Towards continuity: Universal frequency-preserving KAM persistence and remaining regularity

TL;DR

This paper establishes a universal frequency-preserving KAM theorem for finitely differentiable Hamiltonian systems, demonstrating persistence and regularity of invariant tori under critical smoothness conditions, with explicit non-Hölder regularity analysis.

Contribution

It introduces a novel KAM theorem with sharp differentiability conditions that preserves universal frequencies and handles critical smoothness, expanding the scope of KAM theory.

Findings

Proves persistence of invariant tori with universal Diophantine frequencies.

Demonstrates non-Hölder regularity of invariant tori and conjugation.

Provides new systems where previous KAM methods fail but this approach succeeds.

Abstract

Beyond H\"{o}lder's type, this paper mainly concerns the persistence and remaining regularity of an individual frequency-preserving KAM torus in a finitely differentiable Hamiltonian system, even allows the non-integrable part being critical finitely smooth. To achieve this goal, besides investigating the Jackson approximation theorem towards only modulus of continuity, we demonstrate an abstract regularity theorem adapting to the new iterative scheme. Via these tools, we obtain a KAM theorem with sharp differentiability hypotheses, asserting that the persistent torus keeps prescribed universal Diophantine frequency unchanged. Further, the non-H\"older regularity for invariant KAM torus as well as the conjugation is explicitly shown by introducing asymptotic analysis. To our knowledge, this is the first approach to KAM on these aspects in a continuous sense, and we also provide two systems, which cannot be studied by previous KAM but by ours.

Equations

$$|⟨\tilde{k},\omega ⟩|\ge {\alpha }_{\ast }{|\tilde{k}|}^{-\tau },\forall 0\ne \tilde{k}\in {\mathbb{Z}}^n$$

$${\int }_0^1\frac{\varpi \left(x\right)}{x}\mathrm{d}x<+\infty$$

$$\left|{\partial }^{\alpha }H\left(x,y,\epsilon \right)-{\partial }^{\alpha }H\left(\xi ,\eta ,\epsilon \right)\right|\le \frac{c}{{\left(-\ln \left(\left|x-\xi \right|+\left|y-\eta \right|\right)\right)}^{\lambda }}$$

$${\left[f\right]}_{\varpi }:=\underset{x,y\in {\mathbb{R}}^n,0<|x-y|\le \delta }{\sup }\frac{\left|f\left(x\right)-f\left(y\right)\right|}{\varpi \left(\left|x-y\right|\right)},{\left|f\right|}_{C^0}:=\underset{x\in {\mathbb{R}}^n}{\sup }\left|f\left(x\right)\right|.$$

$${\Vert f\Vert }_{\varpi }:=\sum _{|\alpha |\le k}\left({\left|{\partial }^{\alpha }f\right|}_{C^0}+{\left[{\partial }^{\alpha }f\right]}_{\varpi }\right)<+\infty .$$

$${\varpi }_{\mathrm{H}}^{\{\ell \}}(x)\sim x^{\ell },x\to 0^{+},$$

$${\varpi }_{\mathrm{LH}}^{\lambda }\left(x\right)\sim 1/{\left(-\ln x\right)}^{\lambda },x\to 0^{+},$$

$${\varpi }_{\mathrm{GLH}}^{\varrho ,\lambda }\left(x\right)\sim \frac{1}{(\ln (1/x))(\ln \ln (1/x))\cdots {(\underset{\varrho }{\underbrace{\ln \cdots \ln }}(1/x))}^{\lambda }},x\to 0^{+},$$

$${\omega }_{f,\delta }\left(x\right):=\underset{y\in \left[0,1\right],0<|x-y|\le \delta }{\sup }\left|f\left(x\right)-f\left(y\right)\right|.$$

$$\psi \left(x\right):=\underset{0<r<\delta /x}{\sup }\frac{\varpi \left(rx\right)}{\varpi \left(r\right)}=\mathcal{O}\left(x\right),x\to +\infty .$$

$$\underset{x\to 0^{+}}{\overline{\lim }}\frac{\varpi \left(x\right)}{\varpi \left(ax\right)}<+\infty .$$

$$S_{r}f\left(x\right)=r^{-n}{\int }_{{\mathbb{R}}^n}K\left(r^{-1}\left(x-y\right)\right)f\left(y\right)\mathrm{d}y,0<r\le 1$$

$$\left|{\partial }^{\alpha }{S}_{r}f\left(x\right)-P_{{\partial }^{\alpha }f,k-|\alpha |}\left(\mathrm{Re}x;\mathrm{i}\mathrm{Im}x\right)\right|\le c\left(n,k\right){\Vert f\Vert }_{\varpi }{r}^{k-|\alpha |}\varpi (r),$$

$$P_{f,k}\left(x;y\right):=\sum _{|\beta |\le k}\frac{1}{\alpha !}{\partial }^{\beta }f\left(x\right)y^{\alpha }.$$

$$\left|{\partial }^{\alpha }\left(S_{r}f\left(x\right)-f\left(x\right)\right)\right|\le c_{\ast }{\Vert f\Vert }_{\varpi }{r}^{k-|\alpha |}\varpi (r)$$

$$\left|{\partial }^{\alpha }{S}_{r}f\left(x\right)\right|\le c^{\ast }{\Vert f\Vert }_{\varpi }$$

$${\int }_0^1\frac{\varpi \left(x\right)}{x^{2\tau +3-k}}\mathrm{d}x<+\infty .$$

$${\Vert H\Vert }_{\varpi }\le M,\left|{\left({\int }_{{\mathbb{T}}^n}{H}_{yy}\left(\xi ,0\right)\mathrm{d}\xi \right)}^{-1}\right|\le M.$$

$$|⟨\tilde{k},\omega ⟩|\ge {\alpha }_{\ast }{|\tilde{k}|}^{-\tau },\forall 0\ne \tilde{k}\in {\mathbb{Z}}^n,|\tilde{k}|:=\sum _{j=1}^n|{\tilde{k}}_j|.$$

$$\displaystyle\sum\limits_{\left|\alpha\right|\leq k}{\left|{{\partial^{\alpha}}\Big{(}{H\left({x,0}\right)-\int_{{\mathbb{T}^{n}}}{H\left({\xi,0}\right){\rm d}\xi}}\Big{)}}\right|{\varepsilon^{\left|\alpha\right|}}}$$

$$+$$

$${\int }_0^1\frac{{\phi }_i\left(x\right)}{x^{k_i^{\ast }+1}}\mathrm{d}x<+\infty ,{\int }_0^1\frac{{\phi }_i\left(x\right)}{x^{k_i^{\ast }+2}}\mathrm{d}x=+\infty .$$

$$x=u\left(\xi \right),y=v\left(\xi \right)$$

$$Du=H_y\left(u,v\right),Dv=-H_x\left(u,v\right),$$

$${\varpi }_i\left(\gamma \right)\sim \gamma {\int }_{L_i\left(\gamma \right)}^{\epsilon }\frac{{\phi }_i\left(t\right)}{t^{k_i^{\ast }+2}}\mathrm{d}t={\mathcal{O}}^{\#}\left({\int }_0^{L_i\left(\gamma \right)}\frac{{\phi }_i\left(t\right)}{t^{k_i^{\ast }+1}}\mathrm{d}t\right),\gamma \to 0^{+},$$

$$Du=H_y\left(u,v\right),Dv=-H_x\left(u,v\right)$$

$${\varpi }_1\left(x\right)\sim \frac{1}{{\left(-\ln x\right)}^{\lambda -1}}\sim {\varpi }_{\mathrm{LH}}^{\lambda -1}\left(x\right),$$

$${\varpi }_2\left(x\right)\sim \{\begin{array}{rlr}& \frac{1}{{\left(-\ln x\right)}^{\lambda -1}}\sim {\varpi }_{\mathrm{LH}}^{\lambda -1}\left(x\right),& n-1<\tau \in {\mathbb{N}}^{+}\text{hfill},\\ & \frac{x^{\left\{\tau \right\}}}{{\left(-\ln x\right)}^{\lambda }}\sim x^{\left\{\tau \right\}}{\varpi }_{\mathrm{LH}}^{\lambda }\left(x\right),& n-1<\tau \notin {\mathbb{N}}^{+}\text{hfill},\end{array}$$

$${\varpi }_{\mathrm{GLH}}^{\varrho ,\lambda }\left(x\right)\sim \frac{1}{(\ln (1/x))(\ln \ln (1/x))\cdots {(\underset{\varrho }{\underbrace{\ln \cdots \ln }}(1/x))}^{\lambda }},x\to 0^{+}.$$

$$Du=H_y\left(u,v\right),Dv=-H_x\left(u,v\right)$$

Full text

Keywords: Hamiltonian system, frequency-preserving KAM torus, modulus of continuity, remaining regularity, Jackson approximation theorem.

2020 Mathematics Subject Classification: 37J40, 70K60.

1. 1 Introduction

2. 2 Statement of results

3. 2.1 Preliminaries for modulus of continuity

4. 2.2 Jackson type approximation theorem beyond Hölder class

5. 2.3 Universal frequency-preserving KAM via modulus of continuity and remaining regularity

6. 3 Applications

7. 3.1 KAM via different continuity

8. 3.1.1 Hölder continuous case

9. 3.1.2 Hölder plus Logarithmic Hölder continuous case

10. 3.1.3 Hölder plus generalized Logarithmic Hölder continuous case

11. 3.2 Two explicit Hamiltonian systems of Logarithmic Hölder’s type

12. 3.2.1 Non-Hölder corner’s type

13. 3.2.2 Nowhere Hölder’s type

14. 4 Proof of Main Theorem 2.2

15. 4.1 Frequency-preserving KAM persistence

16. 4.2 Iteration theorem on regularity beyond Hölder’s type

17. 5 Proof of other results

18. 5.1 Proof of Theorem 3.1

19. 5.2 Proof of Theorem 3.2

20. 5.3 Proof of Theorem 3.3

21. 6 Appendix

22. 6.1 Semi separability and weak homogeneity for modulus of continuity

23. 6.2 Proof of Theorem 2.1

24. 6.3 Proof of Corollary 2.2

25. 6.4 Proof of Corollary 2.3

26. 6.5 Asymptotic analysis

27. 6.6 KAM theorem for quantitative estimates

1 Introduction

The celebrated KAM theory, due to Kolmogorov and Arnold [22, 2, 3, 4], Moser [27, 28], mainly concerns the preservation of invariant tori of a Hamiltonian function $H(y)$ under small perturbations (i.e., $H(y)\to H\left({x,y,\varepsilon}\right)$ of freedom $n\in\mathbb{N}^{+}$ with $\varepsilon>0$ sufficiently small), and indeed has a history of more than sixty years. So far, KAM theory has been well developed and widely applied to a variety of dynamical systems. For some other fundamental developments, see Kuksin [24], Eliasson [12], Pöschel [33], Wayne [40], Bourgain [7, 8] and so on.

As is known to all, for frequency $\omega={H_{y}}\left({y}\right)$ of the unperturbed Hamiltonian system, one often requires it to satisfy the following classical Diophantine condition (or be of Diophantine class $\tau$)

[TABLE]

with respect to Diophantine index $\tau\geq n-1$ and some $\alpha_{*}>0$, where $|\tilde{k}|:=\sum\nolimits_{j=1}^{n}{|{{\tilde{k}}_{j}}|}$. Otherwise, under a Liouville frequency that is not Diophantine, the torus may break no matter how small the perturbation is, and some chaotic behavior can take place simultaneously. The strong (non-universal) Diophantine frequencies of class $\tau=n-1$, are shown to be a continuum but only form a set of zero Lebesgue measure, therefore the KAM preservation based on that is usually said to be non-universal. As a contrast, the class $\tau>n-1$ Diophantine nonresonance KAM persistence is universal because such frequencies are of full Lebesgue measure in $\mathbb{R}^{n}$, and we will focus on this case throughout this paper.

