2.1 Description of the problem

Let $\alpha \in (0,2\pi )$ be noncommensurable with $\pi $ . Consider the following classical problem:

• ○ Suppose $f:\mathbb {S}^1\mapsto \mathbb {S}^1$ is smooth and is close to 0. Is the diffeomorphism $x\mapsto x+\alpha +f(x)$ smoothly conjugate to the rotation $\varrho _{\alpha }$ ?

Of course, an integrablity condition must be posed for f. Recall the classical theorem due to Denjoy:

If $x\mapsto x+\alpha +f(x)$ has rotation number different from $\alpha $ , then by Denjoy’s theorem, it definitely cannot conjugate to $\varrho _{\alpha }$ . On the other hand, Denjoy’s theorem has no implication on smoothness of the conjugation, even if the known diffeomorphism is smooth. Therefore the question is not answered by Denjoy’s theorem. We modify it as follows:

• ○ Suppose $f:\mathbb {S}^1\mapsto \mathbb {S}^1$ is smooth and is close to 0, such that the diffeomorphism $x\mapsto x+\alpha +f(x)$ still has rotation number $\alpha $ . Is it smoothly conjugate to the rotation $\varrho _{\alpha }$ ?

Let us write the hypothetical conjugation as $\eta (x)=x+u(x)$ , and try to solve the superficially more general conjugacy equation

$$ \begin{align*}\eta(x+\alpha)=\eta(x)+\alpha+f\circ\eta(x)-\lambda \end{align*} $$

for $\eta $ and the auxiliary real parameter $\lambda $ Footnote ² . A simple rearrangement converts this equation to

(2.1) $$ \begin{align} \Delta_\alpha u=f\circ(\mathrm{Id}+u)-\lambda, \quad\text{where}\quad \Delta_\alpha u:=u\circ\varrho_{\alpha}-u. \end{align} $$

We may also compose the equation with $(\mathrm {Id}+u)^\iota $ , and reformulate equation (2.1) as

(2.2) $$ \begin{align} \big[\Delta_\alpha u\big]\circ(\mathrm{Id}+u)^\iota=f-\lambda. \end{align} $$

We will discuss the technical difference between (2.1) and (2.2) in Appendix A.

To clarify the obstacles of solving the conjugacy problem, we may linearize either (2.1) or (2.2) at $(u,\lambda )=(0,0)$ along direction $(v,\mu )$ . Neglecting f as well, we obtain the linear homological equation

$$ \begin{align*}\Delta_\alpha v+\mu=h. \end{align*} $$

This linearized equation is solved via Fourier transform: normalizing $\hat v(0)=0$ , the unique solution is

$$ \begin{align*}v(x)=\sum_{k\neq0}\frac{\hat h(k)e^{ikx}}{e^{ik\alpha}-1}, \quad \mu=\operatorname{\mathrm{Avg}} h, \end{align*} $$

Consequently, in order that the conjugacy problem is solvable even at the linear level, an additional number-theoretic condition must be posed for the number $\alpha $ . We require that $\alpha /\pi $ is Diophantine of type $(\tau ,\gamma )$ : there are $\tau>0$ and $\gamma>0$ such that

(2.3) $$ \begin{align} \left|\frac{q\alpha}{\pi}-p\right|\geq\frac{\gamma }{q^{\tau}}, \quad \forall p,q\in\mathbb{Z}\setminus\{0\}. \end{align} $$

Such numbers are abundant. Indeed, Liouville’s inequality asserts that if an algebraic number $\alpha $ is of degree D, then the inequality $|q\alpha -p|\geq c/q^{D-1}$ must hold for some c. A measure-theoretic argument asserts that with $\tau>1$ fixed, the set of $(\tau ,\gamma )$ Diophantine numbers with $\gamma $ exhausting all positive real numbers form a set of full Lebesgue measure and of first Baire category.

Obviously, if $\alpha $ satisfies the Diophantine condition (2.3), then $\Delta _\alpha ^{-1}$ is a well-defined operator mapping functions of mean zero on $\mathbb {S}^1$ to functions of mean zero, satisfying

(2.4) $$ \begin{align} \|\Delta_\alpha^{-1}f\|_{H^s}\leq C\gamma^{-1}\|f\|_{H^{s+\tau}}, \quad \text{if }\operatorname{\mathrm{Avg}} f=0. \end{align} $$

Here C is an absolute constant independent of $\gamma $ or s. This enables one to solve the linearized equation $\Delta _\alpha v+\mu =h$ , at least for very regular right-hand-side. But due to this loss of information on regularity, a usual iterative scheme to solve (2.2) will necessarily terminate after finitely many steps.

2.2 Approximate right inverse

Since the operator $\Delta _\alpha ^{-1}$ causes a loss of regularity that cannot be compensated by any known elliptic technique, it is natural to employ modified Newtonian iterative scheme to solve the conjugacy problem (2.1). At a first glance it appears to be friendlier than (2.2), but in fact its linearized operator only admits an approximate right inverse. This brings more technicality for the application of Nash-Moser scheme. Let us describe the strategy of resolving it.

We set the unknwon $U=(u,\lambda )$ , and define a mapping

$$ \begin{align*}\mathscr{F}(f,U)=\Delta_\alpha u-f\circ(\mathrm{Id}+u)+\lambda. \end{align*} $$

The linearization of $\mathscr {F}$ at $U=(u,\lambda )$ along $V=(v,\mu )$ is

$$ \begin{align*}D_U\mathscr{F}(f,U)V=\Delta_\alpha v-f'\circ(\mathrm{Id}+u)v+\mu. \end{align*} $$

In the literature of dynamical systems, given $U=(u,\lambda )$ , the linearized equation

(2.5) $$ \begin{align} D_U\mathscr{F}(f,U)V=h \end{align} $$

for the unknown $V=(v,\mu )$ with a given right-hand-side h is referred to as the homological equation.

In general, one only expects to solve the homological equation (2.5) approximately, since the operator $\Delta _\alpha -f'\circ (\mathrm {Id}+u)$ is hard to invert. To explain the meaning of being “approximately solvable,” we write down a simple yet crucial identity, which is a direct consequence of the definition of $\mathscr {F}$ :

(2.6) $$ \begin{align} f'\circ(\mathrm{Id}+u)=\frac{\Delta_\alpha u'}{1+u'}-\frac{[\mathscr{F}(f,U)]'}{1+u'}. \end{align} $$

So in fact

(2.7) $$ \begin{align} \begin{aligned} D_U\mathscr{F}(f,U)V &=\Delta_\alpha v-\frac{\Delta_\alpha u'\cdot v}{1+u'} +\frac{[\mathscr{F}(f,U)]'}{1+u'}v+\mu\\ &=(1+u'\circ\varrho_{\alpha})\Delta_\alpha\left(\frac{v}{1+u'}\right)+\frac{[\mathscr{F}(f,U)]'}{1+u'}v+\mu. \end{aligned} \end{align} $$

Thus, defining

$$ \begin{align*}\Psi(U)h=\left((1+u')\Delta_\alpha^{-1}\left[\frac{h-\mu(h)}{1+u'\circ\varrho_{\alpha}}\right],\,\mu(h)\right), \quad \text{where } \mu(h)=\frac{\mathrm{Avg}\big((1+u'\circ\varrho_{\alpha})^{-1}h\big)}{\mathrm{Avg}\big((1+u'\circ\varrho_{\alpha})^{-1}\big)}, \end{align*} $$

there holds

(2.8) $$ \begin{align} D_U\mathscr{F}(f,U)\Psi(U)h-h =[\mathscr{F}(f,U)]'\Delta_\alpha^{-1}\left(\frac{h}{1+u'\circ\varrho_{\alpha}}-\mu(h)\right). \end{align} $$

Equality (2.8) is the property defining “approximate solvability”: the linear operator $\Psi (U)$ is an exact right inverse of $D_U\mathscr {F}(f,U)$ at a precise solution U of $\mathscr {F}(f,U)=0$ .

The loss of regularity caused by $\Delta _\alpha ^{-1}$ is usually identified as the primary difficulty for solving $\mathscr {F}(f,U)=0$ . A modified Newtonian scheme was implemented by Kolmogorov [Reference Kolmogorov55], Arnold [Reference Arnold6, Reference Arnold7, Reference Arnold8] and Moser [Reference Moser68, Reference Moser69]. Being technically different, the general idea shared by them is to define a sequence $U_k$ by inductively solving a sequence of homological equations:

(2.9) $$ \begin{align} U_{k+1}=U_k-S_k\big[\Psi(U_k)\mathscr{F}(f,U_k)\big]. \end{align} $$

Here the operator $S_k$ is either the restriction operator to a smaller domain in case all functions involved are analytic, or a smoothing operator in case all functions are of finite differentiability. The quadratic convergence property of Newtonian scheme cancels the large constants so produced and ensures the convergence of $U_k$ to a genuine solution. Proving this convergence is where the complexity accumulates.

The issue with approximate invertibility was first noticed by Zehnder [Reference Zehnder85, Reference Zehnder86, Reference Zehnder87], who introduced adapted hard implicit function theorems (Nash-Moser type theorems) to fit such conjugacy problems into a unified formalism. The proofs of these hard implicit function theorems still rely on modified Newtonian schemes, essentially being (2.9). For the statement and detailed proof of Nash-Moser theorem with either exactly or approximately invertible linearized operator, see [Reference Zehnder85, Reference Zehnder86, Reference Zehnder87, Reference Hamilton36]. Though taking various forms on various graded spaces, all versions of Nash-Moser type theorems share the common features of restoring regularity through smoothing operators and ensuring convergence through quadratic property of Newtonian algorithm. While being universal and powerful theoretically, the practical disadvantage of this approach is that the allowed magnitude of perturbation is usually extremely small. A different approach is required if the aim is to find solutions of physically reasonable size.

2.3 Quick introduction to paradifferential calculus

Surprisingly, if paradifferential calculus is used to attack on (2.1), we can drastically simplify the proof of existence of solution. For simplicity, we do not present a panorama of paradifferential calculus in this article; instead, we list the propositions that suffice for our exposition, along with brief descriptions of the heuristics behind them. So let us first introduce the paraproduct operators, the central objects of this paper.

Definition 2.1. Fix Littlewood-Paley decomposition as in Definition 1.1. Given a distribution $a=a(x)$ on $\mathbb {T}^n$ , the paraproduct operator $T_a=\mathbf {Op}^{\mathrm {PM}}(a)$ associated to a is defined as, regardless of the meaning of convergence,

(2.10) $$ \begin{align} T_au=\mathbf{Op}^{\mathrm{PM}}(a)u:=\sum_{j\geq0} S_{j-3}a\cdot \Delta_ju. \end{align} $$

For $j\geq 1$ , notice that the summand $S_{j-3}a\cdot \Delta _ju$ in (2.10) has its Fourier support contained in the annulus $\{0.25\cdot 2^j \leq |\xi |\leq 2.25\cdot 2^j\}$ . A standard construction is the paraproduct decomposition:

$$ \begin{align*}au=T_au+T_ua+R_{\mathrm{PM}}(a,u), \quad R_{\mathrm{PM}}(a,u)=\sum_{j,k:|j-k|<3}\Delta_ja\cdot\Delta_ku, \end{align*} $$

as long as the right-hand-side is well-defined in some suitable function space. Such decomposition separates apart high-low, low-high and high-high frequency interactions in a given multiplication $au$ , and is therefore widely used in the study of harmonic analysis and dispersive partial differential equations. We refer the reader to Section 4.4 of [Reference Grafakos34] for a comprehensive description of the role it plays in harmonic analysis. [Reference Kenig, Ponce and Vega48, Reference Kato46, Reference Staffilani80] are typical examples of application of these constructions in dispersive PDEs.

The following results on paraproduct operators are directly cited from [Reference Bony11] (see also Chapters 8-10 of [Reference Hörmander45]). They are the three fundamental ingredients of paradifferential calculus.

Proposition 2.1 (Continuity of paraproduct, rough version).

If $a\in L^\infty (\mathbb {T}^n)$ , then $T_a$ is a bounded linear operator from $H^s$ to $H^s$ for all $s\in \mathbb {R}$ , and in fact

$$ \begin{align*}\|T_a\|_{\mathcal{L}(H^s,H^s)}\lesssim_s|a|_{L^\infty}. \end{align*} $$

Proposition 2.2 (Composition of paraproducts, rough version).

If $a,b\in C_*^r$ with $r>0$ , then $T_aT_b-T_{ab}$ is a bounded linear operator from $H^s$ to $H^{s+r}$ for all $s\in \mathbb {R}$ . Furthermore, $T_aT_b-T_{ab}$ is continuously bilinear in $a,b\in C_*^r$ :

$$ \begin{align*}\big\|T_aT_b-T_{ab}\big\|_{\mathcal{L}(H^s,H^{s+r})} \lesssim_{s,r}|a|_{C^r_*}|b|_{C^r_*}. \end{align*} $$

Proposition 2.3 (Paralinearization, rough version).

Suppose $r=s-n/2>0$ , $u\in H^s(\mathbb {T}^n;\mathbb {R}^L)$ . Let $N_{s+r}$ be the least integer such that $N_{s+r}>s+r$ . Suppose $F=F(x,z)\in C^{N_{s+r}+2}(\mathbb {T}^n\times \mathbb {R}^L)$ . Then there holds the following paralinearization formula:

$$ \begin{align*}F(x,u)-F(x,0) =T_{F^{\prime}_z(x,u)}u +\mathcal{R}_{\mathrm{PL}}\big(F,u\big)\dot u \in H^s+H^{s+r}, \end{align*} $$

where $\mathcal {R}_{\mathrm {PL}}\big (F,u\big )$ is a bounded linear operator from $H^s\mapsto H^{s+r}$ , so that

$$ \begin{align*}\begin{aligned} \left\|\mathcal{R}_{\mathrm{PL}}(F,u)u\right\|_{H^{s+r}} &\leq |F|_{C^{N_{s+r}+2}}\big(1+\|u\|_{H^s}\big)\|u\|_{H^s}. \end{aligned} \end{align*} $$

If furthermore $F\in C^{N_{s+r}+2+k}$ , then $\mathcal {R}_{\mathrm {PL}}\big (F,u\big )$ has $C^k$ dependence on $u\in H^s H^{s+r}$ .

Precise statements of these results can be found in Section 3. Let us describe how Proposition 2.1–2.3 should be interpreted. The reason that Bony introduced paradifferential calculus in [Reference Bony11] is to understand microlocal regularity for solutions of nonlinear partial differential equations. For this purpose, it is necessary to try to separate out the most irregular part out of a given nonlinear expression $F(x,u)$ , which is the content of the paralinearization theorem 2.3. The terminology paralinearzation refers to the observation that the most irregular part, $T_{F^{\prime }_z(x,u)u}u$ out of $F(x,u)$ , is exactly given by the paradifferential operator corresponding to the linearization of F.

Bony observed that $T_a$ exhibits advantages over the usual product operator: the operator $M_a\colon u\mapsto au$ is bounded on all Sobolev spaces $H^s$ only if $a\in C^\infty $ , while the boundedness of $T_a$ on $H^s$ relies on the minimal regularity assumption that $a\in L^\infty $ , as indicated by Proposition 2.1. On the other hand, $T_a$ shares the same algebraic structure as the usual multiplication $M_a$ , up to smoothing remainders. While the mapping $a\mapsto M_a$ obviously is an algebra embedding from the function algebra $C^r_*$ to the operator algebra $\mathcal {L}(H^s,H^s)$ (with $s\leq r$ ), Proposition 2.2 shows that the mapping $a\mapsto T_a$ is an algebra homomorphism from the function algebra $C^r_*$ to the operator algebra $\mathcal {L}(H^s,H^s)$ modulo smoothing operators for any $s\in \mathbb {R}$ .

Therefore, Proposition 2.1–2.3 provides a procedure that enables one to study a nonlinear expression $F(x,u)$ as if it were linear: by paralinearization, one separates out the most irregular part, which obeys the same algebraic laws of the usual (linear) pseudo-differential operators.

