C^{r}(\overline{\mathcal{D}}(\frac{2l}{2m},\frac{2l+1}{2m}))$ and
${\rho_{l}}(\Im\omega^{m})\in
C^{\omega}(\overline{\mathcal{D}}(\frac{2l}{2m},\frac{2l+\alpha}{2m}))$, we
have

$\sum_{0\leq
j<\underline{k}}{\Upsilon_{j,l}}(\omega^{m},\overline{\omega}^{m})\in
C^{r}(\overline{\mathcal{D}}(\tfrac{2l}{2m},\tfrac{2l+\alpha}{2m}))$

and ${\Upsilon_{j,l}}(\omega^{m},\overline{\omega}^{m})$ is a homogenous map
of degree $j-(m-2)<v(r)$ for $0\leq j<\underline{k}$. Then standard arguments
for smooth homogenous maps (see e.g. Lemma A.1 in Appendix) show that
${\Upsilon_{j,l}}=0$ for $0\leq j<m-2$ and
${\Upsilon_{j,l}}(\omega^{m},\overline{\omega}^{m})$ is a homogenous
polynomial of degree $j-(m-2)$ for $m-2\leq j<\underline{k}$. Suppose that

${\Upsilon_{j,l}}(\omega^{m},\overline{\omega}^{m})=\sum_{0\leq i\leq
j-(m-2)}a_{j,i}\omega^{i}\overline{\omega}^{(j-i)-(m-2)}\text{ for }m-2\leq
j<\underline{k}.$

Then

$\displaystyle\sum_{0\leq j<\underline{k}}\frac{1}{j!}\sum_{0\leq i\leq
j}\binom{j}{i}\frac{\partial^{j}f(0,0)}{\partial\omega^{i}\partial\overline{\omega}^{j-i}}\mathfrak{G}^{l}_{(m-1)-i,(m-1)-(j-i)}(u,s)=\mathscr{F}_{f_{<\underline{k}},l}(u,s)$
$\displaystyle=\\!\\!\sum_{0\leq
j<\underline{k}}\\!\\!{\Upsilon_{j,l}}(u,s)\\!+\\!{\rho_{l}}(s)=\\!\\!\sum_{m-2\leq
j<\underline{k}}\sum_{0\leq i\leq
j-(m-2)}\\!\\!{a_{j,i}}G_{l}(u,s)^{i}\overline{G_{l}(u,s)}^{(j-i)-(m-2)}\\!+\\!{\rho_{l}}(s).$

Differentiating with respect $u$, we get

$\displaystyle\sum_{0\leq j<\underline{k}}$
$\displaystyle\frac{1}{j!}\sum_{0\leq i\leq
j}\binom{j}{i}\frac{\partial^{j}f(0,0)}{\partial\omega^{i}\partial\overline{\omega}^{j-i}}G_{l}^{i-(m-1)}\overline{G_{l}}^{(j-i)-(m-1)}$
$\displaystyle=\sum_{m-2\leq j<\underline{k}}\sum_{0\leq i\leq
j-(m-2)}{a_{j,i}}\big{(}\tfrac{i}{m}G_{l}^{i-m}\overline{G_{l}}^{(j-i)-(m-2)}+\tfrac{(j-i)-(m-2)}{m}G_{l}^{i}\overline{G_{l}}^{(j-i)-2(m-1)}\big{)}$
$\displaystyle=\sum_{m-1\leq j<\underline{k}}\Big{(}\sum_{0\leq i\leq
j-(m-1)}{a_{j,i+1}}\tfrac{i+1}{m}G_{l}^{i-(m-1)}\overline{G_{l}}^{(j-i)-(m-1)}$
$\displaystyle\qquad\qquad+\sum_{m-1\leq i\leq
j}{a_{j,i-(m-1)}}\tfrac{(j-i)+1}{m}G_{l}^{i-(m-1)}\overline{G_{l}}^{(j-i)-(m-1)}\Big{)}.$

It follows that $D^{j}f(0,0)=0$ for $0\leq j\leq m-2$ and for every $m-1\leq
j<\underline{k}$ and $0\leq i\leq j$,

$\frac{1}{j!}\binom{j}{i}\frac{\partial^{j}f(0,0)}{\partial\omega^{i}\partial\overline{\omega}^{j-i}}={a_{j,i+1}}\tfrac{i+1}{m}+{a_{j,i-(m-1)}}\tfrac{(j-i)+1}{m},$

here we adhere to the convention that ${a_{j,i}}=0$ if $i<0$ or $i>j-(m-2)$.
It follows that for any $m-1\leq j<\underline{k}$ and $0\leq i\leq m-2$ with
$i\neq j-(m-1)\operatorname{mod}m$,

$\displaystyle\frac{\partial^{j}_{i}(f)}{j!}$ $\displaystyle=\sum_{0\leq
n\leq\frac{j-i}{m}}\frac{\binom{\frac{(m-1)-i}{m}-1}{n}}{\binom{\frac{(j-i)-(m-1)}{m}}{n}}\frac{1}{j!}\binom{j}{mn+i}\frac{\partial^{j}f}{\partial\omega^{mn+i}\partial\overline{\omega}^{j-(mn+i)}}(0,0)$
$\displaystyle=\sum_{n\geq
0}\frac{\binom{\frac{i-(m-1)}{m}+n}{n}}{\binom{-\frac{(j-i)-(m-1)}{m}+(n-1)}{n}}\Big{(}a_{j,i-(m-1)+m(n+1)}(\tfrac{i-(m-1)}{m}+n+1)$
$\displaystyle\qquad\qquad+a_{j,i-(m-1)+mn}(\tfrac{(j-i)-(m-1)}{m}-(n-1))\Big{)}$
$\displaystyle=\sum_{n\geq
0}\frac{\binom{\frac{i-(m-1)}{m}+n+1}{n+1}(n+1)}{\binom{-\frac{(j-i)-(m-1)}{m}+(n-1)}{n}}a_{j,i-(m-1)+m(n+1)}$
$\displaystyle\quad-\sum_{n\geq
1}\frac{\binom{\frac{i-(m-1)}{m}+n}{n}n}{\binom{-\frac{(j-i)-(m-1)}{m}+(n-2)}{n-1}}{a_{j,i-(m-1)+mn}}-{a_{j,i-(m-1)}}(\tfrac{(j-i)-(m-1)}{m}+1).$

As $i-(m-1)<0$, we have ${a_{j,i-(m-1)}}=0$. Hence $\partial^{j}_{i}(f)=0$ for
every $0\leq j<\underline{k}$ and $0\leq i\leq j\wedge(m-2)$ with $i\neq
j-(m-1)\operatorname{mod}m$. ∎

##  5\. Global properties

In this section, by combining previous results for local analysis near
singularity, we finally obtain solutions for cohomological equations with
optimal loss of regularity.

###  5.1. Transition from local to global results

Let $M$ be a compact connected orientable $C^{\infty}$-surface. Let
$\psi_{\mathbb{R}}$ be a locally Hamiltonian $C^{\infty}$-flow on $M$ with
isolated fixed points such that all its saddles are perfect and all saddle
connections are loops. Let $M^{\prime}\subset M$ be a minimal component of the
flow and let $I\subset M^{\prime}$ be a transversal curve. The corresponding
IET $T:I\to I$ exchanges the intervals
$\\{I_{\alpha}:\alpha\in\mathcal{A}\\}$. There exists $0<\varepsilon\leq 1$
such that for every $\sigma\in\mathrm{Sd}(\psi_{\mathbb{R}})$ we have
$\mathcal{D}_{\sigma,\varepsilon}\subset U_{\sigma}$, where
$\mathcal{D}_{\sigma,\varepsilon}$ is the pre-image of the square
$[-\varepsilon,\varepsilon]\times[-\varepsilon,\varepsilon]$ via the map
$z\mapsto z^{m_{\sigma}}$ in local singular coordinates, see Figure 2.
Moreover, we can assume that every orbit starting from $I$ meets at most one
set $\mathcal{D}_{\sigma,\varepsilon}$ (maybe many times) before return to
$I$. For every $0\leq l<2m_{\sigma}$ let
$\mathcal{D}^{l}_{\sigma,\varepsilon}=\overline{\mathcal{D}}_{\sigma,\varepsilon}(\frac{l}{2m_{\sigma}},\frac{l+1}{2m_{\sigma}})$
be the $l$-th closed angular sector of $\mathcal{D}_{\sigma,\varepsilon}$.

Figure 2. The set $D_{\varepsilon}$ and the incoming and outgoing segments of
its boundary in the case of $m_{\sigma}=3$.

######  Remark 5.1.

By Lemma 8.2 in [14], the enter and exit sets of
${\mathcal{D}}_{\sigma,\varepsilon}(\frac{l}{m_{\sigma}},\frac{l+1}{m_{\sigma}})$
are $C^{\infty}$-curves with standard parametrization

$[-\varepsilon,\varepsilon]\ni s\mapsto G_{l}(-\varepsilon-\iota
s)\in{\mathcal{D}}_{\sigma,\varepsilon}\text{ and
}[-\varepsilon,0)\cup(0,\varepsilon]\ni s\mapsto G_{l}(\varepsilon-\iota
s)\in{\mathcal{D}}_{\sigma,\varepsilon}\text{ resp.}$

Every
$\omega\in{\mathcal{D}}_{\sigma,\varepsilon}(\frac{l}{m_{\sigma}},\frac{l+1}{m_{\sigma}})$
lies on the positive semi-orbit of $G_{l}(-\varepsilon-\iota s)$,
$s\in[-\varepsilon,\varepsilon]$ so that
$\psi_{\xi_{l}(\omega)}G_{l}(-\varepsilon-\iota s)=\omega$ for some
$\xi_{l}(\omega)>0$. By the proof of Lemma 8.2 in [14],
$z=\omega^{m_{\sigma}}=u-\iota s$ for some $u\in[-\varepsilon,\varepsilon]$
and for any $f\in C(M)$,

(5.1)
$\displaystyle\begin{split}\int_{0}^{\xi_{l}(\omega)}&f(\psi_{t}G_{l}(-\varepsilon-\iota
s))dt=\frac{1}{m_{\sigma}^{2}}\int_{-\varepsilon}^{u}\frac{(f\cdot
V)(G_{l}(v-\iota s))}{(v^{2}+s^{2})^{\frac{m_{\sigma}-1}{m_{\sigma}}}}dv\\\
&=\frac{\varepsilon^{-\frac{m_{\sigma}-2}{m_{\sigma}}}}{m_{\sigma}^{2}}\int_{-1}^{u/\varepsilon}\frac{(f\cdot
V)(\varepsilon^{\frac{1}{m_{\sigma}}}G_{l}(v-\iota(s/\varepsilon)))}{(v^{2}+(s/\varepsilon)^{2})^{\frac{m_{\sigma}-1}{m_{\sigma}}}}dv\\\
&=\frac{\varepsilon^{-\frac{m_{\sigma}-2}{m_{\sigma}}}}{m_{\sigma}^{2}}\mathscr{F}_{(f\cdot
V)\circ\varepsilon^{1/m_{\sigma}},l}(z/\varepsilon)=\frac{\varepsilon^{-\frac{m_{\sigma}-2}{m_{\sigma}}}}{m_{\sigma}^{2}}{F}_{(f\cdot
V)\circ\varepsilon^{1/m_{\sigma}}}(\varepsilon^{-1/m_{\sigma}}\omega).\end{split}$

