# Power series as Fourier Series

Debraj Chakrabarti  Department of Mathematics, Central Michigan University,
Mount Pleasant, MI 48859, USA<EMAIL_ADDRESS>and  Anirban Dawn  Department
of Mathematics, The University of Tampa, Tampa, FL 33606, USA<EMAIL_ADDRESS>
###### Abstract.

An abstract theory of Fourier series in locally convex topological vector
spaces is developed. An analog of Fejér’s theorem is proved for these series.
The theory is applied to distributional solutions of Cauchy-Riemann equations
to recover basic results of complex analysis. Some classical results of
function theory are also shown to be consequences of the series expansion.

###### 2010 Mathematics Subject Classification:

32A05,22D12

Debraj Chakrabarti was partially supported by Simons Foundation Collaboration
Grant number 706445.

##  1\. Introduction

###  1.1. Motivation

Two types of series expansion ubiquitous in mathematics are the power series
of an analytic function, and the Fourier series of an integrable function on
the circle. In the complex domain, well-known theorems of elementary complex
analysis guarantee the existence of locally uniformly convergent power series
expansions of holomorphic functions in domains with ample symmetry such as
disks and annuli, where “holomorphic” can be taken in the sense of Goursat,
i.e., the function is complex-differentiable at each point. On the other hand,
the convergence of a Fourier series is a subtle matter, and its study has led
to many developments in analysis (see [Zyg02]). The close connection between
these two types of outwardly different series expansions is a recurring theme
in many areas of classical analysis, e.g., in the theory of Hardy spaces.

It is however not difficult to see that at a certain level of abstraction,
Fourier series and power series are in fact two examples of the same
phenomenon, the representation theory of the circle group. The aim of this
article is to take this idea seriously, and use it to recapture some basic
results of complex analysis. The take-home message is that many properties of
holomorphic functions, such as the almost-supernatural regularity phenomena
are profitably thought of as expressions of symmetry, more precisely the
invariance of certain locally convex spaces under the Reinhardt action of the
torus group on ${\mathbb{C}}^{n}$. It is hoped that the point of view taken
here has pedagogical as well as conceptual value, and will be of interest to
students of complex analysis.

###  1.2. Abstract Fourier series

We begin in Section 2 with an account of Fourier series associated to a
continuous representation of the $n$-dimensional torus $\mathbb{T}^{n}$ on
locally convex topological vector spaces, using some ideas of [Joh76]. This
provides the unifying language of sufficient generality to encompass both
classical Fourier expansions and the power series representations of complex
analysis. Even at this very general and “soft” level, one can establish a
version of Fejér’s theorem on the summability in the Cesàro sense (Theorem
2.1), which can be thought of as a “completeness” statement for the
holomorphic monomials.

###  1.3.  Analyticity of holomorphic distributions

Using the abstract framework of Section 2, we recapture in Section 3, in a
novel way, some of the basic classical facts about holomorphic functions. We
show that on a Reinhardt domain, a distribution which satisfies the Cauchy-
Riemann equations (which we call a _holomorphic distribution_) has a complex
power series representation that converges uniformly along with derivatives of
all orders on compact subsets of a “relatively complete” log-convex Reinhardt
domain (see Theorem 3.1 below), and thus a holomorphic distribution is a
function, in fact an analytic function. Traditionally, to prove such an
assertion, one would start by showing that a holomorphic distribution is
actually a (smooth) function. This can be done either by ad hoc arguments for
the Laplacian going back to Weyl (see [Wey40]), or in a more general way, by
noticing that fundamental solution $\frac{1}{\pi z}$ of the Cauchy-Riemann
operator $\frac{\partial}{\partial\overline{z}}$ is smooth, in fact real-
analytic, away from the singularity at the origin. It then follows by standard
arguments about convolutions (see [Hör03, Theorem 4.4.3]), that any
distributional solution of $\frac{\partial}{\partial\overline{z}}u=0$ is real-
analytic. Therefore, a holomorphic distribution is a holomorphic function in
the sense of Goursat, and classical results of elementary complex analysis
give the power series expansion via the Cauchy integral formula. The extension
of the domain of convergence to the envelope of holomorphy can be obtained by
convexity arguments (see [Ran86]).

While this classical argument has the admirable advantage of placing
holomorphic functions in the context of solutions of hypoelliptic equations,
it also has the shortcoming that the crucial property of _complex-analyticity_
(and the associated Hartogs phenomenon in several variables) of holomorphic
distributions is proved not from an analysis of the action of a differential
operator on distributions, but by falling back on the Cauchy integral formula.
After using the real-analytic hypoellipticity of
$\frac{\partial}{\partial\overline{z}}$ to conclude that holomorphic
distributions are real-analytic, we discard all of this information, except
that holomorphic functions are $\mathcal{C}^{1}$ and satisfy the Cauchy-
Riemann equations in the classical sense. In our approach here, however,
complex analyticity of holomorphic distributions is proved directly in a
conceptually straightforward way, by expanding a holomorphic distribution on a
Reinhardt domain in a Fourier series, and then showing that the resulting
series (the Laurent series of the holomorphic function) converges in the
$\mathcal{C}^{\infty}$-topology, not only on the original set where the
distribution was defined, but possibly on a larger domain, thus underlining
the fact that Hartogs phenomenon can be thought of as a regularity property of
the solutions of the same nature as smoothness. Our proof also clearly locates
the origin of the remarkable regularity of holomorphic distributions in

  * (a)

the invariance properties of the space of holomorphic distributions under
Reinhardt rotations and translations,

  * (b)

symmetry and convexity properties of the Laurent monomial functions $z\mapsto
z_{1}^{\alpha_{1}}\dots z_{n}^{\alpha_{n}},$
$\alpha_{j}\in\mathbb{Z}$ and

  * (c)

the fact that radially symmetric holomorphic distributions are constants.

It is also interesting that the fact that $\frac{1}{\pi z}$ is a fundamental
solution of the Cauchy-Riemann operator, which is key to many results of
complex analysis including the Cauchy integral formula, does not play any role
in our approach.

The method of proving analyticity via Fourier expansion can be used in other
contexts. For example, replacing the representation theory of $\mathbb{T}^{n}$
by that of the special orthogonal group $SO(n)$, the method can be used to
show that a harmonic distribution in a ball of ${\mathbb{R}}^{n}$ is in fact a
real analytic function and admits an expansion in solid harmonics (see, e.g.,
[CH53, pp. 316-317]), which converges uniformly along with all derivatives on
compact subsets of the ball. Similarly, one can obtain the Taylor/Laurent
expansion of a monogenic function of a Clifford-algebra variable in “spherical
monogenics”, the analogs for functions of a Clifford-algebra variable of the
monomial functions $z\mapsto z^{n},n\in\mathbb{Z}$ (see [BDS82]).

###  1.4. Analyticity of continuous holomorphic functions

While the theory of generalized functions forms the natural context for
solution of linear partial differential equations such as the Cauchy-Riemann
equations, for aesthetic and pedagogical reasons it is natural to ask whether
it is possible to develop the Fourier approach to power series, as outlined in
Section 3, using only classical notions of functions as continuous mappings,
and derivatives as limits of difference quotients, and without
anachronistically invoking distributions. We accomplish this in the last
Section 5 of this paper. We start from the assumption that a continuous
function on the complex complex plane satisfies the hypothesis of the Morera
theorem, and show directly that it is complex-analytic _without any recourse
to the Cauchy integral representation formula_ (which, being a case of Stoke’s
theorem, requires the differentiability of the function, at least at each
point). Not unexpectedly, one of the steps of the proof uses the classical
triangle-division used in the standard proof of the Cauchy theorem for
triangles.

###  1.5. Acknowledgements

The first author would like to thank Luke Edholm and Jeff McNeal for many
interesting conversations about the topic of power series. He would also like
to thank the students of MTH 636 and MTH 637 at Central Michigan University
over the years for their many questions, which led him to think about the true
significance of power series expansions.

Sections 2 and 4 of this paper are based on part of the Ph.D. thesis of the
second author under the supervision of the first author. The second author
would also like to thank his other committee members, Dmitry Zakharov and
Sonmez Sahutoglu for their support. Other results from the thesis have
appeared in the paper [Daw21].

