Around the Nash-Moser Theorem

Ruoyu Wang ∗

Assessor: Cl´ement Mouhot† May 1, 2017

Declaration: I declare that this essay is work done as part of the Part III Examination. I have read and understood the Statement on Plagiarism for Part III and Graduate Courses issued by the Faculty of Mathematics, and have abided by it. This essay is the result of my own work, and except where explicitly stated otherwise, only includes material undertaken since the publication of the list of essay titles, and includes nothing which was performed in collaboration. No part of this essay has been submitted, or is concurrently being submitted, for any degree, diploma or similar qualiﬁcation at any university or similar institution.

∗DPMMS, CMS, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK; †DPMMS, CMS, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK.

1 Contents

1 Introduction 3

2 Pseudodiﬀerential Operators 4 2.1 Symbols of Type 1, 0 ...... 4 2.2 Asymptotics and Classical Symbols ...... 7 2.3 Operators, Kernels, Adjoints, Quantisation ...... 9 2.4 Symbolic Algebra of Operators ...... 13 2.5 Operators on Sobolev Spaces ...... 14 2.6 Ellipticity and Parametrices ...... 16

3 Nonlinear Dyadic Analysis 18 3.1 Littlewood-Paley Decomposition ...... 18 3.2 Characterisation of Sobolev and H¨olderSpaces ...... 21 3.3 Operators on H¨older-Zygmund Spaces ...... 23 3.4 Dyadic Analysis of Products ...... 29

4 H¨olderScale Nash-Moser Theorem 30 4.1 Preliminaries ...... 30 4.2 Statement of Theorem ...... 31 4.3 Iteration Scheme ...... 32 4.4 Iterative Hypotheses ...... 36 4.5 Existence and Regularity ...... 41

5 Isometric Embedding Theorem 45 5.1 Isometric Embedding Problem ...... 45 5.2 Nash-Moser Approach ...... 46 5.3 G¨unther’s Approach ...... 50

6 Abstract Nash-Moser Theorem 56 6.1 Scale of Banach Spaces ...... 56 6.2 Banach Scale Nash-Moser Theorem ...... 60

7 Epilogue 63 7.1 Review ...... 63 7.2 Acknowledgement ...... 63 7.3 Bibliography ...... 63

2 1 Introduction

In this essay our main target is to build up an implicit function theorem which could be applied to tamed mappings between C∞(M) spaces for some compact manifold M. Note in this case C∞(M) is only a Fréchet space and hence the standard implicit function theorem can not be invoked. Moreover tameness of the mapping allows a fixed order loss in derivatives, that is, if we utilise a standard iteration scheme by assuming information on finite order derivatives, all information will be lost in finite time, which makes a standard iteration impracticable. Those two facts remark the necessity and difficulty of establishing a new implicit function theorem.

This new implicit function theorem, nowadays known as the Nash-Moser theorem, was firstly devised by Nash [19] in order to prove the smooth case of his famous isometric embedding theorem. Further refinements, improvements and new versions were attributed to, not exhaustively listed, Hörmander[10] [11] [12] [14], Zehnder [24], Mather, Sergeraert, Tougeron, Hamilton [8], Hermann, Craig, Dacorogna, Bourgain, Berti and Bolle, Ekeland and Séré[5] [6], and lately by Villani and Mouhot.

The Nash-Moser theorem is a signiﬁcant result in analysis of nonlinear PDEs, for it allows iteration schemes with a loss of regularity in each step to be run to achieve an existence construction. It is, not exhaustively listed, used in the cases of isometric embedding problem, global existence theorems for hyperbolic second order equations, small divisor problems, and various other control and Cauchy problems.

In this essay, we are going through major ideas in the following order: Section 2 is a brief introduction to pseudodifferential operators; Section 3 is intended to introduce use- ful dyadic results via Littlewood-Paley theory; in Section 4 we detailed a very descriptive proof of the Hölderspace based Nash-Moser theorem; Section 5 specialises in tackling the isometric embedding theorem, respectively by Nash’s original approach, which is an immediate consequence of the Nash-Moser theorem, and by Günther’s approach via ellipticity; in Section 6 we prove an abstract version of Nash-Moser theorem by introducing modification of Banach spaces.

3 2 Pseudodiﬀerential Operators

In this chapter, we develop the basic calculus of pseudodiﬀerential operators.

2.1 Symbols of Type 1, 0 Pseudodifferential operators are generalisation of classical differential operators. By treat- ing classical differential operators as a multiplication operator by multiplying a polynomial symbol on the Fourier side, we aim to define classes of symbols larger than polynomials.

Deﬁnition 2.1 (Space of symbols). Given m ∈ R, by Sm(RN ×RN ) we denote all a(x, ξ) ∈ C∞(RN × RN ) such that for any multi-indices α and β we have the global estimate

α β m−|α| N ∂ξ ∂x a(x, ξ) ≤ Cα,β hξi , ∀x, ξ ∈ R . (1)

2 1/2 −∞ m m Here hξi = (1 + |ξ| ) . And denote S = ∩mS . We call a ∈ S a symbol of order m. Remark. (i) We don’t usually distinguish between estimate by hξim and (1 + |ξ|)m, for it is easy to verify 1 1/2 √ (1 + |ξ|) ≤ 1 + |ξ|2 ≤ (1 + |ξ|) . 2 They both denote growth control of a m-th order polynomial if m is an integer. (ii) Symbols of order m ∈ N have growth control by an order m polynomial, and they resemble polynomial symbols of classical linear diﬀerential operators. The most important part of this deﬁnition is that it gives a growth control when |ξ| is large, and it is exactly the case when |ξ| is large is worth consideration, because symbols are always bounded bounded inside any compactum.

P α Example. (i) Constant coefficient linear differential operators, |α|≤m aαD has differen- P α tial symbol as |α|≤m aαξ , where D = −i∂. For β ≤ α, its β-derivatives in ξ are bounded |α|−|β| by Cβ hξi . Hence the symbols of m-th order constant coefficient linear differential operators are in Sm. P α (ii) For a more general linear differential operators |α|≤m aα(x)D , if its coefficients ∞ aα(x) ∈ C are bounded in derivatives in x of all order, then its differential symbol can have a global control in derivatives in ξ, hence in Sm. (iii) Given a(ξ) ∈ C∞(RN \ 0), which is positive homogeneous of order m. It is easy to deduce that a(k)(ξ), the k-th order derivative of a(ξ) is homogeneous of order m − k. ∞ N Now set a cutoff function χ(ξ) ∈ Cc (R ) and is 1 in a neighbourhood of 0; and set a˜(ξ) = (1 − χ(ξ))a(ξ), which is smooth on whole RN and excludes singularity of a at 0. Hence we have the estimate

α |α|−m α m−|α| ∂ξ a˜(ξ) ≤ |p| ∂ξ a (p) hξi ,

for a ﬁxed p outside support of χ, when |ξ| > 1. Hencea ˜ ∈ Sm.

4 (iv) Given a(ξ) ∈ S(RN ). Because at inﬁnity, Schwartz class function a and all its derivatives are vanishing faster than polynomials of order −m for all m ∈ N, hence the estimate (1) holds for all m. So a ∈ S−∞. We now establish some properties of the family of Sm: m α β m−|α| Proposition 2.2 (Algebra of symbols). (i) If a ∈ S , then ∂ξ ∂x a ∈ S . (ii) Ss ⊂ St for s < t. (iii) If a ∈ Sm1 , b ∈ Sm2 then a + λb ∈ Smax(m1,m2) and ab ∈ Sm1+m2 for any scalar λ ∈ R. Proof. (i) Clear via γ θ α β m−|α+γ| ˜ (m−|α|)−|γ| ∂ξ ∂x ∂ξ ∂x a(x, ξ) ≤ Cα+γ,β+θ hξi ≤ Cγ,θ hξi . (ii) Given a ∈ Ss we have

α β s−|α| t−|α| ∂ξ ∂x a(x, ξ) ≤ Cα,β hξi ≤ Cα,β hξi for hξis−t ≤ 1. m1 (iii) Without loss of generality assume m1 > m2. By (ii) we have b ∈ S and hence

α β 1 2 m1−|α| ∂ξ ∂x (a + λb) ≤ Cα,β + λCα,β hξi . For the second part we have α β X X α β γ θ (α−γ) (β−θ) ∂ ∂ (ab) ≤ ∂ ∂ a ∂ ∂ b ξ x γ θ ξ x ξ x γ≤α θ≤β

X X m1−|γ| m2−|α|+|γ| ≤ Cα,β,γ,θ hξi hξi γ≤α θ≤β

m1+m2−|α| = Cα,β hξi as required. We also quote a lemma for an easier calculus of symbols of order 0: 0 Lemma 2.3 (Function of symbols). Given symbol a1, . . . , ak ∈ S , a smooth function ∞ k 0 F ∈ C (C ), then F (a1, . . . , ak) ∈ S . 0 Proof. Since we also know the real and imaginary parts of aj are in S , we reduce the ∞ k problem without loss of generality to the case of real symbols aj and F ∈ C (R ). Consider the formulae of diﬀerentiation: ∂ X ∂F ∂ X ∂F F (a) = ∂ a , F (a) = ∂ a . ∂ ∂ xj q ∂ ∂ ξj q xj q aq ξj q aq Prove by induction on p that estimate (1) is valid for any α + β ≤ p. Case p = 0 is clear, and for general |α| + |β| = p + 1, pick a component j of α or β which is positive, then ∂ ∂ α β applying the Leibniz’s formula to F (a) or F (a) reduces ∂ξ ∂x F (a) to a ﬁnite sum of ∂ξj ∂xk

multi-indices-sum p estimate on ∂ξj aq or ∂xj aq, which is obvious to establish the required estimate.

5 Remark (Topological structures of Sm). Sm is a Fr´echet space under the semi-norms

Sm n −m+|α| α β o ka(x, ξ)kα,β = sup hξi ∂ξ ∂x a(x, ξ) . RN ×RN

Sm And we say an → a in this topology if kan − akα,β → 0 for all multi-indices α, β.

Lemma 2.4 (Approximation of symbols). Given symbol a ∈ S0(RN ×RN ), have the zoom 0 m function in frequencies aε = a(x, εξ). Then aε ∈ S and when ε → 0, aε → a0 in S for any m > 0.

s t Proof. Firstly we show that for s < t the injection S ⊂ S is continuous, that is, if an → 0 s t in S then so is an in S . Clear we have

St n −s−(t−s)+|α| α β o ka(x, ξ)kα,β = sup hξi ∂ξ ∂x a(x, ξ) RN ×RN n −s+|α| α β o Ss ≤ sup hξi ∂ξ ∂x a(x, ξ) = ka(x, ξ)kα,β RN ×RN

−(t−s) m for hξi ≤ 1. Hence if we can show for 0 < m ≤ 1 we have aε → a0 in S , the convergence for all positive m is evident. α β Now consider 0 < m ≤ 1, and ε < 1 small. For α > 0 we have immediately ∂ξ ∂x a0 = 0 and the estimate

α β |α| α β |α| −|α| ∂ξ ∂x a(x, εξ) = ε ∂εξ∂x a(x, εξ) . ε (1 + ε |ξ|) . And hence we see

Sm n −m+|α| |α| −|α|o kaε − a0kα,β . sup (1 + |ξ|) ε (1 + ε |ξ|) RN ×RN n o = εm sup (1 + |ξ|)−m+|α| ε|α|−m (1 + ε |ξ|)−|α|+m sup (1 + ε |ξ|)−m ≤ εm RN ×RN RN ×RN for the two suprema are both no greater than 1. For α = 0 we see a0 6= 0, but instead we have Z 1 Z ε|ξ| β α dτ m m ∂x (aε − a0) = ε ξ. ∇ξ∂x a(x, tεξ) dt ≤ C = C log(1 + ε |ξ|) ≤ Cmε |ξ| , 0 0 1 + τ m by log(1 + x) ≤ Cmx for x ≥ 0 and m > 0. Then

Sm −m m m m kaε − a0k0,β = sup (1 + |ξ|) Cmε |ξ| ≤ Cmε RN ×RN for (1 + |ξ|)−m |ξ|m ≤ 1. Hence we obtain the control for all α, β:

Sm m kaε − a0kα,β . ε → 0

m as ε → 0, so for 0 < m ≤ 1 we have aε → a0 in S , and hence for all positive m.

6 2.2 Asymptotics and Classical Symbols The definition of symbol only contains its information on its highest order term, that is its highest order of growth in frequencies at infinity. For symbols of classical linear differential P α operators, |α|≤m aαξ , we have the clear layering of symbols of order 0, 1, . . . , m, each part of which is respectively homogeneous of 0, 1, . . . , m. We can further extract the principal P α symbol, |α|=m aαξ , non-trivial zeroes of which determine ellipticity. We want to restore this layering feature on our pseudodifferential symbols in this section, via introducing asymptotics and classical pseudodifferential symbols.

mj Deﬁnition 2.5 (Asymptotics). Given a sequence of symbols aj ∈ S with mj & −∞. P We say aj is the asymptotics of some function a(x, ξ), as of behaviour when |ξ| → ∞, written X a ∼ aj if for any k ≥ 0 we have k X mk+1 a − aj ∈ S . j=0

Remark. For a symbol a(x, ξ) ∈ Sm(RN × RN ) we see it is asymptotically itself, the fact of which does not validate why we need to introduce an asymptotic expansion. Actually, symbols a(x, ξ) which only depend on location and frequency, are only one type of a more general class of symbols, a(x, y, ξ) ∈ Sm(RN × RN × RN ), which aside from being a multiplication operator on the Fourier side, also perturbs the Fourier transform to some degree. For such kind of more general symbol, if it is properly supported, we can asymptotically m N N associate σa(x, ξ) ∈ S (R × R ) with the general symbol a(x, y, ξ) by formula X 1 σ (x, ξ) ∼ ∂αDα a(x, y, ξ)| . a α ξ y x=y α This process is called the left quantisation. Furthermore we can express adjoints and com- positions of the pseudodifferential operators introduced by symbols via their asymptotics, which we will cover later. We firstly see that given any asymptotics we can find an equivalent pseudodifferential symbol, unique up to class S−∞.

mj Proposition 2.6 (Symbol determined by asymptotics). Given aj ∈ S with mj & −∞, m0 P −∞ there exists a ∈ S such that a ∼ aj, unique up to class S . Furthermore the choice can be made such that supp a ⊂ ∪ supp aj. Proof. The uniqueness part is obvious. If a anda ˜ are both asymptotically equivalent to P aj, then for any k > 0 we have

k ! k ! X X mk+1 a − a˜ = a − aj − a˜ − aj ∈ S j=0 j=0

7 mk+1 −∞ as the sum of two S symbols. As mk & −∞ we have a − a˜ ∈ S . ∞ N For the existence part, ﬁrstly set up a standard cutoﬀ function χ(ξ) ∈ Cc (R ) which is 1 near 0. Now construct ∞ ∞ X X a(x, ξ) = a˜j(x, ξ) = (1 − χ(εjξ)) aj(x, ξ), j=0 j=0 in which εj are chosen smartly such that they vanish to zero fast enough for some estimate ona ˜j to hold. In fact, we have α β X X α β γ θ (α−γ) (β−θ) ∂ ∂ a˜j ≤ ∂ ∂ (1 − χ (εjξ)) ∂ ∂ aj ξ x γ θ ξ x ξ x γ≤α θ≤β

X X |γ|−1 γ θ 1+mj −|α| ≤ Cα,β,γ,θ hξi ∂ξ ∂x (1 − χ (εjξ)) hξi . γ≤α θ≤β

Consider that 1−χ (εξ) is in fact a zoom function of 1−χ (ξ) ∈ S0, we have 1−χ (εξ) → 0 1 in S as ε → 0 by the aid of Lemma 2.4, that is we can pick εj small enough such that for any γ ≤ α, θ ≤ β we have ! ! !−1 S1 n |γ|−1 γ θ o −j X X k1 − χ (εjξ)kγ,θ = sup hξi ∂ξ ∂x (1 − χ (εjξ)) ≤ 2 1 1 Cα,β,γ,θ . γ≤α θ≤β So we have α β −j 1+mj −|α| ∂ξ ∂x a˜j ≤ 2 hξi , the estimate which is achieved exactly by choosing εj as above. P Now we verify that a is asymptotically equivalent to aj. Assume we are given arbitrary α and β, pick p ≥ |α| + |β| and pick k such that mp + 1 ≤ mk+1. We ﬁrstly establish !

α β X X −j 1+mj −|α| mk+1−|α| ∂ξ ∂x a − a˜j ≤ 2 hξi ≤ hξi , j≤p−1 j≥p

1+mj mk+1 because hξi ≤ hξi for 1 + mj ≤ 1 + mp ≤ mk+1. Now by the decomposition ! X X X X a − aj = a − a˜j + a˜j + (aj − a˜j) j≤k j≤p−1 k+1≤j≤p−1 j≤k we obtain the full estimate ! !

