Global solutions for the gravity water waves system in 2d

Abstract

We consider the gravity water waves system in the case of a one dimensional interface, for sufficiently smooth and localized initial data, and prove global existence of small solutions. This improves the almost global existence result of Wu (Invent Math 177(1):45–135, 2009). We also prove that the asymptotic behavior of solutions as time goes to infinity is different from linear, unlike the three dimensional case (Germain et al., Ann Math 175(2):691–754, 2012; Wu, Invent Math 184(1):125–220, 2011). In particular, we identify a suitable nonlinear logarithmic correction and show modified scattering. The solutions we construct in this paper appear to be the first global smooth nontrivial solutions of the gravity water waves system in 2D.

Appendices

Appendix A: Supporting material

In this first appendix we gather some useful Lemmas that are used several times in Sects. 9 and 10 and in the course of the energy estimates. First, in Appendix A1 we give some variants of estimates proven in [45] related to the Hilbert transform on curves. In A2 we first recall some Theorems about multilinear operators of “Calderón’s commutators”-type, and then prove some additional bounds on them that are used for the energy estimates.

1.1 A1: Estimates for the Cauchy integral

In what follows we will always be under the a priori assumption that (2.9) holds.

Lemma 11.1

Let ${\mathcal {H} } = {\mathcal {H} } _{\zeta }$. Then, for any $f$ in $X_k$ with $0 \le k \le N_0$, we have

$$\begin{aligned} {\Vert {\mathcal {H} } f \Vert }_{H^k} + {\left\| {\mathcal {H} } \frac{1}{{\zeta }_{\alpha }} f \right\| }_{H^k}&\lesssim {\Vert f \Vert }_{H^k} + {\Vert {\zeta }_{\alpha }- 1 \Vert }_{H^k} {\Vert f \Vert }_{W^{\frac{k}{2},\infty }} , \end{aligned}$$

(11.1)

$$\begin{aligned} {\Vert {\mathcal {H} } f \Vert }_{X_k} + {\left\| {\mathcal {H} } \frac{1}{{\zeta }_{\alpha }} f \right\| }_{X_k}&\lesssim {\Vert f \Vert }_{X_k} + {\Vert {\zeta }_{\alpha }- 1 \Vert }_{X_k} {\Vert f \Vert }_{W^{\frac{k}{2},\infty }} . \end{aligned}$$

(11.2)

In particular, if $k \le N_1+5$, one has

$$\begin{aligned} {\Vert {\mathcal {H} } f \Vert }_{H^k} + {\left\| {\mathcal {H} } \frac{1}{{\zeta }_{\alpha }} f \right\| }_{H^k} \lesssim {\Vert f \Vert }_{H^k} . \end{aligned}$$

(11.3)

Furthermore, for any $0 \le k \le N_0$

$$\begin{aligned} {\Vert ( {\mathcal {H} } + \overline{ {\mathcal {H} } }) f \Vert }_{X_k}&\lesssim {\Vert {\zeta }_{\alpha }- 1 \Vert }_{W^{\frac{k}{2},\infty }} {\Vert f \Vert }_{X_k} + {\Vert {\zeta }_{\alpha }- 1 \Vert }_{X_k} {\Vert f \Vert }_{W^{\frac{k}{2}, \infty }} , \end{aligned}$$

(11.4)

and for $0 \le k \le N_1$

$$\begin{aligned} {\Vert ( {\mathcal {H} } + \overline{ {\mathcal {H} } }) f \Vert }_{H^k}&\lesssim {\Vert \text{ Im }{\zeta }_{\alpha }\Vert }_{W^{k,\infty }} {\Vert f \Vert }_{H^k} . \end{aligned}$$

(11.5)

Proof

The $L^2$ case in (11.1) follows directly from Theorem 11.3. The $H^k$, respectively $X_k$, estimates can be proven by induction using the commutation identities (11.39b) to distribute derivatives, respectively (11.39c) to distribute the vector field $S$, and the bounds given in Theorem 11.3 for operators of the type $C_1$, as defined in (11.16).

To prove (11.4) one notices that

$$\begin{aligned} ( {\mathcal {H} } + \overline{ {\mathcal {H} } }) f = -\frac{2}{\pi } \int \frac{\text{ Im }{\zeta }({\alpha }) - \text{ Im }{\zeta }({\beta })}{ {|{\zeta }({\alpha }) - {\zeta }({\beta })|}^2 } \, f({\beta }) {\zeta }_{\beta }({\beta }) d{\beta }+ \frac{2}{\pi } \int \frac{f({\beta }) \text{ Im }{\zeta }_{\beta }({\beta }) }{\overline{{\zeta }}({\alpha }) - \overline{{\zeta }}({\beta })} d{\beta },\nonumber \\ \end{aligned}$$

(11.6)

which is the sum of two operator of the form $C_1(H, \text{ Im }{\zeta }, f{\zeta }_{\alpha })$ and $C_1(H, \text{ id }, f\text{ Im }{\zeta }_{\alpha })$, for some smooth $H$, see (11.16) below. Applying the commutation identities (11.39b) and (11.39c), followed by the $L^2$-estimates of Theorem 11.3, one can then verify the validity of (11.4) and (11.5). $\Box$

The next Lemma is a variant of Lemma 3.8 in [45] and gives estimates of real valued functions $f$ in terms of the norms of $(I- {\mathcal {H} } )f$. $\square$

Lemma 11.2

Let $f \in X_k$, $0 \le k \le N_0$, be real-valued with

$$\begin{aligned} (I - {\mathcal {H} } ) f = g . \end{aligned}$$

Then, for $0 \le k \le N_1+5$, one has

$$\begin{aligned} {\Vert f \Vert }_{H^k}&\lesssim {\Vert g \Vert }_{H^k} . \end{aligned}$$

(11.7)

Furthermore, for $0 \le k \le N_0$

$$\begin{aligned} {\Vert f \Vert }_{H^k}&\lesssim {\Vert g \Vert }_{H^k} + {\Vert {\zeta }_{\alpha }- 1 \Vert }_{H^k} \left( {\Vert g \Vert }_{ W^{\frac{k}{2},\infty }} + {\Vert \text{ Im }{\zeta }_{\alpha }\Vert }_{W^{\frac{k}{2}+1,\infty }} {\Vert g \Vert }_{H^{\frac{k}{2}+1}} \right) \end{aligned}$$

(11.8)

$$\begin{aligned} {\Vert f \Vert }_{X_k}&\lesssim {\Vert g \Vert }_{X_k} + {\Vert {\zeta }_{\alpha }- 1 \Vert }_{X_k} \left( {\Vert g \Vert }_{ W^{\frac{k}{2},\infty }} + {\Vert \text{ Im }{\zeta }_{\alpha }\Vert }_{W^{\frac{k}{2}+1,\infty }} {\Vert g \Vert }_{H^{\frac{k}{2}+1}} \right) . \end{aligned}$$

(11.9)

Moreover, for $0 \le k \le N_1$, we have

$$\begin{aligned} {\Vert f \Vert }_{W^{k,\infty }} \lesssim {\Vert \text{ Re }\, g \Vert }_{W^{k,\infty }}&+ {\Vert \text{ Im }{\zeta }_{\alpha }\Vert }_{W^{k+1,\infty }} {\Vert g \Vert }_{H^{k + 1}} , \end{aligned}$$

(11.10)

and a similar estimate for $\partial _{\alpha }f$:

$$\begin{aligned} {\Vert \partial _{\alpha }f \Vert }_{W^{k,\infty }} \lesssim {\Vert \text{ Re }\, \partial _{\alpha }g \Vert }_{W^{k,\infty }} + {\Vert {\zeta }_{\alpha }- 1\Vert }_{W^{k+1,\infty }} {\Vert \partial _{\alpha }g \Vert }_{H^{k + 1}} . \end{aligned}$$

(11.11)

Proof

Since $f$ is real-valued we have $(I - {\mathcal {K} } ) f = \text{ Re }g$, where ${\mathcal {K} } = \text{ Re } {\mathcal {H} }$. Then

$$\begin{aligned} (I - {\mathcal {K} } ) \partial _{\alpha }^j f = \text{ Re }\partial _{\alpha }^j g - \left[ {\mathcal {K} } , \partial _{\alpha }^j \right] f = \text{ Re }\partial _{\alpha }^j g - \sum _{k=1}^j \partial _{\alpha }^{j-k} \left[ {\mathcal {K} } , \partial _{\alpha }\right] \partial _{\alpha }^{k-1} f . \end{aligned}$$

Notice that

$$\begin{aligned} {[\partial _{\alpha }, {\mathcal {H} } ]} f = [{\zeta }_{\alpha }, {\mathcal {H} } ] \frac{f_{\alpha }}{{\zeta }_{\alpha }} = C_2(H, {\zeta }_{\alpha }-1, f) , \end{aligned}$$

(11.12)

for some smooth $H$, and where $C_2$ is defined in (11.17). We can then use the fact that the inverse of $I - {\mathcal {K} }$ is bounded on $L^2$ with an operator norm depending only on ${\varepsilon }_1$, (11.12), and Theorem 11.3, to obtain

$$\begin{aligned} {\Vert \partial _{\alpha }^j f \Vert }_{L^2}&\lesssim {\left\| \partial _{\alpha }^j g \right\| }_{L^2} + {\Vert {\zeta }_{\alpha }- 1 \Vert }_{ H^{\frac{j}{2} + 1} } {\Vert f \Vert }_{H^j} + {\Vert {\zeta }_{\alpha }- 1 \Vert }_{H^j} {\Vert f \Vert }_{W^{\frac{j}{2},\infty }} \\&\lesssim {\left\| \partial _{\alpha }^j g \right\| }_{L^2} + {\varepsilon }_1 {\Vert f \Vert }_{H^j} + {\Vert {\zeta }_{\alpha }- 1 \Vert }_{H^j} {\Vert f \Vert }_{W^{\frac{j}{2},\infty }} . \end{aligned}$$

After summing over $j$, the second term in the right-hand side above can be absorbed to the left hand-side for ${\varepsilon }_1$ small enough. We have therefore obtained that for any $0 \le k \le N_0$

$$\begin{aligned} {\Vert f \Vert }_{H^k}&\lesssim {\left\| g \right\| }_{H^k} + {\Vert {\zeta }_{\alpha }- 1 \Vert }_{H^k} {\Vert f \Vert }_{W^{\frac{k}{2}, \infty }} . \end{aligned}$$

(11.13)

If $k \le N_1+5$ the last term above can be also absorbed to the left hand-side thus yielding (11.7).

In order to prove (11.8) let us focus on the term ${\Vert f \Vert }_{W^{\frac{k}{2}, \infty }}$. Using the identity $f = {\mathcal {K} } f + \text{ Re }g$, Sobolev’s embedding, the estimate (11.5), and (11.3), we get

$$\begin{aligned} {\Vert f \Vert }_{W^{\frac{k}{2}, \infty }} \le {\Vert {\mathcal {K} } f \Vert }_{H^{\frac{k}{2} + 1}} + {\Vert \text{ Re }g \Vert }_{W^{\frac{k}{2}, \infty }}&\lesssim {\Vert \text{ Im }{\zeta }_{\alpha }\Vert }_{W^{\frac{k}{2}+1,\infty }} {\Vert f \Vert }_{H^{\frac{k}{2} + 1}} + {\Vert \text{ Re }g \Vert }_{W^{\frac{k}{2}, \infty }} \\&\lesssim {\Vert \text{ Im }{\zeta }_{\alpha }\Vert }_{W^{\frac{k}{2}+1,\infty }} {\Vert g \Vert }_{H^{\frac{k}{2} + 1}} + {\Vert \text{ Re }g \Vert }_{W^{\frac{k}{2}, \infty }}. \end{aligned}$$

Plugging this last inequality into (11.13) gives (11.8). Substituting $k$ with $2k$ we obtain (11.10). (11.11) can be obtained similarly.

From above we see that (11.9) would follow if we show

$$\begin{aligned} {\Vert S f \Vert }_{H^k}&\lesssim {\left\| S g \right\| }_{H^k} + {\Vert S({\zeta }_{\alpha }- 1) \Vert }_{H^k} {\Vert f \Vert }_{W^{k, \infty }} . \end{aligned}$$

(11.14)

Starting from $(I- {\mathcal {H} } )f = g$ one can commute derivatives using (11.12) and commute $S$ by using

$$\begin{aligned}{}[S, {\mathcal {H} } ] f = [S{\zeta }- {\zeta }, {\mathcal {H} } ] \frac{f_{\alpha }}{{\zeta }_{\alpha }} = C_2(H, S{\zeta }- {\zeta }, f) . \end{aligned}$$

(11.15)

Applying Theorem 11.3 one can then obtain

$$\begin{aligned} {\Vert \partial _{\alpha }^j S f \Vert }_{L^2}&\lesssim {\left\| \partial _{\alpha }^j S g \right\| }_{L^2} + {\Vert {\zeta }_{\alpha }- 1 \Vert }_{H^j} {\Vert S f \Vert }_{H^j} + {\Vert S({\zeta }_{\alpha }- 1) \Vert }_{H^j} {\Vert f \Vert }_{W^{j,\infty }} \\&\lesssim {\left\| S g \right\| }_{H^j} + {\varepsilon }_1 {\Vert S f \Vert }_{H^j} + {\Vert S({\zeta }_{\alpha }- 1) \Vert }_{H^j} {\Vert f \Vert }_{W^{j,\infty }} . \end{aligned}$$

Summing over $j$ and absorbing the second summand above in the left-hand side, we obtain (11.14) and hence (11.9). $\square$

1.2 A2: Estimates for multilinear operators

In this section we study some singular integrals that appear when performing the energy estimates. These integral operators are well known objects, which are usually referred to as Calderón’s commutators. We first state some $L^2$-bounds like the ones already given in [45].

Let $H \in C^1$, $A_i \in C^1$ for $i=1,\dots ,m$, and $F \in C^\infty$. Using the same notation in [45] we define

$$\begin{aligned} C_1 (H,A,f)&:= \text{ p.v. } \int F \left( \frac{ H(x) - H(y) }{x-y} \right) \frac{ \prod _{i=1}^m ( A_i(x) - A_i(y) ) }{ {(x-y)}^{m+1} } f(y) \, dy \end{aligned}$$

(11.16)

$$\begin{aligned} C_2 (H,A,f)&:= \text{ p.v. } \int F \left( \frac{ H(x) \!-\! H(y) }{x-y} \right) \frac{ \prod _{i=1}^m ( A_i(x) \!-\! A_i(y) ) }{ {(x-y)}^{m} } \partial _y f(y) \, dy . \end{aligned}$$

(11.17)

Integrals like the ones above are always to be understood in the principal value sense, but, as before, for simplicity we will often omit the p.v. notation.

We also define the quadratic bilinear operators

$$\begin{aligned}&Q_0 (f,g) := \int \frac{ f({\alpha }) - f({\beta }) }{ {\zeta }({\alpha })-{\zeta }({\beta }) } g({\beta }) \, d{\beta }, \end{aligned}$$

(11.18)

$$\begin{aligned}&Q_1 (f,g) := \int \frac{ f({\alpha }) - f({\beta }) }{ {({\zeta }({\alpha })-{\zeta }({\beta }))}^2 } g({\beta }) \, d{\beta }, \end{aligned}$$

(11.19)

$$\begin{aligned}&Q_2 (f,g) := \int \frac{ f({\alpha }) - f({\beta }) }{ {|{\zeta }({\alpha })-{\zeta }({\beta })|}^2 } g({\beta }) \, d{\beta }. \end{aligned}$$

(11.20)

We denote by $\mathbf{Q}$ indistinctly any scalar multiple of the operators $Q_1$ or $Q_2$:

$$\begin{aligned} \mathbf{Q}(f,g) := c_i Q_i(f,g) \end{aligned}$$

(11.21)

for $c_i \in {\mathbb {C}}$, $i=1,2$. $Q_0$ causes some difficulties because it does not admit standard $L^2 \times L^\infty \rightarrow L^2$ estimates. Moreover, it does not admit $L^\infty$ type estimates like those in Lemma 11.5 below for $Q_1$ and $Q_2$; in order to bound it we need to resort to a stronger space than $L^\infty$.