##  2\. Fourier Series in Locally Convex spaces

###  2.1. Nets, series and integrals in LCTVS

We begin by recalling some notions and facts about functional analysis in
topological vector spaces. See the textbooks [Trè67, Rud91, Bou04] for more
details on these matters.

Let $X$ be a locally convex Hausdorff topological vector space (we use the
standard abbreviation _LCTVS_). Recall that the topology of $X$ can also be
described by prescribing the family of _continuous seminorms_ on $X$: a net
$\\{x_{j}\\}$ in $X$ converges to $x$, if and only if for each continuous
seminorm $p$ on $X$, we have $p(x-x_{j})\to 0$ as a net of real numbers. In
practice, we describe the topology of an LCTVS by specifying a _generating
family_ of seminorms (analogous to describing a topology by a subbasis): a
collection of continuous seminorms $\\{p_{k}:k\in K\\}$ on $X$ is said to
_generate the topology_ of $X$ if for every continuous seminorm $q$ on $X$,
there exists a finite subset $\\{k_{1},\dots,k_{n}\\}\subset K$ and a $C>0$
such that

$q(x)\leq C\cdot\max\\{p_{k_{1}}(x),\dots,p_{k_{n}}(x)\\}\quad\textrm{for all
$x\in X$},$ (2.1)

and further, for every nonzero $x\in X$ there exists at least one $k\in K$
such that $p_{k}(x)\neq 0$ (this _separating property_ ensures that the
topology of $X$ is Hausdorff). Then clearly a net $\\{x_{j}\\}$ converges in
$X$ if and only if $p_{k}(x_{j}-x)\rightarrow 0$ for each $k\in K$.

Let $\Gamma$ be a directed set with order $\geq$. Recall that a net
$\\{x_{\alpha}\\}_{\alpha\in\Gamma}$ in $X$ is said to be _Cauchy_ if for
every $\epsilon>0$ and every continuous seminorm $p$ on $X$, there exists
$\gamma\in\Gamma$ such that whenever $\alpha,\beta\in\Gamma$ and
$\alpha,\beta\geq\gamma$, we have $p(x_{\alpha}-x_{\beta})<\epsilon$. The
space $X$ is said to be _complete_ if every Cauchy net in $X$ converges.
Observe that in the above definition we can use a generating family of
seminorms rather than all continuous seminorms.

If $\\{S_{k}\\}_{k\in\mathbb{N}}$ is a sequence in a vector space, we can
define the sequence of the corresponding _Cesàro means_
$\\{C_{n}\\}_{n\in\mathbb{N}}$ by

$\displaystyle{C_{n}=\frac{1}{n+1}\sum\limits_{k=0}^{n}S_{k}}.$

The following is the analog for sequences in LCTVS of an elementary fact well-
known for sequences of numbers:

######  Proposition 2.1.

Let $\\{S_{k}\\}_{k\in\mathbb{N}}$ be a convergent sequence in an LCTVS $X$.
Then the sequence of Cesàro means $\\{C_{n}\\}_{n\in\mathbb{N}}$ is also
convergent, and has the same limit.

###### Proof.

Let $S=\lim\limits_{k\to\infty}S_{k}$. If $p$ is a continuous seminorm on $X$
and $\epsilon>0$, there is $N_{1}$ such that $p(S_{k}-S)<\epsilon/2$ for
$k\geq N_{1}$. Set $s=\sum_{k=0}^{N_{1}}p(S_{k}-S)$. Then for $n>N_{1}$, we
have

$\displaystyle
p(C_{n}-S)=p\left(\frac{1}{n+1}\sum\limits_{k=0}^{n}S_{k}-S\right)\leq\frac{1}{n+1}\sum\limits_{k=0}^{n}p(S_{k}-S)<\frac{s}{n+1}+\frac{n-N_{1}}{n+1}\cdot\frac{\epsilon}{2}.$

Therefore, if we choose $N>N_{1}$ so large that
$\frac{s}{n+1}<\frac{\epsilon}{2}$, then for $n\geq N$ we have
$p(C_{n}-S)<\epsilon$, so $C_{n}\to S$ in $X$. ∎

For a formal series $\displaystyle{\sum_{j=0}^{\infty}x_{j}}$ in an LCTVS $X$,
convergence is defined in the usual way, i.e. the sequence of partial sums
converges in $X$. A formal sum ${\sum_{\alpha\in\mathfrak{A}}x_{\alpha}}$ over
a countable index set $\mathfrak{A}$, where $x_{\alpha}$ are vectors in an
LCTVS $X$, is said to be _absolutely convergent_ if there exists a bijection
$\tau:\mathbb{N}\rightarrow\mathfrak{A}$ such that for every continuous
seminorm $p$ on $X$, the series of non-negative real numbers
$\displaystyle{\sum_{j=0}^{\infty}p(x_{\tau(j)})}$ is convergent (see [KK97]).
To check that a series is absolutely convergent, we only need to check the
convergence of the above series for seminorms in a fixed generating family. If
$X$ is a locally compact Hausdorff space, absolute convergence in the Fréchet
space $\mathcal{C}(X)$ of continuous complex valued functions on $X$ is what
is classically called _normal convergence_ (see [Rem91, pp. 104 ff.]).
Absolute convergence is typical for many spaces of holomorphic functions, e.g.
in the space of holomorphic functions on a Reinhardt domain smooth up to the
boundary (see [Daw21]).

The following result, whose proof mimics the corresponding result for numbers,
shows that absolutely convergent series in LCTVS behave very much like
absolutely convergent series of numbers:

######  Proposition 2.2.

Let $X$ be a _complete_ LCTVS, and let
${\sum_{\alpha\in\mathfrak{A}}x_{\alpha}}$ be an absolutely convergent series
of elements of $X$. Then the series is _unconditionally convergent_ : there is
an $s\in X$ such that for _every_ bijection
$\theta:\mathbb{N}\rightarrow\mathfrak{A}$, the series
$\sum_{j=0}^{\infty}x_{\theta(j)}$ converges in $X$ to $s$.

The element $s\in X$ is naturally called the _sum_ of the series, and we write
$s=\sum_{\alpha\in\mathfrak{A}}x_{\alpha}$.

###### Proof.

By definition, there exists a bijection
$\sigma:\mathbb{N}\rightarrow\mathfrak{A}$, such that for each continuous
seminorm $p$ on $X$, the series $\sum_{j=0}^{\infty}p(x_{\sigma(j)})$
converges. Let $y_{j}=x_{\sigma(j)}$ and $s_{k}=\sum_{j=0}^{k}y_{j}$. Since
$\sum_{j=0}^{\infty}p(y_{j})$ converges, for $\epsilon>0$ there exists
$N_{0}\in\mathbb{N}$ such that whenever $m,\ell\in\mathbb{N}$ with
$m\geq\ell\geq N_{0}$, $\sum_{j=\ell+1}^{m}p(y_{j})<\epsilon$. Therefore for
$m\geq\ell\geq N_{0}$,

$p(s_{m}-s_{\ell})=p\Big{(}\sum_{j=\ell+1}^{m}y_{j}\Big{)}\leq\sum_{j=\ell+1}^{m}p(y_{j})<\epsilon.$
(2.2)

Therefore $\\{s_{k}\\}$ is Cauchy sequence in the complete LCTVS $X$, and
therefore converges to an $s\in X$. In order to complete the proof, it
suffices to show that for every bijection
$\tau:\mathbb{N}\rightarrow\mathbb{N}$, the series
$\sum_{j=0}^{\infty}y_{\tau(j)}$ converges to the same sum $s$. Let
$s_{k}^{\tau}=\sum_{j=0}^{k}y_{\tau(j)}$. Choose $u\in\mathbb{N}$ such that
the set of integers $\\{0,1,2,\cdots,N_{0}\\}$ is contained in the set
$\\{\tau(0),\tau(1),\cdots,\tau(u)\\}$. Then, if $k>u$, the elements
$y_{1},\cdots,y_{N_{0}}$ get cancelled in the difference $s_{k}-s_{k}^{\tau}$
and we have $p(s_{k}-s_{k}^{\tau})<\epsilon$ by (2.2). This proves that the
sequences $\\{s_{k}\\}$ and $\\{s_{k}^{\tau}\\}$ converge to the same limit.
So, $s_{k}^{\tau}\rightarrow s$ as $k\rightarrow\infty$. ∎

By a famous theorem of Dvoretzky and Rogers, the converse of the above result
fails when $X$ is an infinite dimensional Banach space ([KK97, Chapter 4]).