α β X mk+1−|α| X −j 1+mj −|α| ∂ξ ∂x a − aj ≤ hξi + 2 hξi j≤k k+1≤j≤p−1

X α β mk+1−|α| + ∂ξ ∂x (aj − a˜j) ≤ Cα,β hξi , j≤k

8 −∞ P because aj −a˜j have compact support and hence are in S . Hence aj is the asymptotic expansion of a. Furthermore, we have

supp a ⊂ ∪ suppa ˜j ⊂ ∪ supp aj, by how we constructed a. Now we are in a position to define classical pseudodifferential symbols, symbols which retains the layering feature like polynomial symbols induced by classical linear differential operators. Definition 2.7 (Classical symbol). We say a pseudodifferential symbol a(x, ξ) ∈ Sm is P∞ classical if a ∼ j=0 aj for some aj which are respectively homogeneous of degree m−j for m−j |ξ| ≥ 1, that is aj(x, ξ) = λ aj(x, ξ) for |ξ| ≥ 1 and λ ≥ 1. Denote the class of classical pseudodifferential operators by CSm ⊂ Sm. P Definition 2.8 (Principal symbol). For a classical symbol a ∼ j aj we call a0 the principal symbol of a. P α P α Example. Symbol |α|≤m aαξ are classical symbol of degree m, and |α|=m aαξ is the principal symbol.

2.3 Operators, Kernels, Adjoints, Quantisation In the last few sections we have defined and talked about properties of pseudodifferential symbols. Now in this section we will define pseudodifferential operators. Definition 2.9 (Pseudodifferential operators on S). Given a ∈ Sm(R2N ) and u ∈ S(RN ), the space of Schwartz class functions, define Z Op(a)u(x) = ei(x−y).ξa(x, ξ)u(y) dy d¯ξ, (2) R2N in which d¯ξ = (2π)−N dξ. Here Op(a) acting on S is called the pseudodifferential operator of order m, associated with symbol a. Denote the class of pseudodifferential operators of m −∞ m order m by Ψ , and let Ψ = ∩mΨ . Definition 2.10 (Classical pseudodifferential operators). We say Op(a) is classical of order m, if a ∈ CSm. Remark. (i) Op(a) is also written as a(x, D) by substituting the frequencies ξ by Fourier D in the symbol a(x, ξ). Also for concerns about simplicity of expression, in the context we conventionally refer A to the pseudodifferential operator associated with symbol a. We denote the symbol of operator A by σA. (ii) We also can write the formula (2) as Z Op(a)u(x) = F −1 [a(x, ξ)F [u]] = eix.ξa(x, ξ)û(ξ) d¯ξ, R2N which makes perfect sense for u ∈ S. This illustrates the nature of pseudodifferential operator, as a multiplication operator on the Fourier side.

9 P α Example. (i) Classical linear differential operators |α|≤m aα(x)D with all derivatives of aα(x) bounded, are pseudodifferential operators of order m, associated with symbol P α |α|≤m aα(x)ξ . (ii) For N ≥ 2, Laplace operator 4 = PN D2 is pseudodifferential operator of order k=1 xk 2 1 N N 2, with symbol σ4 = |ξ| . Consider the operator |D| induced by symbol |ξ| ∈ S (R ×R ), we see |D| ∈ Ψ1 and moreover

2 −1 −1 −1 |D| = F ◦ m|ξ| ◦ F ◦ F ◦ m|ξ| ◦ F = F ◦ m|ξ|2 ◦ F = 4,

in which ma denotes multiplication operator of factor a. Hence we see |D| is a square root of 4, and this gives us a hint about how to deﬁne fractional power of operators. (iii) Consider 2D Dirac operator acting on u : R2 → R2 with u ∈ S: 0 i∂x1 + ∂x2 0 ξ1 − iξ2 W = , σW = i∂x1 − ∂x2 0 ξ1 + iξ2 0

1 2 2 1 Easy to see σW ∈ S (R ×R ) and hence W ∈ Ψ . Because the symbol is location-invariant, that is the multiplication operator does only depend on ξ, we have

2 2 2 ξ1 + ξ2 0 σW 2 = σW = 2 2 = σ4 0 ξ1 + ξ2 We see the classical result that W 2 = 4, where 4 is understood to be a two-component operator. (iv) For N ≥ 2, deﬁne hDim ∈ Ψm by its symbol hξim, for any m. This is an isometry between Hm and L2. We show this operator is well-deﬁned on S.

Proposition 2.11 (Pseudodiﬀerential Operators on S). For a ∈ Sm(R2N ) we have Op(a): S → S is well-deﬁned, and the mapping (a, u) 7→ Op(a)u is continuous. Moreover Op : a 7→ Op(a) is injective. Proof. For u ∈ S, we have the estimate n o Z |Op(a)u(x)| ≤ sup a(x, ξ) hξi−m sup hξim+N+1 |uˆ| hξi−N−1 d¯ξ RN   Sm X S ≤ kak0,0 C  Cα kuˆkα,0 , |α|≤m+N+1

S where kukα,β are the semi-norms on S:

S α β kukα,β = sup x ∂x u(x) . RN

10 Hence Op(a)u is continuous and bounded with respect to a, and with respect tou ˆ hence with respect to u, by the well-known fact that Fourier transform is continuous between two copies of S. On the injectivity of Op, because Op is clearly linear, we only need to show that if Op(a) = 0 then a = 0. Let Z eix.ξa(x, ξ)ˆu(ξ) d¯ξ = 0 RN for all x ∈ RN and all u ∈ S. With x ﬁxed, consider the term b(ξ) = a(x, ξ) hξi−m hξi−N/2−1/2 . Indeed b ∈ L2, as Z Z −m2 −N−1 −m 2 −N−1 Sm kbkL2 = a hξi hξi dξ ≤ sup a hξi hξi dξ = C kak0,0 < ∞. RN RN Set v(ξ) = e−ix.ξ hξim+N/2+1/2 uˆ(ξ), and note that between u and v this is a bijection between two copies of S. We see that (b(ξ), v(ξ)) = 0 for all v ∈ S. Hence b = 0 in S and by the fact S is dense in L2, we see b = 0 in L2 and hence a(x, ξ) = 0 almost everywhere. As a is continuous, we see a(x, ξ) = 0. Remark. Here we use the trivial fact that Z hξik dξ < +∞ RN for any k < −N. Easy to verify in spherical coordinates.

Proposition 2.12 (Kernel and symbol). If a ∈ Sm ⊂ S0, then −N KOp(a)(x, y) = (2π) [Fξ→ya(x, ξ)] (x, y − x). Furthermore, this mapping as a bijection S0 → S0 has an inverse a(x, ξ) = Fy→ξ KOp(a)(x, x − y) . Proof. By definition of pseudodifferential operators we have Z Op(a)u(x) = ei(x−y).ξa(x, ξ)u(y) dy d¯ξ. R2N Hence we have straightforward Z i(x−y).ξ KOp(a)(x, y) = e a(x, ξ) d¯ξ RN −N = (2π) [Fξ→ya(x, ξ)] (x, y − x). It is easy to show that this formula actually defines a bijection between two copies of S0. By inverting it we have the second formula immediately.

11 Now consider the adjoint A∗ of pseudodiﬀerential operator A on S:

(Au, v) = (u, A∗v) for any u, v ∈ S, where Z (u, v) = u(x)v(x) dx RN is the Hermitian inner product. The uniqueness of A∗ follows from linearity of Hermitian inner product. Hence A∗ is well-deﬁned, if A∗ exists. Till this point it is yet unclear about the existence of A∗; and to answer this question, we quote a result from Alinhac and G´erard [2, Section I.8.2] instead of replicating the long-winded proof.

Proposition 2.13. Given a ∈ Sm, we have that a∗, the symbol of Op(a)∗, is also a symbol of order m, that is, a∗ ∈ Sm. Furthermore we have asymptotics for a∗:

X 1 a∗(x, ξ) ∼ ∂αDαa(x, ξ). α! ξ x α

Remark. (i) Hence if A is a pseudodiﬀerential operator of order m, its adjoint A∗ is also a pseudodiﬀerential operator of order m, and A admits a natural extension to an operator S0 → S0. (ii) If a(ξ) depends only on ξ, that is, it is location-invariant, we have the formula in the special case a∗(ξ) ∼ a(ξ). Furthermore, by a direct computation we see in fact a∗(ξ) = a(ξ).

Definition 2.14 (Pseudodifferential operators on S0). Given u ∈ S0(RN ) and a ∈ Sm(RN × RN ), define A = Op(a) by hAu, ϕi = (Au, ϕ) = (u, A∗ϕ) = u, A∗ϕ for any ϕ ∈ S, where h·, ·i denotes pairing between S and S0. Here A∗ : S → S is the adjoint of A.

We also wish to very brieﬂy introduce the notation of right quantisation.

Deﬁnition 2.15 (Right quantisation). Given symbol a(x, ξ) ∈ Sm(RN × RN ) we deﬁne its right quantisation, Opr(a) or a(D, y), to be the operator acting on S: Z a(D, y)u(x) = ei(x−y).ξa(y, ξ)u(y) dy d¯ξ. R2N It is also an operator in Ψm. It similarly admits an extension to S0.

12 Remark. (i) The standard operator a(x, D) is called left quantisation, in contrast. (ii) Each symbol a(x, ξ) has two influences to the function it acts on: perturbation ξ to the frequency, and perturbation x to the location. In the standard left quantisation, the perturbation in frequency comes firstly, and thereafter follows that in location. However in the right quantisation, the perturbation in location comes firstly. (iii) To examplify (ii), consider the quantisation of symbol xξ. The left quantisation gives xDx, and the right quantisation gives Dxx. The order of changes is very clear: between left and right quantisation, there is a switch. (iv) Reversely, given an operator of specific class, then one can either associate a left symbol and a right symbol, respectively which yield the operator as left and right quantisation. For example, given operator xDx, its left symbol is xξ and right symbol is xξ − 1. (v) The right symbol occurs naturally in the process of taking the adjoint of a left symbol, as of in the omitted proof of Proposition 2.13.

2.4 Symbolic Algebra of Operators We now discuss about the composition of two pseudodifferential operators. When the symbols are location-invariant, the composition is simply the multiplication of two multi- pliers which depends on ξ only, on the Fourier side, and its order is the sum of orders of two symbols. However the case becomes complicated when symbol of the first operator is x-dependent. We refer to a result from Alinhac and Gérard[2, Section I.8.2]:

m1 m2 Theorem 2.16 (Composition). Given a1 ∈ S , a2 ∈ S , we have Op(a1) Op(a2) = m1+m2 Op(b) for b ∈ S denoted by b = a1#a2. Moreover b has asymptotics X 1 b ∼ ∂αa Dαa . α! ξ 1 x 2 α

Example. (i) When a2 depends only on ξ, or a1 depends only on x, we see the simple relation a1#a2 ∼ a1(x).a2(ξ), furthermore they are exactly equal, via a direct computation. (ii) When a1(x, D) and a2(x, D) are linear diﬀerential operators, via Leibniz’s formula we immediately have X 1 a #a = ∂αa Dαa . 1 2 α! ξ 1 x 2 α The asymptotic formula is also exact in this case.

m1 m2 Corollary 2.17. Given a1 ∈ S and a2 ∈ S we have the commutator

[A1,A2] = A1A2 − A2A1

is a pseudodiﬀerential operator of order m1 + m2 − 1. Its symbol is

m1+m2−2 −i {a1, a2} mod S ,

13 where the Poisson brackets denote X ∂f ∂g ∂f ∂g {f, g} = − . ∂ξ ∂x ∂x ∂ξ j j j j j Proof. Write X 1 b = a #a ∼ ∂αa Dαa , 1 1 2 α! ξ 1 x 2 α X 1 b = a #a ∼ ∂αa Dαa , 2 2 1 α! ξ 2 x 1 α then we see the symbol of the commutator is X 1 b − b ∼ ∂αa Dαa − ∂αa Dαa 1 2 α! ξ 1 x 2 ξ 2 x 1 α6=0

X m1+m2−2 = ∂ξj a1Dxj a2 − ∂ξj a2Dxj a1 mod S j

m1+m2−2 = −i {a1, a2} mod S , clearly.

2.5 Operators on Sobolev Spaces Consider the fact that H0 = L2 ⊂ S0, we see a pseudodiﬀerential operator is naturally deﬁned on L2 by a restriction of itself on S. We want to show a fact that if a ∈ S0 then a(x, D) maps L2 into L2, before which we need to prove a lemma: Lemma 2.18 (Operators of negative order are continuous on L2). Given A ∈ Ψm for m < 0, then A is a linear bounded operator from L2 to L2.

2 ∗ Proof. Clear that A is linear. By kAukL2 = (A Au, u), and the fact that kAkL2→L2 = ∗ ∗ 1/2 ∗ 2m kA kL2→L2 as operator norm, we see kAkL2→L2 ≤ kA AkL2→L2 , where A A ∈ Ψ by Theorem 2.16. Iteratively we have

−p kAk ≤ kA∗Ak1/2 ≤ k(A∗A)∗ (A∗A)k1/4 ≤ · · · ≤ R∗R 2 , (3) L2→L2 L2→L2 L2→L2 p p L2→L2

∗ 2pm t in which RpRp ∈ Ψ , for any positive integer p. That is, if we can prove in some Ψ with t negative enough, all operators are bounded from L2 to L2, then there will be some ∗ 2pm p ∗ t 2 2 RpRp ∈ Ψ with 2 m ≤ t hence RpRp ∈ Ψ is bounded from L to L and so is A by (3). We now prove that operators in Ψ−N−1 are all bounded from L2 to L2. Given a(x, D) ∈ Ψ−N−1, write down its kernel and by the estimate Z Z i(x−y).ξ −N−1 |K(x, y)| = e a(x, ξ)d¯ξ ≤ hξi d¯ξ, RN RN 14 |K(x, y)| is bounded by a constant on R2N . Furthermore, it is easy to verify that (x − α |α| α −N−1−|α| −N−1 y) K(x, y) is the kernel of (i) ∂ξ a(x, D) ∈ S ⊂ S , which by the previous estimate is bounded by a constant for all α. Hence by summing over |α| ≤ N + 1 we see

hx − yiN+1 |K(x, y)| ≤ (1 + |x − y|)N+1 |K(x, y)|

is bounded by a constant, for (1 + |x − y|)N+1 can be expanded to a finite sum of terms bounded by a constant by the previous estimate. Hence Z Z N+1 −N−1 |K(x, y)| dx ≤ sup hx − yi K(x, y) hx − yi dx ≤ C RN RN for some constant C, by a change of variable in the last integral. Similarly we have R |K(x, y)| dy ≤ C. Hence we have the estimate RN Z Z Z 2 2 2 |Au(x)| ≤ K(x, y)u (y) dy |K(x, y)| dy ≤ C K(x, y)u (y) dy RN RN RN and the final estimate Z Z 2 2 2 2 kAukL2 ≤ C |K(x, y)| dx |u(y)| dy ≤ C kukL2 RN RN establishes the fact that any operator in Ψ−N−1 are linear bounded operators from L2 to L2. Apply the argument in the first paragraph and the result follows immediately. Theorem 2.19 (Operators of order 0 maps L2 into L2). Given a ∈ S0, then Op(a): L2 → L2 is bounded. Proof. Pick M = 2 sup |a(x, ξ)|2, and take

1/2 c(x, ξ) = M − |a(x, ξ)|2 ,

which is in S0 by Lemma 2.3. From Proposition 2.13 and Theorem 2.16 we have

∗ ∗ ∗ c #c = c c + r1 = c.c + r2 = M − a.a + r2 = M − a #a + r,

where r ∈ S−1 is an asymptotic remainder. Hence we have

C∗C = M − A∗A + R

for remainder R ∈ Ψ−1. Now we have

2 ∗ 2 2 kAukL2 = (A Au, u) = M kukL2 − kCukL2 + (Ru, u) 2 2 ≤ M kukL2 + (Ru, u) ≤ (M + kRkL2→L2 ) kukL2 where the operator norm of R is known to be ﬁnite by the previous lemma.

15 Now for s ≥ 0, set the L2 based Sobolev spaces Hs to be endowed with norm

Z 1/2 2s 2 kukHs = hξi |uˆ| d¯ξ . RN Easy to verify L2 = H0 with the same norm, by Plancherel’s theorem. We now investigate the action of pseudodiﬀerential operators on Hs.

Lemma 2.20 (Isometry between Hs and L2). For s > 0, the bijection hDis : Hs → L2 is isometric.

Proof. Consider for given u ∈ Hs, Z 2 s s s 2 kukHs = (hξi uˆ) (hξi uˆ) d¯ξ = khDi ukL2 . RN It is trivial to verify that hDi−s : L2 → Hs is the inverse to hDis.

Proposition 2.21 (Operators on Hs). Given A ∈ Ψm, then we have A : Hs → Hs−m, for any s > 0 and s − m > 0. Moreover A is bounded.

Proof. Consider u ∈ Hs, then by the fact hDi−s hDis = id we see

hDis−m Au = hDis−m A hDi−s hDis u.

Note that hDis u ∈ L2 by Lemma 2.20, and that hDis−m A hDi−s ∈ Ψ0 is bounded by Theorem 2.16. Apply Theorem 2.19 to see hDis−m Au ∈ L2. Apply Lemma 2.20 again to see Au ∈ Hs−m. The boundedness of A follows immediately from boundedness of hDis−m A hDi−s and isometric nature of hDis and hDis−m.

Remark. (i) Note that here we can take m < 0. Then A : Hs → Hs+(−m) for any s > 0. In plain english: operators of class Ψ−k of negative order −k < 0, improve differentiability by k, in the weak sense. In contrast, operators of class Ψk reduce differentiability by k. This coincides with our sense of classical differential operators of order k, on Hm, in the weak sense. (ii) Pseudodifferential operators have the same mapping properties (boundedness, sharp regularity), between Hölder/Hölder-Zygmund spaces. A detailed discussion about this topic is attached in Section 3. (iii) One can extend this result from Sobolev spaces to weighted Sobolev spaces, or more general modulation spaces. See Molahajloo and Pfander [16].

2.6 Ellipticity and Parametrices In this section we want to define an analogy of elliptic differential operators, that is, ellipticity conditions for pseudodifferential operators.