2.4 Parahomological equation

In this subsection, we illustrate how the conjugacy equation (2.1) can be solved in an extremely simple manner, provided that the paradifferential tools from Subsection 2.3 are admitted as standard, as in the field of harmonic analysis and dispersive partial differential equations. To the best of the authors’ knowledge, the proof presented below stands as the shortest and simplest proof of KAM theorem for the circular map model available in literature.

Proof. Simple solution for (2.1).

We notice that the general Dirichlet approximate theorem implies $\tau \geq 1$ . Fix $s\geq \tau +1.5+\varepsilon $ . Suppose in a priori $u\in H^s$ , so that by Sobolev embedding, we have $u\in C^r_*$ with $r=s-0.5>1$ . Suppose f is sufficiently smooth, for example $f\in C^{N_{s+r}+2}$ , where $N_{s+r}$ is the least integer strictly greater than $s+r$ .

By Proposition 2.3, we have the paralinearization formula for $\mathscr {F}(f,U)$ :

(2.11) $$ \begin{align} \begin{aligned} \mathscr{F}(f,U) &=\Delta_\alpha u-f-T_{f'\circ(\mathrm{Id}+u)}u -\mathcal{R}_{\mathrm{PL}}(f,u)\cdot u+\lambda\\ &=\Delta_\alpha u-T_{\Delta_\alpha u'/(1+u')}u-f +T_{[\mathscr{F}(f,U)]'/(1+u')}u -\mathcal{R}_{\mathrm{PL}}(f,u)\cdot u+\lambda\\ &=T_{(1+u'\circ\varrho_{\alpha})}\Delta_\alpha T_{1/(1+u')}u -f +T_{[\mathscr{F}(f,U)]'/(1+u')}u -\mathcal{R}_{\mathrm{PL}}(f,u)\cdot u+R_1(u)+\lambda. \end{aligned} \end{align} $$

Here we used the key identity (2.6) again. The last equality in (2.11) is valid since

$$ \begin{align*}[T_{1/(1+u')}u]\circ\varrho_{\alpha}=T_{1/(1+u'\circ\varrho_{\alpha})}(u\circ\varrho_{\alpha}). \end{align*} $$

Note that $\varrho _{\alpha }$ commutes with any Fourier multiplier, which is the reason that this equality holds. The remainder

$$ \begin{align*}\begin{aligned} R_1(u)&=\mathcal{R}_{\mathrm{CM}}(1+a,1+b_1)u+\mathcal{R}_{\mathrm{CM}}(1+a,b_2)u\\ &=\mathcal{R}_{\mathrm{CM}}(a,b_1)u+\mathcal{R}_{\mathrm{CM}}(a,b_2)u, \end{aligned} \end{align*} $$

where the last step is because that $\mathcal {R}_{\mathrm {CM}}(\cdot ,\cdot )$ is bilinear, and

$$ \begin{align*}a=u'\circ\varrho_{\alpha},\quad b_1=\frac{-u'\circ\varrho_{\alpha}}{1+u'\circ\varrho_{\alpha}},\quad b_2=-\frac{\Delta_\alpha u'}{(1+u')(1+u'\circ\varrho_{\alpha})} \quad\text{are all in }H^{s-1}\subset C_*^{\tau+\varepsilon}. \end{align*} $$

is produced when replacing $T_{ab}$ by $T_aT_b$ in view of Proposition 2.2. Formally, we obtain (2.11) simply by replacing all products with paraproducts in (2.7).

We then consider the parahomological equation for the unknown $U=(u,\lambda )\in H^s\times \mathbb {R}$ :

(2.12) $$ \begin{align} \begin{aligned} T_{(1+u'\circ\varrho_{\alpha})}\Delta_\alpha T_{1/(1+u')}u &=f+\mathcal{R}_{\mathrm{PL}}(f,u)\cdot u-R_1(u)-\lambda. \end{aligned} \end{align} $$

Equation (2.12) collects all terms but the one linear in $\mathscr {F}(f,U)$ in (2.11). It can be directly reduced to fixed point form, which we shall refer as the parainverse equation following Hörmander [Reference Hörmander44]:

(2.13) $$ \begin{align} \begin{aligned} u &=T_{1/(1+u')}^{-1}\Delta_\alpha^{-1}T_{(1+u'\circ\varrho_{\alpha})}^{-1}\left[f+\mathcal{R}_{\mathrm{PL}}(f,u)\cdot u-R_1(u)-\lambda\right]. \end{aligned} \end{align} $$

The equation itself uniquely determines $\lambda $ : it should balance out the mean value to make $\Delta _\alpha ^{-1}$ applicable.

By Proposition 2.2, the remainder

$$ \begin{align*}R_1(u)\in H^{s+r-1}\subset H^{s+\tau+\varepsilon}, \end{align*} $$

is continuous in $u\in H^s$ , and vanishes quadratically as $u\mapsto 0\ni H^s$ . By proposition 2.3,

$$ \begin{align*}\big\|\mathcal{R}_{\mathrm{PL}}(f,u)\cdot u\big\|_{H^{s+r}} \leq C_s|f|_{C^{N_{s+r}+2}}\|u\|_{H^s}. \end{align*} $$

Due to Proposition 2.1 and (2.4), we find that the right-hand-side of (2.13) is a well-defined continuous mapping from $H^s$ to $H^{s+\varepsilon }$ . The paradifferential remainder estimates further ensure that if ${\|u\|_{H^s}\leq \delta \ll 1}$ , then the $H^{s+\varepsilon }$ norm of the right-hand-side of (2.13) is controlled by

$$ \begin{align*}C\gamma^{-1}\left(|f|_{C^{N_{s+r}+2}}\delta+\delta^2\right). \end{align*} $$

Thus if we further require $\delta \ll \min (\gamma ,1)$ and $|f|_{C^{N_{s+r}+2}}\ll \delta $ , the right-hand-side of (2.13) will be a continuous mapping from the closed ball $\bar B_\delta (0)\subset H^s$ to itself, with range actually in $H^{s+\varepsilon }$ . By the Schauder fixed point theorem, such a mapping has a fixed point u. We thus have a solution $U=(u,\lambda )$ of the parainverse equation (2.13), hence the parahomological equation (2.12).

Given this solution U, the paralinearization formula (2.11) yields

$$ \begin{align*}\mathscr{F}(f,U) -T_{[\mathscr{F}(f,U)]'/(1+u')}u=0. \end{align*} $$

If we consider the left-hand-side as a linear operator acting on $\mathscr {F}(f,U)\in C^1$ , then by Proposition 2.1, it follows that

$$ \begin{align*}\big\|T_{[\mathscr{F}(f,U)]'/(1+u')}u\big\|_{H^{s}} \lesssim \big|\mathscr{F}(f,U)\big|_{C^1}\|u\|_{H^s}. \end{align*} $$

So if $|f|_{C^{N_{s+r}+2}}$ is sufficiently small (hence the solution u has small $H^s$ norm), a Neumann series argument forces $\mathscr {F}(f,U)=0$ .

Remark 2.1. Schauder’s fixed point theorem implies the existence of solution in a nonconstructive manner. If we require more differentiability of f, for example $f\in C^{N_{s+r}+3}$ instead of $C^{N_{s+r}+2}$ , then the remainder $\mathcal {R}_{\mathrm {PL}}(f,u)\cdot uu$ will be continuously differentiable in u. In fact, we can directly compute (see, e.g., [Reference Hörmander44]) the linearization of paralinearization remainder along an increment v as

$$ \begin{align*}\begin{aligned} D_u\left[\mathcal{R}_{\mathrm{PL}}(F,u)\cdot u\right]v &=F'(u)v-T_{F"(u)v}u-T_{F'(u)}v\\ &=T_v\left(\mathcal{R}_{\mathrm{PL}}(F',u)\cdot u\right)+\text{smoother terms}. \end{aligned} \end{align*} $$

If the proof above, if $f\in C^{N_{s+r}+3}$ , then $u\mapsto \mathcal {R}_{\mathrm {PL}}(f,u)\cdot u$ is a $C^1$ mapping from $H^s$ to $H^{s+r}$ . Given that f is close to $0$ in $C^{N_{s+r}+3}$ , a Banach fixed point argument is then applicable, since the right-hand-side of (2.13) will have Lipschitz constant $<1$ in u. The solution u is thus the limit of a Banach fixed point iteration sequence. Furthermore, if we assume $f\in C^\infty $ , then by differentiating the fixed point equation (2.1), we obtain a linear iterative sequence that converges to the derivative of u. This easily yields the additional regularity $u\in C^\infty $ .

Formally, (2.12) is the paradifferential counterpart of the linearized problem (2.5). Nevertheless, the gain of regularity through paralinearization balances with the loss, and thus renders unnecessary any convergence-improvement technique. This significantly differs from the prevailing practice in the existing KAM theory literature, with perhaps the only exception being Herman’s Schwarz derivative technique; see the discussion in Subsection 1.2.

We note that the usual KAM/Nash-Moser iteration (2.9) relies on smoothing operators that are essentially Fourier truncations, which are also used in the construction of paradifferential operators as seen in (2.1). However, smoothing operators are not exploited to its maximal potential in KAM/Nash-Moser iteration since interactions of different frequencies are not used to balance the regularity loss caused by $\Delta _\alpha ^{-1}$ . This is exactly captured by paraproduct operators and paralinearization.

Furthermore, we emphasize that paraproduct operators, together with the remainders in paralinearization, all admit very explicit expressions, essentially involving only convolutions. Consequently, the size of the perturbation f in (2.13), hence in the original conjugacy problem (2.1), does not have to be unreasonably small, as the proof of Banach or Schauder fixed point theorem indicates.

2.5 Refinement in regularity

If we take into account that $x+u(x)$ is a diffeomorphism of $\mathbb {S}^1$ , then the loss of regularity can be managed more delicately using paracomposition operator introduced by Alinhac [Reference Alinhac5].

Definition 2.2. Let $\chi :\mathbb {T}^n\mapsto \mathbb {T}^n$ be a Lipschitz diffeomorphism. The paracomposition operator $\chi ^\star $ associated to $\chi $ is defined by

$$ \begin{align*}\chi^\star F:=\sum_{j\geq0}(S_{j+N}-S_{j-N})\big((\Delta_jF)\circ\chi\big). \end{align*} $$

Here N depends only on $|\partial \chi |_{L^\infty }$ .

In fact, the paracomposition operator admits several equivalent definitions, see for example Appendix A of Chapter 2 in [Reference Taylor82] or Section 5 of [Reference Alazard and Métivier2]; but we do not aim to elaborate them within our current paper. We directly cite several results concerning regularity of paracomposition operators. The proof can be found in, for example, [Reference Alinhac5], Appendix A of Chapter 2 in [Reference Taylor82], Section 3 of [Reference Nguyen72], or Section 5 of [Reference Said76]. Roughly speaking, the paracomposition operator captures the “irregularity” of the paralinearization remainder $F(u)-T_{F'(u)}u$ .

Proposition 2.4. If $\chi :\mathbb {T}^n\mapsto \mathbb {T}^n$ is a Lipschitz diffeomorphism, then $\chi ^\star $ is a bounded linear operator from $H^s$ to $H^s$ for all $s\in \mathbb {R}$ :

$$ \begin{align*}\|\chi^\star\|_{\mathcal{L}(H^s,H^s)} \leq K_s\big(|\partial\chi|_{L^\infty},|\partial\chi^{-1}|_{L^\infty}\big). \end{align*} $$

Furthermore, given $F\in H^s(\mathbb {T}^n)$ , the mapping $\chi \mapsto \chi ^\star F$ is continuous from $W^{1,\infty }$ to $H^s$ .

Proposition 2.5. Suppose $\rho>1$ , $\chi :\mathbb {T}^n\mapsto \mathbb {T}^n$ is a $C^\rho _*$ diffeomorphism, and $F\in H^{s+1}(\mathbb {S}^1)$ with $r:=s-n/2>0$ . Then the following paralinearization formula holds:

$$ \begin{align*}F\circ\chi=\chi^\star F+T_{F'\circ\chi}\chi+\mathcal{R}_{A}(\chi)F, \end{align*} $$

where

$$ \begin{align*}\big\|\mathcal{R}_{A}(\chi)F\big\|_{H^{\rho+\min(\rho,r)}} \leq K_s\big(|\chi|_{C^\rho_*},|\chi^{-1}|_{C^1}\big)\|F\|_{H^{s+1}}. \end{align*} $$

Furthermore, given $F\in H^s(\mathbb {T}^n)$ and $\varepsilon>0$ , the mapping $\chi \mapsto \mathcal {R}_{A}(\chi )F$ is continuous from $C^{\rho }_*$ to $H^{s-\varepsilon }$ .

We can give a refined proof of sovability of (2.1). We still assume in a priori that $u\in H^s$ , where $s\geq \tau +1.5+\varepsilon $ , and $f\in H^{s+1}$ . We then pick $r=\rho =s-1/2$ . By Proposition 2.5, with $\chi =\mathrm {Id}+u$ , we have the paralinearization formula for $\mathscr {F}(f,U)$ :

(2.14) $$ \begin{align} \begin{aligned} \mathscr{F}(f,U) &=\Delta_\alpha u-\chi^\star f-T_{f'\circ\chi}\chi -\mathcal{R}_A(\chi)f+\lambda\\ &=\Delta_\alpha u-T_{\Delta_\alpha u'/(1+u')}u-\chi^\star f +T_{[\mathscr{F}(f,U)]'/(1+u')}u -\mathcal{R}_A(\chi)f+\lambda\\ &=T_{(1+u'\circ\varrho_{\alpha})}\Delta_\alpha T_{1/(1+u')}u -\chi^\star f +T_{[\mathscr{F}(f,U)]'/(1+u')}u -\mathcal{R}_A(\chi)f+R_1(u)+\lambda. \end{aligned} \end{align} $$

Here the computations are almost the same as those in (2.11): the remainder $R_1(u)\in H^{s+r-1}\subset H^{s+\tau +\varepsilon }$ and vanishes bilinearly when $H^s\ni u\mapsto 0$ , while the paracomposition remainder $\mathcal {R}_A(\chi )f\in H^{2s-1}$ by Proposition 2.5.

We then consider the parahomological equation for the unknown $U=(u,\lambda )\in H^s\times \mathbb {R}$ :

(2.15) $$ \begin{align} \begin{aligned} T_{(1+u'\circ\varrho_{\alpha})}\Delta_\alpha T_{1/(1+u')}u &=\chi^\star f+\mathcal{R}_A(\chi)f-R_1(u)-\lambda. \end{aligned} \end{align} $$

Equation (2.15) is in fact just (2.12) in a delicate form. The parainverse equation (2.13) then takes an equivalent form

(2.16) $$ \begin{align} \begin{aligned} u &=T_{1/(1+u')}^{-1}\Delta_\alpha^{-1}T_{(1+u'\circ\varrho_{\alpha})}^{-1}\big(\chi^\star f+\mathcal{R}_A(\chi)f-R_1(u)-\lambda\big). \end{aligned} \end{align} $$

Here the $\lambda $ is still uniquely determined to balance out the mean value, making $\Delta _\alpha ^{-1}$ applicable.

Due to Proposition 2.4 and 2.5, we find that the right-hand-side of (2.16) is a well-defined continuous mapping from $H^s$ to $H^{s+\varepsilon }$ . The paradifferential remainder estimates further ensure that if $\|u\|_{H^s}\leq \delta $ and $|u'|_{L^\infty }\leq \rho $ , then the $H^{s+\varepsilon }$ norm of the right-hand-side of (2.13) is controlled by

$$ \begin{align*}K(\delta,\rho)\|f\|_{H^{s+\tau+\varepsilon}}+C\delta^2, \end{align*} $$

where K is an increasing function. Thus if $\|f\|_{H^{s+\tau +\varepsilon }}$ is sufficiently small, the right-hand-side of (2.16) will be a continuous mapping from the closed ball $\bar B(0,\delta )\subset H^s$ to itself, with compact range. By the Schauder fixed point theorem, such a mapping has a fixed point. The rest of the argument will then be identical with Subsection 2.4.

