# Solving the cohomological equation for locally hamiltonian flows, part I -
local obstructions

Krzysztof Frączek  Faculty of Mathematics and Computer Science, Nicolaus
Copernicus University, ul. Chopina 12/18, 87-100 Toruń, Poland
<EMAIL_ADDRESS>and  Minsung Kim  Centro di Ricerca Matematica Ennio De
Giorgi, Scuola Normale Superiore, Piazza dei Cavalieri 3, 56126 Pisa, Italy
<EMAIL_ADDRESS>
###### Abstract.

We study the cohomological equation $Xu=f$ for smooth locally Hamiltonian
flows on compact surfaces. The main novelty of the proposed approach is that
it is used to study the regularity of the solution $u$ when the flow has
saddle loops, which has not been systematically studied before. Then we need
to limit the flow to its minimal components. We show the existence and
(optimal) regularity of solutions regarding the relations with the associated
cohomological equations for interval exchange transformations (IETs). Our main
theorems state that the regularity of solutions depends not only on the
vanishing of the so-called Forni’s distributions (cf. [9, 11]), but also on
the vanishing of families of new invariant distributions (local obstructions)
reflecting the behavior of $f$ around the saddles. Our main results provide
some key ingredient for the complete solution to the regularity problem of
solutions (in cohomological equations) for a.a. locally Hamiltonian flows
(with or without saddle loops) to be shown in [15].

The main contribution of this article is to define the aforementioned new
families of invariant distributions $\mathfrak{d}^{k}_{\sigma,j}$,
$\mathfrak{C}^{k}_{\sigma,l}$ and analyze their effect on the regularity of
$u$ and on the regularity of the associated cohomological equations for IETs.
To prove this new phenomenon, we further develop local analysis of $f$ near
degenerate singularities inspired by tools from [14] and [17]. We develop new
tools of handling functions whose higher derivatives have polynomial
singularities over IETs.

###### Key words and phrases:

locally Hamiltonian flows, cohomological equation, invariant distributions

###### 2000 Mathematics Subject Classification:

37E35, 37A10, 37C40, 37C83, 37J12

##  1\. Introduction

Let $M$ be a smooth compact connected orientable surface of genus $g\geq 1$.
We deal with smooth flows $\psi_{\mathbb{R}}=(\psi_{t})_{t\in{\mathbb{R}}}$ on
$M$ (associated to a vector field $X:M\to TM$) preserving a smooth positive
measure $\mu$, i.e. such that for any (orientable) choice of local coordinates
$(x,y)$ we have $d\mu=V(x,y)dx\wedge dy$ with $V$ positive and smooth. These
flows are called _locally Hamiltonian flows_. Indeed, for any (orientable)
choice of local coordinates $(x,y)$ such that $d\mu=V(x,y)dx\wedge dy$, the
flow $\psi_{\mathbb{R}}$ is a local solution to the Hamiltonian equation

$\frac{dx}{dt}=\frac{\frac{\partial H}{\partial
y}(x,y)}{V(x,y)},\quad\frac{dy}{dt}=-\frac{\frac{\partial H}{\partial
x}(x,y)}{V(x,y)}$

for a smooth real-valued function $H$, or equivalently
$\frac{dz}{dt}=-2\iota\frac{\frac{\partial
H}{\partial\overline{z}}(z,\overline{z})}{V(z,\overline{z})}$. For general
introduction to locally Hamiltonian flows, we refer readers to [17, 14, 25,
31].

For any smooth observable $f:M\to{\mathbb{C}}$ we are interested in
understanding the smoothness of the solution $u:M\to{\mathbb{C}}$ of the
cohomological equation

(1.1) $u(\psi_{t}x)-u(x)=\int_{0}^{t}f(\psi_{s}x)\,ds\text{ for all }x\in M,\
t\in{\mathbb{R}},$

or equivalently $Xu=f$, where $Xu(x)=\frac{d}{dt}u(\psi_{t}x)|_{t=0}$.

Cohomological equations are an important area of study in dynamical systems
since they are related to smooth conjugacy problems via Kolmogorov-Arnold-
Moser techniques. Here it is worth to mention the pioneering work of Marmi-
Moussa-Yoccoz [22] on the conjugacy problem for small perturbations of
translation flows. The problem of solving cohomological equations for other
classes of smooth dynamical systems of parabolic nature and the regularity of
solutions using invariant distributions has a rich literature and has been
studied, among other places, in [2, 3, 10, 7, 8, 12, 13, 18, 19, 27, 32].

We always assume that all fixed points of the locally Hamiltonian flow
$\psi_{\mathbb{R}}$ are isolated, so the set of fixed points of
$\psi_{\mathbb{R}}$, denoted by $\mathrm{Fix}(\psi_{\mathbb{R}})$, is finite.
For $g\geq 2$, $\mathrm{Fix}(\psi_{\mathbb{R}})$ is non-empty. As
$\psi_{\mathbb{R}}$ is area-preserving, fixed points are either centers,
simple saddles or multi-saddles (saddles with $2k$ prongs with $k\geq 2$). We
will deal only with _perfect_ saddles defined as follows: a fixed point
$\sigma\in\mathrm{Fix}(\psi_{\mathbb{R}})$ is a (perfect) saddle of
multiplicity $m=m_{\sigma}\geq 2$ if there exists a chart $(x,y)$ (called _a
singular chart_) in a neighborhood $U_{\sigma}$ of $\sigma$ such that
$d\mu=V(x,y)dx\wedge dy$ and $H(x,y)=\Im(x+\iota y)^{m}$ ($(0,0)$ are
coordinates of $\sigma$). Then the corresponding local Hamiltonian equation in
$U_{\sigma}$ is of the form

$\frac{dx}{dt}=\frac{\frac{\partial H}{\partial
y}(x,y)}{V(x,y)}=\frac{m\Re(x+\iota
y)^{m-1}}{V(x,y)},\quad\frac{dy}{dt}=-\frac{\frac{\partial H}{\partial
x}(x,y)}{V(x,y)}=-\frac{m\Im(x+\iota y)^{m-1}}{V(x,y)},$

or equivalently $\frac{dz}{dt}=\frac{m\overline{z}^{m-1}}{V(z,\overline{z})}$.
The set of perfect saddles of $\psi_{\mathbb{R}}$ we denote by
$\mathrm{Sd}(\psi_{\mathbb{R}})$.

We call a _saddle connection_ an orbit of $\psi_{\mathbb{R}}$ running from a
saddle to a saddle. A _saddle loop_ is a saddle connection joining the same
saddle. We will deal only with flows such that all their saddle connections
are loops. The set consisting of all saddle loops of the flow we denote by
$\operatorname{SL}(\psi_{{\mathbb{R}}})$.

Recall that if every fixed point in $\mathrm{Fix}(\psi_{\mathbb{R}})$ is
isolated, $M$ splits into a finite number of $\psi_{\mathbb{R}}$-invariant
surfaces (with boundary) so that every such surface is a _minimal component_
of $\psi_{\mathbb{R}}$ (every orbit, except of fixed points and saddle loops,
is dense in the component) or is a periodic component (filled by periodic
orbits, fixed points and saddle loops). The boundary of each component
consists of saddle loops and fixed points.

The interest in the study of locally Hamiltonians flows in higher genus and,
in particular, in their ergodic properties, was highlighted by Novikov [24] in
connection with problems arising in solid-state physics as well as in pseudo-
periodic topology (see e.g. the survey [33] by A. Zorich). The simplest
examples of locally Hamiltonian flows with fixed points on the torus, flows
with one center and one simple saddle, motivated Arnold in [1] to ask about
mixing properties on minimal components. We should mention that mixing
properties of locally Hamiltonian flows with simple saddles restricted to
their minimal components are fully described in a series of papers [5, 6, 25,
28, 29, 30, 31]. When perfect multi-saddles appear, mixing was proved already
by Kochergin in [20].

The problem of existence and regularity of solutions for the cohomological
equation (1.1) was essentially solved in two seminal articles [9, 11] by
Forni. Forni considered the case when the flow $\psi_{\mathbb{R}}$ is minimal
over the whole surface $M$ and the function $f$ belongs to a certain weighted
Sobolev space. More precisely, choose a non-negative smooth function
$W:M\to{\mathbb{R}}_{\geq 0}$ (with zeros at $\mathrm{Sd}(\psi_{\mathbb{R}})$)
and an Abelian $1$-form $\omega$ on $M$ (with zeros at
$\mathrm{Sd}(\psi_{\mathbb{R}})$) such that $X=WS$ and $S$ is the unit
horizontal vector field on the translation surface $(M,\omega)$. In singular
local coordinates around any $\sigma\in\mathrm{Sd}(\psi_{\mathbb{R}})$ we have
$W(z,\overline{z})=|z|^{2(m_{\sigma}-1)}/V(z,\overline{z})$. Then for any
$s>0$, $f\in H_{W}^{s}(M)$ iff $W^{-1}f\in H_{\omega}^{s}(M)$, where
$H_{\omega}^{s}(M)$ is the fractional weighted Sobolev space associated to the
Abelian form $\omega$ and the related area form. For a formal definition of
$H_{\omega}^{s}(M)$ and useful characterization of its smooth elements we
refer the reader to Section 2 in [11].

In [9, 11], for a.e. flow, Forni proved the existence of fundamental invariant
distributions on $H_{W}^{s}(M)$ which are responsible for the degree of
smoothness of the solution of $\eqref{eq:coheqfl}$ for $f\in H_{W}^{s}(M)$.
Roughly speaking, Forni’s distributions are related to the Lyapunov exponents
of the Kontsevich-Zorich cocycle on the absolute $1$-cohomological bundle. If
all Forni’s distributions at $f\in H_{W}^{s}(M)$ are zero then the solution
$u\in H_{\omega}^{s^{\prime}}(M)$ for some $s^{\prime}<s$ with $s^{\prime}$
not too far away from $s$. Forni’s beautiful approach is based on a very deep
analysis of the Kontsevich-Zorich cocycle acting on various kinds of abstract
objects related to translation surfaces. An alternative approach to
constructing invariant distributions was also presented by Bufetov in [4]. A
different approach, based on moving to a special representation and studying
renormalization behavior for piecewise smooth functions over interval exchange
translations, was initiated by Marmi-Moussa-Yoccoz in [21] and later developed
in [23, 17, 14].

The main goal of this article (and the subsequent one [15]) is to go beyond
the case of a minimal flow on the whole surface $M$ and beyond the case of
functions $f$ belonging to a weighted Sobolev space. We deal with locally
Hamiltonian flows restricted to any minimal component and $f:M\to{\mathbb{C}}$
is any smooth function. The study of locally Hamiltonian flows in such a
context gives a rise to new invariant distributions, which, unlike Forni’s
distributions, are local in nature. The first two new families of such
invariant distributions, defined in Section 1.4, read local behaviour of
functions around saddle points. The last family, which is a counterpart of
Forni’s distributions, is defined in [15] using renomalization techniques
inspired by the approach developed in [21, 23, 17, 14].

All three families of invariant distributions affect the degree of smoothness
of the solution of the cohomological equation. However, in the present article
we focus only on the first two families and the main results of the paper are
contained in Theorems 1.1, 1.2 and 1.3. The methods for studying their effect
on the degree of smoothness are purely analytical, in contrast to the
dynamical arguments left to [15], where the last family play a central role.

###  1.1. Special representation and IETs

Locally Hamiltonian flows restricted to their minimal components are
represented as special flows over interval exchange transformations. Let us
consider a restriction of a locally Hamiltonian flow $\psi_{\mathbb{R}}$ on
$M$ to its minimal component $M^{\prime}\subset M$. Let $I\subset M^{\prime}$
be any transversal smooth curve with its standard parametrization
$\gamma:[0,|I|]\to I$, i.e. $\int_{0}^{\gamma(s)}\eta=s$ for $s\in[0,|I|]$,
where $\eta$ is the closed $1$-form given by $\eta=\frac{\partial H}{\partial
x}dx+\frac{\partial H}{\partial y}dy$ in local coordinates. By minimality, $I$
is a global transversal and the first return map $T:I\to I$ is an interval
exchange transformation (IET) in standard coordinates on $I$. We will denote
by $I_{\alpha}$, $\alpha\in\mathcal{A}$ the subintervals translated by $T$. In
order to minimize the number of exchanged intervals, we will always assume
that each end of $I$ is the first meeting point of a separatrix (that is not a
saddle connection) emanating by a fixed point (incoming or outgoing) with the
set $I$.

Let $\tau:I\to{\mathbb{R}}_{>0}\cup\\{+\infty\\}$ be the first return time
map. Then each point in
$M^{\prime}\setminus(\mathrm{Sd}(\psi_{\mathbb{R}})\cup\operatorname{SL}(\psi_{{\mathbb{R}}}))$
is uniquely represented as $\psi_{t}x$ for some $x\in I$ and $0\leq
t<\tau(x)$. The function $\tau:I\rightarrow{\mathbb{R}}_{>0}\cup\\{+\infty\\}$
is smooth on the interior of any exchanged interval and has _singularities_ at
discontinuities of $T$. Each such discontinuity is the first hitting point
(forward or backward) of a separatrix emanated by a saddle with the curve
(interval) $I$. Moreover, degenerate saddles ($m_{\sigma}>2$) of
$\psi_{\mathbb{R}}$ are responsible for the appearance of singularities of
_polynomial type_ and simple saddles ($m_{\sigma}=2$) are responsible for the
appearance of _logarithmic type_ singularities.

###  1.2. Two crucial operators and two cohomological equations

For any smooth observable $f:M\to{\mathbb{C}}$ we deal with the corresponding
map $\varphi_{f}:I\to{\mathbb{C}}\cup\\{\infty\\}$ given by

$\varphi_{f}(x)=\int_{0}^{\tau(x)}f(\psi_{t}x)dt.$

The function $\varphi_{f}$ is smooth on the interior of any interval
$I_{\alpha}$ and can have polynomial or logarithmic type singularities at
discontinuities of $T$ depending on the vanishing of some invariant
distributions on $f$ defined in [14] and based on partial derivatives of $f$
at saddles in $M^{\prime}$. One of the aim of this paper is a deeper
understanding of the operator $f\mapsto\varphi_{f}$ on the kernel of all
invariant distributions coming from [14]. Then $\varphi_{f}$ has no
singularities, but its derivatives can have. In this paper we define an
infinite sequence of new (a little bit more sophisticated) invariant
distributions (based on partial derivatives at saddles) which are responsible
for understanding the regularity of $\varphi_{f}$.

For solving the cohomological equation (1.1) we also need to study another
operator $g\mapsto u_{g,f}$. Suppose that $g:I\to{\mathbb{C}}$ is a smooth
solution (at least continuous) of the another cohomological equation

(1.2) $g(Tx)-g(x)=\varphi_{f}(x)\text{ on }I.$

This is an obvious necessary condition for the existence of a smooth solution
of the equation (1.1). Indeed, if $u$ is smooth and satisfies (1.1), then the
map $g:I\to{\mathbb{C}}$ defined as the restriction of $u$ to $I$ is smooth
and satisfies (1.2). A natural problem is: when is this also a sufficient
condition?

Suppose that $g:I\to{\mathbb{C}}$ is a smooth solution of (1.2). Then the
corresponding solution
$u_{g,f}:M^{\prime}\setminus(\mathrm{Sd}(\psi_{\mathbb{R}})\cup\operatorname{SL}(\psi_{{\mathbb{R}}}))\to{\mathbb{C}}$
is defined as follows. If $\psi_{t}x\in I$ for some $t\in{\mathbb{R}}$ then

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

By the proof of Lemma 6.3 in [16], the function $u_{g,f}$ is well defined on
$M^{\prime}\setminus(\mathrm{Sd}(\psi_{\mathbb{R}})\cup\operatorname{SL}(\psi_{{\mathbb{R}}}))$.
Moreover, if $M$ is a $C^{\infty}$-surface, $\psi_{\mathbb{R}}$ is a
$C^{\infty}$-flow and $f$ is a $C^{\infty}$-observable, then $u_{g,f}$ is as
regular as $g$. Indeed, by the absence of saddle connections joining different
saddles, for every $x_{0}\in
M^{\prime}\setminus(\mathrm{Sd}(\psi_{\mathbb{R}})\cup\operatorname{SL}(\psi_{{\mathbb{R}}}))$
there exists $t_{0}\in{\mathbb{R}}$ such that
$\psi_{t_{0}}x_{0}\in\operatorname{Int}I$. For simplicity, assume that
$t_{0}\leq 0$. Then choose $\varepsilon>0$ such that
$[\psi_{t_{0}}x_{0}-\varepsilon,\psi_{t_{0}}x_{0}+\varepsilon]\subset\operatorname{Int}I$
and let

(1.3) $R(x_{0},t_{0},\varepsilon):=\bigcup_{-\varepsilon\leq t\leq-
t_{0}+\varepsilon}\psi_{t}[\psi_{t_{0}}x_{0}-\varepsilon,\psi_{t_{0}}x_{0}+\varepsilon].$

If $\varepsilon>0$ is small enough then
$\nu:[-\varepsilon,-t_{0}+\varepsilon]\times[\psi_{t_{0}}x_{0}-\varepsilon,\psi_{t_{0}}x_{0}+\varepsilon]\to
R(x_{0},t_{0},\varepsilon)$ given by $\nu(t,x)=\psi_{t}x$ is a
$C^{\infty}$-diffeomorphism. Moreover,

$u_{g,f}\circ\nu(t,x)=g(x)-\int_{0}^{t}f\circ\nu(s-t,x)ds=g(x)+\int_{0}^{t}f\circ\nu(s,x)ds.$

It follows that the regularity of $u_{g,f}$ restricted to
$R(x_{0},t_{0},\varepsilon)$ coincides with the regularity of $g$ on
$[\psi_{t_{0}}x_{0}-\varepsilon,\psi_{t_{0}}x_{0}+\varepsilon]$. Since
$x_{0}\in\operatorname{Int}R(x_{0},t_{0},\varepsilon)$, we obtain our claim.

However, the solution $u_{g,f}$ of the cohomological equation is not fully
satisfactory because it is defined only on an open (dense) subset of the
minimal component, without fixed points and saddle loops. Our main goal is to
find necessary and sufficient conditions for the existence of a smooth
solution (of the cohomological equation) defined over all of $M^{\prime}$.
More precisely, instead of $M^{\prime}$ we will study smooth solutions defined
on the end compactification $M_{e}^{\prime}$ of
$M^{\prime}\setminus\mathrm{Sd}(\psi_{\mathbb{R}})$. Roughly speaking, if a
saddle $\sigma$ emanates $l\geq 2$ loops, then $\sigma$ is the $l$-fold end of
the set $M^{\prime}\setminus\mathrm{Sd}(\psi_{\mathbb{R}})$. For this reason,
$\sigma$ splits in $M^{\prime}_{e}$ into $l$ different end points
$\sigma_{1},\ldots,\sigma_{l}$, see Figure 1.

Figure 1. The minimal component $M^{\prime}$ before and after separation
procedure.

We will look for smooth solutions $u:M^{\prime}_{e}\to{\mathbb{C}}$ of (1.1).
If a smooth solution $u:M^{\prime}_{e}\to{\mathbb{C}}$ exists then it is
smooth in a neighborhood (in $M^{\prime}_{e}$) of any version $\sigma_{i}$ of
the saddle point $\sigma$, but it does not even have to be continuous at
$\sigma$, whenever the limits of $u$ at $\sigma$ with respect to different
neighborhood sectors (connected components) are different. Of course, if each
saddle emanates at most one saddle loop then $M^{\prime}_{e}$ coincides with
$M^{\prime}$ and the problem of regularity of
$u:M^{\prime}_{e}\to{\mathbb{C}}$ and $u:M^{\prime}\to{\mathbb{C}}$ are
equivalent.

###  1.3. Grading of smoothness

Let $M$ be a $C^{\infty}$-manifold with a boundary. For any
$n\in{\mathbb{Z}}_{\geq 0}$ and $0<a<1$ denote by $C^{n+a}(M)$ the space of
$C^{n}$-functions on $M$ such that their $n$-th derivative is $a$-Hölder. Let
$\eta:{\mathbb{R}}_{\geq 0}\to{\mathbb{R}}_{\geq 0}$ be given by
$\eta(x)=-x\log x$ for $x\in[0,e^{-1}]$ and $\eta(x)=e^{-1}$ for $x\geq
e^{-1}$. For any $n\in{\mathbb{Z}}_{\geq 0}$ denote by $C^{n+\eta}(M)$ the
space of $C^{n}$-functions on $M$ such that their $n$-th derivative is
continuous so that a positive multiple of $\eta$ is its modulus of continuity.
For every non-natural real $r>0$ we will write $C^{r}$ for $C^{\lfloor
r\rfloor+\\{r\\}}$.

Let
${\mathbb{R}}_{\eta}:=({\mathbb{R}}_{>-1}\setminus{\mathbb{Z}})\cup({\mathbb{Z}}_{\geq-1}+\\{\eta\\})$
and let $v:{\mathbb{R}}_{\eta}\to{\mathbb{R}}$ be given by $v(r)=r$ if
$r\in({\mathbb{R}}_{>-1}\setminus{\mathbb{N}})$ and $v(n+\eta)=n+1$. Then
$0\leq v(r)\leq v(r^{\prime})$ iff $C^{r}\subset C^{r^{\prime}}$.

###  1.4. Invariant distributions

To solve our main problem, in the present paper we introduce a family of
invariant distributions $f\mapsto\mathfrak{d}^{k}_{\sigma,j}(f)$ for all
$\sigma\in\mathrm{Sd}(\psi_{\mathbb{R}})$, $k\geq 0$ and $0\leq j\leq
k\wedge(m_{\sigma}-2)$. Throughout the article we use the notation $x\vee
y=\max\\{x,y\\}$ and $x\wedge y=\min\\{x,y\\}$ for any pair of real numbers
$x,y$. Recall that a linear bounded functional $f\mapsto\mathfrak{D}(f)$ is an
_invariant distribution_ if $\mathfrak{D}(Xu)=0$ for any $u\in C^{\infty}(M)$.
The distributions are defined locally around saddles and are obstructions to
the existence of smooth solutions to the cohomological equation. The invariant
distributions $\mathfrak{d}^{k}_{\sigma,j}$ are defined based on the higher-
order partial derivatives of the function $f$ in saddles or they are linear
combinations of partial derivatives (if $k>m_{\sigma}-2$). We also introduce
alternative versions of such invariant distributions, i.e.
$f\mapsto\mathfrak{C}^{k}_{\sigma,l}(f)$ for $0\leq l<2m_{\sigma}$, which have
a more geometric interpretation, and generate the same space of invariant
distributions as $\mathfrak{d}^{k}_{\sigma,j}$.