16 Proposition 2.22 (Ellipticity and parametrices). Let a ∈ Sm. Two conditions (i) ∃b ∈ S−m that a(x, D)b(x, D) − id ∈ Ψ−∞, (ii) ∃b ∈ S−m that b(x, D)a(x, D) − id ∈ Ψ−∞ are equivalent, and they imply that (iii) |a(x, ξ)| ≥ c |ξ|m for some c > 0 and |ξ| ≥ C. Conversely, if (iii) is satisfied, then there is b ∈ S−m satisfying (i) and (ii), where b is unique up to a S−∞ difference. We call such b the parametrix of a, and we say Op(a) is elliptic if a satisfies (iii). Proof. Given b0, b00 ∈ S−m satisfying AB0 − id ∈ Ψ−∞ and B00A − id ∈ Ψ−∞, we have

B00 − B0 = B00(id −AB0) + (B00A − id)B0 ∈ Ψ−∞ and hence b00−b0 ∈ S−∞, the uniqueness of parametrix up to class Ψ−∞ is proved. Moreover, (i) or (ii) implies a(x, ξ)b(x, ξ) − 1 ∈ S−1 and

1/2 ≤ |a(x, ξ)| |b(x, ξ)| ≤ C |a(x, ξ)| |ξ|−m

for large |ξ| and we see Op(a) is elliptic. Conversely, if a is an elliptic symbol we take

b = hξi−m F a hξi−m , where F ∈ C∞(C) and F (z) = 1/z for |z| ≥ c0, with some c0. Then Lemma 2.3 implies b ∈ S−m and ab = 1 + χ(ξ) with χ(ξ) = 0 for |ξ| ≥ C0 with some C0. Theorem 2.16 tells that a(x, D)b(x, D) = id −r(x, D), r ∈ S−1. k −m−k 0 −m Set for k ≥ 0, bk(x, D) = b(x, D)r(x, D) ∈ Ψ and choose b ∈ S such that 0 P b ∼ k≥0 bk (Proposition 2.6 tells that we can do this). Then ! 0 0 X X j X j −k k −k −k AB = A B − Bj +AB R = (id −R) R +Ψ = id −R +Ψ = id +Ψ j

17 3 Nonlinear Dyadic Analysis

3.1 Littlewood-Paley Decomposition In this section we will establish the Littlewood-Paley decomposition via pseudodiﬀerential ∞ N operators. Set a cutoﬀ function ψ ∈ Cc (R ), 0 ≤ ψ ≤ 1, with ψ(ξ) = 1 for |ξ| ≤ 1/2, ψ(ξ) = 0 for |ξ| ≥ 1. Set ϕ(ξ) = ψ(ξ/2)−ψ(ξ), we then see ϕ is supported in 1/2 ≤ |ξ| ≤ 2, and for all ξ we have

X −p 1 = ψ(ξ) + ϕp(ξ), ϕp(ξ) = ϕ(2 ξ), ϕ−1(ξ) = ψ(ξ) p≥0 which is locally ﬁnite near ξ, and moreover is pointwise a sum of up to two non-zero terms, by the fact that

p−1 p+1 supp ψ ⊂ [−1, 1] , supp ϕp ⊂ ξ : 2 ≤ |ξ| ≤ 2 . It is an immediate result that

ϕpϕq = 0, if |p − q| ≥ 2, (4)

ψϕp = 0, if p ≥ 0.

0 N Also note that 0 ≤ ϕp ≤ 1. Now given any u ∈ S (R ), set

u−1 = ψ(D)u

up = ϕp(D)u, p ≥ 0 We obtain the so-called Littlewood-Paley decomposition:

∞ X u = up. p=−1 Denote the partial sum by k−1 X Sku = up. p=−1 ∞ −∞ We remark here that ψ(ξ) and ϕp(ξ) are in Cc ⊂ S , hence the pseudodiﬀerential operators induced by them make perfect sense. Lemma 3.1 (Almost-orthogonality of terms). Given the Littlewood-Paley decomposition, we have 2 X 2 1/2 ≤ ψ (ξ) + ϕp(ξ) ≤ 1, (5) p≥0 and for any u ∈ L2 we have

X 2 2 X 2 kupkL2 ≤ kukL2 ≤ kupkL2 . p≥−1 p≥−1

18 Proof. For the ﬁrst inequality, consider that !2 X 2 X 2 X X 2 X 2 1 = ψ(ξ) + ϕp(ξ) = ψ + ϕp + 2ψ ϕp + 2 ϕpϕq ≥ ψ + ϕp, p≥0 p p p

X 2 2 X 2 kupkL2 ≤ kukL2 ≤ 2 kupkL2 p≥−1 p≥−1 immediately. Lemma 3.2 (Sensitivity of terms to diﬀerentiation). With given Littlewood-Paley de- P composition u = p≥−1 up, we have the following estimates independent of p and u, independent of parameter α only: (i)For any α, p ≥ −1 we have

α p|α| k∂ upkH0 . 2 kukH0 (6) α p|α| k∂ SpukH0 . 2 kukH0 (7) and

α p|α| k∂ upkL∞ . 2 kukL∞ (8) α p|α| k∂ SpukL∞ . 2 kukL∞ . (9) (ii) For any s ∈ R and p ≥ 0 we have constant C > 0 such that −1 ps ps C 2 kupkH0 ≤ kupkHs ≤ C2 kupkH0 . (10) (iii)For any k ∈ N and p ≥ 0, we have constant C > 0 such that −1 pk X α pk C 2 kupkL∞ ≤ k∂ upkL∞ ≤ C2 kupkL∞ . (11) |α|=k

19 p p+1 Proof. Consider the fact for p ≥ 0, ϕp is non-zero only when 2 ≤ |ξ| ≤ 2 , we see at −1 2ps 2s 2ps points on which ϕp is non-zero, for any s we have C 2 ≤ hξi ≤ C2 by a direct inequality and with increment in C if necessary. We see by integration Z −1 2ps 2 2s 2 2 2ps 2 C 2 kukH0 ≤ hξi ϕp(ξ) |uˆ(ξ)| d¯ξ = kupkHs ≤ C2 kukH0 . RN Immediately we have α 2p|α| k∂ upkH0 ≤ kupkH|α| ≤ C kukH0 and Z p !2 α 2|α| X 2 2p|α| k∂ SpukH0 ≤ kSpukH|α| = hξi ϕp |uˆ(ξ)| d¯ξ ≤ C kukH0 N R k≥−1 Pp p+1 for the support of k ϕp is with in ball of radius 2 . This proves (6), (7), (10). 1 N We utilise the result that given arbitrary v ∈ L R we have kv ∗ ukL∞ . kukL∞ for ∞ ∞ N any u ∈ L , by H¨olderinequality. Fix a w ∈ Cc R , set wµ(ξ) = w(µξ), one have −1 1 −1 F w ∈ S ⊂ L and hence we obtain the estimate kF wµ ∗ ukL∞ . kukL∞ . We set then we estimate α −|α| k∂ (wµ (D) u)kL∞ . µ kukL∞ .

Note that the constant of estimation do not depend on α. Take w(ξ) = ϕ0(ξ) we get w2−p (ξ) = ϕ(ξ) and hence α p|α| k∂ upkL∞ . 2 kukL∞ , −p verifying (8). Take w(ξ) = ψ(ξ) and we get w2−p = Sp(ξ) = ψ(2 ξ), then we see α p|α| k∂ SpukL∞ . 2 kukL∞ , α ∞ verifying (9). Finally we pick w = ξ χ(ξ) for some standard cutoff function χ(ξ) ∈ Cc equal to 1 near support of ϕ and see α α −p p|α| −p α −p p|α| −p ∂ uˆp = Cξ ϕ(2 ξ)û(ξ) = C2 2 ξ ϕ(2 ξ)û(ξ) = C2 w(2 ξ)ûp(ξ), and hence α p|α| −1 −p ˜ p|α| k∂ upkL∞ = C2 F w(2 ξ) ∗ up L∞ ≤ C2 kupkL∞ . Sum it with |α| = k we get the right hand inequality of (11). ∞ To obtain left hand of (11), choose χ ∈ Cc with χ ≡ 1 near support of ϕ. Write −2 P α α P α ∞ ϕ(ξ) = |α|=k ξ χα ϕ, where χα = (ξ χ(ξ)) |α|=k ξ ∈ Cc . Then see

X −p α −p −pk X −p α uˆp = 2 ξ χα(2 ξ)ˆup(ξ) = 2 χα(2 ξ)D[up(ξ) |α|=k |α|=k and take the inverse Fourier transform and multiply by 2pk to see

pk X pN −1 p α α 2 kupkL∞ ≤ 2 F χα(2 ξ) ∗ D up(ξ) L∞ . kD up(ξ)kL∞ |α|=k by H¨older inequality.

20 3.2 Characterisation of Sobolev and H¨olderSpaces

Proposition 3.3 (Characterisation of Sobolev Spaces). (i) If u ∈ Hs(RN ), then there is constant C for all p ≥ −1, we have

−ps kupkH0 ≤ CkukHs Cp2 P 2 for some Cp that satisﬁes Cp ≤ 1. −ps P 2 s (ii) Conversely if for p ≥ −1, kupkH0 ≤ CCp2 with Cp ≤ 1, then u ∈ H and kukHs . C. s s Proof. For (i) consider the fact (hDi u)p = hDi up by a direct computation. We then see

−ps −ps s −ps s −ps kupkH0 ≤ C2 kupkHs = C2 (hDi u)p = CCp2 khDi ukH0 = CCp2 kukHs H0 P 2 by Lemma 2.20, and for ﬁrst inequality by Lemma 3.2(ii). The estimate Cp ≤ 1 follows from applying Lemma 3.1 on hDis u. For (ii) see

2 s 2 X 2 X 2 2 −2ps 2s+1 2 kukHs = khDi ukH0 ≤ 2 kupkHs ≤ 2 C Cp 2 ≤ 2 C , p≥−1 p≥−1 by the ﬁrst inequality by Lemma 3.2(ii).

k if k < α ≤ k + 1, deﬁne the remainder norm for u ∈ Cb :

0 X γ |u|Cα = |∂ u|Cα−k . |γ|=k and deﬁne the H¨oldernorm of u of index α:

0 kukCα(K) = kukL∞ + |u|Cα . Finiteness of H¨oldernorm of index α characterises H¨olderspace of index α, denoted by Cα. We also take C0 to be the space of bounded continuous functions, where L∞ norm can be used.

Remark. When α = p ∈ N, in general Cα 6= Cp, the former of which is Hölderspace, the latter of which is the classical space of p-th order continuously differentiable functions. Across the context of this section we will avoid the situation when α ∈ N, exactly because when at integer indices, Littlewood-Paley theory does not characterise Cα in the Hölder sense, however, characterises a larger class in Definition 3.8.

21 Proposition 3.5 (Characterisation of H¨olderSpaces of non-integer indices). (i) If u ∈ Cα(RN ), α∈ / N, then there is constant C for all p ≥ −1 we have −pα kupkL∞ ≤ CkukCα 2 . −pα α (ii) Conversely, if for p ≥ −1 we have kupkL∞ ≤ M2 , α∈ / N, then u ∈ C and kukCα . M. α α Proof. By Lemma 3.2 (iii) and the fact that (∂ u)p = ∂ up permit an immediate reduction to the case 0 < α < 1. For (i), since ku−1kL∞ . kukL∞ , it suﬃces to consider Z pN −1 p up(x) = 2 F ϕ (2 (x − y)) u(y) dy RN for p ≥ 0. From the fact that R (F −1ϕ)(z) dz = Cϕ(0) = 0 we also have RN Z pN −1 p up(x) = 2 F ϕ (2 (x − y)) (u(y) − u(x)) dy RN whence Z pN −1 p −pα |up(x)| ≤ kukCα 2 F ϕ (2 (x − y)) |x − y| dy . kukCα 2 RN and (i) is proved. (ii) Conversely for some p to be determined, set X u = Spu + Rpu, Rpu = uq. q≥p Then we have X X −qα −pα kRpukL∞ ≤ kuqkL∞ . M2 ≤ M2 . q≥p q≥p Moreover we have p−1 X |Spu(x) − Spu(y)| ≤ |x − y| k∇uqkL∞ . q=−1 By Lemma 3.2 (iii) we have q(1−α) k∇uqkL∞ ≤ M2

and k∇u−1kL∞ . M. Hence if 0 < α < 1 we have p(1−α) |Spu(x) − Spu(y)| . M |x − y| 2 , P for the series k∇uqkL∞ is then geometrically divergent. Regrouping the estimates for Rpu and Spu we ﬁnd |u(x) − u(y)| ≤ CM |x − y| 2p(1−α) + 2M2−pα. Take p to be the largest integer such that 2p ≤ |x − y|−1, and we obtain α |u(x) − u(y)| . M |x − y| , and (ii) is proved.

22 Proposition 3.6 (Sobolev injection into H¨older spaces). Given s > N/2 and s−N/2 ∈/ N, we have Hs ⊂ Cs−N/2 and the injection is continuous.

N where σN denote the volume of unit ball in R , by Proposition 3.3 (i), with Cp ≤ 1 used. ˜ −p(s−N/2) It is also written as kupkC0 ≤ CkupkHs 2 and we apply Proposition 3.5 (ii).

Proposition 3.7 (Convexity ienqualities). (i) If s = λs0 + (1−λ)s1 for 0 ≤ λ ≤ 1, s0 ≤ s1 ∞ are real numbers, for u ∈ Cc we have the inequality

λ 1−λ kukHs ≤ CkukHs0 kukHs1 . (12)

α1 (ii) If α = λα0 + (1 − λ)α1 for 0 ≤ λ ≤ 1, α0 ≤ α1 are positive non-integers, for u ∈ C we have the inequality λ 1−λ kukCα ≤ CkukCα0 kukCα1 . (13) Proof. For (i), observe Z 2 λs0 2λ (1−λ)s1 2(1−λ) −N λ 1−λ kukHs = hξi |uˆ| hξi |uˆ| d¯ξ ≤ (2π) kukHs0 kukHs1 ,

by H¨older inequality. For (ii) we have

λ 1−λ λ −pα0 λ 1−λ −pα1 1−λ λ 1−λ −pα kupkL∞ = kupkL∞ kupkL∞ ≤ CkukCα0 2 kukCα1 2 = CkukCα0 kukCα1 2 ,

by Proposition 3.5 (i) and then apply Proposition 3.5 (ii).

3.3 Operators on Hölder-Zygmund Spaces In this section we want to establish results to aid our proofs in later sections: Theorem 3.13 will be used in Günther’s approach to solve the isometric embedding problem in Section 5.3; Proposition 3.14 will be used to prove our Hölderscale Nash-Moser theorem in Section 4.

Definition 3.8 (Hölder-Zygmund spaces). Given non-negative α ∈ R, the Hölder-Zygmund α ∞ space of index α, denoted by C∗ , is set to be the class of u ∈ L finite under the norm

pα kukCα = sup 2 kupkL∞ . ∗ p≥−1

23 Remark. (i) Using Littlewood-Paley decomposition, one should also be able to give a dyadic characterisation of Besov spaces. α α (ii) When α > 0 is non-integer, then C∗ = C , the Hölderspace, by Proposition 3.5. By the trivial fact induced by definition of Hölder-Zygmund spaces:

−pα kupk ∞ ≤ kuk α 2 . (14) L C∗ We have hence the immediate modiﬁcations of previous propositions:

α s s−N/2 Proposition 3.9 (Properties of C∗ ). (i) For s > N/2, H ⊂ C∗ . α1 (ii) Given α = λα0 + (1 − λ)α1, with 0 ≤ λ ≤ 1, then for all u ∈ C∗ we have

λ 1−λ kuk α ≤ Ckuk α1 kuk α2 . C∗ C∗ C∗ Proof. Exactly the same in previous propositions, except instead of facilitating characterisation of Proposition 3.5 we use the definition of Hölder-Zygmund norm directly. We now want to establish mapping properties of pseudodifferential operators between Hölder-Zygmund spaces. We firstly need three lemmata then will be able to state our main result.

Lemma 3.10 (Estimates of kernels). Let a ∈ Sm(RN ×RN ) for m ∈ R, and take a(x, D) = P p≥−1 ap(x, D) to be the Littlewood-Paley decomposition of a. Let kp(x, x−y) denote the Schwartz kernel of ap(D), as in Proposition 2.12, that is

−1 kp(x, y) = Fξ→y (ap(x, ξ)) .

Then for any multi-indices α, β, and any M ∈ N we have

β α −M p(N+m−M+|α|) ∂x ∂ξ kp(x, y) ≤ Cα,β,M |y| 2 . (15)

Proof. Firstly from the deﬁnition of kp we have for any α, β, γ, Z γ β α iy.ξ γ α β y ∂x Dy kp(x, y) = e Dξ ξ ∂x ap(x, ξ) d¯ξ RN by a symbolic rewriting. Observe that the integrand is supported within ball {|ξ| ≤ 2p+1} Np+N which has volume 2 σN . Furthermore, since the support is even contained in the p−1 p+1 p p annulus {2 ≤ |ξ| ≤ 2 } (when p 6= −1) on which C12 ≤ hξi ≤ C22 , we have

γ α β p(m+|α|−|γ|) Dξ ξ ∂x ap(x, ξ) ≤ Cα,β,γ2

α β by the property of ξ ∂x ap(x, ξ) as a symbol of order m + |α|, via applying Proposition α β 2.2(iii) to product of symbols ξ and ∂x ap(x, ξ). Hence for |γ| = M we have

γ β α p(N+m+|α|−M) y Dx Dy ap(x, y) ≤ Cα,β,γ2

Let Cα,β,M = sup|γ|=M Cα,β,γ and we have the inequality (15).