In particular, if $\omega=G_{l}(\varepsilon-\iota s)$ for
$s\in[-\varepsilon,\varepsilon]\setminus\\{0\\}$ then
$\tau_{l}(s):=\xi_{l}(G_{l}(-\varepsilon-\iota s))$ is the transit time of
$G_{l}(-\varepsilon-\iota s)$ through the set
${\mathcal{D}}_{\sigma,\varepsilon}(\frac{l}{m_{\sigma}},\frac{l+1}{m_{\sigma}})$,
$u-\iota s=G_{l}(\varepsilon-\iota s)^{m_{\sigma}}=\varepsilon-\iota s$ and

(5.2)
$\displaystyle\begin{split}\int_{0}^{\tau_{l}(s)}f(\psi_{t}G_{l}(-\varepsilon-\iota
s))dt&=\frac{\varepsilon^{-\frac{m_{\sigma}-2}{m_{\sigma}}}}{m_{\sigma}^{2}}\mathscr{F}_{(f\cdot
V)\circ\varepsilon^{1/m_{\sigma}},l}((\varepsilon-\iota s)/\varepsilon)\\\
&=\frac{\varepsilon^{-\frac{m_{\sigma}-2}{m_{\sigma}}}}{m_{\sigma}^{2}}\varphi_{(f\cdot
V)\circ\varepsilon^{1/m_{\sigma}},l}(-s/\varepsilon).\end{split}$

######  Remark 5.2.

Recall that $m\geq 2$ is the maximal multiplicity of saddles in
$\mathrm{Sd}(\psi_{\mathbb{R}})\cap M^{\prime}$ and for any
$r\geq-\frac{m-2}{m}$,

$k_{r}=\left\\{\begin{array}[]{cl}\lceil mr+(m-1)\rceil&\text{if
}-\frac{m-2}{m}\leq r\leq-\frac{m-3}{m},\\\ \lceil mr+(m-2)\rceil&\text{if
}-\frac{m-3}{m}<r.\end{array}\right.$

Note that for any $r\geq-\frac{m-2}{m}$ we have $\lceil r\rceil+1\leq k_{r}$.
Indeed, if $-\frac{m-2}{m}\leq r\leq-\frac{m-3}{m}$ then $-\frac{m-2}{m-1}\leq
r$. Hence $r+1\leq mr+(m-1)$, which yields $\lceil r\rceil+1\leq\lceil
mr+(m-1)\rceil=k_{r}$.

If $-\frac{m-3}{m}<r$ with $m\geq 3$ or $1\leq r$ with $m=2$ then
$-\frac{m-3}{m-1}\leq r$. Hence $r+1\leq mr+(m-2)$, which yields $\lceil
r\rceil+1\leq\lceil mr+(m-2)\rceil=k_{r}$. Suppose that $m=2$ and
$\frac{1}{2}=-\frac{m-3}{m}<r<1$. Then $\lceil r\rceil+1=2\leq\lceil
2r\rceil=\lceil mr+(m-2)\rceil=k_{r}$.

######  Remark 5.3.

For any $r\geq-\frac{m-2}{m}$ and $\sigma\in\mathrm{Sd}(\psi_{\mathbb{R}})\cap
M^{\prime}$ let $k\geq 0$ be such that
$\mathfrak{o}(\sigma,k-1)<r\leq\mathfrak{o}(\sigma,k)$. It follows that
$n:=\lceil\mathfrak{o}(\sigma,k)\rceil=\lceil r\rceil$. In view of (1.8),
$k\leq\lceil mr+(m-2)\rceil\leq k_{r}$. Moreover, by Remark 5.2, $n+1=\lceil
r\rceil+1\leq k_{r}$. Therefore, $k\vee(n+1)\leq k_{r}$.

###### Proof of Theorem 1.1.

Let $\tau:I\to{\mathbb{R}}_{>0}\cup\\{+\infty\\}$ be the first return time map
for the flow $\psi_{\mathbb{R}}$ restricted to $M^{\prime}$. For any interval
(set) $J\subset I$ avoiding the set $disc(T)$ of discontinuities of $T$ let
$J^{\tau}=\\{\psi_{t}s:s\in J,0\leq t\leq\tau(s)\\}$. If an interval $J$
contains some elements of $disc(T)$ then $J^{\tau}$ is the closure of
$(J\setminus disc(T))^{\tau}$.

Case 1. Suppose that $J\subset I_{\alpha}$ is a closed interval such that
$\sup\tau(J)<\infty$ and $\max\tau(J)<2\min\tau(J)$. Choose any
$t_{J}<\min\tau(J)$ so that $2t_{J}>\max\tau(J)$. Let us consider the set
$J^{\tau}$ and its two subsets

(5.3) $J_{+}^{\tau}=\\{\psi_{t}s:s\in J,0\leq t\leq t_{J}\\},\quad
J_{-}^{\tau}=\\{\psi_{-t}(Ts):s\in J,0\leq t\leq t_{J}\\}.$

By assumption, $J_{+}^{\tau}\cup J_{-}^{\tau}=J^{\tau}$. Let
$\rho_{+},\rho_{-}:J^{\tau}\to[0,1]$ be the corresponding
$C^{\infty}$-partition of unity, i.e. $\rho_{\pm}$ are $C^{\infty}$-maps such
that $\rho_{+}+\rho_{-}=1$ and $\rho_{\pm}=0$ on $J^{\tau}\setminus
J_{\pm}^{\tau}$. Let $v_{\pm}:J\times[0,t_{J}]\to J^{\tau}_{\pm}$ be given by
$v_{+}(s,t)=\psi_{t}s$ and $v_{-}(s,t)=\psi_{-t}(Ts)$. Then

$\varphi_{f}(s)=\int_{0}^{t_{J}}(\rho_{+}\cdot f)\circ
v_{+}(s,t)dt+\int_{0}^{t_{J}}(\rho_{-}\cdot f)\circ v_{-}(s,t)dt.$

Since $v_{\pm}$ are of class $C^{\infty}$, it follows that for every $q>0$ if
$f\in C^{q}(M)$ then $\varphi_{f}\in C^{q}(J)$ and there exists $C_{J}^{q}>0$
such that $\|\varphi_{f}\|_{C^{q}(J)}\leq C_{J}^{q}\|f\|_{C^{q}(M)}$ for any
$f\in C^{q}(M)$. Suppose that $f\in C^{k_{r}}(M)$. In view of Remark 5.3,
$n+1\leq k_{r}$, and hence $\varphi_{f}\in C^{n+1}(J)$ with

(5.4) $\|\varphi_{f}\|_{C^{n+1}(J)}\leq C_{J}^{n+1}\|f\|_{C^{k_{r}}(M)}\text{
for any }f\in C^{k_{r}}(M).$

Case 2. Suppose that $J\subset I_{\alpha}$ is of the form
$J=[l_{\alpha},l_{\alpha}+\varepsilon]$. Suppose that $l_{\alpha}$ is the
first backward meeting point of a separatrix incoming to
$\sigma\in\mathrm{Sd}(\psi_{\mathbb{R}})\cap M^{\prime}$. It follows that the
orbits starting from $J$ meet the set $\mathcal{D}_{\sigma,\varepsilon}$
before return to $I$. Suppose that each such orbit meets
$\mathcal{D}_{\sigma,\varepsilon}$ only once and it meets a sector
$\mathcal{D}_{\sigma,\varepsilon}^{2l+1}$ for some $0\leq l<m_{\sigma}$. In
general, the orbits of $J$ can meet $\mathcal{D}_{\sigma,\varepsilon}$ several
times in different sectors. This case arises when the saddle $\sigma$ has
saddle loops, but this situation is discussed later.

For every $s\in J$ denote by $\tau_{+}(s)$ the first forward entrance time of
the orbit of $s$ to $\mathcal{D}_{\sigma,\varepsilon}$ and by $\tau_{-}(s)$
the first backward entrance time of the orbit of $Ts$ to
$\mathcal{D}_{\sigma,\varepsilon}$. Then
$\psi_{\tau_{+}(s)}(s)=G_{l}(-\varepsilon-\iota(s-l_{\alpha}))$. Since
$\tau(s)\to+\infty$ as $s\to l_{\alpha}$ and $\tau_{\pm}$ are bounded,
decreasing $\varepsilon$, if necessary, we can assume that
$\min\tau(J)>\max\tau_{\pm}(J)$. Choose $\max\tau_{\pm}(J)<t_{J}<\min\tau(J)$
and let us consider two subsets $J_{\pm}^{\tau}\subset J^{\tau}$ given by
(5.3). Then
$J^{\tau}=J_{+}^{\tau}\cup\mathcal{D}_{\sigma,\varepsilon}^{2l+1}\cup
J_{-}^{\tau}$. Let us consider the corresponding $C^{\infty}$-partition of
unity $\rho_{+},\rho_{\sigma},\rho_{-}:J^{\tau}\to[0,1]$, i.e. $\rho_{+}$,
$\rho_{\sigma}$, $\rho_{-}$ are $C^{\infty}$-maps such that
$\rho_{+}+\rho_{\sigma}+\rho_{-}=1$, $\rho_{\pm}=0$ on $J^{\tau}\setminus
J_{\pm}^{\tau}$ and $\rho_{\sigma}=0$ on
$J^{\tau}\setminus\mathcal{D}_{\sigma,\varepsilon}$. Then

$\varphi_{f}(s)=\int_{0}^{t_{J}}(\rho_{+}\cdot f)\circ
v_{+}(s,t)dt+\int_{0}^{t_{J}}(\rho_{-}\cdot f)\circ
v_{-}(s,t)dt+\int_{\tau_{+}(s)}^{\tau(s)-\tau_{-}(s)}(\rho_{\sigma}\cdot
f)(\psi_{t}s)dt.$

Repeating the arguments used in Case 1, for any $q>0$ we get $C^{q}_{J}>0$
such that

(5.5) $\Big{\|}\int_{0}^{t_{J}}(\rho_{+}\cdot
f)(\psi_{t}\,\cdot\,)dt+\int_{0}^{t_{J}}(\rho_{-}\cdot
f)(\psi_{-t}(T\,\cdot\,)dt\Big{\|}_{C^{q}(J)}\leq C_{J}^{q}\|f\|_{C^{q}(M)}$

for any $f\in C^{q}(M)$.