Suppose that $\sigma\in\mathrm{Sd}(\psi_{\mathbb{R}})$ is a saddle of
multiplicity $m_{\sigma}\geq 2$. Fix a singular chart $(x,y)$ in a
neighborhood $U_{\sigma}$ of $\sigma$. Then the local Hamiltonian is of the
form $H(x,y)=\Im(x+\iota y)^{m_{\sigma}}$ and the
$\psi_{\mathbb{R}}$-invariant area-measure is $d\mu=V(x,y)dx\wedge dy$, where
$V$ is positive and smooth. Then for every $k\geq 0$ and $0\leq j\leq
k\wedge(m_{\sigma}-2)$ with $j\neq
k-(m_{\sigma}-1)\operatorname{mod}m_{\sigma}$ we define the functional
$\mathfrak{d}^{k}_{\sigma,j}:C^{k}(M)\to{\mathbb{C}}$ as follows:

(1.4) $\mathfrak{d}^{k}_{\sigma,j}(f)=\sum_{0\leq
n\leq\frac{k-j}{m_{\sigma}}}\frac{\binom{k}{j+nm_{\sigma}}\binom{\frac{(m_{\sigma}-1)-j}{m_{\sigma}}-1}{n}}{\binom{\frac{(k-j)-(m_{\sigma}-1)}{m_{\sigma}}}{n}}\frac{\partial^{k}(f\cdot
V)}{\partial z^{j+nm_{\sigma}}\partial\overline{z}^{k-j-nm_{\sigma}}}(0,0).$

Note that for $k\leq m_{\sigma}-2$ we have
$\mathfrak{d}^{k}_{\sigma,j}(f)=\binom{k}{j}\frac{\partial^{k}(f\cdot
V)}{\partial z^{j}\partial\overline{z}^{k-j}}(0,0)$, so
$\mathfrak{d}^{k}_{\sigma,j}$ are essentially distributions defined already in
[14] to study the deviation spectrum of Birkhoff integrals of $f$. Let us
mention that the non-vanishing of any of these distributions is an obstacle to
the existence of any solution of the cohomological equation (even measurable).
The distributions $\mathfrak{d}^{k}_{\sigma,j}$ for $k\geq m_{\sigma}-1$ are
responsible for determining the regularity of the solution if we already know
that equation (1.1) has a smooth solution. To explain this relation in a
better way, we need to introduce another family of distributions
$\mathfrak{C}^{k}_{\sigma,l}:C^{k}(M)\to{\mathbb{C}}$ for $0\leq
l<2m_{\sigma}$,

(1.5) $\mathfrak{C}^{k}_{\sigma,l}(f):=\\!\sum_{\begin{subarray}{c}0\leq i\leq
k\\\ i\neq m_{\sigma}-1\operatorname{mod}m_{\sigma}\\\ i\neq
k-(m_{\sigma}-1)\operatorname{mod}m_{\sigma}\end{subarray}}\\!\theta_{\sigma}^{l(2i-k)}\binom{k}{i}\mathfrak{B}(\tfrac{(m_{\sigma}-1)-i}{m_{\sigma}},\tfrac{(m_{\sigma}-1)-k+i}{m_{\sigma}})\frac{\partial^{k}(f\cdot
V)}{\partial z^{i}\partial\overline{z}^{k-i}}(0,0),$

where $\theta_{\sigma}$ is the principal $2m_{\sigma}$-th root of unity and
the (beta-like) function $\mathfrak{B}(x,y)$ is defined for any pair $x,y$ of
real numbers such that $x,y\notin{\mathbb{Z}}$ as follows

$\mathfrak{B}(x,y)=\frac{\pi
e^{\iota\frac{\pi}{2}(y-x)}}{2^{x+y-2}}\frac{\Gamma(x+y-1)}{\Gamma(x)\Gamma(y)},$

where we adopt the convention $\Gamma(0)=1$ and $\Gamma(-n)=1/(-1)^{n}n!$.

For any saddle $\sigma\in\mathrm{Sd}(\psi_{\mathbb{R}})$ its neighbourhood
splits into $2m_{\sigma}$ angular sectors (see (1.7)) bounded by separatrices
emanated from $\sigma$. As we will see in Corollary 4.8, the values of
$\mathfrak{C}^{k}_{\sigma,l}(f)$ for $k\geq 0$, roughly speaking, measure how
the appearance of orbits (of $\psi_{\mathbb{R}}$) in the $l$-th angular sector
affects on the value of ergodic integrals of the function $f$. Unfortunately,
the geometric interpretation of the value of the distributions
$\mathfrak{d}^{k}_{\sigma,j}$ is difficult to understand.

The functionals $\mathfrak{C}^{k}_{\sigma,l}$ for $0\leq l<2m_{\sigma}$ are
not linearly independent, in contrast to the family of functionals
$\mathfrak{d}^{k}_{\sigma,j}$. Indeed,

(1.6)
$\displaystyle\mathfrak{C}^{k}_{\sigma,l+m_{\sigma}}=(-1)^{k}\mathfrak{C}^{k}_{\sigma,l}\text{
for }0\leq l<m_{\sigma}\text{ and}$ $\displaystyle\sum_{0\leq
l<2m_{\sigma}}\theta_{\sigma}^{(k-2j)l}\mathfrak{C}^{k}_{\sigma,l}=0\text{ if
}j=m_{\sigma}-1\text{ or }j=k-(m_{\sigma}-1).$

The element of ${\mathbb{R}}_{\eta}$ given by

$\mathfrak{e}(\mathfrak{d}^{k}_{\sigma,j})=\mathfrak{e}(\mathfrak{C}^{k}_{\sigma,l})=\mathfrak{e}(\sigma,k)=\left\\{\begin{array}[]{cl}\frac{k-(m_{\sigma}-2)}{m_{\sigma}}&\text{
if }\frac{k-(m_{\sigma}-2)}{m_{\sigma}}\notin{\mathbb{Z}},\\\
\frac{k-2(m_{\sigma}-1)}{m_{\sigma}}+\eta&\text{ if
}\frac{k-(m_{\sigma}-2)}{m_{\sigma}}\in{\mathbb{Z}}.\end{array}\right.$

is called the _exponent_ of $\mathfrak{d}^{k}_{\sigma,j}$ or
$\mathfrak{C}^{k}_{\sigma,l}$. Then

$\mathfrak{o}(\mathfrak{d}^{k}_{\sigma,j})=\mathfrak{o}(\mathfrak{C}^{k}_{\sigma,l})=\mathfrak{o}(\sigma,k):=v(\mathfrak{e}(\sigma,k))=\frac{k-(m_{\sigma}-2)}{m_{\sigma}}$

is called the _order_ of $\mathfrak{d}^{k}_{\sigma,j}$ or
$\mathfrak{C}^{k}_{\sigma,l}$. Finally, let

$\displaystyle\widehat{\mathfrak{e}}(\mathfrak{d}^{k}_{\sigma,j})$
$\displaystyle=\widehat{\mathfrak{e}}(\sigma,k)=k-(m_{\sigma}-1)+\eta\text{
and}$ $\displaystyle\widehat{\mathfrak{o}}(\mathfrak{d}^{k}_{\sigma,j})$
$\displaystyle=\widehat{\mathfrak{o}}(\sigma,k)=v(\widehat{\mathfrak{e}}(\mathfrak{d}^{k}_{\sigma,j}))=k-(m_{\sigma}-2).$

Referring to Section 1.3, $\mathfrak{e}$ and $\widehat{\mathfrak{e}}$ should
be regarded as the degree of (modified Hölder) regularity in the scale
introduced in Section 1.3, while $\mathfrak{o}$ and $\widehat{\mathfrak{o}}$
should be regarded as the numerical values of the degrees of regularity that
are easy to compare.

For any saddle $\sigma\in\mathrm{Sd}(\psi_{\mathbb{R}})$ its (singular)
neighbourhood $U_{\sigma}$ splits into $2m_{\sigma}$ (angular) sectors bounded
by separatrices emanated from $\sigma$. In singular coordinates $z=(x,y)$ they
are of the form

(1.7) $U_{\sigma,l}:=\\{z\in U_{\sigma}:\operatorname{Arg}z\in(\tfrac{\pi
l}{m_{\sigma}},\tfrac{\pi(l+1)}{m_{\sigma}})\\}\text{ for }0\leq
l<2m_{\sigma}.$

Each such sector is either included in a minimal component $M^{\prime}$ of
$\psi_{\mathbb{R}}$ or is disjoint from $M^{\prime}$. In the problem of
studying the regularity of the solutions of the cohomological equation, only
non-zero values of invariant distributions $\mathfrak{C}^{k}_{\sigma,l}(f)$
such that $U_{\sigma,l}\cap M^{\prime}\neq\emptyset$ turn out to be relevant.

###  1.5. Main results

The first main theorem describes the smoothness of the function $\varphi_{f}$
depending on the values of the functionals described in Section 1.4. To
precisely describe the regularity of $\varphi_{f}$, in Section 2, for any
$n\in{\mathbb{Z}}_{\geq 0}$ and $0\leq a<1$ we introduce the space
$C^{n+\operatorname{P_{a}}}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$ (and its
geometric version $C^{n+\operatorname{P_{a}G}}$) of functions whose $n$-th
derivative has polynomial singularities of order at most $-a$ at the ends of
the intervals translated by the IET $T$. We should mention that for any
$n\in{\mathbb{N}}$ we have $C^{n+\operatorname{P_{a}}}\subset C^{(n-1)+(1-a)}$
if $0<a<1$ and $C^{n+\mathrm{P_{0}}}\subset C^{(n-1)+\eta}$.

Recall that we always assume that $M$ is a compact connected orientable
$C^{\infty}$-surface and $\psi_{\mathbb{R}}$ is a locally Hamiltonian
$C^{\infty}$-flow on $M$ with isolated fixed points and 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 $I_{\alpha}$, $\alpha\in\mathcal{A}$.

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

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

Note that

(1.8) $\max\\{k\geq 0:\exists_{\sigma\in\mathrm{Sd}(\psi_{\mathbb{R}})\cap
M^{\prime}}\mathfrak{o}(\sigma,k)<r\\}+1=\lceil mr+(m-2)\rceil.$

Denote by $\mathscr{TD}$ the set of triples
$(\sigma,k,j)\in(\mathrm{Sd}(\psi_{\mathbb{R}})\cap
M^{\prime})\times{\mathbb{Z}}_{\geq 0}\times{\mathbb{Z}}_{\geq 0}$ such that
$0\leq j\leq k\wedge(m_{\sigma}-2)$ and $j\neq
k-(m_{\sigma}-1)\operatorname{mod}m_{\sigma}$ and by $\mathscr{TC}$ the set of
triples $(\sigma,k,l)\in(\mathrm{Sd}(\psi_{\mathbb{R}})\cap
M^{\prime})\times{\mathbb{Z}}_{\geq 0}\times{\mathbb{Z}}_{\geq 0}$ such that
$0\leq l<2m_{\sigma}$ and $U_{\sigma,l}\cap M^{\prime}\neq\emptyset$.

######  Theorem 1.1.

Fix $r\geq-\frac{m-2}{m}$. Suppose that $f\in C^{{k}_{r}}(M)$ is such that
$\mathfrak{C}^{k}_{\sigma,l}(f)=0$ for all $(\sigma,k,l)\in\mathscr{TC}$ such
that $\mathfrak{o}(\mathfrak{C}^{k}_{\sigma,l})<r$. Then $\varphi_{f}\in
C^{n+\operatorname{P_{a}G}}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$ with
$n=\lceil r\rceil$ and $a=\lceil r\rceil-r$. Moreover, the operator

$C^{{k}_{r}}(M)\cap\bigcap_{\begin{subarray}{c}(\sigma,k,l)\in\mathscr{TC}\\\
\mathfrak{o}(\mathfrak{C}^{k}_{\sigma,l})<r\end{subarray}}\ker(\mathfrak{C}^{k}_{\sigma,l})\ni
f\mapsto\varphi_{f}\in
C^{n+\operatorname{P_{a}G}}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$

is bounded.

This result provides a descending filtration of the space
$\Phi^{k}:=\\{\varphi_{f}:f\in C^{k}(M)\\}$,
$k\in{\mathbb{N}}\cup\\{\infty\\}$ that is the basis for proving a spectral
theorem (in [15]) for the so-called Kontsevich-Zorich cocycle on $\Phi^{k}$.
Using renormalization techniques, the aforementioned spectral result allows
understanding the regularity of the solution of the cohomological equation
(1.2) (see also [15]) for a.e. IET $T$.

The second main theorem solves the problem regarding the regularity of the
solutions of (1.1), provided we know the degree of smoothness for the solution
of (1.2). This result is another ingredient in the proof of the final theorem
on the regularity of the solution of the cohomological equation (1.1)
presented in [15].

######  Theorem 1.2.

Fix $r\in{\mathbb{R}}_{\eta}$ so that $v(r)>0$. Assume that $f\in
C^{{k}_{v(r)}}(M)$ is such that

  * •

$\mathfrak{d}^{k}_{\sigma,j}(f)=0$ for all $(\sigma,k,j)\in\mathscr{TD}$ with
$\widehat{\mathfrak{o}}(\mathfrak{d}^{k}_{\sigma,j})<v(r)$;

  * •

$\mathfrak{C}^{k}_{\sigma,l}(f)=0$ for all $(\sigma,k,l)\in\mathscr{TC}$ with
${\mathfrak{o}}(\mathfrak{C}^{k}_{\sigma,l})<v(r)$.

Suppose that $g\in C^{r}(I)$ is a solution of the cohomological equation
$\varphi_{f}=g\circ T-g$. Then there exists $u_{g,f}\in C^{r}(M^{\prime}_{e})$
satisfying $Xu_{g,f}=f$ on $M^{\prime}_{e}$. Moreover, there exists a constant
$C_{r}>0$ such that

$\|u_{g,f}\|_{C^{r}(M^{\prime}_{e})}\leq
C_{r}(\|g\|_{C^{r}(I)}+\|f\|_{C^{{k}_{v(r)}}(M)}).$

The following theorem states that the regularity of the solution obtained in
Theorem 1.2 is optimal.

######  Theorem 1.3 (optimal regularity).

Let $r\in{\mathbb{R}}_{\eta}$ with $v(r)>0$ and let $f\in C^{{k}_{v(r)}}(M)$.
If there exists $u\in C^{r}(M^{\prime}_{e})$ such that $Xu=f$ on
$M^{\prime}_{e}$ then

  * •

$\mathfrak{d}^{k}_{\sigma,j}(f)=0$ for all $(\sigma,k,j)\in\mathscr{TD}$ with
$\widehat{\mathfrak{o}}(\mathfrak{d}^{k}_{\sigma,j})<v(r)$;

  * •

$\mathfrak{C}^{k}_{\sigma,l}(f)=0$ for all $(\sigma,k,l)\in\mathscr{TC}$ with
${\mathfrak{o}}(\mathfrak{C}^{k}_{\sigma,l})<v(r)$.

In summary, all three main results provide an analytical background necessary
to fully solve the regularity problem of solving the cohomological equation
for locally Hamiltonian flows. The dynamical component, using mainly
renormalization techniques, the authors left to [15].

#### Discussion on the minimal case

Assume that the locally Hamiltonian flow $\psi_{\mathbb{R}}$ has no saddle
connection, so it is minimal on the whole surface $M$. Then $\mathscr{TC}$ is
the set of all triples
$(\sigma,k,l)\in\mathrm{Sd}(\psi_{\mathbb{R}})\times{\mathbb{Z}}_{\geq
0}\times{\mathbb{Z}}_{\geq 0}$ such that $0\leq l<2m_{\sigma}$, that is, there
are no restrictions on $l$ coming from the excluded sectors. It follows that
for any $k\geq 0$ the functionals $\mathfrak{C}^{k}_{\sigma,l}$ and
$\mathfrak{d}^{k}_{\sigma,j}$ generate the same space of invariant
distributions. In general, the former space is a subspace of the latter. Since
${\mathfrak{o}}(\sigma,k)<\widehat{\mathfrak{o}}(\sigma,k)$, the conditions
involving the functionals $\mathfrak{d}^{k}_{\sigma,j}$ (in Theorems 1.2 and
1.3) can be removed. Then only sectorial distributions
$\mathfrak{C}^{k}_{\sigma,l}$ matter and our main results (Theorems 1.2 and
1.3) have the following simplified form.

######  Corollary 1.4.

Suppose $\psi_{\mathbb{R}}$ is a locally Hamiltonian flow on $M$ that has no
saddle connections. Fix $r\in{\mathbb{R}}_{\eta}$ so that $v(r)>0$. Assume
that $f\in C^{{k}_{v(r)}}(M)$ and $g\in C^{r}(I)$ is a solution of the
cohomological equation $\varphi_{f}=g\circ T-g$. Then the existence of
$u_{g,f}\in C^{r}(M)$ satisfying $Xu_{g,f}=f$ is equivalent to

$\mathfrak{C}^{k}_{\sigma,l}(f)=0\text{ for all
}\sigma\in{\rm{Sd}}(\psi_{\mathbb{R}}),\ 0\leq l<m_{\sigma}\text{ and
}k<m_{\sigma}v(r)+(m_{\sigma}-2).$

Note that the above simplified version of the main results remains true also
in the presence of some saddle loops. But not in every case, only when the
excluded sectors do not affect the size of the space generated by the
sectorial distributions. This is due to the absence of linear independence of
the distribution $\mathfrak{C}^{k}_{\sigma,l}$. For example, if for each pair
$U_{l,\sigma}$, $U_{l+m_{\sigma},\sigma}$, $0\leq l<m_{\sigma}$ of sectors at
most one is excluded then, by (1.6), for any $k\geq 0$ the functionals
$\mathfrak{C}^{k}_{\sigma,l}$ and $\mathfrak{d}^{k}_{\sigma,j}$ generate the
same space of invariant distributions and Corollary 1.4 remains still true.

Let us mention that local $C^{\infty}$-solutions of cohomological equations
for flows without saddle loops around saddles were studied by Roussarie in
[26]. We should emphasize that our results are new (even for flows without
saddle loops) because they involve solutions with finite differentiability,
which causes significant technical complications. In this case, Forni has
suggested us an alternative strategy that potentially simplifies the complex
techniques used in this article.

However, the main advantage and novelty of the local tools introduced in this
article is the ability to study solutions in closed angular sectors (so-called
semi-solutions), which makes it possible to apply our tools to flows that have
saddle loops. These types of problems has not been systematically studied
before. Under an assumption that some saddles have (many) loops, for every $k$
large enough the functionals $\mathfrak{C}^{k}_{\sigma,l}$ generate smaller
space than that generated by $\mathfrak{d}^{k}_{\sigma,j}$. Then some
functionals $\mathfrak{d}^{k}_{\sigma,j}$ begin to have an independent effect
on the regularity of solutions, but their influence has less intensity than
the functionals $\mathfrak{C}^{k}_{\sigma,l}$, even though both types of
functionals (for fixed $k$) have the same order of regularity. This seems to
be a completely new phenomenon, not previously observed in the study of the
regularity of solutions to cohomological equations in parabolic dynamics.

###  1.6. Structure of the paper

The paper is organized as follows. In Section 2, we define one-parameter
family of Banach spaces of functions whose (higher order) derivatives have
polynomial singularities at the ends of intervals exchanged by an IET. We
establish their basic properties necessary in next sections of the article. In
Section 3, for any continuous function $f$ defined around a saddle, we define
three types of functions: $\varphi_{f,l}$, $\mathscr{F}_{f,l}$ and $F_{f}$.
The map $\varphi_{f,l}$ is a local version of the function $\varphi_{f}$
defined in Section 1.2 and is necessary to study the local behavior of
$\varphi_{f}$ near the ends of intervals exchanged by an IET. The map $F_{f}$
is (in a sense) a local solution to the cohomological equation $Xu=f$ in open
angular sectors $U_{\sigma,l}$ around the saddle. The map $\mathscr{F}_{f,l}$
is a covering of $F_{f}$ and is a technical tool for showing basic properties
of the other two. In Section 3, we prove basic properties of
$\mathscr{F}_{f,l}$, which are used to understand the behavior of $F_{f}$ on
open angular sectors $U_{\sigma,l}$. In Section 4, using the tools introduced
in Section 3, we determine precisely the form of $\varphi_{f,l}$ and
$\mathscr{F}_{f,l}$ on some angular sectors. Both of these results are then
used to prove that $F_{f}$ has a smooth extension to closed angular sectors
$\overline{U_{\sigma,l}}$ and to establish necessary and sufficient conditions
(expressed in the language of local invariant distributions) for such an
extension. Finally, in Section 5, we use the contents of all previous sections
to prove Theorem 1.1, 1.2 and 1.3.

##  2\. Functions whose (higher order) derivatives have polynomial
singularities

In this section we introduce one-parameter family of Banach spaces of
functions whose (higher order) derivatives have polynomial singularities at
the ends of intervals exchanged by an IET. The new spaces simply generalize
Banach spaces $\operatorname{P_{a}}$ studied in [14].

###  2.1. Space $C^{n+\operatorname{P_{a}}}$

Fix $0\leq a<1$ and an IET $T:I\to I$ satisfying so called Keane’s condition.
Denote by $I_{\alpha}=[l_{\alpha},r_{\alpha})$, $\alpha\in\mathcal{A}$ all
subintervals exchanged by $T$. The IET is determined by a pair
$(\pi,\lambda)$, where
$\lambda=(\lambda_{\alpha})_{\alpha\in\mathcal{A}}\in{\mathbb{R}}_{>0}^{\mathcal{A}}$
is the vector of lengths of exchanged intervals, i.e.
$\lambda_{\alpha}=r_{\alpha}-l_{\alpha}$, and $\pi=(\pi_{0},\pi_{1})$ is the
pair of bijections $\pi_{\varepsilon}:\mathcal{A}\to\\{1,\ldots,d\\}$ for
$\varepsilon=0,1$ ($d=|\mathcal{A}|$ is the number of exchanged intervals)
such that $\pi_{0}(\alpha)$ is the item of $I_{\alpha}$ before the translation
and $\pi_{1}(\alpha)$ after the translation.

For every $\alpha\in\mathcal{A}$, denote by $m_{\alpha}$ the middle point of
$I_{\alpha}$, i.e. $m_{\alpha}=(l_{\alpha}+r_{\alpha})/2$. For every
$\varphi\in
C^{1}(\sqcup_{\alpha\in\mathcal{A}}\operatorname{Int}I_{\alpha},{\mathbb{C}})$
let us consider

$\displaystyle p_{a}(\varphi):=$
$\displaystyle\max_{\alpha\in\mathcal{A}}\Big{\\{}\sup_{x\in(l_{\alpha},m_{\alpha}]}|D\varphi(x)(x-l_{\alpha})^{1+a}|,\sup_{x\in[m_{\alpha},r_{\alpha})}|D\varphi(x)(r_{\alpha}-x)^{1+a}|\Big{\\}}.$

######  Definition 1.

For every integer $n\geq 0$, we denote by
$C^{n+\operatorname{P_{a}}}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$ the
space of functions $\varphi\in
C^{n+1}(\sqcup_{\alpha\in\mathcal{A}}\operatorname{Int}I_{\alpha},{\mathbb{C}})$
such that $p_{a}(D^{n}\varphi)<+\infty$ and for every $\alpha\in\mathcal{A}$
the limits

$\displaystyle C_{\alpha,n}^{a,+}(\varphi)$
$\displaystyle=(-1)^{n}C_{\alpha}^{+}(D^{n}\varphi):=(-1)^{n+1}\lim_{x\searrow
l_{\alpha}}D^{n+1}\varphi(x)(x-l_{\alpha})^{1+a},$ $\displaystyle
C_{\alpha,n}^{a,-}(\varphi)$
$\displaystyle=C_{\alpha}^{-}(D^{n}\varphi):=\lim_{x\nearrow
r_{\alpha}}D^{n+1}\varphi(x)(r_{\alpha}-x)^{1+a}$

exist. We denote by
$C^{n+\operatorname{P_{a}G}}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})\subset
C^{n+\operatorname{P_{a}}}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$ the
subset (the union of subspaces) of functions $\varphi\in
C^{n+\operatorname{P_{a}}}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$ of
_geometric type_ , i.e. such that

$C_{\pi_{0}^{-1}(d),n}^{a,-}(\varphi)\cdot
C_{\pi_{1}^{-1}(d),n}^{a,-}(\varphi)=0\quad\text{and}\quad
C_{\pi_{0}^{-1}(1),n}^{a,+}(\varphi)\cdot
C_{\pi_{1}^{-1}(1),n}^{a,+}(\varphi)=0.$

For every $0\leq a<1$ and every integer $n\geq 0$, by Lemma 4.3 in [14], if
$\varphi\in
C^{n+\operatorname{P_{a}}}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$ then
$D^{n}\varphi\in L^{1}(I)$. Let us consider the norm on
$C^{n+\operatorname{P_{a}}}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$ given by

(2.1)
$\|\varphi\|_{C^{n+\operatorname{P_{a}}}}:=\sum_{k=0}^{n}\|D^{k}\varphi\|_{L^{1}(I)}+p_{a}(D^{n}\varphi).$

Recall that, by Lemma 4.2 in [14], for $n=0$ the space
$C^{n+\operatorname{P_{a}}}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$ equipped
with the norm $\|\,\cdot\,\|_{C^{n+\operatorname{P_{a}}}}$ is Banach. This
gives Banach’s condition also for all $n\geq 1$. Moreover,
$C^{n+\operatorname{P_{a}G}}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$ is a
closed subspace of
$C^{n+\operatorname{P_{a}}}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$ for any
$n\geq 0$.