We will also need the notion of the weak (or Pettis) integral of a LCTVS
valued function which we will now recall (see [Bou04, p. INT III.32-39] for
details). Let $K$ be a compact Hausdorff space and let $\mu$ be a Borel
measure on $K$, and let $f$ be a continuous map from $K$ to an LCTVS $X$. An
element $x\in X$ is called a _Pettis integral_ of $f$ on $K$ with respect to
$\mu$ if for all $\phi\in X^{\prime}$,

$\phi(x)=\int_{K}(\phi\circ f)\,d\mu,$ (2.3)

where $X^{\prime}$ denotes the dual space of $X$ (the space of continuous
linear functionals on $X$), and the right hand side of (2.3) is an integral of
a continuous function. If $X$ is complete, one can show that there exists a
unique $x\in X$ such that (2.3) holds and we denote the Pettis integral of $f$
on $K$ with respect to $\mu$ by $\displaystyle{\int_{K}f\hskip
2.84526ptd\mu=x}$. In fact, the integral exists uniquely, as soon as the space
$X$ is _quasi-complete_ , i.e. if each _bounded_ Cauchy net in $X$ converges,
where a net $\\{x_{\alpha}\\}_{\alpha\in\Gamma}$ is bounded if for each
continuous seminorm $p$, the net of real numbers
$\\{p(x_{\alpha})\\}_{\alpha\in\Gamma}$ is bounded. While there are situations
in which this more refined existence theorem for Pettis integrals is useful
(e.g. when the space $X$ is a dual space with weak-* topology), in this paper,
we only consider integrals in complete LCTVS. Since each Hausdorff TVS has a
unique Hausdorff completion, we can define Pettis integrals in any LCTVS,
provided we allow the integral to have a value in the completion.

If $T:X\to Y$ is a continuous linear map of complete LCTVSs, and $f:K\to X$ is
continuous then we have

$T\left(\int_{K}f\,d\mu\right)=\int_{K}T\circ f\,d\mu,$ (2.4)

since for any linear functional $\psi\in Y^{\prime}$,

$\psi\left(T\left(\int_{K}f\,d\mu\right)\right)=(\psi\circ
T)\left(\int_{K}f\,d\mu\right)=\int_{K}(\psi\circ T)(f)\,d\mu$

using the fact that $\psi\circ T\in X^{\prime}$ and the definition (2.3) of
the Pettis integral.

###  2.2. Representation of the torus on an LCTVS $X$.

Let

$\mathbb{T}^{n}=\\{\lambda=(\lambda_{1},...,\lambda_{n})\in{\mathbb{C}}^{n}:\left|\lambda_{j}\right|=1\hskip
2.84526pt\text{for every $1\leq j\leq n$}\\}$

be the $n$-dimensional unit torus. With the subspace topology inherited from
${\mathbb{C}}^{n}$ and binary operation defined as
$(\lambda,\xi)\mapsto\lambda{\cdot}\xi=(\lambda_{1}\xi_{1},\cdots,\lambda_{n}\xi_{n})$,
is a compact abelian topological group. For a LCTVS $X$, and a continuous
function $f:\mathbb{T}^{n}\to X$, we denote the Pettis integral of $f$ with
respect to the Haar measure of $\mathbb{T}^{n}$ (normalized to be a
probability measure) by

$\int_{T^{n}}f(\lambda)d\lambda.$

Let $X$ be an LCTVS, and let $\lambda\mapsto\sigma_{\lambda}$ be a continuous
representation of $\mathbb{T}^{n}$ on $X$. Recall that this means that for
each $\lambda\in\mathbb{T}^{n}$, the map $\sigma_{\lambda}$ is an automorphism
(i.e. linear self-homeomorphism) of $X$ as a topological vector space, the map
$\lambda\mapsto\sigma_{\lambda}$ is a group homomorphism from $\mathbb{T}^{n}$
to $\mathrm{Aut}(X)$, and the associated map

$\sigma:\mathbb{T}^{n}\times X\to
X,\quad\sigma(\lambda,x)=\sigma_{\lambda}(x)$

is continuous.

Given a representation $\sigma$ of the group $\mathbb{T}^{n}$ on an LCTVS $X$,
a continuous seminorm $p$ on $X$ is said to be invariant (with respect to
$\sigma$) if $p(\sigma_{\lambda}(x))=p(x)$ for all $x\in X$ and
$\lambda\in\mathbb{T}^{n}$.

######  Proposition 2.3.

A representation $\sigma$ of $\mathbb{T}^{n}$ on an LCTVS $X$ is continuous if
and only if the following two conditions are both satisfied:

  1. (a)

the topology of $X$ is generated by a family of invariant seminorms, and

  2. (b)

for each $x\in X$ the function from $\mathbb{T}^{n}$ to $X$ given by
$\lambda\mapsto\sigma_{\lambda}(x)$ is continuous at the identity element of
$\mathbb{T}^{n}$.

###### Proof.

Assume that $\sigma$ is continuous, i.e., $\sigma:\mathbb{T}^{n}\times X\to X$
is continuous, so for $x\in X$, the function
$\lambda\mapsto\sigma_{\lambda}(x)$ is continuous on $\mathbb{T}^{n}$, in
particular at the identity. Therefore (b) follows.

For a continuous seminorm $q$ on $X$, define
$p(x)=\sup_{\lambda\in\mathbb{T}^{n}}q(\sigma_{\lambda}(x))$, which is finite
since $\mathbb{T}^{n}$ is compact, and which is easily seen to be a seminorm.
To show $p$ is invariant, we note that

$p(\sigma_{\lambda}(x))=\sup\limits_{\mu\in\mathbb{T}^{n}}q(\sigma_{\mu}(\sigma_{\lambda}(x)))=\sup\limits_{\mu\in\mathbb{T}^{n}}q(\sigma_{\mu{\cdot}\lambda}(x))=\sup\limits_{\xi\in\mathbb{T}^{n}}q(\sigma_{\xi}(x))=p(x).$

It remains to show that $p$ is continuous. For $x,y\in X$, we have

$p(x)=\sup_{\lambda\in\mathbb{T}^{n}}q\left(\sigma_{\lambda}(y)+\sigma_{\lambda}(x-y)\right)\leq\sup_{\lambda\in\mathbb{T}^{n}}\left(q(\sigma_{\lambda}(y))+q(\sigma_{\lambda}(x-y))\right)\leq
p(y)+p(x-y),$

so that $\left|p(x)-p(y)\right|\leq p(x-y)$ for all $x,y\in X$, and it follows
that the seminorm $p$ is continuous on $X$ if and only if it is continuous at
$0$. We show that for each $\epsilon>0$, there exists a neighbourhood $V$ of
$0$ in $X$ such that for all $\lambda\in\mathbb{T}^{n}$,
$q\circ\sigma_{\lambda}<\epsilon$ on $V$. For each $\xi\in\mathbb{T}^{n}$,
since $q$ and $\sigma$ are continuous, there exists a neighborhood $U_{\xi}$
of $\xi$ in $\mathbb{T}^{n}$ and a neighbourhood $V_{\xi}$ of $0$ in $X$ such
that $q(\sigma_{\lambda}(x))<\epsilon$ for all $x\in V_{\xi}$ and $\lambda\in
U_{\xi}$. The collection $\\{U_{\xi}\\}_{\xi\in\mathbb{T}^{n}}$ forms an open
cover of $\mathbb{T}^{n}$. Since $\mathbb{T}^{n}$ is compact, let
$\\{U_{\xi_{1}},...,U_{\xi_{k}}\\}$ be a finite subcover of $\mathbb{T}^{n}$
corresponding to the open cover. Then for all
$x\in\bigcap_{j=1}^{k}V_{\xi_{j}}$ and $\lambda\in\mathbb{T}^{n}$, we have
$q(\sigma_{\lambda}(x))<\epsilon$.