Sharing the same spirit with Subsection 2.4, the paracomposition approach just presented obviously requires less regularity for f when $\tau \leq 2.5$ , at the price of being technically more complicated. To keep our narration as simple as possible, we do not elaborate this approach within this paper. See Appendix A for more discussion.

2.6 Heuristics about generality

It turns out that the aforementioned method is valuable not only in one-dimensional scenarios. In fact, it encompasses all the features of general conjugacy problems of interest in dynamical systems. Roughly speaking, this is the paradifferential calculus version of Zehnder’s heuristics in Section 5 of [Reference Zehnder86]. We refer the reader to Zehnder’s paper for his original argument, and present here the paradifferential version.

The formalism of conjugacy problems is as follows. Given an open set $\mathfrak {B}$ in a Banach space and an infinite dimensional Lie group $\mathfrak {G}$ , suppose there is a differentiable group action (“dynamical system”)

$$ \begin{align*}\Phi:\mathfrak{B}\times\mathfrak{G}\mapsto\mathfrak{B}. \end{align*} $$

Then $\Phi $ is subjected to the algebraic relation

(2.17) $$ \begin{align} \Phi(f,\mathrm{Id})=f, \quad \Phi(f,g_2\circ g_1)=\Phi\big(\Phi(f,g_2),g_1\big). \end{align} $$

This is simply a result of definition of group homomorphism.

Now suppose $\mathfrak {N}\subset \mathfrak {B}$ is a closed linear submanifold, usually being the normal forms of the dynamical system. The conjugacy problem then reads: is $\mathfrak {N}$ structurally stable in this dynamical system? That is, if $f\in \mathfrak {B}$ is close to $\mathfrak {N}$ , is it possible to find an element $g\in \mathfrak {G}$ close to the identity, so that $\Phi (f,g)\in \mathfrak {N}$ ?

To answer this question, introduce the unknwon $u=(\gamma ,\mathfrak {n})$ , where $\gamma $ belongs to the Lie algebra of $\mathfrak {G}$ and $\mathfrak {n}\in \mathfrak {N}$ . Suppose $\mathcal {E}$ is a local chartFootnote ³ near the identity of the group $\mathfrak {G}$ . Define a mapping

$$ \begin{align*}\mathscr{F}(f,u)=\Phi(f,\mathcal{E}(\gamma))-\mathfrak{n}. \end{align*} $$

The conjugacy problem is then equivalent to finding zeroes of $\mathscr {F}$ . Suppose that an approximate solution $u_0=(0,\mathfrak {n}_0)$ is already found, that is $\mathscr {F}(f,u_0)$ is close to 0. We then aim to look for u close to 0 such that $\mathscr {F}(f,u)=0$

Introduce the map $\chi _\gamma $ on the Lie algebra of $\mathfrak {G}$ by $\mathcal {E}(\chi _\gamma (\gamma _1))=\mathcal {E}(\gamma )\circ \mathcal {E}(\gamma _1)$ . Then (2.17) becomes

$$ \begin{align*}\Phi\big(f,\chi_\gamma(\gamma_1)\big) =\mathscr{F}\big(\Phi(f,\gamma),\gamma_1\big). \end{align*} $$

Denote by $v=(\psi ,\mathfrak {m})$ the increment of $u=(\gamma ,\mathfrak {n})$ , and set $L_\gamma v=(D\chi _\gamma (0)\psi ,\mathfrak {m})$ . Differentiating with respect to u, evaluating at $\gamma _1=0$ , we obtain

$$ \begin{align*}D_u\mathscr{F}(f,u)L_\gamma v =D_u\mathscr{F}\big(\Phi(f,\gamma),(0,\mathfrak{n})\big)v. \end{align*} $$

Thus by Taylor’s formula,

$$ \begin{align*}\begin{aligned} D_u\mathscr{F}(f,u)L_\gamma v &=D_u\mathscr{F}\big(\mathfrak{n},(0,\mathfrak{n})\big)v +\left(D_u\mathscr{F}\big(\Phi(f,\gamma),(0,\mathfrak{n})\big) -D_u\mathscr{F}\big(\mathfrak{n},(0,\mathfrak{n})\big)\right)v\\ &=D_u\mathscr{F}\big(\mathfrak{n},(0,\mathfrak{n})\big)v +\boldsymbol{B}\big(\mathscr{F}(f,u),v\big), \end{aligned} \end{align*} $$

where $\boldsymbol {B}$ vanishes bilinearly near (0,0). Equivalently,

(2.18) $$ \begin{align} \begin{aligned} D_u\mathscr{F}(f,u)v &=D_u\mathscr{F}\big(\mathfrak{n},(0,\mathfrak{n})\big)L_\gamma^{1}v +\boldsymbol{B}\big(\mathscr{F}(f,u),L_\gamma^{1}v\big). \end{aligned} \end{align} $$

The idea of paradifferential approach to the conjugacy equation $\mathscr {F}(f,u)=0$ is then based on the algebraic identity (2.18). Suppose the Banach space $\mathfrak {B}$ and Lie group $\mathfrak {G}$ are realized as functions and (finite-dimensional) diffeomorphisms, and the mapping $\mathscr {F}(f,u)$ is a classical nonlinear differential operator acting on u, in the sense that it is a function of $(u,\partial u,\partial ^2u,\cdots )$ . Then the paralinearization theorem of Bony [Reference Bony11] should imply

(2.19) $$ \begin{align} \begin{aligned} \mathscr{F}(f,u) &=\mathscr{F}(f,u_0)+T_{D_u\mathscr{F}(f,u)}(u-u_0) +\boldsymbol{R}_1(u-u_0)\\ &=T_{D_u\mathscr{F}\big(\mathfrak{n},(0,\mathfrak{n})\big)}T_{L_\gamma^{1}}(u-u_0) +\mathscr{F}(f,u_0)+\boldsymbol{R}_2(u-u_0) +\boldsymbol{B}\big(T_{\mathscr{F}(f,u)},T_{L_\gamma^{1}}(u-u_0)\big). \end{aligned} \end{align} $$

Here $u=(\gamma ,\mathfrak {n})\simeq (0,\mathfrak {n}_0)$ , and $\boldsymbol {R}_1,\boldsymbol {R}_2$ are smoother remainders produced by paralinearization. The somewhat abused notation $\boldsymbol {B}\big (T_{\mathscr {F}(f,u)},T_{L_\gamma ^{1}}(u-u_0)\big )$ stands for the replacement of “product” $\boldsymbol {B}\big ({\mathscr {F}(f,u)},L_\gamma ^{1}(u-u_0)\big )$ by the corresponding “paraproduct” $\boldsymbol {B}\big (T_{\mathscr {F}(f,u)},T_{L_\gamma ^{1}}(u-u_0)\big )$ , should $\boldsymbol {B}$ involve only usual functional operations.

The following step is to make sure that the parahomological equation

(2.20) $$ \begin{align} T_{D_u\mathscr{F}\big(\mathfrak{n},(0,\mathfrak{n})\big)}T_{L_\gamma^{1}}(u-u_0) +\mathscr{F}(f,u_0)+\boldsymbol{R}_2(u-u_0)=0 \end{align} $$

is solvable. We will actually consider its equivalent fixed point form, refered as the parainverse equation:

$$ \begin{align*}(u-u_0) =-T_{L_\gamma^{1}}^{-1}T_{D_u\mathscr{F}\big(\mathfrak{n},(0,\mathfrak{n})\big)}^{-1}\big(\mathscr{F}(f,u_0)+\boldsymbol{R}_2(u-u_0)\big), \end{align*} $$

and make sure that it is solvable.

For most conjugacy problems, this really is the case, even if inverting $D_u\mathscr {F}\big (\mathfrak {n},(0,\mathfrak {n})\big )$ causes a loss of regularity due to small denominator. In fact, that loss of regularity can be compensated by the additional smoothness of the paradifferential remainder $\boldsymbol {R}_2(u-u_0)$ . Thus, if $\mathscr {F}(f,u_0)$ is sufficiently smooth, the right-hand-side of the fixed point equation defines a mapping with no loss of regularity (e.g., from $H^s$ to $H^s$ ). We note that this is the paradifferential version of Zehnder’s “finding approximate right inverse.” Consequently, assuming $\mathscr {F}(f,u_0)$ is close enough to 0, a standard fixed point argument should produce a solution $u\simeq u_0$ of (2.20).

Once a solution u to the parahomological equation (2.20) is found, the first three terms in the right-hand-side of (2.19) are then cancelled, and the paralinearization formula then becomes

$$ \begin{align*}\mathscr{F}(f,u) =\boldsymbol{B}\big(T_{\mathscr{F}(f,u)},L_\gamma^{1}(u-u_0)\big) \approx T_{\mathscr{F}(f,u)}L_\gamma^{1}(u-u_0). \end{align*} $$

Now since a paradifferential operator $T_a$ depends only on very mild regularity of the symbol a, this equation must imply $\mathscr {F}(f,u)=0$ by a Neumann series argument, provided that u is close to $u_0$ .

To conclude, solution of the parahomological equation in fact solves the original conjufacy problem.

Of course, the steps outlined above are merely heuristics. In practice, each of the assumptions they rely upon must be validated in a case-by-case fashion. Nevertheless, given that these intuitive deductions pertain to very general scenarios, it is reasonable to expect that they adequately cover several well-studied conjugacy problems. We point out, omitting most details, that the following model problems all fit into this formalism: conjugation of standard map (see, e.g., Section 2 of [Reference de la Llave60]), structural stability of parallel vector field on torus (see, e.g., [Reference Pöschel74]), existence of invariant torus of symplectic map (see, e.g., [Reference de la Llave, González, Jorba and Villanueva59]), and conjugation of Hamiltonian vector fields. We will elaborate on the last one in Section 4.

The idea of replacing a linearized equation by its paradifferential version dates back to Hörmander’s paper [Reference Hörmander44], where Hörmander proposed the terminology parainverse. Hörmander only considered nonlinear mappings with exactly invertible linearized operators, instead of approximately invertible ones. Nevertheless, it is still appropriate to attribute the core idea of the aforementioned heuristics to [Reference Hörmander44] (and [Reference Zehnder86]).

3.1 Meyer multipliers

Before providing the quantitative paradifferential operator estimates, let us first state some auxiliary results relating to Meyer multipliers. Formally, given a sequence $\{m_j(x)\}_{j\geq 0}$ of functions on $\mathbb {T}^n$ , we call the linear operator

$$ \begin{align*}f(x)\mapsto\sum_{j\geq0} m_j(x)\cdot(\Delta_jf)(x) \end{align*} $$

the Meyer multiplier associated to $\{m_j\}_{j\geq 0}$ . Here the $\Delta _j$ ’s are the Littlewood-Paley building block operators introduced in (1.10). The fundamental theorem for such a linear operator is the following:

Proposition 3.1 (Meyer multiplier).

Let $s,r\in \mathbb {R}$ , and suppose that $s+r>0$ . Let $N_{s+r}$ be the smallest integer such that $N_{s+r}>s+r$ . Suppose $(m_j)_{j\geq 0}$ is a sequence in $L^\infty (\mathbb {T}^n)$ , such that for multi-indices $\alpha $ with $|\alpha |\leq N_{s+r}$ , there holds

$$ \begin{align*}\big|\partial^\alpha m_j\big|_{L^\infty} \leq M_\alpha 2^{j(|\alpha|-r)}. \end{align*} $$

Then the linear operator

$$ \begin{align*}f\mapsto\sum_{j=0}^\infty m_j\Delta_j f \end{align*} $$

is bounded from $H^s(\mathbb {T}^n)$ to $H^{s+r}(\mathbb {T}^n)$ . The norm of this operator is estimated by

$$ \begin{align*}C_s\sum_{\beta:|\beta|\leq N_{s+r}}M_\beta. \end{align*} $$

We need a lemma concerning Sobolev regularity of series to aid the proof of Proposition 3.1. It is of some independent interest per se.

Lemma 3.1. Let $s>0$ , let $N_s$ be the smallest integer such that $N_s>s$ . There is a constant $C_s$ with the following properties. If $\{f_k\}_{k\geq 0}$ is a sequence in $H^{N_s}(\mathbb {T}^n)$ , such that for any multi-index $\alpha $ with $|\alpha |\leq N_s$ , there holds

$$ \begin{align*}\|\partial^\alpha f_k\|_{L^2}\leq 2^{k(|\alpha|-s)}c_k,\quad (c_k)\in \ell^2(\mathbb{N}), \end{align*} $$

then $\sum _k f_k\in H^s(\mathbb {R}^n)$ and

$$ \begin{align*}\left\|\sum_{k\geq0}f_k\right\|_{H^s}^2 \leq C_s\sum_{k\geq0}c_k^2. \end{align*} $$

Proof. If $j\leq k$ , then by taking $\alpha =0$ , obviously

$$ \begin{align*}\|\Delta_jf_k\|_{L^2}\leq\|f_k\|_{L^2}\leq 2^{-ks}c_k. \end{align*} $$

If $j>k$ , then by Bernstein’s inequality,

$$ \begin{align*}\|\Delta_jf_k\|_{L^2} \leq C_s2^{-N_sj}\sum_{|\alpha|=N_s}\|\partial^\alpha f_k\|_{L^2} \leq C_s2^{-ks}2^{N_s(k-j)}c_k. \end{align*} $$

Thus, for $f=\sum _kf_k$ , we have

(3.1) $$ \begin{align} 2^{js}\|\Delta_j f\|_{L^2} \leq C_s\sum_{k:k<j}2^{(N_s-s)(k-j)}c_k +\sum_{k:k\geq j}2^{(j-k)s}c_k. \end{align} $$

Given that $s>0$ and $N_s>s$ , the right-hand-side is the convolution of the $\ell ^1$ sequences $(2^{-(N_s-s)k}),(2^{-sk})$ with the $\ell ^2$ sequence $(c_k)$ . Therefore, the right-hand-side forms an $\ell ^2$ sequence (with index j), with norm controlled by $C_s\|(c_k)\|_{\ell ^2}$ . Thus $f\in H^s$ .

Proof of Proposition 3.1.

In fact, for $|\alpha |\leq N_{s+r}$ , each summand $m_j\Delta _j f$ satisfies

$$ \begin{align*}\begin{aligned} \big\|\partial^\alpha(m_j\Delta_jf)\big\|_{L^2} &\leq C_\alpha \sum_{\beta:\beta\leq\alpha} \big|\partial^\beta m_j\big|_{L^\infty}\big\|\partial^{\alpha-\beta}\Delta_jf\big\|_{L^2} \\ &\leq C_\alpha 2^{j(|\alpha|-r-s)}\left(\sum_{\beta:\beta\leq\alpha}M_\beta\right)2^{js}\|\Delta_jf\|_{L^2}. \end{aligned} \end{align*} $$

We can then directly apply Lemma 3.1 to conclude.

Corollary 3.1.1. If the sequence $\{f_k\}$ in Lemma 3.1 in addition satisfies the spectral condition

$$ \begin{align*}\operatorname{\mathrm{supp}}\hat{f}_k\subset\{|\xi|\geq \delta 2^k\} \end{align*} $$

for some $\gamma>0$ , then the restriction $s>0$ in Lemma 3.1 can be relaxed to any real index s:

$$ \begin{align*}\left\|\sum_{k\geq0}f_k\right\|_{H^s}^2 \leq C_{s,\delta}\sum_{k\geq0}c_k^2. \end{align*} $$

Remark 3.1. Suppose the multipliers $m_j=m_j(\lambda )$ in Proposition 3.1 depend on a parameter $\lambda $ varying in some Banach space, such that

$$ \begin{align*}\max_{\alpha:|\alpha|\leq N_{s+r}}\sup_j2^{-j(|\alpha|-r)}\big|\partial^\alpha D_\lambda m_j(\lambda)\big|_{L^\infty} <\infty. \end{align*} $$

Then we can simply repeat the proof of Proposition 3.1 to conclude the following: the operator-valued function

$$ \begin{align*}\lambda\mapsto\sum_{j=0}^\infty m_j(\lambda)\Delta_j \in\mathcal{L}(H^s,H^{s+r}) \end{align*} $$

is differentiable in $\lambda $ under the norm topology of $\mathcal {L}(H^s,H^{s+r})$ , whose differential is simply

$$ \begin{align*}\sum_{j=0}^\infty D_\lambda m_j(\lambda)\Delta_j. \end{align*} $$

3.2 Quantitative estimates for paraproduct operators

With the aid of Meyer multiplier estimates, we are now able to provide a quantitative proof of boundedness of paraproduct operators. We will first list several properties related to Littlewood-Paley characterization of the Zygmund space. The proof is quite elementary using the definition of Zygmund spaces, as in (1.11).