Let $\eta:{\mathbb{R}}_{\geq 0}\to{\mathbb{R}}_{\geq 0}$ be given by
$\eta(x)=-x\log x$ for $x\in[0,e^{-1}]$ and $\eta(x)=e^{-1}$ for $x\geq
e^{-1}$. Denote by $C^{\eta}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$ the
space of functions $f:I\to{\mathbb{C}}$ such that

$|f|_{C^{\eta}}:=\max_{\alpha\in\mathcal{A}}\sup\left\\{\frac{|f(x)-f(y)|}{\eta(|x-y|)}:x,y\in\operatorname{Int}I_{\alpha},x\neq
y\right\\}<+\infty.$

Then $C^{\eta}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$ equipped with the
norm $\|f\|_{C^{\eta}}=\|f\|_{L^{1}}+|f|_{C^{\eta}}$ is a Banach space. For
every $0<a<1$ denote by $C^{a}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$ the
space of piecewise $a$-Hölder continuous functions, i.e. such that

$|f|_{C^{a}}:=\max_{\alpha\in\mathcal{A}}\sup\left\\{\frac{|f(x)-f(y)|}{|x-y|^{a}}:x,y\in\operatorname{Int}I_{\alpha},x\neq
y\right\\}<+\infty,$

equipped with the Banach norm $\|f\|_{C^{a}}=\|f\|_{L^{1}}+|f|_{C^{a}}$.

For every $n\geq 0$ we also deal with the Banach spaces
$C^{n+\eta}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$,
$C^{n+a}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$ equipped with the norms

$\|\varphi\|_{C^{n+\eta}}=\sum_{k=0}^{n}\|D^{k}\varphi\|_{L^{1}}+|D^{n}\varphi|_{C^{\eta}},\quad\|\varphi\|_{C^{n+a}}=\sum_{k=0}^{n}\|D^{k}\varphi\|_{L^{1}}+|D^{n}\varphi|_{C^{a}},\text{
resp.}$

For every non-natural real number $r>0$ we will write $C^{r}$ for $C^{\lfloor
r\rfloor+\\{r\\}}$.

######  Remark 2.1.

In view of Lemma 4.5 in [14], for every $\varphi\in
C^{0+\operatorname{P_{a}}}(\sqcup_{\alpha\in\mathcal{A}}I_{\alpha})$ and
$x\in\operatorname{Int}I_{\alpha}$,

$\displaystyle\begin{split}|\varphi(x)|&\leq\frac{\left\|\varphi\right\|_{L^{1}}}{|I|}+p_{a}(\varphi)\Big{(}\frac{1}{a\min\\{x-l_{\alpha},r_{\alpha}-x\\}^{a}}+\frac{2^{a+2}}{a(1-a)|I_{\alpha}|^{a}}\Big{)}\text{
if }0<a<1,\\\
|\varphi(x)|&\leq\frac{\left\|\varphi\right\|_{L^{1}}}{|I|}+p_{a}(\varphi)\Big{(}\log\frac{|I_{\alpha}|}{2\min\\{x-l_{\alpha},r_{\alpha}-x\\}}+2\Big{)}\text{
if }a=0.\end{split}$

It follows that if $\varphi\in C^{n+\operatorname{P_{a}}}$ for some $n\geq 1$,
then

$\displaystyle\varphi\in C^{(n-1)+(1-a)}$ $\displaystyle\text{ with
}\|\varphi\|_{C^{(n-1)+(1-a)}}\leq\frac{2^{2+a}\max_{\alpha\in\mathcal{A}}|I_{\alpha}|^{1-2a}}{a(1-a)}\|\varphi\|_{C^{n+\operatorname{P_{a}}}}\text{
if }0<a<1,$ $\displaystyle\varphi\in C^{(n-1)+\eta}$ $\displaystyle\text{ with
}\|\varphi\|_{C^{(n-1)+\eta}}\leq(|I|^{-1}+3)\|\varphi\|_{C^{n+\operatorname{P_{a}}}}\text{
if }a=0.$

######  Remark 2.2.

For any $0\leq a<1$ and any interval $J\subset I_{\alpha}$ let

$p_{a}(\varphi,J):=\sup\\{(\min\\{x-l_{\alpha},r_{\alpha}-x\\})^{1+a}|\varphi^{\prime}(x)|:x\in
J\\}.$

Moreover, for any $n\geq 0$ let

$\|\varphi\|_{C^{n+\operatorname{P_{a}}}(J)}:=\sum_{k=0}^{n}\|D^{k}\varphi\|_{L^{1}(J)}+p_{a}(D^{n}\varphi,J).$

In view of Lemma 4.3 in [14], if $J=(l_{\alpha},l_{\alpha}+\varepsilon]$ or
$J=[r_{\alpha}-\varepsilon,r_{\alpha})$ with $\varepsilon\leq|I_{\alpha}|/2$,
then for every $0\leq b<1$ we have

(2.2)
$p_{b}(\varphi,J)\leq\|\varphi^{\prime}\|_{L^{1}(J)}+\frac{p_{a}(\varphi^{\prime},J)}{1-a}.$

Let $n,n^{\prime}\geq 0$ and $0\leq a,a^{\prime}<1$ such that $n-a\leq
n^{\prime}-a^{\prime}$. In view of (2.2),
$C^{n^{\prime}+\mathrm{P_{a^{\prime}}}}\subset C^{n+\operatorname{P_{a}}}$ and
$\|\varphi\|_{C^{n+\operatorname{P_{a}}}(J)}\leq\frac{1}{1-a^{\prime}}\|\varphi\|_{C^{n^{\prime}+\mathrm{P_{a^{\prime}}}}(J)}$.

If $J\subset[l_{\alpha}+\varepsilon,r_{\alpha}-\varepsilon]$ for some
$\varepsilon>0$ then for any $n\geq 0$ and $0\leq a<1$,
$\|\varphi\|_{C^{n+\operatorname{P_{a}}}(J)}\leq\|\varphi\|_{C^{n+1}(J)}$.

##  3\. Local analysis around saddles

In this section, for any continuous function $f$ defined around a saddle
$\sigma$, we introduce three types of functions: $\varphi_{f,l}$,
$\mathscr{F}_{f,l}$ for $0\leq l<m_{\sigma}$ (related to the behavior of the
$f$ function on angular sectors bounded by outgoing separatrices) and $F_{f}$.
The map $\varphi_{f,l}$ is a local version of the function $\varphi_{f}$
defined in Section 1.2 and it is responsible for the behavior of the ergodic
integrals for $f$ along the flow orbits as they pass around the saddle and
through the $l$-th angular sector. The map $F_{f}$ is an ersatz of a local
solution to the cohomological equation $Xu=f$ around the saddle $\sigma$,
which is defined only on the union of open angular sectors. The map
$\mathscr{F}_{f,l}$ is a covering of $F_{f}$ and is a technical tool for
showing basic properties of the other two. In Section 3, we prove basic
properties of $\mathscr{F}_{f,l}$, which are used to understand the behavior
of $F_{f}$ on $U_{\sigma,l}$. In Section 4, using the tools introduced in
Section 3, we determine precisely the form of $\varphi_{f,l}$ and
$\mathscr{F}_{f,l}$ on some angular sectors.

To analyse these three classes of functions we will use a change of variables
around $\sigma$ by the covering map $z\mapsto z^{m_{\sigma}}$. This is a very
useful strategy because in the new coordinates the trajectories of the locally
Hamiltonian flow are straight lines. Furthermore, we will use the Taylor
expansion of the function $f$ at the point $\sigma$ with respect to the
complex variables $z$ and $\bar{z}$. Since the operators
$f\mapsto\varphi_{f,l}$, $f\mapsto\mathscr{F}_{f,l}$ and $f\mapsto F_{f}$ are
linear, our analysis boils down to the study of functions $f$ that are
polynomials in the variables $z$, $\bar{z}$. Moreover, such polynomials are
well-behaved for composition with the map $z\mapsto z^{m_{\sigma}}$ and its
inverse branches.

Let $m\geq 2$ be the multiplicity of a saddle point. Let
$G_{0}:{\mathbb{C}}\to{\mathbb{C}}$ be the principal branch of the $m$-th root
$G_{0}(re^{\iota t})=r^{1/m}e^{\iota t/m}$ if $t\in[0,2\pi$) and let $\theta$
and $\theta_{0}$ be the principal $m$-th and $2m$-th root of unity
respectively. Then $G=G_{l}:{\mathbb{C}}\to{\mathbb{C}}$ given by
$G_{l}=\theta^{l}G_{0}$ for $0\leq l<m$ is the $l$-th branch of the $m$-th
root.

Let $f:\mathcal{D}\to{\mathbb{C}}$ be a bounded Borel map where
$\mathcal{D}=\mathcal{D}_{m}$ is the pre-image of the square
$[-1,1]\times[-1,1]$ by the map
${\mathbb{C}}\ni\omega\mapsto\omega^{m}\in{\mathbb{C}}$. We will usually treat
$f$ as a function depending on a pair of complex variables
$(\omega,\bar{\omega})$. The purpose of this and the next section is to
understand the properties of two types of functions
$\varphi_{f,l}:[-1,0)\cup(0,1]\to{\mathbb{C}}$ and
$\mathscr{F}_{f,l}:[-1,1]^{2}\setminus([0,1]\times\\{0\\})\to{\mathbb{C}}$ for
$0\leq l<m$ associated with $f$, which are crucial in proving the main results
of this article. They are given by

(3.1)
$\varphi_{f,l}(s)=\int_{-1}^{1}\frac{f(G_{l}(u,s))}{(u^{2}+s^{2})^{\frac{m-1}{m}}}\,du,\quad\mathscr{F}_{f,l}(u,s)=\int_{-1}^{u}\frac{f(G_{l}(v,s))}{(v^{2}+s^{2})^{\frac{m-1}{m}}}\,dv.$

Then $\varphi_{f,l}(s)=\mathscr{F}_{f,l}(1,s)$ for $s\neq 0$. We will usually
treat $\mathscr{F}_{f,l}$ as a function depending on a pair of complex
variables $(z,\bar{z})$, where $z=u+\iota s$.

For any $0\leq\alpha<\beta\leq 1$ let
$\mathcal{D}(\alpha,\beta):=\\{\omega\in\mathcal{D}\setminus\\{0\\}:\operatorname{Arg}(\omega)\in(2\pi\alpha,2\pi\beta)\\}$.
We denote its closure by $\overline{\mathcal{D}}(\alpha,\beta)$. For any
$A\subset{\mathbb{C}}$ denote by $A^{1/m}$ the pre-image of $A$ for the map
$\omega\mapsto\omega^{m}$. We will also need third type of associated function
$F_{f}:\mathcal{D}\setminus([0,1]\times\\{0\\})^{1/m}\to{\mathbb{C}}$. As
$\mathcal{D}\setminus([0,1]\times\\{0\\})^{1/m}=\bigcup_{0\leq
l<m}\mathcal{D}(\frac{l}{m},\frac{l+1}{m})$, the map $F_{f}$ is defined by

$F_{f}(\omega,\overline{\omega}):=\mathscr{F}_{f,l}(\omega^{m},{\overline{\omega}}^{m})\text{
on }\mathcal{D}(\tfrac{l}{m},\tfrac{l+1}{m}).$

Note that $F_{f\cdot V}$ is (in a sense) a local solution to the cohomological
equation $Xu=f$ in any angular sector
$\mathcal{D}(\tfrac{l}{m},\tfrac{l+1}{m})$. Indeed, since
$d\omega/dt=m\bar{\omega}^{m-1}/V$, we have

$Xu=m\Big{(}\overline{\omega}^{m-1}\frac{\partial
u}{\partial\omega}+{\omega}^{m-1}\frac{\partial
u}{\partial\overline{\omega}}\Big{)}/V.$

By definition,

$\displaystyle\frac{\partial\mathscr{F}_{f\cdot V,l}(z,\bar{z})}{\partial
z}+\frac{\partial\mathscr{F}_{f\cdot V,l}(z,\bar{z})}{\partial\bar{z}}$
$\displaystyle=\frac{\partial\mathscr{F}_{f\cdot V,l}(u,s)}{\partial u}$
$\displaystyle=\frac{(f\cdot
V)(G_{l}(u,s))}{(u^{2}+s^{2})^{\frac{m-1}{m}}}=\frac{(f\cdot
V)(G_{l}(z,\overline{z}))}{|z|^{2\frac{m-1}{m}}}.$

Then for any $\omega\in\mathcal{D}(\tfrac{l}{m},\tfrac{l+1}{m})$,

$\displaystyle XF_{f\cdot V}(\omega,\overline{\omega})$
$\displaystyle=\frac{m(\overline{\omega}^{m-1}\frac{\partial\mathscr{F}_{f\cdot
V,l}(\omega^{m},{\overline{\omega}}^{m})}{\partial\omega}+{\omega}^{m-1}\frac{\partial\mathscr{F}_{f\cdot
V,l}(\omega^{m},{\overline{\omega}}^{m})}{\partial\overline{\omega}})}{V(\omega,{\overline{\omega}})}$
$\displaystyle=\frac{m^{2}|\omega|^{2(m-1)}(\frac{\partial\mathscr{F}_{f\cdot
V,l}(\omega^{m},{\overline{\omega}}^{m})}{\partial
z}+\frac{\partial\mathscr{F}_{f\cdot
V,l}(\omega^{m},{\overline{\omega}}^{m})}{\partial\overline{z}})}{V(\omega,{\overline{\omega}})}=m^{2}f(\omega,\overline{\omega}).$

The map $F_{f}$ is well defined and smooth on every open angular sector
$\mathcal{D}(\tfrac{l}{m},\tfrac{l+1}{m})$. One of the most important
technical challenges of this article is to answer the question of when and how
the map $F_{f}$ extends smoothly into the closure
$\overline{\mathcal{D}}(\tfrac{l}{m},\tfrac{l+1}{m})$.

Some key properties of the three functions are taken in Theorems 3.12, 4.7 and
4.10. Since their proofs are very technical, long and intertwined, we precede
them with a long list of auxiliary results, which should be regarded as
intermediate steps in the proof of the main theorems.

###  3.1. Preliminary calculations

For every $r>0$ let $\mathscr{S}(r)$ be the circular sector

$\mathscr{S}(r)=\\{(u,s)\neq(0,0):u\leq r|s|\\}.$

For any $0<s\leq 1$ and any $a\in{\mathbb{R}}$ let

$\langle
s\rangle^{a}=\left\\{\begin{array}[]{ccl}\frac{s^{a}-1}{-a}+1&\text{if}&a<0,\\\
1-\log s&\text{if}&a=0,\\\ 1&\text{if}&a>0.\end{array}\right.$

######  Remark 3.1.

Note that

  * •

for any $0<s\leq 1$ and any pair of real number $a\leq b$ we have $\langle
s\rangle^{a}\geq\langle s\rangle^{b}$;

  * •

for any $a\geq 1$ we have $s^{-a}/a\leq\langle s\rangle^{-a}\leq s^{-a}$;

  * •

for any $0<a\leq 1$ we have $s^{-a}\leq\langle s\rangle^{-a}\leq s^{-a}/a$;

  * •

for any $m\geq 1$ and $a\in{\mathbb{R}}$ we have $\langle s^{m}\rangle^{a}\leq
m\langle s\rangle^{am}$.

######  Lemma 3.2.

For every $a\in{\mathbb{R}}$ and $r>0$ there exist $C_{a},C_{a,r}>0$ such that

(3.2) $\int_{-1}^{u}\frac{1}{(v^{2}+s^{2})^{a}}\,dv\leq
C_{a}\langle|s|\rangle^{1-2a}\text{ for all }s\in[-1,1]\setminus\\{0\\}$

and

(3.3) $\int_{-1}^{u}\frac{1}{(v^{2}+s^{2})^{a}}\,dv\leq
C_{a,r}\langle\sqrt{\tfrac{u^{2}+s^{2}}{2}}\rangle^{1-2a}\text{ for all
}(u,s)\in[-1,1]^{2}\cap\mathscr{S}(r).$

If $f:[-1,1]^{2}\to{\mathbb{C}}$ is continuous at $(0,0)$, $f(0,0)=0$ and
$a\geq 1/2$ then

(3.4)
$\int_{-1}^{u}\frac{f(v,s)}{(v^{2}+s^{2})^{a}}\,dv=o(\langle|s|\rangle^{1-2a}).$

###### Proof.

Case 1. Suppose that $a>1/2$. If $s\neq 0$ then

$\int_{-1}^{u}\frac{1}{(v^{2}+s^{2})^{a}}\,dv=|s|^{1-2a}\int_{-1/|s|}^{u/|s|}\frac{1}{(t^{2}+1)^{a}}\,dt\leq|s|^{1-2a}\int_{-\infty}^{+\infty}\frac{1}{(t^{2}+1)^{a}}\,dt,$

which gives (3.2).

If $s=0$ and $u<0$ then

(3.5)
$\int_{-1}^{u}\frac{1}{(v^{2}+s^{2})^{a}}\,dv=\int_{|u|}^{1}v^{-2a}\,dv=\frac{1}{2a-1}(|u|^{1-2a}-1)\leq\frac{1}{2a-1}|u|^{1-2a}.$

Let us consider the function $\nu:(-\infty,+\infty)\to{\mathbb{R}}_{+}$ given
by $\nu(x):=\int_{-\infty}^{x}\frac{1}{(t^{2}+1)^{a}}\,dt$. If $s\neq 0$ then
$\int_{-1}^{u}(v^{2}+s^{2})^{-a}\,dv\leq|s|^{1-2a}\nu(u/|s|)$. As

$\lim_{x\to-\infty}\frac{\nu^{\prime}(x)}{\frac{d}{dx}(x^{2}+1)^{1/2-a}}=\lim_{x\to-\infty}\frac{(x^{2}+1)^{-a}}{(1-2a)x(x^{2}+1)^{-a-1/2}}=\frac{1}{2a-1},$

we have $\nu(x)/(x^{2}+1)^{1/2-a}\to 1/(2a-1)$ as $x\to-\infty$. Therefore
there exists $C_{a,r}>0$ such that $\nu(x)\leq
C_{a,r}(\frac{x^{2}+1}{2})^{1/2-a}$ for $x\leq r$. It follows that for every
$(u,s)\in\mathscr{S}(r)$ with $s\neq 0$,

$\int_{-1}^{u}\frac{dv}{(v^{2}+s^{2})^{a}}\leq|s|^{1-2a}\nu(u/|s|)\leq
C_{a,r}|s|^{1-2a}\big{(}\tfrac{(u/s)^{2}+1}{2}\big{)}^{1/2-a}=C_{a,r}\big{(}\tfrac{u^{2}+s^{2}}{2}\big{)}^{1/2-a},$

which (together with (3.5)) gives (3.3).

Case 2. Suppose that $a=1/2$. If $s\neq 0$ then

$\int_{-1}^{u}(v^{2}+s^{2})^{-1/2}\,dv=\log\frac{u+\sqrt{u^{2}+s^{2}}}{-1+\sqrt{1+s^{2}}}\leq-2\log\frac{s}{3},$

which gives (3.2). If $s=0$ and $u<0$ then

(3.6) $\int_{-1}^{u}(v^{2}+s^{2})^{-1/2}\,dv=-\log|u|.$

Moreover, for any $(u,s)\in\mathscr{S}(r)$ with $s\neq 0$,

$\int_{-1}^{u}(v^{2}+s^{2})^{-1/2}\,dv=\log\frac{1+\sqrt{1+s^{2}}}{-u+\sqrt{u^{2}+s^{2}}}\leq\log\frac{6(r^{2}+1)}{\sqrt{u^{2}+s^{2}}},$

which (together with (3.6)) gives (3.3).

Case 3. Suppose that $a<1/2$. If $0<a<1/2$ then

$\int_{-1}^{u}(v^{2}+s^{2})^{-a}\,dv\leq
2\int_{0}^{1}v^{-2a}\,dv=\frac{2}{1-2a}.$

If $a\leq 0$ then

$\int_{-1}^{u}(v^{2}+s^{2})^{-a}\,dv\leq 2^{1-a},$

which gives (3.2) and (3.3).

Last claim. Suppose that $f:[-1,1]^{2}\to{\mathbb{C}}$ is continuous at
$(0,0)$, $f(0,0)=0$ and $a\geq 1/2$. For any $\varepsilon>0$ choose $\delta>0$
such that $|f(v,s)|\leq\varepsilon$ if $|v|,|s|<\delta$. It follows that if
$|s|<\delta$ then

$\displaystyle\Big{|}\int_{-1}^{u}\frac{f(v,s)}{(v^{2}+s^{2})^{a}}\,dv\Big{|}$
$\displaystyle\leq\int_{-\delta}^{\delta}\frac{|f(v,s)|}{(v^{2}+s^{2})^{a}}\,dv+2\int_{\delta}^{1}\frac{\|f\|_{\sup}}{(v^{2}+s^{2})^{a}}\,dv$
$\displaystyle\leq\varepsilon\int_{-1}^{1}\frac{1}{(v^{2}+s^{2})^{a}}\,dv+2\int_{\delta}^{1}\frac{\|f\|_{\sup}}{v^{2a}}\,dv$
$\displaystyle\leq\varepsilon
C_{a}\langle|s|\rangle^{1-2a}+2\|f\|_{\sup}\langle|\delta|\rangle^{1-2a}.$

This gives (3.4). ∎

######  Remark 3.3.

For ${z}=u+\iota s$, the followings hold:

(3.7) $\displaystyle\frac{\partial}{\partial
u}=\left(\frac{\partial}{\partial{z}}+\frac{\partial}{\partial\overline{z}}\right),\quad\frac{\partial}{\partial
s}$
$\displaystyle=\iota\left(\frac{\partial}{\partial{z}}-\frac{\partial}{\partial\overline{z}}\right),$
(3.8)
$\displaystyle\frac{\partial}{\partial{z}}=\frac{1}{2}\left(\frac{\partial}{\partial
u}-\iota\frac{\partial}{\partial
s}\right),\quad\frac{\partial}{\partial\overline{z}}$
$\displaystyle=\frac{1}{2}\left(\frac{\partial}{\partial
u}+\iota\frac{\partial}{\partial s}\right).$

For any $n_{1},n_{2},a_{1},a_{2}\in{\mathbb{Z}}_{\geq 0}$ and any $f\in
C^{n}(\mathcal{D})$ ($n=n_{1}+n_{2}$), we will deal with some auxiliary
functions
$F_{n_{1},n_{2},a_{1},a_{2}},G_{n_{1},n_{2},a_{1},a_{2}}:[-1,1]^{2}\setminus([0,1]\times\\{0\\})$
given by

$\displaystyle F_{n_{1},n_{2},a_{1},a_{2}}({z},\overline{z})$
$\displaystyle=F_{n_{1},n_{2},a_{1},a_{2}}(u,s)=\frac{\partial^{n}f}{\partial{\omega}^{n_{1}}\partial\overline{\omega}^{n_{2}}}(G(u,s))\cdot{G(u,s)}^{-a_{1}}\overline{G(u,s)}^{-a_{2}},$
$\displaystyle G_{n_{1},n_{2},a_{1},a_{2}}({z},\overline{z})$
$\displaystyle=G_{n_{1},n_{2},a_{1},a_{2}}(u,s)=\int_{-1}^{u}F_{n_{1},n_{2},a_{1},a_{2}}(v,s)dv.$

The functions $F_{n_{1},n_{2},a_{1},a_{2}}$ and $G_{n_{1},n_{2},a_{1},a_{2}}$
will be called _$F$ -type_ and _$G$ -type_ functions.

The main use of $F,G$-type functions is to estimate higher-order derivatives
of the functions $\mathscr{F}_{f,l}$ and ${F}_{f}$ in a convenient way. This
will then be used to study the effect that the remainder term in the Taylor
expansion of $f$ has on the behavior of the functions $\mathscr{F}_{f,l}$ and
${F}_{f}$ around zero (see Lemma 3.7 and 3.8).