Note that for every $s\in(0,\varepsilon]$,

$\displaystyle\varphi^{\sigma}_{f}(l_{\alpha}+s):$
$\displaystyle=\int_{\tau_{+}(l_{\alpha}+s)}^{\tau(l_{\alpha}+s)-\tau_{-}(l_{\alpha}+s)}(\rho_{\sigma}\cdot
f)(\psi_{t}(l_{\alpha}+s))dt$
$\displaystyle=\int_{0}^{\tau_{l}(s)}(\rho_{\sigma}\cdot
f)(\psi_{t}G_{l}(-\varepsilon-\iota s))dt.$

By (5.2), it follows that for any $s\in(0,1]$,
$\varphi^{\sigma}_{f}(l_{\alpha}+\varepsilon
s)=\frac{\varepsilon^{-\frac{m_{\sigma}-2}{m_{\sigma}}}}{m_{\sigma}^{2}}\varphi_{\tilde{f},l}(-s)$,
where $\tilde{f}(\omega,\overline{\omega})=(\rho_{\sigma}\cdot f\cdot
V)(\varepsilon^{\frac{1}{m_{\sigma}}}\omega,\varepsilon^{\frac{1}{m_{\sigma}}}\overline{\omega})$.

Suppose that $f\in C^{k_{r}}(M)$ for some $r\geq-\frac{m-2}{m}$. Choose $k\geq
0$ such that $\mathfrak{o}(\sigma,k-1)<r\leq\mathfrak{o}(\sigma,k)$. By Remark
5.3, we have $\lceil\mathfrak{o}(\sigma,k)\rceil=\lceil r\rceil=n$ and
$k\vee(n+1)\leq k_{r}$. Assume that $\mathfrak{C}^{j}_{\sigma,2l+1}(f)=0$ for
all $0\leq j<k$, or equivalently for all $j\geq 0$ such that
$\mathfrak{o}(\mathfrak{C}^{j}_{\sigma,2l+1})<r$. Since $\rho_{\sigma}=1$ in a
neighborhood of $\sigma$, it follows that

$\mathscr{C}^{j}_{2l+1}(\tilde{f})=\varepsilon^{\frac{j}{m_{\sigma}}}\mathscr{C}^{j}_{2l+1}(f\cdot
V)=\varepsilon^{\frac{j}{m_{\sigma}}}\mathfrak{C}^{j}_{\sigma,2l+1}(f)=0\text{
for all }0\leq j<k.$

Let
$a_{0}:=\lceil\mathfrak{o}(\sigma,k)\rceil-\mathfrak{o}(\sigma,k)=n-\mathfrak{o}(\sigma,k)$.
Then $n-a_{0}=\mathfrak{o}(\sigma,k)\geq r=n-a$.

As $k\vee(n+1)\leq k_{r}$ and both $f$ and $\tilde{f}$ are of class
$C^{k_{r}}$, in view of Theorem 4.7, $\varphi_{\tilde{f},l}\in
C^{n+\mathrm{P}_{a_{0}}}([-1,0))$ and there exists $C^{r}_{\sigma,l}>0$ such
that

$\|\varphi_{\tilde{f},l}\|_{C^{n+\mathrm{P}_{a_{0}}}([-1,0))}\leq
C^{r}_{\sigma,l}\|\tilde{f}\|_{C^{k\vee(n+1)}(\mathcal{D})}\leq
C^{r}_{\sigma,l}\|\rho_{\sigma}\cdot
V\|_{C^{k_{r}}(\mathcal{D})}\|f\|_{C^{k_{r}}(\mathcal{D})}.$

As $\varphi^{\sigma}_{f}(l_{\alpha}+\varepsilon
s)=\frac{\varepsilon^{-\frac{m_{\sigma}-2}{m_{\sigma}}}}{m_{\sigma}^{2}}\varphi_{\tilde{f},l}(-s)$
and $n-a\leq n-a_{0}$, in view of Remark 2.2, for any $f\in C^{k_{r}}(M)$ with
$\mathfrak{C}^{j}_{\sigma,2l+1}(f)=0$ for all $0\leq j<k$,

$\varphi^{\sigma}_{f}\in C^{n+\operatorname{P_{a}}}(J)\text{ and
}\|\varphi^{\sigma}_{f}\|_{C^{n+\operatorname{P_{a}}}(J)}\leq\widetilde{C}^{r}_{\sigma,l}\|f\|_{C^{k_{r}}(\mathcal{D})}.$

In view of (5.5) and Remark 2.2, it follows that for any $f\in
C^{k_{r}}(M)\cap\displaystyle\bigcap_{0\leq
j<k}\ker(\mathfrak{C}^{j}_{\sigma,2l+1})$,

$\displaystyle\|\varphi_{f}\|_{C^{n+\operatorname{P_{a}}}(J)}$
$\displaystyle\leq\widetilde{C}^{r}_{\sigma,l}\|f\|_{C^{k_{r}}(M)}+C_{J}^{n+1}\|f\|_{C^{n+1}(M)}$
$\displaystyle\leq(\widetilde{C}^{r}_{\sigma,l}+C_{J}^{n+1})\|f\|_{C^{k_{r}}(M)}.$

Figure 3. The decomposition of $J^{\tau}$ along the bouquet of $N-1$ saddle
loops.

Case 3. Suppose that $J\subset I_{\alpha}$ is of the form
$J=[l_{\alpha},l_{\alpha}+\varepsilon]$, where $l_{\alpha}$ is the first
backward meeting point of a separatrix incoming to
$\sigma\in\mathrm{Sd}(\psi_{\mathbb{R}})\cap M^{\prime}$. Suppose that
$\sigma$ has some saddle loops and $J^{\tau}$ meets $\sigma$ $N$-times
($1<N=N_{J}<m_{\sigma}$). Then all orbits starting from $\operatorname{Int}J$
meet the set $\mathcal{D}_{\sigma,\varepsilon}$ $N$-times before return to
$I$. Assume that each such orbit meets its sectors
$\mathcal{D}_{\sigma,\varepsilon}^{2l_{i}+1}$ for $1\leq i\leq N$
consecutively. Then $\sigma$ has $N-1$ saddle loops $sl_{i}$ connecting the
sector $\mathcal{D}_{\sigma,\varepsilon}^{2l_{i}+1}$ with
$\mathcal{D}_{\sigma,\varepsilon}^{2l_{i+1}+1}$ for $1\leq i\leq N-1$. In
particular,

(5.6) $J^{\tau}=J_{+}^{\tau}\cup
J_{-}^{\tau}\cup\bigcup_{i=1}^{N}\mathcal{D}_{\sigma,\varepsilon}^{2l_{i}+1}\cup\bigcup_{i=1}^{N-1}J_{i}^{\tau},$

where
$J^{\tau}_{i}=\\{\psi_{t}\gamma_{i}(s):s\in[0,\varepsilon],t\in[0,t_{i}]\\}$
is a rectangle whose base is a $C^{\infty}$-curve
$\gamma_{i}([0,\varepsilon])\subset\mathcal{D}_{\sigma,\varepsilon}^{2l_{i}+1}$
with a standard parametrization while its left side
$\\{\psi_{t}\gamma_{i}(0):t\in[0,t_{i}]\\}$ is a part of the loop $sl_{i}$,
see Figure 3. Using a partition of unity associated to the cover (5.6) and
repeating the arguments used in Case 1 and 2, for every $r>0$ we get
$C_{J}^{r}>0$ such that

(5.7) $\|\varphi_{f}\|_{C^{n+\operatorname{P_{a}}}(J)}\leq
C_{J}^{r}\|f\|_{C^{k_{r}}(M)}\text{ for }f\in C^{k_{r}}(M)\cap\bigcap_{1\leq
i\leq N_{J}}\bigcap_{0\leq j<k}\ker(\mathfrak{C}^{j}_{\sigma,2l_{i}+1}).$

Case 4. Suppose that $J\subset I_{\alpha}$ is of the form
$J=[r_{\alpha}-\varepsilon,r_{\alpha}]$, where $r_{\alpha}$ is the first
backward meeting point of a separatrix incoming to
$\sigma\in\mathrm{Sd}(\psi_{\mathbb{R}})$. Suppose that $J^{\tau}$ meets
$\sigma$ $N$-times ($N=N_{J}$) and the orbits starting from
$\operatorname{Int}J$ meet the set $\mathcal{D}_{\sigma,\varepsilon}^{2l_{i}}$
for $1\leq i\leq N$ consecutively before return to $I$. Then repeating the
arguments used in Case 1, 2 and 3, for every $r>0$ we get $C_{J}^{r}>0$ such
that

(5.8) $\|\varphi_{f}\|_{C^{n+\operatorname{P_{a}}}(J)}\leq
C_{J}^{r}\|f\|_{C^{k_{r}}(M)}\text{ for }f\in C^{k_{r}}(M)\cap\bigcap_{1\leq
i\leq N_{J}}\bigcap_{0\leq j<k}\ker(\mathfrak{C}^{j}_{\sigma,2l_{i}}).$

Final step. We can find a finite family of closed subintervals
$\\{J_{q}\\}_{q=1}^{Q}$ of $I$ which covers the whole interval $I$ and such
that every $J_{q}$ is of the form $[l_{\alpha},l_{\alpha}+\varepsilon]$ (or
$[r_{\alpha}-\varepsilon,r_{\alpha}]$) with
$\min\tau_{\pm}(J_{q})>\max\tau(J_{q})$, or
$J_{q}\subset\operatorname{Int}I_{\alpha}$ with
$2\min\tau(J_{q})>\max\tau(J_{q})$.

If $J_{q}$ is an interval of the form $[l_{\alpha},l_{\alpha}+\varepsilon]$ or
$[r_{\alpha}-\varepsilon,r_{\alpha}]$ then by (5.7) and (5.8),

$\displaystyle\|\varphi_{f}\|_{C^{n+\operatorname{P_{a}}}(J_{q})}\leq
C_{J_{q}}^{r}\|f\|_{C^{k_{r}}(M)}\text{ for }f\in
C^{k_{r}}(M)\cap\bigcap_{\begin{subarray}{c}(\sigma,j,l)\in\mathscr{TC}\\\
\mathfrak{o}(\sigma,j)<r\end{subarray}}\ker(\mathfrak{C}^{j}_{\sigma,l}).$

If $J_{q}\subset\operatorname{Int}I_{\alpha}$ then, by Remark 2.2 and (5.4),

$\displaystyle\|\varphi_{f}\|_{C^{n+\mathrm{P_{a}}}(J_{q})}$
$\displaystyle\leq\|\varphi_{f}\|_{C^{n+1}(J_{q})}\leq
C_{J_{q}}^{n+1}\|f\|_{C^{n+1}(M)}\leq C_{J_{q}}^{n+1}\|f\|_{C^{k_{r}}(M)}$

for all $f\in C^{k_{r}}(M)$. This yields $\varphi_{f}\in
C^{n+\operatorname{P_{a}}}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$ and

$\|\varphi_{f}\|_{C^{n+\mathrm{P_{a}}}}\leq\sum_{q=1}^{Q}\|\varphi_{f}\|_{C^{n+\mathrm{P_{a}}}(J_{q})}\leq
C\|f\|_{C^{k_{r}}(M)}$

for all $f\in C^{k_{r}}(M)$ such that $\mathfrak{C}^{j}_{\sigma,l}(f)=0$ for
$(\sigma,j,l)\in\mathscr{TC}$ with $\mathfrak{o}(\sigma,j)<r$.