We recall the following:

Theorem 11.3

(Coifman-McIntosh-Meyer [10], Wu [45]) There exists $c = c(F, {\Vert H^{\prime }\Vert }_{L^\infty })$ such that the operators $C_j$, for $j=1,2$, satisfy the bounds

$$\begin{aligned} {\Vert C_j (H,A,f) \Vert }_{L^2}&\le c \prod _{i=1}^m {\Vert \partial A_i \Vert }_{L^\infty } {\Vert f \Vert }_{L^2} \end{aligned}$$

(11.22)

$$\begin{aligned} {\Vert C_j (H,A,f) \Vert }_{L^2}&\le c {\Vert \partial A_1 \Vert }_{L^2} \prod _{i=2}^m {\Vert \partial A_i \Vert }_{L^\infty } {\Vert f \Vert }_{L^\infty } . \end{aligned}$$

(11.23)

From the above Theorem we can infer the following bounds on the operators of the type $\mathbf{T}$ defined in (7.28) and $\mathbf{Q}$ in (11.21):

Corollary 11.4

There exists a constant $c = c({\Vert \partial _{\alpha }{\zeta }\Vert }_{L^\infty })$ such that

$$\begin{aligned} {\Vert \mathbf{Q}(f, g) \Vert }_{L^2}&\le c {\Vert \partial _{\alpha }f \Vert }_{L^\infty } {\Vert g \Vert }_{L^2} \\ {\Vert \mathbf{Q}(f, g) \Vert }_{L^2}&\le c {\Vert \partial _{\alpha }f \Vert }_{L^2} {\Vert g \Vert }_{L^\infty } \end{aligned}$$

and

$$\begin{aligned} {\Vert \mathbf{T}(f, g, h_{\alpha }) \Vert }_{L^2}&\le c {\Vert \partial _{\alpha }f \Vert }_{L^\infty } {\Vert \partial _{\alpha }g \Vert }_{L^\infty } {\Vert h \Vert }_{L^2} \\ {\Vert \mathbf{T}(f, g, h_{\alpha }) \Vert }_{L^2}&\le c {\Vert \partial _{\alpha }f \Vert }_{L^2} {\Vert \partial _{\alpha }g \Vert }_{L^\infty } {\Vert h \Vert }_{L^\infty } . \end{aligned}$$

In Appendix A.2.2 we will prove the following simple Lemma:

Lemma 11.5

There exists a constant $c = c( {\Vert \partial _{\alpha }{\zeta }\Vert }_{W^{1,\infty }} )$ such that the operators $\mathbf{Q}$ satisfy the bound

$$\begin{aligned} {\Vert \mathbf{Q}(f, g) \Vert }_{L^\infty }&\le c {\Vert f \Vert }_{W^{2,\infty }} {\Vert g \Vert }_{W^{1,\infty }} . \end{aligned}$$

(11.24)

We will also need to bound operators of the type $Q_0$ in $L^\infty$. However, they will only appear with a derivative in front, so that we can use the following Lemma:

Lemma 11.6

There exists a constant $c = c( {\Vert {\zeta }_{\alpha }- 1 \Vert }_{H^3} )$ such that

$$\begin{aligned} {\Vert \partial _{\alpha }Q_0 (f, g) \Vert }_{L^\infty }&\le c {\Vert f \Vert }_{W^{2,\infty }} \left( {\Vert {\mathcal {H} } g \Vert }_{L^\infty } + {\Vert g \Vert }_{W^{1,\infty }} \right) . \end{aligned}$$

(11.25)

The above results, together with some commutation identities, will give us the following Proposition:

Proposition 11.7

Recall the definitions

$$\begin{aligned} L := ({\zeta }_{\alpha }- 1, u, w, \text{ Im }{\zeta }, \partial _{\alpha }\chi , v) \in {\mathbb {C}}^6 \end{aligned}$$

(11.26)

and

$$\begin{aligned} L^- := ({\zeta }_{\alpha }- 1, u, w, \partial _{\alpha }\chi , v) \in {\mathbb {C}}^5 . \end{aligned}$$

(11.27)

Let $\mathbf{Q}$ and $\mathbf{T}$ be given by (11.21) and (7.27)–(7.28). Then

• (1) There exists a constant $c = c({\Vert {\zeta }_{\alpha }-1 \Vert }_{H^{N_1+4}})$ such that for any integer $k \le N_1$

  $$\begin{aligned}&{\Vert \mathbf{Q}(f, g) \Vert }_{W^{k,\infty }} \le c {\Vert f \Vert }_{W^{k+2,\infty }} {\Vert g \Vert }_{W^{k+2,\infty }} . \end{aligned}$$

  (11.28)

  In particular

  $$\begin{aligned}&{\Vert \mathbf{Q}(L_i, L_j) \Vert }_{W^{[\frac{m}{2}]+1,\infty }} \le c {\Vert L \Vert }^2_{W^{[\frac{m}{2}]+3,\infty }} \end{aligned}$$

  (11.29)

  for any $i,j \in \{ 1, \dots , 6 \}$ and $0 \le m \le N_0$.

• (2) There exists a constant $c = c({\Vert {\zeta }_{\alpha }-1 \Vert }_{H^{N_1+4}})$ such that for any integer $0 \le k \le N_0$

  $$\begin{aligned} \nonumber {\Vert Q_0 (f, g) \Vert }_{X_k}&\lesssim _c {\Vert f \Vert }_{W^{\frac{N_0}{2}+1,\infty }} {\Vert g \Vert }_{X_k} + {\Vert (g, {\mathcal {H} } g) \Vert }_{W^{\frac{N_0}{2}+1,\infty }} {\Vert f \Vert }_{X_k} \\&\quad + {\Vert f \Vert }_{W^{\frac{N_0}{2}+1,\infty }} {\Vert g \Vert }_{W^{\frac{N_0}{2}+1,\infty }} {\Vert {\zeta }_{\alpha }-1 \Vert }_{X_k} \end{aligned}$$

  (11.30)

  and

  $$\begin{aligned} \nonumber {\Vert Q_0 (f, \partial _{\alpha }g) \Vert }_{X_k}&\lesssim _c {\Vert \partial _{\alpha }f \Vert }_{W^{\frac{N_0}{2}+1,\infty }} {\Vert g \Vert }_{X_k} + {\Vert g \Vert }_{W^{\frac{N_0}{2}+2,\infty }} {\Vert f \Vert }_{X_k} \\&\quad + {\Vert f \Vert }_{W^{\frac{N_0}{2}+1,\infty }} {\Vert g \Vert }_{W^{\frac{N_0}{2}+1,\infty }} {\Vert {\zeta }_{\alpha }-1 \Vert }_{X_k} . \end{aligned}$$

  (11.31)

  Furthermore, for $k \le N_1$,

  $$\begin{aligned} {\Vert \partial _{\alpha }Q_0 (f, g) \Vert }_{W^{k,\infty }}&\le c {\Vert f \Vert }_{W^{k+2,\infty }} \left( {\Vert {\mathcal {H} } g \Vert }_{W^{k,\infty }} + {\Vert g \Vert }_{W^{k+1,\infty }} \right) , \end{aligned}$$

  (11.32)

  so that

  $$\begin{aligned} {\Vert \partial _{\alpha }Q_0 (L_i, L_j^-) \Vert }_{W^{[\frac{m}{2}]+1,\infty }}&\le c {\Vert L \Vert }_{W^{[\frac{m}{2}]+3,\infty }} \left( {\Vert {\mathcal {H} } L^- \Vert }_{W^{[\frac{m}{2}]+2,\infty }} + {\Vert L^- \Vert }_{W^{[\frac{m}{2}]+2,\infty }} \right) \end{aligned}$$

  (11.33)

  for any $i \in \{ 1, \dots , 6 \}$, $j \in \{ 1, \dots , 5 \}$ and $0 \le m \le N_0$.

• (3) There exists a constant $c$ as above such that for any triple $(f,g,h)$ with ${\Vert (f,g,h) \Vert }_{H^{N_1-2}} \le 1$, and any integer $m$, one has

  $$\begin{aligned}&{\Vert \mathbf{T}(f, g, h) \Vert }_{X_m} + {\Vert \mathbf{T}(f, g, \partial _{\alpha }h) \Vert }_{X_m} \le c {\Vert (f,g,h,{\zeta }_{\alpha }-1) \Vert }_{X_m}\nonumber \\&\quad {\Vert (f,g,h) \Vert }_{W^{\frac{N_0}{2}+3,\infty }}^2 . \end{aligned}$$

  (11.34)

  In particular

  $$\begin{aligned} {\Vert \mathbf{T}(L_i, L_j, L_k) \Vert }_{X_m} + {\Vert \mathbf{T}(L_i, L_j, \partial _{\alpha }L_k) \Vert }_{X_m}&\le c {\Vert L \Vert }_{X_m} {\Vert L \Vert }_{W^{\frac{N_0}{2}+3,\infty }}^2 \end{aligned}$$

  (11.35)

  for any $i,j,k \in \{ 1, \dots , 6 \}$ and $0 \le m \le N_0$.

The proof of the above Proposition is given in Appendix A.2.4. We will also need the following simple Lemma:

Lemma 11.8

Let ${\mathcal {H} } = {\mathcal {H} } _{\zeta }$ denote the Hilbert transform along a curve ${\zeta }$ satisfying ${\Vert {\zeta }_{\alpha }- 1\Vert }_{H^{N_1+4}} \le \frac{1}{2}$. Then for any $f$ with ${\Vert f \Vert }_{H^{k+2}} \le 1$, and $k \le N_1$, we have

$$\begin{aligned}&{\Vert {\mathcal {H} } \partial _{\alpha }f(t) \Vert }_{W^{k,\infty }} + {\left\| {\mathcal {H} } \frac{1}{{\zeta }_{\alpha }} \partial _{\alpha }f(t) \right\| }_{W^{k,\infty }} \lesssim {\Vert f(t) \Vert }_{W^{k+2,\infty }} \, \end{aligned}$$

(11.36)

and for any $2 \le p < \infty$

$$\begin{aligned}&{\Vert {\mathcal {H} } f(t) \Vert }_{W^{k,\infty }} + {\left\| {\mathcal {H} } \frac{1}{{\zeta }_{\alpha }} f(t) \right\| }_{W^{k,\infty }} \lesssim {\Vert f(t) \Vert }_{W^{k+1,p}} + {\Vert f(t) \Vert }_{W^{k+1,\infty }} . \end{aligned}$$

(11.37)

1.2.1 A.2.1 Commutator identities

Let $\mathbf{K}$ be an integral operator of the form

$$\begin{aligned} \mathbf{K}f ({\alpha },t) = \text{ p.v. } \int K({\alpha },{\beta };t) f({\beta },t) \, d{\beta }\end{aligned}$$

(11.38)

with kernel $K({\alpha },{\beta };t)$ or $({\alpha }-{\beta })K({\alpha },{\beta };t)$ continuous and bounded, and $K$ smooth away from the diagonal ${\alpha }={\beta }$. One can easily verify that

$$\begin{aligned} {[\partial _t, \mathbf{K}]} f({\alpha },t)&= \int \partial _t K({\alpha },{\beta };t) f({\beta },t) \, d{\beta },\end{aligned}$$

(11.39a)

$$\begin{aligned} {[\partial _{\alpha }, \mathbf{K}]} f({\alpha },t)&= \int (\partial _{\alpha }+ \partial _{\beta }) K({\alpha },{\beta };t) f({\beta },t) \, d{\beta },\end{aligned}$$

(11.39b)

$$\begin{aligned} {[S, \mathbf{K}]} f({\alpha },t)&= \int \left( {\alpha }\partial _{\alpha }+ {\beta }\partial _{\beta }+ \frac{1}{2}t \partial _t \right) K({\alpha },{\beta };t) f({\beta },t) \, d{\beta }+ \mathbf{K}f ({\alpha },t) , \end{aligned}$$

(11.39c)

for any sufficiently smooth and decaying $f$.

1.2.2 A.2.2 Proof of Lemma 11.5

It is enough to just look at the case of $Q_1$, as the treatment of $Q_2$ is identical. Expanding out the denominator in (11.19) we can write

$$\begin{aligned} \frac{1}{{({\zeta }({\alpha })-{\zeta }({\beta }))}^2} = F \left( \frac{ {\zeta }({\alpha }) - {\alpha }- ({\zeta }({\beta }) - {\beta }) }{{\alpha }-{\beta }} \right) \frac{1}{ {({\alpha }-{\beta })}^2 } \end{aligned}$$

where $F(x) = \sum _{k\ge 0} {(-1)}^k (k+1) x^k$. Then one can see that proving (11.24) can be reduced to proving the following estimate for operators of the type $C_1$ as in (11.16):

$$\begin{aligned} {\left\| \text{ p.v. } \!\int \! F \left( \frac{ H(x) \!-\! H(y) }{x-y} \right) \frac{ A(x) \!-\! A(y) }{ {(x-y)}^2 } f(y) \, dy \right\| }_{L^\infty }&\!\lesssim \! \prod _{i=1}^m {\Vert A \Vert }_{W^{2,\infty }} {\Vert f \Vert }_{W^{1,\infty }} , \end{aligned}$$

(11.40)

where the implicit constant depends on ${\Vert H^{{\prime }{\prime }} \Vert }_{L^\infty }$. To show this we split the integral into two pieces:

$$\begin{aligned}&\int F \left( \frac{ H(x) - H(y) }{x-y} \right) \frac{ A(x) - A(y) }{ {(x-y)}^2 } f(y) \, dy = I_1(x) + I_2(x) \\&I_1(x) = \int _{|x-y| \le 1} F \left( \frac{ H(x) - H(y) }{x-y} \right) \frac{ A(x) - A(y) }{ {(x-y)}^2 } f(y) \, dy \\&I_2(x) = \int _{|x-y| \ge 1} F \left( \frac{ H(x) - H(y) }{x-y} \right) \frac{ A(x) - A(y) }{ {(x-y)}^2 } f(y) \, dy . \end{aligned}$$

We write

$$\begin{aligned} I_1(x)&= \int _{|x-y| \le 1} \left[ F \left( \frac{ H(x) - H(y) }{x-y} \right) - F(H^{\prime }(x)) \right] \frac{ A(x) - A(y)}{ {(x-y)}^2 } f(y) \, dy \\&\quad + F(H^{\prime }(x)) \int _{|x-y| \le 1} \frac{ A(x) - A(y) - A^{\prime }(x) (x-y)}{ {(x-y)}^2 } f(y) \, dy \\&\quad + F(H^{\prime }(x)) \int _{|x-y| \le 1} \frac{A^{\prime }(x)}{x-y} ( f(y) - f(x) )\, dy =: I_{1,1}(x) + I_{1,2}(x) + I_{1,3}(x) . \end{aligned}$$

It is then easy to see that we can then estimate

$$\begin{aligned} | I_{1,1}(x) |&\lesssim {\Vert F^{\prime }\Vert }_{L^\infty } {\Vert H^{{\prime }{\prime }} \Vert }_{L^\infty } {\Vert A^{\prime }\Vert }_{L^\infty } {\Vert f \Vert }_{L^\infty } \\ | I_{1,2}(x) |&\lesssim {\Vert F \Vert }_{L^\infty } {\Vert A^{{\prime }{\prime }} \Vert }_{L^\infty } {\Vert f \Vert }_{L^\infty } \\ | I_{1,3}(x) |&\lesssim {\Vert F^{\prime }\Vert }_{L^\infty } {\Vert A^{\prime }\Vert }_{L^\infty } {\Vert f^{\prime }\Vert }_{L^\infty } \end{aligned}$$

so that ${\Vert I_1 \Vert }_{L^\infty } \lesssim c\left( {\Vert F \Vert }_{W^{1,\infty }}, {\Vert H^{{\prime }{\prime }} \Vert }_{L^\infty } \right) {\Vert A^{\prime }\Vert }_{W^{1,\infty }} {\Vert f \Vert }_{W^{1,\infty }}$. Since $|x-y|^{-2}$ is integrable for $|x-y| \ge 1$ one has

$$\begin{aligned} {\Vert I_2 \Vert }_{L^\infty }&\lesssim c\left( {\Vert F \Vert }_{L^\infty } \right) {\Vert A \Vert }_{L^\infty } {\Vert f \Vert }_{L^\infty } . \end{aligned}$$

The bound (11.40) follows. $\Box$

1.2.3 A.2.3 Proof of Lemma 11.6

We start by calculating

$$\begin{aligned} \partial _{\alpha }Q_0 (f,g)&= \partial _{\alpha }\int \frac{ f({\alpha }) - f({\beta }) }{ {\zeta }({\alpha })-{\zeta }({\beta }) } g({\beta }) \, d{\beta }= - \partial _{\alpha }{\zeta }({\alpha }) \int \frac{ f({\alpha }) - f({\beta }) }{ {({\zeta }({\alpha })-{\zeta }({\beta }))}^2 } g({\beta }) \, d{\beta }\\&\quad + \int \frac{ \partial _{\alpha }f({\alpha }) }{ {\zeta }({\alpha })-{\zeta }({\beta }) } g({\beta }) \, d{\beta }=: Q_0^1({\alpha }) + Q_0^2({\alpha }) . \end{aligned}$$

Since the integral operators in $Q^0_1$ is of the type $\mathbf{Q}$, we can use Lemma 11.5 to bound

$$\begin{aligned} {\Vert Q_0^1 \Vert }_{L^\infty }&\lesssim {\Vert {\zeta }_{\alpha }\Vert }_{L^\infty } {\Vert f \Vert }_{W^{2,\infty }} {\Vert g \Vert }_{W^{1,\infty }} . \end{aligned}$$

The second contribution to $\partial _{\alpha }Q_0 (f,g)$ is

$$\begin{aligned} Q_0^2 = \partial _{\alpha }f \left( {\mathcal {H} } \frac{1}{{\zeta }_{\alpha }} g \right) = \partial _{\alpha }f {\mathcal {H} } g + \partial _{\alpha }f {\mathcal {H} } \left( \frac{1}{{\zeta }_{\alpha }} - 1\right) g \end{aligned}$$

Thus, using also (11.1), we see that

$$\begin{aligned} {\Vert Q_0^2 \Vert }_{L^\infty }&\lesssim {\Vert \partial _{\alpha }f \Vert }_{L^\infty } {\Vert {\mathcal {H} } g \Vert }_{L^\infty } + {\Vert \partial _{\alpha }f \Vert }_{L^\infty } {\left\| {\mathcal {H} } \left( \frac{1}{{\zeta }_{\alpha }} - 1\right) g \right\| }_{H^1} \\&\lesssim {\Vert \partial _{\alpha }f \Vert }_{L^\infty } {\Vert {\mathcal {H} } g \Vert }_{L^\infty } + {\Vert \partial _{\alpha }f \Vert }_{L^\infty } c( {\Vert {\zeta }_{\alpha }- 1 \Vert }_{H^3}) {\Vert g \Vert }_{W^{1,\infty }}. \end{aligned}$$

We conclude that $Q_0^2$ satisfies the desired bound and so does $\partial _{\alpha }Q_0 (f,g)$. $\Box$

1.2.4 A.2.4 Proof of Proposition 11.7

Proof of (11.28). We want to show that for any two functions $f$ and $g$

$$\begin{aligned} {\Vert \mathbf{Q}(f, g) \Vert }_{W^{k,\infty }}&\le c {\Vert f \Vert }_{W^{k+2,\infty }} {\Vert g \Vert }_{W^{k+2,\infty }} . \end{aligned}$$

(11.41)