######  Lemma 3.4.

For any $f\in C^{n+1}(\mathcal{D})$ we have

(3.9) $\displaystyle\frac{\partial F_{n_{1},n_{2},a_{1},a_{2}}}{\partial{z}}$
$\displaystyle=\frac{1}{m}F_{n_{1}+1,n_{2},a_{1}+m-1,a_{2}}-\frac{a_{1}}{m}F_{n_{1},n_{2},a_{1}+m,a_{2}},$
(3.10) $\displaystyle\frac{\partial
F_{n_{1},n_{2},a_{1},a_{2}}}{\partial\overline{z}}$
$\displaystyle=\frac{1}{m}F_{n_{1},n_{2}+1,a_{1},a_{2}+m-1}-\frac{a_{2}}{m}F_{n_{1},n_{2},a_{1},a_{2}+m},$
(3.11) $\displaystyle\frac{\partial G_{n_{1},n_{2},a_{1},a_{2}}}{\partial{z}}$
$\displaystyle=\frac{1}{2}F_{n_{1},n_{2},a_{1},a_{2}}+\frac{1}{2m}(G_{n_{1}+1,n_{2},a_{1}+m-1,a_{2}}-a_{1}G_{n_{1},n_{2},a_{1}+m,a_{2}})$
$\displaystyle\quad-\frac{1}{2m}(G_{n_{1},n_{2}+1,a_{1},a_{2}+m-1}-a_{2}G_{n_{1},n_{2},a_{1},a_{2}+m}),$
(3.12) $\displaystyle\frac{\partial
G_{n_{1},n_{2},a_{1},a_{2}}}{\partial\overline{z}}$
$\displaystyle=\frac{1}{2}F_{n_{1},n_{2},a_{1},a_{2}}-\frac{1}{2m}(G_{n_{1}+1,n_{2},a_{1}+m-1,a_{2}}-a_{1}G_{n_{1},n_{2},a_{1}+m,a_{2}})$
$\displaystyle\quad+\frac{1}{2m}(G_{n_{1},n_{2}+1,a_{1},a_{2}+m-1}-a_{2}G_{n_{1},n_{2},a_{1},a_{2}+m}).$

###### Proof.

Since $\frac{\partial G}{\partial z}=\frac{1}{m}G^{1-m}$, $\frac{\partial
G}{\partial\overline{z}}=0$,
$\frac{\partial\overline{G}}{\partial\overline{z}}=\frac{1}{m}\overline{G}^{1-m}$
and $\frac{\partial\overline{G}}{\partial z}=0$, we obtain

$\displaystyle\frac{\partial F_{n_{1},n_{2},a_{1},a_{2}}}{\partial{z}}$
$\displaystyle=\frac{1}{m}\frac{\partial^{n_{1}+n_{2}+1}}{\partial{\omega}^{n_{1}+1}\partial\overline{\omega}^{n_{2}}}f(G,\overline{G})\cdot{G}^{-a_{1}+1-m}\overline{G}^{-a_{2}}$
$\displaystyle\quad-\frac{a_{1}}{m}\frac{\partial^{n_{1}+n_{2}}}{\partial{\omega}^{n_{1}}\partial\overline{\omega}^{n_{2}}}f(G,\overline{G})\cdot{G}^{-a_{1}-m}\overline{G}^{-a_{2}}$
$\displaystyle=\frac{1}{m}F_{n_{1}+1,n_{2},a_{1}+m-1,a_{2}}-\frac{a_{1}}{m}F_{n_{1},n_{2},a_{1}+m,a_{2}}.$

We also verify (3.10) in the same manner.

To obtain (3.11), in view of (3.7) and (3.8), we get

$\displaystyle\frac{\partial G_{n_{1},n_{2},a_{1},a_{2}}}{\partial{z}}$
$\displaystyle=\frac{1}{2}\left(\frac{\partial}{\partial
u}-\iota\frac{\partial}{\partial
s}\right)\int_{-1}^{u}F_{n_{1},n_{2},a_{1},a_{2}}(v,s)\,dv$
$\displaystyle=\frac{1}{2}F_{n_{1},n_{2},a_{1},a_{2}}-\frac{\iota}{2}\frac{\partial}{\partial
s}\int_{-1}^{u}F_{n_{1},n_{2},a_{1},a_{2}}(v,s)\,dv$
$\displaystyle=\frac{1}{2}F_{n_{1},n_{2},a_{1},a_{2}}+\frac{1}{2}\int_{-1}^{u}\left(\frac{\partial}{\partial{z}}-\frac{\partial}{\partial\overline{z}}\right)F_{n_{1},n_{2},a_{1},a_{2}}(v,s)\,dv.$

Therefore, in view of (3.9) and (3.10), this gives (3.11). Likewise, we repeat
the same for (3.12). ∎

The quantities

$d(F_{n_{1},n_{2},a_{1},a_{2}})=\frac{n_{1}+n_{2}+a_{1}+a_{2}}{m},\
d(G_{n_{1},n_{2},a_{1},a_{2}})=\frac{n_{1}+n_{2}+a_{1}+a_{2}}{m}-1$

we call the _degrees_ of the functions $F_{n_{1},n_{2},a_{1},a_{2}}$ and
$G_{n_{1},n_{2},a_{1},a_{2}}$. In view of (3.9)-(3.12), we have the following
conclusion.

######  Corollary 3.5.

For any $(l_{1},l_{2})\in{\mathbb{Z}}^{2}_{\geq 0}$ the partial derivative
$\frac{\partial^{l}G_{n_{1},n_{2},a_{1},a_{2}}}{\partial{z}^{l_{1}}\partial\overline{z}^{l_{2}}}$
is a linear combination of $F$-type and $G$-type functions of degree
$d(G_{n_{1},n_{2},a_{1},a_{2}})+l$, where $l=l_{1}+l_{2}$. Moreover, each
component of the linear combination is of the form
$F_{n^{\prime}_{1},n^{\prime}_{2},a^{\prime}_{1},a^{\prime}_{2}}$ and
$G_{n^{\prime}_{1},n^{\prime}_{2},a^{\prime}_{1},a^{\prime}_{2}}$ such that
$n\leq n^{\prime}_{1}+n^{\prime}_{2}\leq n+l$.

######  Lemma 3.6.

Let $n,\in{\mathbb{Z}}_{\geq 0}$. Assume that $f\in C^{k\vee n}(\mathcal{D})$
and $D^{j}f(0,0)=0$ for $0\leq j<k$. Then

$|F_{n_{1},n_{2},a_{1},a_{2}}(z,\overline{z})|\leq\|f\|_{C^{k\vee
n}}|z|^{-d(F_{n_{1},n_{2},a_{1},a_{2}})+\frac{k\vee n}{m}}.$

Moreover, there exists $C=C_{a_{1},a_{2},n,k}>0$ such that

(3.13) $\displaystyle|G_{n_{1},n_{2},a_{1},a_{2}}(z,\overline{z})|\leq
C\|f\|_{C^{k\vee n}}\langle|\Im
z|\rangle^{-d(G_{n_{1},n_{2},a_{1},a_{2}})+\frac{k\vee n}{m}}$

for any $z\in[-1,1]^{2}\setminus([-1,1]\times\\{0\\})$. For every $r>0$ there
exists $C_{r}=C_{a_{1},a_{2},n,k,r}>0$ such that

(3.14) $\displaystyle|G_{n_{1},n_{2},a_{1},a_{2}}(z,\overline{z})|\leq
C_{r}\|f\|_{C^{k\vee
n}}\langle|z|/\sqrt{2}\rangle^{-d(G_{n_{1},n_{2},a_{1},a_{2}})+\frac{k\vee
n}{m}}$

on $[-1,1]^{2}\cap\mathscr{S}(r)$.

If additionally $n\leq k$ and $D^{k}f(0,0)=0$ then

(3.15)
$\displaystyle\begin{split}|F_{n_{1},n_{2},a_{1},a_{2}}(z,\overline{z})|&=o(|\Im
z|^{-d(F_{n_{1},n_{2},a_{1},a_{2}})+\frac{k}{m}})\text{ if }\tfrac{k}{m}\leq
d(F_{n_{1},n_{2},a_{1},a_{2}})\text{ and}\\\
|G_{n_{1},n_{2},a_{1},a_{2}}(z,\overline{z})|&=o(\langle|\Im
z|\rangle^{-d(G_{n_{1},n_{2},a_{1},a_{2}})+\frac{k}{m}})\text{ if
}\tfrac{k}{m}\leq d(G_{n_{1},n_{2},a_{1},a_{2}}).\end{split}$

###### Proof.

As $D^{j}f(0,0)=0$ for $0\leq j<k$,

$\left|\frac{\partial^{n}f}{\partial\omega^{n_{1}}\partial{\bar{\omega}}^{n_{2}}}(\omega,\bar{\omega})\right|\leq\|D^{k\vee
n}f\|_{C^{0}}|\omega|^{(k\vee n)-n}.$

Hence

$|F_{n_{1},n_{2},a_{1},a_{2}}(z,\overline{z})|\leq\|f\|_{C^{k\vee
n}}|z|^{-\frac{a_{1}+a_{2}+n}{m}+\frac{k\vee n}{m}}=\|f\|_{C^{k\vee
n}}|z|^{-d(F_{n_{1},n_{2},a_{1},a_{2}})+\frac{k\vee n}{m}}.$

If $d_{F}=d(F_{n_{1},n_{2},a_{1},a_{2}})=\frac{n+a_{1}+a_{2}}{m}$ then

$|G_{n_{1},n_{2},a_{1},a_{2}}({z},\overline{z})|\leq\int_{-1}^{u}|F_{n_{1},n_{2},a_{1},a_{2}}(v,s)|dv=\|f\|_{C^{k\vee
n}}\int_{-1}^{u}(v^{2}+s^{2})^{\frac{-d_{F}+\frac{k\vee n}{m}}{2}}dv.$

As $d_{G}=d(G_{n_{1},n_{2},a_{1},a_{2}})=d_{F}-1$, the inequalities (3.13) and
(3.14) follow directly from Lemma 3.2.

Suppose that $0\leq n\leq k$, $f\in C^{k}(\mathcal{D})$ and $D^{j}f(0,0)=0$
for $0\leq j\leq k$. Then
$\left|\frac{\partial^{n}f}{\partial\omega^{n_{1}}\partial{\bar{\omega}}^{n_{2}}}(\omega,\bar{\omega})\right|=o(|\omega|^{k-n})$.
Hence

$|F_{n_{1},n_{2},a_{1},a_{2}}(z,\overline{z})|=o(|z|^{-\frac{a_{1}+a_{2}+n}{m}+\frac{k}{m}})=o(|\Im
z|^{-d_{F}+\frac{k}{m}})\text{ if }\frac{k}{m}\leq d_{F}.$

Moreover,

$|G_{n_{1},n_{2},a_{1},a_{2}}({z},\overline{z})|\leq\int_{-1}^{u}|F_{n_{1},n_{2},a_{1},a_{2}}(v,s)|dv=\int_{-1}^{u}\frac{\xi(v,s)}{(v^{2}+s^{2})^{\frac{d_{G}+1-\frac{k}{m}}{2}}}dv,$

where $\lim_{(v,s)\to(0,0)}\xi(v,s)=0$. If $d_{G}\geq\frac{k}{m}$ then the
second line of (3.15) follows directly from (3.4). ∎

###  3.2. Higher derivatives of functions $\mathscr{F}$ and $F$

In this section, using the results proved in Section 3.1, we study the
behaviour around zero of the higher order partial derivatives for the
functions $\mathscr{F}_{f,l}$ and $F_{f}$. For any $a\in{\mathbb{Z}}_{\geq 0}$
and any bounded Borel map $f:\mathcal{D}\to{\mathbb{C}}$ let us consider
$\mathscr{F}=\mathscr{F}_{l}=\mathscr{F}_{f,l}:[-1,1]^{2}\setminus([0,1]\times\\{0\\})\to{\mathbb{C}}$
given by

$\mathscr{F}_{f,l}(u,s)=\int_{-1}^{u}\frac{f(G_{l}(v,s))}{(v^{2}+s^{2})^{\frac{a}{m}}}\,dv.$

######  Lemma 3.7.

Assume that $f\in C^{k\vee n}(\mathcal{D})$ and $D^{j}f(0,0)=0$ for $0\leq
j<k$. Then there exists $C=C_{a,n,k}>0$ such that for every
$(n_{1},n_{2})\in{\mathbb{Z}}^{2}_{\geq 0}$ with $n_{1}+n_{2}=n$ we have

(3.16)
$\displaystyle\left|\frac{\partial^{n}{\mathscr{F}}(z,{\overline{z}})}{\partial
z^{n_{1}}\partial{\overline{z}}^{n_{2}}}\right|\leq C\|f\|_{C^{k\vee
n}}\langle|\Im z|\rangle^{-(\frac{2a}{m}+(n-1)-\frac{k}{m})}\text{ if }\Im
z\neq 0.$

For every $r>0$ there exists $C_{r}=C_{a,n,k,r}>0$ such that

(3.17)
$\displaystyle\left|\frac{\partial^{n}{\mathscr{F}}(z,{\overline{z}})}{\partial
z^{n_{1}}\partial{\overline{z}}^{n_{2}}}\right|\leq C_{r}\|f\|_{C^{k\vee
n}}\langle|z|/\sqrt{2}\rangle^{-(\frac{2a}{m}+(n-1)-\frac{k}{m})}\text{ on
}\mathscr{S}(r).$

If additionally $0\leq n\leq k$ and $D^{k}f(0,0)=0$ then

(3.18)
$\displaystyle\left|\frac{\partial^{n}{\mathscr{F}}(z,{\overline{z}})}{\partial
z^{n_{1}}\partial{\overline{z}}^{n_{2}}}\right|=o(\langle|\Im
z|\rangle^{-(\frac{2a}{m}+(n-1)-\frac{k}{m})})\text{ if
}\frac{2a}{m}+(n-1)\geq\frac{k}{m}.$

###### Proof.

By definition, $\mathscr{F}=G_{0,0,a,a}$. In view of Corollary 3.5, the
partial derivative
$\frac{\partial^{n}G_{0,0,a,a}}{\partial{z}^{n_{1}}\partial\overline{z}^{n_{2}}}$
is a linear combination of $F$-type and $G$-type functions of the form
$F_{n^{\prime}_{1},n^{\prime}_{2},a^{\prime}_{1},a^{\prime}_{2}}$ and
$G_{n^{\prime}_{1},n^{\prime}_{2},a^{\prime}_{1},a^{\prime}_{2}}$ such that
their degree is $2a/m+n-1$ and $0\leq
n^{\prime}:=n^{\prime}_{1}+n^{\prime}_{2}\leq n$.

Suppose that $n\leq k$. Then (3.16) and (3.17) follow directly from Lemma 3.6.
The same arguments combined with (3.15) yield (3.18).

Suppose that $n>k$. Then $k\leq n^{\prime}\vee k<n$. Therefore,
$\|f\|_{C^{n^{\prime}\vee k}}\leq\|f\|_{C^{n}}$ and for any $0\leq s\leq 1$
and $d\in{\mathbb{R}}$ we have $\langle s\rangle^{-d+\frac{n^{\prime}\vee
k}{m}}\leq\langle s\rangle^{-d+\frac{k}{m}}$. In view of Lemma 3.6 this gives
(3.16) and (3.17). ∎

By change of coordinates, we obtain the bound of higher derivatives of the map
$F=F_{f}:\mathcal{D}\setminus([0,1]\times\\{0\\})^{1/m}\to{\mathbb{C}}$ given
by
$F_{f}(\omega,\overline{\omega})=\mathscr{F}_{f,l}(\omega^{m},{\overline{\omega}}^{m})$
on $\mathcal{D}(\frac{l}{m},\frac{l+1}{m})$.

######  Lemma 3.8.

Assume that $f\in C^{k\vee n}(\mathcal{D})$ and $D^{j}f(0,0)=0$ for $0\leq
j<k$. Then for any $r>0$ there exists $C_{r,n}>0$ such that for every
$(n_{1},n_{2})\in{\mathbb{Z}}^{2}_{\geq 0}$ with $n_{1}+n_{2}=n$,

(3.19)
$\displaystyle\left|\frac{\partial^{n}{F}(\omega,{\overline{\omega}})}{\partial\omega^{n_{1}}\partial{\overline{\omega}}^{n_{2}}}\right|\leq
C_{r,n}\|f\|_{C^{k\vee n}}(1+|\log|\omega||)|\omega|^{(-2a+m+k-n)\wedge 0}$

for $\omega\in\mathcal{D}\cap\mathscr{S}(r)^{1/m}$.

###### Proof.

Recall that
${F}(\omega,{\overline{\omega}})=\mathscr{F}(\omega^{m},{\overline{\omega}}^{m})$.
By Faà di Bruno’s formula,

$\displaystyle\begin{split}\frac{\partial^{n}F(\omega,\overline{\omega})}{\partial\omega^{n_{1}}\partial\overline{\omega}^{n_{2}}}&=\frac{d^{n}\mathscr{F}(\omega^{m},\overline{\omega}^{m})}{d\omega^{n_{1}}d\overline{\omega}^{n_{2}}}\\\
&=\sum_{\bar{p},\bar{q}}C_{\bar{p},\bar{q}}\frac{\partial^{|\bar{p}|+|\bar{q}|}\mathscr{F}}{\partial
z^{|\bar{p}|}\partial\overline{z}^{|\bar{q}|}}(\omega^{m},\overline{\omega}^{m})\cdot\prod_{j=1}^{n_{1}\wedge
m}(\omega^{m-j})^{p_{j}}\prod_{j=1}^{n_{2}\wedge
m}(\overline{\omega}^{m-j})^{q_{j}},\end{split}$

where the sum is over all $n_{1}$-tuples $\bar{p}=(p_{1},\ldots,p_{n_{1}})$
and $n_{2}$-tuples $\bar{q}=(q_{1},\ldots,q_{n_{2}})$ of non-negative integers
satisfying the constraints

$\sum_{j=1}^{n_{1}}jp_{j}=n_{1},p_{j}=0\text{ for }j>n_{1}\wedge m\text{ and
}\sum_{j=1}^{n_{2}}jq_{j}=n_{2},q_{j}=0\text{ for }j>n_{1}\wedge m,$

and we use the notation $|\bar{p}|=\sum_{j=1}^{n_{1}}p_{j}$ and
$|\bar{q}|=\sum_{j=1}^{n_{2}}q_{j}$. Let

$\displaystyle P:=\\{|\bar{p}|:\sum_{j=1}^{n_{1}}jp_{j}=n_{1},p_{j}=0\text{
for }j>n_{1}\wedge m\\},$ $\displaystyle
Q:=\\{|\bar{q}|:\sum_{j=1}^{n_{2}}jq_{j}=n_{2},q_{j}=0\text{ for
}j>n_{2}\wedge m\\}.$

Then

$\displaystyle\frac{\partial^{n}F(\omega,\overline{\omega})}{\partial\omega^{n_{1}}\partial\overline{\omega}^{n_{2}}}$
$\displaystyle=\sum_{\bar{p},\bar{q}}C_{\bar{p},\bar{q}}\frac{\partial^{|\bar{p}|+|\bar{q}|}\mathscr{F}}{\partial
z^{|\bar{p}|}\partial\overline{z}^{|\bar{q}|}}(\omega^{m},\overline{\omega}^{m})\cdot\omega^{m|\bar{p}|-n_{1}}{\overline{\omega}}^{m|\bar{q}|-n_{2}}$
$\displaystyle=\sum_{p\in P,q\in
Q}C^{\prime}_{p,q}\frac{\partial^{p+q}\mathscr{F}}{\partial
z^{p}\partial\overline{z}^{q}}(\omega^{m},\overline{\omega}^{m})\cdot\omega^{mp-
n_{1}}{\overline{\omega}}^{mq-n_{2}}.$

In view of Lemma 3.7 and Remark 3.1, for every
$\omega\in\mathcal{D}\cap\mathscr{S}(r)^{1/m}$,

$\displaystyle\left|\frac{\partial^{p+q}\mathscr{F}}{\partial
z^{p}\partial\overline{z}^{q}}(\omega^{m},\overline{\omega}^{m})\right|\leq
mC_{r}\|f\|_{C^{k\vee
n}}\langle|\omega|/\sqrt[2m]{2}\rangle^{-(2a+(p+q-1)m-k)}.$

Therefore, taking $l=p+q\in P+Q$, for every
$\omega\in\mathcal{D}\cap\mathscr{S}(r)^{1/m}$,

(3.20)
$\displaystyle\left|\frac{\partial^{n}{F}(\omega,{\overline{\omega}})}{\partial\omega^{n_{1}}\partial{\overline{\omega}}^{n_{2}}}\right|\leq\sum_{l\in
P+Q}mC_{r}\|f\|_{C^{k\vee
n}}\langle|\omega/\sqrt[2m]{2}|\rangle^{-(2a+(l-1)m-k)}|\omega|^{ml-n}.$

Moreover, for $l=p+q\in P+Q$ we have

$ml-n=\sum_{j=1}^{n_{1}\wedge m}(m-j)p_{j}+\sum_{j=1}^{n_{2}\wedge
m}(m-j)q_{j}\geq 0.$

If $2a+(l-1)m-k>0$ then

$\langle|\omega/\sqrt[2m]{2}|\rangle^{-(2a+(l-1)m-k)}|\omega|^{ml-n}=O(|\omega|^{-(2a+n-m-k)}).$

If $2a+(l-1)m-k=0$ then

$\langle|\omega/\sqrt[2m]{2}|\rangle^{-(2a+(l-1)m-k)}|\omega|^{ml-n}=O((1+|\log|\omega||)|\omega|^{-(2a+n-m-k)}).$

If $2a+(l-1)m-k<0$ then

$\langle|\omega/\sqrt[2m]{2}|\rangle^{-(2a+(l-1)m-k)}|\omega|^{ml-n}=|\omega|^{ml-n}=O(1).$

In view of (3.20), this gives (3.19). ∎

###  3.3. Preliminary results necessary to define invariant distributions

In this section we introduce new functions $\mathfrak{G}^{l}_{a_{1},a_{2}}$
which are useful for computing $\mathscr{F}_{f,l}$ when $f$ is a polynomial.
Their recursive properties are crucial for studying new invariant
distributions.

For any pair of integers $(a_{1},a_{2})$ and any $0\leq l<m$, let
$\mathfrak{F}_{a_{1},a_{2}}=\mathfrak{F}_{a_{1},a_{2}}^{l}:[-1,1]^{2}\setminus([0,1]\times\\{0\\})\to{\mathbb{C}}$
and
$\mathfrak{G}_{a_{1},a_{2}}=\mathfrak{G}_{a_{1},a_{2}}^{l}:[-1,1]^{2}\setminus([0,1]\times\\{0\\})\to{\mathbb{C}}$
be given by

(3.21) $\displaystyle\mathfrak{F}_{a_{1},a_{2}}({z},\overline{z})$
$\displaystyle=\mathfrak{F}_{a_{1},a_{2}}(u,s)={G_{l}(u,s)}^{-a_{1}}\overline{G_{l}(u,s)}^{-a_{2}},$
(3.22) $\displaystyle\mathfrak{G}_{a_{1},a_{2}}({z},\overline{z})$
$\displaystyle=\mathfrak{G}_{a_{1},a_{2}}(u,s)=\int_{-1}^{u}\mathfrak{F}_{a_{1},a_{2}}(v,s)dv.$

Then
$\mathfrak{G}^{l}_{a_{1},a_{2}}=\theta^{l(a_{2}-a_{1})}\mathfrak{G}^{0}_{a_{1},a_{2}}$
for every $0\leq l<m$. As

$G_{0}(-u-\iota s)=\theta_{0}G_{0}(u+\iota s)\text{ and }G_{0}(u-\iota
s)=\theta^{2}_{0}\overline{G_{0}(u+\iota s)}\text{ for }s>0,$

it follows that

(3.23)
$\mathfrak{G}^{l}_{a_{1},a_{2}}(1,s)=\left\\{\begin{array}[]{rl}\theta_{0}^{2l(a_{2}-a_{1})}\mathfrak{G}^{0}_{a_{1},a_{2}}(1,|s|)&\text{
if}\quad s\in(0,1],\\\
\theta_{0}^{(2l+1)(a_{2}-a_{1})}\mathfrak{G}^{0}_{a_{1},a_{2}}(1,|s|)&\text{
if}\quad s\in[-1,0)\end{array}\right.$

and

(3.24)
$\mathfrak{G}^{l}_{a_{1},a_{2}}(u,s)=\left\\{\begin{array}[]{rl}\theta_{0}^{2l(a_{2}-a_{1})}\mathfrak{G}^{0}_{a_{1},a_{2}}(u,|s|)&\text{
if}\quad s\in(0,1],\\\
\theta_{0}^{(2l+2)(a_{2}-a_{1})}\overline{\mathfrak{G}^{0}_{a_{1},a_{2}}(u,|s|)}&\text{
if}\quad s\in[-1,0).\end{array}\right.$

For any set $A\subset{\mathbb{R}}^{d}$ denote by $C^{\omega}(A)$ the space of
complex-valued real-analytic maps on $A$ which have analytic extention to the
closure $\bar{A}$. If $f,g:A\to{\mathbb{C}}$ are such that $f-g\in
C^{\omega}(A)$ then we write $f=g+C^{\omega}(A)$. We denote by
$C^{\omega,m}_{l}$ the space of functions
$f:[-1,1]^{2}\setminus([0,1]\times\\{0\\})\to{\mathbb{C}}$ such that
$\omega\mapsto f(\omega^{m},\overline{\omega}^{m})$ has a real analytic
extension on $\overline{\mathcal{D}}(\frac{l}{m},\frac{l+1}{m})$. For example,
$\mathfrak{F}_{a_{1},a_{2}}\in C^{\omega,m}_{l}$ if $a_{1},a_{2}$ are non-
positive.