Recall that, by assumption, the right end of $I$ is the first meeting point of
a separatrix (that is not a saddle connection) emanating by a fixed point
$\sigma$ (incoming or outgoing) with the interval $I$. Suppose that the right
end is the first backward meeting point of a separatrix incoming to $\sigma$.

Let $\alpha=\pi_{1}^{-1}(d)$, i.e. the interval
$I_{\alpha}=[l_{\alpha},r_{\alpha})$ is the latest after the exchange. It
follows that for every $0<\varepsilon<|I_{\alpha}|$ the strip
$[r_{\alpha}-\varepsilon,r_{\alpha}]^{\tau}$ avoids all fixed points, so
$\sup\tau([r_{\alpha}-\varepsilon,r_{\alpha}])<\infty$. By the continuity of
$\tau$, we can choose $\varepsilon>0$ so that
$\max\tau([r_{\alpha}-\varepsilon,r_{\alpha}])<2\min\tau([r_{\alpha}-\varepsilon,r_{\alpha}])$.
In view of Case 1, $\varphi_{f}\in
C^{n+1}([r_{\alpha}-\varepsilon,r_{\alpha}])$. Hence,

$C_{\alpha,n}^{a,-}(\varphi_{f})=\lim_{x\nearrow
r_{\alpha}}D^{n+1}\varphi_{f}(x)(r_{\alpha}-x)^{1+a}=0.$

The same argument shows that if the right end is the first forward meeting
point of a separatrix outgoing from $\sigma$, then
$C_{\alpha,n}^{a,-}(\varphi_{f})=0$ for $\alpha=\pi_{0}^{-1}(d)$. Finally we
have $C_{\pi_{0}^{-1}(d),n}^{a,-}(\varphi_{f})\cdot
C_{\pi_{1}^{-1}(d),n}^{a,-}(\varphi_{f})=0$. Analyzing the orbit of the left
end in the same way, we also get
$C_{\pi_{0}^{-1}(1),n}^{a,+}(\varphi_{f})\cdot
C_{\pi_{1}^{-1}(1),n}^{a,+}(\varphi_{f})=0$, which shows that $\varphi_{f}\in
C^{n+\operatorname{P_{a}G}}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$. ∎

For all $(\sigma,k,j)\in\mathscr{TD}$ let
${\chi}^{k}_{\sigma,j}:M\to{\mathbb{C}}$ be a $C^{\infty}$-map such that
${\chi}^{k}_{\sigma,j}(\omega,\overline{\omega})=\omega^{j}\overline{\omega}^{k-j}/(k!V(\omega,\overline{\omega}))$
on $U_{\sigma}$ and it is equal to zero on all $U_{\sigma^{\prime}}$ for
$\sigma^{\prime}\neq\sigma$. By definition,
$\mathfrak{d}^{k}_{\sigma,j}({\chi}^{k}_{\sigma,j})=1$ and
$\mathfrak{d}^{k^{\prime}}_{\sigma^{\prime},j^{\prime}}({\chi}^{k}_{\sigma,j})=0$
if $(\sigma^{\prime},k^{\prime},j^{\prime})\neq(\sigma,k,j)$.

In view of Theorem 1.1, we get the following result.

######  Corollary 5.4.

For every $r\geq-\frac{m-2}{m}$ and any $f\in C^{k_{r}}(M)$ we have a
decomposition

(5.9) $f=\sum_{\begin{subarray}{c}(\sigma,k,j)\in\mathscr{TD}\\\
\mathfrak{o}(\sigma,k)<r\end{subarray}}\mathfrak{d}^{k}_{\sigma,j}(f)\chi^{k}_{\sigma,j}+\mathfrak{R}_{r}(f)$

such that $\varphi_{\mathfrak{R}_{r}(f)}\in
C^{n+\operatorname{P_{a}G}}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$ with
$n=\lceil r\rceil$ and $a=n-r$. Moreover, the operators
$\mathfrak{R}_{r}:C^{k_{r}}(M)\to C^{k_{r}}(M)$ and $C^{k_{r}}(M)\ni
f\mapsto\varphi_{\mathfrak{R}_{r}(f)}\in
C^{n+\operatorname{P_{a}G}}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$ are
bounded.

Let us consider an equivalence relation $\sim$ on $\mathscr{TC}$ as follows:
$(\sigma,k,l)\sim(\sigma,k,l^{\prime})$ if the angular sectors $U_{\sigma,l}$
and $U_{\sigma,l^{\prime}}$ are connected through a chain of saddle loops
emanating from the saddle $\sigma$. For every equivalence class
$[(\sigma,k,l)]\in\mathscr{TC}/\sim$, let

$\mathfrak{C}_{[(\sigma,k,l)]}(f):=\sum_{(\sigma,k,l^{\prime})\sim(\sigma,k,l)}\mathfrak{C}^{k}_{\sigma,l}(f).$

For any $[(\sigma,k,l)]\in\mathscr{TC}/\sim$ there exists
$\alpha\in\mathcal{A}$ and an interval $J$ of the form
$[l_{\alpha},l_{\alpha}+\varepsilon]$ or $[r_{\alpha}-\varepsilon,r_{\alpha}]$
such that $l_{\alpha}$ or $r_{\alpha}$ is the first backward meeting point of
a separatrix incoming to $\sigma\in\mathrm{Sd}(\psi_{\mathbb{R}})$ and
$J^{\tau}$ contains all angular sectors $U_{\sigma,l^{\prime}}$ for which
$(\sigma,k,l^{\prime})\sim(\sigma,k,l)$. Let
$\xi_{[(\sigma,k,l)]}:I\to{\mathbb{R}}$ be given as follows:

  * •

$\xi_{[(\sigma,k,l)]}$ is zero on any interval $I_{\beta}$ with
$\beta\neq\alpha$;

  * •

if $J=[l_{\alpha},l_{\alpha}+\varepsilon]$ then for any $s\in I_{\alpha}$,

$\displaystyle\xi_{[(\sigma,k,l)]}(s)$
$\displaystyle=\frac{(s-l_{\alpha})^{\frac{k-(m_{\sigma}-2)}{m_{\sigma}}}}{m_{\sigma}^{2}k!}\text{
if }k\neq m_{\sigma}-2\ \operatorname{mod}m_{\sigma},$
$\displaystyle\xi_{[(\sigma,k,l)]}(s)$
$\displaystyle=-\frac{(s-l_{\alpha})^{\frac{k-(m_{\sigma}-2)}{m_{\sigma}}}\log(s-l_{\alpha})}{m_{\sigma}^{2}k!}\text{
if }k=m_{\sigma}-2\ \operatorname{mod}m_{\sigma};$

  * •

if $J=[r_{\alpha}-\varepsilon,r_{\alpha}]$ then for any $s\in I_{\alpha}$,

$\displaystyle\xi_{[(\sigma,k,l)]}(s)$
$\displaystyle=\frac{(r_{\alpha}-s)^{\frac{k-(m_{\sigma}-2)}{m_{\sigma}}}}{m_{\sigma}^{2}k!}\text{
if }k\neq m_{\sigma}-2\ \operatorname{mod}m_{\sigma},$
$\displaystyle\xi_{[(\sigma,k,l)]}(s)$
$\displaystyle=-\frac{(r_{\alpha}-s)^{\frac{k-(m_{\sigma}-2)}{m_{\sigma}}}\log(r_{\alpha}-s)}{m_{\sigma}^{2}k!}\text{
if }k=m_{\sigma}-2\ \operatorname{mod}m_{\sigma}.$

Of course, $\xi_{[(\sigma,k,l)]}\in
C^{n+\operatorname{P_{a}G}}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$ with
$n:=\lceil\mathfrak{o}(\sigma,k)\rceil$ and $a:=n-\mathfrak{o}(\sigma,k)$.

In view of the proof of Theorem 1.1 we also have the following.

######  Corollary 5.5.

Fix $\sigma\in\mathrm{Sd}(\psi_{\mathbb{R}})\cap M^{\prime}$, $k\geq 0$ and
let $n:=\lceil\mathfrak{o}(\sigma,k)\rceil$ and $a:=n-\mathfrak{o}(\sigma,k)$.
Suppose that $f\in C^{k\vee(n+1)}(M)$ is such that it is equal to zero on
$U_{\sigma^{\prime}}$ for $\sigma^{\prime}\neq\sigma$. Then

(5.10)
$\varphi_{f}=\sum_{\begin{subarray}{c}[(\sigma,j,l)]\in\mathscr{TC}/\sim\\\
0\leq
j<k\end{subarray}}\mathfrak{C}_{[(\sigma,j,l)]}(f)\xi_{[(\sigma,j,l)]}+C^{n+\operatorname{P_{a}G}}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha}).$

###### Proof.

The proof proceeds in the same way as the proof of Theorem 1.1, except that we
use Corollary 4.8 instead of Theorem 4.7 in the key reasoning. For example,
using the notations introduced in the proof of the Theorem 1.1, for any
$s\in(0,1]$

$\varphi^{\sigma}_{f}(l_{\alpha}+\varepsilon
s)=\frac{\varepsilon^{-\frac{m_{\sigma}-2}{m_{\sigma}}}}{m_{\sigma}^{2}}\varphi_{\tilde{f},l}(-s)\text{
with }\tilde{f}(\omega,\overline{\omega})=(\rho_{\sigma}\cdot f\cdot
V)(\varepsilon^{\frac{1}{m_{\sigma}}}\omega,\varepsilon^{\frac{1}{m_{\sigma}}}\overline{\omega})$

and
$\mathscr{C}^{j}_{2l+1}(\tilde{f})=\varepsilon^{\frac{j}{m_{\sigma}}}\mathscr{C}^{j}_{2l+1}(f\cdot
V)=\varepsilon^{\frac{j}{m_{\sigma}}}\mathfrak{C}^{j}_{\sigma,2l+1}(f)$. In
view of Corollary 4.8, for $s\in[-1,0)$,