This can be shown by induction, using (11.24) as the base of the induction. Again, it is enough to just look at the case of $Q_1$. Let us assume that (11.41) holds true for some $1 \le k\le \frac{N_0}{2}+1$. We want to show the estimate for $k+1$. Notice that we can write $\mathbf{Q}$ as an operator of the form $\mathbf{K}$, see (11.38), with Kernel

$$\begin{aligned} K({\alpha },{\beta };t) = \frac{f({\alpha }) - f({\beta }) }{ {({\zeta }({\alpha })-{\zeta }({\beta }))}^2 } . \end{aligned}$$

Using the commutation identity (11.39b) we see that

$$\begin{aligned} \partial _{\alpha }\mathbf{Q}(f,g)&= \mathbf{Q}(f, \partial _{\alpha }g) + \mathbf{Q}(\partial _{\alpha }f,g) + I({\alpha }) \end{aligned}$$

where

$$\begin{aligned} I({\alpha }) = - \int \frac{ (f({\alpha }) - f({\beta }))( \partial _{\alpha }{\zeta }({\alpha }) - \partial _{\beta }{\zeta }({\beta })) }{ {({\zeta }({\alpha })-{\zeta }({\beta }))}^3 } g({\beta }) \,d{\beta }. \end{aligned}$$

Using the inductive hypothesis we have

$$\begin{aligned}&{\Vert \mathbf{Q}(\partial _{\alpha }f, g) \Vert }_{W^{k,\infty }} + {\Vert \mathbf{Q}(f, \partial _{\alpha }g) \Vert }_{W^{k,\infty }} \le c {\Vert \partial _{\alpha }f \Vert }_{W^{k+2,\infty }} {\Vert g \Vert }_{W^{k+2,\infty }}\\&\quad + c {\Vert f \Vert }_{W^{k+2,\infty }} {\Vert \partial _{\alpha }g \Vert }_{W^{k+2,\infty }}. \end{aligned}$$

By expanding the denominator in the integral defining $I$, we see that $I$ is an operator of the form $C_1(H , A, g )$, see (11.16), with $A = (f, {\zeta }_{\alpha }-1)$ and $H = {\zeta }- \text{ id }$. Letting $k+1 = k_1 + k_2 + k_3$, and using (11.39b), we see that $D^{k+1} I$ is a sum of operators of the form

$$\begin{aligned} C_1 \left( {\zeta }- {\alpha }, A_{k_2,k_3}, D^{k_1} g) \right) \end{aligned}$$

where $A_{k_2,k_3} = (\widetilde{A}_{k_3}, D^{k_2} f)$, and $\widetilde{A}_{k_3}$ is a vector with at most $k_3$ components satisfying

$$\begin{aligned} {\Vert A_{k_3}^{\prime }\Vert }_{L^p} \lesssim {\Vert D^{k_3 + 1} {\zeta }_{\alpha }\Vert }_{L^p} \end{aligned}$$

for $p=2,\infty$. Applying Theorem 11.3 we see that:

$$\begin{aligned} {\Vert I \Vert }_{H^{k+2,\infty }}&\!\le \! c {\Vert \partial _{\alpha }f \Vert }_{W^{k+2,\infty }} {\Vert {\zeta }_{\alpha }\!-\!1\Vert }_{H^{k+2}} {\Vert g \Vert }_{W^{k+2,\infty }} \!\le \! c {\Vert f \Vert }_{W^{k+3,\infty }} {\Vert g \Vert }_{W^{k+2,\infty }} , \end{aligned}$$

where the constant $c$ depends only on ${\Vert {\zeta }_{\alpha }\Vert }_{H^{N_1}}$. We can then deduce

$$\begin{aligned}&{\Vert \mathbf{Q}(f,g) \Vert }_{W^{k+1,\infty }} \le {\Vert \mathbf{Q}(\partial _{\alpha }f,g) \Vert }_{W^{k+1,\infty }} + {\Vert \mathbf{Q}(f, \partial _{\alpha }g) \Vert }_{W^{k+1,\infty }} + {\Vert I \Vert }_{W^{k+1,\infty }}\\&\quad \le c {\Vert f \Vert }_{W^{k+3,\infty }} {\Vert g \Vert }_{W^{k+3,\infty }} \end{aligned}$$

which is exactly (11.41) with $k+1$ replacing $k$.

Proof of (11.30). Let us first look at the $H^k$ component of the $X_k$ norm. Since

$$\begin{aligned} Q_0(f,g) = {\mathcal {H} } \frac{1}{{\zeta }_{\alpha }} (fg) - f {\mathcal {H} } \frac{1}{{\zeta }_{\alpha }} g \end{aligned}$$

(11.42)

we can use product Sobolev estimates and the $H^k$ bounds on the Hilbert transform (11.1) to obtain

$$\begin{aligned} {\Vert Q_0(f,g) \Vert }_{H^k}&\lesssim {\Vert f g \Vert }_{H^k} + {\Vert {\zeta }_{\alpha }- 1 \Vert }_{H^k} {\Vert f g \Vert }_{W^{\frac{k}{2}+1,\infty }} + {\Vert f \Vert }_{H^k} {\Vert {\mathcal {H} } g \Vert }_{L^\infty } \\&\quad + {\Vert f \Vert }_{L^\infty } {\Vert {\mathcal {H} } g \Vert }_{H^k} \lesssim {\Vert f \Vert }_{H^k} {\Vert ( {\mathcal {H} } g, g) \Vert }_{L^\infty } \\&\quad + {\Vert f \Vert }_{L^\infty } {\Vert g \Vert }_{H^k} + {\Vert {\zeta }_{\alpha }- 1 \Vert }_{H^k} {\Vert f \Vert }_{W^{\frac{k}{2}+1,\infty }} {\Vert g \Vert }_{W^{\frac{k}{2}+1,\infty }} \end{aligned}$$

where the implicit constants depend only on ${\Vert {\zeta }_{\alpha }-1\Vert }_{H^{N_1}}$.

A similar argument can be used to bound the $S^{-1} H^k$ norm of $Q_0(f,g)$ for $0 \le k \le \frac{N_0}{2}$. First we observe that for any $0 \le k \le \frac{N_0}{2}$ one has

$$\begin{aligned} \nonumber {\Vert S Q_0(f,g) \Vert }_{H^k}&\le {\left\| S {\mathcal {H} } \frac{1}{{\zeta }_{\alpha }} (f g) \right\| }_{H^k} + {\left\| S f {\mathcal {H} } \frac{1}{{\zeta }_{\alpha }} g \right\| }_{H^k} + {\left\| f S {\mathcal {H} } \frac{1}{{\zeta }_{\alpha }} g \right\| }_{H^k} \\&\le {\left\| {\mathcal {H} } \frac{1}{{\zeta }_{\alpha }} (f g) \right\| }_{X_k} + {\Vert f \Vert }_{X_k} {\left\| {\mathcal {H} } \frac{1}{{\zeta }_{\alpha }} g \right\| }_{W^{k,\infty }} + {\Vert f \Vert }_{W^{k,\infty }} {\left\| {\mathcal {H} } \frac{1}{{\zeta }_{\alpha }} g \right\| }_{X_k} . \end{aligned}$$

(11.43)

We can then use (11.2) to obtain

$$\begin{aligned} {\left\| {\mathcal {H} } \frac{1}{{\zeta }_{\alpha }} (f g) \right\| }_{X_k}&\le c {\Vert f g \Vert }_{X_k} + {\Vert {\zeta }_{\alpha }- 1 \Vert }_{X_k} {\Vert f g \Vert }_{W^{\frac{k}{2}+1,\infty }} \\ {\left\| {\mathcal {H} } \frac{1}{{\zeta }_{\alpha }} g \right\| }_{X_k}&\le c {\Vert g \Vert }_{X_k} + {\Vert {\zeta }_{\alpha }- 1 \Vert }_{X_k} {\Vert g \Vert }_{W^{\frac{k}{2}+1,\infty }}. \end{aligned}$$

Since we also have

$$\begin{aligned} {\Vert f g \Vert }_{X_k}&\le c {\Vert f \Vert }_{X_k} {\Vert g \Vert }_{W^{k,\infty }} + {\Vert f \Vert }_{W^{k,\infty }} {\Vert g \Vert }_{X_k} \end{aligned}$$

we can plug the above bounds in (11.43) and get the desired conclusion.

Proof of (11.31). Let us start again with the $H^k$ component of the $X_k$ norm. First of all observe that $Q_0 (f, \partial _{\alpha }g)$ is an operator of the form $C_2({\zeta }-{\alpha }, f ,g)$, see (11.17). Distributing derivatives on $Q_0(f, \partial _{\alpha }g)$ by using (11.39b), we see that for any integer $k = k_1 + k_2 + k_3$, we have that $D^k Q_0(f, \partial _{\alpha }g)$ is a sum of operators of the form

$$\begin{aligned} C_2 \left( {\zeta }- {\alpha }, A_{k_2,k_3}, D^{k_1} g) \right) \end{aligned}$$

where $A_{k_2,k_3} = (\widetilde{A}_{k_3}, D^{k_2} f)$, and $\widetilde{A}_{k_3}$ is a vector with at most $k_3$ components satisfying

$$\begin{aligned} {\Vert A_{k_3}^{\prime }\Vert }_{L^p} \lesssim {\Vert D^{k_3 + 1} {\zeta }_{\alpha }\Vert }_{L^p} \end{aligned}$$

for $p=2,\infty$. One can then apply Theorem 11.3 to deduce that the $H^k$-norm of $Q_0(f, \partial _{\alpha }g)$ is bounded by the right-hand side of (11.31) for any $0 \le k \le N_0$.

The estimate for ${\Vert S Q_0(f, \partial _{\alpha }g) \Vert }_{H^k}$, for $0 \le k \le \frac{N_0}{2}$, follows similarly by using the commutation identity (11.39c). Indeed, applying $S$ to $Q_0(f, \partial _{\alpha }g) \sim C_2 ({\zeta }-{\alpha }, f, g)$, and commuting $S$ and $\partial _{\alpha }$ when $S$ falls on $\partial _{\alpha }g$, one obtains operators of the form $C_2 ({\zeta }-{\alpha }, S f, g)$, $C_2 ({\zeta }-{\alpha }, (f, S {\zeta }_{\alpha }), g)$, $C_2 ({\zeta }-{\alpha }, f, S g)$ or $C_2 ({\zeta }-{\alpha }, f, g)$ itself. Applying and distributing $k$ derivatives as above, one can then estimate the resulting expressions in $L^2$ via Theorem 11.3, eventually obtaining the desired bound.

Proof of (11.32). This estimate follows from the same proof of Lemma 11.6, which is the case $l=0$, after applying and commuting $k$ derivatives similarly to what has been already done before. Since the proof is straightforward, we skip it.

Proof of (11.34). Let us start by showing the $H^m$ estimate

$$\begin{aligned}&{\Vert \mathbf{T}(f, g, h) \Vert }_{H^m} + {\Vert \mathbf{T}(f, g, \partial _{\alpha }h) \Vert }_{H^m} \le c {\Vert (f,g,h,{\zeta }_{\alpha }-1) \Vert }_{H^m}\nonumber \\&\quad {\Vert (f,g,h,{\zeta }_{\alpha }-1) \Vert }_{W^{\frac{N_0}{2}+3,\infty }}^2 , \end{aligned}$$

(11.44)

for all integers $0 \le m \le N_0$. Again we will us induction and commutation identities. The base for the induction is given by the estimates

$$\begin{aligned} {\Vert \mathbf{T}(f, g, h) \Vert }_{L^2}&\le c {\Vert (f,g,h) \Vert }_{L^2} {\Vert (f,g,h) \Vert }_{W^{\frac{N_0}{2}+3,\infty }}^2 , \end{aligned}$$

(11.45)

$$\begin{aligned} {\Vert \mathbf{T}(f, g, \partial _{\alpha }h) \Vert }_{L^2}&\le c {\Vert (f,g,h)\Vert }_{L^2} {\Vert (f,g,h) \Vert }_{W^{\frac{N_0}{2}+3,\infty }}^2 . \end{aligned}$$

(11.46)

To verify these we cannot use directly Theorem 11.3. We instead write

$$\begin{aligned} \mathbf{T}(f, g, h)&= f \mathbf{Q}(g, h) - \mathbf{Q}(g, f h) \\ \mathbf{T}(f, g, \partial _{\alpha }h)&= f \mathbf{Q}(g, \partial _{\alpha }h) - \mathbf{Q}(g, f \partial _{\alpha }h) . \end{aligned}$$

From Theorem 11.4 we have

$$\begin{aligned} {\Vert \mathbf{Q}(a, b) \Vert }_{L^2} \le c {\Vert \partial _{\alpha }a \Vert }_{L^\infty } {\Vert b \Vert }_{L^2} . \end{aligned}$$

Thus, using (11.29), we obtain

$$\begin{aligned} {\Vert \mathbf{T}(f,g,h) \Vert }_{L^2}&\le {\Vert f \mathbf{Q}(g, h) \Vert }_{L^2} + {\Vert \mathbf{Q}(g, f h) \Vert }_{L^2} \lesssim {\Vert f \Vert }_{L^2} {\Vert \mathbf{Q}(g, h) \Vert }_{L^\infty } \\&+ {\Vert \partial _{\alpha }g \Vert }_{L^\infty } {\Vert f h \Vert }_{L^2} \lesssim {\Vert (f,g,h) \Vert }_{L^2} {\Vert (f,g,h) \Vert }^2_{W^{2,\infty }} . \end{aligned}$$

Similarly we have

$$\begin{aligned} {\Vert \mathbf{T}(f,g,\partial _{\alpha }h) \Vert }_{L^2}&\le {\Vert f \mathbf{Q}(g, \partial _{\alpha }h) \Vert }_{L^2} + {\Vert \mathbf{Q}(g, f \partial _{\alpha }h) \Vert }_{L^2} \lesssim {\Vert f \Vert }_{L^2} {\Vert \mathbf{Q}(g, \partial _{\alpha }h) \Vert }_{L^\infty } \\&+ {\Vert \partial _{\alpha }g \Vert }_{L^\infty } {\Vert f \partial _{\alpha }h \Vert }_{L^2} \lesssim {\Vert (f,g,h) \Vert }_{L^2} {\Vert (f,g,h) \Vert }^2_{W^{3,\infty }} . \end{aligned}$$

Now let us assume that (11.44) holds true for some integer $0 \le l \le m-1$. Using the commutation identity (11.39b) we see that

$$\begin{aligned} \partial _{\alpha }\mathbf{T}(f,g,h)&= \mathbf{T}(\partial _{\alpha }f,g,h) + \mathbf{T}(f, \partial _{\alpha }g, h) + \mathbf{T}(f, g, \partial _{\alpha }h) + J_1({\alpha }) \end{aligned}$$

where

$$\begin{aligned} J_1({\alpha }) = - \int \frac{ (f({\alpha }) - f({\beta }))(g({\alpha }) - g({\beta })) ( \partial _{\alpha }{\zeta }({\alpha }) - \partial _{\beta }{\zeta }({\beta })) }{ {({\zeta }({\alpha })-{\zeta }({\beta }))}^3 } h({\beta }) \,d{\beta }. \end{aligned}$$

Since

$$\begin{aligned} \mathbf{T}(\partial _{\alpha }f,g,h) = \partial _{\alpha }f \mathbf{Q}(g, h) - \mathbf{Q}(g, \partial _{\alpha }f h) \end{aligned}$$

we have

$$\begin{aligned} {\Vert \mathbf{T}(\partial _{\alpha }f,g,h) \Vert }_{H^l}&\le {\Vert \partial _{\alpha }f \Vert }_{H^l} {\Vert \mathbf{Q}(g, h) \Vert }_{L^\infty } + {\Vert \partial _{\alpha }f \Vert }_{L^\infty } {\Vert \mathbf{Q}(g, h) \Vert }_{H^l}\nonumber \\&\quad + {\Vert \mathbf{Q}(g, \partial _{\alpha }f h) \Vert }_{H^l} . \end{aligned}$$

(11.47)

From Theorem 11.4, and commutation identities, it is not hard to see that

$$\begin{aligned}&{\Vert \mathbf{Q}(a, b) \Vert }_{H^l} \le c {\Vert a \Vert }_{W^{\frac{l}{2}+2,\infty }} {\Vert b \Vert }_{H^l} + c {\Vert \partial _{\alpha }a \Vert }_{H^l} {\Vert b \Vert }_{W^{\frac{l}{2}+2,\infty }}\nonumber \\&\quad + {\Vert a \Vert }_{W^{\frac{l}{2}+2,\infty }} {\Vert b \Vert }_{W^{\frac{l}{2}+2,\infty }} {\Vert {\zeta }_{\alpha }- 1 \Vert }_{H^l} . \end{aligned}$$

(11.48)

We can then use the above estimate and (11.29) to bound the right-hand side of (11.47) and obtain

$$\begin{aligned}&{\Vert \mathbf{T}(\partial _{\alpha }f, g, h) \Vert }_{H^l} \\&\quad \lesssim {\Vert \partial _{\alpha }f \Vert }_{H^l} {\Vert (f,g,h) \Vert }^2_{W^{2,\infty }} + {\Vert \partial _{\alpha }f \Vert }_{L^\infty } {\Vert (g,h) \Vert }_{H^{l+1}} {\Vert (g,h) \Vert }_{W^{\frac{l}{2}+2,\infty }} \\&\quad \quad + {\Vert (\partial _{\alpha }f,g,h) \Vert }^2_{W^{\frac{l}{2}+2,\infty }} {\Vert {\zeta }_{\alpha }- 1 \Vert }_{H^l} + {\Vert g \Vert }_{W^{\frac{l}{2}+2,\infty }} {\Vert \partial _{\alpha }f h \Vert }_{H^l} \\&\qquad + {\Vert \partial _{\alpha }g \Vert }_{H^l} {\Vert \partial _{\alpha }f h \Vert }_{W^{\frac{l}{2}+2,\infty }}\lesssim {\Vert (f,g,h,{\zeta }_{\alpha }-1) \Vert }_{H^{l+1}} {\Vert (f,g,h) \Vert }^2_{W^{\frac{l}{2}+3,\infty }} . \end{aligned}$$