######  Lemma 3.9.

For any pair of integers $(a_{1},a_{2})$,

(3.25)
$a_{1}\mathfrak{G}^{l}_{a_{1}+m,a_{2}}(1,s)+a_{2}\mathfrak{G}^{l}_{a_{1},a_{2}+m}(1,s)\in
C^{\omega}((0,1])\cap C^{\omega}([-1,0)).$

If $a_{1},a_{2}$ are additionally both non-positive then

(3.26)
$a_{1}\mathfrak{G}^{l}_{a_{1}+m,a_{2}}+a_{2}\mathfrak{G}^{l}_{a_{1},a_{2}+m}\in
C^{\omega,m}_{l}.$

###### Proof.

Note that, by (3.7) and (3.22),

$\displaystyle\mathfrak{F}_{a_{1},a_{2}}(z,\overline{z})-\mathfrak{F}_{a_{1},a_{2}}(-1+\iota\Im
z,-1-\iota\Im
z)=\mathfrak{F}_{a_{1},a_{2}}(u,s)-\mathfrak{F}_{a_{1},a_{2}}(-1,s)$
$\displaystyle=\int_{-1}^{u}\frac{\partial}{\partial
v}\mathfrak{F}_{a_{1},a_{2}}(v,s)\,dv=\int_{-1}^{u}\left(\frac{\partial}{\partial
z}+\frac{\partial}{\partial\overline{z}}\right)\mathfrak{F}_{a_{1},a_{2}}(v,s)\,dv$
$\displaystyle=-\int_{-1}^{u}\left(\frac{a_{1}}{m}\mathfrak{F}_{a_{1}+m,a_{2}}(v,s)+\frac{a_{2}}{m}\mathfrak{F}_{a_{1},a_{2}+m}(v,s)\right)\,dv$
$\displaystyle=-\frac{1}{m}({a_{1}}\mathfrak{G}_{a_{1}+m,a_{2}}(z,\overline{z})+{a_{2}}\mathfrak{G}_{a_{1},a_{2}+m}(z,\overline{z})).$

Taking $(z,\overline{z})=(1,s)$, by (3.21), it follows that

$\displaystyle
a_{1}\mathfrak{G}_{a_{1}+m,a_{2}}(1,s)+a_{2}\mathfrak{G}_{a_{1},a_{2}+m}(1,s)$
$\displaystyle=m(G_{l}(-1,s)^{-a_{1}}\overline{G_{l}(-1,s)}^{-a_{2}}-G_{l}(1,s)^{-a_{1}}\overline{G_{l}(1,s)}^{-a_{2}}).$

Since the maps $[-1,1]\ni s\mapsto G_{l}(-1,s)\in{\mathbb{C}}$, $[0,1]\ni
s\mapsto G_{l}(1,s)\in{\mathbb{C}}$ and $[-1,0)\ni s\mapsto
G_{l}(1,s)\in{\mathbb{C}}$ are analytic and the latter has an analytic
extension to $[-1,0]$, this gives (3.25). Moreover, for any
$\omega\in\mathcal{D}(\frac{l}{m},\frac{l+1}{m})$,

$\displaystyle
a_{1}\mathfrak{G}_{a_{1}+m,a_{2}}({\omega}^{m},\overline{\omega}^{m})+a_{2}\mathfrak{G}_{a_{1},a_{2}+m}({\omega}^{m},\overline{\omega}^{m})$
$\displaystyle=m(G_{l}(-1+\iota\Im\omega^{m})^{-a_{1}}\overline{G_{l}(-1+\iota\Im\omega^{m})}^{-a_{2}}-G_{l}(\omega^{m})^{-a_{1}}\overline{G_{l}(\omega^{m})}^{-a_{2}})$
$\displaystyle=m(G_{l}(-1+\iota\Im\omega^{m})^{-a_{1}}\overline{G_{l}(-1+\iota\Im\omega^{m})}^{-a_{2}}-{\omega}^{-a_{1}}\overline{\omega}^{-a_{2}}).$

Since $-a_{1},-a_{2}$ are non-negative integers, all functions on the RHS are
analytic which completes the proof. ∎

As a conclusion we obtain that for any integer $k\neq m$,

$\mathfrak{G}_{0,k}(1,s),\mathfrak{G}_{k,0}(1,s)\in C^{\omega}((0,1])\cap
C^{\omega}([-1,0))$

and for any integer $k<m$,

(3.27) $\mathfrak{G}_{0,k},\mathfrak{G}_{k,0}\in C^{\omega,m}_{l}.$

Moreover,

$\displaystyle\mathfrak{G}_{0,m}(1,s)$
$\displaystyle=\int_{-1}^{1}\frac{1}{v-\iota s}dv=\int_{-1}^{1}\frac{v+\iota
s}{v^{2}+s^{2}}dv=\iota\operatorname{sgn}(s)\int_{-1/|s|}^{1/|s|}\frac{1}{x^{2}+1}dx$
$\displaystyle=\iota\operatorname{sgn}(s)(\arctan(1/|s|)-\arctan(-1/|s|))$
$\displaystyle=\iota(\operatorname{arccot}(s)-\operatorname{arccot}(-s))+\iota\operatorname{sgn}(s)\pi.$

Hence, $\mathfrak{G}_{0,m}(1,s),\mathfrak{G}_{m,0}(1,s)\in
C^{\omega}((0,1])\cap C^{\omega}([-1,0))$. Using (3.25) again, we have

(3.28) $\mathfrak{G}_{k,-lm}(1,s),\mathfrak{G}_{-lm,k}(1,s)\in
C^{\omega}((0,1])\cap C^{\omega}([-1,0))\text{ for all
}k\in{\mathbb{Z}},l\in{\mathbb{Z}}_{\geq 0}.$

###  3.4. Invariant distributions $\partial^{k}_{j}$ and their effect on the
regularity of $\mathscr{F}$ and $F$

For every $m\geq 2$, $0\leq l<m$, $k\geq 0$ and $f\in C^{k}(\mathcal{D})$ we
deal with three associated functions $\mathscr{F}_{f,l}$, $\varphi_{f,l}$ and
$F_{f}$. Recall that
$\mathscr{F}_{f,l}:[-1,1]^{2}\setminus([0,1]\times\\{0\\})\to{\mathbb{C}}$ is
given by

$\mathscr{F}_{f,l}(z,\overline{z})=\mathscr{F}_{f,l}(u,s)=\int_{-1}^{u}\frac{f(G_{l}(v,s))}{(v^{2}+s^{2})^{\frac{m-1}{m}}}dv,$

$\varphi_{f,l}:[-1,0)\cup(0,1]\to{\mathbb{C}}$ is given by
$\varphi_{f,l}(s)=\mathscr{F}_{f,l}(1,s)$ and
$F_{f}:\mathcal{D}\setminus([0,1]\times\\{0\\})^{1/m}\to{\mathbb{C}}$ is given
by
$F_{f}(\omega,\overline{\omega})=\mathscr{F}_{f,l}(\omega^{m},{\overline{\omega}}^{m})$
on $\mathcal{D}(\frac{l}{m},\frac{l+1}{m})$.

For every $k\geq 0$ and let us consider functionals
$\partial^{k}_{j}:C^{k}(\mathcal{D})\to{\mathbb{C}}$ for $0\leq j\leq
k\wedge(m-1)$ given by

(3.29) $\partial^{k}_{j}(f)=\sum_{0\leq
n\leq\frac{k-j}{m}}\frac{\binom{k}{j+nm}\binom{\frac{(m-1)-j}{m}-1}{n}}{\binom{\frac{(k-j)-(m-1)}{m}}{n}}\frac{\partial^{k}f}{\partial\omega^{j+nm}\partial\overline{\omega}^{k-j-
nm}}(0,0).$

Comparing with (1.4), functionals $\partial^{k}_{j}$ will play a key role in
understanding the meaning of distribution $\mathfrak{d}^{k}_{\sigma,j}$. If
$0\leq k\leq m-2$ then
$\partial^{k}_{j}(f)=\binom{k}{j}\frac{\partial^{k}f}{\partial\omega^{j}\partial\overline{\omega}^{k-j}}(0,0)$.
If $k\geq m-1$ then as we will see in the following lemma, only $m-2$
functionals matter. More precisely, $\partial^{k}_{j}$ is irrelevant if
$j=m-1$ or $j=k-(m-1)\operatorname{mod}m$, this is, its value does not affect
the smoothness of $\varphi_{f,l}$ and $F_{f}$, see also Remark 3.11. Note that
if $k=m-2\operatorname{mod}m$ then, in this exceptional case, we have $m-1$
relevant functionals.

Recall that for any $0\leq\alpha<\beta\leq 1$ let
$\mathcal{D}(\alpha,\beta):=\\{\omega\in\mathcal{D}\setminus\\{0\\}:\operatorname{Arg}(\omega)\in(2\pi\alpha,2\pi\beta)\\}$.
We denote its closure by $\overline{\mathcal{D}}(\alpha,\beta)$.

######  Lemma 3.10.

Suppose that $f\in{\mathbb{C}}[\omega,\overline{\omega}]$ is a polynomial of
degree at most $k$ such that $\partial^{j}_{i}(f)=0$ for all $0\leq j\leq k$
and $0\leq i\leq j\wedge(m-2)$ with $i\neq j-(m-1)\operatorname{mod}m$. Then

$F_{f}\in C^{\omega}(\mathcal{D}(\tfrac{l}{m},\tfrac{l+1}{m}))\ \text{ and }\
\varphi_{f,l}\in C^{\omega}([-1,0))\cap C^{\omega}((0,1])\text{ for }0\leq
l<m.$

Moreover, for any $n\geq 0$ there exists a constant $C^{n}_{k}>0$ such that

(3.30)
$\displaystyle\|F_{f}\|_{C^{n}(\mathcal{D}(\frac{l}{m},\frac{l+1}{m}))}\leq
C^{n}_{k}\|f\|_{C^{k}(\mathcal{D})}\text{ and
}\|\varphi_{f,l}\|_{C^{n}([-1,0)\cup(0,1])}\leq
C^{n}_{k}\|f\|_{C^{k}(\mathcal{D})}.$

###### Proof.

First note that (3.30) follows directly from the first part of the lemma.
Indeed, $f\mapsto F_{f}\in C^{\omega}(\mathcal{D}(\frac{l}{m},\frac{l+1}{m}))$
and $f\mapsto\varphi_{f,l}\in C^{\omega}([-1,0))\cap C^{\omega}((0,1])$ are
linear operators on a finite-dimensional space, so they are bounded. This
gives (3.30).

By assumption, $f=\sum_{0\leq j\leq k}f_{j}$ with

$f_{j}(\omega,\overline{\omega})=\frac{1}{j!}\sum_{0\leq i\leq
j}\binom{j}{i}\frac{\partial^{j}f}{\partial\omega^{i}\partial\overline{\omega}^{j-i}}(0,0)\omega^{i}\overline{\omega}^{j-i}.$

We will show that if $\partial^{j}_{i}(f)=0$ for all $0\leq i\leq
j\wedge(m-2)$ with $i\neq j-(m-1)\operatorname{mod}m$, then

$F_{f_{j}}\in C^{\omega}(\mathcal{D}(\tfrac{l}{m},\tfrac{l+1}{m}))\ \text{ and
}\ \varphi_{f_{j},l}\in C^{\omega}([-1,0))\cap C^{\omega}((0,1])\text{ for
}0\leq l<m.$

This gives our claim.

Note that

$\displaystyle\begin{split}\mathscr{F}_{f_{j},l}&=\frac{1}{j!}\sum_{0\leq
i\leq
j}\binom{j}{i}\frac{\partial^{j}f}{\partial\omega^{i}\partial\overline{\omega}^{j-i}}(0,0)\mathfrak{G}_{(m-1)-i,(m-1)-(j-i)}=\frac{1}{j!}\sum_{0\leq
i<m}\xi_{i},\end{split}$

where

$\xi_{i}(z,\overline{z})=\\!\\!\sum_{0\leq
n\leq\frac{j-i}{m}}\\!\\!\binom{j}{i+nm}\frac{\partial^{j}f}{\partial\omega^{i+nm}\partial\overline{\omega}^{(j-i)-nm}}(0,0)\mathfrak{G}_{(m-1)-i-nm,(m-1)-(j-i)+nm}(z,\overline{z}).$

For every $1\leq n\leq\frac{j-i}{m}$ we have

$a_{1}=(m-1)-i-nm<0\text{ and }a_{2}=(m-1)-(j-i)+(n-1)m<0.$

In view of (3.26),

$\displaystyle((m-1)-i-nm)\mathfrak{G}_{(m-1)-i-(n-1)m,(m-1)-(j-i)+(n-1)m}$
$\displaystyle\quad+((m-1)-(j-i)+(n-1)m)\mathfrak{G}_{(m-1)-i-nm,(m-1)-(j-i)+nm}\in
C^{\omega,m}_{l},$

so

$\displaystyle\mathfrak{G}_{(m-1)-i-nm,(m-1)-(j-i)+nm}$
$\displaystyle\quad=\frac{(\frac{(m-1)-i}{m}-n)}{(\frac{(j-i)-(m-1)}{m}-(n-1))}\mathfrak{G}_{(m-1)-i-(n-1)m,(m-1)-(j-i)+(n-1)m}+C^{\omega,m}_{l},$

It follows that for every $0\leq n\leq\frac{j-i}{m}$,

(3.31)
$\mathfrak{G}_{(m-1)-i-nm,(m-1)-(j-i)+nm}=\frac{\binom{\frac{(m-1)-i}{m}-1}{n}}{\binom{\frac{(j-i)-(m-1)}{m}}{n}}\mathfrak{G}_{(m-1)-i,(m-1)-(j-i)}+C^{\omega,m}_{l}.$

It follows that for every $0\leq i\leq m-1$,

$\xi_{i}=\partial^{j}_{i}(f)\mathfrak{G}_{(m-1)-i,(m-1)-(j-i)}+C^{\omega,m}_{l}.$

If $i=m-1$ then by (3.27), $\mathfrak{G}_{(m-1)-i,(m-1)-(j-i)}\in
C^{\omega,m}_{l}$ so $\xi_{i}\in C^{\omega,m}_{l}$. If
$i=j-(m-1)\operatorname{mod}m$, then $(m-1)-(j-i)+nm=0$ with
$n=\lfloor\frac{j-i}{m}\rfloor=\frac{j-i-(m-1)}{m}$. Again by (3.27),
$\mathfrak{G}_{(m-1)-i-nm,(m-1)-(j-i)+nm}\in C^{\omega,m}_{l}$. In view of
(3.31), it follows that $\mathfrak{G}_{(m-1)-i,(m-1)-(j-i)}\in
C^{\omega,m}_{l}$ and again $\xi_{i}\in C^{\omega,m}_{l}$. Hence

(3.32) $\mathscr{F}_{f_{j},l}=\frac{1}{j!}\sum_{\begin{subarray}{c}0\leq i\leq
j\wedge(m-2)\\\ i\neq
j-(m-1)\operatorname{mod}m\end{subarray}}\partial^{j}_{i}(f)\mathfrak{G}_{(m-1)-i,(m-1)-(j-i)}+C^{\omega,m}_{l}.$

As $\partial^{j}_{i}(f_{j})=0$ for $i\neq m-1$ and $i\neq j-(m-1)\
\operatorname{mod}m$, this yields $\mathscr{F}_{f_{j},l}\in C^{\omega,m}_{l}$
and $F_{f_{j}}\in C^{\omega}(\mathcal{D}(\frac{l}{m},\frac{l+1}{m}))$.

Using (3.25) instead of (3.26), the same arguments show
$\varphi_{f_{j},l}{(s)}=\mathscr{F}_{f_{j},l}(1,s)\in C^{\omega}([-1,0))\cap
C^{\omega}((0,1])$. ∎

######  Remark 3.11.

Let $k\geq m-1$ and let us consider
$P(\omega,\overline{\omega})=\omega^{m-1}\overline{\omega}^{k-(m-1)}$ and
$Q(\omega,\overline{\omega})=\omega^{k-(m-1)}\overline{\omega}^{m-1}$. Then

$\mathscr{F}_{P,l}=\mathfrak{G}^{l}_{0,2(m-1)-k}\text{ and
}\mathscr{F}_{Q,l}=\mathfrak{G}^{l}_{2(m-1)-k,0}.$

As $2(m-1)-k<m$, in view of (3.27), $\mathscr{F}_{P,l},\mathscr{F}_{Q,l}\in
C^{\omega,m}_{l}$, so $F_{P}$ and $F_{Q}$ are analytic on each closed angular
sector. On the other hand, by (3.29)

$\partial^{k}_{m-1}(P)=k!\text{ and
}\partial^{k}_{j}(Q)=k!\binom{\frac{(m-1)-j}{m}-1}{\frac{(k-j)-(m-1)}{m}},$

where $0\leq j<m$ is such that $j=k-(m-1)\operatorname{mod}m$. It follows that
the smoothness of $F_{f}$ is not related to the vanishing of distributions
$\partial^{k}_{m-1}(f)$ and $\partial^{k}_{j}(f)$.

We finish this section by showing a smooth extension of $F_{f}$ on angular
sectors.

######  Theorem 3.12.

Let $k\geq m-1$. Suppose that $f\in C^{k}(\mathcal{D})$ and
$\partial^{j}_{i}(f)=0$ for all $0\leq j<k$ and $0\leq i\leq j\wedge(m-2)$
with $i\neq j-(m-1)\operatorname{mod}m$. Then for every $r>0$ the map $F_{f}$
on every angular sector of $\mathcal{D}\cap\mathscr{S}(r)^{1/m}$ has a
$C^{\widehat{\mathfrak{e}}(\sigma,k)}$-extension at $(0,0)$ (recall that
$\widehat{\mathfrak{e}}(\sigma,k)=k-(m-1)+\eta$). Moreover, there exists
$C_{r}>0$ such that
$\|F_{f}\|_{C^{\widehat{\mathfrak{e}}(\sigma,k)}(\mathcal{D}\cap\mathscr{S}(r)^{1/m})}\leq
C_{r}\|f\|_{C^{k}}$.

###### Proof.

Let $\mathscr{A}$ be an angular sector of
$\mathcal{D}\cap\mathscr{S}(r)^{1/m}$. Let us decompose $f=f_{<k}+e_{f}$ with

$f_{<k}(\omega,\overline{\omega})=\sum_{0\leq j<k}\frac{1}{j!}\sum_{0\leq
i\leq
j}\binom{j}{i}\frac{\partial^{j}f}{\partial\omega^{i}\partial\overline{\omega}^{j-i}}(0,0)\omega^{i}\overline{\omega}^{j-i}.$

Note that the operator $C^{k}(\mathcal{D})\ni f\mapsto f_{<k}\in
C^{k}(\mathcal{D})$ is bounded. By Lemma 3.10, $F_{f_{<k}}$ is analytic and
for every $n\geq 0$ there exists $C^{n}_{k}>0$ such that
$\|F_{f_{<k}}\|_{C^{n}(\mathcal{D}(\frac{l}{m},\frac{l+1}{m}))}\leq
C^{n}_{k}\|f\|_{C^{k}(\mathcal{D})}$ for every $0\leq l<m$ . On the other
hand, $D^{j}(e_{f})=0$ for every $0\leq j<k$. In view of Lemma 3.8, if
$(n_{1},n_{2})\in{\mathbb{Z}}^{2}_{\geq 0}$ is such that
$n_{1}+n_{2}=n=k-(m-2)$ then for every $\omega\in\mathscr{A}$,

$\left|\frac{\partial^{n}{F_{e_{f}}}(\omega,{\overline{\omega}})}{\partial\omega^{n_{1}}\partial{\overline{\omega}}^{n_{2}}}\right|\leq
C_{r,n}\|e_{f}\|_{C^{k}(\mathcal{D})}(1+|\log|\omega||)|\omega|^{(k-n-(m-2))\wedge
0}.$

As $\|D^{k-(m-2)}F_{e_{f}}(\omega,{\overline{\omega}})\|\leq
C_{r,n}\|e_{f}\|_{C^{k}(\mathcal{D})}(1+|\log|\omega||)$,
$D^{k-(m-1)}F_{e_{f}}$ on $\mathscr{A}$ has a continuous extension on
$\overline{\mathscr{A}}=\mathscr{A}\cup\\{(0,0)\\}$ with the modulus of
continuity bounded by a multiplicity of $\eta$. Therefore, $F_{e_{f}}$ can be
extended to a $C^{k-(m-1)+\eta}$-function on $\overline{\mathscr{A}}$ and
$\|F_{e_{f}}\|_{C^{k-(m-1)+\eta}(\overline{\mathscr{A}})}\leq
C\|e_{f}\|_{C^{k}(\mathcal{D})}\leq C^{\prime}\|f\|_{C^{k}(\mathcal{D})}$. As
$F_{f}=F_{f_{<k}}+F_{e_{f}}$, this gives our claim. ∎

##  4\. Local analysis of $\varphi_{f}$

This section is devoted to computing a limiting behavior of higher derivatives
of $\varphi_{f}$ related to singularities on angular sectors of $\mathcal{D}$.
We introduce a family of functionals $\mathscr{C}^{k}_{l}$ which are
responsible for the asymptotic behaviour of $\varphi_{f,l}$ around zero. The
new result is inspired by the approach for multi-saddles (related to
polynomial singularities) in [14]. The main results of this section (Theorem
4.7) plays a central role in proving Theorem 1.1 in §5 as well as is applied
to extend the regularity of $F_{f}$ (obtained in Theorem 3.12) to the closure
of any sector $\mathcal{D}(\frac{l}{m},\frac{l+1}{m})$.

###  4.1. Preliminary properties of $\mathfrak{G}_{a_{1},a_{2}}(1,s)$

Firstly we present limiting behaviour of
$\frac{d^{n}}{ds^{n}}\mathfrak{G}_{a_{1},a_{2}}(1,s)$ around zero. We show
that for large enough higher derivatives their asymptotic is polynomial with a
weight factor established by the Beta-like function $\mathfrak{B}$. This is
further used in evaluating asymptotics of $D^{n+1}\varphi_{f,l}$ in §4.2.