24 m Lemma 3.11 (Boundedness of decomposition). (i) Given a ∈ S1,0, let ap(x, ξ) = a(x, ξ)ϕk(ξ) be the Littlewood-Paley decomposition of a. Then we have

pm kap(x, D)kLρ→Lρ ≤ C2 (16) for 1 ≤ ρ ≤ ∞, where C depends not on p. m N N (ii) Let b ∈ S (R × R ) and let bq(D, x) = ϕq(D)b(D, x), where b(D, x) means Z Z b(D, y)u(x) = ei(x−y)b(y, ξ)u(y) dyd¯ξ, RN RN that is, a right-quantised operator. Then for all l we have

qm kbq(D, x)kLρ→Lρ ≤ C2 (17) for 1 ≤ ρ ≤ ∞, where C depends not on p.

Proof. For (i), we represent ap in form of integral kernel kp: Z ap(x, D)u = kp(x, x − y)u(y) dy. RN We know from (15) that

|kp(x, y)| ≤ C1 , −N−1 p(m−1) |kp(x, y)| ≤ C2 |y| 2

by taking α = β = 0 and M to be 0 and N + 1 respectively. Then Z Z Z ˜ p(N+m) −N−1 p(m−1) |kp(x, y)| dy ≤ C 2 dy + |y| 2 dy RN |y|≤2−p |y|>2−p ˜ pm 0 p(m−1) pm ≤ C 2 σN + C 2 ≤ C2 .

And see Z Z pm kp(x, x − y)u(y) dy ≤ kukLρ sup |kp(x, y)| dy ≤ C2 kukLρ RN Lρ x∈RN RN by Young’s inequality. For (ii), we represent

Z N bq(D, x)u = kq(y, x − y)u(y)dy R

−1 where kq(x, y) = Fξ→y (bq(x, ξ)) is the same kernel for the left quantisation bq(x, D). By a very similar estimate the result (ii) holds.

25 m N N Lemma 3.12. Let a ∈ S (R × R ) for m ∈ R, and let ap(x, D) be the Littlewood-Paley decomposition of a(x, D). Then for all l we have

min{p,q}m −|p−q|l kϕq(D)ap(x, D)kLρ→Lρ ≤ Cl2 2 (18) for any p, q ≥ −1, for any 1 ≤ ρ ≤ ∞ where Cl is independent of p, q.

Proof. Firstly we decompose 1 such that

n !l 1 X ξj X α 1 = + ξj = cα(ξ)ξ , 1 + |ξ|2 1 + |ξ|2 j=1 |α|≤l

−2l+|α| −l for cα ∈ S ⊂ S by virtue of coeﬃcients of order 1 and 2 in the binomial expansion. Hence

X α X α ϕq(D)ap(x, D) = ϕq(D) cα(D)Dx ap(x, D) = cα(D)ϕq(D)Dx ap(x, D) |α|≤l |α|≤l by commutativity of cα(D) and ϕq(D) for their symbols are location-invariant. Further- α more by considering that Dx a(x, D) is an operator of order m + |α| by Theorem 2.16, and hence α α α Dx ap(x, D) = Dx a(x, D)ϕp(x) = (Dx a)p (x, D). α Invoke (16) for (cα)q and (Dx a)p to see

X α −ql p(l+m) kϕq(D)ap(x, D)kLρ→Lρ ≤ kcα(D)ϕq(D)kLρ→Lρ (Dx a)p (x, D) ≤ C2 2 Lρ→Lρ |α|≤l which is what we want when p ≤ q. When p > q, see that

a(x, D) = (a (x, D)∗)∗ = (a∗(x, D))∗ = b(D, x), b(y, ξ) = a∗(y, ξ), and hence decompose

X α ϕq(D)ap(x, D) = ϕq(D)b(D, x)ϕp(D) = bq(D, x)Dx cα(D)ϕp(D). |α|≤l

α Now invoke (17) for bq(D, x)Dx and (cα)p we shall get

X α −pl q(l+m) kϕq(D)ap(x, D)kLρ→Lρ ≤ kbq(D, x)Dx kLρ→Lρ kcα(D)ϕp(D)kLρ→Lρ ≤ C2 2 , |α|≤l as required for p > q.

26 Theorem 3.13 (Operators on H¨older-Zygmund spaces). Let a ∈ Sm(RN × RN ), m > 0, s+m N s N s > 0 such that s + m > 0. Then a(x, D): C∗ (R ) → C∗ (R ) is a bounded linear operator.

Proof. Firstly decompose ∞ X u = up, p=−1 0 where up = ϕp(D)u and u ∈ S . Since by (4) we have ϕp.ϕq = 0 for |p − q| ≥ 2, and

∞ ∞ ∞ X X X a(x, D)u = ap(x, D)u = ap(x, D)(up−1 + up + up+1) = ap(x, D)˜up p=−1 p=−1 p=−1

s+m foru ˜p = up−1 + up + up+1 and u−2 = 0. Suppose u ∈ C∗ then

(s+m)p (s+m)p 2 ku˜ k ∞ ≤ 2 ku k ∞ + ku k ∞ + ku k ∞ ≤ Ckuk s+m p L p−1 L p L p+1 L C∗ by (14) where C = 2−(s+m) + 1 + 2s+m. Apply (18) to get

sq sq+min{p,q}m−|p−q|l−p(s+m) p(s+m) 2 kϕq(D)ap(x, D)˜upkL∞ ≤ Cl2 2 ku˜pkL∞ , here after simplication, ( 2−|p−q|(l−s), p ≤ q 2sq+min{p,q}m−|p−q|l−p(s+m)2p(s+m) = 2−|p−q|(m+s+l), p > q.

Now we choose l large such that l ≥ s + 1 and m + s + l ≥ 1, we obtain

sq X −|p−q| p(s+m) ka(x, D)uk s = sup 2 kϕ (D)a(x, D)uk ∞ ≤ C sup 2 2 ku˜ k ∞ ≤ Ckuk s+m . C q L p L C∗ ∗ q≥−1 q≥−1 p≥−1 We hereby see a(x, D) is bounded.

Remark. (i) When s and s + m are restricted to non-integers, then we see the mapping properties of pseudodiﬀerential operators between H¨olderspaces. s (ii) This theorem has an straightforward adaption to Besov spaces Bq,r, with reference to Abels [1].

Proposition 3.14 (Regularisation operators). There exists a family of operators, for θ ≥ 1, α β Sθ : ∪α≥0C∗ → ∩β≥0C∗ with the following properties:

(i) kSθuk α ≤ Ckuk β , for α ≤ β, C∗ C∗

α−β (ii) kSθuk α ≤ θ Ckuk β , for α ≥ β, C∗ C∗

α−β (iii) kSθu − uk α ≤ Cθ kuk β , for α ≤ β, C∗ C∗

27 d α−β−1 (iv) Sθu α ≤ Cθ kuk β , for all α, β. dθ C∗ C∗

∞ Proof. Fix a cutoﬀ χ ∈ Cc (R), with χ = 1 in |ξ| ≤ 1 and 0 on |ξ| ≥ 2, and 0 ≤ χ ≤ 1. Set regularisation operators X p Sθu = χ(2 /θ)up. p≥−1 β p Given u ∈ C∗ , immediately we have (Sθu)p = 0 when 2 ≥ 2θ, and

−pβ −pα (Sθu)p ≤ kukCβ 2 ≤ CkukCβ 2 L∞ ∗ ∗ for α < β (C is used to compensate the case p = −1): in this case we have

kSθuk α ≤ Ckuk β , C∗ C∗ proving (i); when α > β, we have

p(α−β) −pα α−β −pα (Sθu)p ≤ 2 kukCβ 2 ≤ Cθ kukCβ 2 , L∞ ∗ ∗

p for 2 ≤ θ or (Sθu)p = 0, and hence

α−β kSθuk α ≤ Cθ kuk β , C∗ C∗

P p p implying (ii). Observe (Sθ −1)u = p (χ(2 /θ) − 1) up, and for 2 ≤ 2θ we have (Sθ −1)u = 0, and for 2p ≥ 2θ

p(α−β) −pα α−β −pα (Sθu − u)p ≤ 2 kukCβ 2 ≤ Cθ kukCβ 2 L∞ ∗ ∗ and hence α−β kSθu − uk α ≤ Cθ kuk β C∗ C∗ d −1 P p 0 proving (iii). Eventually we see Sθu = θ χ1(2 /θ)up, where χ1(ξ) = −ξχ (ξ). Note dθ p p d that the support of χ1 is within 1 ≤ |ξ| ≤ 2, hence θ ≤ 2 ≤ 2θ or Sθu = 0. Then dθ similarly

d −1 p(α−β) −pα α−β−1 −pα Sθu ≤ θ 2 kukCβ 2 ≤ Cθ kukCβ 2 dθ ∗ ∗ p L∞ and d α−β−1 Sθu ≤ Cθ kuk β , C∗ dθ α C∗ as in (iv).

Remark. We will use this family of regularisation operators with restriction α, β∈ / N. α β α β β β ∞ Note in our case, C∗ , C∗ are C , C respectively. Also note that ∩β≥0C∗ = ∩β≥0C = C .

28 3.4 Dyadic Analysis of Products

Lemma 3.15. Let (aq)q≥−1 be a sequence of functions such that q suppa ˆq ⊂ {ξ, |ξ| ≤ C02 } . −qα P α Suppose that kaqk ∞ ≤ M2 for some α > 0, then u = aq ∈ C with kuk α ≤ L q≥−1 ∗ C∗ CM. Proof. Observe that for some N we have X up = (aq)p q≥p−N and hence

X X X −qα −pα kupkL∞ ≤ (aq)p ≤ C kaqkL∞ ≤ C M2 ≤ CM2 , L∞ q≥p−N q≥p−N q≥p−N by (8). Clearly we have what is required. α Theorem 3.16 (Estimates for products). Given u, v ∈ C∗ , we have kuvk α ≤ C kuk 0 kvk α + kuk α kvk 0 . (19) C∗ C∗ C∗ C∗ C∗ Proof. Decompose u and v in Littlewood-Paley formulation: X X u = up, v = vp. p p Then we have X X X uv = upvq = (Squ) vq + up (Sp+1v) = Σ1 + Σ2. p,q q p p Note that supports of Fourier transforms of Σ1 and Σ2 are both in the ball {|ξ| ≤ C02 } for some C0. This enables us to apply the lemma, by the help of the following estimates: −qα k(Squ) vqk ∞ ≤ kSquk ∞ kvqk ∞ ≤ Mkuk ∞ kvk α 2 , L L L L C∗ −pα kup (Sp+1v)k ∞ ≤ kupk ∞ kSp+1vk ∞ ≤ Mkuk α 2 kvk ∞ , L L L C∗ L by (8), and hence we have by previous lemma

kΣ1k α Mkuk ∞ kvk α , C∗ . L C∗

kΣ2k α Mkuk α kvk ∞ C∗ . C∗ L and kuvk α ≤ kΣ1k α + kΣ2k α ≤ C kuk 0 kvk α + kuk α kvk 0 C∗ C∗ C∗ C∗ C∗ C∗ C∗ as required. Remark. There is an immediate adaption of this theorem to Cα for α∈ / N, by restricting α C∗ to α∈ / N. We will apply this dyadic result repeatedly in Section 4 and Section 5.

29 4 H¨olderScale Nash-Moser Theorem

4.1 Preliminaries 4.1.1 Fr´echet Derivative In the setting for the Nash-Moser theorem we set M to be a compact C∞ manifold, and Φ a mapping from an open set U ⊂ C∞(M, Rp) to C∞(M, Rq). We would like to talk about diﬀerentiability of Φ as a mapping. Given two topological vector spaces E1 and E2 on R, U 1 open in R, we say a continuous mapping Φ : U → E2 is of class C if there is a continuous mapping, linear in second variable:

0 0 Φ : U × E1 → E2;(u, v) 7→ Φ (u)v, such that −1 0 ∀u ∈ U, ∀v ∈ E2, lim t (Φ(x + tv) − Φ(x)) = Φ (u)v. t→0 Hence we deﬁne Ck mappings recursively, via that Φ is Ck is Φ0(u) is Ck−1 for all u ∈ U. Write ∂k Φ(k)(u)(v , . . . , v ) = Φ(u + t v + ··· + t v )| . 1 k 1 1 k k t1=···=tk=0 ∂t1 ··· ∂tk

4.1.2 Tame Mappings We also need to introduce the concept of a tame mapping. Firstly, note that a neighbour- ∞ µ hood of u0 in C (M) is a neighbourhood of u0 in some C (M), we call the neighbourhood a µ-neighbourhood of u0. Furthermore we need to ﬁx a decreasing family Es with norm ∞ s |·|s, such that ∩sEs = C , for the very moment Es = C . Deﬁnition 4.1 (Tame mapping). Let u → Φ(u) be a continuous mapping from an open ∞ p ∞ q set U in C (M, R ) to C (M, R ). Then Φ is said to be tame if, for any u0 ∈ U, there is a neighbourhood V of u0, numbers b ≥ 0 and r ≥ 0, and constants Cs such that for all u ∈ V , for all s ≥ b we have |Φ(u)|s ≤ Cs 1 + |u|s+r , where the constant Cs depend on s, u0 and V , numbers b, r may only depend on u0 and V .

4.1.3 Preliminary Lemma

∞ Lemma 4.2. Given a family of C functions (pθ) with parameter θ > θ0, and given that

bj −1 kpθkCaj ≤ Mθ , j = 1, 2 for some b1 < 0 < b2, a1 < a2, then if we interpolate in between a1 and a2 by

a = νa1 + (1 − ν) a2

30 with respect to parameter ν deﬁned by νb1 + (1 − ν)b2 = 0, we have Z ∞ a p = pθ dθ ∈ C , kpkCa ≤ CaM θ0 for a1, a2, a∈ / N.

Proof. It suﬃces to prove the theorem for a1, a2 between 0 and 1 (indeed, we can apply convexity inequality (13) to reduce the general case ﬁrstly to that between two adjacent integers k and k + 1, then by considering all k-th order derivatives to reduce the problem back to case 0 and 1). Let Z θ Z ∞ vθ = pτ dτ, wθ = pτ dτ. θ0 θ Integrate the given estimates we have

b2 b1 kvθkCa2 ≤ Mθ /b2, kwθkCa1 ≤ −Mθ /b1. If |x − y| = δ it follows that

b1 a1 b2 a2 |p(x) − p(y)| ≤ M −θ δ /b1 + θ δ /b2 .

For suﬃciently small δ we can choose θ > θ0 so that two terms in the parentheses are equal and it then follows that a −λ λ−1 |u(x) − u(y)| ≤ 2Mδ |b1| |b2| −λ λ−1 and we see kukCa ≤ 2M |b1| |b2| as required. Remark. This important lemma fails when a ∈ N. This is exactly why we have to exclude H¨olderspaces of integer indices.

4.2 Statement of Theorem 4.2.1 Assumptions

∞ p We make two assumptions on operator Φ, deﬁned on a µ-neighbourhood of u0 in C (M, R ): 2 (A1) Φ is of class C in the previous settings for operators, in a µ-neighbourhood of u0 and satisﬁes the tame estimate

00 kΦ (u)(v1, v2)kCα ≤ C {kv1kCa kv2kCa (1 + kukCα+b )

+kv1kCa kv2kCα+c + kv1kCα+c kv2kCa } , (20) for some a, b, c ≥ 0 and all α ≥ 0.

0 ∞ p (A2) For u in a µ -neighbourhood of u0 in C (M, R ), there is a linear mapping ψ(u): C∞(M, Rq) 7→ C∞(M, Rp) such that Φ0(u)ψ(u) = id, satisfying the tame estimate

kψ(u)gkCα ≤ C {kgkCα+λ + kgkCλ (1 + kukCα+d )} , (21) for some numbers λ, d ≥ 0 and all α ≥ 0.

31 4.2.2 Statement We now can state our main theorem of the essay, the Nash-Moser theorem, based on H¨older spaces.

Theorem 4.3 (H¨olderscale Nash-Moser theorem). Suppose Φ satisﬁes (A1) and (A2), and α is such that α ≥ µ, α ≥ µ0, α ≥ d, α > λ + a + c, α > a + 1/2 sup(λ + b, 2a), α∈ / N. Then the followings hold:

(i) There is a (α + λ)-neighbourhood W of the orignin such that for f ∈ W , the equation

Φ(u) = Φ(u0) + f (22)

has a solution u = u(f) ∈ Cα. Moreover we have

ku(f) − u0kCα ≤ CkfkCα+λ .

(ii) If there is α0 > α, a non-integer, such that f ∈ W ∩ Cα0+λ, then the solution u(f) constructed is in Cα0 . In particular if f ∈ W ∩ C∞ then u(f) ∈ C∞.

∞ p Here (22) means there is a sequence uj ∈ C (M, R ) such that for all ε > 0 we have α−ε α+λ−ε uj → u in C and Φ(uj) → Φ(u0) + f in C . Remark. What makes this theorem novel? (i) It is applied to a mapping between two Fr´echet spaces with a local right inverse, (ii) where the mapping and its right inverse are tame, that is, they are allowed to have ﬁxed losses in derivatives. Those facts make a standard implicit function impracticable, and thereby lie the novelty of Nash-Moser theorem.