An identical bound clearly holds for $\mathbf{T}(f, \partial _{\alpha }g, h)$. Since $l \le N_0$ we have then obtained

$$\begin{aligned}&{\Vert \partial _{\alpha }\mathbf{T}(f, g, h) - \mathbf{T}(f, g, \partial _{\alpha }h) - J_1 \Vert }_{H^l}\\&\quad \le c {\Vert (f,g,h,{\zeta }_{\alpha }-1) \Vert }_{H^{l+1}} {\Vert (f,g,h) \Vert }_{W^{\frac{N_0}{2}+3,\infty }}^2 . \end{aligned}$$

To estimate

$$\begin{aligned} \mathbf{T}(f, g, \partial _{\alpha }h) = \int \frac{ (f({\alpha }) - f({\beta }))(g({\alpha }) - g({\beta })) }{ {({\zeta }({\alpha })-{\zeta }({\beta }))}^2 } \partial _{\beta }h({\beta }) \,d{\beta }\end{aligned}$$

we need to get rid of the extra derivative falling on $h$. Integrating by parts in ${\beta }$ we have

$$\begin{aligned} \mathbf{T}(f, g, \partial _{\alpha }h)&= \mathbf{Q}(g, h \partial _{\alpha }f) + \mathbf{Q}(f, h \partial _{\alpha }g) + J_2({\alpha }) \end{aligned}$$

where

$$\begin{aligned} J_2({\alpha })&= - 2 \int \frac{ (f({\alpha }) - f({\beta }))(g({\alpha }) - g({\beta })) }{ {({\zeta }({\alpha })-{\zeta }({\beta }))}^3 } {\zeta }_{\beta }({\beta }) h({\beta }) \,d{\beta }. \end{aligned}$$

Using (11.48) we can bound

$$\begin{aligned} {\Vert \mathbf{Q}(g, h\partial _{\alpha }f) \Vert }_{H^l} + {\Vert \mathbf{Q}(f, h\partial _{\alpha }g) \Vert }_{H^l} \le c {\Vert (f,g,h,{\zeta }_{\alpha }-1) \Vert }_{H^{l+1}} {\Vert (f,g,h) \Vert }_{W^{\frac{N_0}{2}+3,\infty }}^2 \end{aligned}$$

as desired. To bound $J_2$, which is an operator of the form $C_1({\zeta }-\text{ id }, (f,g), {\zeta }_{\alpha }h)$, we can again commute derivatives via (11.39b) and apply Theorem 11.3 to obtain:

$$\begin{aligned} {\Vert J_2 \Vert }_{H^l}&\le c {\Vert (f,g,h,{\zeta }_{\alpha }-1)\Vert }_{H^{l+1}} {\Vert (f,g,h) \Vert }_{W^{\frac{N_0}{2}+3,\infty }}^2 . \end{aligned}$$

We have then shown

$$\begin{aligned} {\Vert \partial _{\alpha }\mathbf{T}(f, g, h) - J_1 \Vert }_{H^l}&\le c {\Vert (f,g,h,{\zeta }_{\alpha }-1) \Vert }_{H^{l+1}} {\Vert (f,g,h) \Vert }_{W^{\frac{N_0}{2}+3,\infty }}^2 . \end{aligned}$$

To eventually estimate $J_1$ we notice that

$$\begin{aligned} J_1({\alpha }) = - {\zeta }_{\alpha }\mathbf{T}(f, g, h) + \mathbf{T}(f, g, {\zeta }_{\alpha }h) , \end{aligned}$$

so that

$$\begin{aligned} {\Vert J_1 \Vert }_{H^l} \le c {\Vert \mathbf{T}(f, g, h) \Vert }_{H^l} + {\Vert \mathbf{T}(f, g, {\zeta }_{\alpha }h) \Vert }_{H^l} . \end{aligned}$$

Using the inductive hypotheses we see that

$$\begin{aligned} {\Vert J_1 \Vert }_{H^l}&\le c {\Vert (f,g,h,{\zeta }_{\alpha }-1) \Vert }_{H^l} {\Vert (f,g,h) \Vert }_{W^{\frac{N_0}{2}+3,\infty }}^2\\&\quad \quad + c {\Vert (f,g, {\zeta }_{\alpha }h,{\zeta }_{\alpha }-1) \Vert }_{H^l} {\Vert (f,g, {\zeta }_{\alpha }h) \Vert }_{W^{\frac{N_0}{2}+3,\infty }}^2 \\&\le c {\Vert (f,g,h,{\zeta }_{\alpha }-1) \Vert }_{H^l} {\Vert (f,g,h)\Vert }_{W^{\frac{N_0}{2}+3,\infty }}^2 \end{aligned}$$

where the constant $c$ depends only on lower Sobolev norms of $(f,g,h,{\zeta }_{\alpha }-1)$, which are uniformly bounded by assumption. It follows that

$$\begin{aligned} {\Vert \partial _{\alpha }\mathbf{T}(f, g, h) \Vert }_{H^l}&\le c {\Vert (f,g,h,{\zeta }_{\alpha }-1) \Vert }_{H^{l+1}} {\Vert (f,g,h) \Vert }_{W^{\frac{N_0}{2}+3,\infty }}^2 , \end{aligned}$$

which gives the bound on the first term on the left-hand side of (11.44).

To complete the proof of (11.44) we need to show

$$\begin{aligned} {\Vert \mathbf{T}(f, g, \partial _{\alpha }h) \Vert }_{H^m}&\le c {\Vert (f,g,h,{\zeta }_{\alpha }-1) \Vert }_{H^m} {\Vert (f,g,h) \Vert }^2_{W^{\frac{N}{2}+3,\infty }} . \end{aligned}$$

Again we proceed by induction, the base being given by (11.46) which has already been verified. The argument is similar to those above. Applying a derivative to $\mathbf{T}(f, g, \partial _{\alpha }h)$ we get

$$\begin{aligned} \partial _{\alpha }\mathbf{T}(f, g, \partial _{\alpha }h)&= \mathbf{T}(\partial _{\alpha }f, g, \partial _{\alpha }h) + \mathbf{T}(f, \partial _{\alpha }g, \partial _{\alpha }h) + \mathbf{T}(f, g, \partial ^2_{\alpha }h) + J_3({\alpha }) \end{aligned}$$

where

$$\begin{aligned} J_3({\alpha }) = - 2 \int \frac{ (f({\alpha }) - f({\beta }))(g({\alpha }) - g({\beta })) ( \partial _{\alpha }{\zeta }({\alpha }) - \partial _{\beta }{\zeta }({\beta })) }{ {({\zeta }({\alpha })-{\zeta }({\beta }))}^3 } \partial _{\beta }h({\beta }) \,d{\beta }. \end{aligned}$$

Since

$$\begin{aligned} \mathbf{T}(\partial _{\alpha }f, g, \partial _{\alpha }h) = \partial _{\alpha }f \mathbf{Q}(g, \partial _{\alpha }h) - \mathbf{Q}(g, \partial _{\alpha }f \partial _{\alpha }h) \end{aligned}$$

this term can be directly estimated using (11.28) and (11.48). One can bound similarly $\mathbf{T}(f, \partial _{\alpha }g, \partial _{\alpha }h)$.

To control $\mathbf{T}(f, g, \partial ^2_{\alpha }h)$ we need to resort again to an integration by parts to remove the presence of the extra derivative. More precisely we have

$$\begin{aligned} \mathbf{T}(f, g, \partial _{\alpha }^2 h)&= \mathbf{Q}(g, \partial _{\alpha }h \partial _{\alpha }f) + \mathbf{Q}(f, \partial _{\alpha }h \partial _{\alpha }g) + J_4({\alpha }) \end{aligned}$$

where

$$\begin{aligned} J_4({\alpha })&= - 2 \int \frac{ (f({\alpha }) - f({\beta }))(g({\alpha }) - g({\beta })) }{ {({\zeta }({\alpha })-{\zeta }({\beta }))}^3 } {\zeta }_{\beta }({\beta }) \partial _{\beta }h({\beta }) \,d{\beta }. \end{aligned}$$

The terms $\mathbf{Q}(g, \partial _{\alpha }h \partial _{\alpha }f)$ and $\mathbf{Q}(f, \partial _{\alpha }h \partial _{\alpha }g)$ can be estimated via (11.48):

$$\begin{aligned}&{\Vert \mathbf{Q}(g, \partial _{\alpha }h \partial _{\alpha }f) \Vert }_{H^l} + {\Vert \mathbf{Q}(f, \partial _{\alpha }h \partial _{\alpha }g) \Vert }_{H^l}\\&\quad \le c {\Vert (f,g,h,{\zeta }_{\alpha }-1) \Vert }_{H^{l+1}} {\Vert (f,g,h) \Vert }^2_{W^{\frac{N_0}{2}+3,\infty }} . \end{aligned}$$

Similarly to what has been done before, we can expand the factor $({\zeta }({\alpha })-{\zeta }({\beta }))^{-3}$, and write $J_4$ as an operator of the type $C_1$ as in (11.16). By using the commutation identity (11.39b) we can then bound it by

$$\begin{aligned} {\Vert J_4 \Vert }_{H^l}&\le c {\Vert \partial _{\alpha }(f,g,h,{\zeta }_{\alpha }-1) \Vert }_{H^l} {\Vert \partial _{\alpha }(f,g,h) \Vert }^2_{W^{\frac{l}{2}+2,\infty }}\\&\le c {\Vert (f,g,h,{\zeta }_{\alpha }-1) \Vert }_{H^{l+1}} {\Vert (f,g,h) \Vert }^2_{W^{\frac{N_0}{2}+3,\infty }} . \end{aligned}$$

This shows that

$$\begin{aligned} {\Vert \partial _{\alpha }\mathbf{T}(f, g, \partial _{\alpha }h) - J_3 \Vert }_{H^l}&\le c {\Vert (f,g,h,{\zeta }_{\alpha }-1) \Vert }_{H^{l+1}} {\Vert (f,g,h) \Vert }_{W^{\frac{N_0}{2}+3,\infty }}^2 . \end{aligned}$$

To eventually bound $J_3$ in $H^l$ notice that it can be written as follows:

$$\begin{aligned} J_3 = - 2 \partial _{\alpha }{\zeta }\mathbf{T}(f, g, \partial _{\alpha }h) - J_4 . \end{aligned}$$

Using the inductive hypothesis for the first summand above, and the bound we have already obtained for $J_4$, one can easily see how the desired bound for $J_3$ follows. This eventually yields

$$\begin{aligned} {\Vert \partial _{\alpha }\mathbf{T}(f, g, \partial _{\alpha }h) \Vert }_{H^l}&\le c {\Vert (f,g,h,{\zeta }_{\alpha }-1) \Vert }_{H^{l+1}} {\Vert (f,g,h) \Vert }_{W^{\frac{N_0}{2}+3,\infty }}^2 , \end{aligned}$$

thereby completing the proof of (11.44).

We now prove the estimate in the space $S^{-1} H^k$ with $k := [\frac{m}{2}]$, $0 \le m \le N_0$:

$$\begin{aligned} {\Vert S \mathbf{T}(f, g, h) \Vert }_{H^k} + {\Vert S \mathbf{T}(f, g, \partial _{\alpha }h) \Vert }_{H^k}&\le c {\Vert S L \Vert }_{H^k} {\Vert L \Vert }_{W^{\frac{N_0}{2}+2,\infty }}^2 . \end{aligned}$$

(11.49)

For simplicity we just show the proof of the bound for the second term in the above right-hand side. The first term can be bounded similarly, and it is actually easier to estimate, since there is one less derivative on $h$ to worry about. Let us start by computing $S \mathbf{T}(f, g, \partial _{\alpha }h)$ in $L^2$. By using the commutation identity (11.39c), and $[S, \partial _{\alpha }] = - \partial _{\alpha }$, we see that

$$\begin{aligned} S \mathbf{T}(f, g, \partial _{\alpha }h)&= \mathbf{T}(S f, g, \partial _{\alpha }h) + \mathbf{T}(f, S g, \partial _{\alpha }h) + \mathbf{T}(f, g, \partial _{\alpha }S h) + K_1({\alpha }) \end{aligned}$$

where

$$\begin{aligned} K_1 ({\alpha })&= - 2 \int \frac{ (f({\alpha }) - f({\beta }))(g({\alpha }) - g({\beta })) (S{\zeta }({\alpha }) - S{\zeta }({\beta })) }{ {({\zeta }({\alpha })-{\zeta }({\beta }))}^3 } \partial _{\beta }h({\beta }) \,d{\beta }. \end{aligned}$$

Notice that we can write

$$\begin{aligned} \mathbf{T}(S f, g, \partial _{\alpha }h) = S f \mathbf{Q}(g,\partial _{\alpha }h) - \mathbf{Q}(g, S f \partial _{\alpha }h) . \end{aligned}$$

Then, using the $W^{l,\infty }$ estimate (11.29) and the $H^l$ estimate (11.48) for $\mathbf{Q}$, we see that for any $l \le k$:

$$\begin{aligned} {\Vert \mathbf{T}(S f, g, \partial _{\alpha }h) \Vert }_{H^l}&\le {\Vert S f \Vert }_{H^l} {\Vert \mathbf{Q}(g,\partial _{\alpha }h) \Vert }_{W^{l,\infty }} + {\Vert \mathbf{Q}(g, S f \partial _{\alpha }h) \Vert }_{H^l} \\&\le c {\Vert S f \Vert }_{H^l} {\Vert (g ,\partial _{\alpha }h) \Vert }_{W^{l+2,\infty }}^2 + {\Vert \partial _{\alpha }g \Vert }_{W^{l,\infty }} {\Vert S f \partial _{\alpha }h \Vert }_{H^l}\\&\lesssim {\Vert S f \Vert }_{H^l} {\Vert (g,h) \Vert }_{W^{l+3,\infty }}^2 . \end{aligned}$$

An analogous bound holds for $\mathbf{T}(f, S g, \partial _{\alpha }h)$.

To control $\mathbf{T}(f, g, \partial _{\alpha }S h)$ we need to integrate by parts in order to remove the derivative from $S h$. This integration by parts gives:

$$\begin{aligned} \mathbf{T}(f, g, \partial _{\alpha }S h) = \mathbf{Q}(g, \partial _{\alpha }f S h) + \mathbf{Q}(f, \partial _{\alpha }g S h) + K_2({\alpha }) \end{aligned}$$

where

$$\begin{aligned} K_2 ({\alpha })&= - 2 \int \frac{ (f({\alpha }) - f({\beta }))(g({\alpha }) - g({\beta })) }{ {({\zeta }({\alpha })-{\zeta }({\beta }))}^3 } \partial _{\beta }{\zeta }({\beta }) h({\beta }) \,d{\beta }. \end{aligned}$$

The $\mathbf{Q}$ terms can be treated as before, and therefore satisfy the desired bound. Thus, so far we have obtained

$$\begin{aligned} {\Vert S \mathbf{T}(f, g, \partial _{\alpha }h) - K_1 - K_2 \Vert }_{H^l}&\le c {\Vert S (f,g,h) \Vert }_{H^l} {\Vert (f,g,h) \Vert }_{W^{l+2,\infty }}^2 \end{aligned}$$

for any $l \le k$.

To conclude we notice that $K_1$, respectively $K_2$, are operators of the form $C_1(H,A,f)$ as in (11.16), for some smooth $F$, $H = {\zeta }- \text{ id }$, and $(A,f) = (f,g,S{\zeta }, \partial _{\alpha }h)$, respectively $(A,f)=(f,g, {\zeta }_{\alpha }h)$. Commuting derivatives, using Theorem 11.3 and the assumptions, we can deduce that

$$\begin{aligned} {\Vert K_1 \Vert }_{H^l} + {\Vert K_2 \Vert }_{H^l}&\le c {\Vert \partial _{\alpha }f \Vert }_{H^l} {\Vert \partial _{\alpha }g \Vert }_{W^{l,\infty }} {\Vert \partial _{\alpha }S {\zeta }\Vert }_{H^l} {\Vert \partial _{\alpha }h \Vert }_{W^{l,\infty }}\\&\quad + c {\Vert \partial _{\alpha }f \Vert }_{H^l} {\Vert \partial _{\alpha }g \Vert }_{W^{l,\infty }} {\Vert {\zeta }_{\alpha }h \Vert }_{W^{l,\infty }} \\&\le c {\Vert (f,g,h) \Vert }_{W^{l+1,\infty }}^2 {\Vert (\partial _{\alpha }f, S({\zeta }_{\alpha }-1)) \Vert }_{H^l} \end{aligned}$$

where $c$ depends only on the $H^{l+2}$-norm of $(f,g,h, {\zeta }_{\alpha }-1)$, which is uniformly bounded by assumptions. Here we have used $\partial _{\alpha }S {\zeta }= S ({\zeta }_{\alpha }- 1) + \partial _{\alpha }{\zeta }$ for the second inequality. We can then conclude that

$$\begin{aligned} {\Vert S \mathbf{T}(f, g, \partial _{\alpha }h) \Vert }_{H^l}&\le c {\Vert S (f,g,h,{\zeta }_{\alpha }-1) \Vert }_{H^l} {\Vert (f,g,h) \Vert }_{W^{l+3,\infty }}^2 \end{aligned}$$

(11.50)

for any $l \le k$. This shows the validity of (11.49) and finishes the proof of (11.34). $\square$

1.2.5 A.2.5 Proof of Lemma 11.8

We want to show that for any $f$ with ${\Vert f \Vert }_{H^{k+2}} \le 1$, $0\le k \le N_1$, we have

$$\begin{aligned} {\Vert {\mathcal {H} } \partial _{\alpha }f \Vert }_{W^{k,\infty }} \lesssim {\Vert f \Vert }_{W^{k+2,\infty }} . \end{aligned}$$

(11.51)

From the definition of ${\mathcal {H} }$ we can write $i\pi {\mathcal {H} } \partial _{\alpha }f = I_1 + I_2$ with

$$\begin{aligned} I_1 ({\alpha })&= \int \frac{ \partial _{\beta }f({\beta }) }{ {\zeta }({\alpha })-{\zeta }({\beta }) } \,d{\beta }\\ I_2 ({\alpha })&= \int \frac{ \partial _{\beta }f({\beta }) }{ {\zeta }({\alpha })-{\zeta }({\beta }) } (\partial _{\beta }{\zeta }({\beta }) - 1) \,d{\beta }= {\mathcal {H} } \frac{\partial _{\alpha }f ({\zeta }_{\alpha }-1)}{{\zeta }_{\alpha }} . \end{aligned}$$

$I_2$ is a quadratic term and can be directly estimated using Sobolev’s embedding and the boundedness of ${\mathcal {H} }$ on Sobolev spaces (11.1):

$$\begin{aligned} {\Vert I_2 \Vert }_{W^{k,\infty }}&\lesssim {\Vert I_2 \Vert }_{H^{k+1}} \lesssim {\Vert \partial _{\alpha }f ({\zeta }_{\alpha }-1) \Vert }_{H^{k+1}} + {\Vert {\zeta }_{\alpha }- 1 \Vert }_{H^{k+1}} {\Vert \partial _{\alpha }f ({\zeta }_{\alpha }-1) \Vert }_{W^{\frac{k}{2}+1,\infty }} \\&\lesssim {\Vert {\zeta }_{\alpha }-1 \Vert }_{H^{k+1}} {\Vert f \Vert }_{W^{k+2,\infty }} \lesssim {\Vert f \Vert }_{W^{k+2,\infty }} \end{aligned}$$

having used ${\Vert f \Vert }_{H^{k+2}} \le 1$ and the assumption on the Sobolev norm of ${\zeta }_{\alpha }-1$.