Note that for any pair of integers $a_{1},a_{2}$,

(4.1)
$\displaystyle\begin{split}\frac{d}{ds}\mathfrak{G}_{a_{1},a_{2}}(1,s)&=-\frac{2\iota
a_{1}}{m}\mathfrak{G}_{a_{1}+m,a_{2}}(1,s)+C^{\omega}((0,1])\cap
C^{\omega}([-1,0))\\\ &=\frac{2\iota
a_{2}}{m}\mathfrak{G}_{a_{1},a_{2}+m}(1,s)+C^{\omega}((0,1])\cap
C^{\omega}([-1,0)).\end{split}$

Indeed,

$\displaystyle\frac{d}{ds}\mathfrak{G}_{a_{1},a_{2}}(u,s)$
$\displaystyle=\int_{-1}^{u}\frac{d}{ds}\mathfrak{F}_{a_{1},a_{2}}(v,s)dv=\int_{-1}^{u}\iota\left(\frac{\partial}{\partial
z}-\frac{\partial}{\partial\overline{z}}\right)\mathfrak{F}_{a_{1},a_{2}}(v,s)dv$
$\displaystyle=\int_{-1}^{u}\iota\left(-\frac{a_{1}}{m}\mathfrak{F}_{a_{1}+m,a_{2}}(v,s)+\frac{a_{2}}{m}\mathfrak{F}_{a_{1},a_{2}+m}(v,s)\right)\,dv$
$\displaystyle=\frac{\iota}{m}(-{a_{1}}\mathfrak{G}_{a_{1}+m,a_{2}}(u,s)+{a_{2}}\mathfrak{G}_{a_{1},a_{2}+m}(u,s)).$

In view of (3.25), this gives (4.1). It follows that for every $n\geq 1$,

(4.2)
$\frac{d^{n}}{ds^{n}}\mathfrak{G}_{a_{1},a_{2}}(1,s)=n!(-2\iota)^{n}\binom{-\frac{a_{2}}{m}}{n}\mathfrak{G}_{a_{1},a_{2}+nm}(1,s)+C^{\omega}((0,1])\cap
C^{\omega}([-1,0))$

and

(4.3) $\frac{d^{n}}{ds^{n}}\mathfrak{G}_{a_{1},a_{2}}=\iota^{n}n!\sum_{0\leq
j\leq
n}(-1)^{n-j}\binom{-\frac{a_{1}}{m}}{j}\binom{-\frac{a_{2}}{m}}{n-j}\mathfrak{G}_{a_{1}+jm,a_{2}+(n-j)m}.$

Suppose that $a_{1},a_{2}$ are integers such that $a_{1}+a_{2}>m$. Then for
every $s\in(0,1]$,

$\displaystyle\mathfrak{G}^{0}_{a_{1},a_{2}}(1,s)$
$\displaystyle=\int_{-1}^{1}G_{0}(v+\iota s)^{-a_{1}}\overline{G_{0}(v+\iota
s)}^{-a_{2}}dv$
$\displaystyle=s^{-\frac{a_{1}+a_{2}-m}{m}}\int_{-1/s}^{1/s}G_{0}(x+\iota)^{-a_{1}}\overline{G_{0}(x+\iota)}^{{-a_{2}}}dx.$

Since
$|G_{0}(x+\iota)^{-a_{1}}\overline{G_{0}(x+\iota)}^{{-a_{2}}}|=\sqrt{x^{2}+1}^{-\frac{a_{1}+a_{2}}{m}}$
with $\frac{a_{1}+a_{2}}{m}>1$, we have

$\displaystyle\lim_{s\to
0^{+}}s^{\frac{a_{1}+a_{2}-m}{m}}\mathfrak{G}^{0}_{a_{1},a_{2}}(1,s)=\mathfrak{B}(\tfrac{a_{1}}{m},\tfrac{a_{2}}{m}):=\int_{{\mathbb{R}}}G_{0}(x+\iota)^{-a_{1}}\overline{G_{0}(x+\iota)}^{{-a_{2}}}dx.$

Note that, making the substitution $x=\cot t$, we obtain

$\mathfrak{B}(\tfrac{a_{1}}{m},\tfrac{a_{2}}{m})=\int_{0}^{\pi}\frac{e^{\iota\frac{a_{2}-a_{1}}{m}t}}{\sin^{-\frac{a_{1}+a_{2}}{m}+2}t}dt$

and for $\frac{a_{1}}{m},\frac{a_{2}}{m}\notin{\mathbb{Z}}_{\leq 0}$,

$\int_{0}^{\pi}\frac{e^{\iota\frac{a_{2}-a_{1}}{m}t}}{\sin^{-\frac{a_{1}+a_{2}}{m}+2}t}dt=\frac{\pi
e^{\iota\frac{a_{2}-a_{1}}{2m}\pi}}{2^{\frac{a_{1}+a_{2}-2m}{m}}\frac{a_{1}+a_{2}-m}{m}}\frac{\Gamma(\frac{a_{1}}{m}+\frac{a_{2}}{m}-1)}{\Gamma(\frac{a_{1}}{m})\Gamma(\frac{a_{2}}{m})}.$

For any pair $x,y$ of real numbers such that $x,y\notin{\mathbb{Z}}_{\leq 0}$
and $x+y\notin{\mathbb{Z}}_{\leq 1}$ let

$\mathfrak{B}(x,y)=\frac{\pi
e^{\iota\frac{\pi}{2}(y-x)}}{2^{x+y-2}(x+y-1)B(x,y)}=\frac{\pi
e^{\iota\frac{\pi}{2}(y-x)}}{2^{x+y-2}}\frac{\Gamma(x+y-1)}{\Gamma(x)\Gamma(y)}\neq
0.$

Note that

(4.4)
$\overline{\mathfrak{B}(x,y)}=\mathfrak{B}(y,x)=e^{-\iota{\pi}(y-x)}\mathfrak{B}(x,y).$

By (3.23), for any pair of integers $a_{1},a_{2}$ such that $a_{1}+a_{2}>m$
and $\frac{a_{1}}{m},\frac{a_{2}}{m}\notin{\mathbb{Z}}_{\leq 0}$,

(4.5) $\displaystyle\begin{split}\lim_{s\to
0^{+}}|s|^{\frac{a_{1}+a_{2}-m}{m}}\mathfrak{G}^{l}_{a_{1},a_{2}}(1,s)&=\theta_{0}^{2l(a_{2}-a_{1})}\mathfrak{B}(\tfrac{a_{1}}{m},\tfrac{a_{2}}{m}),\\\
\lim_{s\to
0^{-}}|s|^{\frac{a_{1}+a_{2}-m}{m}}\mathfrak{G}^{l}_{a_{1},a_{2}}(1,s)&=\theta_{0}^{(2l+1)(a_{2}-a_{1})}\mathfrak{B}(\tfrac{a_{1}}{m},\tfrac{a_{2}}{m}).\end{split}$

In view of (3.28), if $\frac{a_{1}}{m}\in{\mathbb{Z}}_{\leq 0}$ or
$\frac{a_{2}}{m}\in{\mathbb{Z}}_{\leq 0}$ then the limit is zero. For this
reason, we extend the definition of the function $\mathfrak{B}$ by letting

(4.6) $\mathfrak{B}(x,y)=0\text{ if }x\in{\mathbb{Z}}_{\leq 0}\text{ or
}y\in{\mathbb{Z}}_{\leq 0}.$

In the next three lemmas, we present a crucial decomposition of the map
$\mathfrak{G}^{l}_{a_{1},a_{2}}$ depending on the value of $a=a_{1}+a_{2}$. We
deal with three cases: $a>m$, $a=m$ and $a<m$.

######  Lemma 4.1.

Suppose that $a=a_{1}+a_{2}>m$. For every $0<r<1$ there exist
$\rho^{\pm},\varrho^{\pm}\in C^{\omega}([0,r])$ such that if $0<u\leq 1$ and
$0<|s|\leq ru$ then

(4.7)
$\displaystyle\begin{split}\mathfrak{G}^{l}_{a_{1},a_{2}}(u,s)&\\!=\\!\theta_{0}^{2l(a_{2}-a_{1})}\\!\big{(}\mathfrak{B}(\tfrac{a_{1}}{m},\tfrac{a_{2}}{m})|s|^{-\frac{a-m}{m}}+\rho^{+}(|s|)+u^{-\frac{a-m}{m}}\varrho^{+}(\tfrac{|s|}{u})\big{)}\text{
if }s>0,\\\
\mathfrak{G}^{l}_{a_{1},a_{2}}(u,s)&\\!=\\!\theta_{0}^{(2l+1)(a_{2}-a_{1})}\\!\big{(}\mathfrak{B}(\tfrac{a_{1}}{m},\tfrac{a_{2}}{m})|s|^{-\frac{a-m}{m}}\\!+\\!\rho^{-}(|s|)\\!+\\!u^{-\frac{a-m}{m}}\\!\varrho^{-}(\tfrac{|s|}{u})\big{)}\text{
if }s<0.\end{split}$

###### Proof.

In view of (3.24) and (4.4), it suffices to show the first line of (4.7) for
$l=0$.

By change of variables used twice, for every $s,u\in(0,1]$,

$\displaystyle\mathfrak{G}^{0}_{a_{1},a_{2}}(u,s)$
$\displaystyle=s^{-\frac{a-m}{m}}\int_{-1/s}^{u/s}G_{0}(x+\iota)^{-a_{1}}\overline{G_{0}(x+\iota)}^{{-a_{2}}}dx$
$\displaystyle=s^{-\frac{a-m}{m}}\Big{(}\int_{-\infty}^{+\infty}-\int_{-\infty}^{-1/s}-\int_{u/s}^{+\infty}\Big{)}G_{0}(x+\iota)^{-a_{1}}\overline{G_{0}(x+\iota)}^{{-a_{2}}}dx$
$\displaystyle=s^{-\frac{a-m}{m}}\mathfrak{B}(\tfrac{a_{1}}{m},\tfrac{a_{2}}{m})+s^{1-\frac{a}{m}}\Big{(}\int_{0}^{s/u}\frac{\xi_{+}(t)}{t^{2-\frac{a}{m}}}dt+\int_{0}^{s}\frac{\xi_{-}(t)}{t^{2-\frac{a}{m}}}dt\Big{)},$

where $\xi_{\pm}:{\mathbb{R}}\to{\mathbb{C}}$, $\xi_{\pm}(t)=-G_{0}(\pm
1+\iota t)^{-a_{1}}\overline{G_{0}(\pm 1+\iota t)}^{{-a_{2}}}$ is an analytic
map with the radius of convergence at $0$ equal to $1$. Then for every $0<r<1$
let $\sum_{n\geq 0}|c^{\pm}_{n}|r^{n}<+\infty$ such that $\sum_{n\geq
0}c^{\pm}_{n}t^{n}$ tends to $\xi_{\pm}(t)$ uniformly on $[0,r]$. As
$\frac{a}{m}>1$,

$\frac{1}{t^{2-\frac{a}{m}}}\sum_{n\geq 1}c^{\pm}_{n}t^{n}=\sum_{n\geq
1}c^{\pm}_{n}t^{(n-1)+\frac{a}{m}-1}\text{ tends on $[0,r]$ uniformly to
}\frac{\xi_{\pm}(t)-c^{\pm}_{0}}{t^{2-\frac{a}{m}}}.$

It follows that

$s^{1-\frac{a}{m}}\int_{0}^{s}\frac{(\xi_{\pm}(t)-c^{\pm}_{0})}{t^{2-\frac{a}{m}}}dt=s^{1-\frac{a}{m}}\sum_{n\geq
1}c^{\pm}_{n}\int_{0}^{s}t^{(n-1)+\frac{a}{m}-1}dt=\sum_{n\geq
1}\frac{c^{\pm}_{n}s^{n}}{n+\frac{a}{m}-1}.$

Since $\sum_{n\geq 0}|c^{\pm}_{n}|r^{n}<+\infty$, the map
$s^{1-\frac{a}{m}}\int_{0}^{s}\frac{(\xi_{\pm}(t)-c^{\pm}_{0})}{t^{2-\frac{a}{m}}}dt\in
C^{\omega}([0,r])$. Moreover,

$s^{1-\frac{a}{m}}\int_{0}^{s}\frac{\xi_{\pm}(t)}{t^{2-\frac{a}{m}}}dt=\frac{c^{\pm}_{0}}{\frac{a}{m}-1}+s^{1-\frac{a}{m}}\int_{0}^{s}\frac{(\xi_{\pm}(t)-c^{\pm}_{0})}{t^{2-\frac{a}{m}}}dt,$

so
$\widetilde{\xi}_{\pm}(s)=s^{1-\frac{a}{m}}\int_{0}^{s}\frac{\xi_{\pm}(t)}{t^{2-\frac{a}{m}}}dt\in
C^{\omega}([0,r])$. As

$\mathfrak{G}^{0}_{a_{1},a_{2}}(u,s)=s^{-\frac{a-m}{m}}\mathfrak{B}(\tfrac{a_{1}}{m},\tfrac{a_{2}}{m})+u^{-\frac{a-m}{m}}\widetilde{\xi}_{+}(s/u)+\widetilde{\xi}_{-}(s)\text{
if }0\leq s/u\leq r,$

this completes the proof of (4.7). ∎

By definition, for every natural number $n$ if $x,y\notin{\mathbb{Z}}$ and
$x+y\notin{\mathbb{Z}}_{\leq 1}$ then

(4.8)
$(2\iota)^{n}\tbinom{-y}{n}\mathfrak{B}(x,y+n)=\tbinom{x+y+n-2}{n}\mathfrak{B}(x,y)=(-2\iota)^{n}\tbinom{-x}{n}\mathfrak{B}(x+n,y).$

We can extend again the domain of the function $\mathfrak{B}$ by adding the
pairs $(x,y)$ such that $x,y\notin{\mathbb{Z}}$ and $x+y\in{\mathbb{Z}}_{\leq
1}$. For every such pair we let $\mathfrak{B}(x,y)=\frac{\pi
e^{\iota\frac{\pi}{2}(y-x)}}{2^{x+y-2}}\frac{\Gamma(x+y-1)}{\Gamma(x)\Gamma(y)}$,
where we adopt the convention

$\Gamma(0):=\lim_{x\to 0}x\Gamma(x)=1\text{ and
}\Gamma(-n):=\frac{\Gamma(0)}{(-1)\cdots(-n)}=\frac{1}{(-1)\cdots(-n)}$

for any $n\in{\mathbb{N}}$. Then we also have

(4.9)
$\tbinom{-x}{n}\mathfrak{B}(x+n,y-n)=(-1)^{n}\tbinom{-y+n}{n}\mathfrak{B}(x,y).$

The extended $\Gamma$-function satisfies $\Gamma(x+1)=x\Gamma(x)$ for all
$x\in{\mathbb{R}}\setminus\\{0\\}$ and $\Gamma(1)=\Gamma(0)=1$. It follows
that (4.8) holds even when $x+y+n\in{\mathbb{Z}}_{\leq 1}$.

Finally note that, if $x+y=1$ then we also have

(4.10) $\mathfrak{B}(x,y)=\frac{2\pi
e^{\iota\pi(y-1/2)}}{\Gamma(1-y)\Gamma(y)}=-2\iota e^{\iota\pi y}\sin(\pi
y)=1-e^{2\iota\pi y}=1+e^{\iota\pi(y-x)}.$

######  Lemma 4.2.

Suppose that $a=a_{1}+a_{2}=m$. There exist $\rho^{\pm},\varrho^{\pm}\in
C^{\omega}([0,1])$ such that for all $0<u\leq 1$ and $0<|s|\leq u$,

(4.11)
$\displaystyle\begin{split}&\mathfrak{G}^{l}_{a_{1},a_{2}}(u,s)\\!=\\!\theta_{0}^{2l(a_{2}-a_{1})}\\!\big{(}\\!-\\!\mathfrak{B}(\tfrac{a_{1}}{m},\tfrac{a_{2}}{m})\\!\log|s|\\!+\\!\log
u\\!+\\!\rho^{+}(|s|)\\!+\\!\varrho^{+}(\tfrac{|s|}{u})\big{)}\\!\text{ if
}s>0,\\\
&\mathfrak{G}^{l}_{a_{1},a_{2}}(u,s)\\!=\\!\theta_{0}^{(2l+1)(a_{2}-a_{1})}\\!\big{(}\\!-\\!\mathfrak{B}(\tfrac{a_{1}}{m},\tfrac{a_{2}}{m})\\!\log|s|\\!+\\!\theta_{0}^{a_{2}-a_{1}}\log
u\\!+\\!\rho^{-}(|s|)\\!+\\!\varrho^{-}(\tfrac{|s|}{u})\big{)}\\!\text{ if
}s<0.\end{split}$

###### Proof.

In view of (3.24) and (4.4), it suffices to show the first line of (4.11) for
$l=0$.

By change of variables, for every $u,s\in(0,1]$,

$\displaystyle\psi(u,s)$ $\displaystyle:=\int_{0}^{u}G_{0}(v+\iota
s)^{-a_{1}}\overline{G_{0}(v+\iota s)}^{-a_{2}}dv-\int_{0}^{u}\frac{1}{v-\iota
s}dv$ $\displaystyle=\int_{0}^{u}\left(\left(\frac{\overline{G_{0}(v+\iota
s)}}{G_{0}(v+\iota s)}\right)^{a_{1}}-1\right)\frac{1}{v-\iota
s}dv=\int_{0}^{u/s}\frac{\left(\left(\frac{\overline{G_{0}(x+\iota)}}{G_{0}(x+\iota)}\right)^{a_{1}}-1\right)}{x-\iota}dx.$

It follows that $\psi(u,s)=\widetilde{\psi}(s/u)$, where

$\widetilde{\psi}^{\prime}(x)=-\frac{1}{x^{2}}\frac{\left(\frac{\overline{G_{0}(1/x+\iota)}}{G_{0}(1/x+\iota)}\right)^{a_{1}}-1}{1/x-\iota}=\frac{\left(\frac{\overline{G_{0}(1+\iota
x)}}{G_{0}(1+\iota x)}\right)^{a_{1}}-1}{x}\frac{1}{\iota x-1}.$

As the map $x\mapsto\left(\frac{\overline{G_{0}(1+\iota x)}}{G_{0}(1+\iota
x)}\right)^{a_{1}}-1$ is real analytic and vanishes at $0$,
$\widetilde{\psi}^{\prime}$ is also analytic. Hence $\widetilde{\psi}\in
C^{\omega}({\mathbb{R}})$. Moreover,

$\displaystyle\int_{0}^{u}\frac{1}{v-\iota s}dv=\int_{0}^{u}\frac{v+\iota
s}{{v^{2}+s^{2}}}dv=\log\sqrt{u^{2}+s^{2}}-\log
s+\iota\operatorname{arccot}(s/u).$

Hence

$\int_{0}^{u}G_{0}(v+\iota s)^{-a_{1}}\overline{G_{0}(v+\iota
s)}^{-a_{2}}dv=-\log(s/u)+\varrho^{+}(s/u),$

where
$\varrho^{+}(x)=\log\sqrt{1+x^{2}}+\widetilde{\psi}(x)+\iota\operatorname{arccot}(x)$
is analytic. In particular,

(4.12) $\int_{0}^{1}G_{0}(v+\iota s)^{-a_{1}}\overline{G_{0}(v+\iota
s)}^{-a_{2}}dv=-\log s+\varrho^{+}(s).$

Since

$\int_{-1}^{0}G_{0}(v+\iota s)^{-a_{1}}\overline{G_{0}(v+\iota
s)}^{-a_{2}}dv=\int_{0}^{1}G_{0}(-v+\iota s)^{-a_{1}}\overline{G_{0}(-v+\iota
s)}^{-a_{2}}dv$

and $G_{0}(-v+\iota s)=\theta_{0}\overline{G_{0}(v+\iota s)}$ if
$s,v\in(0,1]$, we get

$\int_{-1}^{0}G_{0}(v+\iota s)^{-a_{1}}\overline{G_{0}(v+\iota
s)}^{-a_{2}}dv=\theta_{0}^{(a_{2}-a_{1})}\int_{0}^{1}G_{0}(v+\iota
s)^{-a_{2}}\overline{G_{0}(v+\iota s)}^{-a_{1}}dv.$

In view of (4.12), this gives

$\mathfrak{G}^{0}_{a_{1},a_{2}}(u,s)=-(1+\theta_{0}^{(a_{2}-a_{1})})\log
s+\log
u+\theta_{0}^{(a_{2}-a_{1})}\overline{\varrho^{+}(s)}+\varrho^{+}(s/u).$

Since, by (4.10),
$1+\theta_{0}^{(a_{2}-a_{1})}=1+e^{\pi\iota(\frac{a_{2}}{m}-\frac{a_{1}}{m})}=\mathfrak{B}(\tfrac{a_{1}}{m},\tfrac{a_{2}}{m})$,
which gives the first line of (4.11). ∎

######  Lemma 4.3.

Suppose that $a=a_{1}+a_{2}<m$ and
$\frac{a_{1}}{m},\frac{a_{2}}{m}\notin{\mathbb{Z}}$. If
$\frac{a}{m}\notin{\mathbb{Z}}$ then for every $0<r<1$ there exist
$\rho^{\pm},\varrho^{\pm}\in C^{\omega}([0,r])$ such that if $0<u\leq 1$ and
$0<|s|\leq ru$ then

(4.13)
$\displaystyle\begin{split}\mathfrak{G}^{l}_{a_{1},a_{2}}(u,s)&\\!=\\!\theta_{0}^{2l(a_{2}-a_{1})}\big{(}\mathfrak{B}(\tfrac{a_{1}}{m},\tfrac{a_{2}}{m})|s|^{\frac{m-a}{m}}\\!+\\!\rho^{+}(|s|)\\!+\\!u^{\frac{m-a}{m}}\varrho^{+}(\tfrac{|s|}{u})\big{)}\\!\text{
if }s>0,\\\
\mathfrak{G}^{l}_{a_{1},a_{2}}(u,s)&\\!=\\!\theta_{0}^{(2l+1)(a_{2}-a_{1})}\big{(}\mathfrak{B}(\tfrac{a_{1}}{m},\tfrac{a_{2}}{m})|s|^{\frac{m-a}{m}}\\!+\\!\rho^{-}(|s|)\\!+\\!u^{\frac{m-a}{m}}\varrho^{-}(\tfrac{|s|}{u})\big{)}\\!\text{
if }s<0.\end{split}$

If $\frac{a}{m}\in{\mathbb{Z}}$ then there exist $\rho^{\pm},\varrho^{\pm}\in
C^{\omega}([0,1])$ and $c_{\pm}\in{\mathbb{C}}$ such that if $0<u\leq 1$ and
$0<|s|\leq u$ then

(4.14)
$\displaystyle\begin{split}\mathfrak{G}^{l}_{a_{1},a_{2}}(u,s)&=\theta_{0}^{2l(a_{2}-a_{1})}\Big{(}-\mathfrak{B}(\tfrac{a_{1}}{m},\tfrac{a_{2}}{m})|s|^{\frac{m-a}{m}}\log|s|\\\
&\quad+c_{+}|s|^{\frac{m-a}{m}}\log
u+\rho^{+}(|s|)+u^{\frac{m-a}{m}}\varrho^{+}(\tfrac{|s|}{u})\Big{)}\text{ if
}s>0,\\\
\mathfrak{G}^{l}_{a_{1},a_{2}}(u,s)&=\theta_{0}^{(2l+1)(a_{2}-a_{1})}\Big{(}-\mathfrak{B}(\tfrac{a_{1}}{m},\tfrac{a_{2}}{m})|s|^{\frac{m-a}{m}}\log|s|\\\
&\quad+c_{-}|s|^{\frac{m-a}{m}}\log
u+\rho^{-}(|s|)+u^{\frac{m-a}{m}}\varrho^{-}(\tfrac{|s|}{u})\Big{)}\text{ if
}s<0.\end{split}$

###### Proof.

In view of (3.24) and (4.4), it suffices to show the first line of (4.13) and
(4.14) for $l=0$. Let $n=\lceil\frac{m-a}{m}\rceil$. By (4.3), for every
$k\geq 0$,

$\frac{d^{k}}{ds^{k}}\mathfrak{G}^{0}_{a_{1},a_{2}}=\iota^{k}\sum_{0\leq j\leq
k}k!(-1)^{k-j}\binom{-\frac{a_{1}}{m}}{j}\binom{-\frac{a_{2}}{m}}{k-j}\mathfrak{G}^{0}_{a_{1}+jm,a_{2}+(k-j)m}.$

A direct computation shows that if $a=a_{1}+a_{2}<m$ and $u\in[0,1]$ then

$\mathfrak{G}^{0}_{a_{1},a_{2}}(u,0)=\frac{m}{m-a}(u^{\frac{m-a}{m}}+\theta_{0}^{a_{2}-a_{1}}).$

It follows that if $k<\frac{m-a}{m}$ (i.e. $k<n$) then there exist
$c_{k,1},c_{k,0}\in{\mathbb{C}}$ such that

(4.15)
$\frac{d^{k}}{ds^{k}}\mathfrak{G}^{0}_{a_{1},a_{2}}(u,0)=c_{k,1}u^{\frac{m-a}{m}-k}+c_{k,0}.$

If $\frac{a}{m}\notin{\mathbb{Z}}$ then for any $0\leq j\leq n$ we have
$a_{1}+jm+a_{2}+(n-j)m=a+nm>m$. Hence, by Lemma 4.1, there exist
$\rho_{n}^{+},\varrho_{n}^{+}\in C^{\omega}([0,r])$ such that for all $0<u\leq
1$ and $0<s\leq ru$,

