, we simply have}

|∂α∆ju(x)−∂α∆ju(y)| ≤2k∂α∆jukL∞ ≤2j([s]−s)k∂αuk_Cs−[s]. Putting everything together we see that

|∂αu(x)₋∂αu(y)_{| ≤}C22j′(1−s+[s])_k∂αu_k_Cs−[s]|x−y|+ 2j

′_([_s_]₋_s₎

k∂αu_k_Cs−[s]

≤C2_|x₋y_|s−[s]_k∂αu_k_Cs−[s] +|x−y|s−[s]k∂αuk_Cs−[s] which is the desired inequality. This concludes the proof of the claim.

2.4 Paradifferential Calculus

A central theme of our approach is to develop a theory of integration for distributions driven against irregular noise. For this purpose it is first necessary to make sense of the products between distributions. One way to do this is to consider paraproduct.

2.4.1 The Bony Decomposition

Consider two tempered distributions uand v. It is natural to study simple operations between them. In particular, since we would like distribution to generalise the space of functions, it would be useful to have a notion of addition and multiplication. Addition is easily understood. However, it is generally difficult to make sense of the product

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uv, and paraproduct is the mathematical tool of doing that via duality. We have already developed a representation ofuand v that involves duality, namely, the dyadic decomposition u= X j≥−1 ∆ju and v = X j′_≥−₁ ∆j′v

which are defined using Fourier transform. Formally, we could write

uv =Tuv+Tvu+R(u, v), with Tuv def= X j≥−1 Sj−1u∆jv and R(u, v) def= X |k−j|≤1 ∆ku∆jv.

Constructed this way, we say that Tuv is the paraproduct of v by u, and R(u, v) is the

remainder of u and v. Such a decomposition is called the Bony decomposition. The main continuity properties of the paraproduct and the remainder are described below.

Proposition 2.4.1. For any couple of real numbers (s, r) we have T :L∞_{× C}s_{→ C}s: (u, v)_7−→ X j≥−1 Sj−1u∆jv, s∈R, T :_Cs_{× C}r _{→ C}s+r: (u, v)_7−→ X j≥−1 Sj−1∆jv, s <0, r∈R, R:_Cs_{× C}r _{→ C}s+r: (u, v)_7−→ X |k−j|≤1 ∆ku∆jv, s+r >0, are bounded bilinear operators.

Proof. We remark that for j, j′ _≥₀

∆j′u∆_jv =F−1 ϕ(2−j′uˆ(ξ))∗ϕ(2−jξ)ˆv(ξ) and for j′ ₌₋_1, ∆j′u∆_jv =F−1 χ(2−j′uˆ(ξ))_∗ϕ(2−jξ)ˆv(ξ).

Assume u is in L∞ _and _v _{is in} _Cs_{. Then by the above identities} _F₍_Sj

−1u∆jv) is

supported in 2j_C_˜_{. Since}

kSj−1u∆jvkL∞ ≤ kSj₋₁uk_L∞k∆_jvk_L∞ ≤C2−jskuk_L∞kvk_Cs.

By Lemma 2.3.2 the first bound follows. If we have (u, v)_{∈ C}s_{× C}r _with _{r <}_{0. Then}

the second bound follows from

kSj−1v∆jukL∞ ≤C_r2(−j+1)rk∆_jvk_L∞kvk_Cr ≤C_s,r2−j(r+s)kuk_Cskvk_Cr If s+r >0, then write R(u, v) = X j Rj(u, v) with Rj(u, v) def= X |ν|≤1 ∆j−νu∆jv.

Notice there exists a ball B′ _{such that}_F_R

j is uniformly supported in 2jB′ and kRj(u, v)kL∞ ≤ X |ν|≤1 2(j−ν)s_k∆j−νukL∞2jrk∆_jvk_L∞2(−j+ν)s2−jr ≤2−j(s+r)_ku_kCskvk_Cr X |ν|≤1 2−νs <_∞.

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2.4. Paradifferential Calculus 21

2.4.2 Actions on Besov Spaces and Paralinearisation

In this section we will study various actions on the H¨older-Besov spaces and highlight a number of important calculations. Our main result will concern the effects of left composition by sufficiently smooth functions.

Proposition 2.4.2. Let s be a positive real number and σ be the smallest positive integer such that σ ≥ s. If u be long to Cs_{, then so does} _f _◦_u _{for every} _f _∈_Cσ

b, and we have

kf_◦u_kCs ≤C(s,kuk_L∞)kfk_Cσ bkukCs for some constantC(s,_ku_kL∞) depending on s and kuk_L∞. Proof. We introduce formally the series

X

j≥−2

fj with fj def= f(Sj+1u)−f(Sju) and f−2 def= f|x=0. Forj _{≥ −}1, it is clear that

fj =mj∆ju with mj def=

Z 1

0 f

′₍_Sj_u₊_t′_∆

ju)dt′. (2.7)

We will need to make use of the following lemma.