Proposition 3.2. Suppose $r>0$ and $f\in C^r_*$ . Then for any multi-index $\alpha $ , there holds

$$ \begin{align*}\big|\partial^\alpha S_jf\big|_{L^\infty} \lesssim_{r,\alpha} \left\{ \begin{aligned} & 2^{j(|\alpha|-r)_+}|f|_{C^r_*} & \quad |\alpha|\neq r \\ & j|f|_{C^r_*} & \quad |\alpha|=r \end{aligned} \right. \end{align*} $$

where $s_+=\max (s,0)$ . Furthremore, there holds

$$ \begin{align*}|f-S_jf|\lesssim_r 2^{-jr}|f|_{C^r_*}. \end{align*} $$

We are now at the place to state and prove the continuity of paraproduct operators, which appeared as Proposition 2.1.

Proposition 3.3. For all $a\in L^\infty (\mathbb {T}^n)$ and all $s\in \mathbb {R}$ , the paraproduct $T_a$ is a bounded linear operator from $H^s(\mathbb {T}^n)$ to $H^s(\mathbb {T}^n)$ :

$$ \begin{align*}\left\Vert T_a\right\Vert{}_{\mathcal{L}(H^s,H^s)}\leq C_s|a|_{L^\infty}. \end{align*} $$

Proof. In fact, in the defining equality (2.10), each summand has Fourier support in the annulus $\{0.25\cdot 2^j \leq |\xi |\leq 2.25\cdot 2^j\}$ . We then simply notice that the Meyer multipliers $m_j=S_{j-3}a$ satisfy

$$ \begin{align*}\big|\partial^\alpha m_j\big|_{L^\infty} \leq C_\alpha 2^{j|\alpha|}|a|_{L^\infty} \end{align*} $$

by Proposition 3.2. Thus we can repeat the proof of Corollary 3.1.1 and conclude.

The nontrivial fact about paraproduct operators is that they behave as if they were genuine multiplication operators: the multiplication $T_aT_b$ is the same as $T_{ab}$ modulo smoothing operators. Formally, this means that $a\mapsto T_a$ is an algebra homomorphism from the algebra of functions to the algebra of bounded operators (modulo smoothing ones). We provide the precise version of Proposition 2.2:

Proposition 3.4. Fix $r>0$ , and suppose $a,b\in C^r_*$ . Then

$$ \begin{align*}\mathcal{R}_{\mathrm{CM}}(a,b):=T_a\circ T_b-T_{ab} \end{align*} $$

is a bounded linear operator from $H^s(\mathbb {T}^n)$ to $H^{s+r}(\mathbb {T}^n)$ for all $s\in \mathbb {R}$ , where CM is the abbreviation for composition de paramultiplication, satisfying the tame estimate

$$ \begin{align*}\left\Vert \mathcal{R}_{\mathrm{CM}}(a,b)\right\Vert{}_{\mathcal{L}(H^s, H^{s+r})}\leq C_{s,r}\big(|a|_{L^\infty}|b|_{C^r_*}+|a|_{C^r_*}|b|_{L^\infty}\big). \end{align*} $$

In other words, $\mathcal {R}_{\mathrm {CM}}:C^r_* \times C^r_* \mapsto \mathcal {L}(H^s, H^{s+r})$ is a bounded bilinear mapping.

Proof. The proof here is exactly the same as Theorem 2.3 of [Reference Bony11]. Suppose $a,b\in C^r_*$ and $u\in H^s$ are given. We set $v=T_bu$ , $v_q=S_{q-3}b\cdot \Delta _qu$ . By Corollary 3.1.1 and the definition (1.11) of Zygmund space norm, the error

$$ \begin{align*}R_1v:=T_av-\sum_{p}S_{p-5}a\cdot \Delta_pv =\sum_{p}(\Delta_{p-4}a+\Delta_{p-3}a)\Delta_pu \end{align*} $$

satisfies $\|R_1v\|_{H^{s+r}}\leq C_{s,r}|a|_{C^r_*}|b|_{L^\infty }\|u\|_{H^{s}}$ , since each summand $(\Delta _{p-4}a+\Delta _{p-3}a)\Delta _pv$ has Fourier support contained in the annulus $\{0.25\cdot 2^p\leq |\xi |\leq 2.25\cdot 2^p\}$ . We then compute

(3.2) $$ \begin{align} \begin{aligned} T_aT_bu-\sum_{q}(S_{q-5}a)(S_{q-3}b)\Delta_qu &= T_aT_bu-\sum_{q}\sum_{p:p\leq q-5}\Delta_pa\cdot v_q\\ &=\sum_{q}\sum_{p:p\leq q-5}\Delta_pa\cdot \big(\Delta_qv-v_q\big)+R_1T_bu\\ &=\sum_{p\geq-5}\Delta_pa\sum_{q:p+5\leq q \leq p+7} \big(\Delta_qv-v_q\big)+R_1T_bu. \end{aligned} \end{align} $$

The last step is because $\sum _q v_q=\sum _q \Delta _qv$ and $v_q$ has Fourier support contained in the annulus $\{0.25\cdot 2^q\leq |\xi |\leq 2.25\cdot 2^q\}$ . Using Corollary 3.1.1 once again, we estimate (3.2) as

(3.3) $$ \begin{align} \left\|T_aT_bu-\sum_{q}(S_{q-5}a)(S_{q-3}b)\Delta_qu\right\|_{H^{s+r}} \leq C_{s,r}|a|_{C^r_*}|b|_{L^\infty}\|u\|_{H^s}. \end{align} $$

On the other hand, if we set

$$ \begin{align*}c_q=(S_{q-5}a)(S_{q-3}b), \end{align*} $$

then $c_q$ has Fourier support contained in the ball $|\xi |\leq 0.25\cdot 2^q$ , and

(3.4) $$ \begin{align} \begin{aligned} |ab-c_q|_{L^\infty} &\leq|\big((\mathrm{Id}-S_{q-5})a\big)b|_{L^\infty} +|(S_{q-5}a)\big((\mathrm{Id}-S_{q-5})b\big)|_{L^\infty}\\ &\leq \sum_{p>q-5}2^{-pr}|a|_{C^r_*}|b|_{L^\infty} +|a|_{L^\infty}\sum_{p>q-5}2^{-pr}|b|_{C^r_*}\\ &\leq C_r2^{-q}\big(|a|_{C^r_*}|b|_{L^\infty}+|a|_{L^\infty}|b|_{C^r_*}\big). \end{aligned} \end{align} $$

Noticing that

$$ \begin{align*}T_{ab}u-\sum_q c_q\Delta_qu =\sum_{q}\big(S_{q-3}(ab)-c_q\big)\cdot\Delta_qu =\sum_{q}\big(S_{q-3}(ab)-ab+ab-c_q\big)\cdot\Delta_qu, \end{align*} $$

using Corollary 3.1.1 and (3.4), we find

(3.5) $$ \begin{align} \left\|T_{ab}u-\sum_{q}c_q\cdot\Delta_qu\right\|_{H^{s+r}} \leq C_{s,r}\big(|a|_{C^r_*}|b|_{L^\infty}+|a|_{L^\infty}|b|_{C^r_*}\big)\|u\|_{H^s}. \end{align} $$

Combining (3.3) and (3.5), the proof is complete.

3.3 Bony’s paralinearization theorem

In this subsection, we present a quantitative refinement of Bony’s paralinearization theorem in [Reference Bony11], which asserts that given a nonlinear composition $F(u)$ , the paraproduct $T_{F'(u)}u$ carries most of the irregularity, and

$$ \begin{align*}F(u)-T_{F'(u)}u \end{align*} $$

is more regular than $F(u)$ . This is a far-reaching generalization of the paraproduct decomposition.

While Bony’s theorem is well-established and extensively applied in modern nonlinear analysis, we still need to manage the size of the smoothing remainder to effectively study the parahomological equation. We state the following refined version of Proposition 2.3:

Proposition 3.5. Fix $s,r>0$ , and suppose $u\in (H^s\cap C^r_*)(\mathbb {T}^n;\mathbb {R}^L)$ . Let $N_{s+r}$ be the smallest integer $>s+r$ . Suppose $F=F(x,z)\in C^{N_{s+r}+2}(\mathbb {T}^n\times \mathbb {R}^L)$ . Then there holds the following paralinearization formula:

$$ \begin{align*}\begin{aligned} F(x,u)-F(x,0) &=T_{F^{\prime}_z(x,u)}u +\mathcal{R}_{\mathrm{PL}}(F,u)\cdot u\\ &=:T_{F^{\prime}_z(x,u)}u +\mathcal{R}_{\mathrm{PL};1}(F)\cdot u +\mathcal{R}_{\mathrm{PL};2}(F,u)\cdot u \in H^s+H^{s+r}. \end{aligned} \end{align*} $$

Here $\mathcal {R}_{\mathrm {PL};1}(F):H^s\mapsto H^{s+r}$ and $\mathcal {R}_{\mathrm {PL};2}(F,u):H^s\mapsto H^{s+r}$ are bounded linear operators, so that for some increasing function $K_{r,s}$ depending only on $r,s$ ,

$$ \begin{align*}\begin{aligned} \left\|\mathcal{R}_{\mathrm{PL};1}(F)\right\|_{\mathcal{L}(H^s,H^{s+r})} &\leq C_{r,s}\big|F^{\prime}_z(x,0)-\operatorname{\mathrm{Avg}} F^{\prime}_z(x,0)\big|_{C^{r}_*}\\ \left\|\mathcal{R}_{\mathrm{PL};2}(F,u)\right\|_{\mathcal{L}(H^s,H^{s+r})} &\leq K_{r,s}\big(|u|_{C^r_*}\big)|F|_{C^{N_{s+r}+2}}|u|_{C^r_*} \end{aligned} \end{align*} $$

Moreover, the operator $\mathcal {R}_{\mathrm {PL};2}\big (F,u\big )\in \mathcal {L}(H^s,H^{s+r})$ depends continuously on $u\in H^s\cap C^r_*$ in the operator norm.

If in addition, $F=F(x,z)$ is of better regularity $C^{N_{s+r}+3}(\mathbb {T}^n\times \mathbb {R}^L)$ instead of merely $C^{N_{s+r}+2}(\mathbb {T}^n\times \mathbb {R}^L)$ , then both operators $T_{F^{\prime }_z(x,u)}\in \mathcal {L}(H^s,H^{s})$ and $\mathcal {R}_{\mathrm {PL};2}\big (F(x,\cdot ),u\big )\in \mathcal {L}(H^s,H^{s+r})$ are continuously differentiable in $u\in H^s\cap C^{r}_*$ with respect to operator norms.

Proof. The proof is a refinement of the standard telescope series argument; see for example, Chapter 10 of [Reference Hörmander45]. We write

(3.6) $$ \begin{align} \begin{aligned} F(x,u)-F(x,0) &=F(x,S_0u)+\sum_{j=1}^\infty F(x,S_ju)-F(x,S_{j-1}u)\\ &=\left(\int_0^1F^{\prime}_z(x,\tau \Delta_0u)\operatorname{\mathrm{d}}\!\tau\right)\Delta_0u +\sum_{j=1}^\infty\left(\int_0^1 F^{\prime}_z\big(x,S_{j-1}u+\tau\Delta_j u\big)\operatorname{\mathrm{d}}\!\tau\right)\Delta_ju. \end{aligned} \end{align} $$

Defining the multipliers

$$ \begin{align*}\begin{aligned} m_j^1&=(1-S_{j-3})\big(F^{\prime}_z(x,0)-\operatorname{\mathrm{Avg}} F^{\prime}_z(x,0)\big),\quad j\geq0\\ m_0^2&=\int_0^1 \Big[F^{\prime}_z\big(x,\tau\Delta_0 u\big)-F^{\prime}_z(x,0)\Big]\operatorname{\mathrm{d}}\!\tau -S_{-3}\big(F^{\prime}_z(x,u)-F^{\prime}_z(x,0)\big),\\ m_j^2&=\int_0^1 \Big[F^{\prime}_z\big(x,S_{j-1}u+\tau\Delta_j u\big)-F^{\prime}_z(x,0)\Big]\operatorname{\mathrm{d}}\!\tau -S_{j-3}\big(F^{\prime}_z(x,u)-F^{\prime}_z(x,0)\big), \quad j\geq1 \end{aligned} \end{align*} $$

we can rewrite the telescope series (3.6) as

$$ \begin{align*}F(x,u)-F(x,0) =T_{F^{\prime}_z(x,u)}u+\sum_{j=0}^\infty m_j^1\Delta_ju +\sum_{j=0}^\infty m_j^2\Delta_ju. \end{align*} $$

Let us then just define, given $u\in H^s\cap C^r_*$ , the linear operators

$$ \begin{align*}\mathcal{R}_{\mathrm{PL};1}(F)\cdot v :=\sum_{j=0}^\infty m_j^1\Delta_jv, \quad \mathcal{R}_{\mathrm{PL};2}(F,u)\cdot v :=\sum_{j=0}^\infty m_j^2\Delta_jv. \end{align*} $$

In view of Proposition 3.1, it suffices to estimate the two sets of Meyer multipliers $\{m_j^1\}$ and $\{m_j^2\}$ .

For the multipliers $\{m_j^1\}$ , we estimate them by the simple decay property in Proposition 3.2 for the Zygmund class $C^r_*$ . Since $F^{\prime }_z(x,0)\in C^{N_{s+r}+1}$ by assumption, we obtain by Proposition 3.2 that

$$ \begin{align*}|\partial^\alpha m_j^1|_{L^\infty} \leq C_{s,\alpha} 2^{j(|\alpha|-r)}\big|F^{\prime}_z(x,0)-\operatorname{\mathrm{Avg}} F^{\prime}_z(x,0)\big|_{C^{r}}, \quad |\alpha|\leq N_{s+r}. \end{align*} $$

Proposition 3.1 then ensures the estimate for the operator $\mathcal {R}_{\mathrm {PL};1}(F)$ .

The estimate for multipliers $\{m_j^2\}$ follows by the same idea but is technically more involved. In view of Proposition 3.1, it suffices to show that

(3.7) $$ \begin{align} |\partial^\alpha m_j^2|_{L^\infty} \leq K_{s,\alpha}\big(|u|_{C^r_*}\big) 2^{j(|\alpha|-r)}|F|_{C^{N_{s+r}+2}}|u|_{C^r_*}, \quad |\alpha|\leq N_{s+r}. \end{align} $$

To obtain (3.7) with $\alpha =0$ , we notice that for $j\geq 1$ ,

(3.8) $$ \begin{align} \begin{aligned} m_j^2&=\int_0^1 \Big[\left(F^{\prime}_z\big(x,S_{j-1}u+\tau\Delta_j u\big)-F^{\prime}_z(x,u)\right)\Big]\operatorname{\mathrm{d}}\!\tau +(1-S_{j-3})\big(F^{\prime}_z(x,u)-F^{\prime}_z(x,0)\big) \\ &=\int_{[0,1]^2}F^{\prime\prime}_{zz}\big(x,\tau_2(S_{j-1}u+\tau_1\Delta_j u)\big)\big((1-S_{j-1})u+\tau_1\Delta_ju\big)\operatorname{\mathrm{d}}\!\tau_1\operatorname{\mathrm{d}}\!\tau_2 \\ &\quad+(1-S_{j-3})\int_0^1F^{\prime\prime}_{zz}(x,\tau u)u\operatorname{\mathrm{d}}\!\tau. \end{aligned} \end{align} $$

The first term in the right-hand-side of (3.8) is directly seen (by Proposition 3.2) to be controlled by $2^{-jr}|F|_{C^{2}}|u|_{C^r_*}$ . For the second term, we simply notice that $F\in C^{N_{s+r}+2}\subset C^{r+2}$ , so

$$ \begin{align*}\big|F^{\prime\prime}_{zz}(x,\tau u)u\big|_{C^r_*} \leq K_{s}\big(|u|_{C^r_*}\big) |F|_{C^{r+2}}|u|_{C^r_*}, \end{align*} $$

which then implies the desired estimate by implementing Proposition 3.2 once again. The estiamte for $m_0^2$ is similar.