$\displaystyle\varphi_{\tilde{f},l}(s)$
$\displaystyle=-\sum_{\begin{subarray}{c}0\leq j<k\\\
j=m_{\sigma}-2\operatorname{mod}m_{\sigma}\end{subarray}}\frac{\mathscr{C}^{j}_{2l+1}(\tilde{f})}{j!}(-s)^{\frac{j-(m_{\sigma}-2)}{m_{\sigma}}}\log(-s)$
$\displaystyle\quad+\sum_{\begin{subarray}{c}0\leq j<k\\\ j\neq
m_{\sigma}-2\operatorname{mod}m_{\sigma}\end{subarray}}\frac{{\mathscr{C}}^{j}_{2l+1}(\tilde{f})}{j!}(-s)^{\frac{j-(m_{\sigma}-2)}{m_{\sigma}}}+C^{n+\operatorname{P_{a}}}([-1,0)).$

It follows that for $s\in(l_{\alpha},l_{\alpha}+\varepsilon]$,

$\displaystyle\varphi^{\sigma}_{f}(s)$
$\displaystyle=\frac{\varepsilon^{-\frac{m_{\sigma}-2}{m_{\sigma}}}}{m_{\sigma}^{2}}\varphi_{\tilde{f},l}((l_{\alpha}-s)/\varepsilon)$
$\displaystyle=-\frac{\varepsilon^{-\frac{m_{\sigma}-2}{m_{\sigma}}}}{m_{\sigma}^{2}}\sum_{\begin{subarray}{c}0\leq
j<k\\\
j=m_{\sigma}-2\operatorname{mod}m_{\sigma}\end{subarray}}\frac{\varepsilon^{\frac{j}{m_{\sigma}}}\mathfrak{C}^{j}_{\sigma,2l+1}(f)}{j!}\Big{(}\frac{s-l_{\alpha}}{\varepsilon}\Big{)}^{\frac{j-(m_{\sigma}-2)}{m_{\sigma}}}\log\Big{(}\frac{s-l_{\alpha}}{\varepsilon}\Big{)}$
$\displaystyle\quad+\frac{\varepsilon^{-\frac{m_{\sigma}-2}{m_{\sigma}}}}{m_{\sigma}^{2}}\\!\\!\sum_{\begin{subarray}{c}0\leq
j<k\\\ j\neq
m_{\sigma}-2\operatorname{mod}m_{\sigma}\end{subarray}}\frac{\varepsilon^{\frac{j}{m_{\sigma}}}\mathfrak{C}^{j}_{\sigma,2l+1}(f)}{j!}\Big{(}\frac{s-l_{\alpha}}{\varepsilon}\Big{)}^{\frac{j-(m_{\sigma}-2)}{m_{\sigma}}}+C^{n+\operatorname{P_{a}}}((l_{\alpha},l_{\alpha}+\varepsilon])$
$\displaystyle=-\sum_{\begin{subarray}{c}0\leq j<k\\\
j=m_{\sigma}-2\operatorname{mod}m_{\sigma}\end{subarray}}\\!\\!\mathfrak{C}^{j}_{\sigma,2l+1}(f)\frac{(s-l_{\alpha})^{\frac{j-(m_{\sigma}-2)}{m_{\sigma}}}\log(s-l_{\alpha})}{m_{\sigma}^{2}j!}$
$\displaystyle\quad+\sum_{\begin{subarray}{c}0\leq j<k\\\ j\neq
m_{\sigma}-2\operatorname{mod}m_{\sigma}\end{subarray}}\mathfrak{C}^{j}_{\sigma,2l+1}(f)\frac{(s-l_{\alpha})^{\frac{j-(m_{\sigma}-2)}{m_{\sigma}}}}{m_{\sigma}^{2}j!}+C^{n+\operatorname{P_{a}}}((l_{\alpha},l_{\alpha}+\varepsilon]).$

This key observation makes it possible to get (5.10) proceeding further as in
the proof of Theorem 1.1. ∎

######  Theorem 5.6.

For any $r\geq-\frac{m-2}{m}$ let $n=\lceil r\rceil$ and $a=n-r$. Then for any
$f\in C^{k_{r}}(M)$ we have

(5.11) $\displaystyle\mathfrak{s}_{r}(f)$
$\displaystyle=\varphi_{f}-\sum_{\begin{subarray}{c}[(\sigma,k,l)]\in\mathscr{TC}/\sim\\\
\mathfrak{o}(\sigma,k)<r\end{subarray}}\mathfrak{C}_{[(\sigma,k,l)]}(f)\xi_{[(\sigma,k,l)]}\in
C^{n+\operatorname{P_{a}}}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$

and the operator $\mathfrak{s}_{r}:C^{k_{r}}(M)\to
C^{n+\operatorname{P_{a}}}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$ is
bounded.

###### Proof.

Let $\\{\rho_{\sigma}:\sigma\in\mathrm{Sd}(\psi_{\mathbb{R}})\cap
M^{\prime}\\}$ be a $C^{\infty}$-partition of unity of $M$ such that
$\rho_{\sigma}=1$ on $U_{\sigma}$. For any
$\sigma\in\mathrm{Sd}(\psi_{\mathbb{R}})\cap M^{\prime}$ choose
$k_{\sigma}\geq 1$ so that
$\mathfrak{o}(\sigma,k_{\sigma}-1)<r\leq\mathfrak{o}(\sigma,k_{\sigma})$. Let
$n_{\sigma,k_{\sigma}}=\lceil\mathfrak{o}(\sigma,k_{\sigma})\rceil$ and
$a_{\sigma,k_{\sigma}}=n_{\sigma,k_{\sigma}}-\mathfrak{o}(\sigma,k_{\sigma})$.
By Remark 5.3, $k_{\sigma}\vee(n_{\sigma,k_{\sigma}}+1)\leq k_{r}$. Therefore,
Corollary 5.5 applied to $f\cdot\rho_{\sigma}$, shows that

$\varphi_{f\cdot\rho_{\sigma}}=\sum_{\begin{subarray}{c}[(\sigma,j,l)]\in\mathscr{TC}/\sim\\\
0\leq
j<k_{\sigma}\end{subarray}}\mathfrak{C}_{[(\sigma,j,l)]}(f\cdot\rho_{\sigma}){\xi}_{[(\sigma,j,l)]}+C^{n_{\sigma,k_{\sigma}}+\mathrm{P}_{a_{\sigma,k_{\sigma}}}\mathrm{G}}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha}).$

As $r\leq\mathfrak{o}(\sigma,k_{\sigma})$, by Remark 2.2,
$C^{n_{\sigma,k_{\sigma}}+\mathrm{P}_{a_{\sigma,k_{\sigma}}}\mathrm{G}}\subset
C^{n+\operatorname{P_{a}G}}$. Since
$\mathfrak{C}_{[(\sigma,j,l)]}(f\cdot\rho_{\sigma})=\mathfrak{C}_{[(\sigma,j,l)]}(f)$,
this gives

$\varphi_{f\cdot\rho_{\sigma}}=\sum_{\begin{subarray}{c}[(\sigma,j,l)]\in\mathscr{TC}/\sim\\\
0\leq
j<k_{\sigma}\end{subarray}}\mathfrak{C}_{[(\sigma,j,l)]}(f){\xi}_{[(\sigma,j,l)]}+C^{n+\operatorname{P_{a}G}}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha}).$

When summed against $\sigma$, this yields (5.11).

To prove that the operator $\mathfrak{s}_{r}$ is bounded, we use the
decomposition (5.9). Indeed,

$\displaystyle\mathfrak{s}_{r}(f)$
$\displaystyle=\sum_{\begin{subarray}{c}(\sigma,k,j)\in\mathscr{TD}\\\
\mathfrak{o}(\sigma,k)<r\end{subarray}}\mathfrak{d}^{k}_{\sigma,j}(f)\mathfrak{s}_{r}(\chi^{k}_{\sigma,j})+\mathfrak{s}_{r}(\mathfrak{R}_{r}(f))$
$\displaystyle=\sum_{\begin{subarray}{c}(\sigma,k,j)\in\mathscr{TD}\\\
\mathfrak{o}(\sigma,k)<r\end{subarray}}\mathfrak{d}^{k}_{\sigma,j}(f)\mathfrak{s}_{r}(\chi^{k}_{\sigma,j})+\varphi_{\mathfrak{R}_{r}(f)}.$

Since the functionals $\mathfrak{d}^{k}_{\sigma,j}$ and the operator
$f\mapsto\varphi_{\mathfrak{R}_{r}(f)}$ (by Corollary 5.4) are bounded, this
gives that $\mathfrak{s}_{r}$ is bounded. ∎

###### Proof of Theorem 1.2.

Arguments presented in Section 1.2 show that if $g\in C^{r}(I)$ is a solution
of the cohomological equation $g\circ T-g=\varphi_{f}$, then the corresponding
function
$u=u_{g,f}:M^{\prime}\setminus(\mathrm{Sd}(\psi_{\mathbb{R}})\cup\operatorname{SL}(\psi_{{\mathbb{R}}}))\to{\mathbb{C}}$
given by

$u(x):=g(\psi_{t}x)-\int_{0}^{t}f(\psi_{s}x)\,ds$

whenever $\psi_{t}x\in I$ for some $t\in{\mathbb{R}}$, is of class $C^{r}$ on
$M^{\prime}\setminus(\mathrm{Sd}(\psi_{\mathbb{R}})\cup\operatorname{SL}(\psi_{{\mathbb{R}}}))$.
We need to show that if $\mathfrak{d}^{k}_{\sigma,j}(f)=0$ for all
$(\sigma,k,j)\in\mathscr{TD}$ such that
$\widehat{\mathfrak{o}}(\mathfrak{d}^{k}_{\sigma,j})<v(r)$ and
$\mathfrak{C}^{k}_{\sigma,l}(f)=0$ for all $(\sigma,k,l)\in\mathscr{TC}$ such
that ${\mathfrak{o}}(\mathfrak{C}^{k}_{\sigma,l})<v(r)$ then $u$ has a
$C^{r}$-extension to $M^{\prime}_{e}$ and

(5.12) $\|u\|_{C^{r}(M^{\prime}_{e})}\leq
C(\|g\|_{C^{r}(I)}+\|f\|_{C^{k_{v(r)}}(M)}).$

We split the proof of our claim into several steps. In fact, we split
$M^{\prime}_{e}$ into subsets of two kinds: subsets which are far from saddles
and saddle loops, and sets surrounding saddles or saddle loops.