4.3 Iteration Scheme 4.3.1 Intuition From Original Newton Scheme We aim to solve the perturbation problem

Φ(u) = Φ(u0) + f

The normal Newton scheme implies in iteration scheme

0 Φ(uk+1) = Φ(uk) + Φ (uk)∆uk + εk (23)

a delicate control over ∆uk = uk+1−uk is needed, so that the accumulated error is controlled by latest εk. That it, by controlling ∆un we let the error at each step n

Φ(un+1) = Φ(u0) + f + εn. (24)

32 0 By denoting each γk = Φ (uk)∆uk, by a recursive substitution of Φ(uk) by Φ(uk−1) in formula (23) we see representation

n n X X Φ(un+1) = Φ(u0) + γk + εk. k=0 k=0 And hence the control is exactly

n n X X γk + εk = f + εn, (25) k=0 k=0 and we construct iteratively n−1 n−1 X X γn = f − γk − εk k=0 k=0 hence

Φ(un+1) = Φ(u0) + f + εn → Φ(u0) + f

γn = f + Φ(u0) − Φ(un) = εn−1 → 0 as εn → 0. Remark (Failure of Newton scheme). Unfortunately in our case, the simple Newton + scheme does not work, for even if we assume information kγ0kCβ for β ∈ [0, β ], however, in the process of inverting γk to ∆uk = ψ(uk), by estimate (A2) (21) we can only + obtain a control over k∆ukkCβ for β ∈ [0, β − λ]: in plain english, in each iteration, we lose λ order of information, and eventually lose all in ﬁnite time. Moreover there is another loss of b order derivatives in the process of obtain ek, generated by Φ in (A1) (20). It is exactly the loss of all information in ﬁnite time that prevents us from running the Newton scheme.

4.3.2 Symbolism and Analysis of Nash-Moser Scheme However a modiﬁed Newton scheme can be used. We split our argument into several steps for a more approachable explanation.

The ﬁrst step is to decompose the perturbation f. From Proposition 3.14, there exists a family of regularisation operators Sθ with following properties

kSθukCα ≤ CkukCβ , for α ≤ β, (26) α−β kSθukCα ≤ θ CkukCβ , for α ≥ β, (27) α−β kSθu − ukCα ≤ Cθ kukCβ , for α ≤ β, (28)

d α−β−1 Sθu ≤ Cθ kukCβ , for all α, β. (29) dθ Cα

33 Note that here we replace the H¨older-Zygmund spaces by H¨olderspaces, this makes perfect sense when α, β are non-integers. Pick the sequence

ε 1/ε θn = θ0 + n , (30) where we choose parameter θ0 smartly later, we have the decomposition X ˙ f = Sθ0 f + ∆kfk, k≥0 in which

∆k = θk+1 − θk, (31) ˙ 1 fk = Sθk+1 − Sθk f. (32) ∆k

n Note that by picking θn = C2 we would obtain the usual Littlewood-Paley decomposition of f, but be alerted that here we certainly can decompose f in a diﬀerent way. A smart choice of ∆k, the step from θk to θk=1, is essential to the success of this iteration scheme, ˙ discussed later. Here one should also note we have an estimate for each block fk:

Z θk+1 d ˙ −1 s−α−1 fk = ∆k Sθf dθ ≤ Cθk kfkCα , Cs dθ θk Cs by property (29).

The second step is explication of three alterations to our original iteration formula and control of error in Newton scheme. (i) The ﬁrst alteration, to the iteration formula, is that, 0 0 in each iteration when we use Φ(uk) and Φ (uk) to search for ∆uk, we use Φ (vk) in place 0 of Φ (uk), where vk = Sθk uk is a regularisation of uk, that is, the iteration formula becomes

0 Φ(un+1) = Φ(un) + Φ (vn)∆un + εn.

This clever alteration moves the losses in uk to vk, all order derivatives of which are controlled by ﬁxed order derivatives of uk; that is, assume information of β order derivative

of uk and we can control the β + λ + b order derivative of vk+1 by the properties of Sθk , and the loss of derivatives in vk will tell us that β order derivative of uk is given by β + λ + b order derivative of vk, controlled by β order derivative of uk. The loss of derivatives are recovered! (ii) The second change, to the control of error, is that, in our old scheme, we try find iteration for un such that each Φ(un) comes closer to the same target, Φ(u0) + f, as in (24). In the new scheme, we set a different target for each un, that is, we start with easy targets, and after each round of iteration we increase the difficulty of next target. To be specific, we want to control

th Φ(un+1) = Φ(u0) + Sθn f + n remainder.

34 (iii) The last modiﬁcation, to the control of error, is that for the nth remainder, instead of a simple estimate with εn only, we ask the control to be

Φ(un+1) = Φ(u0) + Sθn f + εn + (1 − Sθn ) En, where n−1 X En = εk k=0 is the accumulated error up to step n − 1. That is, not like the situation of (25), in which Pn all error of previous steps are cancelled by k=0 γk, we keep a very small bit of the accumulated errors.

The third step is to write down the iteration scheme explicitly. Starting with known u0, . . . , un, we want to determine un+1 via solving

0 Φ(un+1) = Φ(un) + Φ (vn)∆un + εn,

0 where ∆un = un+1−un, via choosing γn = Φ (vn)∆un smartly and then by ∆un = ψ (vn) γn. We want to pick γn to be such that

Φ(un+1) = Φ(u0) + Sθn f + εn + (1 − Sθn ) En, which is for each n, n X γk + Sθn En = Sθn f. k=0 More explicitly, we want γn = Sθn − Sθn−1 (f − En−1) − Sθn εn−1,

for each n ≥ 1. By iterating we can ﬁnd uk corresponding to errors εk for all k, and we want to pick those εk in a smart way.

The fourth step is a rough analysis of errors. In this modiﬁed Newton scheme, we have two sources of errors, the ﬁrst of which is from the errors we set deliberately, that is, usual quadratic error in original Newton scheme; the second of which is the error occurring in 0 0 the process of replacing Φ (uk) by Φ (vk). We want our errors εk to be decomposed into 0 00 such two categories of errors, that, εk = εk + εk:

0 0 0 εk = {Φ (uk) − Φ (vk)} ∆uk, 00 0 εk = Φ(uk+1) − Φ(uk) − Φ (uk)∆uk,

0 00 where εk is the error by regularising uk by vk and εk is the quadratic error in original New- 0 ton scheme. We should take into account the nature of the modiﬁcation εk when picking εk.

35 The fifth step is a clarification of symbolism: we want to define a set of variables according to the style of decomposition (31) and (32). Set

−1 −1 −1 u˙ n = ∆n ∆un, gn = ∆n γn, en = ∆n εn, 0 −1 0 0 0 en = ∆n εn = {Φ (un) − Φ (vn)} u˙ n, (33) 00 −1 00 −1 0 en = ∆n εn = ∆n {Φ(un+1) − Φ(un) − Φ (un)∆un} . (34)

with

∆n = θn+1 − θn, ˙ 1 fn = Sθn+1 − Sθn f. ∆n and introduce the same decomposition of accumulated error

˙ 1 En = Sθn+1 − Sθn En. ∆n Till this point, the setting of the iteration scheme is done.

4.3.3 Explicit Iteration Scheme Here we rewrite our Nash-Moser scheme in a clear and simpliﬁed algorithmic way. For the initial setup we write

−1 g0 = ∆0 Sθ0 f, and for the iteration we write with k ≥ 0:

vk = Sθk uk

u˙ k = ψ(vk)gk

uk+1 = uk + ∆ku˙ k −1 0 ek = ∆k (Φ(uk+1) − Φ(uk)) − Φ (vk)u ˙ k −1 gk+1 = ∆k+1 Sθk+1 Sθk (f − Ek) − ∆kSθk+1 ek . This scheme is just a reader-friendly version of the Nash-Moser scheme.

4.4 Iterative Hypotheses We now want to make a family of hypotheses, to make sure that we have convergence of

the sequence {un} under the hypothesis that kfkCα+λ is suﬃciently small, where α is given in statement of Theorem 4.3 and λ is given in assumption (A2) (21). Firstly we are given some small δ > 0 and someα ˜ > α, both of which will be ﬁxed later. Set the hypothesis

36 (Hn) For 0 ≤ k ≤ n and s ∈ [0, α˜], we have

s−α−1 ku˙ kkCs ≤ δθk .

We now prove the iteration hypothesis (Hn) for all n, and we use an iterative argument, by assuming (Hn) holds in our case and try to derive (Hn+1). Note that it is trivial that (Hn) =⇒ (Hn−1).

4.4.1 Validity of Tame Estimates

0 Firstly, we claim given α ≥ µ, α ≥ µ , 0 < δ < δ0, δ0 sufficiently small and θ0 sufficiently large, (Hn) implies that un+1 and vn+1 are both in a sufficiently small ball around u0, where we can use assumptions (A1) and (A2). This is done in the following lemma.

Lemma 4.4. (Hn) implies the followings:

(i) kun+1 − u0kCα ≤ Cαδ,

(a−α)+ (ii) Sθn+1 (un+1 − u0) Ca ≤ Cαδθn+1 , for a in a ﬁnite interval, a−α (iii) 1 − Sθn+1 (un+1 − u0) Ca ≤ Cαδθn+1 , for 0 ≤ a ≤ α˜, a−α (iv) kun+1 − vn+1kCa ≤ Cαθn+1 , for 0 ≤ a ≤ α˜,

(a−α)+ (v) kvn+1kCa ≤ Cαθn+1 , for a in a ﬁnite interval,

where x+ = sup {x, 0}. Pn Proof. Set U = un+1 − u0 = k=0 ∆ju˙ j. Set a continuous family of functions pθ with ( u˙ k, θk ≤ θ < θk+1, k ≤ n pθ = 0, θn+1 ≤ θ.

Moreover we can establish an estimate on kpθkCs :

ε(α+1) s−α−1 kpθkCs ≤ 1 + 2 δθ ; (35)

s−α−1 s−α−1 indeed, we have kpθkCs ≤ δθk ≤ δθ when s − α − 1 ≥ 0, and

ε(α+1) s−α−1 ε(α+1) s−α−1 kpθkCs ≤ 2 δθk+1 ≤ 2 δθ

ε by the fact that θk < θk+1 < 2 θk as an immediate result from (30), and that s − α − 1 ≥ −α − 1. By (35) invoke the previous lemma with

a1 = α − ρ, a2 = α + (1 − ρ), b1 = −ρ, b2 = 1 − ρ. (36)

37 for some ρ ∈ (0, 1) such that a1, a2 ∈/ N. Set ν = 1 − ρ and see a = α, then we have n Z θn+1 X α U = ∆ju˙ j = pθ dθ ∈ C k=0 θ0

and kUkCα ≤ Cδ, hence (i) is clear. By property of the regularisation operators (28) we have for a ≤ α: a−α 1 − Sθn+1 U Ca ≤ Cδθn+1 . Consider the case when α < a ≤ α˜: n Z θn+1 X a−α−1 a−α−1 0 a−α kUkCa ≤ δ ∆kθk = δ pθ ≤ C δθn+1 . k=0 θ0 Furthermore by (28) we have 00 a−α 1 − Sθn+1 U Ca ≤ C kUkCa ≤ Cδθn+1 , and (iii) is complete. However by virtue of decomposition un+1 − vn+1 = u0 + U − Sθn+1 (u0 + U) 1 − Sθn+1 u0 + 1 − Sθn+1 U we have kun+1 − vn+1kCa ≤ 1 − Sθn+1 u0 Ca + 1 − Sθn+1 U Ca 0 00 a−α 000 a−α a−α ≤ (C ku0kCα + C ku0kCa¯ ) θn+1 + C δθn+1 = Cθn+1 , by virtue of (iii); indeed by (28) we have 0 a−α 1 − Sθn+1 u0 Ca ≤ C θn+1 ku0kCα , when a ≤ α, 00 a−a˜ 00 a−α 1 − Sθn+1 u0 Ca ≤ C θn+1ku0kCa˜ ≤ C θn+1 ku0kCa˜ , when α < a ≤ a,˜ a−a˜ a−α by virtue of θn+1 ≤ θn+1 . However note that we need information ona ¯ which is arbitrary but ﬁnite, and the informative interval for a is (0, a˜]. This completes (iv). By (21) and (27) (ii) follows immediately, and by considering

vn+1 = Sθn+1 (u0 + U) and estimate

0 00 (a−α)+ kvn+1kCa ≤ Sθn+1 u Ca + Sθn+1 u0 Ca ≤ (C δ + C ) θn+1 by (21) and (27).

Remark. Here in this lemma we see from (i) that kun+1 − u0kCµ , kun+1 − u0kCµ0 are both less than Cδ by virtue of µ, µ0 ≤ α, arbitrarily small if we pick δ small enough. Also we have kvn+1 − u0kCα ≤ Sθn+1 (un+1 − u0) Cα + 1 − Sθn+1 u0 Cα 0 −1 0 −1 ≤ Cδ + C θn+1ku0kCα+1 ≤ Cδ + C ku0kCα+1 θ0

is arbitrarily small if we pick small δ and large θ0. Hence we can make use of conditions (A1) and (A2) on un+1 and vn+1.

38 4.4.2 Estimates on ek

0 Lemma 4.5 (Estimate on ek). For 0 ≤ k ≤ n, s ∈ [0, α˜ − sup {b, c}], we have 0 L(s)−1 kekkCs ≤ Cδθk , (37) where L(s) = sup s + a + c − 2α, (s + b − α)+ + 2a − 2α and note that a, b and c are ﬁxed parameters in tame hypothesis (A1) (20). Proof. We have Z 1 0 0 0 00 ek = {Φ (uk) − Φ (vk)} u˙ k = Φ (vk + t(uk − vk)) (u ˙ k, uk − vk) dt. 0

By (A1) we have Z 1 0 00 kekkCs ≤ kΦ (vk + t(uk − vk)) (u ˙ k, uk − vk)kCs dt ≤ Cku˙ kkCa kuk − vkkCa 0 0 00 1 + sup kvk + t(uk − vk)kCs+b + C ku˙ kkCa kuk − vkkCs+c + C ku˙ kkCs+c kuk − vkkCa t ˜ L(s)−1 ≤ Cδθk by virtue of

s+a+c−2α−1 ku˙ kkCa kuk − vkkCs+c . δθk , s+a+c−2α−1 ku˙ kkCs+c kuk − vkkCa . δθk using (Hn) and Lemma 4.4 (iv) for (Hn−1), and by a−α−1 a−α ku˙ kkCa kuk − vkkCa 1 + sup kvk + t(uk − vk)kCs+b ≤ Cδθk θk t

2a−2α−1 (s+b−α)+ s+b−α (kvkkCs+b + kuk − vkkCs+b ) ≤ Cδθk C1θk + C2θk

(s+b−α)++2a−2α−1 ≤ C3δθk , for large θ0, obtained by invoking Lemma 4.4 (v) for (Hn−1). 00 Remark (Negligibility of ek). We see Z 1 00 −1 0 00 ek = ∆k {Φ(uk+1) − Φ(uk) − Φ (uk)∆uk} = ∆k (1 − t)Φ (uk + t∆ku˙ k)(u ˙ k, u˙ k) dt. 0 00 and we estimate ek similarly:

00 L(s)−1 kekkCs ≤ C∆kδθk 1−ε−1 0 is negligible for ∆k ≤ θk , when ε and δ small enough, compared to ek. Hence we can use (37) to estimate whole of ek.

39 4.4.3 Estimates on gn+1

Proposition 4.6 (Estimates on gn+1). For s ∈ [0, s0], with any s0 > 0 ﬁnite, we have

L(s)−1 s−α−λ−1 kgn+1kCs ≤ C δθn+1 + θn+1 kfkCα+λ . Proof. Consider the iterative formula

∆n n ˙ ˙ o gn+1 = fn − En − Sθn+1 en . ∆n+1 By (37) we have

˙ s−α−λ−1 fn ≤ Cθn kfkCα+λ (38) Cs and ˙ s−r−1 (En)n ≤ Cθn kEnkCr , Cs where r =α ˜ − sup {b, c}. Now pickα ˜ large enough such that L(r) > 0 and L0(s) = 1 on

(r, s0] (that is, when s ≥ α − b or s + c − a ≥ (s + b − α)+). We then have n−1 n−1 n−1 X X L(r)−1 L(r)−1 X L(r) kEnkCr ≤ ∆kkekkCr ≤ ∆kCδθk ≤ Cδθn ∆k = Cδθn (39) k=0 k=0 k=0

and hence ˙ s−r−1+L(r) L(s)−1 (En)n ≤ Cδθn = Cδθn Cs for L(r) ≤ L(s) + r − s when s ≤ r, and L(r) = L(s) + r − s when s > r for L0(s) = 1. We also have 0 L(s)−1 Sθn+1 en Cs ≤ C kenkCs ≤ Cδθn

by (27) and previous estimate on en. Summing up three estimates we have the required estimates for gn+1.

4.4.4 Estimates on u˙ n+1

Proposition 4.7 (Estimates onu ˙ n+1). For s ∈ [0, α˜], we have the estimate (Hn+1), that s−α−1 ku˙ n+1kCs ≤ δθk ,

if δ, kfkCα+λ , ε are suﬃciently small, andα, ˜ θ0 suﬃciently large.