Considering Diophantine nonresonance, the invariant tori are shown in classical KAM theorems to be preserved under analytic settings. It should be pointed out that, if a constant Diophantine frequency is prescribed in advance, one could obtain KAM persistence with frequency being unchanged by proposing certain nondegeneracy or transversality conditions, which preserves more dynamics from the perturbed one, see Salamon [35], Du et al [11] and Tong et al [38] for instance, respectively, and this analytic torus is interestingly shown to be never isolated by Eliasson et al in [13]. Therefore, beyond analyticity, it is still interesting to touch the minimum initial regularity for Hamiltonian to which KAM could applied. On this aspect, the first breakthrough was made by Moser in [26], where he studied twist maps (could correspond to an iso-energetic KAM theorem for $n=2$) admitting finite smoothness of integral order sufficiently large. Much effort has been devoted on this technical problem in terms of Hölder continuity, including constructing counterexamples and reducing the differentiability hypotheses. For some classic fundamental work, see Moser [29], Jacobowitz [19], Zehnder [41, 42], Mather [25], Herman [16, 17], Pöschel [31, 32] and etc. It is worth mentioning that, very recently Pöschel announced in his preprint [34] a KAM theorem on the $n$-dimensional torus $\mathbb{T}^{n}$ (without action variables) based on a non-universal frequency being of Diophantine class $\tau=n-1$ in (1.1). Specially, he pointed out that the derivatives of order $n$ need not be continuous, but rather $L^{2}$ in a certain strong sense, by introducing a new norm over $\mathbb{T}^{n}$.

Back to our concern on general $n$-freedom Hamiltonian systems having action-angular variables, it is always conjectured that the minimum regularity requirement for the Hamiltonian function $H(x,y)$ is at least $C^{2n}$ due to the $C^{2n-\epsilon}$ (with any $\epsilon$ close to $0^{+}$) counterexample constructed by Cheng and Wang [9], which allows for all frequencies in $\mathbb{R}^{n}$, see also Wang [39]. However, via Diophantine nonresonance, a more precise counterexample does not seem to have appeared. As to reducing initial regularity, along with the idea of Moser [29], the best known Hölder case $C^{\ell}$ with $\ell>2\tau+2$ has been established by Salamon in [35], the prescribed universal frequency under consideration is of Diophantine class $\tau>n-1$ in (1.1), and the remaining regularity of the frequency-preserving KAM torus as well as the conjugation is also showed to be Hölder’s type. More precisely, the KAM torus is at least of class $C^{1}$, and the conjugation from the dynamic on it to the linear flow is at least of class $C^{\tau+1}$. Besides, the $C^{\ell}$ ($\ell>2\tau+2>2n$) differentiability hypotheses is sharp due to the counterexamples of Herman [16, 17] and Cheng and Wang [9]. In the aspect of Hölder’s type, see Khesin et al [21], Bounemoura [6] and Koudjinan [23] for some other new developments. To strictly weaker than Hölder continuity, a KAM theorem via non-universal Diophantine nonresonance of class $\tau=n-1$ in (1.1) was proved by Albrecht in [1], claiming that $C^{2n}$ plus certain modulus of continuity $\varpi$ satisfying the classical Dini condition

[TABLE]

is enough for the non-universal KAM persistence. To the best of our knowledge, there is no other work on KAM via only modulus of continuity except for [1].

Concerning about universal KAM persistence for finitely differentiable Hamiltonians of freedom $n$, the best result so far, obtained by Salamon [35], still requires $C^{2n}$ plus certain Hölder continuity depending on the Diophantine index. It is therefore natural that ones should consider the following fundamental questions successively, in order to further touch the criticality in this long standing finitely differentiable KAM since Moser:

• •

Can non-Hölder regularity allow for KAM persistence with universal frequency-preserving?

• •

What kind of smoothness shall the invariant KAM torus admit, if it is indeed persistent? How about the conjugation?

• •

Does there exist a Dini type integrability condition similar to (1.2) that reveals the explicit relation between nonresonance and regularity?

Usually there are at least two ways reaching finitely smooth KAM: abstract implicit function method began in Nash [30] and Zehnder [41, 42], as well as the analytic smoothing approach due to Moser [29]. We also refer to Hamilton [14] and Hörmander [18] for some other versions, and we shall use the latter to touch the case with only continuity, because it provides an iterative scheme which could derive accurate remaining regularity for KAM torus and conjugation. As a consequence, towards continuity, there are at least four difficulties to overcome. Firstly, note that the Jackson approximation theorem for classical Hölder continuity is no longer valid at present, hence it must be developed to approximate the perturbed Hamiltonian function $H\left({x,y,\varepsilon}\right)$ in the sense of modulus of continuity, as a crucial step. It needs to be emphasized that the analysis uses different approaches in comparison with the Hölder one, by proposing the semi separability on modulus of continuity. Secondly, it is also basic how to establish a corresponding regularity iteration theorem to study the remaining regularity of the invariant torus and the conjugation without Hölder’s type. Thirdly, we have to set up a new KAM iterative scheme and prove its uniform convergence in the $C^{1}$ topology via these tools. Fourthly, it is somewhat difficult to extract an integrability condition of universal nonresonance and initial regularity from KAM iteration, keeping the frequency-preserving KAM persistence. Indeed, to achieve Main Theorem 2.2 having sharp differentiability hypotheses and frequency-preserving, we apply Theorem 2.1 to construct a series of analytic approximations to Hamiltonian $H\left({x,y,\varepsilon}\right)$ with modulus of continuity, and prove the persistence and regularity of invariant torus via a modified KAM iteration as well as a generalized Dini type condition. As some new efforts, our Main Theorem 2.2 applies to a wide range in a continuous sense (even allows the non-integrable part to be non-Hölder), and reveals the sharp integral relation between regularity and universal Diophantine nonresonance for the first time. Apart from above, it is well known that small divisors must lead to certain loss of smoothness (such as reducing the radius of analyticity), and only Hölder class has been investigated so far, see Salamon [35]. If only continuity for highest order derivatives is assumed, obviously ones should not expect the KAM remaining regularity still be of Hölder’s type. On this aspect, our Main Theorem 2.2 gives the first approach to the regularity of invariant KAM torus and conjugation being non-Hölder, explicitly shown by certain asymptotic analysis. Besides, as shown by Theorem 3.2, the KAM torus and conjugation may interestingly admit a completely different class of regularity, due to the affect of the Diophantine index. Particularly, as a direct application, our Main Theorem 2.2 could deal with the case of general modulus of continuity for $H\left({x,y,\varepsilon}\right)$, such as Logarithmic Hölder continuity case, that is, for all $0<\left|{x-\xi}\right|+\left|{y-\eta}\right|\leq 1/2$,

[TABLE]

with respect to all $\alpha\in{\mathbb{N}^{2n}}$ with $\left|\alpha\right|={2n}$, where $n\geq 2$, $\lambda>1$, $n-1<\tau\in\mathbb{N}^{+}$, $c,\varepsilon>0$ are sufficiently small, $\left({x,y}\right)\in{\mathbb{T}^{n}}\times G$ with ${\mathbb{T}^{n}}:={\mathbb{R}^{n}}/\mathbb{Z}^{n}$, and $G\subset{\mathbb{R}^{n}}$ is a connected closed set with interior points. One shall notice that, this Hamiltonian system with non-Hölder continuity cannot be studied by any available KAM theorems so far, see Section 3 for more details.

This paper is organized as follows. In Section 2, by introducing some basic notions and properties for modulus of continuity, we establish a Jackson type approximation theorem in a continuous sense, and the proof will be postponed to Section 6.2. Then we state our main result in this paper. Namely, considering that the highest order derivatives of Hamiltonian function $H$ with respect to the action-angular variables $(x,y)$ are only continuous, we present a KAM theorem (Main Theorem 2.2) with sharp differentiability hypotheses under certain assumptions, involving a generalized Dini type integrability condition (H1). The applications of this result are given in Section 3, including Hölder, Hölder plus Logarithmic Hölder and a more complicated circumstance, aiming to show the importance and universality of Main Theorem 2.2. In particular, two explicit Hamiltonian systems are constructed, which cannot be studied by any KAM theorems for finite smoothness via classical Hölder continuity, namely having a non-Hölder corner and even being of nowhere Hölder’s type, while our Main Theorem 2.2 still works. Section 4 provides the proof of Main Theorem 2.2 and is mainly divided into two parts: the first part deals with the modified KAM steps via only modulus of continuity, while the second part is devoted to giving an iteration theorem (Theorem 4.1) on regularity, which is essential in analyzing the KAM remaining smoothness for the invariant torus as well as the conjugation. Finally Section 5 presents the proof of Theorems 3.1, 3.2 and 3.3 in Section 3, respectively.

2 Statement of results

2.1 Preliminaries for modulus of continuity

We first give some notions, including the modulus of continuity along with the norm based on it, the semi separability which will be used in Theorem 2.1, as well as the weak homogeneity which will appear in Theorem 2.2.

Denote by $|\cdot|$ the sup-norm in $\mathbb{R}^{d}$ and the dimension $d\in\mathbb{N}^{+}$ may vary throughout this paper. We formulate that in the limit process, $f_{1}(x)=\mathcal{O}^{\#}\left(f_{2}(x)\right)$ means there are absolute positive constants $\ell_{1}$ and $\ell_{2}$ such that ${\ell_{1}}{f_{2}}\left(x\right)\leq{f_{1}}\left(x\right)\leq{\ell_{2}}{f_{2}}\left(x\right)$, and $f_{1}(x)=\mathcal{O}\left(f_{2}(x)\right)$ implies that there exists an absolute positive constant $\ell_{3}$ such that $|f_{1}(x)|\leq\ell_{3}f_{2}(x)$, and finally $f_{1}(x)\sim f_{2}(x)$ indicates that $f_{1}(x)$ and $f_{2}(x)$ are equivalent.

Definition 2.1** (Modulus of continuity).**

A nondecreasing continuous function $\varpi(t)>0$ on the interval $\left({0,\delta}\right]$ with respect to some $\delta>0$ is said to be a modulus of continuity, if $\mathop{\lim}\limits_{x\to{0^{+}}}\varpi\left(x\right)=0$ and $\mathop{\overline{\lim}}\limits_{x\to{0^{+}}}x/{\varpi}\left(x\right)<+\infty$. Next, we define the following semi norm and norm for a continuous function $f$ on ${\mathbb{R}^{n}}$:

[TABLE]

We call $f$ $C_{k,\varpi}$ continuous if $f$ has partial derivatives ${{\partial^{\alpha}}f}$ for $\alpha=(\alpha_{1},\ldots,\alpha_{n})\in\mathbb{N}^{n},\left|\alpha\right|:=\sum\nolimits_{i=1}^{n}{\left|{{\alpha_{i}}}\right|}\leq k\in\mathbb{N}$ and satisfies

[TABLE]

Denote by ${C_{k,\varpi}}\left({{\mathbb{R}^{n}}}\right)$ the space composed of all functions $f$ satisfying (2.1).

Remark 2.1**.**

For $f:{\mathbb{R}^{n}}\to\Omega\subset{\mathbb{R}^{d}}$ with a modulus of continuity $\varpi$, we modify the above designation to ${C_{k,\varpi}}\left({{\mathbb{R}^{n}},\Omega}\right)$.