To obtain (3.7) with $|\alpha |=N_{s+r}$ , we rewrite $m_j^2$ in a different manner: for $j\geq 1$ ,

(3.9) $$ \begin{align} \begin{aligned} m_j^2 &=\int_{[0,1]^2}F^{\prime\prime}_{zz}\big(x,\tau_2(S_{j-1}u+\tau_1\Delta_j u)\big)\big(S_{j-1}u+\tau_1\Delta_j u\big)\operatorname{\mathrm{d}}\!\tau_1\operatorname{\mathrm{d}}\!\tau_2 -S_{j-3}\int_0^1F^{\prime\prime}_{zz}(x,\tau u)u\,\operatorname{\mathrm{d}}\!\tau. \end{aligned} \end{align} $$

We then use $\big |F^{\prime\prime }_{zz}(x,\tau u)u\big |_{C^r_*} \leq K_{s}\big (|u|_{C^r_*}\big ) |F|_{C^{r+2}}|u|_{C^r_*}$ again: for $|\alpha |=N_{s+r}$ , by Proposition 3.2, the second term in (3.9) satisfies

$$ \begin{align*}\left|\partial^\alpha S_{j-3}\int_0^1F^{\prime\prime}_{zz}(x,\tau u)u\,\operatorname{\mathrm{d}}\!\tau\right| \leq 2^{j(N_{s+r}-r)}K_{s}\big(|u|_{C^r_*}\big) |F|_{C^{r+2}}|u|_{C^r_*}. \end{align*} $$

As for the first term in (3.9), by the Faà di Bruno formula, the partial derivative

$$ \begin{align*}\partial^\alpha\Big[F^{\prime\prime}_{zz}\big(x,\tau_2(S_{j-1}u+\tau_1\Delta_j u)\big)\big(S_{j-1}u+\tau_1\Delta_j u\big)\Big] \end{align*} $$

is a linear combination of terms of the form

(3.10) $$ \begin{align} (\tau_2D_z)^{2+I}F\big(x,\tau_2(S_{j-1}u+\tau_1\Delta_j u)\big)\cdot \left[\partial^{\beta_1}\big(S_{j-1}u+\tau_1\Delta_j u\big)\right]^{i_1} \cdots \left[\partial^{\beta_l}\big(S_{j-1}u+\tau_1\Delta_j u\big)\right]^{i_l}, \end{align} $$

where the scalar indices $i_1,\cdots ,i_l,I\leq N_{s+r}$ and multi-indices $\beta _1,\cdots ,\beta _l$ satisfy

$$ \begin{align*}\begin{aligned} N_{s+r}=i_1|\beta_1|+\cdots+i_l|\beta_l|, \quad I=i_1+\cdots+i_l. \end{aligned} \end{align*} $$

By the Littlewood-Paley characterization of Zygmund functions, such a term has an upper bound

(3.11) $$ \begin{align} C_s|F|_{C^{N_{s+r}+2}} \prod_{p:\beta_p>r}\big(|u|_{C^r_*}2^j\big)^{i_p(|\beta_p|-r)} \prod_{p:\beta_p\leq r}\big(|u|_{C^r_*}j\big)^{i_p} \leq K_{s}\big(|u|_{C^r_*}\big) |F|_{C^{N_{s+r}+2}}|u|_{C^r_*}2^{j(N_{s+r}-r)}. \end{align} $$

Note that the expression vanishes at least linearly as $u\mapsto 0$ since at least one $i_p\neq 0$ . The estimate for $\partial ^\alpha m_0^2$ is similar.

Therefore, (3.7) holds for $|\alpha |=N_{s+r}$ . By interpolation, it remains valid for those multi-indices $\alpha $ with length between $0$ and $N_{s+r}$ . This proves the operator norm estimates of the paradifferential remainders.

To prove continuous dependence on $u\in H^s\cap C^r_*$ of the operator $\mathcal {R}_{\mathrm {PL};2}\big (F(x,\cdot ),u\big )\in \mathcal {L}(H^s,H^{s+r})$ , in view of Proposition 3.1, it suffices to show that when $u_1,u_2$ are close in $C^r_*$ , then the Meyer multipliers $m_j^2$ corresponding to $u_1$ and $u_2$ will be close to each other, that is, for $|\alpha |\leq N_{s+r}$ , the quantities

(3.12) $$ \begin{align} 2^{-j(|\alpha|-r)}\big|\partial^\alpha\big(m_j^2(u_1)-m_j^2(u_2)\big)\big|_{L^\infty} \end{align} $$

will be close to zero. This is easily verified if we take into account the assumption $F\in C^{N_{s+r}+2}$ . For example, in the summand (3.10), the worst scenario is when all the indices $i_p\equiv 1$ , so that (3.10) becomes a “highest order derivative”:

$$ \begin{align*}(\tau_2D_z)^{2+N_{s+r}}F\big(x,\tau_2(S_{j-1}u+\tau_1\Delta_j u)\big)\cdot \left[D\big(S_{j-1}u+\tau_1\Delta_j u\big)\right]^{\otimes N_{s+r}}. \end{align*} $$

When evaluated at $u_1,u_2\in C^r_*$ that are close in the $C^r_*$ topology, we find that

$$ \begin{align*}\left[(\tau_2D_z)^{2+N_{s+r}}\Delta_{12}F\big(x,\tau_2(S_{j-1}u+\tau_1\Delta_j u)\big)\right] \end{align*} $$

will be close to zero since $D_z^{2+N_{s+r}}F(x,z)$ is continuous, while using Proposition 3.2, we estimate

$$ \begin{align*}\left|\Delta_{12}\left[D\big(S_{j-1}u+\tau_1\Delta_j u\big)\right]^{\otimes N_{s+r}}\right|{}_{L^\infty} \end{align*} $$

similarly as in (3.11), with a factor $|\Delta _{12}u|_{C^r_*}$ . In conclusion, we obtain that (3.12) is close to zero when $u_1$ is close to $u_2$ in $C^r_*$ .

Finally, let us suppose F has better regularity, namely $C^{N_{s+r}+3}(\mathbb {T}^n\times \mathbb {R}^L)$ . Continuous differentiability of the operator $\mathcal {R}_{\mathrm {PL};2}\big (F(x,\cdot ),u\big )$ in u is a direct consequence of the equality

$$ \begin{align*}(D_um_j^2)v=\int_0^1 \Big[F^{\prime\prime}_{zz}\big(x,S_{j-1}u+\tau\Delta_j u\big)(S_{j-1}+\tau\Delta_j)v\Big]\operatorname{\mathrm{d}}\!\tau -S_{j-3}\big(F^{\prime\prime}_{zz}(x,u)v\big) \end{align*} $$

and Remark 3.1.

4.1 Algebraic set-up

Let us recall that we are working with the phase space $\mathbb {T}^n\times \mathbb {R}^n$ . We are interested in Hamiltonian functions $h(x,y)$ as in (1.2) and embeddings of $\mathbb {T}^n$ into the phase space close to the “flat” one (1.1)Footnote ⁴ . The torus $\{y=0\}$ is supposed to be “approximately invariant” under the phase flow of Hamilton vector field $X_h$ , and the restriction of $X_h$ is approximately the rotation generated by a constant frequency vector field $\omega \in \mathbb {R}^n$ .

Let us define the right inverse of $\nabla _\omega $ as follows:

$$ \begin{align*}\nabla_\omega^{-1}f(\theta) :=\sum_{k\in\mathbb{Z}^n\setminus\{0\}}\frac{\hat f(k)e^{ik\cdot \theta}}{i(k\cdot\omega)}. \end{align*} $$

Clearly, when restricted to the subspace of vanishing average, the operator $\nabla _\omega ^{-1}$ is also the left inverse of $\nabla _\omega $ .

The Diophantine condition (1.3) of $\omega $ enables us to estimate the operator bound of $\nabla _\omega ^{-1}$ :

(4.1) $$ \begin{align} \big\|\nabla_\omega ^{-1}f\big\|_{H^s} \leq \gamma^{-1}\|f\|_{H^{s+\tau}}, \quad \text{if }\hat f(0)=0. \end{align} $$

We now study the conjugacy equations (1.6) and (1.9), that is,

$$ \begin{align*}X_h(u)-\nabla_\omega u=0 \end{align*} $$

and

$$ \begin{align*}X_{h_\xi}(u)-\nabla_\omega u=0. \end{align*} $$

To employ the general heuristics discussed in Subsection 2.6, we define a mapping

(4.2) $$ \begin{align} \mathscr{F}(h,u):=X_h(u)-\nabla_\omega u, \quad \text{whose value we denote as }E. \end{align} $$

It is the “error function” measuring whether u is indeed an invariant torus of $X_h$ or not. The problem is then reduced to looking for zero $u\simeq \zeta _0$ of $\mathscr {F}(h,u)$ under the condition of Theorem 1.1, or looking for $\xi \in \mathbb {R}^n$ and zero $u\simeq \zeta _0$ of $\mathscr {F}(h_\xi ,u)$ under the condition of Theorem 1.2.

We claim that, to solve $\mathscr {F}(h,u)=0$ , it suffices to solve a slightly “softer” equation

(4.3) $$ \begin{align} \mathscr{F}(h,u)+\begin{pmatrix}0 \\ \mu\end{pmatrix}=0 \end{align} $$

for an auxiliary constant vector $\mu \in \mathbb {R}^n$ . We directly quote the following geometric lemma from [Reference Berti and Bolle10]:

Lemma 4.1 ([Reference Berti and Bolle10], Lemma 3).

There holds

$$ \begin{align*}\mu=\operatorname{\mathrm{Avg}}\Big((\partial u^y)^{\mathsf{T}} \mathscr{F}^x(h,u)-(\partial u^x)^{\mathsf{T}} \big(\mathscr{F}^y(h,u)-\mu\big)\Big). \end{align*} $$

In particular, $\mathscr {F}(h,u)=\begin {pmatrix}0 \\ \mu \end {pmatrix}$ necessarily implies $\mu =0$ .

In this subsection, our main task is to compute the linearization of $\mathscr {F}(h,u)$ at a given u, as in [Reference de la Llave, González, Jorba and Villanueva59]. For convenience in notation, write

(4.4) $$ \begin{align} \begin{aligned} A[u]:=(DX_h)(u) =\begin{pmatrix} D_x\nabla_yh(u) & D_y\nabla_yh(u) \\ -D_x\nabla_xh(u) & -D_y\nabla_xh(u) \end{pmatrix} \in\boldsymbol{M}_{2n\times 2n}, \end{aligned} \end{align} $$

so that

$$ \begin{align*}D_u\mathscr{F}(h,u)v=A[u]v-\nabla_\omega v. \end{align*} $$

We also write

(4.5) $$ \begin{align} N[u]=\big((\partial u)^{\mathsf{T}} \cdot\partial u\big)^{-1}\in\boldsymbol{M}_{n\times n}, \quad M[u]=\big( \partial u \quad (J\partial u)\cdot N[u] \big)\in\boldsymbol{M}_{2n\times2n}, \end{align} $$

provided that $\partial u^{\mathsf {T}} \cdot \partial u$ is invertible. This is true when u is $C^1$ close to $\zeta _0$ , whence $N[u]=I_n+O\big (\partial (u-\zeta _0)\big )$ . This furthermore ensures that

$$ \begin{align*}M[u]= \begin{pmatrix}I_n & \\ & -I_n\end{pmatrix}+O\big(\partial(u-\zeta_0)\big), \quad M[u]^{-1}=\begin{pmatrix}I_n & \\ & -I_n\end{pmatrix}+O\big(\partial(u-\zeta_0)\big). \end{align*} $$

We then introduce a function $L[u]$ , reflecting the “lack of being Lagrangian” for the submanifold $u(\mathbb {T}^n)\subset \mathbb {T}^n\times \mathbb {R}^n$ , denoted by

(4.6) $$ \begin{align} L[u]=(\partial u)^{\mathsf{T}} \cdot J\cdot\partial u \in\boldsymbol{M}_{n\times n}. \end{align} $$

Then $L[u]$ is nothing but the matrix representation of the pull-back of the symplectic form on $\mathbb {T}^n\times \mathbb {R}^n$ . The submanifold $u(\mathbb {T}^n)\subset \mathbb {T}^n\times \mathbb {R}^n$ is Lagrangian if and only if $L[u]=0$ . In particular, this is the case when u does solve the conjugacy equation $\mathscr {F}(h,u)=0$ .

The linearization formula for $\mathscr {F}(h,u)$ . is summarized as the following lemmaFootnote ⁵ :

Lemma 4.2. Suppose u is close to the trivial embedding $\zeta _0$ . Let $N[u],M[u],L[u]$ be as in (4.5)(4.6). Define the $n\times n$ matrix function

(4.7) $$ \begin{align} \begin{aligned} S[u] &=\big(I_n+(N[u]L[u])^2\big)^{-1}N[u]\cdot (\partial u)^{\mathsf{T}}\cdot\big[A[u],J\big]\cdot(\partial u)\cdot N[u]. \end{aligned} \end{align} $$

Then the linearization of $\mathscr {F}(h,u)$ with respect to u satisfies

(4.8) $$ \begin{align} D_u\mathscr{F}(h,u)\big(M[u]v\big) =M[u]\begin{pmatrix} 0_n & S[u] \\ 0_n & 0_n \end{pmatrix}v -M[u]\nabla_\omega v +\boldsymbol{B}\big(\partial\mathscr{F}(h,u),L[u]\big)v. \end{align} $$

Here in (4.8), the matrix function $\boldsymbol {B}(\partial E,L)$ is rationalFootnote ⁶ in $\partial E$ and L, vanishing linearly as $\partial E,L\to 0$ , with coefficients being rational functions of $A[u]$ , $\partial u$ and $\partial ^2u$ . We may then rewrite (4.8) as the equivalent form

(4.9) $$ \begin{align} D_u\mathscr{F}(h,u)v =M[u]\begin{pmatrix} 0_n & S[u] \\ 0_n & 0_n \end{pmatrix}M[u]^{-1}v -M[u]\nabla_\omega \big(M[u]^{-1}v\big) +\boldsymbol{B}\big(\partial\mathscr{F}(h,u),L[u]\big)M[u]^{-1}v, \end{align} $$

Proof. Throughout this proof, for simplicity in notation, we shall drop the $[u]$ (indicating dependence on u), write, for example, A for $A[u]$ , and denote $E:=\mathscr {F}(h,u)$ . For a given u and its increment v, we first write

$$ \begin{align*}\begin{aligned} D_u\mathscr{F}(h,u)(M[u]v) &=AMv -(\nabla_\omega M)v-M\nabla_\omega v\\ &=: \big(C_1\quad C_2\big)v-M\nabla_\omega v \end{aligned} \end{align*} $$

Here we write the matrix $AM -(\nabla _\omega M)\in \boldsymbol {M}_{2n\times 2n}$ into block form $\big (C_1\quad C_2\big )$ . The task is to determine the blocks $C_1,C_2\in \boldsymbol {M}_{2n\times n}$ .