Proof. With formula foru ˙ n+1, u˙ n+1 = ψ(vn+1)gn+1,

apply tame estimate (A2) (21) to obtain

ku˙ n+1kCs ≤ C {kgn+1kCs+λ + kgn+1kCλ (1 + kvn+1kCs+d )}

0 L(s+λ)−1 0 s−α−1 00 L(λ)−1 −α−1 (s+d−α)+ ≤ C δθn+1 + C θn+1 kfkCα+λ + C δθn+1 + kfkCα+λ θn+1 θn+1 . (40) It is immediate to see if the followings hold:

40 (i) L(s + λ) < s − α,

(ii) kfkCα+λ /δ is small,

(iii) L(λ) + (s + d − α)+ < s − α,

(iv) (s + d − α)+ − α ≤ s − α,

(v) θ0 is suﬃciently large, then (Hn+1) is satisﬁed. Note that θ0 is made large here to cancel out the constants with one order of polynomial decreases, to meet the form of (Hn+1). Condition α ≥ d validates (iv). Conditions α > λ + a + c and α > a + 1/2 sup {λ + b, 2a} imply L(λ) < −α, hence

implies (i), and together with (iv) imply (iii). All what is left is to pick kfkCα+λ small. Remark. Note here the parameters are chosen depending on a, b, c, d, α only, not depending on un. Till now every parameter is ﬁxed, except kfkCα+λ .

4.4.5 Validation of Case (H0)

Proposition 4.8 (Validation of case (H0)). We have (H0) holds, that is, for any s ∈ [0, α˜] we have s−α−1 ku˙ 0kCs ≤ δθ0 ,

when kfkCα+λ is small enough. −1 −1 Proof. By initiation g0 = ∆0 Sθ0 f, and hence haveu ˙ 0 = ∆0 ψ(Sθ0 u0)Sθ0 f and see

−1 ku˙ 0kCs ≤ C∆0 {kSθ0 fkCs+λ + kSθ0 fkCλ (1 + kSθ0 u0kCs+d )}

−1 n (s−α)+ s o ≤ C∆0 C1θ0 kfkCα+λ + C2kfkCα+λ (1 + C3θ0ku0kCd ) n o ˜ −1 (s−α)+ s ≤ C∆0 θ0 + 1 + θ0ku0kCd kfkCα+λ ,

by virtue of tame estimate (A2) (21) and property of regularisation operator (26). Note s−α−1 everything before kfkCα+λ are already ﬁxed. Easy to see ku˙ 0kCs ≤ δθ0 when −α−1 ˜−1 kfkCα+λ ≤ δθ0 ∆0C (2 + ku0kCd ) , as required.

Remark. (i) Here we have proved (Hn) hold for all n, if δ, kfkCα+λ , ε are suﬃciently small, andα, ˜ θ0 suﬃciently large.

(ii) Here we can take kfkCα+λ arbitrarily small because f is our small perturbation term in the statement of Nash-Moser theorem.

4.5 Existence and Regularity We claim the following results:

41 4.5.1 Existence of Solution in Cα

α Proposition 4.9 (Existence of C -limit). If (Hn) hold for all n, then we have un = Pn−1 α u0 + k=0 ∆ku˙ k converges pointwise to u, moreover, u ∈ C .

Proof. From assumption that (Hn) hold for all n, we deduce that

s−α−1 ku˙ kkCs ≤ δθk for all k. To apply Lemma 4.2 we ﬁrstly need to convertu ˙ k into a family of parameter continuum θ from the discrete nature of θk. We deﬁne

pθ =u ˙ k, if θk ≤ θ < θk+1, and (p ) is a family of C∞-functions. Clearly we still have (35): θ θ>θ0

ε(α+1) s−α−1 kpθkCs ≤ 1 + 2 δθ . Invoke Lemma 4.2 with (36) and we see Z ∞ α u = u0 + pθ dθ ∈ C , θ0 ∞ by u0 ∈ C . Note there u is written in the continuous integral form instead of discrete sum for ∞ X Z ∞ u = u0 + (θk+1 − θk)u ˙ k = u0 + pθ dθ, k=0 θ0 α by virtue of the nature of pθ. Hence the limit solution is in C .

α−γ Proposition 4.10 (Convergence in C ). By assuming (Hn) hold for all n, we have α−γ un → u in all C for γ > 0, where u is the limit solution in previous proposition. Proof. With n > m, consider that

n−1 n−1 X X ku − u k = ∆ u˙ ≤ ∆ ku˙ k n m Cα−γ k k k k Cα−γ k=m Cα−γ k=m Z θn Z θn ε(α+1) −γ−1 −1 −γ −γ 2C −γ = kpθkCα−γ dθ ≤ 1 + 2 δ θ dθ = Cγ θm − θn ≤ θm → 0 θm θm γ

α−γ as m → ∞. Hence {un} are Cauchy hence convergent in C , as of the nature of Banach spaces Cα−γ. Proposition 4.11 (Existence of Solution). For every γ > 0, we have

Φ(uk) → Φ(u0) + f,

α+λ−γ α+λ in C . Moreover, limit of Φ(uk) is bounded in C .

42 Proof. See

Φ(uk+1) − Φ(u0) − f = (Sθn − 1) f + (1 − Sθn ) En + ∆nen.

A direct computation shows L(α + λ) = −a. Consider kfkCα+λ is small and ﬁxed; as n → ∞, via (39) we have −a kEnkCα+λ ≤ Cδθn → 0 and via (37) we have

−a−1 ˜ −a k∆nenkCα+λ ≤ Cδθn ∆n ≤ Cδθn → 0.

α+λ Hence Φ(uk) are bounded in C . Furthermore, see

0 −γ 00 −γ 000 −γ kΦ(uk+1) − Φ(u0) − fkCα+λ−γ ≤ C θn kfkCα+λ +C θn kEnkCα+λ +C θn k∆nenkCα+λ → 0

as n → ∞.

Remark. Here we showed that if the iteration hypothesis (Hn) hold for all n, then the existence of the solution and the convergence in lower regularity spaces are proved. However we remark here the convergence is based on Lemma 4.2, which is a unique result which is valid to H¨olderspaces only. Later when we drop the restriction of Nash-Moser theorem on H¨olderspaces, we can not make use of this lemma.

4.5.2 Regularity We now want to prove the regularity part, (ii) of Theorem 4.3. We now verify the modiﬁed hypotheses for α0 < α˜ when we assume additional information on data f that f ∈ W ∩ Cα0+λ:

0 (Hn) For 0 ≤ k ≤ n and s ∈ [0, α˜], we have

s−α0−1 ku˙ kkCs ≤ δθk . (41)

To see this, we ﬁrstly consider that in (40) that terms in which kfkCα+λ does not occur has θn+1-exponents strictly less than s − α − 1 by the fact

L(s + λ) − 1 < s − α − 1,L(λ) − 1 + (s + d − α)+ ≤ L(λ) − 1 + s+ < s − α − 1.

Also, the kfkCβ only originates from (38). If we replace (38) by

0 ˙ s−α −λ−1 fn ≤ Cθn kfkCα0+λ , Cs we have then recovered an estimate

s−α−γ−1 s−α0−1 ku˙ n+1kCs ≤ C θn+1 + θn+1 kfkCα0+λ

43 for some small γ > 0. In fact, ignoring the trivial case n = 0, we have for all n that

s−ρ−1 ku˙ nkCs ≤ Cθn , where ρ = min {α + γ, α0}. Here we remark that γ depends only on a lower bound for α, and hence we can repeat the process above by k times to get for all n, within informative interval s ∈ [0, α˜] we have s−ρk−1 ku˙ nkCs ≤ Cθn , 0 0 where ρk = min {α + kγ, α }. When k is suﬃciently large we have ρk = α and we obtain 0 iterative hypotheses (Hn) (41) for all n.

0 0 Note that (Hn) are exactly iterative hypotheses for index α replaced by α , with constant possibly altered; however in the proof of Proposition 4.9 the scale of the constant is not 0 important (it is important to run the iterative hypotheses (Hn), and here we derive (Hn) from (Hn) directly without an iterative process). We apply Proposition 4.9 to see the regularity result u ∈ Cα0 . The desired result is achieved.

Remark. (i) Till this point, our main result Theorem 4.3 is fully proved. One should be alerted that this theorem evades all integer indices of Hölderspaces. (ii) The Nash-Moser theorem can also be achieved via Bony’s paradifferential calculus. See Bony [4] and Hörmander[14].

44 5 Isometric Embedding Theorem

5.1 Isometric Embedding Problem 5.1.1 Statement of Problem In this essay, we only pay attention to the embedding for compact manifolds. The problem reads:

Can every smooth compact manifold be smoothly embedded into RN for some N?

We ask, whether for any manifold M with smooth metric g of dimension n, we can ﬁnd an embedding u : M → RN for some N, such that ∂u ∂u gij = . , ∂xi ∂xj for any local coordinates near all x ∈ M.

Remark. (i) We will present two methods for tackling the isometric embedding problem for compact manifolds: in Section 5.2 we see the isometric embedding theorem as a natural consequence of our Nash-Moser theorem via the original proof of Nash, presented in the form of Alinhac and G´erard[2, Chapter III], and in Section 5.3 we will go through G¨unther’s approach [7]. (ii) The case for non-compact manifolds is also provable. See Nash [19].

5.1.2 Reduction to Embedding of Tn We want to reduce the embedding problem of a general smooth compact (M, g) of dimension n back to that of a torus. This is done by smoothly embedding M into TN for some N > n. We will explain why we can do this. Firstly we invoke Whitney embedding theorem, without proof:

Theorem 5.1 (Strong Whitney embedding theorem). Every smooth manifold M of dimension n can be smoothly embedded into R2n. We ﬁrstly invoke the Whitney embedding theorem and get a smooth embedding ι : M → ι(M) ⊂ R2n. ι is continuous, and M is compact, hence ι(M) is compact in R2n, where we invoke Heine-Borel theorem to get that ι(M) is bounded. Hence we can cut a 2d-cube in R2n containing ι(M), and treat this cube as a torus. Hereby M is smoothly embedded into T2n. Hence we reduced the case for general compact M back to that for Tn, for any n.

Remark. (i) We remark that in the process of embedding M into T2n, the Riemannian metric g is not necessarily invariant under the embedding. However, it does not matter because we can apply our theorem to new g after the embedding extended to the whole Tn

45 via Tietze extension theorem and the solution embedding composited with inclusion map on the right is the isometric embedding, which is associated with (M, g). (ii) The beneﬁt of the reduction to torus is that on torus we have a global chart.

5.1.3 Free Embedding Deﬁnition 5.2 (Free embedding). For a smooth manifold M of dimension n, an embedding N u0 : M → R is called a free embedding if for each x ∈ M we have that, the n(n + 3)/2 N vectors, ∂iu0(x) and ∂i∂ju0(x) in R , are linearly independent in local coordinates. Proposition 5.3 (Existence of free embedding of Tn). There exists a free embedding n 2n+n(2n+1) u0 : T → R . n Proof. First embed Tn = S1 × · · · × S1 into (R2) in the normal way, viau ˜:

u˜(x1 + k12π, . . . , xn + kn2π) = (cos(x1), sin(x1),..., cos(xn), sin(xn)) .

Write ϕ : R2n → R2n+n(2n+1):

ϕ (x1, . . . , xn) 7→ (x1, . . . , xn, x¯) , x¯ = (xixj)1≤i≤j≤2n .

By a direct computation one sees ∂iϕ and ∂i∂jϕ are linearly independent, and by the fact thatu ˜ is an embedding, it is easy to see

n 2n+n(2n+1) u = ϕ ◦ u˜ : T → R is a free embedding.

Remark. A free embedding enables one to establish an invertible linear system near u0, and then we can ﬁnd a simple right inverse to the objective operator in the perturbation problem, which allows us to run Nash-Moser scheme.

5.2 Nash-Moser Approach

Theorem 5.4 (Local embedding theorem). Let g0 be a metric induced by a free embedding n ρ u0 of T . For ρ > 2 and ρ∈ / N, and any metric g of class C near g0, there is a map u : Tn → RN of class Cρ for which we have

∂u ∂u t 0 0 gij = . = u .u ij . ∂xi ∂xj Proof. We will apply Nash-Moser theorem in this proof. Construct the objective operator

φ(u) = tu0.u0.

The problem is reduced to solving perturbation problem

φ(u) = φ(u0) + (g − g0) (42)

46 where g−g0 is known to be suﬃciently small, and symmetric for g and g0 being Riemannian metric. Observe that

φ0(u)(v) = tu0.v0 + tv0.u0 φ00(u)(v, w) = tw0.v0 + tv0.w0 and 00 t 0 0 kφ (u)(v, w)kCα ≤ C w .v Cα ≤ C (kwkC1 kvkCα+1 + kwkCα+1 kvkC1 ) (43) show that φ is tame, in the sense that φ satisﬁes assumption (A1) (20). Moreover we want to consider whether there is a right inverse to objective operator φ, near u0. We consider 0 the problem to solve φ (u)(v) = r, given some symmetric metric r, and u near u0. The problem reads φ0(u)(v) = tu0.v0 + tv0.u0 = r. By choosing v such that

tu0.v = 0, (44) tu00.v = −r/2, (45) then we have tv.u0 = 0 and tv.u00 = −r/2 by symmetry, and furthermore

tu0.v0 + tv0.u0 = tu0.v0 − tu00.v + tv.u00 − tv.u00 = r then the problem is solved. However, the choosing of v by (44) and (45) is in fact to solve

M(u0, u00).v = t (u0, u00) .v = t (0, −r/2) where we dropped the transpose of 0 and −r/2 for their symmetry. Recall that by definition t 0 00 of free embedding u0 we have det (u0, u0) > 0. However by continuity of determinant t 0 00 we know det (u , u ) > 0 for u sufficiently close to u0, which is exactly the assumption 0 00 we made on u. Hence M(u , u ) is invertible hence define for u sufficiently near u0, ψ(u)(r) = M(u0, u00)−1.t (0, −r/2) and we see φ(u)ψ(u)(r) = r, hence ψ is the right inverse of φ. Moreover we see

˜ 00 kψ(u)(r)kCα = M(u ).r Cα where M˜ is −1/2 times the matrix M −1 with ﬁrst d rows and columns deleted, note

˜ 00 ˜ that entries of M depend only on u with M ≤ CkukCs+2 . Hereby for u in some Cs 0 0 µ -neighbourhood of u0, with µ > 2, we have ˜ ˜ kψ(u)(r)kCα ≤ C1 M krkCα + M krkC0 ≤ C0 Cα

C2 (kukC2 krkCα + kukCα+2 krkC0 ) ≤ C (krkCα + krkC0 kukCα+2 ) (46)

47 shows ψ is tame with loss of two derivatives in u hence satisﬁes (A2) (21). Note in the last

inequality we used the fact that kukC2 is bounded by a constant, exactly because u is in 0 0 some µ -neighbourhood of u0, with µ > 2. By the previous arguments, we set the parameters a = 1, b = 0, c = 1, d = 2, λ = 0, µ = µ0 = 2 + 1/2 (ρ − 2)

for g ∈ Cρ with ρ > 2 and ρ∈ / N. Take α = ρ, it is trivial to verify α ≥ µ, α ≥ µ0, α > d, α > λ + a + c, α > a + 1/2 sup(λ + b, 2a).

Invoke Nash-Moser Theorem 4.3 with assumptions (A1) on φ and (A2) on ψ, we get when

kg − g0kCρ is suﬃciently small, we obtain

φ(u) = φ(u0) + g − g0 = g has a solution u ∈ Cρ. Remark. (i) This is the important theorem in which we invoke the Nash-Moser Theorem 4.3. Note that here we have not finished the proof for embedding of arbitrary metric tensor g; we have only proved metrics sufficiently near to our freely embedded g0 admit a solution to the embedding problem. The rest is done in next few pages. (ii) We stick to a freely embedded metric g0 exactly because near g0 we have a nice inverse for φ0. (iii) Note that we obtain (43) and (46) by the dyadic analysis result (19). (iv) Note the injectivity of the solution is not proved. However by the upcoming proposition and lemma we can see increasing the dimension N suffices the injectivity condition.

1 n Nj Lemma 5.5. (i) Given two C -embedding uj : T → R for j = 1, 2, deﬁne embedding n N1+N2 u : T → R , u(x) = (t1u1(x), t2u2(x)), with scalars t1, t2. Denote by g, g1, g2 the metrics induced by embedding u, u1, u2 respectively, then we have 2 2 g = t1g1 + t2g2. (ii) Assume every metric g can be represented by an u : Tn → RN , then every metric g can be represented by an injectiveu ˜ : Tn → RN+n. Proof. (i) Let id (·, ·) denote the normal inner product in Euclidean spaces. Then we see immediate properties id ((a, b), (c, d)) = id (a, c) id (b, d) . Now consider ∗ ∂ ∂ 0 ∂ 0 ∂ 0 ∂ 0 ∂ gij = (u id) , = id u , u = id t1u1 , t1u1 ∂xi ∂xj ∂xi ∂xj ∂xi ∂xj 0 ∂ 0 ∂ 2 ∗ ∂ ∂ 2 ∗ ∂ ∂ + id t2u2 , t2u2 = t1 (u1 id) , + t2 (u2 id) , ∂xi ∂xj ∂xi ∂xj ∂xi ∂xj 2 2 = t1 (g1)ij + t2 (g2)ij ,

48 as required. ∞ n n (ii) Firstly we see there is an injective C mapping u1 : T → R given by the ordinary t 0 0 2 2 embedding. Let g1 = u1.u1, and ﬁx t such that t g1 < g; then g − t g1 can be represented n N by some u2 : T → R by our assumption. Easy to checku ˜(x) = (tu1, u2) represents g by virtue of (i). We quote a proposition with an abbreviated proof borrowed from Tao [21]. Proposition 5.6 (Density of metrics induced by embedding). On a torus, the set of metrics g representable as ∂u ∂u gij = . . ∂xi ∂xj for some C∞-mappings u : Tn → RN , is dense in the set of all metrics. Proof. It suﬃces to prove that given any metric g on Tn, we have a smooth symmetric ∞ perturbation metric h on T such that for all ε there are uε which are C ,

2 ∂uε ∂uε g + ε h ij = . ∂xi ∂xj is representable. By the spectral theorem for finite-dimensional matrices, we see all positive definite 0 tensors gij can be decomposed as finite positive linear sum of symmetric rank one tensors N vivj where v ∈ R . If necessary add additional rank one tensors then one can make sure 0 that any nearby metric gij is also a positive linear combination of vivj. Write vi = ∂iψ for some linear ψ : RN → RN and we see m X k2 k k gij = η ∂iψ ∂jψ k=1 by compactness of T and smooth partition of unity, where ψk is ψ restricted to partition and ηk is the cutoff for partition, both mapping from RN to R. For any ε > 0 consider the k n 2 map uε : T → R by k k k k uε (x) = εη (x). cos ψ (x)/ε , sin ψ (x)/ε . Compute k k ∂uε ∂uε k2 k k 2 k k . = η ∂iψ ∂jψ + ε ηi ηj . ∂xi ∂xj 1 2 m Take uε = (uε, uε, . . . , uε ) and we see

∂uε ∂uε 2 . = gij + ε hij ∂xi ∂xj where m X k k h = ηi ηj , k=1 as required.