$\displaystyle\frac{d^{n}}{ds^{n}}$
$\displaystyle\mathfrak{G}^{0}_{a_{1},a_{2}}(u,s)=\rho_{n}^{+}(s)+u^{\frac{m-a}{m}-n}\varrho_{n}^{+}(\tfrac{s}{u})$
$\displaystyle+\iota^{n}\sum_{0\leq j\leq
n}n!(-1)^{n-j}\binom{-\frac{a_{1}}{m}}{j}\binom{-\frac{a_{2}}{m}}{n-j}\mathfrak{B}(\tfrac{a_{1}}{m}+j,\tfrac{a_{2}}{m}+n-j)s^{\frac{m-a}{m}-n}.$

If $\frac{a}{m}\in{\mathbb{Z}}$ then $n=\frac{m-a}{m}$ and
$a_{1}+jm+a_{2}+(n-j)m=a+nm=m$. Hence, by Lemma 4.2, there exist
$\rho_{n}^{+},\varrho_{n}^{+}\in C^{\omega}([0,1])$ such that for all $0<u\leq
1$ and $0<s\leq u$,

$\displaystyle\frac{d^{n}}{ds^{n}}$
$\displaystyle\mathfrak{G}^{0}_{a_{1},a_{2}}(u,s)=\rho_{n}^{+}(s)+\varrho_{n}^{+}(\tfrac{s}{u})$
$\displaystyle+\iota^{n}\sum_{0\leq j\leq
n}n!(-1)^{n-j}\binom{-\frac{a_{1}}{m}}{j}\binom{-\frac{a_{2}}{m}}{n-j}\big{(}-\mathfrak{B}(\tfrac{a_{1}}{m}+j,\tfrac{a_{2}}{m}+n-j)\log
s+\log u).$

By (4.8) (and its extension in the integer case),

$\displaystyle\iota^{n}\sum_{0\leq j\leq
n}n!(-1)^{n-j}\binom{-\frac{a_{1}}{m}}{j}\binom{-\frac{a_{2}}{m}}{n-j}\mathfrak{B}(\tfrac{a_{1}}{m}+j,\tfrac{a_{2}}{m}+n-j)$
$\displaystyle=(-1/2)^{n}\sum_{0\leq j\leq
n}n!\binom{n}{j}\binom{\frac{a}{m}+n-2}{n}\mathfrak{B}(\tfrac{a_{1}}{m},\tfrac{a_{2}}{m})=(-1)^{n}n!\binom{\frac{a}{m}+n-2}{n}\mathfrak{B}(\tfrac{a_{1}}{m},\tfrac{a_{2}}{m}).$

Therefore, in the non-integer case, for all $0<u\leq 1$ and $0<s\leq ru$,

$\frac{d^{n}}{ds^{n}}\mathfrak{G}^{0}_{a_{1},a_{2}}(u,s)=\rho_{n}^{+}(s)+u^{\frac{m-a}{m}-n}\varrho_{n}^{+}(\tfrac{s}{u})+(-1)^{n}n!\tbinom{-(\frac{m-a}{m}-(n-1))}{n}\mathfrak{B}(\tfrac{a_{1}}{m},\tfrac{a_{2}}{m})s^{\frac{m-a}{m}-n}.$

In the integer case, for all $0<u\leq 1$ and $0<s\leq u$,

$\frac{d^{n}}{ds^{n}}\mathfrak{G}^{0}_{a_{1},a_{2}}(u,s)=\rho_{n}^{+}(s)+\varrho_{n}^{+}(\tfrac{s}{u})-n!\mathfrak{B}(\tfrac{a_{1}}{m},\tfrac{a_{2}}{m})\log
s+c_{n}\log u.$

Since

$\frac{d^{k}}{ds^{k}}\mathfrak{G}^{0}_{a_{1},a_{2}}(u,s)=\frac{d^{k}}{ds^{k}}\mathfrak{G}^{0}_{a_{1},a_{2}}(u,0)+\int_{0}^{s}\frac{d^{k+1}}{ds^{k+1}}\mathfrak{G}^{0}_{a_{1},a_{2}}(u,t)dt\text{
for all }0\leq k<n,$

using the formulae for $\frac{d^{n}}{ds^{n}}\mathfrak{G}^{0}_{a_{1},a_{2}}$
together with (4.15) and induction, we obtain (4.13) and (4.14). ∎

######  Remark 4.4.

To summarize, by Lemmas 4.1, 4.2 and 4.3, for any pair of integer numbers
$a_{1},a_{2}$ such that $\frac{a_{1}}{m},\frac{a_{2}}{m}\notin{\mathbb{Z}}$ if
$\frac{m-a}{m}\notin{\mathbb{Z}}$ ($a=a_{1}+a_{2}$) or
$\frac{m-a}{m}\in{\mathbb{Z}}_{<0}$ then

(4.16)
$\displaystyle\begin{split}\mathfrak{G}^{l}_{a_{1},a_{2}}(1,s)&=\theta_{0}^{2l(a_{2}-a_{1})}\mathfrak{B}(\tfrac{a_{1}}{m},\tfrac{a_{2}}{m})|s|^{\frac{m-a}{m}}+C^{\omega}((0,1]),\\\
\mathfrak{G}^{l}_{a_{1},a_{2}}(1,s)&=\theta_{0}^{(2l+1)(a_{2}-a_{1})}\mathfrak{B}(\tfrac{a_{1}}{m},\tfrac{a_{2}}{m})|s|^{\frac{m-a}{m}}+C^{\omega}([-1,0)).\end{split}$

If $\frac{m-a}{m}\in{\mathbb{Z}}_{\geq 0}$ then

(4.17)
$\displaystyle\begin{split}\mathfrak{G}^{l}_{a_{1},a_{2}}(1,s)&=-\theta_{0}^{2l(a_{2}-a_{1})}\mathfrak{B}(\tfrac{a_{1}}{m},\tfrac{a_{2}}{m})|s|^{\frac{m-a}{m}}\log|s|+C^{\omega}((0,1]),\\\
\mathfrak{G}^{l}_{a_{1},a_{2}}(1,s)&=-\theta_{0}^{(2l+1)(a_{2}-a_{1})}\mathfrak{B}(\tfrac{a_{1}}{m},\tfrac{a_{2}}{m})|s|^{\frac{m-a}{m}}\log|s|+C^{\omega}([-1,0)).\end{split}$

Indeed, in the non-integer case, we obtain the analyticity of the remainder
only on intervals $[-r,0]$ and $[0,r]$ for any $0<r<1$. Nevertheless, for any
choice of integer $a_{1},a_{2}$, the function
$\mathfrak{G}^{l}_{a_{1},a_{2}}(1,s)$ is analytic on $[r,1]$ and $[-1,-r]$ for
any $0<r<1$. This gives our claim.

###  4.2. Evaluation of asymptotic factors for $\varphi_{f,l}$

The behaviour of higher derivatives of $\varphi_{f,l}$ at zero is evaluated by
linear combinations of invariant distributions $\partial_{j}^{k}$. For this
reason, we define a list of new functionals
$\mathscr{C}^{k}_{l}:C^{k}(\mathcal{D})\to{\mathbb{C}}$ for $k\geq 0$ and
$0\leq l<2m$ given by

$\displaystyle\mathscr{C}^{k}_{l}(f)=\sum_{\begin{subarray}{c}0\leq j\leq
k\wedge(m-2)\\\ j\neq
k-(m-1)\operatorname{mod}m\end{subarray}}\theta_{0}^{l(2j-k)}\mathfrak{B}(\tfrac{(m-1)-j}{m},\tfrac{(m-1)-(k-j)}{m})\partial_{j}^{k}(f).$

As we will see in the next theorem, the values of $\mathscr{C}^{k}_{l}(f)$ for
$k\geq 0$ measure the effect that the appearance of orbits in the $l$-th
angular sector has on the value of ergodic integrals of the function $f$.
Comparing with (1.5), functionals $\mathscr{C}^{k}_{l}$ play a key role in
understanding the meaning of distribution $\mathfrak{C}^{k}_{\sigma,l}$.

From now on, we adopt the convention $\tbinom{0}{n}:=\lim_{x\to
0}\tbinom{x}{n}/x=\frac{(-1)^{n-1}}{n}$.

######  Theorem 4.5.

For any $k\geq 0$ let $n=\lceil\frac{k-(m-2)}{m}\rceil$ and
$b=n-\frac{k-(m-2)}{m}$. Suppose that $f\in C^{k\vee(n+1)}(\mathcal{D})$ is
such that $\partial^{j}_{i}(f)=0$ for all $0\leq j<k$ and $0\leq i\leq
j\wedge(m-2)$ with $i\neq j-(m-1)\operatorname{mod}m$. Then $\varphi_{f,l}\in
C^{n+\operatorname{P_{b}}}([-1,0)\cup(0,1])$ and there exists $C>0$ such that
$\|\varphi_{f,l}\|_{C^{n+\operatorname{P_{b}}}([-1,0)\cup(0,1])}\leq
C\|f\|_{C^{k\vee(n+1)}(\mathcal{D})}$. Moreover, for every $0\leq l<m$,

(4.18) $\displaystyle\lim_{s\to 0^{+}}|s|^{b+1}D^{n+1}\varphi_{f,l}(s)$
$\displaystyle=(-1)^{n+1}\frac{(n+1)!}{k!}\binom{b}{n+1}\mathscr{C}^{k}_{2l}(f),$
(4.19) $\displaystyle\lim_{s\to 0^{-}}|s|^{b+1}D^{n+1}\varphi_{f,l}(s)$
$\displaystyle=\frac{(n+1)!}{k!}\binom{b}{n+1}\mathscr{C}^{k}_{2l+1}(f).$

######  Remark 4.6.

Before the proof, let us note that

$k\vee(n+1)=\left\\{\begin{array}[]{cl}k+1&\text{if }k=0\text{ or }(k=1\text{
with }m=2),\\\ k&\text{otherwise.}\end{array}\right.$

Indeed, the inequality $\frac{k-(m-2)}{m}+1\leq k$ is equivalent to $2\leq
k(m-1)$. It follows that if $k\geq 1$ with $m\geq 3$ or $k\geq 2$ then $n<k$,
so $k\vee(n+1)=k$.

###### Proof.

Let us decompose $f=f_{<k}+f_{k}+e_{f}$ with

$\displaystyle f_{<k}(\omega,\overline{\omega})=\sum_{0\leq
j<k}\frac{1}{j!}\sum_{0\leq i\leq
j}\binom{j}{i}\frac{\partial^{j}f}{\partial\omega^{i}\partial\overline{\omega}^{j-i}}(0,0)\omega^{i}\overline{\omega}^{j-i},$
$\displaystyle f_{k}(\omega,\overline{\omega})=\frac{1}{k!}\sum_{0\leq i\leq
k}\binom{k}{i}\frac{\partial^{k}f}{\partial\omega^{i}\partial\overline{\omega}^{k-i}}(0,0)\omega^{i}\overline{\omega}^{k-i}.$

By Lemma 3.10,

(4.20) $\displaystyle\varphi_{f_{<k},l}\in C^{\omega}([-1,0))\cap
C^{\omega}((0,1])\text{ for }0\leq l<m,$ (4.21)
$\displaystyle\|\varphi_{f_{<k},l}\|_{C^{n+1}([-1,0)\cup(0,1])}\leq
C^{n+1}_{k}\|f\|_{C^{k}(\mathcal{D})}.$

Since $D^{j}(f_{k}+e_{f})=0$ for every $0\leq j<k$, in view of (3.16), if
$(j_{1},j_{2})\in{\mathbb{Z}}^{2}_{\geq 0}$ is such that $j_{1}+j_{2}=j\leq
n+1$ then

$\displaystyle\left|\frac{\partial^{j}{\mathscr{F}_{f_{k}+e_{f},l}}(z,{\overline{z}})}{\partial
z^{j_{1}}\partial{\overline{z}}^{j_{2}}}\right|$
$\displaystyle=O\big{(}\|f_{k}+e_{f}\|_{C^{k\vee(n+1)}(\mathcal{D})}\langle|\Im
z|\rangle^{-(\frac{(m-2)-k}{m}+j)}\big{)}$
$\displaystyle=O\big{(}\|f\|_{C^{k\vee(n+1)}(\mathcal{D})}\langle|\Im
z|\rangle^{(n+1-j)-(b+1)}\big{)}.$

As
$\frac{d^{j}}{ds^{j}}=\left(\iota\left(\frac{\partial}{\partial{z}}-\frac{\partial}{\partial\overline{z}}\right)\right)^{j}$,
this gives

(4.22) $\displaystyle|D^{j}\varphi_{f_{k}+e_{f},l}(s)|$
$\displaystyle=O\big{(}\|f\|_{C^{k\vee(n+1)}(\mathcal{D})}\big{)}\text{ if
}0\leq j\leq n-1,$ (4.23) $\displaystyle|D^{n}\varphi_{f_{k}+e_{f},l}(s)|$
$\displaystyle=O\big{(}\|f\|_{C^{k\vee(n+1)}(\mathcal{D})}\langle|s|\rangle^{-b}\big{)},$
(4.24) $\displaystyle|D^{n+1}\varphi_{f_{k}+e_{f},l}(s)|$
$\displaystyle=O\big{(}\|f\|_{C^{k\vee(n+1)}(\mathcal{D})}|s|^{b+1}\big{)}.$

By (4.23),

$\|D^{n}\varphi_{f_{k}+e_{f},l}\|_{L^{1}}=O\big{(}\|f\|_{C^{k\vee(n+1)}(\mathcal{D})}\big{)}.$

In view of (4.21), (4.22) and (4.24), this gives

$\|\varphi_{f,l}\|_{C^{n+\operatorname{P_{b}}}}\leq\|\varphi_{f_{<k},l}\|_{C^{n+\operatorname{P_{b}}}}+\|\varphi_{f_{k}+e_{f},l}\|_{C^{n+\operatorname{P_{b}}}}=O\big{(}\|f\|_{C^{k\vee(n+1)}(\mathcal{D})}\big{)}.$

Since $D^{j}(e_{f})=0$ for every $0\leq j\leq k$, we also have

(4.25) $|D^{n+1}\varphi_{e_{f},l}(s)|=o(|s|^{-(b+1)}).$

Indeed, if $k\vee(n+1)=k+1$, i.e. $k=n$ then, again by (3.16),

$\displaystyle\left\|D^{n+1}{\mathscr{F}_{e_{f},l}}(z,{\overline{z}})\right\|=O(\langle|\Im
z|\rangle^{-(\frac{(m-2)-(k+1)}{m}+n+1)})=O(|\Im z|^{-(b+1)+\frac{1}{m}}).$

If $k\vee(n+1)=k$, i.e. $n+1\leq k$ then, by (3.18),

$\displaystyle\left\|D^{n+1}{\mathscr{F}_{e_{f},l}}(z,{\overline{z}})\right\|=o(|\Im
z|^{-(\frac{(m-2)-k}{m}+n+1)})=o(|\Im z|^{-(b+1)}).$

Both yield (4.25).

Therefore, by (4.20) and (4.25),

(4.26)
$|s|^{b+1}D^{n+1}\varphi_{f,l}(s)=|s|^{b+1}D^{n+1}\varphi_{f_{k},l}(s)+o(1).$

By (3.32) (see the proof of Lemma 3.10),

$\mathscr{F}_{f_{k},l}(1,s)\\!=\\!\frac{1}{k!}\\!\sum_{\begin{subarray}{c}0\leq
j\leq k\wedge(m-2)\\\ j\neq
k-(m-1)\operatorname{mod}m\end{subarray}}\\!\partial_{j}^{k}(f)\mathfrak{G}^{l}_{(m-1)-j,(m-1)-(k-j)}(1,s)+C^{\omega}((0,1])\cap
C^{\omega}([-1,0)).$

As $\frac{m-((m-1)-j+(m-1)-(k-j))}{m}=\frac{k-(m-2)}{m}=n-b$, by the
definition of ${\mathscr{C}}^{k}_{l}(f)$ and (4.16),

(4.27)
$\varphi_{f_{k},l}(s)=\frac{{\mathscr{C}}^{k}_{2l}(f)}{k!}|s|^{n-b}+C^{\omega}((0,1]),\
\varphi_{f_{k},l}(s)=\frac{{\mathscr{C}}^{k}_{2l+1}(f)}{k!}|s|^{n-b}+C^{\omega}([-1,0))$

if $0<b<1$ and, by (4.17),

(4.28)
$\displaystyle\begin{split}\varphi_{f_{k},l}(s)&=-\frac{\mathscr{C}^{k}_{2l}(f)}{k!}|s|^{n}\log|s|+C^{\omega}((0,1]),\\\
\varphi_{f_{k},l}(s)&=-\frac{{\mathscr{C}}^{k}_{2l+1}(f)}{k!}|s|^{n}\log|s|+C^{\omega}([-1,0))\end{split}$

if $b=0$. After $n+1$ times differentiation, it follows that

$\displaystyle\begin{split}&D^{n+1}\varphi_{f_{k},l}(s)=|s|^{-(b+1)}(-1)^{n+1}\frac{(n+1)!}{k!}\binom{b}{n+1}\mathscr{C}^{k}_{2l}(f)+C^{\omega}((0,1]),\\\
&D^{n+1}\varphi_{f_{k},l}(s)=|s|^{-(b+1)}\frac{(n+1)!}{k!}\binom{b}{n+1}\mathscr{C}^{k}_{2l+1}(f)+C^{\omega}([-1,0)).\end{split}$

Finally, by (4.20) and (4.26), this yields (4.18) and (4.19). ∎

We also verify a similar result when we restrict our attention only to an
individual $l$-th sector. Then we assume only the vanishing of functionals
$\mathscr{C}^{j}_{l}$.

######  Theorem 4.7.

Let $k\geq 0$, $0\leq l<m$ and $\epsilon\in\\{0,1\\}$. Suppose that $f\in
C^{k\vee(n+1)}(\mathcal{D})$ and $\mathscr{C}^{j}_{2l+\epsilon}(f)=0$ for all
$0\leq j<k$. Then $\varphi_{f,l}\in
C^{n+\operatorname{P_{b}}}((0,(-1)^{\epsilon}])$ with

(4.29) $\lim_{\begin{subarray}{c}s\to 0\\\
s\in(0,(-1)^{\epsilon}]\end{subarray}}|s|^{b+1}D^{n+1}\varphi_{f,l}(s)=(-1)^{(1-\epsilon)(n+1)}\frac{(n+1)!}{k!}\binom{b}{n+1}\mathscr{C}^{k}_{2l+\epsilon}(f)$

and

(4.30) $\text{there exists $C>0$ such that
}\|\varphi_{f,l}\|_{C^{n+\operatorname{P_{b}}}((0,(-1)^{\epsilon}])}\leq
C\|f\|_{C^{k\vee(n+1)}(\mathcal{D})}.$

In particular, if $k\geq m-1$ then $\varphi_{f,l}\in
C^{\mathfrak{e}(\sigma,k)}((0,(-1)^{\epsilon}])$ and there exists $C>0$ such
that $\|\varphi_{f,l}\|_{C^{\mathfrak{e}(\sigma,k)}((0,(-1)^{\epsilon}])}\leq
C\|f\|_{C^{k\vee(n+1)}(\mathcal{D})}$.

On the other hand, if $f\in C^{k\vee(n+1)}(\mathcal{D})$ is such that
$\varphi_{f,l}\in C^{r}((0,(-1)^{\epsilon}])$ for some
$r\in{\mathbb{R}}_{\eta}$ with $0<v(r)\leq\mathfrak{o}(\sigma,k)$ then
$\mathscr{C}^{j}_{2l+\epsilon}(f)=0$ for all $j\geq 0$ such that
$\mathfrak{o}(\sigma,j)<v(r)$.

###### Proof.

We will focus only on the even case, when $\epsilon=0$. The proof in the odd
case proceeds in the same way. Let us decompose $f=f_{<k}+f_{k}+e_{f}$, where
$f_{<k}=\sum_{0\leq j<k}f_{j}$ with

$\displaystyle f_{j}(\omega,\overline{\omega})=\frac{1}{j!}\sum_{0\leq i\leq
j}\binom{j}{i}\frac{\partial^{j}f}{\partial\omega^{i}\partial\overline{\omega}^{j-i}}(0,0)\omega^{i}\overline{\omega}^{j-i}.$

By (4.27), (4.28),

(4.31)
$\displaystyle\begin{split}\varphi_{f_{j},l}(s)&=\frac{{\mathscr{C}}^{j}_{2l}(f)}{j!}s^{\frac{j-(m-2)}{m}}+C^{\omega}((0,1])\text{
if }j\neq m-2\ \operatorname{mod}m,\\\
\varphi_{f_{j},l}(s)&=-\frac{\mathscr{C}^{j}_{2l}(f)}{j!}s^{\frac{j-(m-2)}{m}}\log
s+C^{\omega}((0,1]))\text{ if }j=m-2\ \operatorname{mod}m.\end{split}$

Since the operator $f\mapsto f_{j}$ takes values in the finite-dimensional
space of homogenous polynomials of degree $j$, for every $0\leq j<k$ there
exists $C_{j}>0$ such that

$\displaystyle\|\varphi_{f_{j},l}(s)-\frac{{\mathscr{C}}^{j}_{2l}(f)}{j!}s^{\frac{j-(m-2)}{m}}\|_{C^{n+\operatorname{P_{b}}}((0,1])}$
$\displaystyle\leq C_{j}\|f\|_{C^{k}(\mathcal{D})}\text{ or }$
$\displaystyle\|\varphi_{f_{j},l}(s)+\frac{\mathscr{C}^{j}_{2l}(f)}{j!}s^{\frac{j-(m-2)}{m}}\log
s\|_{C^{n+\operatorname{P_{b}}}((0,1])}$ $\displaystyle\leq
C_{j}\|f\|_{C^{k}(\mathcal{D})}.$

If $\mathscr{C}^{j}_{2l}(f)=0$ for all $0\leq j<k$, then

(4.32) $\varphi_{f_{<k},l}\in C^{\omega}((0,1])\text{ and
}\|\varphi_{f_{<k},l}\|_{C^{n+\operatorname{P_{b}}}((0,1])}\leq\sum_{0\leq
j<k}C_{j}\|f\|_{C^{k}(\mathcal{D})}.$

Again, by Theorem 4.5 applied to $f_{k}+e_{f}$, we have
$\varphi_{f_{k}+e_{f},l}\in C^{n+\operatorname{P_{b}}}((0,1])$,

$\|\varphi_{f_{k}+e_{f},l}\|_{C^{n+\operatorname{P_{b}}}((0,1])}\leq
C\|f_{k}+e_{f}\|_{C^{k\vee(n+1)}(\mathcal{D})}\leq
C^{\prime}\|f\|_{C^{k\vee(n+1)}(\mathcal{D})}$

and

$\lim_{s\to
0^{+}}s^{b+1}D^{n+1}\varphi_{f_{k}+e_{f},l}(s)=(-1)^{(n+1)}\frac{(n+1)!}{k!}\binom{b}{n+1}\mathscr{C}^{k}_{2l}(f).$

Since $\varphi_{f,l}=\varphi_{f_{<k},l}+\varphi_{f_{k}+e_{f},l}$, in view of
(4.32), this yields (4.29) and (4.30). As $n-b=\mathfrak{o}(\sigma,k)$, by
Remark 2.1, this gives $\varphi_{f,l}\in C^{\mathfrak{e}(\sigma,k)}((0,1])$.