Now assume the two conditions (a) and (b). Let $(\Gamma,\geq)$ be a directed
set and let $(\lambda_{\alpha})_{\alpha\in\Gamma}$ and
$(x_{\alpha})_{\alpha\in\Gamma}$ be nets in $\mathbb{T}^{n}$ and $X$
respectively with $(\lambda_{\alpha},x_{\alpha})\rightarrow(\lambda,x)$ in
$\mathbb{T}^{n}\times X$. We need to show
$\sigma_{\lambda_{\alpha}}(x_{\alpha})\rightarrow\sigma_{\lambda}(x)$ in $X$,
i.e., $p(\sigma_{\lambda_{\alpha}}(x_{\alpha})-\sigma_{\lambda}(x))\to 0$ for
each invariant continuous seminorm $p$ of $X$. But we have, by the invariance
of $p$:

$p\left(\sigma_{\lambda_{\alpha}}(x_{\alpha}\right)-\sigma_{\lambda}(x))=p\left(x_{\alpha}-\sigma_{\lambda_{\alpha}^{-1}\lambda}(x)\right)\leq
p\left(x_{\alpha}-x\right)+p\left(x-\sigma_{\lambda_{\alpha}^{-1}\lambda}(x)\right).$

The term $p\left(x_{\alpha}-x\right)$ goes to zero since $x_{\alpha}\to x$,
and the term $p\left(x-\sigma_{\lambda_{\alpha}^{-1}\lambda}(x)\right)$ also
goes to zero since $\lambda_{\alpha}\to\lambda$ and
$\mu\mapsto\sigma_{\mu}(x)$ is continuous at $\mu=\mathbf{1}$. The result
follows. ∎

###  2.3. Abstract Fejér Theorem

Let $X$ be a complete LCTVS and let $\sigma$ be a continuous representation of
$\mathbb{T}^{n}$ on $X$. For each
$\alpha=(\alpha_{1},...,\alpha_{n})\in\mathbb{Z}^{n}$ and $x\in X$, define

$\bm{\pi}_{\alpha}^{\sigma}(x)=\int_{\mathbb{T}^{n}}\lambda^{-\alpha}\sigma_{\lambda}(x)\,d\lambda,$
(2.5)

the Pettis integral of the continuous function
$\lambda\mapsto\lambda^{-\alpha}\sigma_{\lambda}(x)$ on $\mathbb{T}^{n}$ with
respect to the Haar probability measure of $\mathbb{T}^{n}$. We will say that
$\bm{\pi}^{\sigma}_{\alpha}(x)$ is the $\alpha$-th _Fourier component_ of $x$
with respect to the representation $\sigma$. We use the standard convention
with respect to multi-index powers, i.e.,
$\lambda^{\alpha}=\lambda_{1}^{\alpha_{1}}\dots\lambda_{n}^{\alpha_{n}}$. We
will say that the subspace of $X$ defined as

$[X]^{\sigma}_{\alpha}=\\{x\in X:\sigma_{\lambda}(x)=\lambda^{\alpha}\cdot
x\text{ for all }\lambda\in\mathbb{T}^{n}\\}$ (2.6)

is the $\alpha$-th _Fourier mode_ of the space $X$, and we will call the map
$\bm{\pi}^{\sigma}_{\alpha}$ the $\alpha$-th _Fourier projection_ , both with
respect to the representation $\sigma$. We note the following facts:

######  Proposition 2.4.

As above, let $X$ be a complete LCTVS and $\sigma$ be a continuous
representation of $\mathbb{T}^{n}$ on $X$.

  1. (1)

For each $\alpha\in\mathbb{Z}^{n}$, the $\alpha$-th Fourier mode
$[X]^{\sigma}_{\alpha}$ is a closed $\sigma$-invariant subspace of $X$, and
the Fourier projection $\bm{\pi}^{\sigma}_{\alpha}$ is a continuous linear
projection from $X$ onto $[X]^{\sigma}_{\alpha}$.

  2. (2)

Let $Y$ be another complete LCTVS, and let $\tau$ be a continuous
representation of $\mathbb{T}^{n}$ on $Y$, and let $j:Y\to X$ be a continuous
linear map intertwining $\sigma$ and $\tau$, i.e., for each
$\lambda\in\mathbb{T}^{n}$, $j\circ\tau_{\lambda}=\sigma_{\lambda}\circ j.$
Then for each $\alpha\in\mathbb{Z}^{n}$, we have

$j\circ\bm{\pi}^{\tau}_{\alpha}=\bm{\pi}^{\sigma}_{\alpha}\circ j.$ (2.7)

###### Proof.

  1. (1)

Linearity of $\bm{\pi}_{\alpha}^{\sigma}$ follows from the linearity of
$\sigma_{\lambda}$. Recall from Proposition 2.3 that there exists a family
$\mathscr{P}$ of continuous invariant seminorms that generates the locally
convex topology of $X$. To see the continuity of $\bm{\pi}_{\alpha}^{\sigma}$,
observe that for each $p\in\mathscr{P}$, we have

$p(\bm{\pi}_{\alpha}^{\sigma}(x))=p\left(\int_{\mathbb{T}^{n}}\lambda^{-\alpha}\sigma_{\lambda}(x)\,d\lambda\right)\leq\int_{\mathbb{T}^{n}}p\left(\lambda^{-\alpha}\sigma_{\lambda}(x)\right)\,d\lambda=\int_{\mathbb{T}^{n}}p(x)\,d\lambda=p(x),$
(2.8)

where the inequality is due to Proposition 6 in [Bou04, p. INT III.37]. Now,
if $x\in[X]^{\sigma}_{\alpha}$, then

$\bm{\pi}_{\alpha}^{\sigma}(x)=\int_{\mathbb{T}^{n}}\lambda^{-\alpha}\sigma_{\lambda}(x)\hskip
2.84526ptd\lambda=\int_{\mathbb{T}^{n}}\lambda^{-\alpha}\cdot\lambda^{\alpha}x\hskip
2.84526ptd\lambda=x\cdot\bigg{(}\int_{\mathbb{T}^{n}}\hskip
2.84526ptd\lambda\bigg{)}=x,$

so that $[X]^{\sigma}_{\alpha}\subset\bm{\pi}_{\alpha}^{\sigma}(X)$. To prove
that $\bm{\pi}_{\alpha}^{\sigma}(X)\subset[X]^{\sigma}_{\alpha}$, notice that
for each $\alpha\in\mathbb{Z}^{n}$, each $\lambda\in\mathbb{T}^{n}$ and $x\in
X$,

$\displaystyle\sigma_{\lambda}(\bm{\pi}^{\sigma}_{\alpha}(x))$
$\displaystyle=\sigma_{\lambda}\left(\int_{\mathbb{T}^{n}}\mu^{-\alpha}\sigma_{\mu}(x)d\mu\right)=\int_{\mathbb{T}^{n}}\mu^{-\alpha}\sigma_{\lambda\cdot\mu}(x)d\mu$
using (2.4)
$\displaystyle=\lambda^{\alpha}\cdot\int_{\mathbb{T}^{n}}(\lambda\mu)^{-\alpha}\sigma_{\lambda\mu}(x)d\mu=\lambda^{\alpha}\cdot\bm{\pi}^{\sigma}_{\alpha}(x)$
$\displaystyle\text{by Haar invariance}.$

  2. (2)

For $x\in X$, we have

$j\circ\bm{\pi}^{\tau}_{\alpha}(x)=j\left(\int_{\mathbb{T}^{n}}\lambda^{-\alpha}\tau_{\lambda}(x)d\lambda\right)=\int_{\mathbb{T}^{n}}\lambda^{-\alpha}j\circ\tau_{\lambda}(x)d\lambda=\int_{\mathbb{T}^{n}}\lambda^{-\alpha}\sigma_{\lambda}\circ
j(x)d\lambda=\bm{\pi}^{\sigma}_{\alpha}\circ j(x),$

where we have used (2.4) to go from the second to the third step.