Step 1. Sets far from saddles and saddle loops. We will show that for any
compact subset $A\subset
M^{\prime}\setminus(\mathrm{Sd}(\psi_{\mathbb{R}})\cup\operatorname{SL}(\psi_{{\mathbb{R}}}))$
there exists $C_{A}>0$ such that

(5.13) $\|u\|_{C^{r}(A)}\leq C_{A}(\|g\|_{C^{r}(I)}+\|f\|_{C^{k_{v(r)}}(M)}).$

Recall that, by arguments from Section 1.2, for any $x_{0}\in
M^{\prime}\setminus(\mathrm{Sd}(\psi_{\mathbb{R}})\cup\operatorname{SL}(\psi_{{\mathbb{R}}}))$
there exist closed intervals $[\tau_{1},\tau_{2}]$ and
$J\subset\operatorname{Int}I$ such that the set $R(x_{0})=\\{\psi_{t}x:x\in
J,t\in[\tau_{1},\tau_{2}]\\}$ is a rectangle in $M^{\prime}$, i.e. the map

$J\times[\tau_{1},\tau_{2}]\ni(x,t)\mapsto\nu(x,t)=\psi_{t}x\in R(x_{0})$

is a $C^{\infty}$-diffeomorphism and $x_{0}\in\operatorname{Int}R(x_{0})$.
Moreover,

$u\circ\nu(x,t)=g(x)+\int_{0}^{t}f\circ\nu(x,s)ds\text{ on
}J\times[\tau_{1},\tau_{2}].$

By Remark 5.2, it follows that there exists $C_{x_{0}}>0$ such that

$\|u\|_{C^{r}(R(x_{0}))}\leq C_{x_{0}}(\|g\|_{C^{r}(I)}+\|f\|_{C^{r}(M)})\leq
C_{x_{0}}(\|g\|_{C^{r}(I)}+\|f\|_{C^{k_{v(r)}}(M)}).$

Covering $A$ by a finite number of rectangles, this yields (5.13).

Step 2. Some sets far from saddles. Suppose that $\gamma:[a,b]\to
M\setminus\rm{Fix}(\psi_{\mathbb{R}})$ is a standard
$C^{\infty}$-parametrization of a curve and $\xi:[a,b]\to{\mathbb{R}}_{>0}$ is
a $C^{\infty}$ map such that

$[a,b]^{\xi}\ni(x,t)\mapsto\nu(x,t)=\psi_{t}x\in\nu([a,b]^{\xi})=:(\gamma[a,b])^{\xi}$

is a $C^{\infty}$-diffeomorphism, where $[a,b]^{\xi}=\\{(x,t):x\in[a,b],0\leq
t\leq\xi(x)\\}$. Then the arguments used in Step 1 show that if
$u\circ\gamma\in C^{r}([a,b])$ then $u\in C^{r}([a,b]^{\gamma})$ and there
exists $C_{\gamma,\xi}>0$ such that

(5.14) $\|u\|_{C^{r}(\gamma([a,b])^{\xi})}\leq
C_{\gamma,\xi}(\|u\circ\gamma\|_{C^{r}([a,b])}+\|f\|_{C^{k_{v(r)}}(M)}).$

Step 3. Strips touching saddles and saddle loops and their decomposition. From
now on we will use a notation introduced in the proof of Theorem 1.1. Let
$\tau:I\to{\mathbb{R}}_{>0}\cup\\{+\infty\\}$ be the first return time map.
Suppose that $J\subset I_{\alpha}$ is of the form
$J=[l_{\alpha},l_{\alpha}+\varepsilon]$, where $l_{\alpha}$ is the first
backward meeting point of a separatrix incoming to
$\sigma\in\mathrm{Sd}(\psi_{\mathbb{R}})\cap M^{\prime}$. Suppose that
$J^{\tau}$ meets $\sigma$ exactly $N$-times ($1\leq N=N_{J}<m_{\sigma}$) and
the orbits starting from $\operatorname{Int}J$ meet
$\mathcal{D}_{\sigma,\varepsilon}$ in its sectors
$\mathcal{D}_{\sigma,\varepsilon}^{2l_{i}+1}$ for $1\leq i\leq N$
consecutively before return to $I$. Then $\sigma$ has $N-1$ saddle loops
$sl_{i}$ connecting the sector $\mathcal{D}_{\sigma,\varepsilon}^{2l_{i}+1}$
with $\mathcal{D}_{\sigma,\varepsilon}^{2l_{i+1}+1}$ for $1\leq i\leq N-1$.
Recall that $J^{\tau}$ is the closure of $(\operatorname{Int}J)^{\tau}$. Then

(5.15)
$J^{\tau}=\bigcup_{i=1}^{N}\mathcal{D}_{\sigma,\varepsilon}^{2l_{i}+1}\cup\bigcup_{i=0}^{N}E_{i},$

where each $E_{i}$ is of the form $\gamma_{i}([0,\varepsilon])^{\xi_{i}}$ with

  * •

$\gamma_{0}(s)=\gamma(l_{\alpha}+s)$ (here $\gamma$ is the parametrization of
$I$) and $\xi_{0}(s)$ is the time spent to go from $J$ to
$\mathcal{D}_{\sigma,\varepsilon}^{2l_{1}+1}$;

  * •

for $0\leq i\leq N-1$, $\gamma_{i}(s)=G_{l_{i}}(\varepsilon-\iota s)$ and
$\xi_{i}(s)$ is the time spent to go from
$\mathcal{D}_{\sigma,\varepsilon}^{2l_{i}+1}$ to
$\mathcal{D}_{\sigma,\varepsilon}^{2l_{i+1}+1}$;

  * •

$\gamma_{N}(s)=G_{l_{N}}(\varepsilon-\iota s)$ and $\xi_{N}(s)$ is the time
spent to go from $\mathcal{D}_{\sigma,\varepsilon}^{2l_{N}+1}$ to $I$.

Step 4.0. The set $E_{0}$. In view of (5.14) in Step 2,

(5.16) $u\in C^{r}(E_{0})\text{ and }\|u\|_{C^{r}(E_{0})}\leq
C_{\gamma_{0},\xi_{0}}(\|g\|_{C^{r}(I)}+\|f\|_{C^{k_{v(r)}}(M)}).$

Step 4.1. The sets $\mathcal{D}_{\sigma,\varepsilon}^{2l_{i}+1}$ surrounding
the saddle $\sigma$. We will show that for every $1\leq i\leq N$ there exist
$C_{i},C^{\prime}_{i}>0$ such that if $u$ has a $C^{r}$-extension on $E_{i-1}$
then it has $C^{r}$-extension on $\mathcal{D}_{\sigma,\varepsilon}^{2l_{i}+1}$
and

(5.17) $\|u\|_{C^{r}({\mathcal{D}}_{\sigma,\varepsilon}^{2l_{i}+1})}\leq
C_{i}\|u\|_{C^{r}(E_{i-1})}+C^{\prime}_{i}\|f\|_{C^{k_{v(r)}}(M)}.$

This is the main inductive step running to the proof of (5.12) restricted to
$J^{\tau}$.

By Remark 5.1, for every
$\omega\in\mathcal{D}_{\sigma,\varepsilon}(\frac{l_{i}}{m_{\sigma}},\frac{l_{i}+1}{m_{\sigma}})$
we have $\psi_{\xi_{l}(\omega)}G_{l_{i}}(-\varepsilon-\iota s)=\omega$ for
$s=-\Im\omega^{m_{\sigma}}\in[0,\varepsilon]$ and

$u(\omega)-u(G_{l_{i}}(-\varepsilon-\iota
s))=\int_{0}^{\xi_{l}(\omega)}f(\psi_{t}G_{l_{i}}(-\varepsilon-\iota s))dt.$

In view of (5.1), for
$\omega\in\mathcal{D}_{\sigma,\varepsilon}(\frac{2l_{i}+1}{2m_{\sigma}},\frac{2l_{i}+2}{2m_{\sigma}})$,

(5.18)
$u(\omega)-u(G_{l_{i}}(-\varepsilon+\iota\Im\omega^{m_{\sigma}}))=\frac{\varepsilon^{-\frac{m_{\sigma}-2}{m_{\sigma}}}}{m_{\sigma}^{2}}{F}_{(f\cdot
V)\circ\varepsilon^{1/m_{\sigma}}}(\varepsilon^{-1/m_{\sigma}}\omega).$

Choose $m-1\leq\underline{k}\leq k\leq k_{v(r)}$ such that

$\mathfrak{o}(\sigma,{k}-1)<v(r)\leq\mathfrak{o}(\sigma,{k})\text{ and
}\widehat{\mathfrak{o}}(\sigma,\underline{k}-1)<v(r)\leq\widehat{\mathfrak{o}}(\sigma,\underline{k}).$

Then
$\widehat{\mathfrak{o}}(\sigma,\underline{k})=\lceil\mathfrak{o}(\sigma,{k})\rceil$.
Moreover, by Remark 5.3, $n:=\lceil
v(r)\rceil=\lceil\mathfrak{o}(\sigma,{k})\rceil$ and $k\vee(n+1)\leq
k_{v(r)}$.

By assumption, for every $0\leq j<k$ and $0\leq i\leq j\wedge(m_{\sigma}-2)$
with $i\neq j-(m_{\sigma}-1)\operatorname{mod}m_{\sigma}$ we have
$\partial_{i}^{j}(f\cdot V)=\mathfrak{d}_{\sigma,i}^{j}(f)=0$ and
$\mathscr{C}^{j}_{l}(f\cdot V)=\mathfrak{C}_{\sigma,l}^{j}(f)=0$ for all
$0\leq j<k$ and $l=2l_{i}+1$, $1\leq i\leq N$.