Now suppose that $f\in C^{k\vee(n+1)}(\mathcal{D})$ is such that
$\varphi_{f,l}\in C^{r}((0,1])$ for some $r\in{\mathbb{R}}_{\eta}$ with
$0<v(r)\leq\mathfrak{o}(\sigma,k)$. Choose $m-2<j_{0}\leq k$ such that
$\mathfrak{o}(\sigma,j_{0}-1)<v(r)\leq\mathfrak{o}(\sigma,j_{0})$. By the
first part of the theorem, $\varphi_{f-f_{<j_{0}},l}\in
C^{\mathfrak{e}(\sigma,j_{0})}((0,1])$. As $\varphi_{f,l}\in C^{r}((0,1])$ and
$v(r)\leq\mathfrak{o}(\sigma,j_{0})$, it follows that
$\varphi_{f_{<j_{0}},l}\in C^{r}((0,1])$. In view of (4.31),

$\displaystyle\varphi_{f_{<j_{0}},l}(s)$
$\displaystyle=\\!\\!\\!\sum_{\begin{subarray}{c}0\leq j<j_{0}\\\ j\neq
m-2\operatorname{mod}m\end{subarray}}\\!\\!\\!\frac{{\mathscr{C}}^{j}_{2l}(f)}{j!}s^{\frac{j-(m-2)}{m}}+\\!\\!\\!\sum_{\begin{subarray}{c}0\leq
j<j_{0}\\\
j=m-2\operatorname{mod}m\end{subarray}}\\!\\!\\!\frac{\mathscr{C}^{j}_{2l}(f)}{j!}s^{\frac{j-(m-2)}{m}}(-\log
s)+C^{\omega}((0,1]).$

Therefore,

$\sum_{\begin{subarray}{c}0\leq j<j_{0}\\\ j\neq
m-2\operatorname{mod}m\end{subarray}}\frac{{\mathscr{C}}^{j}_{2l}(f)}{j!}s^{\frac{j-(m-2)}{m}}-\sum_{\begin{subarray}{c}0\leq
j<j_{0}\\\
j=m-2\operatorname{mod}m\end{subarray}}\frac{\mathscr{C}^{j}_{2l}(f)}{j!}s^{\frac{j-(m-2)}{m}}\log
s\in C^{r}((0,1])$

with $\frac{j-(m-2)}{m}\leq\mathfrak{o}(\sigma,j_{0}-1)<v(r)$ for $0\leq
j<j_{0}$. It follows that ${\mathscr{C}}^{j}_{2l}(f)=0$ for $0\leq j<j_{0}$. ∎

By the proof of Theorem 4.7, we also have the following.

######  Corollary 4.8.

Let $k\geq 0$, $0\leq l<m$ and $\epsilon\in\\{0,1\\}$. Suppose that $f\in
C^{k\vee(n+1)}(\mathcal{D})$. Then

(4.33)
$\displaystyle\begin{split}\varphi_{f,l}(s)&=-\sum_{\begin{subarray}{c}0\leq
j<k\\\
j=m-2\operatorname{mod}m\end{subarray}}\frac{\mathscr{C}^{j}_{2l+\epsilon}(f)}{j!}|s|^{\frac{j-(m-2)}{m}}\log|s|\\\
&\quad+\sum_{\begin{subarray}{c}0\leq j<k\\\ j\neq
m-2\operatorname{mod}m\end{subarray}}\frac{{\mathscr{C}}^{j}_{2l+\epsilon}(f)}{j!}|s|^{\frac{j-(m-2)}{m}}+C^{n+\operatorname{P_{b}}}((0,(-1)^{\epsilon}]).\end{split}$

###  4.3. Basic properties of $\mathscr{C}^{k}_{l}$

Recall that $\mathscr{C}^{k}_{l}:C^{k}(\mathcal{D})\to{\mathbb{C}}$ for $k\geq
0$ and $0\leq l<2m$ are given by

$\displaystyle\mathscr{C}^{k}_{l}(f)=\sum_{\begin{subarray}{c}0\leq j\leq
k\wedge(m-2)\\\ j\neq
k-(m-1)\operatorname{mod}m\end{subarray}}\theta_{0}^{l(2j-k)}\mathfrak{B}(\tfrac{(m-1)-j}{m},\tfrac{(m-1)-(k-j)}{m})\partial_{j}^{k}(f).$

The functionals $\mathscr{C}^{k}_{l}$, $0\leq l<2m$ are not independent. By
definition,

(4.34) $\displaystyle\mathscr{C}^{k}_{l+m}=(-1)^{k}\mathscr{C}^{k}_{l}\text{
for any }0\leq l<m.$

Moreover, we can also get back the value of $\partial^{k}_{j}$ from
$\mathscr{C}^{k}_{l}$. Indeed, for every $0\leq j\leq k\wedge(m-2)$ with
$j\neq k-(m-1)\operatorname{mod}m$,

$\displaystyle\mathfrak{B}(\tfrac{(m-1)-j}{m},\tfrac{(m-1)-(k+1)+j}{m})\partial_{j}^{k}=\frac{1}{2m}\sum_{0\leq
l<2m}\theta_{0}^{l(k-2j)}\mathscr{C}^{k}_{l}=\frac{1}{m}\sum_{0\leq
l<m}\theta_{0}^{l(k-2j)}\mathscr{C}^{k}_{l}.$

Similarly, if $k\wedge(m-2)<j\leq m-2$ or $j=m-1$ or
$j=k-(m-1)\operatorname{mod}m$, then

$\displaystyle\sum_{0\leq
l<2m}\theta_{0}^{l(k-2j)}\mathscr{C}^{k}_{l}=2\sum_{0\leq
l<m}\theta_{0}^{l(k-2j)}\mathscr{C}^{k}_{l}=0.$

Together with (4.34) this gives all linear relations involving the functionals
$\mathscr{C}^{k}_{l}$.

Moreover, using (3.29), we obtain an elegant formula for $\mathscr{C}^{k}_{l}$
depending on the partial derivatives of the function $f$. Indeed, if $0\leq
j\leq m-2$, $j\neq k-(m-1)\operatorname{mod}m$ and $0\leq n\leq\frac{j-i}{m}$
then, by (4.9),

$\displaystyle\mathfrak{B}$
$\displaystyle(\tfrac{(m-1)-j}{m},\tfrac{(m-1)-(k-j)}{m})\frac{\binom{\frac{(m-1)-j}{m}-1}{n}}{\binom{\frac{(k-j)-(m-1)}{m}}{n}}$
$\displaystyle=\mathfrak{B}(\tfrac{(m-1)-j-nm}{m}+n,\tfrac{(m-1)-(k-j)+nm}{m}-n)(-1)^{n}\frac{\binom{-\frac{(m-1)-j-nm}{m}}{n}}{\binom{-\frac{(m-1)-(k-j)+nm}{m}+n}{n}}$
$\displaystyle=\mathfrak{B}(\tfrac{(m-1)-j-nm}{m},\tfrac{(m-1)-(k-j)+nm}{m}).$

By the definition of $\partial^{k}_{j}$, it follows that

$\displaystyle\mathscr{C}^{k}_{l}(f)$
$\displaystyle=\sum_{\begin{subarray}{c}0\leq j\leq k\wedge(m-2)\\\ j\neq
k-(m-2)\operatorname{mod}m\end{subarray}}\Big{(}\theta_{0}^{l(2j-k)}\mathfrak{B}(\tfrac{(m-1)-j}{m},\tfrac{(m-1)-(k-j)}{m})$
$\displaystyle\qquad\sum_{0\leq
n\leq\frac{k-j}{m}}\frac{\binom{k}{j+nm}\binom{\frac{(m-1)-j}{m}-1}{n}}{\binom{\frac{(k-j)-(m-1)}{m}}{n}}\frac{\partial^{k}f}{\partial\omega^{j+nm}\partial\overline{\omega}^{k-j-
nm}}(0,0)\Big{)}$ $\displaystyle=\sum_{\begin{subarray}{c}0\leq i\leq k\\\
i\neq m-1\operatorname{mod}m\\\ i\neq
k-(m-1)\operatorname{mod}m\end{subarray}}\theta_{0}^{l(2i-k)}\binom{k}{i}\mathfrak{B}(\tfrac{(m-1)-i}{m},\tfrac{(m-1)-(k-i)}{m})\frac{\partial^{k}f}{\partial\omega^{i}\partial\overline{\omega}^{k-i}}(0,0).$

According to (4.6),

(4.35) $\displaystyle\mathscr{C}^{k}_{l}(f)=\sum_{0\leq i\leq
k}\theta_{0}^{l(2i-k)}\binom{k}{i}\mathfrak{B}(\tfrac{(m-1)-i}{m},\tfrac{(m-1)-(k-i)}{m})\frac{\partial^{k}f}{\partial\omega^{i}\partial\overline{\omega}^{k-i}}(0,0).$

######  Remark 4.9.

This formula generalizes the one for
$C_{\alpha}^{\pm}(\varphi_{f,l}),\alpha\in\mathcal{A}$ in [14, Theorem 9.1] by
replacing new functionals for higher order derivatives.

We now strengthen Theorem 3.12 by proving that $F_{f}$ is also smooth (with
some drop of regularity) on the closed sectors
$\overline{\mathcal{D}}(\frac{l}{2m},\frac{l+1}{2m})$.

######  Theorem 4.10.

Fix $k\geq m-1$ and $0\leq l<2m$. Let $m-1\leq\underline{k}\leq k$ be the
natural number given by
$\widehat{\mathfrak{o}}(\sigma,\underline{k})=\underline{k}-(m-2)=\lceil\frac{k-(m-2)}{m}\rceil=\lceil\mathfrak{o}(\sigma,k)\rceil=:n$.
Suppose that $f\in C^{k\vee(n+1)}(\mathcal{D})$ is such that
$\partial^{j}_{i}(f)=0$ for all $0\leq j<\underline{k}$ and $0\leq i\leq
j\wedge(m-2)$ with $i\neq j-(m-1)\operatorname{mod}m$ and
$\mathscr{C}^{j}_{l}(f)=0$ for all $0\leq j<k$. Then the map
$F_{f}:\mathcal{D}(\frac{l}{2m},\frac{l+1}{2m})\to{\mathbb{C}}$ has a
$C^{\mathfrak{e}(\sigma,k)}$-extension on
$\overline{\mathcal{D}}(\frac{l}{2m},\frac{l+1}{2m})$ and there exists $C>0$
such that
$\|F_{f}\|_{C^{\mathfrak{e}(\sigma,k)}(\overline{\mathcal{D}}(\frac{l}{2m},\frac{l+1}{2m}))}\leq
C\|f\|_{C^{k\vee(n+1)}(\mathcal{D})}$.

###### Proof.

We focus only on the even sectors
${\mathcal{D}}(\frac{2l}{2m},\frac{2l+1}{2m})$. The proof in the odd case
proceeds in the same way. By Theorem 3.12, for every $0<\varepsilon<1/2$ the
map $F_{f}$ has a $C^{\widehat{\mathfrak{e}}(\sigma,\underline{k})}$-extension
on
$\overline{\mathcal{D}}(\frac{2l+\varepsilon}{2m},\frac{2l+1}{2m})\subset\overline{\mathcal{D}}(\frac{2l+\varepsilon}{2m},\frac{2l+2-\varepsilon}{2m})$
and there exists $C_{\varepsilon}>0$ so that
$\|F_{f}\|_{C^{\widehat{\mathfrak{e}}(\sigma,\underline{k})}(\overline{\mathcal{D}}(\frac{2l+\varepsilon}{2m},\frac{2l+1}{2m}))}\leq
C_{\varepsilon}\|f\|_{C^{\underline{k}}(\mathcal{D})}$. Moreover,

$\displaystyle\mathscr{F}_{f,l}(u,s)$
$\displaystyle=\int_{-1}^{u}\frac{f(G_{l}(v,s))}{(v^{2}+s^{2})^{\frac{m-1}{m}}}dv=\varphi_{f,l}(s)-\int_{u}^{1}\frac{f(G_{l}(v,s))}{(v^{2}+s^{2})^{\frac{m-1}{m}}}dv$
$\displaystyle=\varphi_{f,l}(s)-\int_{-1}^{-u}\frac{f(G_{l}(-v,s))}{(v^{2}+s^{2})^{\frac{m-1}{m}}}dv.$

As $G_{l}(-v,s)=\theta_{0}^{-1}G_{l}(v,-s)$ for $s>0$, this gives

$\mathscr{F}_{f,l}(z,\overline{z})=\varphi_{f,l}(\Im
z)-\mathscr{F}_{f\circ\theta_{0}^{-1},l}(-z,-\overline{z})\text{ if }\Im z>0.$

It follows that

(4.36)
${F}_{f}(\omega,\overline{\omega})=\varphi_{f,l}(\Im\omega^{m})-{F}_{f\circ\theta_{0}^{-1}}(\theta_{0}\omega,\theta_{0}^{-1}\overline{\omega})\text{
on }\mathcal{D}(\tfrac{2l}{2m},\tfrac{2l+1}{2m}).$

Note that

$\frac{\partial^{j}(f\circ\theta_{0}^{-1})}{\partial\omega^{i}\partial\overline{\omega}^{j-i}}(0,0)=\theta_{0}^{-(2i-j)}\frac{\partial^{j}f}{\partial\omega^{i}\partial\overline{\omega}^{j-i}}(0,0).$

By (3.29), it follows that
$\partial_{i}^{j}(f\circ\theta_{0}^{-1})=\theta_{0}^{-(2i-j)}\partial_{i}^{j}(f)$.
Therefore, by assumption, $\partial^{j}_{i}(f\circ\theta_{0}^{-1})=0$ for all
$0\leq j<\underline{k}$ and $0\leq i\leq j\wedge(m-2)$ with $i\neq
j-(m-1)\operatorname{mod}m$. Using Theorem 3.12 again, we obtain the map
$F_{f\circ\theta_{0}^{-1}}$ has a
$C^{\widehat{\mathfrak{e}}(\sigma,\underline{k})}$-extension on
$\overline{\mathcal{D}}(\frac{2l+1}{2m},\frac{2l+2-\varepsilon}{2m})\subset\overline{\mathcal{D}}(\frac{2l+\varepsilon}{2m},\frac{2l+2-\varepsilon}{2m})$
and
$\|F_{f\circ\theta_{0}^{-1}}\|_{C^{\widehat{\mathfrak{e}}(\sigma,\underline{k})}(\overline{\mathcal{D}}(\frac{2l+1}{2m},\frac{2l+2-\varepsilon}{2m}))}\leq
C_{\varepsilon}\|f\circ\theta_{0}^{-1}\|_{C^{\underline{k}}(\mathcal{D})}$. In
particular,

(4.37)
$\displaystyle\begin{split}&\text{${F}_{f\circ\theta_{0}^{-1}}(\theta_{0}\omega,\theta_{0}^{-1}\overline{\omega})$
is of the class $C^{\widehat{\mathfrak{e}}(\sigma,\underline{k})}$ on
$\overline{\mathcal{D}}(\tfrac{2l}{2m},\tfrac{2l+1-\varepsilon}{2m})$ and}\\\
&\|F_{f\circ\theta_{0}^{-1}}(\theta_{0}\omega,\theta_{0}^{-1}\overline{\omega})\|_{C^{\widehat{\mathfrak{e}}(\sigma,\underline{k})}(\overline{\mathcal{D}}(\frac{2l}{2m},\frac{2l+1-\varepsilon}{2m}))}\leq
C_{\varepsilon}\|f\|_{C^{\underline{k}}(\mathcal{D})}.\end{split}$

By Theorem 4.7, $\varphi_{f,l}$ has a $C^{\mathfrak{e}(\sigma,k)}$-extension
on $[0,1]$ with

$\|\varphi_{f,l}\|_{C^{\mathfrak{e}(\sigma,k)}([0,1])}\leq
C\|f\|_{C^{k\vee(n+1)}(\mathcal{D})}.$

Therefore, $\omega\to\varphi_{f,l}(\Im\omega^{m})$ has a
$C^{\mathfrak{e}(\sigma,k)}$-extension on
$\overline{\mathcal{D}}(\tfrac{2l}{2m},\tfrac{2l+1}{2m})$ and

$\displaystyle\|\varphi_{f,l}(\Im\omega^{m})\|_{C^{\mathfrak{e}(\sigma,k)}(\overline{\mathcal{D}}(\frac{2l}{2m},\frac{2l+1}{2m}))}\leq
C^{\prime}\|f\|_{C^{k\vee(n+1)}(\mathcal{D})}.$

As $F_{f}$ is a $C^{\widehat{\mathfrak{e}}(\sigma,\underline{k})}$-map on
$\overline{\mathcal{D}}(\frac{2l+\varepsilon}{2m},\frac{2l+1}{2m})$ with
$\|F_{f}\|_{C^{\widehat{\mathfrak{e}}(\sigma,\underline{k})}(\overline{\mathcal{D}}(\frac{2l+\varepsilon}{2m},\frac{2l+1}{2m}))}\leq
C_{\varepsilon}\|f\|_{C^{\underline{k}}(\mathcal{D})}$,
$\mathfrak{o}(\sigma,k)\leq\lceil\mathfrak{o}(\sigma,k)\rceil=\widehat{\mathfrak{o}}(\sigma,\underline{k})$
and $\underline{k}\leq k$, in view of (4.36) and (4.37), this gives our claim.
∎

We now show that the regularity of $F_{f}$ obtained in Theorem 4.10 is
optimal.

######  Theorem 4.11.

Fix $k\geq m-1$ and $0\leq l<2m$. If $f\in C^{k\vee(n+1)}(\mathcal{D})$ is
such that $F_{f}\in
C^{r}(\overline{\mathcal{D}}(\frac{l}{2m},\frac{l+1}{2m}))$ for some
$r\in{\mathbb{R}}_{\eta}$ with $0<v(r)\leq\mathfrak{o}(\sigma,k)$ then
$\mathscr{C}^{j}_{l}(f)=0$ for all $j\geq 0$ such that
$\mathfrak{o}(\sigma,j)<v(r)$ and $\partial^{j}_{i}(f)=0$ for all $j\geq 0$
with $\widehat{\mathfrak{o}}(\sigma,j)<v(r)$ and $0\leq i\leq j\wedge(m-2)$
with $i\neq j-(m-1)\operatorname{mod}m$.

###### Proof.

We will focus only on the even sectors
${\mathcal{D}}(\frac{2l}{2m},\frac{2l+1}{2m})$. The proof in the odd case
proceeds in the same way.

By definition, $\varphi_{f,l}(s)=\mathscr{F}_{f,l}(1,s)=F_{f}(G_{l}(1+\iota
s),\overline{G_{l}(1+\iota s)})$ on $(0,1]$. As $F_{f}\in
C^{r}(\overline{\mathcal{D}}(\frac{2l}{2m},\frac{2l+1}{2m}))$, it follows that
$\varphi_{f,l}\in C^{r}((0,1])$. In view of Theorem 4.7,
$\mathscr{C}^{j}_{2l}(f)=0$ for all $j\geq 0$ such that
$\mathfrak{o}(\sigma,j)<v(r)$.

The proof of the vanishing of $\partial^{j}_{i}$ is much more involved. Choose
$m-1\leq\underline{k}\leq\overline{k}<k$ such that
$\mathfrak{o}(\sigma,\overline{k}-1)<v(r)\leq\mathfrak{o}(\sigma,\overline{k})$
and
$\widehat{\mathfrak{o}}(\sigma,\underline{k}-1)<v(r)\leq\widehat{\mathfrak{o}}(\sigma,\underline{k})$.
By the first part of the theorem, $\mathscr{C}^{j}_{2l}(f)=0$ for all $0\leq
j<\overline{k}$. Let us decompose $f=f_{<\underline{k}}+e_{f}$, where
$f_{<\underline{k}}=\sum_{0\leq j<\underline{k}}f_{j}$ with

$\displaystyle f_{j}(\omega,\overline{\omega})=\frac{1}{j!}\sum_{0\leq i\leq
j}\binom{j}{i}\frac{\partial^{j}f}{\partial\omega^{i}\partial\overline{\omega}^{j-i}}(0,0)\omega^{i}\overline{\omega}^{j-i}.$

Then for every $0\leq j<\underline{k}$ we have $D^{j}e_{f}(0,0)=0$ and
$\mathscr{C}^{j}_{2l}(e_{f})=0$ and for $\underline{k}\leq j<\overline{k}$ we
have $\mathscr{C}^{j}_{2l}(e_{f})=\mathscr{C}^{j}_{2l}(f)=0$. Since
$\widehat{\mathfrak{o}}(\sigma,\underline{k})=\lceil\mathfrak{o}(\sigma,\overline{k})\rceil$,
in view of Theorem 4.10, this gives $F_{e_{f}}\in
C^{\mathfrak{e}(\sigma,\overline{k})}(\overline{\mathcal{D}}(\frac{2l}{2m},\frac{2l+1}{2m}))$.
As $F_{f}\in C^{r}(\overline{\mathcal{D}}(\frac{2l}{2m},\frac{2l+1}{2m}))$ and
$v(r)\leq\mathfrak{o}(\sigma,\overline{k})$, this yields
$F_{f_{<\underline{k}}}=F_{f}-F_{e_{f}}\in
C^{r}(\overline{\mathcal{D}}(\frac{2l}{2m},\frac{2l+1}{2m}))$.

For every $0<a<1$ let $\Delta_{a}=\\{(u,s):0<u\leq 1,0<s\leq au\\}$. By Lemmas
4.1, 4.2, 4.3 and (4.35), for every $0<a<1$, there exist ${\varrho_{j,l}}\in
C^{\omega}([0,a])$ and ${c_{j,l}}\in{\mathbb{C}}$ for $0\leq j<\underline{k}$
and ${\rho_{l}}\in C^{\omega}([0,a])$ such that for any $(u,s)\in\Delta_{a}$,

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

Let $\alpha\in(0,1/4)$ so that $\tan(\pi\alpha)=a$. Fix any $0<\beta<\alpha$
and let $\omega_{0}=e^{2\pi\iota\frac{2l+\beta}{2m}}$. Then for any
$t\in(0,a]$,

$\displaystyle\mathscr{F}_{f_{<\underline{k}},l}((t\omega_{0})^{m},\overline{t\omega_{0}}^{m})$
$\displaystyle=\mathscr{F}_{f_{<\underline{k}},l}(t^{m}\cos(\pi\beta),t^{m}\sin(\pi\beta))$
$\displaystyle=\sum_{0\leq
j<\underline{k}}\cos(\pi\beta)^{\frac{j-(m-2)}{m}}{\varrho_{j,l}}(\tan(\pi\beta))t^{j-(m-2)}+{\rho_{l}}(t^{m}\sin(\pi\beta))$
$\displaystyle\quad+\sum_{0\leq
j<\underline{k}}{c_{j,l}}\sin(\pi\beta)^{\frac{j-(m-2)}{m}}t^{j-(m-2)}\log(t^{m}\cos(\pi\beta)).$

Since $(0,a]\ni
t\mapsto\mathscr{F}_{f_{<\underline{k}},l}((t\omega_{0})^{m},\overline{t\omega_{0}}^{m})\in{\mathbb{C}}$
is of class $C^{r}$, $[0,a]\ni
t\mapsto{\rho_{l}}(t^{m}\sin(\pi\beta))\in{\mathbb{C}}$ is analytic and
$v(r)>\widehat{\mathfrak{o}}(\sigma,\underline{k}-1)=\underline{k}-1-(m-2)\geq
j-(m-2)$ for every $0\leq j<\underline{k}$, it follows that ${c_{j,l}}=0$ for
all $0\leq j<\underline{k}$, so
$\mathscr{F}_{f_{<\underline{k}},l}(u,s)=\sum_{0\leq
j<\underline{k}}u^{\frac{j-(m-2)}{m}}{\varrho_{j,l}}(s/u)+{\rho_{l}}(s)$.

For every $0\leq j<\underline{k}$ let
${\Upsilon_{j,l}}:\Delta_{a}\to{\mathbb{C}}$ be a real analytic homogenous map
of degree $\frac{j-(m-2)}{m}$ given by
${\Upsilon_{j,l}}(u,s)=u^{\frac{j-(m-2)}{m}}{\varrho_{j,l}}(s/u)$. Then

$\mathscr{F}_{f_{<\underline{k}},l}(z,\overline{z})=\sum_{0\leq
j<\underline{k}}{\Upsilon_{j,l}}(z,\overline{z})+{\rho_{l}}(\Im z)\text{ on
}\Delta_{a}$

and

${F}_{f_{<\underline{k}},l}(\omega,\overline{\omega})=\mathscr{F}_{f_{<\underline{k}},l}(\omega^{m},\overline{\omega}^{m})=\sum_{0\leq
j<\underline{k}}{\Upsilon_{j,l}}(\omega^{m},\overline{\omega}^{m})+{\rho_{l}}(\Im\omega^{m})\text{
on }{\mathcal{D}}(\tfrac{2l}{2m},\tfrac{2l+\alpha}{2m}).$

Since $F_{f_{<\underline{k}}}\in