∎

######  Remark.

The inequality (2.8)

$p\left(\bm{\pi}_{\alpha}^{\sigma}(x)\right)\leq p(x),$ (2.9)

which holds for each $\alpha$ for an invariant seminorm $p$ can be thought of
as an abstract form of the familiar _Cauchy inequalities_ of complex analysis.

If $X$ is $L^{1}(\mathbb{T})$, the Banach space of integrable functions on
$\mathbb{T}$, and if $\sigma$ is the continuous representation of $\mathbb{T}$
on $L^{1}(\mathbb{T})$ given by

$\sigma_{\lambda}(f)(\mu)=f(\lambda\cdot\mu),\quad\lambda,\mu\in\mathbb{T},f\in
L^{1}(\mathbb{T}),$

an easy computation shows that for $\phi\in{\mathbb{R}}$, and $f\in
L^{1}(\mathbb{T})$, we have

$\bm{\pi}_{\alpha}^{\sigma}(f)(e^{i\phi})=e^{i\alpha\phi}\cdot\hat{f}(\alpha),\quad\text{with
}\hat{f}(\alpha)=\frac{1}{2\pi}\cdot\int_{0}^{2\pi}e^{-i\alpha\theta}f(e^{i\theta})d\theta,$

the $\alpha$-th term of the Fourier series of $f$.

It is therefore natural to define, for $x\in X$, the _Fourier series_ of $x$
with respect to $\sigma$ to be the formal series

$x\sim\sum\limits_{\alpha\in\mathbb{Z}^{n}}\bm{\pi}_{\alpha}^{\sigma}(x).$
(2.10)

For an integer $k\geq 0$, define the $k$-th _square partial sum_ of the
Fourier series in (2.10) by

$S_{k}^{\sigma}(x)=\sum\limits_{\left|\alpha\right|_{\infty}\leq
k}\bm{\pi}_{\alpha}^{\sigma}(x),$ (2.11)

where
$\left|\alpha\right|_{\infty}\coloneqq\max\big{\\{}\left|\alpha_{j}\right|,1\leq
j\leq n\big{\\}}$. We are ready to state an abstract version of Fejér’s
theorem.

######  Theorem 2.1.

Let $\sigma$ be a continuous representation of $\mathbb{T}^{n}$ on an LCTVS
$X$ and let $x\in X$. Then the Cesàro means of the square partial sums of the
Fourier series of $x$ (with respect to $\sigma$) converge to $x$ in the
topology of $X$.

###### Proof.

Write the Cesàro means of the square partial sums of the Fourier series of $x$
as,

$\displaystyle C_{N}^{\sigma}(x)$
$\displaystyle=\frac{1}{N+1}\sum\limits_{k=0}^{N}S_{k}^{\sigma}(x)=\frac{1}{N+1}\sum\limits_{k=0}^{N}\sum\limits_{\left|\alpha\right|_{\infty}\leq
k}\bm{\pi}_{\alpha}^{\sigma}(x)=\frac{1}{N+1}\sum\limits_{k=0}^{N}\sum\limits_{\left|\alpha\right|_{\infty}\leq
k}\left(\int\limits_{\mathbb{T}^{n}}\lambda^{-\alpha}\sigma_{\lambda}(x)\,d\lambda\right)$
$\displaystyle=\frac{1}{N+1}\int\limits_{\mathbb{T}^{n}}\sum\limits_{k=0}^{N}\left(\sum\limits_{\left|\alpha\right|_{\infty}\leq
k}\lambda^{-\alpha}\right)\sigma_{\lambda}(x)\,d\lambda=\int\limits_{\mathbb{T}^{n}}F_{N}(\lambda)\hskip
2.84526pt\sigma_{\lambda}(x)\,d\lambda$

where

$F_{N}(\lambda)=\displaystyle{\frac{1}{N+1}\sum\limits_{k-0}^{N}\left(\sum\limits_{\left|\alpha\right|_{\infty}\leq
k}\lambda^{-\alpha}\right)}$

is the classical Fejér kernel. Introducing polar coordinates,
$\lambda_{j}=e^{i\theta_{j}}$ on $\mathbb{T}^{n}$, and summing, we obtain the
classical representation

$F_{N}(e^{i\theta_{1}},\cdots,e^{i\theta_{n}})=\frac{1}{N+1}\prod_{j=1}^{n}\frac{\sin^{2}\left(\left(\frac{N+1}{2}\right)\theta_{j}\right)}{\sin^{2}\left(\frac{\theta_{j}}{2}\right)}$

It is well-known that the Fejér kernel has the properties that

  * (a)

$F_{N}\geq 0$ for all $N$,

  * (b)

$\int\limits_{\mathbb{T}^{n}}F_{N}(\lambda)\,d\lambda=1$ and

  * (c)

For each $\delta>0$, $F_{N}\rightarrow 0$ uniformly on
$\mathbb{T}^{n}\setminus B(\mathbf{1},\delta)$, where $B(\mathbf{1},\delta)$
is the $n$-dimensional ball centered at $\mathbf{1}=(1,1,...,1)$ and radius
$\delta$.

Let $p$ be a continuous $\sigma$-invariant seminorm on $X$. Then for $x\in X$,
we have

$\displaystyle p(C_{N}(x)-x)$
$\displaystyle=p\left(\int\limits_{\mathbb{T}^{n}}F_{N}(\lambda)\cdot\sigma_{\lambda}(x)\,d\lambda-x\cdot\int\limits_{\mathbb{T}^{n}}F_{N}(\lambda)\,d\lambda\right)$
using property (b) of $F_{N}$
$\displaystyle=p\left(\int\limits_{\mathbb{T}^{n}}F_{N}(\lambda)\cdot\sigma_{\lambda}(x)\,d\lambda-\int\limits_{\mathbb{T}^{n}}x\cdot
F_{N}(\lambda)\,d\lambda\right)$ using (2.4)
$\displaystyle=p\left(\int\limits_{\mathbb{T}^{n}}\big{(}\sigma_{\lambda}(x)-x\big{)}\cdot
F_{N}(\lambda)\,d\lambda\right)$
$\displaystyle\leq\int\limits_{\mathbb{T}^{n}}p(\sigma_{\lambda}(x)-x)\cdot
F_{N}(\lambda)\,d\lambda$ (2.12) using [Bou04, Prop. 6, p. INT III.37] and the
positivity of $F_{N}(\lambda)$

Since $\lambda\mapsto\sigma_{\lambda}(x)$ is continuous and $p$ is a
continuous seminorm, there exists $\delta>0$ such that
$p(\sigma_{\lambda}(x)-\sigma_{\mathbf{1}}(x))=p(\sigma_{\lambda}(x)-x)<\frac{\epsilon}{2}$
whenever $\lambda\in\mathbb{T}^{n}\cap B(\mathbf{1},\delta).$ Then on the set
$\mathbb{T}^{n}\setminus B(\mathbf{1},\delta)$ we have

$p(\sigma_{\lambda}(x)-x)\leq p(\sigma_{\lambda}(x))+p(x)=2\cdot p(x).$

By property (c) of $F_{N}$, there exists $N_{1}\in\mathbb{N}$ such that
whenever $\lambda\in\mathbb{T}^{n}\setminus B(\mathbf{1},\delta)$, for all
$N\geq N_{1}$ we have,

$F_{N}(\lambda)<\frac{\epsilon}{4\cdot p(x)}$ (2.13)

Now from (2.12), we have

$\displaystyle p(C_{N}(x)-x)$
$\displaystyle\leq\int\limits_{\mathbb{T}^{n}\cap
B(\mathbf{1},\delta)}F_{N}(\lambda)\cdot
p(\sigma_{\lambda}(x)-x)\,d\lambda+\int\limits_{\mathbb{T}^{n}\setminus
B(\mathbf{1},\delta)}F_{N}(\lambda)\cdot p(\sigma_{\lambda}(x)-x)\,d\lambda$
$\displaystyle<\frac{\epsilon}{2}\int\limits_{\mathbb{T}^{n}}F_{n}(\lambda)\,d\lambda+\frac{\epsilon}{4\cdot
p(x)}\cdot 2\cdot p(x)$
$\displaystyle=\frac{\epsilon}{2}+\frac{\epsilon}{2}=\epsilon.$

Since by Proposition 2.3 the topology of $X$ is generated by
$\sigma$-invariant seminorms, the result follows. ∎

######  Corollary 2.5.

Let $X$ be a complete LCTVS and suppose that we are given a continuous
representation of the group $\mathbb{T}^{n}$ on $X$. Then if the Fourier
series of an element $x\in X$ with respect to this representation is
absolutely convergent in $X$, the sum of the Fourier series equals $x$.