To estimate $I_1$ we expand the expression $({\zeta }({\alpha })-{\zeta }({\beta }))^{-1}$ in a geometric sum as follows:

$$\begin{aligned} \frac{1}{ {\zeta }({\alpha })-{\zeta }({\beta }) }&= \frac{1}{{\alpha }-{\beta }} \sum _{k \ge 0} {\left( \frac{{\zeta }({\alpha }) - {\alpha }- ({\zeta }({\beta }) - {\beta })}{{\alpha }-{\beta }} \right) }^k \\&= \frac{1}{{\alpha }-{\beta }} + \frac{H({\alpha })- H({\beta })}{{\alpha }-{\beta }} F \left( \frac{H({\alpha })- H({\beta })}{{\alpha }-{\beta }} \right) , \end{aligned}$$

where $H := {\zeta }- \text{ id }$ and $F$ is a smooth function. We can then write

$$\begin{aligned} I_1 ({\alpha })&= I_0({\alpha }) + C_2 (F,H,f) ({\alpha }) \end{aligned}$$

where $C_2$ is as in (11.17), and

$$\begin{aligned} I_0 ({\alpha })&= \int \frac{ \partial _{\beta }f({\beta }) }{ {\alpha }-{\beta }} \, d{\beta }\end{aligned}$$

(11.52)

is a constant multiple of the (flat) Hilbert transform $H_0 := {\mathcal {H} } _{\text{ id }}$. To estimate $C_2(F,H,\partial _{\alpha }f)$ we can use Sobolev’s embedding, the commutation identity (11.39b), and the bounds provided by Theorem 11.3:

$$\begin{aligned} {\Vert C_2(F,H,\partial _{\alpha }f) \Vert }_{W^{k,\infty }} \! \lesssim \! {\Vert C_2(F,H,\partial _{\alpha }f) \Vert }_{H^{k+1}} \!\lesssim \! {\Vert f \Vert }_{H^{k+1}} {\Vert H_{\alpha }\Vert }_{W^{k+1,\infty }} \!\lesssim \! {\Vert f \Vert }_{W^{k+1,\infty }} . \end{aligned}$$

In the last inequality above we have used again the assumptions ${\Vert f \Vert }_{H^{k+1}} \le 1$ and ${\Vert {\zeta }_{\alpha }-1 \Vert }_{H^{k+1}} \le \frac{1}{2}$. So far we have shown

$$\begin{aligned} {\Vert {\mathcal {H} } \partial _{\alpha }f - H_0 \partial _{\alpha }f \Vert }_{W^{k,\infty }} \lesssim {\Vert f \Vert }_{W^{k+2,\infty }} . \end{aligned}$$

Applying the Littlewood–Paley decomposition to $f$, and using the boundedness of $H_0 P_l$ on $L^\infty$, we see that

$$\begin{aligned} {\Vert H_0 \partial _{\alpha }f \Vert }_{W^{k,\infty }} \lesssim \sum _l {\Vert H_0 P_l \partial _{\alpha }f \Vert }_{W^{k,\infty }} \lesssim \sum _l 2^l {\Vert f \Vert }_{W^{k,\infty }} \lesssim {\Vert f \Vert }_{W^{k+2,\infty }} . \end{aligned}$$

This concludes the proof of (11.51). The bound for the second summand in the left-hand side of (11.36) can be obtain similarly.

To prove (11.37) one can use an argument similar to the one just showed, replacing $\partial _{\alpha }f$ with $f$. The same estimates as above will show:

$$\begin{aligned} {\Vert {\mathcal {H} } f - H_0 f \Vert }_{W^{k,\infty }} \lesssim {\Vert f \Vert }_{W^{k+1,\infty }} . \end{aligned}$$

To conclude it is then enough to observe that for any $2 \le p < \infty$

$$\begin{aligned} {\Vert H_0 f \Vert }_{W^{k,\infty }} \lesssim {\Vert H_0 f \Vert }_{W^{k+1,p}} \lesssim {\Vert f \Vert }_{W^{k+1,p}} . \end{aligned}$$

The second summand in the left-hand side of (11.37) can be estimated analogously. $\Box$

Appendix B: The symbols $c^{\iota _1\iota _2\iota _3}$

In this section we calculate explicitly the symbols $c^{\iota _1\iota _2\iota _3}$ defined in (6.6) and prove the bounds (6.8) and (6.9).

With $V^+(t)=V(t)$ and $V^-(t)=\overline{V}(t)$, recall that

$$\begin{aligned} H(t)=\frac{V^+(t)+V^-(t)}{2},\qquad \Psi (t)=\frac{i[\Lambda ^{-1}V^-(t)-\Lambda ^{-1}V^+(t)]}{2}. \end{aligned}$$

Starting from the formula (4.9),

$$\begin{aligned} M_3(H,H,\Psi )\!=\!- (1/2) {|\partial _x|}\left[ H^2 {|\partial _x|}^2 \Psi \!+\! {|\partial _x|}(H^2 {|\partial _x|}\Psi ) \!- \!2 H {|\partial _x|}(H {|\partial _x|}\Psi ) \right] \!, \end{aligned}$$

we calculate easily

$$\begin{aligned}&\mathcal {F}[M_3(H,H,\Psi )](\xi )=\frac{i}{4\pi ^2}\sum _{(\iota _1\iota _2\iota _3)}\int _{\mathbb {R}^2}c_1^{\iota _1\iota _2\iota _3}(\xi ,\eta ,\sigma )\widehat{V^{\iota _1}}(\xi -\eta ,t)\\&\quad \times \,\widehat{V^{\iota _2}}(\eta -\sigma ,t)\widehat{V^{\iota _3}}(\sigma ,t)\,d\eta d\sigma , \end{aligned}$$

where the sum is taken over $(\iota _1\iota _2\iota _3)\in \{(++-),(--+),(+++),(---)\}$, and

$$\begin{aligned} c^{++-}_1(\xi ,\eta ,\sigma )&=\frac{2|\xi ||\eta -\sigma |^{3/2}-|\xi ||\sigma |^{3/2}+2|\xi |^2|\eta -\sigma |^{1/2}-|\xi |^2|\sigma |^{1/2}}{16} \nonumber \\&\quad +\frac{-|\xi ||\xi -\sigma ||\eta -\sigma |^{1/2}-|\xi ||\eta ||\eta -\sigma |^{1/2}+|\xi ||\eta ||\sigma |^{1/2}}{8}, \nonumber \\ c^{+++}_1(\xi ,\eta ,\sigma )&=\frac{|\xi ||\sigma |^{3/2}+|\xi |^2|\sigma |^{1/2}-2|\xi ||\eta ||\sigma |^{1/2}}{16}, \nonumber \\ c^{--+}_1(\xi ,\eta ,\sigma )&=-c^{++-}_1(\xi ,\eta ,\sigma ),\nonumber \\ c^{---}_1(\xi ,\eta ,\sigma )&=-c^{+++}_1(\xi ,\eta ,\sigma ). \end{aligned}$$

(12.1)

Using now the formula (4.11),

$$\begin{aligned} Q_3(\Psi ,H,\Psi )=|\partial _x|\Psi \left[ H|\partial _x|^2\Psi -|\partial _x|(H|\partial _x|\Psi )\right] \end{aligned}$$

we calculate easily

$$\begin{aligned}&\mathcal {F}[i\Lambda Q_3(\Psi ,H,\Psi )](\xi )\\&\quad =\frac{i}{4\pi ^2}\sum _{(\iota _1\iota _2\iota _3)}\int _{\mathbb {R}^2}c_2^{\iota _1\iota _2\iota _3}(\xi ,\eta ,\sigma )\widehat{V^{\iota _1}}(\xi -\eta ,t)\widehat{V^{\iota _2}}(\eta -\sigma ,t)\widehat{V^{\iota _3}}(\sigma ,t)\,d\eta d\sigma , \end{aligned}$$

where the sum is taken over $(\iota _1\iota _2\iota _3)\in \{(++-),(--+),(+++),(---)\}$, and

$$\begin{aligned}&c^{++-}_2(\xi ,\eta ,\sigma )\nonumber \\&\quad =\frac{|\xi |^{1/2}|\xi -\eta |^{3/2}|\sigma |^{1/2}+|\xi |^{1/2}|\xi -\eta |^{1/2}|\sigma |^{3/2}-|\xi |^{1/2}|\xi -\eta |^{1/2}|\eta -\sigma |^{3/2}}{8} \nonumber \\&\quad \quad +\frac{-|\xi |^{1/2}|\xi -\sigma ||\eta -\sigma |^{1/2}|\sigma |^{1/2}-|\xi |^{1/2}|\xi -\eta |^{1/2}|\eta ||\sigma |^{1/2}+|\xi |^{1/2}|\xi -\eta |^{1/2}|\eta ||\eta -\sigma |^{1/2}}{8}, \nonumber \\&c^{+++}_2(\xi ,\eta ,\sigma )=\frac{-|\xi |^{1/2}|\xi -\eta |^{1/2}|\eta -\sigma |^{3/2}+|\xi |^{1/2}|\xi -\eta |^{1/2}|\eta ||\sigma |^{1/2}}{8}, \nonumber \\&c^{--+}_2(\xi ,\eta ,\sigma )=c^{++-}_2(\xi ,\eta ,\sigma ), \nonumber \\&c^{---}_2(\xi ,\eta ,\sigma )=c^{+++}_2(\xi ,\eta ,\sigma ). \end{aligned}$$

(12.2)

Let

$$\begin{aligned} \widetilde{q}_2(\xi ,\eta )=\Lambda ^{-1}(\xi -\eta )\Lambda ^{-1}(\eta )q_2(\xi ,\eta ),\qquad \widetilde{m}_2(\xi ,\eta )=\Lambda ^{-1}(\eta )m_2(\xi ,\eta ).\nonumber \\ \end{aligned}$$

(12.3)

Using the fact that $A,Q_2$ are symmetric we calculate

$$\begin{aligned}&\mathcal {F}[2A(M_2(H,\Psi ),H)](\xi )=\frac{i}{4\pi ^2}\sum _{(\iota _1\iota _2\iota _3)}\int _{\mathbb {R}^2}c_3^{\iota _1\iota _2\iota _3}(\xi ,\eta ,\sigma )\widehat{V^{\iota _1}}(\xi -\eta ,t)\\&\quad \times \,\widehat{V^{\iota _2}}(\eta -\sigma ,t)\widehat{V^{\iota _3}}(\sigma ,t)\,d\eta d\sigma , \end{aligned}$$

where the sum is taken over $(\iota _1\iota _2\iota _3)\in \{(++-),(--+),(+++),(---)\}$, and

$$\begin{aligned}&c^{++-}_3(\xi ,\eta ,\sigma )\nonumber \\&\quad =\frac{a(\xi ,\eta )\widetilde{m}_2(\eta ,\sigma )\!-\!a(\xi ,\eta )\widetilde{m}_2(\eta ,\eta \!-\!\sigma )\!-\!a(\xi ,\xi \!-\!\sigma )\widetilde{m}_2(\xi -\sigma ,\xi -\eta )}{4}, \nonumber \\&c^{+++}_3(\xi ,\eta ,\sigma )=\frac{-a(\xi ,\eta )\widetilde{m}_2(\eta ,\sigma )}{4}, \nonumber \\&c^{--+}_3(\xi ,\eta ,\sigma )=-c^{++-}_3(\xi ,\eta ,\sigma ), \nonumber \\&c^{---}_3(\xi ,\eta ,\sigma )=-c^{+++}_3(\xi ,\eta ,\sigma ). \end{aligned}$$

(12.4)

Similarly we calculate

$$\begin{aligned}&\mathcal {F}[i\Lambda B(M_2(H,\Psi ),\Psi )](\xi )\nonumber \\&\quad =\frac{i}{4\pi ^2}\sum _{(\iota _1\iota _2\iota _3)}\int _{\mathbb {R}^2}c_4^{\iota _1\iota _2\iota _3}(\xi ,\eta ,\sigma )\widehat{V^{\iota _1}}(\xi -\eta ,t)\widehat{V^{\iota _2}}(\eta -\sigma ,t)\widehat{V^{\iota _3}}(\sigma ,t)\,d\eta d\sigma , \end{aligned}$$

where the sum is taken over $(\iota _1\iota _2\iota _3)\in \{(++-),(--+),(+++),(---)\}$, and

$$\begin{aligned}&c^{++-}_4(\xi ,\eta ,\sigma )\nonumber \\&\quad =\!\frac{|\xi |^{1/2}b(\xi ,\xi \!-\!\eta )|\xi \!-\!\eta |^{-1/2}\widetilde{m}_2(\eta ,\sigma )\!-\!|\xi |^{1/2}b(\xi ,\xi \!-\!\eta )|\xi \!-\!\eta |^{-1/2}\widetilde{m}_2(\eta ,\eta \!-\!\sigma )}{8} \nonumber \\&\quad \quad +\frac{|\xi |^{1/2}b(\xi ,\sigma )|\sigma |^{-1/2}\widetilde{m}_2(\xi -\sigma ,\xi -\eta )}{8}, \nonumber \\&c^{+++}_4(\xi ,\eta ,\sigma )=\frac{-|\xi |^{1/2}b(\xi ,\xi -\eta )|\xi -\eta |^{-1/2}\widetilde{m}_2(\eta ,\sigma )}{8}, \nonumber \\&c^{--+}_4(\xi ,\eta ,\sigma )=c^{++-}_4(\xi ,\eta ,\sigma ), \nonumber \\&c^{---}_4(\xi ,\eta ,\sigma )=c^{+++}_4(\xi ,\eta ,\sigma ). \end{aligned}$$

(12.5)

Finally we calculate

$$\begin{aligned}&\mathcal {F}[i\Lambda B(H,Q_2(\Psi ,\Psi ))](\xi )\nonumber \\&\quad =\frac{i}{4\pi ^2}\sum _{(\iota _1\iota _2\iota _3)}\int _{\mathbb {R}^2}c_5^{\iota _1\iota _2\iota _3}(\xi ,\eta ,\sigma )\widehat{V^{\iota _1}}(\xi -\eta ,t)\widehat{V^{\iota _2}}(\eta -\sigma ,t)\widehat{V^{\iota _3}}(\sigma ,t)\,d\eta d\sigma , \end{aligned}$$

where the sum is taken over $(\iota _1\iota _2\iota _3)\in \{(++-),(--+),(+++),(---)\}$, and

$$\begin{aligned} c^{++-}_5&(\xi ,\eta ,\sigma )=\frac{2|\xi |^{1/2}b(\xi ,\eta )\widetilde{q}_2(\eta ,\sigma )\!-\!|\xi |^{1/2}b(\xi ,\xi -\sigma )\widetilde{q}_2(\xi \!-\!\sigma ,\xi \!-\!\eta )}{8}, \nonumber \\ c^{+++}_5&(\xi ,\eta ,\sigma )=\frac{-|\xi |^{1/2}b(\xi ,\eta )\widetilde{q}_2(\eta ,\sigma )}{8}, \nonumber \\ c^{--+}_5&(\xi ,\eta ,\sigma )=c^{++-}_5(\xi ,\eta ,\sigma ), \nonumber \\ c^{---}_5&(\xi ,\eta ,\sigma )=c^{+++}_5(\xi ,\eta ,\sigma ). \end{aligned}$$

(12.6)

The following lemma gives the desired bounds (6.8) and (6.9).