In view of Theorem 4.10, the map $F_{(f\cdot
V)\circ\varepsilon^{1/m_{\sigma}}}\circ\varepsilon^{-1/m_{\sigma}}:\mathcal{D}_{\sigma,\varepsilon}(\frac{2l_{i}+1}{2m_{\sigma}},\frac{2l_{i}+2}{2m_{\sigma}})\to{\mathbb{C}}$
has a $C^{\mathfrak{e}(\sigma,k)}$-extension on
${\mathcal{D}}_{\sigma,\varepsilon}^{2l_{i}+1}=\overline{\mathcal{D}}_{\sigma,\varepsilon}(\frac{2l_{i}+1}{2m_{\sigma}},\frac{2l_{i}+2}{2m_{\sigma}})$
and there exists $C^{\prime}_{i}>0$ such that

(5.19)
$\displaystyle\Big{\|}\frac{\varepsilon^{-\frac{m_{\sigma}-2}{m_{\sigma}}}}{m_{\sigma}^{2}}{F}_{(f\cdot
V)\circ\varepsilon^{1/m_{\sigma}}}\circ\varepsilon^{-1/m_{\sigma}}\Big{\|}_{C^{\mathfrak{e}(\sigma,k)}({\mathcal{D}}_{\sigma,\varepsilon}^{2l_{i}+1})}\leq
C^{\prime}_{i}\|f\|_{C^{k\vee(n+1)}(M)}.$

Moreover, the map
${\mathcal{D}}_{\sigma,\varepsilon}(\frac{2l_{i}+1}{2m_{\sigma}},\frac{2l_{i}+2}{2m_{\sigma}})\ni\omega\mapsto
G_{l_{i}}(-\varepsilon+\iota\Im\omega^{m_{\sigma}})\in{\mathcal{D}}_{\sigma,\varepsilon}^{2l_{i}+1}\cap
E_{i-1}$ has an obvious analytic extension on
${\mathcal{D}}_{\sigma,\varepsilon}^{2l_{i}+1}$. It follows that there exists
$C_{i}>0$ such that if $u$ is of class $C^{r}$ on $E_{i-1}$ then $u\circ
G_{l_{i}}(-\varepsilon+\iota\Im\omega^{m_{\sigma}})$ has a $C^{r}$-extension
to ${\mathcal{D}}_{\sigma,\varepsilon}^{2l_{i}+1}$ and

(5.20) $\|u\circ
G_{l_{i}}(-\varepsilon+\iota\Im\omega^{m_{\sigma}})\|_{C^{r}({\mathcal{D}}_{\sigma,\varepsilon}^{2l_{i}+1})}\leq
C_{i}\|u\|_{C^{r}(E_{i-1})}.$

As $v(r)\leq\mathfrak{o}(\sigma,k)$ and $k\vee(n+1)\leq k_{v(r)}$, by (5.18),
(5.19) and (5.20), $u$ has a $C^{r}$-extension on
${\mathcal{D}}_{\sigma,\varepsilon}^{2l_{i}+1}$ and (5.17) holds.

Step 4.2. The sets $E_{i}$ surrounding the saddle loops. We will show that for
every $1\leq i\leq N$ there exist
$C^{\prime\prime}_{i},C^{\prime\prime\prime}_{i}>0$ such that if $u$ has a
$C^{r}$-extension on $\mathcal{D}_{\sigma,\varepsilon}^{2l_{i}+1}$ then it has
$C^{r}$-extension on $E_{i}$ and

$\|u\|_{C^{r}(E_{i})}\leq
C^{\prime\prime}_{i}\|u\|_{C^{r}({\mathcal{D}}_{\sigma,\varepsilon}^{2l_{i}+1})}+C^{\prime\prime\prime}_{i}\|f\|_{C^{k_{v(r)}}(M)}.$

This is an easy inductive step leading to the proof of (5.12) restricted to
$J^{\tau}$, which follows directly from (5.14). Indeed, as
$\gamma_{i}:[0,\varepsilon]\to{\mathcal{D}}_{\sigma,\varepsilon}^{2l_{i}+1}$
is an analytic curve, there exists $C>0$ such that if $u$ is of class $C^{r}$
on ${\mathcal{D}}_{\sigma,\varepsilon}^{2l_{i}+1}$ then
$\|u\circ\gamma_{i}\|_{C^{r}([0,\varepsilon])}\leq
C\|u\|_{C^{r}({\mathcal{D}}_{\sigma,\varepsilon}^{2l_{i}+1})}$. As
$E_{i}=\gamma_{i}([0,\varepsilon])^{\xi_{i}}$, in view of (5.14), $u$ has
$C^{r}$-extension on $E_{i}$ and

$\displaystyle\|u\|_{C^{r}(E_{i})}$ $\displaystyle\leq
C_{\gamma_{i},\xi_{i}}(\|u\circ\gamma_{i}\|_{C^{r}([0,\varepsilon])}+\|f\|_{C^{k_{v(r)}}(M)})$
$\displaystyle\leq
C_{\gamma_{i},\xi_{i}}(C\|u\|_{C^{r}({\mathcal{D}}_{\sigma,\varepsilon}^{2l_{i}+1})}+\|f\|_{C^{k_{v(r)}}(M)}).$

Step 4.3. Induction. Starting from Step 4.0 (as the initial inductive step)
and then repeating alternately Steps 4.1 and 4.2 $N$-times, we have that there
exists $C_{J}>0$ such that $u$ has a $C^{r}$-extension on $J^{\tau}$ and

(5.21) $\|u\|_{C^{r}(J^{\tau})}\leq
C_{J}(\|g\|_{C^{r}(J)}+\|f\|_{C^{{k}_{v(r)}}(M)}).$

Step 5. Summary. Using the arguments from Step 4, we obtain (5.21) also in the
case where $J=[r_{\alpha}-\varepsilon,r_{\alpha}]$. Then the strip $J^{\tau}$
touches a saddle on right side. Let $A\subset
M^{\prime}\setminus(\mathrm{Sd}(\psi_{\mathbb{R}})\cup\operatorname{SL}(\psi_{{\mathbb{R}}}))$
be the closure of

$M^{\prime}\setminus\bigcup_{\alpha\in\mathcal{A}}([l_{\alpha},l_{\alpha}+\varepsilon]^{\tau}\cup[r_{\alpha}-\varepsilon,r_{\alpha}]^{\tau}).$

Then by Step 1 applied to $A$ and Step 4 applied to the intervals
$[l_{\alpha},l_{\alpha}+\varepsilon]$ and
$[r_{\alpha}-\varepsilon,r_{\alpha}]$ for all $\alpha\in\mathcal{A}$, we have
that $u$ has a $C^{r}$-extension on $M^{\prime}_{e}$ and (5.12) holds with
$C=C_{A}+\sum_{\alpha\in\mathcal{A}}(C_{[l_{\alpha},l_{\alpha}+\varepsilon]}+C_{[r_{\alpha}-\varepsilon,r_{\alpha}]})$.
∎

Finally, we prove the optimality of the regularity of solutions for
cohomological equation. The proof of Theorem 1.3 reduces our problem to a
local framework and then applies Theorem 4.11 (proved in Section 4) knowing
that $\mathscr{C}^{j}_{l}$ corresponds to $\mathfrak{C}^{j}_{\sigma,l}$ and
$\partial^{j}_{i}$ corresponds to $\mathfrak{d}^{j}_{\sigma,i}$ in the local
context. If $u$ is a $C^{r}$-solution of the cohomological equation on an
angular sector around a saddle, then we compare the value of $u$ at any point
of the sector with the value of $u$ at the entry point of its orbit into the
sector. As both are of class $C^{r}$, their difference is also of class
$C^{r}$. Moreover, this difference is related to $F_{f}$ on the sector. This
is what the mentioned reduction is all about.

###### Proof of Theorem 1.3.

Suppose that there exists $u\in C^{r}(M^{\prime}_{e})$ such that $Xu=f$ for
some $r\in{\mathbb{R}}_{\eta}$ with $v(r)>0$. Choose
$\sigma\in\mathrm{Sd}(\psi_{\mathbb{R}})\cap M^{\prime}$ and $0\leq
l<m_{\sigma}$ such that $U_{\sigma,2l+1}\cap M^{\prime}\neq\emptyset$. We will
show that $\mathfrak{C}^{j}_{\sigma,2l+1}(f)=0$ for all $j\geq 0$ such that
$\mathfrak{o}(\sigma,j)<v(r)$ and $\mathfrak{d}^{j}_{\sigma,i}(f)=0$ for all
$j\geq 0$ such that $\widehat{\mathfrak{o}}(\sigma,j)<v(r)$ and $0\leq i\leq
j\wedge(m_{\sigma}-2)$ with $i\neq
j-(m_{\sigma}-1)\operatorname{mod}m_{\sigma}$. The proof for even sectors
follows the same way as for odd sectors, so we will only focus on the latter.

In view of (5.18), for
$\omega\in\mathcal{D}_{\sigma,\varepsilon}(\frac{2l+1}{2m_{\sigma}},\frac{2l+2}{2m_{\sigma}})$,

$u(\omega)-u(G_{l}(-\varepsilon+\iota\Im\omega^{m_{\sigma}}))=\frac{\varepsilon^{-\frac{m_{\sigma}-2}{m_{\sigma}}}}{m_{\sigma}^{2}}{F}_{(f\cdot
V)\circ\varepsilon^{1/m_{\sigma}}}(\varepsilon^{-1/m_{\sigma}}\omega).$

By assumption, $u$ is of class $C^{r}$ on
$\overline{\mathcal{D}}_{\sigma,\varepsilon}(\frac{2l+1}{2m_{\sigma}},\frac{2l+2}{2m_{\sigma}})$,
and hence $u(G_{l}(-\varepsilon+\iota\Im\omega^{m_{\sigma}}))$ is of class
$C^{r}$ on
$\overline{\mathcal{D}}_{\sigma,\varepsilon}(\frac{2l+1}{2m_{\sigma}},\frac{2l+2}{2m_{\sigma}})$.
Therefore, the map ${F}_{(f\cdot
V)\circ\varepsilon^{1/m_{\sigma}}}\circ\varepsilon^{-1/m_{\sigma}}$ has a
$C^{r}$-extension on
$\overline{\mathcal{D}}(\frac{2l+1}{2m},\frac{2l+2}{2m})$.

Choose $k\geq m_{\sigma}-1$ such that
$\mathfrak{o}(\sigma,k-1)<v(r)\leq\mathfrak{o}(\sigma,k)$. By Remark 5.3, we
have $n:=\lceil v(r)\rceil=\lceil\mathfrak{o}(\sigma,k)\rceil$ and
$k\vee(n+1)\leq k_{v(r)}$. Therefore, by Theorem 4.11,

$\varepsilon^{j/m_{\sigma}}\mathfrak{C}^{j}_{\sigma,2l+1}(f)=\mathscr{C}^{j}_{2l+1}((f\cdot
V)\circ\varepsilon^{1/m_{\sigma}})=0$

for all $j\geq 0$ such that $\mathfrak{o}(\sigma,j)<v(r)$, and

$\varepsilon^{j/m_{\sigma}}\mathfrak{d}^{j}_{\sigma,i}(f)=\partial^{j}_{i}((f\cdot
V)\circ\varepsilon^{1/m_{\sigma}})=0$

for all $j\geq 0$ with $\widehat{\mathfrak{o}}(\sigma,j)<v(r)$ and $0\leq
i\leq j\wedge(m_{\sigma}-2)$ with $i\neq
j-(m_{\sigma}-1)\operatorname{mod}m_{\sigma}$. ∎

## Acknowledgements

The authors would like to thank Alexander Gomilko for his help in
understanding some analytical issues used in Section 4.1 and Giovanni Forni
for an interesting discussion of potentially alternative approaches to solving
cohomological equations. We are grateful to the referees for their helpful
comments and suggestions for improving in the presentation of this work.