49 Remark. (i) This proposition is does not prove the isometric embedding theorem since u does not necessarily converge as ε converges to 0. (ii) It means we can approximate a given metric to an arbitrary accuracy, by good metrics (metrics induced by isometric embedding). Note that we do not have any free embedding structures in these approximations. However we can use these approximations to reduce the distance between given metric and a ﬁxed freely embedded metric, by using an approximation as a stepping stone.

Theorem 5.7 (Isometric embedding theorem). Given any Cρ metric g on Tn, ρ > 2, ρ∈ / N, there exists an Cρ-embedding u : Tn → RN such that ∂u ∂u gij = . . ∂xi ∂xj

2 Proof. We know there is a freely embedded metric g0. Fix t such that t g0 < g. Apply 2 n N2 Proposition 5.6 to approximate g − t g0 we can ﬁnd an smooth mapping u2 : T → R 2 2 such that metric g2 = g − t g0 − t f is represented by u2, with f small. Note here we can make f arbitrarily small by improving approximation u2 by density. Then we have the decomposition of our given metric g:

2 2 2 2 g = t (g0 + f) + g − t g0 − t f = t g1 + g2.

ρ where g1 = g0 + f ∈ C . Note that g1 is suﬃciently close to freely embedded metric g0 ρ n N1 by suﬃciently small f, hence by Theorem 5.4 we have a C representation u1 : T → R n N representing g1. Now set u : T → R for N = N1 + N2 to be

u(x) = (tu1(x), u2(x)) and we see that u is Cρ and u represents g because

2 gu = t g1 + g2 by virtue of Lemma 5.5 (i). By Lemma 5.5 (ii) increase the dimension of u in case injectivity is required, and we obtain the desired isometric embedding.

5.3 Günther’s Approach In this section want to get Theorem 5.4 without invoking Nash-Moser theorem. Instead, we are going to practice a fixed point argument facilitating the ellipticity of the system, to solve the perturbation problem (42), following Günther [7].

We start from the given free embedding u0 with induced metric g0. Write u = u0 + v, ∂ g = g0 + h, ∂i = , the perturbation problem ∂iu.∂ju = gij reads ∂xi

∂iu0.∂jv + ∂ju0.∂iv + ∂iv.∂jv = hij (47) in global coordinates of Tn.

50 ρ Lemma 5.8. The perturbation problem (47) is equivalent to ﬁnding v ∈ C∗ for ρ > 2 such that 2 ∂j (∂iu0.v) + ∂jFi(v) + ∂i (∂ju0.v) + ∂iFj(v) − 2 ∂iju0 .v = hij − Rij, (48) for n ! −1 −1 2 X 2 2 Rij(v) = (1 − 4) rij(v) = (1 − 4) ∂iv.∂jv + 2∂ijv. 4v − 2 ∂ikv.∂jkv , (49) k=1 −1 −1 Fi(v) = (1 − 4) fi(v) = (1 − 4) (− 4v.∂iv) , n P 2 in which 4 is the intrinsic Laplace-Beltrami operator on (T , id), that is, 4 = k ∂kk in global coordinates. Proof. Firstly see

(1 − 4)(∂iv.∂jv) = −∂i (− 4v.∂jv) − ∂j (− 4v.∂iv) n ! 2 X 2 2 + ∂iv.∂jv + 2∂ijv. 4v − 2 ∂ikv.∂jkv = −∂ifj(v) − ∂jfi(v) + rij(v) k=1

by a direct computation via chain rule and commutativity between 4 and ∂i. Note that (1 − 4) is an elliptic pseudodiﬀerential operator of degree 2, hence by Proposition 2.22 we have its inverse (1 − 4)−1

Z −1 (1 − 4)−1 u = eix.ξ 1 + |ξ|2 uˆ(ξ) d¯ξ Rn as a pseudodiﬀerential operator of degree −2. Moreover it commutes with ∂, because both symbols are location-invariant. Hence we have −1 (∂iv.∂jv) = − (1 − 4) (∂ifj(v) − ∂jfi(v)) + Rij(v) = −∂iFj(v) − ∂jFi(v) + Rij(v). And the rest follows immediately.

Lemma 5.9 (Estimates of terms). For kvk ρ ≤ R and kwk ρ ≤ R we have the following C∗ C∗ estimates: ˜r0 krij(v)k ρ−2 ≤ 2RC kvk ρ , C∗ ρ C∗ f kf (v)k ρ−2 ≤ 2RC˜ kvk ρ , i C∗ ρ C∗ ˜r krij(v) − rij(w)k ρ−2 ≤ 2RC kv − wk ρ , C∗ ρ C∗ f kf (v) − f (w)k ρ−2 ≤ 2RC˜ kv − wk ρ , i i C∗ ρ C∗ r0 kRij(v)k ρ ≤ 2RC kvk ρ , C∗ ρ C∗ f 0 kF (v)k ρ ≤ 2RC kvk ρ , i C∗ ρ C∗ r kRij(v) − Rij(w)k ρ ≤ 2RC kv − wk ρ , C∗ ρ C∗ f kF (v) − F (w)k ρ ≤ 2RC kv − wk ρ , i i C∗ ρ C∗ ˜r0 ˜f 0 ˜r ˜f r0 f 0 r f for constants Cρ , Cρ , Cρ , Cρ ,Cρ ,Cρ ,Cρ ,Cρ which depend on ρ only.

51 Proof. By Proposition 3.16 we have

k∂iv.∂jvk ρ−2 k∂ivk 0 k∂jvk ρ−2 + k∂ivk ρ−2 k∂jvk 0 C∗ . C∗ C∗ C∗ C∗

kvk 1 kvk ρ−1 + kvk ρ−1 kvk 0 2kvk ρ kvk ρ 2Rkvk ρ . . C∗ C∗ C∗ C∗ . C∗ C∗ . C∗ Similarly we have

2 ∂ijv. 4v ρ−2 2kvk 2 kvk ρ 2Rkvk ρ , C∗ . C∗ C∗ . C∗ 2 2 ∂ v.∂ v ρ−2 2kvk 2 kvk ρ 2Rkvk ρ ik ik C∗ . C∗ C∗ . C∗ and then by formula (49) we immediately have

˜r0 krij(v)k ρ−2 ≤ 2RC kvk ρ . C∗ ρ C∗

Then for fi(v) we have similarly

˜f 0 kfi(v)k ρ−2 = k− 4v.∂ivk ρ−2 kvk 2 kvk ρ−1 + kvk ρ kvk 1 ≤ 2RC kvk ρ . C∗ C∗ . C∗ C∗ C∗ C∗ ρ C∗

Denote q = v − w, and easy to see kqk ρ ≤ 2R. By Proposition 3.16 we have C∗

k4q.∂iwk ρ−2 k4qk 0 k∂iwk ρ−2 + k4qk ρ−2 k∂iwk 0 kqk 2 kwk ρ−1 C∗ . C∗ C∗ C∗ C∗ . C∗ C∗

+ kqk ρ kwk 1 2kqk ρ kwk ρ 2Rkqk ρ . C∗ C∗ . C∗ C∗ . C∗ Similarly we have

k4w.∂iqk ρ−2 kwk 2 kqk ρ−1 + kwk ρ kqk 1 2Rkqk ρ , C∗ . C∗ C∗ C∗ C∗ . C∗

k4q.∂iqk ρ−2 kqk 2 kqk ρ−1 + kqk ρ kqk 1 4Rkqk ρ , C∗ . C∗ C∗ C∗ C∗ . C∗ whence

f kf (v) − f (w)k ρ−2 ≤ k4q.∂ wk ρ−2 +k4w.∂ qk ρ−2 +k4q.∂ qk ρ−2 ≤ 2RC˜ kv − wk ρ , i i C∗ i C∗ i C∗ i C∗ ρ C∗ as required. Via a similar scheme, we estimate

k∂iq.∂jwk ρ−2 kqk 1 kwk ρ−1 + kqk ρ−1 kwk 1 2Rkqk ρ , C∗ . C∗ C∗ C∗ C∗ . C∗

k∂iw.∂jqk ρ−2 kwk 1 kqk ρ−1 + kwk ρ−1 kqk 1 2Rkqk ρ , C∗ . C∗ C∗ C∗ C∗ . C∗

k∂iq.∂jqk ρ−2 2kqk 1 kqk ρ−1 4Rkqk ρ , C∗ . C∗ C∗ . C∗ and

k∂iv.∂jv − ∂iw.∂jwk ρ−2 ≤ k∂iq.∂jwk ρ−2 +k∂iw.∂jqk ρ−2 +k∂iq.∂jqk ρ−2 2Rkv − wk ρ ; C∗ C∗ C∗ C∗ . C∗

52 estimate

2 ∂ijq. 4w ρ−2 kqk 2 kwk ρ + kqk ρ kwk 2 2Rkqk ρ , C∗ . C∗ C∗ C∗ C∗ . C∗ 2 ∂ijw. 4q ρ−2 kwk 2 kqk ρ + kwk ρ kqk 2 2Rkqk ρ , C∗ . C∗ C∗ C∗ C∗ . C∗ 2 ∂ijq. 4q ρ−2 2kqk 2 kqk ρ 4Rkqk ρ , C∗ . C∗ C∗ . C∗ and

2 2 2 2 2 ∂ijv. 4v − ∂ijw. 4w ρ−2 ≤ ∂ijq. 4w ρ−2 + ∂ijw. 4q ρ−2 + ∂ijq. 4q ρ−2 C∗ C∗ C∗ C∗

2Rkv − wk ρ ; . C∗ estimate

2 2 ∂ikq.∂jkw ρ−2 kqk 2 kwk ρ + kqk ρ kwk 2 2Rkqk ρ , C∗ . C∗ C∗ C∗ C∗ . C∗ 2 2 ∂ikw.∂jkq ρ−2 kwk 2 kqk ρ + kwk ρ kqk 2 2Rkqk ρ , C∗ . C∗ C∗ C∗ C∗ . C∗ 2 2 ∂ikq.∂jkq ρ−2 2kqk 2 kqk ρ 4Rkqk ρ , C∗ . C∗ C∗ . C∗ and

2 2 2 2 2 2 2 2 2 2 ∂ikv.∂jkv − ∂ikw.∂jkw ρ−2 ≤ ∂ikq.∂jkw ρ−2 + ∂ikw.∂jkq ρ−2 + ∂ikq.∂jkq ρ−2 C∗ C∗ C∗ C∗

2Rkv − wk ρ . . C∗ eventually we have

2 2 krij(v) − rij(w)k ρ−2 ≤ k∂iv.∂jv − ∂iw.∂jwk ρ−2 + ∂ijv. 4v − ∂ijw. 4w ρ−2 C∗ C∗ C∗ 2 2 2 2 ˜r + ∂ikv.∂jkv − ∂ikw.∂jkw ρ−2 ≤ 2RCρ kv − wk ρ , C∗ C∗ as required. From Theorem 3.13, we know the pseudodiﬀerential operator (1 − 4)−1 is a bounded ρ−2 ρ linear operator from C∗ to C∗ , hence

−1 (1 − 4) u ρ ≤ Ckuk ρ−2 . C∗ C∗

By letting u be rij(v), fi(v), rij(v)−rij(w) or fi(v)−fi(w), the rest follow immediately.

Theorem 5.10 (Local embedding theorem). Let g0 be a metric induced by a free embed- n ρ n N ding u0 of T . For ρ > 2, and any metric g of class C∗ near g0, there is a map u : T → R ρ of class C∗ for which we have gij = ∂iu.∂ju.

53 Proof. To solve perturbation problem (48), it suﬃces to solve

∂iu0.v = −Fi(v), 1 ∂2 u .v = (R − h ) , ij 0 2 ij ij which is to solve t 0 u0 −Fi(v) M(u0).v = t 00 .v = 1 . u0 2 (Rij(v) − hij)

As u0 is a free embedding, M(u0) is invertible. Construct solution operator G with −1 −Fi(v) G(v) = M(u0) . 1 2 (Rij(v) − hij) and the problem is reduced to ﬁnding a ﬁxed point of G. Consider −1 −Fi(v) + Fi(w) G(v − w) = M(u0) . 1 . 2 (Rij(v) − Rij(w))

ρ ρ ρ ρ We want G to be BC∗ (R) → BC∗ (R), where BC∗ (R) denotes ball of radius R in C∗ around 0. By the estimate

X r0 f 0 kG(v)k ρ kFi(v)k ρ + kRij(v)k ρ + khijk ρ 2RC kvk ρ + 2RC kvk ρ C∗ . C∗ C∗ C∗ . ρ C∗ ρ C∗ i,j 2 + khk ρ ≤ C R + C khk ρ , C∗ 2 3 C∗

where we see kG(v)k rho < R when C∗

−1 q R < C 1 + 1 − 4C C khk ρ /4 2 2 3 C∗ provided khk ρ , the perturbation in metric, is suﬃciently small. By Lemma 5.9, we have C∗ ρ the estimate for v, w ∈ BC∗ (R):

f r kG(v − w)k ρ ≤ 2RC C + C kv − wk ρ . C∗ ρ ρ C∗

−1 −1 f r ρ ρ When R < C Cρ + Cρ /4 we see G : BC∗ (R) → BC∗ (R) is a contraction. However ρ ρ BC∗ (R) is clearly Banach. Invoke Banach contraction theorem and the existence of v ∈ C∗ is straightforward. The mapping constructed u = u0 + v obviously solves the embedding problem. Remark. (i) This is almost the same statement as of Theorem 5.4, and next steps towards the full embedding theorem are exactly the same. (ii) Note that the difference between this theorem is that we no longer have the con- straint that ρ∈ / N. This is because we proved a flawed Nash-Moser theorem with holes within the indices of Hölderspaces (so that we can use Lemma 4.2). Later we will prove a

54 more general theorem which extends to Hölder-Zygmund spaces and the same result can be achieved using a complete Nash-Moser theorem. (iii) Note that in Günther’s original paper, he sticked to two de Rham Laplacians on vector fields and symmetric tensors, however here we simplified his geometrically strict arguments down to plain analytic descriptions. Another improvement is that he invoked interior Schauder estimates to recover information lost due to the second order elliptic operator (1 − 4), however here we use the Hölder-Zygmund (Besov) mapping properties, Theorem 3.13, of parametrix (1 − 4)−1 as a pseudodifferential operator of order −2.

55 6 Abstract Nash-Moser Theorem

6.1 Scale of Banach Spaces We want to generalise a modification process of a family of Banach spaces, just like ex- tending Hölderspaces to Hölder-Zygmund spaces by interpolation. Let Ea be a decreasing family of Banach spaces for indices a ≥ 0, with injections Eb ,→ Ea for b ≥ a, such that operator norm of the injection is no greater than 1. We assume for this family {Ea} we are given a family of regularisation operators:

Deﬁnition 6.1 (Regularisation operators). A family of continuous linear operators Sθ for parameter θ ≥ 1 0 ∞ ∞ a Sθ : E → E = ∩a=1E , are said to be the regularisation operators on family of Banach spaces {Ea} if for a, b < ∞ we have constants not depending on θ only for the conditions to hold:

(A) kSθukEb ≤ CkukEa , for b ≤ a.

d b−a−1 (B) dθ Sθu Eb ≤ Cθ kukEa , for all a, b. 0 0 (C) If u ∈ E and Sθu → v in E as θ → ∞, then v = u. Proposition 6.2 (Properties of regularisation operators). We have the followings properties for Sθ:

0 b 1 (i) θ 7→ Sθ ∈ L(E ,E ) is C with respect to θ for any b < ∞.

−1 b−a (ii) kSθukEb ≤ C(b − a) θ kukEa , for a < b.

−1 b−a (iii) ku − SθukEb ≤ C(a − b) θ kukEa , for a > b.

λ 1−λ (iv) kukEc ≤ C ((b − a)λ(1 − λ)) kukEa kukEb , for c = λa + (1 − λ)b with 0 < λ < 1. Here L(E0,Eb) indicates the space of linear bounded operators from E0 to Eb.