###### Proof.

Since the series $\sum_{\alpha\in\mathbb{Z}^{n}}\pi_{\alpha}^{\sigma}(x)$ is
absolutely convergent, by Proposition 2.2, there exists the sum of the series,
i.e., an $\widetilde{x}\in X$ such that for every bijection
$\theta:\mathbb{N}\rightarrow\mathbb{Z}^{n}$ we have
$\sum_{j=0}^{\infty}\pi_{\theta(j)}^{\sigma}(x)=\widetilde{x}$. Let
$S_{N}=\sum_{j=0}^{N}\pi_{\theta(j)}^{\sigma}(x)$; then the sequence of
partial sums $\\{S_{N}\\}$ converges to $\widetilde{x}$ in $X$. By Proposition
2.1, the sequence of Cesàro means $\\{C_{N}\\}$ of the partial sums converges
to $\widetilde{x}$ as well. However, by Theorem 2.1, the Cesàro means converge
to $x$. Therefore $\widetilde{x}=x$. ∎

##  3\. Recapturing Complex analysis

We now use the machinery developed in the previous section to give a
conceptually simple account of the remarkable regularity properties of
holomorphic distributions. So we will pretend that we have forgotten
everything about complex analysis, but do remember the rudiments of the theory
of distributions, accounts of which can be found in the classic treatises
[Hör03, Trè67, Sch66]. First we clarify notations and recall a few facts.

###  3.1. The basic spaces

For an open $\Omega\subset{\mathbb{R}}^{n}$, the space $\mathscr{D}(\Omega)$
of test functions is the $LF$-space of smooth compactly supported complex
valued functions, topologized as the inductive limit of the Fréchet spaces
$\mathscr{D}_{K}$ consisting, for a given compact $K\subset\Omega$, of those
elements of $\mathscr{D}(\Omega)$ which have support in $K$. Recall that a
subset $B\subset\mathscr{D}(\Omega)$ is bounded, if and only if there is a
compact $K\subset\Omega$ such that $B\subset\mathscr{D}_{K}$, and for each
nonnegative multi-index $\alpha\in\mathbb{N}^{n}$, we have

$\sup_{\begin{subarray}{c}\phi\in B\\\ x\in
K\end{subarray}}\left|\frac{\partial^{\alpha}}{\partial
x^{\alpha}}\phi(x)\right|<\infty,$ (3.1)

where, here and later, we will use standard multi-index conventions such as

$\frac{\partial^{\alpha}}{\partial
x^{\alpha}}=\frac{\partial^{\left|\alpha\right|}}{\partial
x_{1}^{\alpha_{1}}\dots\partial
x_{n}^{\alpha_{n}}},\quad\left|\alpha\right|=\sum_{j}\alpha_{j},\quad\text{for}\quad\alpha\in\mathbb{N}^{n}.$
(3.2)

The space of distributions $\mathscr{D}^{\prime}(\Omega)$ on $\Omega$ is the
dual of $\mathscr{D}(\Omega)$, consisting of continuous linear forms on
$\mathscr{D}(\Omega)$. We denote the value of a distribution
$T\in\mathscr{D}^{\prime}(\Omega)$ at a test function
$\phi\in\mathscr{D}(\Omega)$ by $\left\langle T,\phi\right\rangle.$ The space
$\mathscr{D}^{\prime}(\Omega)$ is endowed with the usual strong dual topology.
Recall that this topology is generated by the family of seminorms

$\left\\{p_{B}:B\subset\mathscr{D}(\Omega)\text{ is
bounded}\right\\},\quad\text{where, }p_{B}(T)=\sup_{\phi\in
B}\left|\left\langle T,\phi\right\rangle\right|.$ (3.3)

In this topology, the space $\mathscr{D}^{\prime}(\Omega)$ is complete.

Given a locally integrable function $f\in L^{1}_{\mathrm{loc}}(\Omega)$, we
can associate to $f$ a distribution $T_{f}$ defined by

$\left\langle T_{f},\phi\right\rangle=\int_{\Omega}f\phi\,dV,\textrm{ for
}\phi\in\mathscr{D}(\Omega),$

where $dV$ denotes the Lebesgue measure of ${\mathbb{R}}^{n}$. Then the
_locally integrable distribution_ $T_{f}$ is said to be generated by $f$, and
as usual we grant ourselves the the right to abuse language by identifying the
distribution $T_{f}\in\mathscr{D}^{\prime}(\Omega)$ with the function $f\in
L^{1}_{\mathrm{loc}}(\Omega)$.

We will use the abbreviations

$\bm{D}_{j}=\frac{\partial}{\partial
z_{j}},\quad\bm{\overline{D}}_{j}=\frac{\partial}{\partial\overline{z_{j}}},$
(3.4)

for the basic constant coefficient differential operators of complex analysis,
acting on functions or distributions on ${\mathbb{C}}^{n}$. If
$\Omega\subset{\mathbb{C}}^{n}$ is an open set, define

$\mathscr{O}(\Omega)=\left\\{f\in\mathscr{D}^{\prime}(\Omega):\bm{\overline{D}}_{j}f=0,1\leq
j\leq n\right\\},$ (3.5)

the space of _holomorphic distributions_ on $\Omega$. The subspace
$\mathscr{O}(\Omega)$ is closed in the space $\mathscr{D}^{\prime}(\Omega)$ by
the continuity of the operators $\bm{\overline{D}}_{j}$, and is therefore a
complete LCTVS in the subspace topology.

###  3.2. The main theorem

We begin with some definitions and notational conventions. For
$\lambda\in\mathbb{T}^{n}$ we denote by $\mathrm{R}_{\lambda}$ the _Reinhardt
rotation_ of ${\mathbb{C}}^{n}$ by the element $\lambda$, the linear
automorphism of the vector space ${\mathbb{C}}^{n}$ given by

$\mathrm{R}_{\lambda}(z)=(\lambda_{1}z_{1},\dots,\lambda_{n}z_{n}).$ (3.6)

A domain $\Omega\subset{\mathbb{C}}^{n}$ is defined to be _Reinhardt_ if and
only if $\mathrm{R}_{\lambda}(\Omega)=\Omega$ for each
$\lambda\in\mathbb{T}^{n}$. Let $Z$ denote the union of the coordinate
hyperplanes of ${\mathbb{C}}^{n}$:

$Z=\bigcup_{j=1}^{n}\\{z\in{\mathbb{C}}^{n}:z_{j}=0\\}.$ (3.7)

Recall that a Reinhardt domain $\Omega\subset{\mathbb{C}}^{n}$ is said to be
_log-convex_ , if whenever $z,w\in\Omega\setminus Z$, the point
$\zeta\in{\mathbb{C}}^{n}$ belongs to $\Omega$ if there is a $t\in[0,1]$ such
that for $1\leq j\leq n$

$\left|\zeta_{j}\right|=\left|z_{j}\right|^{t}\left|w_{j}\right|^{1-t}.$ (3.8)

For $\alpha\in\mathbb{Z}^{n}$, let $e_{\alpha}$ be the monomial function given
by

$e_{\alpha}(z)=z^{\alpha}=z_{1}^{\alpha_{1}}\dots z_{n}^{\alpha_{n}}.$ (3.9)

For a Reinhardt subset $\Omega\subset{\mathbb{C}}^{n}$ and $1\leq j\leq n$,
let

${\Omega}^{(j)}=\\{(z_{1},\dots,\zeta
z_{j},\dots,z_{n}):z\in\Omega,\zeta\in\overline{\mathbb{D}}\\},$ (3.10)

where $\overline{\mathbb{D}}=\\{\left|\zeta\right|\leq
1\\}\subset{\mathbb{C}}$ is the closed disk. This can be thought of as the
result of “completing” $\Omega$ in the $j$-th coordinate. Following [JP08], we
say that the Reinhardt domain $\Omega\subset{\mathbb{C}}^{n}$ is _relatively
complete_ if for each $1\leq j\leq n$, whenever we have
$\Omega\cap\\{z_{j}=0\\}\not=\emptyset$, we also have