It is fairly easy to see that $C_1=\partial E$ . In fact, we may write $M[u]$ into block form as

$$ \begin{align*}M =\big( \partial u \quad (J\partial u)N \big) =:\big(M_1\quad M_2\big). \end{align*} $$

Then obviously

(4.10) $$ \begin{align} \begin{aligned} C_1 &=AM_1-\nabla_\omega M_1\\ &=A\partial u-\nabla_\omega\partial u =\partial\big(\mathscr{F}(h,u)\big) =\partial E. \end{aligned} \end{align} $$

The true difficulty is with the expression of $C_2=A(J\partial u)N-\nabla _\omega \big ((J\partial u)N\big )$ . We try to find $n\times n$ matrices $Q,W$ , such that

(4.11) $$ \begin{align} C_2 =(\partial u)Q+(J\partial u)NW =M_1Q+M_2W. \end{align} $$

In solving this matrix system, we take into account that $L=(\partial u)^{\mathsf {T}} J\partial u$ is supposed to be close to 0. Multiplying (4.11) from left by $(\partial u)^{\mathsf {T}} J$ , using in sequence the Hamiltonian nature $JA=-A^{\mathsf {T}} J$ and the symplectic nature $J^2=-I_{2n}$ , we obtain

(4.12) $$ \begin{align} \begin{aligned} LQ-W &=(\partial u)^{\mathsf{T}} JA(J\partial u)N-(\partial u)^{\mathsf{T}} J\nabla_\omega \big((J\partial u)N\big)\\ &=(\partial u)^{\mathsf{T}} A^{\mathsf{T}}(\partial u) N +(\partial u)^{\mathsf{T}}(\nabla_\omega\partial u) N +(\partial u)^{\mathsf{T}}(\partial u)\nabla_\omega N\\ &=(\partial E)^{\mathsf{T}}(\partial u)N. \end{aligned} \end{align} $$

Here in the last step we differentiated the definition $(\partial u)^{\mathsf {T}}(\partial u) N=I_n$ to deduce

$$ \begin{align*}(\partial u)^{\mathsf{T}}(\nabla_\omega\partial u) N +(\partial u)^{\mathsf{T}}(\partial u)\nabla_\omega N =-\big[\nabla_\omega(\partial u)^{\mathsf{T}}\big](\partial u)N, \end{align*} $$

and then use the last equality of (4.10).

On the other hand, multiplying (4.11) from left by $N(\partial u)^{\mathsf {T}}$ , we obtain

(4.13) $$ \begin{align} \begin{aligned} Q+NLNW &=N(\partial u)^{\mathsf{T}} A\cdot(J\partial u)N -N(\partial u)^{\mathsf{T}}\nabla_\omega \big((J\partial u)N\big)\\ &=N(\partial u)^{\mathsf{T}} \Big[A,J\Big](\partial u)N +N(\partial u)^{\mathsf{T}} JA(\partial u)N\\ &\quad-N(\partial u)^{\mathsf{T}} J(\nabla_\omega\partial u)N -N(\partial u)^{\mathsf{T}} J(\partial u) \nabla_\omega N\\ &=N(\partial u)^{\mathsf{T}} \Big[A,J\Big](\partial u)N +N(\partial u)^{\mathsf{T}} J(\partial E) N -NL\nabla_\omega N. \end{aligned} \end{align} $$

We can then directly solve from (4.12)(4.13) the expression of $Q,W$ :

(4.14) $$ \begin{align} \begin{aligned} Q&=S +\big(I_n+(NL)^2\big)^{-1} \Big(NLN(\partial E)^{\mathsf{T}}(\partial u)N +N(\partial u)^{\mathsf{T}} J(\partial E)N-NL\nabla_\omega N\Big)\\ W&=LQ-(\partial E)^{\mathsf{T}}(\partial u)N. \end{aligned} \end{align} $$

Therefore, we immediately see that both $W-LS$ and $Q-S$ are (matrix) rational functions of L and $\partial E$ , vanishing linearly in L and $\partial E$ , with coefficients being rational functions of $\partial u$ and $\partial ^2u$ .

Combining (4.10) and (4.11), we find that the matrix $AM-(\nabla _\omega M)$ has block-form expression

$$ \begin{align*}\begin{aligned} AM-(\nabla_\omega M) &=\big(\partial E\quad M_1S\big) +\Big(0_{2n\times n} \quad M_1(Q-S)+M_2W\Big)\\ &=M\begin{pmatrix} 0_n & S \\ 0_n & 0_n \end{pmatrix} +\Big(\partial E \quad M_1(Q-S)+M_2W\Big). \end{aligned} \end{align*} $$

Therefore, the remainder $\boldsymbol {B}(\partial E,L)=\big (\partial E \quad M_1(Q-S)+M_2W\big )$ , which, by (4.14), is a matrix rational function of L and $\partial E$ , vanishing linearly in L and $\partial E$ , with coefficients being rational functions of A, $\partial u$ and $\partial ^2u$ .

A key feature of the Lagrangian character $L[u]$ is that it is in proportion to the error $E=\mathscr {F}(h,u)$ . In fact by Lemma 19 of [Reference de la Llave, González, Jorba and Villanueva59] or Lemma 5 of [Reference Berti and Bolle10], if we define the matrix function

$$ \begin{align*}L_1(u,E):=\partial \big((\partial u)^{\mathsf{T}}\cdot JE\big)-\left[\partial\big((\partial u)^{\mathsf{T}}\cdot JE\big)^{\mathsf{T}}\big)\right]^{\mathsf{T}}, \end{align*} $$

then there holds

(4.15) $$ \begin{align} \nabla_\omega L[u] =L_1(u,\mathscr{F}(h,u)) \quad\text{or equivalently}\quad L[u] =\nabla_\omega ^{-1}L_1(u,\mathscr{F}(h,u)). \end{align} $$

Note that due to the presence of $\partial $ , the function $L_1(u,E)$ necessarily has average zero. In other words, the matrix function $\nabla _\omega L[u]$ is nothing but the matrix representation of the exterior differential of the 1-form represented by $(\partial u)^{\mathsf {T}}\cdot J\mathscr {F}(h,u)$ . Of course, this is a refined version of the observation that an invariant torus is necessarily a Lagrangian submanifold. Therefore, the mapping $\boldsymbol {B}\big (\mathscr {F}(h,u),L[u]\big )$ in (4.9) vanishes linearly in E if u is approximately invariant:

Corollary 4.2.1. Suppose $\omega $ satisfies the Diophantine condition (1.3), so that (4.1) holds. Suppose $u\in H^s$ for some $s>\max (\tau +2,n/2+2)$ , while $|u-\zeta _0|_{C^2}$ is sufficiently close to zero. Then for any constant shift $(0,\mu )^{\mathsf {T}}\in \mathbb {R}^{n}\times \mathbb {R}^n$ , there holds

$$ \begin{align*}\big\|\boldsymbol{B}\big(\partial\mathscr{F}(h,u),L[u]\big)\big\|_{H^{s-(\tau+2)}} \leq \gamma^{-1} K_s\big(|h|_{C^{[s]+3}},\|u\|_{H^s}\big)\big\|\mathscr{F}(h,u)+(0,\mu)^{\mathsf{T}}\big\|_{H^{s-1}}. \end{align*} $$

The function $K_s$ is increasing in both arguments and does not depend on the shift $(0,\mu )^{\mathsf {T}}\in \mathbb {R}^{n}\times \mathbb {R}^n$ .

Proof. By Lemma 4.2, we know that $\boldsymbol {B}\big (\partial \mathscr {F}(h,u),L[u]\big )$ is a rational function of $\partial \mathscr {F}(h,u)$ and $L[u]$ , vanishing linearly as the arguments $\to 0$ , with coefficients being rational functions of $A[u],\partial u,\partial ^2 u$ . Since $|u-\zeta _0|_{C^2}$ is close to 0, we find that both $\partial \mathscr {F}(h,u)$ and $L[u]$ are $C^0$ close to 0, since they only involve u and $\partial u$ . Consequently, for example, the matrix $I_n+(N[u]L[u])^2$ is $H^{s-1}$ close to the identity matrix.

Obviously, shifting $\mathscr {F}(h,u)$ by a constant addendum does not affect the value of $\boldsymbol {B}\big (\partial \mathscr {F}(h,u),L[u]\big )$ . Therefore,

$$ \begin{align*}\big\|\boldsymbol{B}\big(\partial\mathscr{F}(h,u),L[u]\big)\big\|_{H^{s-(\tau+2)}} \leq K_s\big(|h|_{C^{[s]+3}},\|u\|_{H^s}\big)\left(\big\|\mathscr{F}(h,u)+(0,\mu)^{\mathsf{T}}\big\|_{H^{s-1}} +\|L[u]\|_{H^{s-(\tau+2)}}\right). \end{align*} $$

Now we make use of (4.15). Note that for the matrix function

$$ \begin{align*}L_1(u,E)=\partial \big((\partial u)^{\mathsf{T}}\cdot JE\big)-\left[\partial\big((\partial u)^{\mathsf{T}}\cdot JE\big)^{\mathsf{T}}\big)\right]^{\mathsf{T}}, \end{align*} $$

shifting E by a constant addendum $(0,\mu )^{\mathsf {T}}$ yields the same value of $L_1(u,E)$ , by a direct computation (using $\partial _i\partial _ju=\partial _j\partial _iu$ ). Therefore, by (4.15), we have

$$ \begin{align*}\begin{aligned} \|L[u]\|_{H^{s-(\tau+2)}} &\leq \gamma^{-1} \big\|L_1\big(u,\mathscr{F}(h,u)+(0,\mu)^{\mathsf{T}}\big)\big\|_{H^{s-2}}\\ &\lesssim \gamma^{-1} K_s\big(\|u\|_{H^s}\big)\big\|\mathscr{F}(h,u)+(0,\mu)^{\mathsf{T}}\big\|_{H^{s-1}}. \end{aligned} \end{align*} $$

This concludes the proof.

4.2 Deriving parahomological equation

So far we have been mostly quoting and rewriting the algebraic findings reported in Section 8 of [Reference de la Llave, González, Jorba and Villanueva59]. However, from this point, we will proceed quite differently from [Reference de la Llave, González, Jorba and Villanueva59], or any other known reference on KAM theory, for example, [Reference Berti and Bolle10] – instead of a Newtonian algorithm involving approximate right inverse, we employ the idea of Subsection 2.6 to directly solve the parahomological equation. We may treat Theorem 1.1 and 1.2 in a unified manner, since for Theorem 1.1 it suffices to require that the parameter $\xi $ must be zero.

So we restrict to a neighbourhood of the already known approximate solution $\zeta _0$ . We notice that $\mathscr {F}(h_\xi ,u)$ is a first order nonlinear partial differential operator acting on the unknown $(u,\xi )\in H^s\times \mathbb {R}^n$ , and in fact no differential in the constant $\xi $ is involved at all. Let us recall from (1.4) and (1.7) that $e_0=\mathscr {F}(h,\zeta _0)$ . Then the paralinearization formula

(4.16) $$ \begin{align} \begin{aligned} \mathscr{F}(h_\xi,u)+\begin{pmatrix} 0 \\ \mu\end{pmatrix} &=e_0+ \begin{pmatrix} \xi \\ \mu\end{pmatrix} +T_{D_u\mathscr{F}(h,u)}(u-\zeta_0) +\mathcal{R}_{\mathrm{PL}}(X_h,u-\zeta_0)\cdot(u-\zeta_0) \end{aligned} \end{align} $$

is valid. The expression $\boldsymbol {B}(E,L[u])$ involves only $\partial E$ , so it does not change if E is shifted by a constant vector. Thus, writing for simplicity

$$ \begin{align*}E:=\mathscr{F}(h_\xi,u)+\begin{pmatrix} 0 \\ \mu\end{pmatrix}, \end{align*} $$

we actually haveFootnote ⁷ , by Lemma 4.2, the following equality:

$$ \begin{align*}\begin{aligned} T_{D_u\mathscr{F}(h,u)}(u-\zeta_0) &= \mathbf{Op}^{\mathrm{PM}}\left[M[u]\begin{pmatrix} 0_n & T_{S[u]} \\ 0_n & 0_n \end{pmatrix}M[u]^{-1}\right](u-\zeta_0)\\ &\quad-\mathbf{Op}^{\mathrm{PM}}\Big({M[u]}\big(\nabla_\omega M[u]^{-1}\big)+\nabla_\omega \Big)(u-\zeta_0) +\mathbf{Op}^{\mathrm{PM}}\left(\boldsymbol{B}(E,L[u]) M[u]^{-1}\right)(u-\zeta_0). \end{aligned} \end{align*} $$

Thus the paralinearization formula (4.16) becomes

(4.17) $$ \begin{align} \begin{aligned} E&= e_0+\begin{pmatrix} \xi \\ \mu\end{pmatrix} +T_{M[u]}\begin{pmatrix} 0_n & T_{S[u]} \\ 0_n & 0_n \end{pmatrix}T_{M[u]^{-1}}(u-\zeta_0) -T_{M[u]}\nabla_\omega T_{M[u]^{-1}}(u-\zeta_0)\\ &\quad +\mathbf{Op}^{\mathrm{PM}}\left(\boldsymbol{B}(E,L[u]) M[u]^{-1}\right)(u-\zeta_0) +\mathcal{R}_1[u](u-\zeta_0) +\mathcal{R}_{\mathrm{PL}}(X_h,u-\zeta_0)\cdot(u-\zeta_0). \end{aligned} \end{align} $$

Here the linear operator

(4.18) $$ \begin{align} \begin{aligned} \mathcal{R}_1[u] &=\mathbf{Op}^{\mathrm{PM}}\left[M[u]\begin{pmatrix} 0_n & T_{S[u]} \\ 0_n & 0_n \end{pmatrix}M[u]^{-1}\right] -T_{M[u]}\begin{pmatrix} 0_n & T_{S[u]} \\ 0_n & 0_n \end{pmatrix}T_{M[u]^{-1}}\\ &\quad-\mathbf{Op}^{\mathrm{PM}}\Big[{M[u]}\big(\nabla_\omega M[u]^{-1}\big)+\nabla_\omega \Big] +T_{M[u]}\nabla_\omega T_{M[u]^{-1}} \end{aligned} \end{align} $$

is produced in view of Proposition 3.4. Note that we used the Leibniz rule $\partial T_au=T_{\partial a}u+T_a\partial u$ .

Lemma 4.3. Fix an index $s>2\tau +2+n/2+\varepsilon $ and set $r=s-n/2$ . Suppose the Hamilton function $h\in C^{N_{s+r}+3} u\in H^s(\mathbb {T}^n;\mathbb {T}^n\times \mathbb {R}^n)$ is close to $\zeta _0$ in that space. Then the operator $\mathcal {R}_1[u]$ defined by (4.18) maps $H^{t}$ to $H^{t+2\tau +\varepsilon }$ for every index $t\in \mathbb {R}$ : with $K_{t,s}$ being some increasing function,

$$ \begin{align*}\|\mathcal{R}_1[u]v\|_{H^{t+r-2}} \leq K_{t,s}\big(|h|_{C^{N_{s+r}+3}},\|u\|_{H^s}\big)\|u-\zeta_0\|_{H^s}\|v\|_{H^{t}}. \end{align*} $$

Furthermore, as such an operator, $\mathcal {R}_1[u]$ has $C^1$ dependence on $u\in H^s$ .