It can be seen that the well-known Lipschitz continuity and Hölder continuity are special cases in the above definition. In particular, for $0<\ell\notin\mathbb{N}^{+}$, we denote by $f\in{C^{\ell}}\left({{\mathbb{R}^{n}}}\right)$ the function space in which the higher derivatives in $\mathbb{R}^{n}$ are $\{\ell\}$-Hölder continuous, that is, the modulus of continuity is of the form

[TABLE]

where $\{\ell\}\in(0,1)$ denotes the fractional part of $\ell$. As a generalization of classical Hölder continuity, we define the Logarithmic Hölder continuity with index $\lambda>0$, where

[TABLE]

and we will omit the the range $0<x\ll 1$ without causing ambiguity for these kind of functions with singularities away from $0^{+}$, because one only needs to focus on the asymptotic behavior of a given modulus of continuity near $0^{+}$. Moreover, one could further consider the generalized Logarithmic Hölder’s type with indices $\varrho\in\mathbb{N}^{+}$ and $\lambda>0$ as follows:

[TABLE]

and we have $\varpi_{\mathrm{GLH}}^{1,\lambda}\left(x\right)\sim\varpi_{\mathrm{LH}}^{\lambda}\left(x\right)$.

Remark 2.2**.**

It is well known that a mapping defined on a bounded connected closed set in a finite dimensional space must have a modulus of continuity, see [15, 20]. For example, for a function $f(x)$ defined on $[0,1]\subset{\mathbb{R}^{1}}$, it automatically admits a modulus of continuity

[TABLE]

In view of Remark 2.2, we therefore only focus on modulus of continuity throughout this paper. Next we introduce the comparison relation between the strength and the weakness of modulus of continuity.

Definition 2.2**.**

Let ${\varpi_{1}}$ and ${\varpi_{2}}$ be modulus of continuity on interval $\left({0,\delta}\right]$. We say that ${\varpi_{1}}$ is weaker (strictly weaker) than ${\varpi_{2}}$ if $\mathop{\overline{\lim}}\limits_{x\to{0^{+}}}{\varpi_{2}}\left(x\right)/{\varpi_{1}}\left(x\right)<+\infty$ ($=0$).

Remark 2.3**.**

Obviously any modulus of continuity is weaker than Lipschitz’s type, and the Logarithmic Hölder’s type $\varpi_{\mathrm{LH}}^{\lambda}\left(x\right)\sim 1/{\left({-\ln x}\right)^{\lambda}}$ with any $\lambda>0$ is strictly weaker than arbitrary Hölder’s type $\varpi_{\mathrm{H}}^{\alpha}\left(x\right)\sim{x^{\alpha}}$ with any $0<\alpha<1$. The generalized Logarithmic Hölder’s type $\varpi_{\mathrm{GLH}}^{\varrho,\lambda}\left(x\right)$ in (2.2) with $\varrho\in\mathbb{N}^{+}$ and $\lambda>0$ is weaker than both of them.

As a consequence, one directly obtains the following corollary by Remark 2.3. This shows that it is indeed very necessary for us to extend Hölder’s continuous KAM to only continuous type.

Corollary 2.1**.**

Give a bounded connected closed set $\Omega\subset\mathbb{R}^{n}$ with interior points. Then for every $k\in\mathbb{N}$, we have ${C_{k,\varpi_{\mathrm{H}}^{\alpha}}}\left(\Omega\right)\subseteq{C_{k,\varpi_{\mathrm{LH}}^{\lambda}}}\left(\Omega\right)\subset{C_{k,\varpi_{\mathrm{GLH}}^{\varrho,\lambda}}}\left(\Omega\right)$ with arbitrary given $0<\alpha<1,\lambda>0$ and $\varrho\in\mathbb{N}^{+}$.

Finally, in order to emphasize the rigor of the KAM analysis throughout this paper, we have to introduce two crucial concepts for modulus of continuity, namely the semi separability and the weak homogeneity.

Definition 2.3** (Semi separability).**

A modulus of continuity $\varpi$ is said to be semi separable, if for $x\geq 1$, there holds

[TABLE]

Remark 2.4**.**

Semi separability directly leads to $\varpi\left({rx}\right)\leq\varpi\left(r\right)\psi\left(x\right)$ for $0<rx\leq\delta$, which will be used in the proof of the Jackson type Theorem 2.1 via only modulus of continuity.

Definition 2.4** (Weak homogeneity).**

A modulus of continuity $\varpi$ is said to admit weak homogeneity, if for fixed $0<a<1$, there holds

[TABLE]

Remark 2.5**.**

The weak homogeneity plays a controlling role in KAM iteration, see details from the proof of Main Theorem 2.2.

It should be emphasized that semi separability and weak homogeneity are universal hypotheses. The Hölder and Lipschitz types automatically admit them. Many modulus of continuity weaker than the Hölder one are semi separable and also admit weak homogeneity, e.g., for the Logarithmic Hölder’s type $\varpi_{\mathrm{LH}}^{\lambda}\left(x\right)\sim 1/{\left({-\ln x}\right)^{\lambda}}$ with any $\lambda>0$, one verifies that $\psi\left(x\right)\sim{\left({\ln x}\right)^{\lambda}}=\mathcal{O}\left(x\right)$ as $x\to+\infty$ in (2.3), and $\mathop{\overline{\lim}}\limits_{x\to{0^{+}}}{\varpi_{\mathrm{LH}}^{\lambda}}\left(x\right)/{\varpi_{\mathrm{LH}}^{\lambda}}\left({ax}\right)=1<+\infty$ with all $0<a<1$ in (2.4). See examples in Lemmas 6.1 and 6.2, in particular, it is pointed out that a convex modulus of continuity naturally possesses these two properties.

2.2 Jackson type approximation theorem beyond Hölder class

This section provides a Jackson type approximation theorem beyond Hölder’s type and some related corollaries based on Definitions 2.1 and 2.3. Their proof will be postponed to Sections 6.2, 6.3 and 6.4, respectively. We shall emphasize that, the Hölder’s conclusion is relatively easy to obtain, because the Hölder continuity $\varpi_{\mathrm{H}}^{\alpha}(x)$ with $0<\alpha<1$ is naturally homogeneous, i.e., $\varpi_{\mathrm{H}}^{\alpha}(bx)=b^{\alpha}\varpi_{\mathrm{H}}^{\alpha}(x)$ holds for any $b>0$, and it is indeed crucial in the analysis. As to the general modulus of continuity, one has to propose semi separability instead of this (in a weaker sense) and the analysis becomes more complicated, namely estimating the integrals separately, under different approaches, see details from Section 6.2.

Theorem 2.1**.**

There is a family of convolution operators

[TABLE]

from ${C^{0}}\left({{\mathbb{R}^{n}}}\right)$ into the space of entire functions on ${\mathbb{C}^{n}}$ with the following property. For every $k\in\mathbb{N}$, there exists a constant $c\left({n,k}\right)>0$ such that, for every $f\in{C_{k,\varpi}}\left({{\mathbb{R}^{n}}}\right)$ with a semi separable modulus of continuity $\varpi$, every multi-index $\alpha\in{\mathbb{N}^{n}}$ with $\left|\alpha\right|\leq k$, and every $x\in{\mathbb{C}^{n}}$ with $\left|{\operatorname{Im}x}\right|\leq r$, we have

[TABLE]

where the Taylor polynomial $P$ is defined as follows

[TABLE]

Moreover, ${{S_{r}}f}$ is real analytic whenever $f$ is real valued.

As a direct consequence of Theorem 2.1, we give the following Corollaries 2.2 and 2.3. These results have been widely used in Hölder’s case, see for instance, [23, 35, 10].

Corollary 2.2**.**

The approximation function ${{S_{r}}f\left(x\right)}$ in Theorem 2.1 satisfies

[TABLE]

and

[TABLE]

for $x\in\mathbb{C}^{n}$ with $\left|{\operatorname{Im}x}\right|\leq r$, $|\alpha|\leq k$, where $c_{*}=c_{*}\left({n,k}\right)>0$ and ${c^{*}}={c^{*}}\left({n,k,\varpi}\right)>0$ are some universal constants.

Corollary 2.3**.**

If the function $f\left(x\right)$ in Theorem 2.1 also satisfies that the period of each variables ${x_{1}},\ldots,{x_{n}}$ is $1$ and the integral over ${\mathbb{T}^{n}}$ is zero, then the approximation function ${S_{r}}f\left(x\right)$ also satisfies these properties.

2.3 Universal frequency-preserving KAM via modulus of continuity and remaining regularity

We are now in a position to state the universal frequency-preserving KAM theorem via only modulus of continuity in this paper. Before this, let’s start with our parameter settings. Let $n\geq 2$ (degree of freedom), $\tau>n-1$ (Diophantine index), $2\tau+2\leq k\in{\mathbb{N}^{+}}$ (differentiable order) and a sufficiently large number $M>0$ be given, throughout all KAM theorems in this paper. Consider a Hamiltonian function with action-angular variables $H(x,y):{\mathbb{T}^{n}}\times G\to\mathbb{R}$ with ${\mathbb{T}^{n}}:={\mathbb{R}^{n}}/\mathbb{Z}^{n}$, and $G\subset{\mathbb{R}^{n}}$ is a connected closed set with interior points. It follows from Remark 2.2 that $H(x,y)$ automatically admits a modulus of continuity $\varpi$. In view of the comments below Definition 2.4, we assume that $\varpi$ admits semi separability and weak homogeneity without loss of generality. Besides, we make the following assumptions:

• (H1)

Dini type integrability condition: Assume that $H(x,y)\in{C_{k,\varpi}}\left({{\mathbb{T}^{n}}\times G}\right)$ with the above modulus of continuity $\varpi$. In other words, $H(x,y)$ has at least derivatives of order $k$, and the highest order derivatives admit the regularity of $\varpi$. Moreover, $\varpi$ satisfies the Dini type integrability condition

[TABLE]

• (H2)

Boundedness and nondegeneracy:

[TABLE]

• (H3)

Diophantine condition: For some $\alpha_{*}>0$, the prescribed frequency $\omega\in{\mathbb{R}^{n}}$ satisfies

[TABLE]

• (H4)

KAM smallness: There holds

[TABLE]

for every $x\in\mathbb{R}^{n}$ and some constant $0<\varepsilon\leq{\varepsilon^{*}}$.

• (H5)

Criticality: For $\varphi_{i}(x):=x^{k-(3-i)\tau-1}\varpi(x)$ with $i=1,2$, there exist critical $k_{i}^{*}\in\mathbb{N}^{+}$ such that

[TABLE]

Let us make some comments on our assumptions.

• (C1)

Our Dini type integrability condition (H1) reveals the deep relationship between the irrationality for frequency $\omega$, order and continuity of the highest order derivatives for the Hamiltonian $H(x,y)$, and appears in universal KAM for the first time. It also admits sharpness, because $H(x,y)$ could be $C^{[2\tau+2]}$ differentiable, close to the counterexample constructed in [9].

• (C2)

According to the properties of Banach algebra, for the Hölder’s type, it is assumed that (H4) only needs the term of $\left|\alpha\right|=0$, and does not need higher order derivatives to satisfy the condition. However, for general modulus of continuity, it seems not easy to establish the corresponding Banach algebraic properties, we thus add higher order derivatives in (H4). Sometimes they can be removed correspondingly.

• (C3)

The existence of $k_{i}^{*}$ in (H5) is directly guaranteed by (H1), actually this assumption is proposed to investigate the higher regularity of the persistent KAM torus as well as the conjugation, that is, the regularity to $C^{k_{i}^{*}}$ plus certain kinds of modulus of continuity. In general, given an explicit modulus of continuity $\varpi$, such $k_{i}^{*}$ in (H5) are automatically determined by using asymptotic analysis, see details from Section 3.