$\Omega^{(j)}\subset\Omega.$

We prove the following well-known structure theorem for holomorphic
distributions on Reinhardt domains, as an application of the ideas of Section
2:

######  Theorem 3.1.

Let $\Omega$ be a Reinhardt domain in ${\mathbb{C}}^{n}$ and let

$\mathcal{S}(\Omega)=\\{\alpha\in\mathbb{Z}^{n}:e_{\alpha}\in\mathcal{C}^{\infty}(\Omega)\\}.$
(3.11)

Let $\widehat{\Omega}$ be the smallest relatively complete log-convex
Reinhardt domain in ${\mathbb{C}}^{n}$ that contains $\Omega$. The for each
$\alpha\in\mathcal{S}(\Omega)$ there is a continuous linear functional
$a_{\alpha}:\mathscr{O}(\Omega)\to{\mathbb{C}}$ such that for each
$T\in\mathscr{O}(\Omega)$, the series

$\sum_{\alpha\in\mathcal{S}(\Omega)}a_{\alpha}(T)e_{\alpha}$ (3.12)

converges absolutely in $\mathcal{C}^{\infty}(\widehat{\Omega})$ to a function
$f\in\mathcal{C}^{\infty}(\widehat{\Omega})$, and $f|_{\Omega}$ generates the
distribution $T$.

Remarks:

  1. (1)

For $n=1$, all Reinhardt domains in the plane, i.e., disks and annuli, are
automatically relatively complete and log-convex. For $n\geq 2$, it is easy to
give examples of Reinhardt domains which are not log-convex, or not relatively
complete, or perhaps both. For such a domain $\Omega$, it follows that each
holomorphic distribution $f\in\mathscr{O}(\Omega)$ extends to a holomorphic
function $F\in\mathscr{O}(\widehat{\Omega})$. This is the simplest example of
_Hartogs phenomenon_ , the compulsory extension of all holomorphic functions
from a smaller domain to a larger one, characteristic of domains in several
complex variables.

  2. (2)

The functionals $a_{\alpha}$ are called the _coefficient functionals_ , and
the series (3.12) is of course the _Laurent series_ of the function $f$ (the
_Taylor series_ if $\mathcal{S}(\Omega)=\mathbb{N}^{n}$).

  3. (3)

It is known (by a direct construction of a plurisubharmonic exhaustion) that
relatively complete log-convex Reinhardt domains are pseudoconvex. This means
that such a domain $\Omega$ admits a holomorphic distribution whose Laurent
expansion converges absolutely precisely on $\Omega$.

As an immediate consequence of Theorem 3.1 we have the following:

######  Corollary 3.1.

Let $\Omega\subset{\mathbb{C}}^{n}$ be open. Then each distribution
$T\in\mathscr{O}(\Omega)$ is _complex-analytic_ , i.e., for each $p\in\Omega$
there is a neighborhood $U$ of $p$, where the function $f$ generating $T$ is
represented by a Taylor series centered at $p$.

###  3.3. Holomorphic functions and maps

A holomorphic distribution $T\in\mathscr{O}(\Omega)$ will be called a
_holomorphic function_ if it is generated by a $\mathcal{C}^{\infty}$ function
$f$. We denote the space of holomorphic functions on $\Omega$ temporarily by
$(\mathscr{O}\cap\mathcal{C}^{\infty})(\Omega)$. Once Theorem 3.1 is proved,
it will follow that
$(\mathscr{O}\cap\mathcal{C}^{\infty})(\Omega)=\mathscr{O}(\Omega)$.

Let $\Omega_{1},\Omega_{2}$ be domains in ${\mathbb{C}}^{n}$. By a
_holomorphic map_ $\Phi:\Omega_{1}\to\Omega_{2}$, we mean a mapping each of
whose components is a holomorphic function on $\Omega_{1}$. A holomorphic map
is a _biholomorphism_ , if it is a bijection, and its set-theoretic inverse is
also a holomorphic map. (It is of course known that the assumption of the
holomorphicity of the inverse map is redundant, but this is a consequence of
complex-analyticity, which is exactly what we are proving here). If
$\Phi:\Omega_{1}\to\Omega_{2}$ is a biholomorphism, then for a distribution
$T\in\mathscr{D}^{\prime}(\Omega_{2})$, we can define in the usual way the
_pullback_ distribution $\Phi^{*}T\in\mathscr{D}^{\prime}(\Omega_{1})$: if $T$
is generated by a test function $f\in\mathscr{D}(\Omega_{2})$, then
$\Phi^{*}T$ is the distribution generated by the function $f\circ\Phi$, and
for general $T$, we extend this definition by continuity, using the density of
test functions in $\mathscr{D}^{\prime}(\Omega)$, see [Hör03, Theorem 6.1.2]
for details. Extending the chain rules for the complex derivative operators
from test functions to distributions, we have the following relations
analogous to [Hör03, formula (6.1.2)] for the Wirtinger derivatives (3.4):

$\bm{D}_{j}(\Phi^{*}T)=\sum_{k=1}^{n}\bm{D}_{j}{\Phi_{k}}\cdot\Phi^{*}\left(\bm{D}_{k}T\right),$
(3.13)

and

$\bm{\overline{D}}_{j}(\Phi^{*}T)=\sum_{k=1}^{n}\bm{\overline{D}}_{j}\overline{\Phi_{k}}\cdot\Phi^{*}\left(\bm{\overline{D}}_{k}T\right),$
(3.14)

where as above, $\Phi:\Omega_{1}\to\Omega_{2}$ is a biholomorphism of domains
in ${\mathbb{C}}^{n}$, written in components as
$\Phi=(\Phi_{1},\dots,\Phi_{n})$, and $T\in\mathscr{D}^{\prime}(\Omega_{2})$.

Therefore, we have the following immediate consequence of (3.14):

######  Proposition 3.2.

If $\Phi:\Omega_{1}\to\Omega_{2}$ is a biholomorphism and
$T\in\mathscr{O}(\Omega_{2})$, then we have
$\Phi^{*}T\in\mathscr{O}(\Omega_{1})$. If
$f\in(\mathscr{O}\cap\mathcal{C}^{\infty})(\Omega_{2})$, then
$\Phi^{*}f\in(\mathscr{O}\cap\mathcal{C}^{\infty})(\Omega_{1})$.

Therefore the spaces of holomorphic distributions and functions are invariant
under pullbacks under biholomorphic maps. In fact, in the proof of Theorem
3.1, only two simple special cases of Proposition 3.2 noted below are needed:

  1. (1)

Translation by a vector $a\in{\mathbb{C}}^{n}$ is the map
$\mathrm{M}_{a}:{\mathbb{C}}^{n}\to{\mathbb{C}}^{n}$

$\mathrm{M}_{a}(z)=z+a,$ (3.15)

which is obviously a biholomorphic automorphism of ${\mathbb{C}}^{n}$. For a
domain $\Omega\subset{\mathbb{C}}^{n}$, we therefore have a pullback
isomorphism of spaces of holomorphic distributions
$\mathrm{M}_{a}^{*}:\mathscr{O}(\mathrm{M}_{a}(\Omega))\to\mathscr{O}(\Omega)$.
This can be thought of as an expression of the fact that the operator
$\overline{\partial}=(\bm{\overline{D}}_{1},\dots,\bm{\overline{D}}_{n})$ is
translation invariant.

  2. (2)

The Reinhardt rotations $\mathrm{R}_{\lambda}$ of (3.6) are clearly
biholomorphic automorphisms of Reinhardt domains, and the pullback operation
induces a representation of the group $\mathbb{T}^{n}$ on the space of
holomorphic distributions (see (3.32) below).

A domain $\Omega$ is said to be _Reinhardt centered at $a$_ for an
$a\in{\mathbb{C}}^{n}$ if there is a Reinhardt domain $\Omega_{0}$ such that
$\Omega=\mathrm{M}_{a}(\Omega_{0})$. Every open set in ${\mathbb{C}}^{n}$ has
local Reinhardt symmetry, in the sense that each point has neighborhood which
is a Reinhardt domain centered at that point.