Proof. For simplicity of notation we write

$$ \begin{align*}a=M[u],\quad b=\begin{pmatrix} 0_n & T_{S[u]} \\ 0_n & 0_n \end{pmatrix},\quad c=M[u]^{-1}. \end{align*} $$

Then (4.18) is rewritten as

$$ \begin{align*}\begin{aligned} \mathcal{R}_1[u] &=T_{abc}-T_aT_bT_c+(T_aT_c-1)\nabla_\omega+T_aT_{\nabla_\omega c}-T_{a\nabla_\omega c}\\ &=\mathcal{R}_{\mathrm{CM}}(ab,c)+\mathcal{R}_{\mathrm{CM}}(a,b)T_c+\mathcal{R}_{\mathrm{CM}}(a,c)\nabla_\omega-\mathcal{R}_{\mathrm{CM}}(a,\nabla_\omega c). \end{aligned} \end{align*} $$

Here we used the fact that $ac$ is the identity matrix. We then recall from (4.5) that $a=\mathrm {diag}(I_n,-I_n)+O\big (\partial (u-\zeta _0)\big )$ , and $S[u]$ is given by (4.7). Therefore, we have

$$ \begin{align*}\big|S[u]\big|_{C^{r-1}_*}\leq K_{s}\big(|h|_{C^{N_{s+r}}},\|u\|_{H^s}\big), \end{align*} $$

and

$$ \begin{align*}\big|a-\text{diag}(I_n,-I_n)\big|_{C^{r-1}_*} +\big|c-\text{diag}(I_n,-I_n)\big|_{C^{r-1}_*} \leq \|u-\zeta_0\|_{H^s}. \end{align*} $$

We then use Proposition 3.4; notice that $\mathcal {R}_{\mathrm {CM}}(\cdot ,\cdot )$ does not change its value when the inputs are shifted by constants. Therefore, $\mathcal {R}_1[u]$ is a smoothing operator of order $r-2$ , vanishing linearly as $u\to \zeta _0$ . This gives the desired operatorial bound.

We can now formulate the parahomological equation in the unknown $(u,\xi ,\mu )$ , where $\xi ,\mu \in \mathbb {R}^n$ are constant parameters to be fixed. The equation reads

(4.19) $$ \begin{align} \begin{aligned} T_{M[u]}&\begin{pmatrix} 0_n & T_{S[u]} \\ 0_n & 0_n \end{pmatrix}T_{M[u]^{-1}}(u-\zeta_0) -T_{M[u]}\nabla_\omega T_{M[u]^{-1}}(u-\zeta_0) +\begin{pmatrix} \xi \\ \mu \end{pmatrix} \\ &=-e_0 -\mathcal{R}_1[u](u-\zeta_0) -\mathcal{R}_{\mathrm{PL}}(X_h,u-\zeta_0)\cdot(u-\zeta_0) \,. \end{aligned} \end{align} $$

In other words, it aims to cancel out all terms in the right-hand-side of (4.17) but the one linear in E.

To solve the parahomological equation (4.19), we need a lemma concerning linear parahomological equations, which is essentially the simple equation $\nabla _\omega v=f$ .

Lemma 4.4. Let $\omega $ be a Diophantine frequency vector satisfying (1.3). Set

$$ \begin{align*}M_Q=\Bigg\{\begin{aligned} \max\Big(\big|(\operatorname{\mathrm{Avg}} Q)^{-1}\big|&,|Q|_{L^\infty}\Big) &\quad\mathrm{under\ the\ assumption\ of\ Theorem}\ 1.1,\\&|Q|_{L^\infty}&\mathrm{under\ the\ assumption\ of\ Theorem}\ 1.2. \end{aligned} \end{align*} $$

There are constants $\rho _1,\rho _2$ depending on $|h|_{C^3}$ and $M_Q$ with the following property. If $|e_0|_{C^1}\leq \rho _1$ , and the embedding $u:\mathbb {T}^n\mapsto \mathbb {T}^n\times \mathbb {R}^n$ is such that $|u-\zeta _0|_{C^1}\leq \rho _2$ , then the linear parahomological equation in the unknown $(v,\xi ,\mu )\in H^s\times \mathbb {R}^n\times \mathbb {R}^n$

(4.20) $$ \begin{align} T_{M[u]}\begin{pmatrix} 0_n & T_{S[u]} \\ 0_n & 0_n \end{pmatrix}T_{M[u]^{-1}}v -T_{M[u]}\nabla_\omega T_{M[u]^{-1}}v +\begin{pmatrix} \xi \\ \mu \end{pmatrix} =f \end{align} $$

has a linear solution operator

$$ \begin{align*}\big( v^x, \, v^y, \, \xi, \, \mu \big) =\big( \mathfrak{L}^x[u]f, \, \mathfrak{L}^y[u]f, \, P^x[u]f, \, P^y[u]f \big), \end{align*} $$

satisfying the following estimates: with $K_s$ being increasing functions in the arguments,

$$ \begin{align*}\begin{aligned} \big\|\mathfrak{L}^x[u]f\big\|_{H^s} &\leq K_s\big(M_Q,|h|_{C^3}\big) \gamma^{-2}\|f\|_{H^{s+2\tau}} \\ \big\|\mathfrak{L}^y[u]f\big\|_{H^s} &\leq K_s\big(M_Q,|h|_{C^3}\big)\gamma^{-1}\|f\|_{H^{s+\tau}} \\ |P^x[u]f|+|P^y[u]f| &\leq K_s\big(M_Q,|h|_{C^3}\big)\gamma^{-1}\|f\|_{H^{s+\tau}}. \end{aligned} \end{align*} $$

Moreover, the four linear operators of concern are all continuously differentiable mappings from $u\in C^1$ to the space of linear operators (with operator norm). In particular, the solution operator $\xi =P^x[u]f$ is fixed as 0 under the assumption of Theorem 1.1.

Proof. Since $M[u]=\mathrm {diag}(I_n,-I_n)+O\big (\partial (u-\zeta _0)\big )$ , it follows that $T_{M[u]}$ and $T_{M[u]^{-1}}$ are both invertible bounded operator from $H^s$ to $H^s$ if $|u-\zeta _0|_{C^1}$ is small. Note further that for a constant $\lambda $ and a function A, there holds $T_A\lambda =(\operatorname {\mathrm {Avg}} A)\lambda $ . Thus, writing

(4.21) $$ \begin{align} T_{M[u]}^{-1}\begin{pmatrix} \xi \\ \mu \end{pmatrix}=\begin{pmatrix} \xi_1 \\ \mu_1 \end{pmatrix}, \quad T_{M[u]}^{-1}f=f_1, \quad T_{M[u]^{-1}}v=v_1 \end{align} $$

for some constant vectors $\xi _1,\mu _1\in \mathbb {R}^n$ to be determined, the linear parahomological equation (4.20) is equivalent to

$$ \begin{align*}\begin{pmatrix} 0_n & T_{S[u]} \\ 0_n & 0_n \end{pmatrix}v_1 -\nabla_\omega v_1 +\begin{pmatrix} \xi_1 \\ \mu_1 \end{pmatrix} =f_1. \end{align*} $$

Furthermore, recalling the expression (1.2) of $h(x,y)$ and (4.7), we have

(4.22) $$ \begin{align} S[u]=Q+\partial^2a_0+O\big(\partial(u-\zeta_0)\big), \end{align} $$

where the implicit constant depends on $|h|_{C^3}$ . Noting that

(4.23) $$ \begin{align} e_0=\mathscr{F}(h,\zeta_0) =\begin{pmatrix} a_1-\omega \\ \partial a_0 \end{pmatrix}, \end{align} $$

it follows that $S[u]$ is $C^0$ -close to Q if $|e_0|_{C^1}$ and $|u-\zeta _0|_{C^1}$ are small.

In components of $v_1$ , the linear parahomological equation becomes

$$ \begin{align*}\begin{aligned} T_{S[u]}v_1^y-\nabla_\omega v_1^x+\xi_1&=f_1^x, \\ -\nabla_\omega v_1^y+\mu_1&=f_1^y. \end{aligned} \end{align*} $$

We discuss how to solve this system under the assumption of Theorem 1.1 and 1.2 separately. For simplicity of notation, we define

$$ \begin{align*} \mathcal{A}f:=f-\operatorname{\mathrm{Avg}} f.\\[-34pt] \end{align*} $$

Case 1: $\mathrm {Avg} Q$ is invertible

This is the nondegeneracy condition in Therorem 1.1. We fix $\xi =0$ in this case, and obtain from (4.21) the following:

(4.24) $$ \begin{align} \begin{pmatrix} 0 \\ \mu \end{pmatrix} =\begin{pmatrix} \xi_1 \\ -\mu_1 \end{pmatrix} +O\big(\partial(u-\zeta_0)\big)\begin{pmatrix} \xi_1 \\ \mu_1 \end{pmatrix}. \end{align} $$

We first solve

(4.25) $$ \begin{align} v_1^y=\operatorname{\mathrm{Avg}} v_1^y-\nabla_\omega ^{-1}\mathcal{A}f_1^y, \quad \mu_1=\operatorname{\mathrm{Avg}} f_1^y, \end{align} $$

where $\operatorname {\mathrm {Avg}} v_1^y$ is yet to be determined. With (4.24), we can solve $\xi _1$ , thus $\mu $ , as linear mappings of $\mu _1$ . Since the norm of $T_{M[u]}^{-1}$ from $H^s$ to $H^s$ depends only on $|u-\zeta _0|_{C^1}$ , this immediately yields

$$ \begin{align*}\big\|\mathcal{A}v_1^y\big\|_{H^s}+|\mu|+|\xi_1| \leq C_s\gamma^{-1}\|f\|_{H^{s+\tau}}. \end{align*} $$

Here we used (4.1).

We then choose $\operatorname {\mathrm {Avg}} v_1^y$ to balance the mean value of the equation along x-direction. By assumption $S[\zeta _0]$ is invertible, so by (4.22), the matrix-valued function $S[u]=Q+O(\partial e_0)+O\big (\partial (u-\zeta _0)\big )$ is still invertible if $|e_0|_{C^1}$ and $|u-\zeta _0|_{C^1}$ are small, and the implicit constant here depends on $|h|_{C^3}$ . In that case, $\big |(\operatorname {\mathrm {Avg}} S[u])^{-1}\big |\lesssim M_Q$ . We can then set

$$ \begin{align*}\operatorname{\mathrm{Avg}} v_1^y=\big(\mathrm{Avg} S[u]\big)^{-1}\operatorname{\mathrm{Avg}}\big(f_1^x-\xi_1-T_{S[u]}\mathcal{A}v_1^y\big), \end{align*} $$

and solve

$$ \begin{align*}v_1^x=\nabla_\omega ^{-1}\mathcal{A} \big(T_{S[u]}\mathcal{A}v_1^y+\xi_1-f_1^x\big). \end{align*} $$

By (4.7), $S[u]$ depends on up to second order derivative of h, so the solutions are immediately estimated as

$$ \begin{align*}\big\|v_1^x\big\|_{H^s} \leq K_s\big(M_Q,|h|_{C^3}\big) \gamma^{-2}\|f\|_{H^{s+2\tau}}. \end{align*} $$

Here we used (4.1) once again.

Case 2: Nontrivial parameter $\xi $

This case matches with Theorem 1.2, whence no requirement about nondegeneracy of Q is posed. In this case the parameter $\xi $ cannot be fixed as 0. But we still obtain from (4.21) the following:

(4.26) $$ \begin{align} \begin{pmatrix} \xi \\ \mu \end{pmatrix} =\begin{pmatrix} \xi_1 \\ -\mu_1 \end{pmatrix} +O\big(\partial(u-\zeta_0)\big)\begin{pmatrix} \xi_1 \\ \mu_1 \end{pmatrix}. \end{align} $$

We then first solve

(4.27) $$ \begin{align} v_1^y=-\nabla_\omega ^{-1}\mathcal{A} f_1^y, \quad \mu_1=\operatorname{\mathrm{Avg}} f_1^y, \end{align} $$

that is, the mean value of $v_1^y$ is fixed as 0. For the equation along x-direction, we solve

$$ \begin{align*}v_1^x=-\nabla_\omega ^{-1}\mathcal{A} \big(f_1^x-T_{S[u]}v_1^y\big), \quad \xi_1=\operatorname{\mathrm{Avg}} \big(f_1^x-T_{S[u]}v_1^y\big). \end{align*} $$

Using (4.1), they immediately yield the similar estimates as in Case 1.

Finally, continuous differentiability of the operators for $u\in C^1$ follows immediately from Proposition 3.3 and 3.4, that is, continuous differentiability of $T_a$ in a and $T_{ab}-T_aT_b$ in $(a,b)$ .

4.3 Proof of Theorem 1.1–1.2

Recall that we set $r=s-n/2$ , and $N_{s+r}$ to be the least integer $>s+r$ . Since we are working with a perturbative problem, we assume that, for example, $\|u-\zeta _0\|_{H^s}\leq 2$ , and the requirement of Lemma 4.4 is also fulfilled: with $\rho _1=\rho _1\big (M_Q,|h|_{C^3}\big )$ , $\rho _2=\rho _2\big (M_Q,|h|_{C^3}\big )$ as in Lemma 4.4, we have

(4.28) $$ \begin{align} \|u-\zeta_0\|_{H^s}\leq2, \quad |e_0|_{C^1}\leq \rho_1,\quad |u-\zeta_0|_{C^1}\leq\rho_2. \end{align} $$

The parahomological equation (4.19) is then converted to the following parainverse equation:

(4.29) $$ \begin{align} \begin{aligned} u-\zeta_0 &=-\begin{pmatrix} \mathfrak{L}^x[u]-P^x[u] \\ \mathfrak{L}^y[u]-P^y[u] \end{pmatrix}e_0\\ &\quad -\begin{pmatrix} \mathfrak{L}^x[u]-P^x[u] \\ \mathfrak{L}^y[u]-P^y[u] \end{pmatrix}\left(\mathcal{R}_1[u](u-\zeta_0) -\mathcal{R}_{\mathrm{PL}}(X_h,u-\zeta_0)\cdot(u-\zeta_0)\right)\\ &=:\mathcal{G}_1(u)+\mathcal{G}_2(u) \end{aligned} \end{align} $$

We analyze the right-hand side of (4.29). The requirements (4.28) meet the conditions of Lemma 4.4, so

(4.30) $$ \begin{align} \begin{aligned} \|\mathcal{G}_1(u)\|_{H^{s+\varepsilon}} &\leq K_s\big(M_Q,|h|_{C^3}\big)\gamma^2\|e_0\|_{H^{s+2\tau+\varepsilon}}. \end{aligned} \end{align} $$

We turn to $\mathcal {G}_2$ . The paraproduct remainder $\mathcal {R}_1[u](u-\zeta _0)$ is estimated by Lemma 4.3:

(4.31) $$ \begin{align} \big\|\mathcal{R}_1[u](u-\zeta_0)\big\|_{H^{s+r-2}} \leq K_s\big(|h|_{C^{N_{s+r}}}\big)\|u-\zeta_0\|_{H^s}^2, \end{align} $$

and $\mathcal {R}_1[u](u-\zeta _0)$ depends continuously on u (with respect to Sobolev norms $H^s$ and $H^{s+r-2}$ ). As for the paralinearization remainder $\mathcal {R}_{\mathrm {PL}}$ , by Proposition 3.5, recalling (4.4),

(4.32) $$ \begin{align} \mathcal{R}_{\mathrm{PL}}(X_h,u-\zeta_0)\cdot(u-\zeta_0) =\mathcal{R}_{\mathrm{PL};1}\big(A[\zeta_0]\big)(u-\zeta_0) +\mathcal{R}_{\mathrm{PL};2}(X_h,u-\zeta_0)\cdot(u-\zeta_0). \end{align} $$

Recalling further the definition (1.2) of $h(x,y)$ , we find

$$ \begin{align*}A[\zeta_0] =\begin{pmatrix} \partial a_1(\theta) & Q(\theta) \\ \partial^2a_0(\theta) & -(\partial a_1)^{\mathsf{T}}(\theta) \end{pmatrix}. \end{align*} $$

On the other hand, recalling the definition (4.23), we find

$$ \begin{align*}\begin{aligned} \|a_1-\omega\|_{H^{s+\tau+\varepsilon}} +\|\partial a_0\|_{H^{s+\tau+\varepsilon}} &=\|e_0\|_{H^{s+2\tau+\varepsilon}}, \end{aligned} \end{align*} $$

that is, $a_0$ is almost a constant, and $a_1$ almost equals the constant vector $\omega $ . As a result, we find $A[\zeta _0]$ is almost a constant:

$$ \begin{align*}\begin{aligned} \big|A[\zeta_0]-\operatorname{\mathrm{Avg}} A[\zeta_0]\big|_{C^r_*} &\leq C_s\big(\|a_1-\omega\|_{H^{s+\tau+\varepsilon}} +\|\partial a_0\|_{H^{s+\tau+\varepsilon}} +|Q-\operatorname{\mathrm{Avg}} Q|_{C^r_*}\big)\\ &\leq C_s\big(\|e_0\|_{H^{s+2\tau+\varepsilon}}+|e_1|_{C^r_*}\big). \end{aligned} \end{align*} $$

Implementing the paralinearization theorem, that is, Proposition 3.5, we estimate (4.32) as (note that $X_h=J\nabla _{x,y}h\in C^{N_{s+r}+2}$ )

(4.33) $$ \begin{align} \begin{aligned} \big\|\mathcal{R}_{\mathrm{PL}}&(X_h,u-\zeta_0)\cdot(u-\zeta_0)\big\|_{H^{s+r}} \\ &\leq C_s\Big(\big(\|e_0\|_{H^{s+2\tau+\varepsilon}}+|e_1|_{C^r_*}\big)\|u-\zeta_0\|_{H^s} +|h|_{C^{N_{s+r}+3}}\|u-\zeta_0\|_{H^s}^2\Big). \end{aligned} \end{align} $$

Furthermore, by Proposition 3.5, the operator $\mathcal {R}_{\mathrm {PL}}\big (X_h(\zeta _0+\cdot \,),u-\zeta _0\big )\in \mathcal {L}(H^s,H^{s+r})$ depends continuously on $u\in H^s$ .