Lemma 2.4.3. Let g be a function from R2 _toR _{such that}_g _∈_Cσ

b. Forj ≥ −1, define mj(g) def= g(Sju,∆ju).

For any bounded function u, there exists positive constant Cα such that ∀α ∈Nn_, _k_∂α_mj₍_g₎_k_L∞ ≤Cα(1 +kuk_L∞)|α|kgk_Cα

b2 j|α|_.

Proof. By the Fa´a di Bruno’s formula, we have

∂αmj(g) = X p1,p2,ν C_pν₁_,p₂ _Y 1≤|β|≤|α| (∂βSju)νβ1(∂β∆_ju)νβ2 ∂p1 1 ∂ p2 2 g(Sju,∆ju),

where the coefficientsCν

p1,p2 are nonnegative integers, and the sum is taken over those

p1, p2 and ν such that 1≤p1+p2 ≤ |α| such that

X 1≤|β|≤|α| νβ_j =p1 for j = 1,2 and X 1≤|β|≤|α| β(νβ1 +νβ2) = α. Note that there exists a constant C′ _{such that}

max_{k∆ju_kL∞,kSjuk_L∞} ≤C′kuk_L∞.

Ifgis sufficiently regular, then its derivatives are bounded up to orderαonB(0, C_ku_kL∞).

By an application of the Bernstein estimate we thus arrive at

k∂αmj(g)_kL∞ ≤ max p1,p2,νC ν p1,p2 X ν Y 1≤|β|≤|α| C2|β|+32j|β|(νβ1+νβ2)kukνβ1+νβ2 L∞ kgkCα b ≤C_ν,p2|α₁|+3_,p₂(1 +_ku_kL∞)|α|kgk_Cα b2 j|α|

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Let us now resume the proof of Proposition 2.4.2. It is clear that f−2 is in Cs. For

j′ ≥ −1 we make the decomposition

∆j′ X j≥−1 fj = X j<j′ ∆j′(mj∆_ju) + X j≥j′ ∆j′(mj∆ju).

For the first sum, we take advantage of Lemma 2.4.3 to get that

k∂αmj_kL∞ ≤ Z 1 0 k∂ α_f′₍_Sj_u₊_t′_∆ ju)kL∞dt′ ≤Cα(1 +kuk_L∞)σkf′k_Cσ b2 jσ

whenever _|α_|=σ. It then follows from the Bernstein lemma2.2.4 and the Leibniz rule that k∆j′ mj(u)∆ju_kL∞ ≤Cσ+12−j ′_σ kDσ∆j′ mj∆ju kL∞ ≤C_σ2σ+22−j′σ X β≤σ Cα,β2j|β|(1 +kukL∞)σkf′k_Cσ b2 j(|α|−|β|)_k_∆ jukL∞ ≤C′ σ(1 +kukL∞)σkf′k_Cσ b2 (j′₋_j₎₍_s₋_σ₎ (2js_k_∆ jukL∞). Therefore, as ∆j′ X j<j′ (mj∆ju) _L∞ ≤C ′ σ(1 +kukL∞)σkf′k_Cσ b X j∈Z 2(j′−j)(s−σ)(2js_k∆jukL∞),

an application of Young’s inequality yields

2j′s ∆j′ X j′_≥−₁ (mj∆ju) L∞ j′_∈Z ℓ∞₍Z₎ ≤C_σ′(1 +_ku_kL∞)σkf′k_Cσ bk(2 j(s−σ)₎ j∈Zk_ℓ1₍Z₎kuk_Cs.

Likewise, estimation for the second sum follows easily from 2j′sk∆j′(mj∆ju)k_L∞ ≤2(j

′₋_j₎_s

kf′kL∞2jsk∆_juk_L∞)≤ kf′k_L∞kuk_Cs.

This proves the claim.

Similarly, if f is sufficiently regular, then the action of f on u can be linearised using paraproduct. Such operation is called paralinearisation.

Proposition 2.4.4. Let s be in (0,1) and r ∈(0, s]. Suppose that f is in C_b1,r/s, then there exists a locally bounded operator

Rf :Cs_{→ C}s+r_:_u_7−→_f _◦_u₋_Tf ′_◦uu in the sense that

kRfu_kCs+r ≤Ckfk

C_b1,r/s(1 +kuk

1+r/s Cs ).

If instead f is in C_b2,r/s, then Rf is locally Lipschitz continuous in the sense that kRfu−RfvkCs+r ≤Ckfk

C2_b,r/s(1 +kukCs +kvkCs)

1+r/s_k_u₋_v_k Cs.