Finally, under sharp differentiability of $C^{[2\tau+2]}$, we state the following Main Theorem 2.2 involving frequency-preserving KAM persistence as well as remaining regualrity for invariant torus and conjugation. To the best of our knowledge, this is the first approach on these two aspects, towards KAM via only continuity.

Theorem 2.2** (Main Theorem).**

(Part I) Assume (H1)-(H4). Then there is a solution

[TABLE]

of the following equation with the operator $D:=\sum\limits_{\nu=1}^{n}{{\omega_{\nu}}\frac{\partial}{{\partial{\xi_{\nu}}}}}=\omega\cdot{\partial_{\xi}}$

[TABLE]

*such that $u\left(\xi\right)-\xi$ and $v\left(\xi\right)$ are of period $1$ in all variables, where $u$ and $v$ are at least $C^{1}$.

(Part II) In addition, assume (H5). Then there exist modulus of continuity ${\varpi_{i}}$ ($i=1,2$) such that $u\in{C_{k_{1}^{*},{\varpi_{1}}}}\left({{\mathbb{R}^{n}},{\mathbb{R}^{n}}}\right)$ and $v\circ{u^{-1}}\in{C_{k_{2}^{*},{\varpi_{2}}}}\left({{\mathbb{R}^{n}},G}\right)$. Particularly, ${\varpi_{i}}$ can be explicitly determined as follows*

[TABLE]

where $L_{i}(\gamma)\to 0^{+}$ are some functions such that the second relation in (2.8) holds for $i=1,2$.

Remark 2.6**.**

We say such a solution $x=u(\xi),y=v(\xi)$ the KAM one. Besides, the regularity for $v\circ{u^{-1}}$ and $u$ represents to the smoothness of the invariant torus and the conjugation from the dynamic on it to the linear flow, respectively.

Remark 2.7**.**

• (a)

With the same as in **[35]**, the unperturbed system under consideration might be non-integrable (e.g., $H=\left\langle{\omega,y}\right\rangle+\left\langle{A\left(x\right)y,y}\right\rangle+\cdots$), which is different from the requirement of having integrable part in **[32, 6]**, and the KAM persistence is of frequency-preserving.

• (b)

A Hamiltonian system having non-integrable unperturbed part is generally not reduced to a nearly integrable one, see Arnold et al **[5]**.

• (c)

Moreover, our non-integrable part here could only be $C_{k,\varpi}$ finitely differentiable with $k\geq 2\tau+2$ (we will see later that the critical case $k=[2\tau+2]$ could be indeed achieved in Theorems 3.1, 3.2 and 3.3), while considering the integrable part, **[1]** and **[32]** require the real-analyticity, and **[6]** requires the Hölder continuity $C^{\varsigma}$ with $\varsigma>2\tau+4$.

Remark 2.8**.**

Actually Theorem 2.2 provides a method for determining $\varpi_{i}$ with $i=1,2$, see (2.8). For the prescribed modulus of continuity to Hamiltonian, such as the Hölder and Logarithmic Hölder types, we have to use asymptotic analysis to derive the remaining continuity explicitly in Section 3, which is somewhat difficult.

As mentioned forego, the Hölder’s type $H\in C^{\ell}(\mathbb{T}^{n},G)$ with $\ell>2\tau+2$ (where $\tau>n-1$ is the Diophantine index) is always regarded as the critical case in the sense of Hölder due to [35, 6, 9]. Now let us consider the non-universal Diophantine nonresonance, i.e., $\tau=n-1$, and such frequencies can only form a set of zero Lebesgue measure. Then one observes that $k=2\tau+2=2n$ is the critical degenerate case in our settings, and our Dini type integrability condition (2.6) in (H1) becomes the classical Dini condition (1.2), which also allows regularity weaker than arbitrary Hölder’s type. This is exactly the same regularity required in [1], concentrating on non-universal KAM persistence without frequency-preserving. However, the non-universal Diophantine frequencies with index $\tau=n-1$ are not enough to represent almost all frequencies in $\mathbb{R}^{n}$.

Back to our concern on universal KAM persistence, we may have to require the generalized Dini condition in (H1) instead of (1.2). Obviously, (H1) automatically holds if the highest differentiable order $k$ of $H(x,y)$ satisfies $k\geq 2\tau+3$ or even larger, because the modulus of continuity $\varpi$ does not have a singularity at [math], and therefore so does the integrand. But our Main Theorem 2.2 still makes sense, because the remaining regularity of the persistent torus as well as the conjuation will be higher correspondingly, as one would expect.

3 Applications

In this section, we show certain detailed regularity about KAM torus as well as conjugation such as Hölder and Logarithmic Hölder ones etc. Denote by $\{a\}$ and $[a]$ the fractional part and the integer part of $a\geq 0$, respectively. It should be emphasized that the Dini type integrability condition (2.6) in (H1) is easy to verify, that is, the frequency-preserving KAM persistence is relatively easy to obtain. However, some complicated techniques of asymptotic analysis are needed to investigate the specific regularity of KAM torus as well as the conjugation, which are mainly reflected in the selection of functions $L_{i}(\gamma)$ ($i=1,2$) in (2.8). In particular, caused by small divisors, we will explicitly see how much regularity will be lost, see, for instance, Theorems 3.1, 3.2, 3.3 and the examples shown in Section 3.2. In what follows, the modulus of continuity under consideration are always convex near $0^{+}$ and therefore automatically admit semi separability as well as weak homogeneity which we forego.

3.1 KAM via different continuity

Recall that we require $\tau>n-1$ and $n\geq 2$ in (H3), that is, the prescribed Diophantine frequency is universal. Under such settings, the known minimum regularity requirement for Hamiltonian $H(x,y)$ is Hölder’s type $C^{\ell}$ with $\ell>2\tau+2$, see Salamon’s KAM in [35] as well as Theorem 3.1 below. Interestingly, if one considers weaker modulus of continuity, such as $C^{2\tau+2}$ plus Logarithmic Hölder’s type, the above regularity $C^{\ell}$ could be weakened, which can be seen from our new Theorems 3.2 and 3.3. It should be noted that weakening the initial regularity does not mean that KAM results with higher initial regularity are fully included, because we also have to consider the remaining KAM regularity, namely the smoothness of the KAM torus and conjugation. Intuitively and obviously, the weaker the initial regularity, the weaker the remaining regularity. One should not expect the KAM torus and conjugation to be Hölder’s type when considering Hamiltonian $H(x,y)$ via only continuous high order derivatives.

3.1.1 Hölder continuous case

Let us discuss the classical Hölder case first, see also the result given in [35]. As we will see that, the remaining regularity is still of Hölder’s type.

Theorem 3.1**.**

Let $H(x,y)\in C^{\ell}(\mathbb{T}^{n},G)$ with $\ell>2\tau+2$, where $\ell\notin\mathbb{N}^{+}$, $\ell-\tau\notin\mathbb{N}^{+}$ and $\ell-2\tau\notin\mathbb{N}^{+}$. That is, $H(x,y)$ is of $C_{k,\varpi}$ with $k=[\ell]$ and $\varpi(x)\sim\varpi_{\mathrm{H}}^{\ell}(x)\sim x^{\{\ell\}}$. Assume (H2), (H3) and (H4). Then there is a solution $x=u\left(\xi\right),y=v\left(\xi\right)$ of the following equation with the operator $D:=\sum\limits_{\nu=1}^{n}{{\omega_{\nu}}\frac{\partial}{{\partial{\xi_{\nu}}}}}=\omega\cdot{\partial_{\xi}}$

[TABLE]

such that $u\left(\xi\right)-\xi$ and $v\left(\xi\right)$ are of period $1$ in all variables. In addition, $u\in{C^{\ell-2\tau-1}}\left({{\mathbb{R}^{n}},{\mathbb{R}^{n}}}\right)$ and $v\circ{u^{-1}}\in{C^{\ell-\tau-1}}\left({{\mathbb{R}^{n}},G}\right)$.

3.1.2 Hölder plus Logarithmic Hölder continuous case

To explicitly show different modulus of continuity strictly weaker than Hölder’s type, we establish the following Theorem 3.2. One will see later that Theorem 3.2 employs very complicated asymptotic analysis, and interestingly, the remaining regularity basically characterized by $\varpi_{1}$ and $\varpi_{2}$ admit different forms.

Theorem 3.2**.**

Let $\tau>n-1$ be given and let $H(x,y)\in C_{[2\tau+2],\varpi}$, where $\varpi\left(x\right)\sim{x^{\{2\tau+2\}}}/{\left({-\ln x}\right)^{\lambda}}$ with $\lambda>1$. Assume (H2), (H3) and (H4). That is, $H(x,y)$ is of $C^{k}$ plus the above $\varpi$ with $k=[2\tau+2]$. Then there is a solution $x=u\left(\xi\right),y=v\left(\xi\right)$ of the following equation with the operator $D:=\sum\limits_{\nu=1}^{n}{{\omega_{\nu}}\frac{\partial}{{\partial{\xi_{\nu}}}}}=\omega\cdot{\partial_{\xi}}$

[TABLE]

such that $u\left(\xi\right)-\xi$ and $v\left(\xi\right)$ are of period $1$ in all variables. In addition, letting

[TABLE]

and

[TABLE]

one has that $u\in{C_{1,{\varpi_{1}}}}\left({{\mathbb{R}^{n}},{\mathbb{R}^{n}}}\right)$ and $v\circ{u^{-1}}\in{C_{[\tau+1],{\varpi_{2}}}}\left({{\mathbb{R}^{n}},G}\right)$.

3.1.3 Hölder plus generalized Logarithmic Hölder continuous case

Actually, in view of the Dini type integrability condition (H1), one notices that the regularity for the given Hamiltonian function could be further weakened, such as the generalized Logarithmic Hölder continuity in (2.2). But the parameter $\lambda>0$ needs to be appropriately chosen, otherwise it may contradict (2.6). We will only present the following Theorem 3.3 for the case where Diophantine index $\tau$ is an integer for the sake of simplicity. One could deal with $\tau\notin\mathbb{N}^{+}$ similar to that in Theorem 3.2.

Theorem 3.3**.**

Let $n-1<\tau\in\mathbb{N}^{+}$, and $H(x,y)\in C_{2\tau+2,{\varpi_{\mathrm{GLH}}^{\varrho,\lambda}}}$ with $\varrho\in\mathbb{N}^{+},\lambda>1$. That is, all $(2\tau+2)$-order derivatives of $H(x,y)$ admit generalized Logarithmic Hölder continuity:

[TABLE]

Assume (H2), (H3) and (H4). Then there is a solution $x=u\left(\xi\right),y=v\left(\xi\right)$ of the following equation with the operator $D:=\sum\limits_{\nu=1}^{n}{{\omega_{\nu}}\frac{\partial}{{\partial{\xi_{\nu}}}}}=\omega\cdot{\partial_{\xi}}$

[TABLE]

such that $u\left(\xi\right)-\xi$ and $v\left(\xi\right)$ are of period $1$ in all variables. Besides, the remaining regularity in Theorem 2.2 is $u\in{C_{1,{\varpi_{1}}}}\left({{\mathbb{R}^{n}},{\mathbb{R}^{n}}}\right)$ and $v\circ{u^{-1}}\in{C_{\tau+1,{\varpi_{2}}}}\left({{\mathbb{R}^{n}},G}\right)$ with modulus of continuity

[TABLE]

Remark 3.1**.**

Particularly (3.1) reduces to the Logarithmic Hölder’s type $\varpi_{\mathrm{LH}}^{\lambda}(x)\sim 1/{(-\ln x)^{\lambda}}$ with $\lambda>1$ as long as $\varrho=1$. As can be seen that, the remaining regularity in (3.2) is much weaker than the initial regularity in (3.1), and it is indeed very weak if $\lambda>1$ is sufficiently close to $1$ (but cannot degenerate to $1$ by (H1)), or $\varrho\in\mathbb{N}^{+}$ is sufficiently large, because the explicit modulus of continuity in (3.2) tends to [math] quite slowly as $x\to 0^{+}$.