Recall that by assumption, $s+r-2=2s-n/2-2\geq s+2\tau +\varepsilon $ . Combining (4.30)(4.31)(4.33), using Lemma 4.4, we conclude that if (4.28) holds, then

(4.34) $$ \begin{align} \begin{aligned} \|\mathcal{G}_1(u)&\|_{H^{s+\varepsilon}} +\|\mathcal{G}_2(u)\|_{H^{s+\varepsilon}}\\ &\leq K_s\big(M_Q,|h|_{C^{N_{s+r}+3}}\big)\gamma^{-2} \Big(\|e_0\|_{H^{s+2\tau+\varepsilon}}+\big(\|e_0\|_{H^{s+2\tau+\varepsilon}}+|e_1|_{C^r_*}\big)\|u-\zeta_0\|_{H^s} +\|u-\zeta_0\|_{H^s}^2\Big). \end{aligned} \end{align} $$

By some elementary algebra with quadratic polynomials, there are decreasing functions $c_0,c_1$ and an increasing function $c_2$ , with the following property: if

$$ \begin{align*}\|e_0\|_{H^{s+2\tau+\varepsilon}} \leq\gamma^{4}c_0\big(M_Q,|h|_{C^{N_{s+r}+3}}\big), \quad |e_1|_{C^r_*} \leq \gamma^{2}c_1\big(M_Q,|h|_{C^{N_{s+r}+3}}\big), \end{align*} $$

then the positive number $\rho =\|e_0\|_{H^{s+\tau +\varepsilon }}\gamma ^{2}c_2\big (M_Q,|h|_{C^{N_{s+r}+3}}\big )$ satisfies the quadratic inequality

$$ \begin{align*}K_s\big(M_Q,|h|_{C^{N_{s+r}+3}}\big)\gamma^{-2} \Big(\|e_0\|_{H^{s+2\tau+\varepsilon}}+\big(\|e_0\|_{H^{s+2\tau+\varepsilon}}+|e_1|_{C^r_*}\big)\rho +\rho^2\Big)\leq\rho. \end{align*} $$

Here $K_s$ is the increasing function in (4.34). In other words,

$$ \begin{align*}\|\mathcal{G}_1(u)\|_{H^{s+\varepsilon}} +\|\mathcal{G}_2(u)\|_{H^{s+\varepsilon}} \leq \rho \quad\text{if }\|u-\zeta_0\|_{H^s}\leq\rho\text{ and (4.28) holds.} \end{align*} $$

In that case, the mapping $\zeta _0+\mathcal {G}_1+\mathcal {G}_2$ in the right-hand-side of (4.29) maps the closed ball $\bar B_{\rho }(\zeta _0)\subset H^s$ to itself. Furthermore it is continuous with respect to $u\in \bar B_{\rho }(\zeta _0)\subset H^s$ and the image is totally bounded (since the embedding $H^{s+\varepsilon }\subset H^s$ is compact). By the Schauder fixed point theorem, the mapping $u\mapsto \zeta _0+\mathcal {G}_1(u)+\mathcal {G}_2(u)$ has a fixed point $u\in \bar B_{\rho }(\zeta _0)$ , and in fact

(4.35) $$ \begin{align} \|u-\zeta_0\|_{H^s} \leq c_2\big(M_Q,|h|_{C^{N_{s+r}+3}}\big)\gamma^{-2}\|e_0\|_{H^{s+2\tau+\varepsilon}}. \end{align} $$

Having found a solution u of the parahomological equation (4.19), we determine the constants $\xi ,\eta $ in terms of u via the operators $P^x[u],P^y[u]$ . We will then be able to cancel all terms but the one linear in E in the paralinearization formula (4.17). Thus the paralinearization formula (4.17) at u reduces to

(4.36) $$ \begin{align} E=\mathbf{Op}^{\mathrm{PM}}\left(\boldsymbol{B}(E,L[u]) M[u]^{-1}\right)(u-\zeta_0). \end{align} $$

Using Corollary 4.2.1, we estimate

(4.37) $$ \begin{align} \begin{aligned} \big\|\mathbf{Op}^{\mathrm{PM}}&\left(\boldsymbol{B}(E,L[u]) M[u]^{-1}\right)(u-\zeta_0)\big\|_{H^s}\\&\overset{\text{Prop. } 3.3}{\leq} C_s\big|\boldsymbol{B}(E,L[u]) M[u]^{-1}\big|_{L^\infty}\|u-\zeta_0\|_{H^s}\\&\overset{(4.35)}{\leq} C_sc_2\big(M_Q,|h|_{C^{N_{s+r}+3}}\big)\gamma^{-2}\|e_0\|_{H^{s+2\tau+\varepsilon}}\big\|\boldsymbol{B}(E,L[u]) M[u]^{-1}\big\|_{H^{s-(\tau+2)}}\\&\overset{\text{Cor. } 4.2.1}{\leq} C_sc_2\big(M_Q,|h|_{C^{N_{s+r}+3}}\big)\gamma^{-2}\|e_0\|_{H^{s+2\tau+\varepsilon}}\|E\|_{H^{s-1}}. \end{aligned} \end{align} $$

Note that it is legitimate to apply Corollary 4.2.1, since we already know that u is close to $\zeta _0$ , and therefore $E=\mathscr {F}(h_\xi ,u)+(0,\mu )^{\mathsf {T}}$ is close to zero. Consequently, if we further require the function $c_0\ll 1/c_2$ , then the factor

$$ \begin{align*}C_sc_2\big(M_Q,|h|_{C^{N_{s+r}+3}}\big)\gamma^{-3}\|e_0\|_{H^{s+2\tau+\varepsilon}} \end{align*} $$

in the right-hand-side of (4.37) will be $\leq 1/2$ (remembering $e_0$ is of size $O(\gamma ^{4}c_0)$ and $\gamma $ is assumed to be small). Therefore, (4.37) yields $\|E\|_{H^s}\leq (1/2)\|E\|_{H^{s-1}}$ , forcing $E=0$ .

To summarize, under the assumption of Theorem 1.1, there is a solution u of the the parahomological equation (4.19) with $\xi =0$ in (4.19), the constant vector $\mu $ is determined by u, and u satisfies

$$ \begin{align*}E=\mathscr{F}(h,u)+\begin{pmatrix} 0 \\ \mu\end{pmatrix}=0. \end{align*} $$

Under the assumption of Theorem 1.2, there is a solution u of the the parahomological equation (4.19), the parameters $\xi ,\mu $ are determined by u, and u satisfies

$$ \begin{align*}E=\mathscr{F}(h_\xi,u)+\begin{pmatrix} 0 \\ \mu\end{pmatrix}=0. \end{align*} $$

By Lemma 4.1, the proof of Theorem 1.1–1.2 is complete.

4.4 Miscellaneous

We discuss the generality of the assumption of Theorem 1.1–1.2. The idea of our discussion closely follows [Reference Berti and Bolle10]. At a first glance, the coverage of Theorem 1.1–1.2 is very limited: the approximate invariant torus around which we seek for a true one has to be the “flat” embedding $\theta \mapsto (\theta ,0)$ . However, under suitable geometric transformations, which are elementary and independent from KAM type results, it is possible to reduce quite general situations to this special case.

To be concrete, let the frequency $\omega $ satisfy (1.3). Suppose $h(x,y)$ is an arbitrary Hamiltonian function, not necessarily brought into the form (1.2). Let

$$ \begin{align*}\zeta(\theta)=\begin{pmatrix} \zeta^x(\theta) \\ \zeta^y(\theta) \end{pmatrix} \end{align*} $$

be an embedding from $\mathbb {T}^n$ to the phase space, not necessarily close to the “flat” embedding, such that $\zeta ^x:\mathbb {T}^n\mapsto \mathbb {T}^n$ is a diffeomorphism, and $\zeta (\mathbb {T}^n)$ is an approximately invariant torus with frequency $\omega $ under the flow of h: that is,

$$ \begin{align*}\mathscr{F}(h,\zeta):=X_h(\zeta)-\nabla_\omega \zeta \end{align*} $$

is close to 0. We aim to find suitable symplectic coordinate transformation that reduces this general approximate solution to the special case discussed in Theorem 1.1–1.2.

Recall (4.6): we introduce

$$ \begin{align*}L[\zeta]=(\partial \zeta)^{\mathsf{T}} \cdot J \partial \zeta\in\boldsymbol{M}_{n\times n} \end{align*} $$

to measure the “lack of isotropy” of the embedding $\zeta $ , since the pull-back of the symplectic form $\operatorname {\mathrm {d}}\!x\wedge \operatorname {\mathrm {d}}\!y$ to $\mathbb {T}^n$ via $\zeta $ has exactly the matrix representation $L[\zeta ]$ . We then have the following, as seen in (4.15):

(4.38) $$ \begin{align} L[\zeta]=\partial\nabla_\omega ^{-1}\left[(\partial \zeta)^{\mathsf{T}}\cdot J\mathscr{F}(h,\zeta)\right] -\partial\nabla_\omega ^{-1}\left[(\partial \zeta)^{\mathsf{T}}\cdot J\mathscr{F}(h,\zeta)\right]^{\mathsf{T}}. \end{align} $$

According to Lemma 6 of [Reference Berti and Bolle10], if one sets

$$ \begin{align*}p(\theta)=\Delta^{-1}\partial\cdot L[\zeta] =\left(\Delta^{-1}\sum_{j=1}^n\partial_jL_{kj}(\theta)\right)_{1\leq k\leq n}, \end{align*} $$

then the embedded torus

$$ \begin{align*}\eta(\theta) =\begin{pmatrix} \eta^x(\theta) \\ \eta^y(\theta) \end{pmatrix} :=\begin{pmatrix} \zeta^x(\theta) \\ \zeta^y(\theta)-\big(\partial\zeta^x(\theta)\big)^{\mathsf{T}} p(\theta) \end{pmatrix} \end{align*} $$

is isotropic. From (4.38), p, hence $\eta -\zeta $ , is linear in $\mathscr {F}(h,\zeta )$ , and for $s>1+n/2$ ,

$$ \begin{align*}\|\eta-\zeta\|_{H^s}\leq C_s\gamma^{-1}\|\zeta\|_{H^{s+\tau+1}}\|\mathscr{F}(h,\zeta)\|_{H^{s+\tau}}, \end{align*} $$

where $C_s$ only depends on s. Since

$$ \begin{align*}\begin{aligned} \mathscr{F}(h,\eta) &=X_h\big(\zeta+(\eta-\zeta)\big)-\nabla_\omega \zeta -\nabla_\omega (\eta-\zeta)\\ &=\mathscr{F}(h,\zeta) +\int_0^1 (DX_h)\big(\zeta+\tau(\eta-\zeta)\big)(\eta-\zeta)\operatorname{\mathrm{d}}\!\tau -\nabla_\omega (\eta-\zeta), \end{aligned} \end{align*} $$

it then follows that

$$ \begin{align*}\|\mathscr{F}(h,\eta)\|_{H^s} \leq C_s\gamma^{-1}\big(1+|h|_{C^{N_{s+r}+3}}\big) \big(1+\|\zeta\|_{H^{s+\tau+2}}\big)\|\mathscr{F}(h,\zeta)\|_{H^{s+\tau+1}}, \end{align*} $$

where $C_s$ only depends on s and $|\omega |$ . In other words, if $\zeta (\mathbb {T}^n)$ is an approximately invariant torus for $X_h$ , then so is the isotropic torus $\eta (\mathbb {T}^n)$ , but the “error of being invariant” is less regular compared to $\zeta (\mathbb {T}^n)$ .

Since $\eta $ is an isotropic embedding, the diffeomorphism in phase space

$$ \begin{align*}G:\begin{pmatrix} x \\ y \end{pmatrix}\mapsto \begin{pmatrix} \eta^x(x) \\ \eta^y(x)+\big(\partial\eta^x(x)\big)^{\mathsf{T}} y \end{pmatrix} \end{align*} $$

is a symplectic one. The isotropic torus $\eta (\mathbb {T}^n)$ is straightened to the “flat” torus $\{y_1=0\}$ under the new symplectic coordinate $(x_1,y_1)=G^\iota (x,y)$ . If we set

$$ \begin{align*}H(x_1,y_1) :=(h\circ G)(x_1,y_1) =a_0(x_1)+\langle a_1(x_1),y_1\rangle +\frac{1}{2}\langle Q(x_1)y_1,y_1\rangle+O(|y_1|^3) \quad\text{near }y=0, \end{align*} $$

then the Hamiltonian vector field $X_h$ is changed to $X_H=(DG)^{-1}X_h\circ G$ . Furthermore, the “error of being invariant” is transformed to

$$ \begin{align*}(DG)^{-1}\mathscr{F}(h,\eta) =X_H(\zeta_0)-\begin{pmatrix}\omega \\ 0 \end{pmatrix} =\begin{pmatrix} a_1(\theta)-\omega \\ \partial a_0(\theta) \end{pmatrix}. \end{align*} $$

Thus $a_0$ is almost constant, and $a_1$ is almost equal to $\omega $ . We then set $S=\nabla _\omega ^{-1}(Q-\operatorname {\mathrm {Avg}} Q)$ , and introduce another canonical variable $(x_2,y_2)$ , defined by

$$ \begin{align*}\begin{aligned} x_1&=x_2+S(x_1)y_2\\ y_1&=y_2-\frac{1}{2}\partial S(x_1)(y_2,y_2). \end{aligned} \end{align*} $$

The symplectic diffeomorphism $\Gamma $ maps the isotropic torus $\{y_2=0\}$ to $\{y_1=0\}$ , while the Hamiltonian function

$$ \begin{align*}\begin{aligned} H(x_1,y_1) &=\left(a_0(x_1)-\operatorname{\mathrm{Avg}} a_0\right) +\langle a_1(x_1)-\omega,y_1\rangle\\ &\quad+\operatorname{\mathrm{Avg}} a_0+\langle \omega,y_2\rangle +\frac{1}{2}\langle (\operatorname{\mathrm{Avg}} Q)y_2,y_2\rangle+O(|y_2|^3). \end{aligned} \end{align*} $$

We thus find that the first three Taylor coefficients of $H(x_1,y_1)$ with respect to $y_2$ around $y_2=0$ will be close to

$$ \begin{align*}\operatorname{\mathrm{Avg}} a_0,\quad \omega,\quad \operatorname{\mathrm{Avg}} Q, \end{align*} $$

respectively. Consequently, under the new symplectic coordinate $(x_2,y_2)$ , the Hamiltonian function and approximately invariant torus are exactly in the form discussed by Theorem 1.1–1.2.

In conclusion, we have successfully reduced the general situation to the special case of Theorem 1.1 without any assumption on smallness of $\zeta ^x,\zeta ^y$ . For Hamiltonian functions depending on parameters, the argument will be exactly the same.