Lemma 12.1

The symbols $c^{\iota _1\iota _2\iota _3}$ satisfy the uniform bounds

$$\begin{aligned}&\big \Vert \mathcal {F}^{-1}[c^{\iota _1\iota _2\iota _3}(\xi ,\eta ,\sigma )\cdot \varphi _l(\xi )\varphi _{k_1}(\xi -\eta )\varphi _{k_2}(\eta -\sigma )\varphi _{k_3}(\sigma )]\big \Vert _{L^1(\mathbb {R}^3)}\nonumber \\&\quad \lesssim 2^{l/2}2^{2\max (k_1,k_2,k_3)}, \end{aligned}$$

(12.7)

for any $(\iota _1\iota _2\iota _3)\in \{(++-),(--+),(+++),(---)\}$ and $l,k_1,k_2,k_3\in \mathbb {Z}$. Moreover, for any $\mathbf {k}=(k_1,k_2,k_3),\mathbf {l}=(l_1,l_2,l_3)\in \mathbb {Z}^3$ let

$$\begin{aligned} c^*_\xi (x,y)&=c^{++-}(\xi ,-x,-\xi -x-y), \\ (\partial _xc^*_\xi )_{\mathbf {k},\mathbf {l}}(x,y)&=(\partial _xc^*_\xi )(x,y)\cdot \varphi _{k_1}(\xi +x)\varphi _{k_2}(\xi +y)\varphi _{k_3}(\xi +x+y)\\&\qquad \varphi _{l_1}(x)\varphi _{l_2}(y) \varphi _{l_3}(2\xi +x+y), \\ (\partial _yc^*_\xi )_{\mathbf {k},\mathbf {l}}(x,y)&=(\partial _yc^*_\xi )(x,y)\cdot \varphi _{k_1}(\xi +x)\varphi _{k_2}(\xi +y)\varphi _{k_3}(\xi +x+y)\\&\qquad \varphi _{l_1}(x)\varphi _{l_2}(y)\varphi _{l_3}(2\xi +x+y). \end{aligned}$$

Then, for any $\mathbf {k},\mathbf {l}\in \mathbb {Z}^3$, and $\xi \in \mathbb {R}$

$$\begin{aligned}&\Vert (\partial _xc^*_\xi )_{\mathbf {k},\mathbf {l}}\Vert _{\mathcal {S}^\infty }\lesssim 2^{-\min (k_1,k_3)}2^{5\max (k_1,k_2,k_3)/2}, \nonumber \\&\Vert (\partial _yc^*_\xi )_{\mathbf {k},\mathbf {l}}\Vert _{\mathcal {S}^\infty }\lesssim 2^{-\min (k_2,k_3)}2^{5\max (k_1,k_2,k_3)/2}. \end{aligned}$$

(12.8)

Proof of Lemma 12.1

Clearly, for any $(\iota _1\iota _2\iota _3)\in \{(++-),(--+),(+++),(---)\}$

$$\begin{aligned} c^{\iota _1\iota _2\iota _3}=\sum _{l=1}^5c^{\iota _1\iota _2\iota _3}_l. \end{aligned}$$

The bound (12.7) follows from the explicit formulas (12.1)–(12.6), the symbol bounds in Lemma 5.3, and the algebra properties in Lemma 5.2 (ii).

Let $\iota :\mathbb {R}\setminus \{0\}\rightarrow \{-1,1\}$, $\iota (x):=x/|x|$. Recalling the formulas in Lemma 5.1 and using (12.1)–(12.6) we calculate

$$\begin{aligned} c^*_\xi (x,y)=c^*_{\xi ,1}(x,y)+c^*_{\xi ,2}(x,y)+c^*_{\xi ,3}(x,y)+c^*_{\xi ,4}(x,y)+c^*_{\xi ,5}(x,y), \end{aligned}$$

where

$$\begin{aligned} c^*_{\xi ,1}(x,y)&=\frac{|\xi ||\xi +y|^{3/2}}{8}-\frac{|\xi ||\xi +x+y|^{3/2}}{16}+\frac{|\xi |^2|\xi +y|^{1/2}}{8}\\&\quad -\frac{|\xi |^2|\xi +x+y|^{1/2}}{16} -\frac{|\xi ||2\xi +x+y||\xi +y|^{1/2}}{8}\\&\quad -\frac{|\xi ||x||\xi +y|^{1/2}}{8}+\frac{|\xi ||x||\xi +x+y|^{1/2}}{8},\\ c^*_{\xi ,2}(x,y)&=\frac{|\xi |^{1/2}|\xi \!+\!x|^{3/2}|\xi \!+\!x+y|^{1/2}}{8}\!+\!\frac{|\xi |^{1/2}|\xi +x|^{1/2}|\xi +x+y|^{3/2}}{8}\\&\quad -\frac{|\xi |^{1/2}|\xi +x|^{1/2}|\xi +y|^{3/2}}{8} \\&\quad -\frac{|\xi |^{1/2}|2\xi +x+y||\xi +y|^{1/2}|\xi +x+y|^{1/2}}{8} \\&\quad -\frac{|\xi |^{1/2}|\xi \!+\!x|^{1/2}|x||\xi \!+\!x+y|^{1/2}}{8}\!+\!\frac{|\xi |^{1/2}|\xi \!+\!x|^{1/2}|x||\xi \!+\!y|^{1/2}}{8},\\ c^*_{\xi ,3}(x,y)&=\frac{|\xi ||\xi +x+y|^{1/2}\iota (\xi +x)[|x|\iota (\xi +x+y)-x]}{8}\\&\quad +\frac{|\xi ||\xi +y|^{1/2}\iota (\xi +x)[|x|\iota (\xi +y)+x]}{8} \\&\quad -\frac{|\xi ||\xi \!+\!x|^{1/2}\iota (\xi \!+\!x\!+\!y)[|2\xi \!+\!x+y|\iota (\xi +x)\!-\!(2\xi +x+y)]}{8},\\ \end{aligned}$$

$$\begin{aligned}&c^*_{\xi ,4}(x,y)=\frac{|\xi |^{1/2}|\xi \!+\!x|^{1/2}|\xi \!+\!x\!+\!y|^{1/2}\iota (\xi )[|x|\iota (\xi \!+\!x\!+\!y)\!-\!x]}{8}\\&\quad \quad \quad \quad \quad \quad +\frac{|\xi |^{1/2}|\xi +x|^{1/2}|\xi +y|^{1/2}\iota (\xi )[|x|\iota (\xi +y)+x]}{8} \\&\quad -\frac{|\xi |^{1/2}|\xi \!+\!x|^{1/2}|\xi \!+\!x\!+\!y|^{1/2}\iota (\xi )[|2\xi \!+\!x\!+\!y|\iota (\xi \!+\!x)\!-\!(2\xi \!+\!x\!+\!y)]}{8}, \end{aligned}$$

and

$$\begin{aligned}&c^*_{\xi ,5}(x,y)\\&\quad \!\!\!\!=-\frac{|\xi |^{1/2}|\xi \!+\!y|^{1/2}|\xi \!+\!x\!+\!y|^{1/2}\iota (\xi )\iota (\xi \!+\!x)|x|[1\!-\!\iota (\xi \!+\!y)\iota (\xi \!+\!x\!+\!y)]}{8} \\&\quad \quad \!\!\!\!\!-\frac{|\xi |^{1/2}|\xi \!+\!y|^{1/2}|\xi \!+\!x|^{1/2}\iota (\xi )\iota (\xi \!+\!x\!+\!y)|2\xi \!+\!x\!+\!y|[1\!+\!\iota (\xi \!+\!y)\iota (\xi \!+\!x)]}{16}. \end{aligned}$$

The desired bounds (12.8) are verified easily for every term in these formulas.

Using these formulas, we also calculate

$$\begin{aligned} c^*_{\xi }(0,0)=-|\xi |^{5/2}/2. \end{aligned}$$

(12.9)

$\square$

Appendix C: Estimate of remainder terms

In the first section below we give estimates for the Dirichlet–Neumann operator ${\mathcal {N}}$ in $L^2$, weighted $L^2$, and $L^1$-based Sobolev spaces. We will the use these to establish several bounds for $R_1$ and $R_2$ in (4.6)–(4.7). We then proceed to estimate all quartic and higher order remainder terms, and in particular prove (6.25).

1.1 C1: Dirichlet-to-Neumann operator: multilinear estimates

Here we recall that if $N$ denotes the outward normal vector of the interface $S_0$

$$\begin{aligned} {\mathcal {N}}(h) \phi := N \cdot \nabla \Phi = N \cdot \nabla \phi _ {\mathcal {H} } , \qquad G(h) \phi = \sqrt{1 + {|h^{\prime }|}^2} {\mathcal {N}}(h) \phi . \end{aligned}$$

(13.1)

We are interested in particular in estimating quartic and higher order terms in the expansion of the Dirichlet–Neumann operator. The $L^2$ and weighted $L^2$ estimates are needed to obtain the improved weighted bounds on $V$ in Proposition 4.3. The $L^1$ estimates are used to bound these higher order terms in the $Z$-norm.

The first Proposition below gives estimates in $L^2$-based spaces and its proven in Appendix C.1.1:

Proposition 13.1

The Dirichlet–Neumann operator $G$ can be expanded in a series

$$\begin{aligned} G(h) f = \sum _{n\ge 0} M_{n+1}(h,\dots , h, f) , \end{aligned}$$

(13.2)

where $M_{n+1}$ is an $n+1$-linear operator satisfying the following $L^2$ bounds:

$$\begin{aligned}&{\Vert M_{n+1} (h_1, \dots , h_n, f) \Vert }_{L^2} \nonumber \\&\quad \le C_0^n \min \left\{ \prod _{i=1}^n {\Vert h_i^{\prime }\Vert }_{L^\infty } {\Vert f^{\prime }\Vert }_{L^2}, \min _{j\in \{1,\dots ,n\}} \prod _{i\ne j} {\Vert h_i^{\prime }\Vert }_{L^\infty } {\Vert h_j^{\prime }\Vert }_{L^2} {\Vert f^{\prime }\Vert }_{L^\infty } \right\} , \end{aligned}$$

(13.3)

for some absolute constant $C_0$.

Moreover, $G$ is invariant under translation and scaling symmetries, and the following identities hold:

$$\begin{aligned} \partial _x M_{n+1}(h_1,\dots , h_n, f)&= \sum _{i=1}^n M_{n+1}(h_1, \dots , \partial _x h_i, \dots , h_n, f) \nonumber \\&\quad + M_{n+1}(h_1,\dots , h_n, \partial _x f)\end{aligned}$$

(13.4)

$$\begin{aligned} S M_{n+1}(h_1,\dots , h_n, f)&= \sum _{i=1}^n M_{n+1}(h_1,\dots , S h_i, \dots , h_n, f) \nonumber \\&\quad + M_{n+1}(h_1,\dots , h_n, S f)- \sum _{i=1}^n M_{n+1} ( h_1, \dots , h_n, f) , \end{aligned}$$

(13.5)

where $S$ denotes the scaling vector field. As a consequence, for any integer $l \ge 0$ one has:

$$\begin{aligned} {\Vert M_{n+1}(h_1,\dots , h_n, f) \Vert }_{H^l}&\lesssim \sum _{i=1}^n {\Vert h_i^{\prime }\Vert }_{H^l} \prod _{j \ne i} {\Vert h_j \Vert }_{W^{N_1,\infty }} {\Vert f_x \Vert }_{W^{N_1,\infty }}\nonumber \\&\quad + \prod _{i=1}^n {\Vert h_i \Vert }_{W^{N_1,\infty }} {\Vert f_x \Vert }_{H^l} ,\end{aligned}$$

(13.6)

$$\begin{aligned} {\Vert S M_{n+1}(h_1,\dots , h_n, f) \Vert }_{H^l}&\lesssim \sum _{i,j = 1, i\ne j}^n {\Vert (S h_i)^{\prime }\Vert }_{H^l} {\Vert h_j^{\prime }\Vert }_{W^{l,\infty }}\nonumber \\&\quad \times \prod _{k\ne i,j} {\Vert h_k \Vert }_{W^{N_1,\infty }} {\Vert f_x \Vert }_{W^{l,\infty }}\nonumber \\&\quad + \sum _{i=1}^n {\Vert h_i^{\prime }\Vert }_{H^l} \prod _{j \ne i} {\Vert h_j \Vert }_{W^{N_1,\infty }}\left( {\Vert {(Sf)}^{\prime }\Vert }_{H^l} \!+\! {\Vert f^{\prime }\Vert }_{H^l} \right) \! , \end{aligned}$$

(13.7)

where the implicit constants are bounded by $C_0^n$ for some absolute constant $C_0$.

Let us denote

$$\begin{aligned} {[ G (h) \phi ]}_{\ge 4}(t) := \sum _{n\ge 3} M_{n+1}(h(t),\dots , h(t), \phi (t)) \end{aligned}$$

(13.8)

to be the quartic and higher order terms (in $h$ and $\phi$) in the expansion of $G$. A corollary of this expansion and Proposition 13.1 is the following:

Corollary 13.2

Under the a priori assumptions (2.8) on $h$ and $\phi$ one has

$$\begin{aligned} {\Vert {[ G (h) \phi ]}_{\ge 4}(t) \Vert }_{H^{N_0 - 2}} + {\Vert S {[ G (h) \phi ]}_{\ge 4} (t) \Vert }_{H^{\frac{N_0}{2} - 2}}&\lesssim {\varepsilon }_1^4 {(1+t)}^{3p_0 -3/2} . \end{aligned}$$

(13.9)

Moreover, for $R_1$ and $R_2$ defined in (4.6)–(4.7) we have

$$\begin{aligned} {\Vert (R_1 + i\Lambda R_2) (t) \Vert }_{H^{N_0-5}} + {\Vert S (R_1 + i\Lambda R_2)(t) \Vert }_{H^{\frac{N_0}{2}-5}}&\lesssim {\varepsilon }_1^4 {(1+t)}^{3p_0 -3/2} . \end{aligned}$$

(13.10)

The next Proposition establishes $L^1$-type estimates:

Proposition 13.3

With the same notations of Proposition 13.1, and for any $n \ge 3$, we have

$$\begin{aligned} {\left\| {|\partial _x|}^\frac{{\beta }}{4} M_{n+1} (h_1, \dots , h_n, f) \right\| }_{W^{1,l}} \lesssim \sum _{i=1}^{n} {\Vert h_i\Vert }_{H^{l+3}} \prod _{j\ne i} {\Vert h_j \Vert }_{W^{N_1,\infty }} {\Vert f_x \Vert }_{H^l} . \end{aligned}$$

(13.11)

As a consequence, under the a priori assumption (2.8) on $h$ and $\phi$,

$$\begin{aligned} {\left\| {|\partial _x|}^\frac{{\beta }}{4} {[ G (h) \phi ]}_{\ge 4}(t) \right\| }_{W^{1,N_0-10}} \lesssim {(1+t)}^{3p_0 - 1} . \end{aligned}$$

(13.12)

Here $\beta = 1/100$ is the parameter that appears in the definition of the $Z$ norm (1.9).

1.1.1 C.1.1 Proof of Proposition 13.1

This Proposition follows from some standard potential theory, arguments similar to those in [20, sec. 7.2], and Theorem 11.3. We sketch the proof below.

Expansion of the Dirichlet–Neumann operator. In order to find an explicit formula for the Dirichlet–Neumann operator we start with an ansatz for the harmonic extension of a function $f$ to the domain $\{ (x,z) \, : \, z \le h(x) \}$:

$$\begin{aligned} \Phi (x,z) = \int \frac{1}{2} \log \left( {|x-y|}^2 + {|z-h(y)|}^2 \right) \rho (y) \, dy . \end{aligned}$$

(13.13)

By standard potential theory one has

$$\begin{aligned} {\mathcal {N}}(h) f(x)&= \lim _{z \rightarrow h(x)} \nabla \Phi (x,z) \cdot N(x)\nonumber \\&= \frac{1}{2} \frac{\rho (x) }{\sqrt{1+{|h^{\prime }(x) |}^2}} + \int \frac{ h(x) - h(y) + h^{\prime }(x) (y-x) }{ {|x-y|}^2 + {|h(x)-h(y)|}^2} \rho (y) \, dy .\qquad \end{aligned}$$

(13.14)

We then aim at determining $\rho$ in terms of $h$ and $f$. Using (13.13) and $f(x) = \Phi (x,h(x))$, one has

$$\begin{aligned} f(x) = \int \frac{1}{2} \log \left( {|x-y|}^2 + {|h(x)-h(y)|}^2 \right) \rho (y) \, dy . \end{aligned}$$

(13.15)

It follows that

$$\begin{aligned} |\partial _x| f(x)&= iH_0 \int \frac{x-y + (h(x) - h(y))h^{\prime }(x)}{{|x-y|}^2 + {|h(x)-h(y)|}^2} \rho (y) \, dy = \rho (x) + \sum _{n = 1}^{\infty } i H_0 \\&\quad \times \int \left( \frac{h(x)-h(y)}{x-y} \right) ^{2n} \frac{x-y + (h(x) - h(y))h^{\prime }(x)}{{(x-y)}^2} \rho (y) \, dy \\&=: \rho (x) + \sum _{n = 1}^{\infty } P_{n}(h) \rho (x) . \end{aligned}$$

One can then invert the above series expansion and write

$$\begin{aligned} \rho = \sum _{k \ge 0} {(-1)}^k {\left[ \sum _{n=1}^\infty P_{n}(h) \right] }^k |\partial _x| f \end{aligned}$$

(13.16)

where

$$\begin{aligned} P_{n}(h)g(x)&:= iH_0 \int \frac{ {\left( h(x) - h(y) \right) }^{2n} h^{\prime }(x) }{ {(x-y)}^{2n+1} } g(y) \, dy\nonumber \\&\quad + iH_0 h^{\prime }(x) \int \frac{ {\left( h(x) - h(y) \right) }^{2n+1} }{ {(x-y)}^{2n+2} } g(y) \, dy . \end{aligned}$$

(13.17)

Expanding the second summand in (13.14) one can write

$$\begin{aligned} \nonumber {\mathcal {N}}(h) f(x)&= \frac{1}{2} \frac{\rho (x) }{\sqrt{1+{|h^{\prime }(x) |}^2}} + \sum _{n=0}^\infty Q_{n}(h) \rho \\ Q_{n}(h)&:= \int \left( \frac{ h(x) - h(y) }{x-y}\right) ^{2n} \frac{h(x) - h(y) + h^{\prime }(x) (y-x)}{{(x-y)}^2} \rho (y) \, dy . \end{aligned}$$

(13.18)

Putting together (13.1), (13.16) and (13.18) we eventually obtain (13.2).