3.2 Two explicit Hamiltonian systems of Logarithmic Hölder’s type

To illustrate the wider applicability of our theorems, we shall present two explicit examples strictly beyond Hölder’s type, involving non-Hölder corner’s type and even nowhere Hölder’s type. Note that the Hölder plus Logarithmic Hölder regularity for $H(x,y)$ in Theorem 3.2 becomes simpler Logarithmic Hölder’s type for $2n<2\tau+2\in\mathbb{N}^{+}$ (because $\{2\tau+2\}=0$), we therefore consider the following critical settings (with sharp differentiability) throughout this section.

Recall Theorem 3.2. Let $n=2,\tau=2,k=6=[2\tau+2],{\alpha_{*}}>0,\lambda>1$ and $M>0$ be given. Assume that $\left({x,y}\right)\in{\mathbb{T}^{2}}\times G$ with $G:=\{{y\in{\mathbb{R}^{2}}:\left|y\right|\leq 1}\}$, and the universal Diophantine frequency $\omega={\left({{\omega_{1}},{\omega_{2}}}\right)^{\top}}\in\mathbb{R}^{2}$ satisfies

[TABLE]

i.e., with full Lebesgue measure.

3.2.1 Non-Hölder corner’s type

Now we shall construct a function for finite smooth perturbations via non-Hölder corner’s type, whose regularity is $C^{6}$ plus Logarithmic Hölder’s type $\varpi_{\mathrm{LH}}^{\lambda}(r)\sim 1/(-\ln r)^{\lambda}$ with index $\lambda>1$. In other words, the highest order derivative admits exact Logarithmic Hölder regularity at a non-Hölder corner, but it may have good regularity away from this corner, such as being sufficiently smooth. Namely, define

[TABLE]

Obviously $\mathcal{P}(r)\in C_{6,\varpi_{\mathrm{LH}}^{\lambda}}([-1,1])$. Let us consider the perturbed Hamiltonian function below with some constant $0<{\varepsilon}<{\varepsilon^{*}}$ sufficiently small (${\varepsilon^{*}}$ depends on the constants given before):

[TABLE]

At this point, we have

[TABLE]

In addition, one can verify that $\mathcal{H}(x,y,\varepsilon)\in{C_{6,{\varpi_{\mathrm{LH}}^{\lambda}}}}({{\mathbb{T}^{2}}\times G})$ with $\varpi_{\mathrm{LH}}^{\lambda}(r)\sim 1/(-\ln r)^{\lambda}$.

However, for $\tilde{\alpha}={\left({0,0,6,0}\right)^{\top}}$ with $\left|{\tilde{\alpha}}\right|=6=k$, we have

[TABLE]

for any $0<\ell\leq 1$, where $c_{\lambda,\ell}>0$ is a constant that only depends on $\lambda$ and $\ell$. This implies that $\mathcal{H}(x,y,\varepsilon)\notin{C_{6,{\varpi_{\mathrm{H}}^{\ell}}}}({{\mathbb{T}^{2}}\times G})$ with ${\varpi_{\mathrm{H}}^{\ell}}(r)\sim{r^{\ell}}$, i.e., $\mathcal{H}(x,y,\varepsilon)\notin C^{6+\ell}({{\mathbb{T}^{2}}\times G})$ with any $0<\ell\leq 1$, because $\varpi_{\mathrm{LH}}^{\lambda}$ is strictly weaker than $\varpi_{\mathrm{H}}^{\ell}$, see also Remark 2.3 and Corollary 2.1.

In other words, the highest order derivatives (of order $k=6$) of $H(x,y)$ in (3.3) can be rigorously proved to be Logarithmic Hölder continuous with index $\lambda>1$, but not any Hölder’s type. Therefore, the finite smooth KAM theorems via classical Hölder continuity cannot be applied. But, all the assumptions of Theorem 3.2 can be verified to be satisfied, then the invariant torus persists, and the frequency $\omega={\left({{\omega_{1}},{\omega_{2}}}\right)^{\top}}$ for the unperturbed system can remain unchanged. Moreover, the remaining regularity for mappings $u$ and $v\circ u^{-1}$ in Theorem 3.2 could also be determined as $u\in{C_{1,{\varpi_{\mathrm{LH}}^{\lambda-1}}}}\left({{\mathbb{R}^{n}},{\mathbb{R}^{n}}}\right)$ and $v\circ{u^{-1}}\in{C_{3,{\varpi_{\mathrm{LH}}^{\lambda-1}}}}\left({{\mathbb{R}^{n}},G}\right)$, where $\varpi_{\mathrm{LH}}^{\lambda-1}(r)\sim 1/(-\ln r)^{\lambda-1}$. More precisely, $u$ is at least $C^{1}$, while $v\circ u^{-1}$ is least $C^{3}$, and the higher regularity for them is still not any Hölder’s type, but Logarithmic Hölder one with index $\lambda-1$, i.e., lower than the original index $\lambda>1$, this is because the small divisors causes the loss of regularity. In KAM terminology, the regularity of invariant torus with frequency-preserving is $C^{3}$ plus $(\lambda-1)$-Logarithmic Hölder, while the conjugation from the dynamic on it to the linear flow is $C^{1}$ plus $(\lambda-1)$-Logarithmic Hölder, see Remark 2.6.

3.2.2 Nowhere Hölder’s type

Although an explicit irregular case has been discussed in Section 3.2.1, it is not universal enough. Notice that the non-integrable part and perturbations for Hamiltonian should at least be of $C^{6}$ under our results (see Theorem 2.2 and Remark 2.7). However, if perturbations are selected at random from the function space $C^{6}$, then the derivatives of $6$-order are likely to be nowhere-differentiable (note that the example given in Section 3.2.1 is $C^{\infty}$ except for the corner point), or even nowhere Hölder from the point of view of Baire category. More precisely, perturbations may be convergent trigonometric series composed of high frequency oscillations. This is also one of our motivations for establishing KAM in the sense of modulus of continuity. We shall focus on this case below, aiming to show the strength of our results.

Given $0<q<\frac{1}{3}$ and $\lambda>1$. Define a sequence $\{\mathcal{Q}_{n}\}_{n\in\mathbb{N}^{+}}$ that satisfies $\left\{{{\mathcal{Q}_{n+1}}\mathcal{Q}_{n}^{-1},{\mathcal{Q}_{n}}}\right\}_{n\in\mathbb{N}^{+}}\subseteq 10{\mathbb{N}^{+}}$ and $\mathcal{Q}_{n}=\mathcal{O}^{\#}\left(\exp(\Theta^{n})\right)$ with $\Theta=\left({\frac{1}{q}}\right)^{\frac{1}{\lambda}}>1$. Then the $1$-periodic function

[TABLE]

is well defined. We shall construct a Hamiltonian system using $\mathscr{P}$ as the highest order derivatives for the perturbation. To this end, consider the following Hamiltonian with high frequency oscillation perturbation:

[TABLE]

provided with some constant $0<{\varepsilon}<{\varepsilon^{*}}$ sufficiently small, and ${\varepsilon^{*}}$ depends on the constants given before. Here the matrix $\mathscr{A}(x_{1},x_{2})$ could only be of class $C_{6,\varpi_{\mathrm{LH}}^{\lambda}}$ (recall Remark 2.7), and assume that (H2) holds. At this case, the unperturbed part $\mathscr{N}_{\mathscr{H}}$ of $\mathscr{H}$ is indeed $C_{6,\varpi_{\mathrm{LH}}^{\lambda}}$ finitely smooth and non-integrable. Let us point out that, the non-integrable part could have weaker regularity, i.e.,

[TABLE]

where ${\tilde{\varepsilon}}\tilde{\mathscr{A}}$ and ${\tilde{\varepsilon}}\bar{\mathscr{A}}$ are small perturbations of $\mathscr{A}$, admitting weaker regularity than $C_{6,\varpi_{\mathrm{LH}}^{\lambda}}$, e.g., $C^{5}$, and $\tilde{\mathscr{A}}-\bar{\mathscr{A}}$ must be of at least $C_{6,\varpi_{\mathrm{LH}}^{\lambda}}$. The analyzability of this case is due to our technique for dealing with finite smooth KAM in this paper, but obviously, the new perturbation of $\tilde{\mathscr{H}}$ is quite special.

Back to our concern on Hamiltonian (3.5). Now, the universal frequency-preserving KAM persistence and remaining regularity same as that in Section 3.2.1 could be obtained if one proves that the perturbation is also $C_{6,\varpi_{\mathrm{LH}}^{\lambda}}$. However, this is not simple. The following proposition verifies this claim, but simultaneously shows that it is indeed nowhere Hölder continuous (and therefore, the Hölder type KAM tools cannot be used at all, at any point). As you will see later, the technique used here is somewhat similar to that in Abstract Iterative Theorem 4.1.

Proposition 3.1**.**

The perturbation part $\varepsilon\mathscr{P}_{\mathscr{H}}$ of Hamiltonian $\mathscr{H}$ in (3.5) is at least $C^{6}$, and the regularity for $6$-order derivatives is $\lambda$-Logarithmic Hölder’s type, but is nowhere Hölder continuous.

Proof.

By direct derivation, it suffices to verify that $\mathscr{P}\left(x\right)$ defined in (3.4) is $\lambda$-Logarithmic Hölder continuous, but is nowhere Hölder continuous. Obivously $\mathscr{P}\left(x\right)$ is continuous on $\mathbb{R}$ because

[TABLE]

Then it immediately follows from Remark 2.2 and the $1$-periodicity of $\mathscr{P}$ that $\mathscr{P}$ automatically admits a modulus of continuity $\varpi_{\mathscr{P}}$. We first prove the $\lambda$-Logarithmic Hölder continuity, i.e., $\varpi_{\mathscr{P}}$ is weaker than $\varpi_{\rm LH}^{\lambda}\sim 1/{\left({-\ln x}\right)^{\lambda}}$ for $\lambda>1$. For every fixed $0<h<1$, define (it is indeed well defined because $\mathcal{Q}_{n}\to+\infty$ as $n\to+\infty$)

[TABLE]

Now one derives

[TABLE]

from the assumptions on $\{\mathcal{Q}_{n}\}_{n\in\mathbb{N}^{+}}$, where $C\left({q,\lambda}\right)>0$ is a generic constant that only depends on $q$ and $\lambda$. Therefore, for $0<h<1$ sufficiently small, we obtain that

[TABLE]

where (3.8) uses the Mean Value Theorem and $\left|{\sin x}\right|\leq 1$, (3.9) is due to $\left\{{{\mathcal{Q}_{n+1}}}\mathcal{Q}_{n}^{-1}\right\}_{n\in\mathbb{N}^{+}}\subseteq 10{\mathbb{N}^{+}}$, (3.10) employs the definition of $\mathscr{N}_{h}$ in (3.6), and finally (3.11) follows from the estimate (3.7). This gives the claim asserting that $\mathscr{P}(x)$ is $\lambda$-Logarithmic Hölder with respect to $\lambda>1$.