Proof. We see (i) is immediate from (B). For (ii) see

Z θ Z θ d b−a−1 −1 b−a kSθukEb ≤ Sτ u dτ ≤ Cτ kukEa dτ ≤ C(b − a) θ kukEa 1 dτ Eb 1 by (B) and dropping the lower index term of the integral, and we can do this only when a < b. Note that here we can interchange norm in and out integral for the integral is with respect to parameter θ instead of u. For (iii) see u = limθ→∞ Sθu via (C) and Z ∞ Z ∞ d b−a−1 −1 b−a ku − SθukEb ≤ Sτ u dτ ≤ Cτ kukEa dτ ≤ C(a − b) θ , θ dτ Eb θ

56 by (B) and for b > a, the integral converges and the upper index term of integral vanishes. Lastly for (iv), observe

−1 c−a −1 c−b kukEc ≤ kSθukEc + ku − SθukEc ≤ C (c − a) θ kukEa + (b − c) θ kukEb

1/(b−a) by (ii) with a < c and (iii) with b > c. Take θ = (kukEb /kukEa ) , which is no smaller than 1 by the contracting injection of Eb ,→ Ea, and (iv) follows immediately.

a 0 Deﬁnition 6.3 (Weak scales). Given a > 0 ﬁxed, we denote by E∗ the set of all u ∈ E such that there is some M, the following conditions hold

(D) kukE0 ≤ M,

d −a−1 (E) dθ Sθu E0 ≤ Mθ ,

d (F) dθ Sθu Ea+1 ≤ M. a And set the norm of E∗ : d a+1 d kukEa = inf kukE0 , Sθu θ , Sθu . (50) ∗ θ≥1 dθ E0 dθ Ea+1

a a a The weak scale E∗ is Banach. We call {E } the strong scale and {E∗ } the weak scale. Example (Examples of regularisation operators). (i) Take the Hölderspaces {Ca} to be the strong scale with respect to regularisation operators defined in Proposition 3.14, and a we obtain the Hölder-Zygmund spaces {C∗ } as the weak scale. That is, strong and weak scales coincide for non-integer indices. (ii) Take the strong scale to be the Sobolev spaces Ha, with respect to the same family a a of regularisation operators in Proposition 3.14, we obtain a new space H∗ . Note that {H∗ } a a a is strictly weaker than {H }, that is, H∗ is strictly larger than H .

a a b Proposition 6.4 (Injections of function spaces). For any b < a, we have E ⊂ E∗ ⊂ E , a b and E∗ ⊂ E∗.

a a a Proof. From (B) the we clearly have E ⊂ E∗ . Given u ∈ C∗ for 0 < b < a + 1, we have from (E) and (F) that

d −a−1 d Sθu ≤ θ kuk a , Sθu ≤ kuk a E∗ E∗ dθ E0 dθ Ea+1 and from Proposition 6.2 (iv) we see

λ 1−λ d d d b−a+1 Sθu ≤ C Sθu Sθu ≤ Cθ kuk a E∗ dθ Eb dθ E0 dθ Ea+1

57 by taking interpolating parameter λ = (a + 1 − b)/(a + 1). Moreover specify 0 < b < n we d −1 see dθ Sθu Eb = o(θ ) and hence Z ∞ Z ∞ d d kSθukEb ≤ kS1ukEb + Sθu dx ≤ kS1ukEb + Sθu dx < ∞ 1 dθ Eb 1 dθ Eb

b 0 b a b Hence Sθu converges in E . However Sθu → u in E , hence u ∈ E and hence E∗ ⊂ E . Naturally for a < b we have a b b E∗ ⊂ E ⊂ E∗, as required.

0 Lemma 6.5. Given [1, ∞) 3 t 7→ u(t) ∈ E such that for some a1, a2 with 0 ≤ a1 < a2 < ∞, we have for t ≥ 1, λj −1 ku(t)kEaj ≤ Ct , j = 1, 2, where λ1 < 0 < λ2. Deﬁne a = (λ2a1 − λ1a2)/(λ2 − λ1), then a ∈ (a1, a2) and Z ∞ a U = u(t) dt ∈ E∗ . 1

0 Moreover we have kUk a ≤ C C and E∗

d b−a−1 SθU ≤ Cbθ dθ Eb for any b ≥ 0.

Proof. By (B) we have

d b−aj −1 λj −1 S(θ)u(t) ≤ Cb min θ t . j=1,2 dθ Eb

The two estimates are equal when tλ2−λ1 = θa2−a1 . When t is below this bound the better estimate is obtained with j = 2 and j = 1 when t is above the bound. Since

b − aj − 1 + (a2 − a1) λj/ (λ2 − λ1) = b − 1 − a.

Integrate with respect to t to see

d b−a−1 SθU ≤ Cbθ . dθ Eb Take b = a to see 0 kUk a ≤ C C E∗ a and U ∈ E∗ .

58 Remark. This lemma exactly restores the facility of Lemma 4.2 in the case of weak scale, and it is not surprising with those new spaces we can restore our main theorem because in the process of generalising the H¨olderscale problem to this general Banach scale problem, the only reasoning we cannot naturally preserve is the Lemma 4.2. Corollary 6.6 (Additional properties of regularisation operators). We have a strengthened version of Deﬁnition 6.1 (B) and Proposition 6.2 (ii) that

0 d b−a−1 (B ) Sθu b ≤ Cbθ kuk a , for all a, b, dθ E E∗

0 b−a (ii ) ku − Sθuk b ≤ Ca,bθ kuk a for a > b. E E∗ where both constants do not depend on θ.

a Proof. For u ∈ E∗ write Z ∞ d u = S1u + Stu dt = S1u + U. 1 dt Invoke Lemma 6.5 with

a1 = 0, λ1 = −a, a2 = a + 1, λ2 = 1,

together with kS1uk a ≤ Ckuk 0 ≤ Ckuk a , given by Proposition 6.2 (ii) to obtain E E E∗

d b−a−1 Sθu ≤ Cbθ kuk a . E∗ dθ Eb

d 0 Furthermore integrate dθ Sθu from 1 to θ with respect to θ to obtain (ii ). We deﬁne the weak convergence:

a Deﬁnition 6.7 (Weak convergence). We say a sequence un ∈ E∗ is weakly convergent to a b u ∈ E∗ if for every b < a we have un → u in E∗.

b Remark. We have an equivalent deﬁnition by convergence un → u in E for every b < a by help of the injection between weak and strong spaces. We end this section with a version of Leray-Schauder ﬁxed point theorem: Lemma 6.8 (Leray-Schauder). Let E be a Banach spcae, K a continuous map mapping the closed unit ball Br ⊂ E into Br/2, with K(Br) contained in a compact set. If v ∈ Br/2 we have that equation u + K(u) = v has a solution u ∈ Br with kukE ≤ kvkE + r/2.

−1 Proof. We extend the definition of K to E by defining K(u) = K(r u/ kukE) whenever kukE > r. If 0 ≤ λ ≤ 1 and u is a fixed point of the map u 7→ λ (v − K(u)), then kukE ≤ kvkE + r/2, which is independent of λ. Hence the Leray-Schauder theorem proves that there is a fixed point for every λ ∈ [0, 1], and for λ = 1 proves this lemma.

59 6.2 Banach Scale Nash-Moser Theorem We need to set some assumptions. Firstly we need two families of Banach spaces, {Ea} , {F a} E F and regularisation operators Sθ ,Sθ on them as described in Deﬁnition 6.1, and denote their a a weak scales by {E∗ } , {F∗ } respectively. We assume further that the injection Fb ,→ Fa is compact for all b > a. We also have assumptions on tame operators. Let α and β > 0 be ﬁxed, [a1, a2] α be an interval with 0 ≤ a1 < α < a2, V be a convex E∗ neighbourhood of 0, and Φ: V ∩ Ea2 → F β. Assume

0 2 (A1) Φ is of class C in the previous settings for operators and satisﬁes the tame estimate for some γ > 0,

00 X kΦ (u)(v, w)k β+γ ≤ C 1 + kuk m0 kvk m00 kwk m000 . F E j E j E j j≤jmax

0 0 ∞ a2 (A2) For v ∈ C ∩ E∞,Φ (v) has a right inverse ψ(v): F → E , continuous as mapping (v, g) → ψ(v)g, such that

kψ(v)gkEa ≤ C (kgkF β+a−α + kgkF 0 kvkF β+a )

for a1 ≤ a ≤ a2. We state the main theorem:

0 00 000 Theorem 6.9 (Abstract Nash-Moser theorem). Let a2 be now less than all mj, mj , mj , and 0 00 000 max mj − α, 0 + max mj , a1 + mj < 2α, (51) β for every j, and α − β < a1. For every f ∈ F with kfk β suﬃciently small, one can ﬁnd ∗ F∗ a2 α β a sequence uj ∈ V ∩ E which has a weak limit u ∈ E∗ and Φ(uj) converges weakly in F∗ to Φ(0) + f. β Proof. Let g ∈ F∗ , take gj = Sθj+1 − Sθ g and we have ∞ X b−β−1 g = ∆jgj, kgjk b ≤ Cbθ kgk β F j F∗ j=0 ε 1/ε for ∆j = θj+1 − θj and θj = θ + n . Assume the perturbation kgk β is small and we 0 F∗ claim that we can invoke the iteration scheme with initial u0 = 0,

uj+1 = uj + ∆ju˙ j, u˙ j = ψ(vj)gj, vj = Sθj uj, and we claim the following estimates, there is constant δ,

a−α−1 ku˙ jk a ≤ δkgk β θ , a1 ≤ a ≤ a2 (52) E F∗ j a−α kvjk a ≤ C2kgk β θ , α < a ≤ a2 (53) E F∗ j a−α kuj − vjk a ≤ C3kgk β θ , a ≤ a2. (54) E F∗ j

60 To validate the estimates, we conduct an iterative argument similar to our iterative hypotheses in the case of H¨olderspaces. Suppose we have constructed uj for all j ≤ k, with (53) and (54) holding for j ≤ k, then automatically (52) holds for j < k by aid of −1 ε u˙ j = ∆j [(uj+1 − vj+1) − (uj − vj) + (vj+1 − vj)] (Note θj+1/θj ≤ 2 ). At the boundary 0 case j = k, use (A2) to derive

a−α−1 −β−1 β+a−α kψ(vk)gkk a ≤ C θ kgk β + θ kgk β C2kgk β θ E k F∗ k F∗ F∗ k

Here we have β + a1 > α by (51) and hence (52) holds for case j = k as well, when δ < C Pk and kgk β is suﬃciently small (δ does not depend on C2). Since uk+1 = ∆ju˙ j we F∗ j=0 obtain from Lemma 6.5 via discretisation that

0 kuk+1k α ≤ C δkgk β . (55) E∗ F∗ From Corollary 6.6 (ii0) we see for a < α we have

0 a−α a−α kuk+1 − vk+1k a ≤ C Ca,αδθ kgk β ≤ C3θ kgk β (56) E k+1 F∗ k+1 F∗ if a is not close to α if C3/δ is large enough. In the limit case a = a2, by adding (52) we obtain (56) at a = a2. By the convexity inequality (12) we obtain (54) for j ≤ k + 1 with same constant in the whole interval a ∈ [0, a2]. By the property of regularisation operators and (55), (53) for j = k + 1 is also clear. Hence the iteration runs on and on such that (52)–(54) hold for all j. We have now proved that the construction of the infinite sequence uj, vj, u˙ j is possible for (55) and (54) are in V when kgk β is sufficiently small. It follows F∗ α from (52) that uk has a weak limit in E∗ . What remains is to examine the limit of Φ(uk). Write 0 00 Φ(uj+1 − Φ(uj)) = ∆j(ej + ej + gj) 0 00 where we adopt the notations ej and ej from (33) and (34). We want to obtain some estimates similar to our error estimate for the case of Hölder spaces. Consider

Z 1 00 00 ej = Φ (vj + t(uj − vj))(u ˙ j, uj − vj) dt 0

0 and (A2), (52)–(54), (51) to derive that if e > 0 is suﬃciently small that

0 00 000 max mj − α, 0 + max mj , a1 + mj + e < 2α we will have 00 −1−e 2 e ≤ Cθ kgk β . j F β+γ j F∗ −N For any N > 0 we can pick ε small to control the growth of θj, that is, ∆j = O θj . For large N we obtain that 0 −1−e 2 e ≤ Cθ kgk β j F β+γ j F∗

61 It follows that Φ(uk) converges weakly to Φ(0) + T (g) + g where

∞ X 0 00 T (g) = ∆j ej + ej . j=0

The sum is uniformly convergent in Fβ+γ norm when kgk β is suﬃciently small. Hence F∗ β β T (g) is a continuous map from a neighbourhood of 0 in F∗ to a compact subset of F∗ , and

2 kT (g)k β ≤ Ckgk β . F∗ F∗ Invoke Leray-Schauder theorem we see g + T (g) takes on all values in a neighbourhood of β 0 and take g + T (g) = f for f ∈ F with kfk β suﬃciently small we have kgk β small ∗ F∗ F∗ enough to extract uk as above such that Φ(uk) converges to Φ(0) + f weakly.

a a a Remark (Consequences of abstract Nash-Moser theorem). (i) If we take E = F = C∗ , the Hölder-Zygmund spaces, then we obtain an extension of Theorem 4.3 to integer indices (Hölder-Zygmund spaces). To get the existence part of Theorem 4.3 then exclude the integer indices. a a a a a (ii) Take E = F = H∗ , where H∗ is the weak scale of Sobolev spaces H as of a modification (50). We can then establish a Nash-Moser theorem for weak scales H∗ , stating

that for a tamed map Φ losing γ derivatives, for perturbation f with kfk a sufficiently H∗ a−γ a−γ small, we can obtain uk ∈ H∗ such that uk converges weakly to u in H∗ and Φ(uk) a a a a converges weakly to Φ(u0) + f in H∗ . Indeed H∗ is a strictly weaker scale that H∗ 6= H a a for any a, that is H∗ is strictly larger than H . By the injection between weak and strong scales, we can obtain an almost-sharp Sobolev space based theorem: for kfkHa small, we α−γ α−γ−ε have uk, u ∈ H∗ for which uk → u in any H with ε > 0, and Φ(uk) converges a weakly to Φ(u0) + f in H∗ . One can obtain a sharp version (ε = 0) with other techniques, for example Baldi and Haus [3] Theorem 2.1, by facilitating an orthogonality property similar to our Proposition 3.3. (iii) In general, one can prove a Nash-Moser theorem between any tame Fréchet spaces, which are graded Fréchet spaces with a tame structure, with reference to Hamilton [8]. (iv) There is a relatively recent result by Ekeland [5], in which an implicit function theorem between two graded Fréchet spaces is devised. Note that the assumptions of this new theorem are much weakened: (i) the objective mapping Φ should be continuous and Gâteaux-differentiable (not necessarily C2, C1 or even Fréchet differentiable); (ii) Fréchet spaces should be graded (not necessarily tame, not necessarily with regularisation operators). Its proof does not utlitise Newton iterations, but uses dominated convergence theorem and Ekeland’s variational principle. The Nash-Moser theorems are special cases of this more general theorem. See also Ekeland and Séré[6].

62 7 Epilogue

7.1 Review This essay has its skeleton and its order of presentation as in Alinhac and Gérard[2]; indeed, presentations of pseudodifferential operators, dyadic analysis results, our major Hölderscale Nash-Moser theorerm (Theorem 4.3), local embedding theorems (Theorem 5.4, Theorem 5.10) are largely attributed to [2]. In Section 6, the modification of spaces and Banach scale Nash-Moser theorem are very much attributed to Hörmander[12] [14]. Map- ping properties of pseudodifferential operators on Hölder-Zygmund spaces are attributed to the textbook of Abels [1]. Auxiliary results around the isometric embedding are well explained in Tao’s online notes [21]. Historic statements of mathematical works around this topic presented in introduction section are borrowed from [3] and [6].

It is also noted, though originality is hard to achieve in very limited time on this histor- ical topic, some originality and deep understanding of this topic are still reflected in essay, by the following evidences: (i) a natural adaption from Hölderspaces to Hölder-Zygmund spaces of most dyadic results and of Günther’s approach; (ii) most of the proofs are with more details for better clarity; (iii) a very detailed estimate, as an original computation, is obtained in Lemma 5.9; (iv) many remarks provide deep intuition into and around the topic and refer to historic and recent developments (see remarks on pp. 16, 17, 24, 27, 44, 45, 54, 62).

7.2 Acknowledgement Firstly I thank Clément Mouhot for kindly setting this essay topic, recommendation of references and assessing the essay. I thank Anthony Ashton for related discussions on pseudodifferential operators. I thank Matteo Capoferri for bringing me to the conflicts between constructiveness of Nash-Moser scheme and poor numerical performance of solving isometric embedding problems. I thank Fritz Hiesmayr for brief discussion on embedding of torus into flat spaces. I thank Ilia Kamotski for discussion on the Nash-Moser theorem and dyadic results in Hölderspaces. I thank Zhuo Min Lim for various discussions around the topic. I thank Nikolai Saveliev and Dmitri Vassiliev for related discussion on the reduction of isometric embedding problem back to the case of a torus.

7.3 Bibliography [1] Abels, H. Pseudodiﬀerential and Singular Integral Operators: An Introduction with Applications. Walter de Gruyter, Jan. 2012.

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[11] Hormander,¨ L. Implicit function theorems : lectures at Stanford University, summer quarter 1977 /. 1977.

[12] Hormander,¨ L. On the Nash-Moser Implicit Function Theorem. Annales Academiae Scientiarum Fennicae. Series A. I. Mathematica 10 (1985), 255–259.

[13] Hormander,¨ L. The Analysis of Linear Partial Diﬀerential Operators I: Distribution Theory and Fourier Analysis. Springer Berlin Heidelberg, Aug. 1990.

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