Symmetries and $L^2$ -bounds. The basic $L^2$-type bounds (13.3) follow directly from the expansion (13.16)–(13.18) and Theorem 11.3. The formulas (13.4) and (13.5) follow from the space translation and scaling invariances of the basic operators $P_n$ and $Q_n$ in (13.17) and (13.18). More precisely, for any ${\delta }\in {\mathbb {R}}$ and ${\lambda }>0$

$$\begin{aligned}{}[G( h (\cdot +{\delta }) ) f(\cdot +{\delta })] (x)&= [G(h) f] (x+{\delta }) , \\ G \left( \frac{1}{{\lambda }} h ({\lambda }\cdot ) \right) f({\lambda }\cdot ) (x)&= {\lambda }[G(h) f] ({\lambda }x) . \end{aligned}$$

These identities hold true for each operator $M_n$ in the expansion (13.2), that is

$$\begin{aligned} M_n ( h_1 (\cdot +{\delta }), \dots , h_n (\cdot +{\delta }) , f(\cdot +{\delta }) ) (x)&= M_n ( h_1,\dots , h_n, f) (x+{\delta }) , \end{aligned}$$

(13.19)

$$\begin{aligned} M_n \left( \frac{1}{{\lambda }} h_1 ({\lambda }\cdot ), \dots , \frac{1}{{\lambda }} h_n ({\lambda }\cdot ), f({\lambda }\cdot ) \right) (x)&= M_n (h_1,\dots , h_n, f) ({\lambda }x) , \end{aligned}$$

(13.20)

and can be verified directly on the operators $P_n$ and $Q_n$ defined above. Differentiating with respect to the parameters in (13.19) and (13.20), one sees that

$$\begin{aligned} \partial _x M_n ( h_1,\dots , h_n, f )&=\! \sum _{i=1}^n M_n ( h_1, \dots , \partial _x h_i, \dots , h_n, f) \!+ \!M_n (h_1, \dots , h_n, \partial _x f) , \\ x \partial _x M_n (h,\dots , h, f) (x)&= \sum _{i=1}^n M_n ( h_1, \dots , x \partial _x h_i, \dots , h_n, f) \\&\quad + M_n \left( h_1, \dots , h_n, x \partial _x f \right) - \sum _{i=1}^n M_n ( h_1, \dots , h_n, f) . \end{aligned}$$

The first identity is (13.4). If $h$ and $f$ depend on time, one can similarly derive (13.5) from the last identity above. The estimates (13.6) and (13.7) follow by repeated applications of (13.4) and (13.5) and the $L^2$ estimate (13.3). $\square$

1.1.2 C.1.2 Proof of Corollary 13.2

The estimate (13.9) is an immediate consequence of the bounds (13.6) and (13.7). To prove (13.10) it then suffices to prove

$$\begin{aligned} {\Vert \Lambda R_2 (t) \Vert }_{H^{N_0-5}} + {\Vert S \Lambda R_2(t) \Vert }_{H^{\frac{N_0}{2}-5}}&\lesssim {\varepsilon }_1^4 {(1+t)}^{3p_0 -3/2} . \end{aligned}$$

(13.21)

From the definition of $R_2$ (4.7) we see that

$$\begin{aligned} R_2&= {\left[ {(|\partial _x|\phi + M_2(h,\phi ) + M_3 (h,h,\phi ) + R_1(h,\phi )+ h_x\phi _x)}^2 \right] }_{\ge 4} \nonumber \\&\quad + {\left[ 2 {(1+{|h_x|}^2)}^{-1/2} \right] }_{\ge 2} (|\partial _x| \phi \!+\! M_2(h,\phi ) + M_3 (h,h,\phi )+ R_1(h,\phi )+ h_x\phi _x)^2 \nonumber \\&= {(M_2(h,\phi ) + M_3 (h,h,\phi ) + R_1(h,\phi )+ h_x\phi _x)}^2+ 2 |\partial _x| \phi (M_3 (h,h,\phi ) \nonumber \\&\quad + R_1(h,\phi )) + {\left[ 2 {(1\!+\!{|h_x|}^2)}^{-1/2} \right] }_{\ge 2} (|\partial _x| \phi \nonumber \\&\quad + M_2(h,\phi ) \!+\! M_3 (h,h,\phi ) + R_1(h,\phi )+ h_x\phi _x)^2 . \end{aligned}$$

(13.22)

To obtain the desired bound it suffices to apply appropriately Hölder’s inequality in combination with the a priori estimates (4.13), the $L^2$ estimates (5.33) for $M_2$, and (5.35) for $M_3$, and the following $L^\infty$ estimates:

$$\begin{aligned} {\Vert P_k M_2(h,\phi )\Vert }_{L^\infty }&\lesssim {\varepsilon }_1^2 (1+t)^{-1} 2^k 2^{-N_0 k_+/2 }, \end{aligned}$$

(13.23)

$$\begin{aligned} {\Vert P_k M_3(h,h,\phi )\Vert }_{L^\infty }&\lesssim {\varepsilon }_1^2 (1+t)^{-3/2} 2^k 2^{-N_0 k_+/2 } . \end{aligned}$$

(13.24)

The last two estimates above can be obtained by inspection of (4.8) and (4.9) using the a priori bounds (5.23).

1.1.3 C.1.3 Proof of Proposition 13.3

Given the expansion of ${\mathcal {N}}$ in (13.16)–(13.18), the already established $L^2$-based estimates, and the commutation property (13.4), it is not hard to see that (13.11) would follow if one can show that operators of the form

$$\begin{aligned} C_1 \left( h_1, \dots , h_n, f \right)&:= \text{ p.v. } \int \frac{ \prod _{i=1}^n ( h_i(x) - h_i(y) ) }{ {(x-y)}^{n+1} } f(y) \, dy \end{aligned}$$

satisfy

$$\begin{aligned} {\left\| {|\partial _x|}^\frac{{\beta }}{4} H_0 C_1 (h_1, \dots , h_n, f) \right\| }_{L^1} \lesssim \min _{i=1,\dots , n} {\Vert h_i\Vert }_{H^3} \prod _{j\ne i} {\Vert h_j \Vert }_{W^{2,\infty }} {\Vert f \Vert }_{H^2} . \end{aligned}$$

(13.25)

The above estimate is in turn implied by

$$\begin{aligned} {\left\| C_1 (h_1, \dots , h_n, f) \right\| }_{W^{1,1}} \lesssim \min _{i=1,\dots , n} {\Vert h_i\Vert }_{H^3} \prod _{j\ne i} {\Vert h_j \Vert }_{W^{2,\infty }} {\Vert f \Vert }_{H^2} . \end{aligned}$$

Since the action of derivatives on operators of the type $C_1$ produces operators of the same type (acting on derivatives of the arguments), it is enough to obtain

$$\begin{aligned} {\left\| C_1 (h_1, \dots , h_n, f) \right\| }_{L^1} \lesssim \min _{i=1,\dots , n} {\Vert h_i\Vert }_{H^2} \prod _{j\ne i} {\Vert h_j \Vert }_{W^{1,\infty }} {\Vert f \Vert }_{H^1} . \end{aligned}$$

We only provide details of the proof of the above estimate in the case $n=1$, that is

$$\begin{aligned} {\left\| \text{ p.v. } \int \frac{ h(y) - h(x) }{ {(y-x)}^{2} } f(y) \, dy \right\| }_{L^1} \lesssim {\Vert h \Vert }_{H^2} {\Vert f \Vert }_{H^1} , \end{aligned}$$

(13.26)

as the case $n \ge 2$ can be treated similarly. Let us write

$$\begin{aligned}&\int \frac{ h(x) - h(y) }{ {(x-y)}^{2} } f(y) \, dy = I_1 + I_2 + I_3\nonumber \\&I_1(x) := \int _{|y-x| \ge 1} \frac{ h(y) - h(x) }{ {(y-x)}^{2} } f(y) \, dy\end{aligned}$$

(13.27)

$$\begin{aligned}&I_2(x) := \int _{|y-x| \le 1} \frac{ h(y) - h(x) - h^{\prime }(x) (y-x)}{ {(y-x)}^{2} } f(y) \, dy\end{aligned}$$

(13.28)

$$\begin{aligned}&I_3(x) := h^{\prime }(x) \int _{|y-x| \le 1} \frac{f(y)}{ y-x } \, dy . \end{aligned}$$

(13.29)

Notice that $I_1$ can be written as $I_1 = K *(hf) - h K *f$, where $K(x) := {|x|}^{-2} \chi _{|x| \ge 1}$ is an $L^1$ kernel. It follows that

$$\begin{aligned}&{\Vert I_1 \Vert }_{L^1} \lesssim {\Vert K *(hf) \Vert }_{L^1} + {\Vert h\Vert }_{L^2} {\Vert K *f\Vert }_{L^2} \lesssim {\Vert h\Vert }_{L^2} {\Vert f\Vert }_{L^2}. \end{aligned}$$

Using Taylor’s formula and a change of variables we can write

$$\begin{aligned}&I_2 = - \int _{|y| \le 1} \int _0^1 t h^{{\prime }{\prime }}(x + ty) \, dt f(y+x) \, dy . \end{aligned}$$

It follows that

$$\begin{aligned}&{\Vert I_2 \Vert }_{L^1} \lesssim \int _0^1 \int _{|y| \le 1} \int |h^{{\prime }{\prime }}(x + ty)| |f(y+x)| \, dx \, dy \, dt \lesssim {\Vert h^{{\prime }{\prime }} \Vert }_{L^2} {\Vert f\Vert }_{L^2} . \end{aligned}$$

For the last term (13.29) we first write

$$\begin{aligned}&I_3 = h^{\prime }(x) \int _{|y-x| \le 1} \frac{f(y)-f(x)}{ y-x } \, dy = h^{\prime }(x) \int _{|y| \le 1} \int _0^1 t f^{{\prime }}(x + ty) \, dt \, dy \end{aligned}$$

and then estimate

$$\begin{aligned}&{\Vert I_3 \Vert }_{L^1} \lesssim \int _0^1 \int _{|y| \le 1} \int |h^{\prime }(x)| |f^{{\prime }}(x + ty)| \, dx \, dy \, dt \lesssim {\Vert h^{\prime }\Vert }_{L^2} {\Vert f^{\prime }\Vert }_{L^2} . \end{aligned}$$

This shows that (13.26) holds and completes the proof of Proposition 13.3. $\square$

1.2 C2: Proof of (6.25)

For $m\in \mathbb {Z}\cap [20,\infty )$, $k \in {\mathbb {Z}}\cap [-m/2,m/50-1000]$, $|\xi |\in [2^k,2^{k+1}]$, $t_1\le t_2\in [2^{m-1},2^{m+1}]\cap [0,T]$, we want to show

$$\begin{aligned} \left| \int _{t_1}^{t_2} e^{iH(\xi ,s)}e^{is\Lambda (\xi )}\widehat{R}(\xi ,s)\,ds\right| \lesssim \varepsilon _1^32^{-p_1m}(2^{\beta k}+2^{(N_1+15)k})^{-1}. \end{aligned}$$

(13.30)

where

$$\begin{aligned} R:=\mathcal {N}_3+\mathcal {N}_4-\widetilde{\mathcal {N}}_3. \end{aligned}$$

(13.31)

with ${\mathcal {N}}_3$, ${\mathcal {N}}_4$ and $\widetilde{{\mathcal {N}}}_3$ defined respectively in (5.8), (5.9) and (6.2). To prove (13.30) we will use Lemma 13.4 and 13.5 below.

Lemma 13.4

Let $R$ be defined as in (13.31). Under the a priori assumptions (5.23) on $h$ and $\phi$, we have for $k\in {\mathbb {Z}}$

$$\begin{aligned} \big | \widehat{P_k R}(\xi ,t) \big | \lesssim {\varepsilon }_1^4 {(1+t)}^{10p_0 - 1} 2^{-(N_0-20)k_+} 2^{-{\beta }k/4} \end{aligned}$$

(13.32)

and

$$\begin{aligned} {\Vert P_k R(t) \Vert }_{L^2} + {\Vert P_k S R(t) \Vert }_{L^2} \lesssim {\varepsilon }_1^4 {(1+t)}^{20p_0 - 3/2} 2^{-(N_0/2-20)k_+} . \end{aligned}$$

(13.33)

Lemma 13.5

Assume that a function $D = D(\xi ,t)$ satisfies for all $t \in [0,T]$

$$\begin{aligned} {\Vert D (t) \Vert }_{L^2} + {\Vert S D(t) \Vert }_{L^2}&\lesssim {\delta }{(1+t)}^{-11/8 } , \nonumber \\ {\Vert \widehat{D} (t) \Vert }_{L^\infty }&\lesssim {\delta }{(1+t)}^{20 p_0 -1} . \end{aligned}$$

(13.34)

It follows that for $k\in {\mathbb {Z}}$, $|\xi | \in [2^k,2^{k+1}]$, $m\in \{1,2,\dots \}$ and $t_1\le t_2 \in [2^m\!-\!2, 2^{m+1}] \cap [0,T]$

$$\begin{aligned} \left| \int _{t_1}^{t_2} e^{i H(\xi ,s)} e^{is\Lambda (\xi )} \widehat{D}(\xi ,s) \, ds \right|&\lesssim {\delta }(1 + 2^{-k}) 2^{-m/16} . \end{aligned}$$

(13.35)

We now show how (13.30) follows from Lemma 13.4 and 13.5.

Proof of (13.30)

From (13.32) we see that for $|\xi | \in [2^k,2^{k+1}]$ one has

$$\begin{aligned}&\left| \int _{t_1}^{t_2} e^{iH(\xi ,s)}e^{is\Lambda (\xi )}\widehat{R}(\xi ,s)\,ds\right| \lesssim 2^m \sup _{s \in [2^m-2,2^{m+1}]} \big | \widehat{R}(\xi ,s) \big | \nonumber \\&\quad \lesssim {\varepsilon }_1^4 2^{10p_0 m} 2^{-(N_0-20)k_+} 2^{-{\beta }k/4} . \end{aligned}$$

Given our choice of $N_0$ and $N_1$, the desired bound (13.30) follows for $k \ge 22p_0/(N_0-80)$ and $k \le - 44p_0/3{\beta }$, with any $p_1 \le p_0$. For the remaining frequencies

$$\begin{aligned} k \in [-44p_0/3{\beta }, 22p_0/(N_0-80)] \end{aligned}$$

(13.36)

we want to apply Lemma 13.5 with

$$\begin{aligned} D(\xi ,t) = \left( 2^{{\beta }k} + 2^{(N_1+15)k} \right) P_k R (\xi ,t) \end{aligned}$$

and ${\delta }= {\varepsilon }_1^4$. From (13.32) and (13.33) we see that

$$\begin{aligned}&\big | \widehat{D} (\xi ,t) \big | \lesssim {\varepsilon }_1^4 {(1+t)}^{10p_0 - 1} 2^{-(N_0/2-50)k_+} 2^{3{\beta }k/4} \lesssim {\varepsilon }_1^4 {(1+t)}^{10p_0 - 1} , \\&{\Vert D (t) \Vert }_{L^2} + {\Vert S D(t) \Vert }_{L^2} \lesssim {\varepsilon }_1^4 {(1+t)}^{20p_0 - 3/2} 2^{40 k_+} \lesssim {(1+t)}^{21p_0 - 3/2} , \end{aligned}$$

under the restriction (13.36). The hypotheses of Lemma 13.5 are then satisfied, and the conclusion (13.35) implies

$$\begin{aligned} \left| \int _{t_1}^{t_2} e^{i H(\xi ,s)} e^{is\Lambda (\xi )} \widehat{R}(\xi ,s) \, ds \right|&\lesssim {\varepsilon }_1^4 {\left( 2^{{\beta }k} + 2^{(N_1 +15) k} \right) }^{-1} \left( 1 + 2^{-k} \right) 2^{-m/16} . \end{aligned}$$

This gives (13.30) in the considered frequency range (13.36), by choosing $p_1 \le 1/16 - 44p_0/3{\beta }$. $\square$

1.2.1 C.2.1 Proof of Lemma 13.4

Since $R =\mathcal {N}_3+\mathcal {N}_4-\widetilde{\mathcal {N}}_3$, from (5.8), (5.9) and (6.2) we can write

$$\begin{aligned} R = \mathcal {N}_4 + \sum _{j=1}^5 \widetilde{\mathcal {N}}_{3,j} \end{aligned}$$

where we recall that

$$\begin{aligned} {\mathcal {N}}_4&= R_1(h,\phi ) +2A(M_3(h,h,\phi )+R_1(h,\phi ),h)+ i \Lambda \big [R_2(h,\phi ) \nonumber \\&\quad + B(h,Q_3(\phi ,h,\phi ) + R_2(h,\phi ))+ B(M_3(h,h,\phi ) + R_1(h,\phi ),\phi ) \big ],\nonumber \\ \end{aligned}$$

(13.37)

and we have defined

$$\begin{aligned} \widetilde{\mathcal {N}}_{3,1}&:= M_3(h,h,\phi ) - M_3(H,H,\Psi ) , \end{aligned}$$

(13.38)

$$\begin{aligned} \widetilde{\mathcal {N}}_{3,2}&:= 2A(M_2(h,\phi ),h) - 2A(M_2(H,\Psi ),H) , \end{aligned}$$

(13.39)

$$\begin{aligned} \widetilde{\mathcal {N}}_{3,3}&:= i \Lambda \left[ Q_3(\phi ,h,\phi ) - Q_3(\Psi ,H,\Psi ) \right] , \end{aligned}$$

(13.40)

$$\begin{aligned} \widetilde{\mathcal {N}}_{3,4}&:= i \Lambda \left[ B(M_2(h,\phi ),\phi ) - B(M_2(H,\Psi ),\Psi ) \right] , \end{aligned}$$

(13.41)

$$\begin{aligned} \widetilde{\mathcal {N}}_{3,5}&:= i \Lambda \left[ B(h, Q_2(\phi ,\phi )) - B(H, Q_2(\Psi ,\Psi )) \right] . \end{aligned}$$

(13.42)