The authors acknowledge the Center of Excellence “Dynamics, mathematical
analysis and artificial intelligence” at the Nicolaus Copernicus University in
Toruń and Centro di Ricerca Matematica Ennio De Giorgi - Scuola Normale
Superiore, Pisa for hospitality during their visits. The Research was
partially supported by the Narodowe Centrum Nauki Grant 2022/45/B/ST1/00179.

## Appendix A

The following technical result is crucial to prove an optimality of regularity
in Theorem 4.11. Its proof is included for reader’s convenience.

######  Lemma A.1.

Let $0\leq\alpha<\beta<1$, $N\in{\mathbb{N}}$ and $r>0$. Suppose that
$f:\overline{\mathcal{D}}(\alpha,\beta)\to{\mathbb{C}}$ is a $C^{r}$-maps such
that

$f(x,y)=\sum_{-N\leq n<r}f_{n}(x,y)\text{ for
}(x,y)\in\overline{\mathcal{D}}(\alpha,\beta)\setminus\\{(0,0)\\},$

where
$f_{n}:\overline{\mathcal{D}}(\alpha,\beta)\setminus\\{(0,0)\\}\to{\mathbb{C}}$
is a $C^{\infty}$-map homogenous of degree $n$ for all $-N\leq n<r$. Then
$f_{-n}\equiv 0$ for $1\leq n\leq N$ and $f_{n}$ is a homogenous polynomial of
degree $n$ for all $0\leq n<r$.

###### Proof.

Take any pair $(i,j)\in{\mathbb{Z}}_{\geq 0}^{2}$ with $i+j\leq r$. Then

$\frac{\partial^{i+j}}{\partial x^{i}\partial y^{j}}f(x,y)=\sum_{-N\leq
n<r}\frac{\partial^{i+j}}{\partial x^{i}\partial y^{j}}f_{n}(x,y)\text{ for
}(x,y)\in\overline{\mathcal{D}}(\alpha,\beta)\setminus\\{(0,0)\\},$

where $\frac{\partial^{i+j}}{\partial x^{i}\partial
y^{j}}f_{n}:\overline{\mathcal{D}}(\alpha,\beta)\setminus\\{(0,0)\\}\to{\mathbb{C}}$
is a $C^{\infty}$-map homogenous of degree $n-i-j$ for all $-N\leq n<r$.

Since $\frac{\partial^{i+j}}{\partial x^{i}\partial y^{j}}f$ is continuous on
$\overline{\mathcal{D}}(\alpha,\beta)$, for any
$(x,y)\in\overline{\mathcal{D}}(\alpha,\beta)\setminus\\{(0,0)\\}$ we have
$\frac{\partial^{i+j}}{\partial x^{i}\partial
y^{j}}f(sx,sy)\to\frac{\partial^{i+j}}{\partial x^{i}\partial y^{j}}f(0,0)$ as
$s\searrow 0$. On the other hand,

$\frac{\partial^{i+j}}{\partial x^{i}\partial y^{j}}f(sx,sy)=\sum_{-N\leq
n<r}s^{n-i-j}\frac{\partial^{i+j}}{\partial x^{i}\partial y^{j}}f_{n}(x,y).$

It follows that for any
$(x,y)\in\overline{\mathcal{D}}(\alpha,\beta)\setminus\\{(0,0)\\}$ we have

$\displaystyle\frac{\partial^{i+j}}{\partial x^{i}\partial
y^{j}}f_{n}(x,y)=0\text{ if }-N\leq n<i+j\text{ and }$ (A.1)
$\displaystyle\frac{\partial^{i+j}}{\partial x^{i}\partial
y^{j}}f_{n}(x,y)=\frac{\partial^{i+j}}{\partial x^{i}\partial
y^{j}}f(0,0)\text{ if }n=i+j.$

Letting $i=j=0$ we get $f_{-k}\equiv 0$ for $1\leq k\leq N$.

Let us consider any $0\leq k<r$. Then for any
$(x,y)\in\overline{\mathcal{D}}(\alpha,\beta)\setminus\\{(0,0)\\}$ and
$0<s\leq 1$ we have $f_{k}(sx,sy)=s^{k}f_{k}(x,y)$. Applying the $k$th-order
derivative with respect to $s$, we get

$\sum_{0\leq i\leq k}\binom{k}{i}\frac{\partial^{k}}{\partial x^{i}\partial
y^{k-i}}f_{k}(sx,sy)x^{i}y^{k-i}=k!f_{k}(x,y).$

In view of (A.1), we obtain

$f_{k}(x,y)=\frac{1}{k!}\sum_{0\leq i\leq
k}\binom{k}{i}\frac{\partial^{k}}{\partial x^{i}\partial
y^{k-i}}f(0,0)x^{i}y^{k-i}.$

∎

## References

  * [1] V.I. Arnold, Topological and ergodic properties of closed $1$-forms with incommensurable periods, (Russian) Funktsional. Anal. i Prilozhen. 25 (1991), 1-12; translation in Funct. Anal. Appl. 25 (1991), 81–90.
  * [2] A. Avila, B. Fayad, A. Kocsard, _On manifolds supporting distributionally uniquely ergodic diffeomorphisms_ , J. Differential Geom. 99 (2015), 191-213.
  * [3] A. Avila, A. Kocsard, _Cohomological equations and invariant distributions for minimal circle diffeomorphisms_ , Duke Math. J. 158 (2011), 501-536.
  * [4] A. Bufetov, _Limit theorems for translation flow_ , Ann. of Math. (2) 179 (2014), 431-499.
  * [5] J. Chaika, A. Wright, _A smooth mixing flow on a surface with nondegenerate fixed points_ , J. Amer. Math. Soc. 32 (2019), 81-117.
  * [6] B. Fayad, A. Kanigowski, R. Zelada, _A non-mixing Arnold flow on a surface_ , preprint arXiv:2308.01247.
  * [7] L. Flaminio, G. Forni, _Invariant distributions and time averages for horocycle flows_ , Duke Math. J. 119 (2003), 465-526.
  * [8] L. Flaminio, G. Forni, _On the cohomological equation for nilflows_ , J. Mod. Dyn. 1 (2007), 37-60.
  * [9] G. Forni, _Solutions of the cohomological equation for area-preserving flows on compact surfaces of higher genus_ , Ann. of Math. (2) 146 (1997), 295-344.
  * [10] G. Forni, _Deviation of ergodic averages for area-preserving flows on surfaces of higher genus_ , Ann. of Math. (2) 155 (2002), 1-103.
  * [11] G. Forni, _Sobolev regularity of solutions of the cohomological equation_ , Ergodic Theory Dynam. Systems 41 (2021), 685-789.
  * [12] G. Forni, _Twisted cohomological equations for translation flows_ , Ergodic Theory Dynam. Systems 42 (2022), 881-916.
  * [13] G. Forni, S. Marmi, C. Matheus _Cohomological equation and local conjugacy class of Diophantine interval exchange maps_ , to appear in Proceedings of the American Mathematical Society, DOI:https://doi.org/10.1090/proc/14538.
  * [14] K. Frączek, M. Kim, _New phenomena in deviation of Birkhoff integrals for locally Hamiltonian flows_ , J. Reine Angew. Math. 807 (2024), 81-149.
  * [15] K. Frączek, M. Kim, _Solving the cohomological equation for locally hamiltonian flows, part II - global obstructions_ , preprint https://arxiv.org/abs/2306.02340.
  * [16] K. Frączek, C. Ulcigrai, _Ergodic properties of infinite extensions of area-preserving flows_ , Math. Ann. 354 (2012), 1289-1367.
  * [17] K. Frączek, C. Ulcigrai, _On the asymptotic growth of Birkhoff integrals for locally Hamiltonian flows and ergodicity of their extensions_ , preprint https://arxiv.org/abs/2112.05939, accepted in Commentarii Mathematici Helvetici, DOI10.4171/CMH/567.
  * [18] P. Giulietti, C. Liverani, _Parabolic dynamics and anisotropic Banach spaces_ , J. Eur. Math. Soc. (JEMS) 21 (2019), 2793-2858.
  * [19] A. Katok, _Combinatorial constructions in ergodic theory and dynamics_. University Lecture Series, 30. American Mathematical Society, Providence, RI, 2003. iv+121 pp.
  * [20] A.V. Kochergin, _Mixing in special flows over a rearrangement of segments and in smooth flows on surfaces_ , Mat. Sb. (N.S.) 96(138) (1975), 471-502.
  * [21] S. Marmi, P. Moussa, J.-C. Yoccoz, _The cohomological equation for Roth-type interval exchange maps_ , J. Amer. Math. Soc. 18 (2005), 823-872.
  * [22] S. Marmi, P. Moussa, J.-C. Yoccoz, _Linearization of generalized interval exchange maps_ , Ann. of Math. (2) 176 (2012), 1583-1646.
  * [23] S. Marmi, J.-C. Yoccoz, _Hölder regularity of the solutions of the cohomological equation for Roth type interval exchange maps_ , Comm. Math. Phys. 344 (2016), 117-139.
  * [24] S.P. Novikov. _The Hamiltonian formalism and a multivalued analogue of Morse theory_ , Uspekhi Mat. Nauk 37 (1982), 3–49; translated in Russian Math. Surveys 37 (1982), 1–56.
  * [25] D. Ravotti, _Quantitative mixing for locally Hamiltonian flows with saddle loops on compact surfaces_ , Ann. Henri Poincaré 18 (2017), 3815-3861.
  * [26] R. Roussarie, _Modèles locaux de champs et de formes._ With an English summary. Astérisqûe, No. 30. Société Mathématique de France, Paris, 1975. 181 pp.
  * [27] J. Tanis, _The cohomological equation and invariant distributions for horocycle maps_ , Ergodic Theory Dynam. Systems 34 (2014), 299-340.
  * [28] C. Ulcigrai, _Mixing of asymmetric logarithmic suspension flows over interval exchange transformations_ , Ergodic Theory Dynam. Systems 27 (2007), 991-1035.
  * [29] C. Ulcigrai, _Weak mixing for logarithmic flows over interval exchange transformations_ , J. Mod. Dyn. 3 (2009), 35-49.
  * [30] C. Ulcigrai, _Absence of mixing in area-preserving flows on surfaces_ , Ann. of Math. (2) 173 (2011), 1743-1778.
  * [31] C. Ulcigrai, _Dynamics and ’arithmetics’ of higher genus surface flows_ , ICM Proceedings 2022.
  * [32] Z.J. Wang, _Cohomological equation and cocycle rigidity of parabolic actions in some higher-rank Lie groups_ , Geom. Funct. Anal. 25 (2015), 1956-2020.
  * [33] A. Zorich, _How do the leaves of a closed 1-form wind around a surface?_ Pseudoperiodic topology, 135–178, Amer. Math. Soc. Transl. Ser. 2, 197, Adv. Math. Sci., 46, Amer. Math. Soc., Providence, RI, 1999.