Next we will show that $\mathscr{P}(x)$ is nowhere Hölder continuous on $\mathbb{R}$. Let $\alpha>0$ be arbitrary given, and consider $\varpi_{\rm H}^{\alpha}\sim x^{\alpha}$ (arbitrary Hölder modulus of continuity). Fix $x>0$, and choose $m=m_{x,\alpha}\in\mathbb{N}^{+}$ sufficiently large such that ${\mathcal{Q}_{m}}x\geq 1$, and

[TABLE]

Note that this is achievable because $\mathcal{Q}_{m}$ is of the superexponential order. Now it reads ${\mathcal{Q}_{m}}x={\mathcal{N}_{m}}+{\mathcal{R}_{m}}$ with ${\mathcal{N}_{m}}\in{\mathbb{N}^{+}}$ and $0<\left|{{\mathcal{R}_{m}}}\right|\leq{2^{-1}}$. Taking

[TABLE]

as $m\to\infty$ yields that ${2^{-1}}\leq\left|{\upsilon_{x,m}}\right|{\mathcal{Q}_{m}}\leq 1$. Then we get

[TABLE]

Firstly, direct calculation as well as (3.13) gives

[TABLE]

Secondly, by applying the Mean Value Theorem we obtain that

[TABLE]

where we use the following fact due to $\left\{{{\mathcal{Q}_{n+1}}}\mathcal{Q}_{n}^{-1}\right\}_{n\in\mathbb{N}^{+}}\subseteq 10{\mathbb{N}^{+}}$:

[TABLE]

Thirdly, note that ${\mathcal{Q}_{n}}\mathcal{Q}_{m}^{-1}\in 10\mathbb{N}^{+}$. Then it follows that

[TABLE]

which leads to

[TABLE]

Finally, substituting (3.15), (3.16), (3.18) into (3.14) and using (3.12) yields that

[TABLE]

In other words, for every $x\in\mathbb{R}$ (note that $\mathscr{P}$ is $1$-periodic), one could construct a sequence $\{\upsilon_{x,m}\}_{m\in\mathbb{N}^{+}}$ such that

[TABLE]

This implies that $\mathscr{P}(x)$ is nowhere Hölder continuous because the Hölder index $0<\alpha<1$ could be arbitrary chosen, and we therefore prove the proposition. ∎

4 Proof of Main Theorem 2.2

Now let us prove Theorem 2.2 by separating two subsections, namely frequency-preserving KAM persistence (Section 4.1) and remaining regularity (Section 4.2) for KAM torus as well as conjugation. For the former, the overall process is similar to that in [35], but the key points to weaken the Hölder regularity to only modulus of continuity are using Jackson type approximation Theorem 2.1 and proving the uniform convergence of the transformation mapping, that is, the convergence of the upper bound series (see (4.11) and (4.13)). As we will see later, the Dini type integrability condition (H1) is crucial for the boundedness. As to the latter, we have to establish a more general regularity iterative theorem (Theorem 4.1), which does not seem simple since the resulting regularity might be somewhat complicated due to asymptotic analysis.

4.1 Frequency-preserving KAM persistence

The proof of the frequency-preserving KAM persistence is organized as follows. Firstly, we construct a series of analytic approximation functions $H^{\nu}(x,y)$ of $H(x,y)$ by using Theorem 2.1 and considering (H1) and (H2). Secondly, we shall construct a sequence of frequency-preserving analytic and symplectic transformations $\psi^{\nu}$ by induction. According to (H2), (H3) and (H4), the first step of induction is established by applying Theorem 6.1 in Section 6.6 (or Theorem 1 in [35]). Then, combining with weak homogeneity and certain specific estimates we complete the proof of induction and obtain the uniform convergence of the composite transformations. Finally, in the light of (H5), the regularity of the KAM torus as well as the conjugation is guaranteed by Theorem 4.1. We shall emphasize that the unperturbed part of Hamiltonian $H(x,y)$ could be non-integrable and finitely smooth due to our strategy.

Step1: In view of Theorem 2.1 (we have assumed that the modulus of continuity $\varpi$ admits semi separability and thus Theorem 2.1 could be directly applied here), one could approximate $H(x,y)$ by a sequence of real analytic functions $H_{\nu}(x,y)$ for $\nu\geq 0$ in the strips

[TABLE]

around $\left|{\operatorname{Re}x}\right|\in{\mathbb{T}^{n}},\left|{\operatorname{Re}y}\right|\leq\rho,$ such that

[TABLE]

for $\left|{\operatorname{Im}x}\right|\leq{r_{\nu}},\;\left|{\operatorname{Im}y}\right|\leq{r_{\nu}}$, and $c_{1}=c(n,k)$ is the constant provided in (2.5).

Fix $\theta=1/\sqrt{2}$. In what follows, we will construct a sequence of real analytic symplectic transformations $z=(x,y),\zeta=(\xi,\eta),z={\phi^{\nu}}\left(\zeta\right)$ of the form

[TABLE]

by induction, such that ${u^{\nu}}\left(\xi\right)-\xi$ and ${v^{\nu}}\left(\xi\right)$ are of period $1$ in all variables, and ${\phi^{\nu}}$ maps the strip $\left|{\operatorname{Im}\xi}\right|,\left|\eta\right|\leq\theta{r_{\nu+1}}$ into $\left|{\operatorname{Im}x}\right|,\left|y\right|\leq{r_{\nu}},\left|{\operatorname{Re}y}\right|\leq\rho$, and the transformed Hamiltonian function ${K^{\nu}}:={H^{\nu}}\circ{\phi^{\nu}}$ satisfies

[TABLE]

i.e., with prescribed universal Diophantine frequency-preserving. Namely by verifying certain conditions we obtain $z=\psi^{\nu}(\zeta)$ of the form (4.2) from Theorem 6.1 by induction, mapping $\left|{\operatorname{Im}\xi}\right|,\left|\eta\right|\leq{r_{\nu+1}}$ into $\left|{\operatorname{Im}x}\right|,\left|y\right|\leq\theta{r_{\nu}}$, and $\psi^{\nu}\left({\xi,0}\right)-\left({\xi,0}\right)$ is of period $1$, and (4.3) holds. Here we denote $\phi^{\nu}:=\phi^{\nu-1}\circ\psi^{\nu}$ with ${\phi^{-1}}:=\mathrm{id}$ (where $\mathrm{id}$ denotes the $2n$-dimensional identity mapping and therefore ${\phi^{0}}={\psi^{0}}$). Further more, Theorem 6.1 will lead to

[TABLE]

on $\left|{\operatorname{Im}\xi}\right|,\left|\eta\right|,\left|{\operatorname{Im}x}\right|\leq r_{\nu+1}$, where ${S^{\nu}}\left({x,\eta}\right)={U^{\nu}}\left(x\right)+\left\langle{{V^{\nu}}\left(x\right),\eta}\right\rangle$ is the generating function for ${\psi^{\nu}}$, and $Q^{\nu}:=K_{\eta\eta}^{\nu-1}$, and $\mathbb{I}$ denotes the $2n\times 2n$-dimensional identity mapping, and

[TABLE]

Step2: Here we show that $\psi^{0}=\phi^{0}$ exists, and it admits the properties mentioned in Step 1. Denote

[TABLE]

Then by the first term in ((H4)), we have

[TABLE]

Note that

[TABLE]

Hence, for $\left|{\operatorname{Im}x}\right|\leq\theta{r_{0}}=\theta\varepsilon$, by using Theorem 2.1, Corollary 2.2 and (4.9) we arrive at

[TABLE]

Now consider the vector valued function $f\left(x\right):={H_{y}}\left({x,0}\right)-\omega$ for $x\in{\mathbb{R}^{n}}$. In view of the second term in ((H4)), we have

[TABLE]

Note that

[TABLE]

Therefore, for $\left|{\operatorname{Im}x}\right|\leq\theta\varepsilon$, by using (4.1) and (4.10) we obtain that

[TABLE]

Recall (4.8). Then it follows from (4.1) that

[TABLE]

and

[TABLE]

Now, by taking $r^{*}=\theta\varepsilon,\delta^{*}={\varepsilon^{k-2\tau-2}}\varpi\left(\varepsilon\right)$ and using Theorem 6.1 there exists a real analytic symplectic transformation $z={\phi^{0}}\left(\zeta\right)$ of the form (4.2) (with $\nu=0$) mapping the strip $\left|{\operatorname{Im}\xi}\right|,\left|\eta\right|\leq{r_{1}}=r_{0}/2$ into $\left|{\operatorname{Im}x}\right|,\left|y\right|\leq\theta{r_{0}}=r_{0}/\sqrt{2}$, such that ${u^{0}}\left(\xi\right)-\xi$ and ${v^{0}}\left(\xi\right)$ are of period $1$ in all variables and the Hamiltonian function ${K^{0}}:={H^{0}}\circ{\phi^{0}}$ satisfies (4.3) (with $\nu=0$). Moreover, (4.4)-(4.6) (with $\nu=0$) hold.

Also assume that

[TABLE]

for $\left|{\operatorname{Im}x}\right|,\left|y\right|\leq{r_{\nu}}$. Finally, define

[TABLE]

with respect to $\left|{\operatorname{Im}x}\right|,\left|y\right|\leq{r_{\nu}}$. One can verify that ${\tilde{H}}$ is well defined.

Next we assume that the transformation $z={\phi^{\nu-1}}\left(\zeta\right)$ of the form (4.2) has been constructed, mapping $\left|{\operatorname{Im}\xi}\right|,\left|\eta\right|\leq\theta{r_{\nu}}$ into $\left|{\operatorname{Im}x}\right|,\left|{\operatorname{Im}y}\right|\leq{r_{\nu-1}},\left|{\operatorname{Re}y}\right|\leq\rho$, and ${u^{\nu-1}}\left(\xi\right)-\xi,{v^{\nu-1}}\left(\xi\right)$ are of period $1$ in all variables, and $K_{\xi}^{\nu-1}\left({\xi,0}\right)=0,K_{\eta}^{\nu-1}\left({\xi,0}\right)=\omega$. In addition, we also assume that (4.4)-(4.7) hold for $0,\ldots,\nu-1$. In the next Step 3, we will verify that the above still hold for $\nu$, which establishes a complete induction.

Step3: We will prove the existence of transformation ${\phi^{\nu}}$ in each step according to the specific estimates below and Theorem 6.1.

Let $\left|{\operatorname{Im}x}\right|\leq\theta{r_{\nu}}$. Then $\phi^{\nu-1}(x,0)$ lies in the region where the estimates in (4.1) hold for both $H^{\nu}$ and $H^{\nu-1}$. Note that $x\mapsto H^{\nu-1}(\phi^{\nu-1}(x,0))$ is constant by (4.3). Then by (4.1), we arrive at the following for $\left|{\operatorname{Im}x}\right|\leq\theta{r_{\nu}}$

[TABLE]

where the weak homogeneity of $\varpi$ with respect to $a=1/2$ (see Definition 2.4) has been used in the last inequality, because $\varpi(r_{\nu-1})=\varpi(2r_{\nu})\leq c\varpi(r_{\nu})$ (thus $c$ is independent of $\nu$). For convenience we may therefore not mention it in the following.