Proof of (13.32). We start by proving that each term in ${\mathcal {N}}_4$ is bounded by the right hand side of (13.32). The bound for $R_1 = {[ G (h) \phi ]}_{\ge 4}(t)$ is an immediate consequence of the $L^1$ estimate (13.12). The bound for $\Lambda R_2$ can be obtained from (13.22) using Cauchy’s inequality and the $L^2$ bounds for $M_2$, $M_3$ and $R_1$ given respectively in (5.33), (5.35) and (13.9). From the definition of $A$ in (5.1)–(5.2) we see that for any integer $l$

$$\begin{aligned} \left| \widehat{P_k A}(F,G) \right| \lesssim 2^{-l k_+} {\Vert F \Vert }_{H^{l+1}} {\Vert G \Vert }_{H^{l+1}} . \end{aligned}$$

(13.43)

Using the $L^2$ bounds (5.35) on $M_3$, (13.9) on $R_1$, and the a priori assumptions, it immediately follows that

$$\begin{aligned} \left| \widehat{P_k A}(M_3(h,h,\phi ) + R_1(h,\phi ),h) \right|&\lesssim 2^{-(N_0-10) k_+} {\Vert M_3 + R_1 \Vert }_{H^{N_0-9}} {\Vert h \Vert }_{H^{N_0-9}} \\&\lesssim {\varepsilon }_1^4 2^{-(N_0-10) k_+} {(1+t)}^{4p_0 -1} . \end{aligned}$$

Similarly, from the definition of $B$ in (5.1)–(5.3) we have

$$\begin{aligned} \left| \widehat{P_k B}(F,G) \right| \lesssim 2^{-l k_+} {\Vert F \Vert }_{H^l} {\Vert \partial _x G \Vert }_{H^l} . \end{aligned}$$

(13.44)

Using again (5.35) and (13.10) we get

$$\begin{aligned}&\left| {\mathcal {F}}\left[ P_k \Lambda B(h,Q_3(\phi ,h,\phi ) + R_2(h,\phi ))+ P_k \Lambda B(M_3(h,h,\phi ) + R_1(h,\phi ),\phi ) \right] \right| \\&\lesssim 2^{-(N_0-15) k_+} \left[ {\Vert h \Vert }_{H^{N_0-10}} {\Vert \partial _x (Q_3 + R_2) \Vert }_{H^{N_0-10}} + {\Vert M_3 \!+\! R_1 \Vert }_{H^{N_0-10}} {\Vert \partial _x \phi \Vert }_{H^{N_0-10}} \right] \\&\lesssim {\varepsilon }_1^4 2^{-(N_0-15) k_+} {(1+t)}^{4p_0 -1} . \end{aligned}$$

We now estimate the terms (13.38)–(13.42). From (4.17) we see that

$$\begin{aligned} -\widetilde{\mathcal {N}}_{3,1}&= M_3(A,h,\phi ) + M_3(H,A,\phi ) + M_3(H,H,B) . \end{aligned}$$

(13.45)

From the definition of $M_3$ in (4.9) we see that for any integer $0\le l\le N_0-10$

$$\begin{aligned}&\left| \widehat{P_k M_3}(E,F,G) \right| \nonumber \\&\quad \lesssim 2^{-l k_+} 2^k \Big [ {\Vert E \Vert }_{W^{N_0/2 - 5,\infty }} {\Vert F \Vert }_{H^{l+2}} {\Vert \partial _x G \Vert }_{H^{l+2}} + {\Vert E \Vert }_{H^{l+2}} {\Vert F \Vert }_{W^{N_0/2 - 5,\infty }} \nonumber \\&\quad {\Vert \partial _x G \Vert }_{H^{l+2}} + {\Vert E \Vert }_{H^{l+2}} {\Vert F \Vert }_{H^{l+2}} {\Vert |\partial _x| G \Vert }_{W^{N_0/2 - 5,\infty }} \Big ] . \end{aligned}$$

(13.46)

Applying this together with the $L^\infty$ bounds (5.27) on $A$ and $\Lambda B$, the a priori bounds (5.23), (4.25) and (4.26), one can obtain the desired bound for each of the three terms in (13.45).

To estimate (13.39) we write

$$\begin{aligned} -\frac{1}{2} \widetilde{\mathcal {N}}_{3,2}&= A(M_2(A,\phi ),h) + A(M_2(H,B),h) + A(M_2(H,\Psi ),A) . \end{aligned}$$

Notice that for any integer $0\le l\le N_0-10$

$$\begin{aligned} {\Vert P_k M_2(F,G) \Vert }_{L^2} \lesssim 2^{-l k_+} 2^k \Big [ {\Vert F \Vert }_{H^l} {\Vert |\partial _x| G \Vert }_{W^{N_0/2-5,\infty }} + {\Vert F \Vert }_{W^{N_0/2-5,\infty }} {\Vert \partial _x G \Vert }_{H^l} \Big ] . \end{aligned}$$

(13.47)

Using (13.43), (13.47), the estimates for $A$ in (5.26) and (5.27), and Proposition 4.2, we get

$$\begin{aligned}&\left| {\mathcal {F}}[ P_k A (M_2(A,\phi ),h) ] \right| \nonumber \\&\quad \lesssim 2^{-(N_0-15) k_+} {\Vert M_2(A,\phi ) \Vert }_{H^{N_0-10}} {\Vert h \Vert }_{H^{N_0-10}}\lesssim 2^{-(N_0-15) k_+} {(1+t)}^{p_0} \\&\quad \Big [ {\Vert A \Vert }_{H^{N_0-8}} {\Vert |\partial _x| \phi \Vert }_{W^{N_0/2-5,\infty }} + {\Vert A \Vert }_{W^{N_0/2-5,\infty }} {\Vert \partial _x \phi \Vert }_{H^{N_0-8}} \Big ] \\&\quad \lesssim {\varepsilon }_1^4 2^{-(N_0-15) k_+} {(1+t)}^{3p_0 -1} . \end{aligned}$$

To bound $\widetilde{N}_{3,3}$ in (13.40) we first write it as

$$\begin{aligned} - \widetilde{{\mathcal {N}}}_{3,3}&= \Lambda \Big [ Q_3(B,h,\phi ) + Q_3(\Psi ,A,\phi ) + Q_3(\Psi ,H,B) \Big ] . \end{aligned}$$

(13.48)

We then notice that for any integer $0\le l\le N_0-10$ one has

$$\begin{aligned} \left| \widehat{P_k Q_3}(E,F,G) \right|&\lesssim 2^{-l k_+} \Big [ {\Vert |\partial _x| E \Vert }_{W^{N_0/2 - 5,\infty }} {\Vert F \Vert }_{H^{l+2}} {\Vert \partial _x G \Vert }_{H^{l+3}} \nonumber \\&\quad + {\Vert \partial _x E \Vert }_{H^l} {\Vert F \Vert }_{W^{N_0/2 - 5,\infty }} {\Vert \partial _x G \Vert }_{H^{l+2}} \Big ] . \end{aligned}$$

(13.49)

One can that then bound each one of the three summands in (13.48) by using the above estimate together with Proposition 4.2, (5.23), (5.26) and (5.27).

(13.41) can be estimated in a similar fashion to what we have done above by writing out the difference as sums of quartic terms, and using (13.44) together with (13.47), (5.26), (5.27) and Proposition 4.2. The term (13.42) can also be estimated similarly by using in addition

$$\begin{aligned} {\Vert P_k Q_2(F,G) \Vert }_{L^2}&\lesssim 2^{-l k_+} \Big [ {\Vert \partial _x F \Vert }_{H^l} ( {\Vert \partial _x G \Vert }_{W^{N_0/2-5,\infty }} + {\Vert |\partial _x| G \Vert }_{W^{N_0/2-5,\infty }} ) \nonumber \\&\quad + ( {\Vert \partial _x F \Vert }_{W^{N_0/2-5,\infty }} + {\Vert |\partial _x| F \Vert }_{W^{N_0/2-5,\infty }} ) {\Vert |\partial _x| G \Vert }_{H^l} \Big ] , \end{aligned}$$

(13.50)

for any $0 \le l \le N_0 -10$.

Proof of (13.33). First observe that from (13.10) we already have the desired bound for $R_1$ and $\Lambda R_2$. To bound the three remaining contributions from $\widetilde{N}_4$ in (13.37) and the five terms (13.38)–(13.42) we first observe that for ${\Gamma }= 1$ or $S$ we have the following $L^2$ estimates:

$$\begin{aligned}&{\Vert {\Gamma }P_k A(P_{k_1}F, P_{k_2}G) \Vert }_{L^2} \nonumber \\&\quad \lesssim 2^k 2^{-(N_0/2 - 10)k_+}\Big [ \left( {\Vert {\Gamma }P_{k_1}F \Vert }_{H^{N_0/2-10}} + {\Vert P_{k_1}F \Vert }_{H^{N_0/2-10}} \right) {\Vert P_{k_2} G \Vert }_{W^{N_0/2 -10,\infty }} \nonumber \\&\quad \quad + {\Vert P_{k_1} F \Vert }_{W^{N_0/2 -10,\infty }} {\Vert {\Gamma }P_{k_2}G \Vert }_{H^{N_0/2-10}} \Big ] ,\end{aligned}$$

(13.51)

$$\begin{aligned}&{\Vert {\Gamma }P_k B(P_{k_1}F, P_{k_2}G) \Vert }_{L^2} \nonumber \\&\quad \lesssim 2^{-(N_0/2 - 10)k_+}\Big [ \left( {\Vert {\Gamma }P_{k_1}F \Vert }_{H^{N_0/2-10}} + {\Vert P_{k_1}F \Vert }_{H^{N_0/2-10}} \right) 2^{k_2} {\Vert P_{k_2} G \Vert }_{W^{N_0/2 -10,\infty }} \nonumber \\&\quad \quad + {\Vert P_{k_1} F \Vert }_{W^{N_0/2 -10,\infty }} 2^{k_2} \left( {\Vert {\Gamma }P_{k_2}G \Vert }_{H^{N_0/2-10}} + {\Vert P_{k_2}G \Vert }_{H^{N_0/2-10}} \right) \Big ] . \end{aligned}$$

(13.52)

We also have the following $L^\infty$ estimates for $M_3$ and $Q_3$:

$$\begin{aligned}&{\Vert P_k M_3(P_{k_1}E, P_{k_2}F, P_{k_3}G) \Vert }_{L^\infty } \lesssim 2^{-(N_0/2 - 15)k_+} 2^k 2^{k_2} 2^{\max (k_1,k_2,k_3)}\nonumber \\&\quad {\Vert P_{k_1}E \Vert }_{W^{N_0/2 -10,\infty }} {\Vert P_{k_2}F \Vert }_{W^{N_0/2 -10,\infty }} {\Vert P_{k_3}G \Vert }_{W^{N_0/2 -10,\infty }} , \end{aligned}$$

(13.53)

$$\begin{aligned}&{\Vert P_k Q_3(P_{k_1}E, P_{k_2}F, P_{k_3}G) \Vert }_{L^\infty } \lesssim 2^{-(N_0/2 - 15)k_+} 2^{k_1} 2^{k_3} 2^{\max (k_2,k_3)}\nonumber \\&\quad {\Vert P_{k_1}E \Vert }_{W^{N_0/2 -10,\infty }} {\Vert P_{k_2}F \Vert }_{W^{N_0/2 -10,\infty }} {\Vert P_{k_3}G \Vert }_{W^{N_0/2 -10,\infty }} . \end{aligned}$$

(13.54)

From the homogeneity of degree $2$ of $M_2$ and $Q_2$, and of degree $3$ of $M_3$ and $Q_3$, one can obtain identities similar to (5.28) for the symbols of these operators, and deduce the following analogues of the commutation identities (5.29):

$$\begin{aligned} S M_2(F,G)&= M_2(SF,G) + M_2(F,SG) - 2M_2(F,G) , \nonumber \\ S Q_2(F,G)&= Q_2(SF,G) + Q_2(F,SG) - 2Q_2(F,G) , \nonumber \\ S M_3(E,F,G)&= M_3(SE,F,G) + M_3(E,SF,G) \nonumber \\&\quad + M_3(E,F,SG) - 3M_2(E,F,G) , \nonumber \\ S Q_3(E,F,G)&= Q_3(SE,F,G) + Q_3(E,SF,G) + Q_3(E,F,SG) \nonumber \\&\quad - 3Q_2(E,F,G) . \end{aligned}$$

(13.55)

One can then use (13.51)–(13.54) together with the commutation identities (5.29) and (13.55), the estimates (5.23), (5.33) and (5.35), (13.10), and arguments similar to those used above and in Sect. 5.4, in particular in the proof of Lemma 5.5, to obtain

$$\begin{aligned}&{\Vert P_k \mathcal {N}_4 \Vert }_{L^2} + {\Vert P_k S \mathcal {N}_4 \Vert }_{L^2} + \sum _{j=1}^5 {\Vert P_k \widetilde{\mathcal {N}}_{3,j} \Vert }_{L^2} + {\Vert P_k S \widetilde{\mathcal {N}}_{3,j} \Vert }_{L^2}\\&\quad \lesssim {\varepsilon }_1^4 {(1+t)}^{20p_0 - 3/2} 2^{-(N_0/2-20)k_+} \end{aligned}$$

which is the desired conclusion. $\square$

1.2.2 C.2.2 Proof of Lemma 13.5

For $t_1 \le t_2 \in [2^m-2, 2^{m+1}]$ and $|\xi | \in [2^k, 2^{k+1}]$ let us define

$$\begin{aligned} F(\xi ) = \int _{t_1}^{t_2} e^{i H(\xi ,s)} e^{is\Lambda (\xi )} \widehat{D} (\xi ,s) \, ds . \end{aligned}$$

We then have

$$\begin{aligned}&| F(\xi ) | \lesssim {\Vert {\mathcal {F}}^{-1} F \Vert }_{L^1(|x| \le 2^{m/2} )} + {\Vert {\mathcal {F}}^{-1} F \Vert }_{L^1(|x| \ge 2^{m/2} )}\\&\quad \lesssim 2^{m/4} {\Vert F \Vert }_{L^2} + 2^{-m/4} 2^{-k} {\Vert \xi \partial _\xi F \Vert }_{L^2} . \end{aligned}$$

Thus, to obtain (13.35) it suffices to show the following two estimates:

$$\begin{aligned}&{\Vert F \Vert }_{L^2} \lesssim {\delta }2^{-3m/8}\end{aligned}$$

(13.56)

$$\begin{aligned}&{\Vert \xi \partial _\xi F \Vert }_{L^2} \lesssim {\delta }2^{21 m p_0} . \end{aligned}$$

(13.57)

(13.56) can be easily verified using the $L^2$ bound in (13.34). To prove (13.57) we write:

$$\begin{aligned} \xi \partial _\xi F(\xi )&= F_1(\xi ) + F_2(\xi ) + F_3(\xi ) \nonumber \\ F_1(\xi )&= \int _{t_1}^{t_2} e^{i H(\xi ,s)} \left( i \xi \partial _\xi H(\xi ,s) \right) e^{is\Lambda (\xi )} \widehat{D}(\xi ,s) \, ds , \nonumber \\ F_2(\xi )&= \int _{t_1}^{t_2} e^{i H(\xi ,s)} S(\xi ) e^{is\Lambda (\xi )} \widehat{D}(\xi ,s) \, ds , \nonumber \\ F_3(\xi )&= \frac{1}{2} \int _{t_1}^{t_2} e^{i H(\xi ,s)} s\partial _s \left( e^{is\Lambda (\xi )} \widehat{D}(\xi ,s) \right) \, ds , \end{aligned}$$

(13.58)

having denoted $S(\xi ) := \xi \partial _\xi - \frac{1}{2} s \partial _s$; notice that $S(\xi ) \widehat{f}(\xi ) = - \widehat{S f}(\xi ) -\widehat{f}(\xi )$, where $S$ is the scaling vector field. Using the definition of $H$ in (6.10) and the a priori assumptions, it is easy to see that for $s \in [2^m-2, 2^{m+1}]$ one has

$$\begin{aligned}&{\Vert \xi \partial _\xi H(\xi ,s) \Vert }_{L^2} \lesssim 2^{m p_0} , \nonumber \\&{\Vert \partial _s H(\xi ,s) \Vert }_{L^\infty } \lesssim 2^{-m} . \end{aligned}$$

(13.59)

Using the first bound above and the $L^\infty$ bound in (13.34) we see that

$$\begin{aligned}&{\Vert F_1 \Vert }_{L^2} \lesssim \int _{t_1}^{t_2} {\Vert \xi \partial _\xi H(s) \Vert }_{L^2} {\Vert \widehat{D}(s) \Vert }_{L^\infty } \, ds \lesssim {\delta }2^m 2^{m p_0} 2^{m(20 p_0 - 1)} \lesssim {\delta }2^{21 m p_0} , \end{aligned}$$

as desired. Since $[\left( \xi \partial _\xi - \frac{1}{2} s \partial _s \right) , e^{is\Lambda (\xi )}] = 0$, we can use the $L^2$ bounds in (13.34) to deduce

$$\begin{aligned}&{\Vert F_2 \Vert }_{L^2} \lesssim \int _{t_1}^{t_2} {\Vert D(s) \Vert }_{L^2} + {\Vert S D(s) \Vert }_{L^2} \, ds \lesssim {\delta }, \end{aligned}$$

which is more than sufficient. To estimate $F_3$ we integrate by parts in $s$, use the second bound in (13.59) and (13.34) to obtain:

$$\begin{aligned}&{\Vert F_3 \Vert }_{L^2} \lesssim 2^m \sup _{s\in [2^m-2, 2^{m+1}] } {\Vert D(s) \Vert }_{L^2} + \int _{t_1}^{t_2} s {\Vert \partial _s H(\xi ,s) \Vert }_{L^\infty } {\Vert D(s) \Vert }_{L^2} \, ds \lesssim {\delta }. \end{aligned}$$

This proves (13.57) and concludes the proof of the Lemma. $\square$