Taking $\eta=0$ in (4.5) we have

[TABLE]

for $\left|{\operatorname{Im}\xi}\right|\leq\theta{r_{\nu}}$, and the Dini type condition (2.6) in (H1) together with Cauchy Theorem are used since $\varepsilon>0$ is sufficiently small. Then it leads to

[TABLE]

Finally, by (4.12) and (4.1) we obtain that

[TABLE]

and

[TABLE]

for $\left|{\operatorname{Im}x}\right|,\left|y\right|\leq\theta{r_{\nu}}$. Then denote $r^{*}:=r_{\nu}$ and $\delta^{*}:=cr_{\nu}^{k-2\tau-2}\varpi\left({{r_{\nu}}}\right)$ in Theorem 6.1, we obtain the analytic symplectic preserving transformation ${\phi^{\nu}}$ of each step, mapping the strip $\left|{\operatorname{Im}\xi}\right|\leq\theta{r_{\nu}},\left|\eta\right|\leq\theta{r_{\nu}}$ into $\left|{\operatorname{Im}x}\right|\leq{r_{\nu}},\left|y\right|\leq{r_{\nu}}$, such that ${u^{\nu}}\left(\xi\right)-\xi$ and ${v^{\nu}}\left(\xi\right)$ are of period $1$ in all variables, and the transformed Hamiltonian function ${K^{\nu}}={H^{\nu}}\circ{\phi^{\nu}}$ satisfies

[TABLE]

Moreover, (4.4)-(4.7) are valid for $\left|{\operatorname{Im}\xi}\right|,\left|\eta\right|,\left|{\operatorname{Im}x}\right|\leq\theta{r_{\nu}}$.

Step4: By (4.5) for $0,\ldots,\nu-1$ and the arguments in (4.11), there holds

[TABLE]

for $\left|{\operatorname{Im}\xi}\right|,\left|\eta\right|\leq\theta{r_{\nu}}$ as long as $\varepsilon>0$ is sufficiently small, which leads to

[TABLE]

for $\left|{\operatorname{Im}\xi}\right|,\left|\eta\right|\leq{r_{\nu+1}}$. Then by Cauchy’s estimate, we obtain that

[TABLE]

It can be proved in the same way that $|{\phi_{\zeta}^{\nu}\left(\zeta\right)}|\leq 2$ for $\left|{\operatorname{Im}\xi}\right|,\left|\eta\right|\leq\theta{r_{\nu+1}}$, which implies

[TABLE]

Besides, we have $\left|{\operatorname{Re}y}\right|\leq\rho$.

Note that

[TABLE]

Recall (4.11). By employing the contraction mapping principle we have $\left|{\operatorname{Im}\xi}\right|\leq{r_{\nu+1}}$ if $\left|{\operatorname{Im}x}\right|\leq\theta{r_{\nu+1}}$ with respect to $x$ defined above. Then from (4.7) and (4.12) one can verify that

[TABLE]

Step5: Finally, we are in a position to prove the uniform convergence of $u^{\nu}$ and $v^{\nu}$, and the regularity of their limit functions. Note (4.14). Then we obtain the following analytic iterative scheme

[TABLE]

and

[TABLE]

And especially, (4.15) and (4.16) hold when $\nu=0$ since ${u^{0-1}}=\mathrm{id}$ and ${v^{0-1}}=0$. It is obvious to see that the uniform limits $u$ and $v\circ u^{-1}$ of $u^{\nu}$ and $v^{\nu}\circ(u^{\nu})^{-1}$ are at least $C^{1}$ (in fact, this is implied by the higher regularity studied later in Section 4.2, or one could derive the lower one using simpler technique than that in Theorem 4.1, which we omit here). In addition, the persistent invariant torus possesses the same universal Diophantine (class $\tau>n-1$) frequency $\omega$ as the unperturbed torus by (4.3).

4.2 Iteration theorem on regularity beyond Hölder’s type

To obtain accurate regularity for $u$ and $v\circ u^{-1}$ from the analytic iterative scheme (4.15) and (4.16), we shall along with the idea of Moser and Salamon to establish an abstract iterative theorem on regularity beyond Hölder’s type, which explicitly shows the modulus of continuity of the integral form. It should be emphasized that, from Hölder to non-Hölder, the estimates here are much more difficult.

Theorem 4.1** (Abstract iterative theorem).**

Let $n\in\mathbb{N}^{+},\varepsilon>0$ and $\{r_{\nu}\}_{\nu\in\mathbb{N}}=\{\varepsilon 2^{-\nu}\}_{n\in\mathbb{N}}$ be given, and denote by $f:{\mathbb{R}^{n}}\to\mathbb{R}$ the limit of a sequence of real analytic functions ${f_{\nu}}\left(x\right)$ in the strips $\left|{\operatorname{Im}x}\right|\leq{r_{\nu}}$ such that

[TABLE]

where $\varphi$ is a nondecreasing continuous function satisfying $\varphi\left(0\right)=0$. Assume that there is a critical $k_{*}\in\mathbb{N}$ such that

[TABLE]

Then there exists a modulus of continuity $\varpi_{*}$ such that $f\in{C_{k_{*},\varpi_{*}}}\left({{\mathbb{R}^{n}}}\right)$. In other words, the regularity of $f$ is at least of $C^{k_{*}}$ plus $\varpi_{*}$. In particular, $\varpi_{*}$ could be determined as

[TABLE]

where $L(\gamma)\to 0^{+}$ is some function such that the second relation in (4.19) holds.

Proof.

Define ${g_{\nu}}(x):={f_{\nu}}\left(x\right)-{f_{\nu-1}}\left(x\right)$ for all $\nu\in\mathbb{N}^{+}$. Determine an integer function $\widetilde{N}(\gamma):[0,1]\to\mathbb{N}^{+}$ (note that $\widetilde{N}(\gamma)$ can be extended to $\mathbb{R}^{+}$ due to the arguments below, we thus assume that it is a continuous function). Then for the given critical $k_{*}\in\mathbb{N}$ and $x,y\in\mathbb{R}^{n}$, we obtain the following for all multi-indices $\alpha=\left({{\alpha_{1}},\ldots,{\alpha_{n}}}\right)\in{\mathbb{N}^{n}}$ with $\left|\alpha\right|=k_{*}$:

[TABLE]

where Cauchy’s estimate and (4.17) are used in the second inequality, and arguments similar to (4.11) are employed in (4.20), $c>0$ is a universal constant. Besides, we similarly get

[TABLE]

Now let us choose $\widetilde{N}(\gamma)\to+\infty$ as $\gamma\to 0^{+}$ such that

[TABLE]

where $\varepsilon{2^{-\widetilde{N}\left(\gamma\right)-1}}:=L\left(\gamma\right)\to 0^{+}$. This is achievable due to assumption (4.18), Cauchy Theorem and The Intermediate Value Theorem. Note that the choice of $L(\gamma)$ (i.e., $\widetilde{N}$) and $\varpi_{*}$ is not unique (may up to a constant), and $\varpi_{*}$ could be continuously extended to some given interval (e.g., $[0,1]$), but this does not affect the qualitative result. Combining (4.20), (4.21) and (4.22) we finally arrive at $f\in{C_{k_{*},\varpi_{*}}}\left({{\mathbb{R}^{n}}}\right)$ because

[TABLE]

∎

Theorem 4.1 can be extended to the case $f:{\mathbb{R}^{n}}\to\mathbb{R}^{m}$ with $n,m\in\mathbb{N}^{+}$ since the analysis is completely same, and the strip $\left|{\operatorname{Im}x}\right|\leq{r_{\nu}}$ can also be replaced by $\left|{\operatorname{Im}x}\right|\leq{r_{\nu+1}}$ (or $\leq\theta r_{\nu+1}$). Theorem 4.1 can also be used to estimate the regularity of solutions of finite smooth homological equations, thus KAM uniqueness theorems in some cases might be derived, see Section 4 in [35] for instance. However, in order to avoid too much content in this paper, it is omitted here.

Finally, by recalling (4.15) and (4.16), one applies Theorem 4.1 on $\{u^{\nu}-\mathrm{id}\}_{\nu}$ (because Theorem 4.1 requires that the initial value vanishes) and $\{v^{\nu}\circ(u^{\nu})^{-1}\}_{\nu}$ to directly analyze the regularity of KAM torus and conjugation according to (H5), i.e., there exist modulus of continuity ${\varpi_{i}}$ ($i=1,2$) such that $u\in{C_{k_{1}^{*},{\varpi_{1}}}}\left({{\mathbb{R}^{n}},{\mathbb{R}^{n}}}\right)$ and $v\circ{u^{-1}}\in{C_{k_{2}^{*},{\varpi_{2}}}}\left({{\mathbb{R}^{n}},G}\right)$. This completes the proof of Theorem 2.2.

5 Proof of other results

5.1 Proof of Theorem 3.1

Note that $\ell\notin{\mathbb{N}^{+}}$ implies $\{\ell\}\in(0,1)$. Then $k=[\ell]$ and $\varpi(x)\sim\varpi_{\mathrm{H}}^{\ell}(x)\sim x^{\{\ell\}}$, i.e., modulus of continuity of Hölder’s type. Consequently, (H1) can be directly verified because of $\ell>2\tau+2$:

[TABLE]

Here and below, let $i$ be $1$ or $2$ for simplicity. Recall that ${\varphi_{i}}\left(x\right)={x^{k-\left({3-i}\right)\tau-1}}\varpi\left(x\right)={x^{\left[\ell\right]-\left({3-i}\right)\tau-1}}\times{x^{\left\{\ell\right\}}}={x^{\ell-\left({3-i}\right)\tau-1}}$, and let

[TABLE]

Then the critical $k_{i}^{*}$ in (H5) could be uniquely chosen as $k_{i}^{*}:=\left[{\ell-\left({3-i}\right)\tau-1}\right]\in\mathbb{N}^{+}$ since $\ell-\left({3-i}\right)\tau-1\notin\mathbb{N}^{+}$. Further, letting ${L_{i}}\left(\gamma\right)=\gamma\to{0^{+}}$ yields

[TABLE]

and

[TABLE]

This leads to modulus of continuity being of Hölder’s type

[TABLE]

due to (2.8) in Theorem 2.1.

Finally by observing the Hölder indices $k_{i}^{*}+\left\{{\ell-\left({3-i}\right)\tau-2}\right\}=\ell-\left({3-i}\right)\tau-1$ for $i=1,2$, we conclude that $u\in{C^{\ell-2\tau-1}}\left({{\mathbb{R}^{n}},{\mathbb{R}^{n}}}\right)$ and $v\circ{u^{-1}}\in{C^{\ell-\tau-1}}\left({{\mathbb{R}^{n}},G}\right)$. This proves Theorem 3.1.

5.2 Proof of Theorem 3.2

Firstly, note that $k=[2\tau+2]$ and $\varpi\left(x\right)\sim{x^{\{2\tau+2\}}}/{\left({-\ln x}\right)^{\lambda}}$ with $\lambda>1$, then (H1) holds since

[TABLE]

Thus the frequency-preserving KAM persistence is obtained. The followings are devoted to the remaining regularity.

Secondly, in view of $\varphi_{i}(x)$ in (H5), we have

[TABLE]

This leads to critical $k_{1}^{*}=1$ and $k_{2}^{*}=[\tau+1]$ in (H5). Here one uses the following fact: for given $\lambda>1$,

[TABLE]

if and only if $\iota\in(0,1]$.

Next, we shall investigate the KAM remaining regularity through certain complicated asymptotic analysis, that is, show the modulus of continuity $\varpi_{1}$ and $\varpi_{2}$ explicitly. We have to discuss this problem in two cases. This is because $(i-1)\tau=0$ when $i=1$ (corresponds to the conjugation), i.e., independent of the Diophantine property; while $(i-1)\tau=\tau$ when $i=2$ (corresponds to the KAM torus). As can be seen later, they lead to different asymptotic analysis.

Case 1: Here we provide the analysis for $\varpi_{1}$ with all $\tau>n-1$, as well as $\varpi_{2}$ with $n-1<\tau\in\mathbb{N}^{+}$. In view of $\varphi_{i}(x)$ in (H5), then by applying Lemma 6.3 with $\varrho=1$ we get

[TABLE]

and by direct calculation one arrives at

[TABLE]

Finally, choosing

[TABLE]

leads to the second relation in (2.8) for $i=1,2$, and substituting $L_{i}(\gamma)$ into (5.3) or (5.4) yields

[TABLE]

and

[TABLE]

in Theorem 2.2, see (2.8).

Case 2: However, the asymptotic analysis for $\varpi_{2}$ becomes more different when $n-1<\tau\notin\mathbb{N}^{+}$. Note that $\{\tau\}\in(0,1)$ and $\left[{\tau+1}\right]-\tau=\left[\tau\right]+1-\tau=1-\left\{\tau\right\}$ at present. Hence, by applying (6.12) in Lemma 6.4 we get

[TABLE]

and similarly according to (6.13) in Lemma 6.4 we have

[TABLE]

Now let us choose $L_{2}(\gamma)\sim\gamma\to 0^{+}$, i.e., different from that when $n-1<\tau\in\mathbb{N}^{+}$ in Case 1, one verifies that the second relation in (2.8) holds for $i=2$, and substituting $L_{2}(\gamma)$ into (5.6) or (5.7) yields the following modulus of continuity

[TABLE]

due to (2.8) in Theorem 2.2.

This finishes the proof of Theorem 3.2.

5.3 Proof of Theorem 3.3

It is sufficient to determine $k_{i}^{*}$ in (H5) and to choose functions $L_{i}(\gamma)\to 0^{+}$ (as $\gamma\to 0^{+}$), obtaining the modulus of continuity $\varpi_{i}$ in (2.8) for $i=1,2$. Obviously $k_{1}^{*}=1$ and $k_{2}^{*}=\tau+1$ due to $\tau\in\mathbb{N}^{+}$ and

[TABLE]

In view of $\varphi_{i}(x)$ in (H5), then by applying Lemma 6.3 we get

[TABLE]

and by direct calculation one derives

[TABLE]

Finally, choosing

[TABLE]

leads to the second relation in (2.8) for $i=1,2$, and substituting $L_{i}(\gamma)$ into (5.8) or (5.9) yields that

[TABLE]

in Theorem 2.2, see (2.8), which proves Theorem 3.3.

6 Appendix

6.1 Semi separability and weak homogeneity for modulus of continuity

Lemma 6.1**.**

Let a modulus continuity $\varpi$ be given. If $\varpi$ is piecewise continuously differentiable and $\varpi^{\prime}\geq 0$ is nonincreasing, then $\varpi$ admits semi separability in Definition 2.3. As a consequence, if $\varpi$ is convex near $0^{+}$, then it is semi separable.

Proof.

Assume that $\varpi$ is continuously differentiable without loss of generality. Then we obtain semi separability due to

[TABLE]

∎

Lemma 6.2**.**

Let a modulus continuity $\varpi$ be given. If $\varpi$ is convex near $0^{+}$, then it admits weak homogeneity in Definition 2.4.

Proof.

For $x>0$ sufficiently small, one verifies that

[TABLE]

for $0<a<1$, which leads to weak homogeneity

[TABLE]

∎

6.2 Proof of Theorem 2.1

Proof.

For the completeness of the analysis we shall give a very detailed proof. An outline of the strategy for the proof is provided: we firstly construct an approximation integral operator by the Fourier transform of a compactly supported function, and then present certain properties of the operator (note that these preparations are classical); finally, we estimate the approximation error in the sense of modulus of continuity.

Let

[TABLE]

be an entire function, whose Fourier transform

[TABLE]

is a smooth function with compact support, contained in the ball $\left|\xi\right|\leq 1$, that satisfies $\widehat{K}\left(\xi\right)=\widehat{K}\left({-\xi}\right)$ and

[TABLE]

Next, we assert that

[TABLE]

Note that we assume $\widehat{K}\in C_{0}^{\infty}\left({{\mathbb{R}^{n}}}\right)$ and $\operatorname{supp}\widehat{K}\subseteq B\left({0,1}\right)$, thus

[TABLE]

where $\mathcal{F}$ represents the Fourier transform. Since {\partial^{\beta+\gamma}}\mathcal{F}\big{(}{\widehat{K}\left(\xi\right)}\big{)}\left(z\right)\in C_{0}^{\infty}({\overline{B\left({0,1}\right)}}) does not change the condition, we only need to prove that

[TABLE]

Obviously

[TABLE]

where $c>0$ is independent of $n$. Then assertion (6.2) is proved by recalling (6.3).

The inequality in (6.2) is usually called the Paley-Wiener Theorem, see also Chapter III in [36]. As we will see later, it plays an important role in the subsequent verification of definitions, integration by parts and the translational feasibility according to Cauchy’s integral formula.

Next we assert that $K:{\mathbb{C}^{n}}\to\mathbb{R}$ is a real analytic function with the following property

[TABLE]

for $u,v\in{\mathbb{R}^{n}}$ and multi-indices $\alpha,\beta\in{\mathbb{N}^{n}}$. In order to prove assertion (6.4), we first consider proving the following for $x\in\mathbb{R}^{n}$:

[TABLE]

Case1: If $\alpha=\beta$, then

[TABLE]

Case2: There exists some ${\alpha_{j}}\leq{\beta_{j}}-1$, let $j=1$ without loss of generality. Then

[TABLE]

Case3: Now we have ${\alpha_{1}}\geq{\beta_{1}}$, and some ${\alpha_{j}}\geq{\beta_{j}}+1$ (otherwise $\alpha=\beta$). Let $j=1$ without loss of generality. At this time we first prove a conclusion according to (6.1). Since

[TABLE]

it follows that

[TABLE]

Hence, we arrive at

[TABLE]

This proves (6.5). Next, we will consider a complex translation of (6.5) and prove that

[TABLE]

Actually one only needs to pay attention to (6.2), and the proof is completed according to Cauchy’s integral formula.

Finally, we only prove that

[TABLE]

In fact, only real polynomials need to be considered

[TABLE]

It can be obtained by straight calculation

[TABLE]

where ${{\varphi_{\gamma}}\left({r,x}\right)}$ are coefficients independent of $z$.

As to the complex case one only needs to perform complex translation to obtain

[TABLE]

The above preparations are classical, see also [35, 10]. Next we begin to prove the Jackson type approximation theorem via only modulus of continuity. We will make use of (6.6) in case of the Taylor polynomial

[TABLE]

of $f$ with $k\in\mathbb{N}$. Note that

[TABLE]

for every $x,y\in{\mathbb{R}^{n}}$.

Define the following domains to partition $\mathbb{R}^{n}$:

[TABLE]

We have to use different estimates in the above two domains, which are abstracted as follows. If $0<\left|y\right|<\delta$, we obtain that

[TABLE]

If $\left|y\right|\geq\delta$, one easily arrives at

[TABLE]

The Hölder inequality has been used in (6.7) and (6.8) with $k\geq 1,{\alpha_{i}}\geq 1,\mu_{i}=k/\alpha_{i}\geq 1$ without loss of generality:

[TABLE]

Now let $x=u+\mathrm{i}v$ with $u,v\in{\mathbb{R}^{n}}$ and $\left|v\right|\leq r$. Fix $p=n+k+2$, and let $c=c\left({n,k}\right)>0$ be a universal constant, then it follows that

[TABLE]

As it can be seen later, $I_{1}$ is the main part while $I_{2}$ is the remainder.

Recall Remark 2.4 and (2.3). Hence the following holds due to (6.7)

[TABLE]

In view of (6.8), we have

[TABLE]

By (6.2) and (6.2), we finally arrive at

[TABLE]

due to $\mathop{\overline{\lim}}\limits_{r\to{0^{+}}}r/{\varpi}\left(r\right)<+\infty$ in Definition 2.1. This proves Theorem 2.1 for the case $|\alpha|=0$. As to $|\alpha|\neq 0$, the result follows from the fact that ${S_{r}}$ commutes with ${\partial^{\alpha}}$. We therefore finish the proof of Theorem 2.1. ∎

6.3 Proof of Corollary 2.2

Proof.

Only the analysis of case $\left|\alpha\right|=0$ is given. In view of Theorem 2.1 and (6.7), we obtain that

[TABLE]

where the constant $c_{*}>0$ depends on $n$ and $k$. Further, by (6.11) we have

[TABLE]

provided a constant $c^{*}>0$ depending on $n,k$ and $\varpi$. This completes the proof. ∎

6.4 Proof of Corollary 2.3

Proof.

It is easy to verify that

[TABLE]

According to Fubini’s theorem, we obtain

[TABLE]

This completes the proof. ∎

6.5 Asymptotic analysis

Here we provide some useful asymptotic results, all of which can be proved by L’Hopital’s rule or by integration by parts, thus the proof is omitted here.

Lemma 6.3**.**

Let $\varrho\in\mathbb{N}^{+}$, $\lambda>1$ and some $M>0$ sufficiently large be fixed. Then as $X\to+\infty$,

[TABLE]

Lemma 6.4**.**

Let $0<\sigma<1$, $\lambda>1$ and some $M>0$ sufficiently large be fixed. Then as $X\to+\infty$,

[TABLE]

and

[TABLE]

6.6 KAM theorem for quantitative estimates

Here we give a KAM theorem for quantitative estimates, which is used in Theorem 2.2 in this paper. See Theorem 1 in Salamon’s paper [35] for details.

Theorem 6.1**.**

Let $n\geq 2,\tau>n-1,0<\theta<1$, and $M\geq 1$ be given. Then there are positive constants $\delta_{*}$ and $c$ such that $c\delta_{*}\leq 1/2$ and the following holds for every $0<r^{*}\leq 1$ and every $\omega\in\mathbb{R}^{n}$ that satisfies (1.1) (for $\tau>n-1$).

Suppose that $H(x,y)$ is a real analytic Hamiltonian function defined in the strip $\left|{\operatorname{Im}x}\right|\leq{r^{*}},\left|y\right|\leq{r^{*}}$, which is of period $1$ in the variables ${x_{1}},\ldots,{x_{n}}$ and satisfies

[TABLE]

for $\left|{\operatorname{Im}x}\right|\leq r^{*},\left|y\right|\leq r^{*}$, where $0<{\delta^{*}}\leq{\delta_{*}}$, and $Q\left({x,y}\right)\in{\mathbb{C}^{n\times n}}$ is a symmetric (not necessarily analytic) matrix valued function in the strip $\left|{\operatorname{Im}x}\right|\leq r,\left|y\right|\leq r$ and satisfies in this domain

[TABLE]

Then there exists a real analytic symplectic transformation $z=\phi\left(\zeta\right)$ of the form

[TABLE]

mapping the strip $\left|{\operatorname{Im}\xi}\right|\leq\theta r^{*},\left|\eta\right|\leq\theta r^{*}$ into $\left|{\operatorname{Im}x}\right|\leq r^{*},\left|y\right|\leq r^{*}$, such that $u\left(\xi\right)-\xi$ and $v\left(\xi\right)$ are of period $1$ in all variables and the Hamiltonian function $K:=H\circ\phi$ satisfies

[TABLE]

Moreover, $\phi$ and $K$ satisfy the estimates

[TABLE]

for $\left|{\operatorname{Im}\xi}\right|\leq\theta r^{*},\left|\eta\right|\leq\theta r^{*}$, and $\left|{\operatorname{Im}x}\right|\leq\theta r^{*}$.

Acknowledgements

This work was supported in part by National Basic Research Program of China (Grant No. 2013CB834100), National Natural Science Foundation of China (Grant No. 12071175, Grant No. 11171132, Grant No. 11571065), Project of Science and Technology Development of Jilin Province (Grant No. 2017C028-1, Grant No. 20190201302JC), and Natural Science Foundation of Jilin Province (Grant No. 20200201253JC).