###  3.4. Mean value property of the monomials

It is easily verified by direct computation that for each $\alpha$, we have
$e_{\alpha}\in(\mathscr{O}\cap\mathcal{C}^{\infty})({\mathbb{C}}^{n}\setminus
Z)$, where $e_{\alpha}$ is the monomial of (3.9). We now note a remarkable
symmetry property (the Mean-Value Property) of the functions $e_{\alpha}$:

######  Lemma 3.3.

Let $\alpha\in\mathbb{Z}^{n}$, let $z\in{\mathbb{C}}^{n}\setminus Z$, and let
$\psi\in\mathscr{D}({\mathbb{C}}^{n}\setminus Z)$ be a test function which has
radial symmetry in each variable around the point $z$, i.e., there are
functions $\rho_{1},\dots,\rho_{n}\in\mathscr{D}({\mathbb{R}})$ such that
$\psi(\zeta)=\prod_{j=1}^{n}\rho_{j}(\left|\zeta_{j}-z_{j}\right|)$, and whose
integral is 1:

$\int_{{\mathbb{C}}^{n}\setminus Z}\psi(\zeta)dV(\zeta)=1.$ (3.16)

Then we have

$\left\langle e_{\alpha},\psi\right\rangle=e_{\alpha}(z)$ (3.17)

###### Proof.

First consider the case $n=1$. If $\alpha\geq 0$, the formula

$\int_{0}^{2\pi}(z+re^{i\theta})^{\alpha}d\theta=2\pi z^{\alpha}$ (3.18)

holds for $r>0$, by expanding the integrand using the binomial formula, and
integrating the finite sum term by term. The formula (3.18) also holds for
$\alpha<0$, provided $z\not=0$, and $0<r<\left|z\right|$. This follows on
noticing that we have an infinite series expansion

$(z+re^{i\theta})^{\alpha}=z^{\alpha}\left(1+\frac{r}{z}e^{i\theta}\right)^{\alpha}=z^{\alpha}\sum_{k=0}^{\infty}\binom{\alpha}{k}\left(\frac{r}{z}\right)^{k}e^{ik\theta},$

where by the $M$-test, the convergence is uniform in $\theta$, and then
integrating the series on the right term by term.

Now $\psi(\zeta)=\rho(\left|\zeta-z\right|)$ and $\psi$ is supported in some
disc $B(z,R)\subset{\mathbb{C}}^{*}={\mathbb{C}}\setminus\\{0\\}$, which means
$R<\left|z\right|$. Then the normalization (3.16) is equivalent to

$2\pi\int_{0}^{R}\rho(r)rdr=1.$ (3.19)

Now

$\displaystyle\left\langle e_{\alpha},\psi\right\rangle$
$\displaystyle=\int_{{\mathbb{C}}^{*}}e_{\alpha}(\zeta)\rho(\left|\zeta-z\right|)dV(\zeta)=\int\limits_{0}^{R}\int\limits_{0}^{2\pi}(z+re^{i\theta})^{\alpha}\rho(r)rd\theta
dr$
$\displaystyle=\int_{0}^{R}\rho(r)r\left(\int\limits_{0}^{2\pi}(z+re^{i\theta})^{\alpha}d\theta\right)dr=2\pi
z^{\alpha}\cdot\int_{0}^{R}\rho(r)rdr$ using (3.18)
$\displaystyle=z^{\alpha}=e_{\alpha}(z),$ $\displaystyle\text{using \eqref{eq-
mvp2}}.$

which establishes the result for $n=1$. In the general case, notice that for
$\alpha\in\mathbb{Z}^{n}$ and $\zeta\in{\mathbb{C}}^{n}\setminus Z$ we have
$e_{\alpha}(\zeta)=\prod_{j=1}^{n}e_{\alpha_{j}}(\zeta_{j}).$ Therefore, since
${\mathbb{C}}^{n}\setminus
Z={\mathbb{C}}^{*}\times\dots\times{\mathbb{C}}^{*}$,

$\displaystyle\left\langle e_{\alpha},\psi\right\rangle$
$\displaystyle=\int_{{\mathbb{C}}^{n}\setminus
Z}\prod_{j=1}^{n}e_{\alpha_{j}}(\zeta_{j})\rho_{j}(\left|\zeta_{j}-z_{j}\right|)\,dV(\zeta)=\prod_{j=1}^{n}\int_{{\mathbb{C}}^{*}}e_{\alpha_{j}}(\zeta_{j})\rho_{j}(\left|\zeta_{j}-z_{j}\right|)\,dV(\zeta_{j})$
$\displaystyle=\prod_{j=1}^{n}e_{\alpha_{j}}(z_{j})=e_{\alpha}(z).$

∎

###  3.5. The representation $\tau$

If $\Omega\subset{\mathbb{C}}^{n}$ is a Reinhardt domain, then for each
$\lambda$, the map $\mathrm{R}_{\lambda}$ of (3.6) is a biholomorphic
automorphism of $\Omega$. Define a representation $\tau$ of $\mathbb{T}^{n}$
on the space $\mathscr{D}(\Omega)$ of test functions by

$\tau_{\lambda}\psi=\psi\circ\mathrm{R}_{\lambda},\quad\psi\in\mathscr{D}(\Omega).$
(3.20)

Recall that a net $\\{\phi_{j}\\}$ converges in the space
$\mathscr{D}(\Omega)$, if each $\phi_{j}$ is supported in a fixed compact
$K\subset\Omega$, and the net $\\{\phi_{j}\\}$ converges in the Fréchet space
$\mathscr{D}_{K},$ i.e. all partial derivatives converge uniformly on $K$.
Using this it is easily verified that $\tau$ is a continuous representation of
$\mathbb{T}^{n}$ in the space $\mathscr{D}(\Omega)$.

Notice that ${\mathbb{C}}^{n}\setminus Z$ is a Reinhardt domain. For a
positive integer $k$ we define the norm $\left\|\psi\right\|_{k}$ with respect
to the polar coordinates for a function $\psi$ in
$\mathscr{D}({\mathbb{C}}^{n}\setminus Z)$ :

$\left\|\psi\right\|_{k}=\max_{\begin{subarray}{c}\beta,\gamma\in\mathbb{N}^{n}\\\
\left|\beta\right|+\left|\gamma\right|\leq
k\end{subarray}}\,\sup_{{\mathbb{C}}^{n}\setminus
Z}\left|\frac{\partial^{\beta}}{\partial
r^{\beta}}\frac{\partial^{\gamma}}{\partial\theta^{\gamma}}\psi\right|,$
(3.21)

where the tuples $r\in({\mathbb{R}}^{+})^{n},\theta\in{\mathbb{R}}^{n}$ are
the polar coordinates on ${\mathbb{C}}^{n}\setminus Z$ specified by
$z_{j}=re^{i\theta_{j}}$, and $\frac{\partial^{\beta}}{\partial
r^{\beta}},\frac{\partial^{\gamma}}{\partial\theta^{\gamma}}$ are partial
derivatives operators in the polar coordinates defined as in (3.2). From the
formulas

${\frac{\partial}{\partial x_{j}}=\cos\theta_{j}\frac{\partial}{\partial
r_{j}}-\frac{\sin\theta_{j}}{r_{j}}\frac{\partial}{\partial\theta_{j}}}\text{
and }{\frac{\partial}{\partial y_{j}}={\sin\theta_{j}}\frac{\partial}{\partial
r_{j}}+\frac{\cos\theta_{j}}{r_{j}}\frac{\partial}{\partial\theta_{j}}},$
(3.22)

we see that for $0<\varrho_{1}<\varrho_{2}<\infty$, there is a constant
$B_{k}(\varrho_{1},\varrho_{2})$ such that for each compact set $K$ such that

$K\subset\\{z\in{\mathbb{C}}^{n}:\varrho_{1}<\left|z_{j}\right|<\varrho_{2}\\},$

we have

$\frac{1}{B_{k}(\varrho_{1},\varrho_{2})}\cdot\left\|\psi\right\|_{k}\leq\left\|\psi\right\|_{\mathcal{C}^{k}}\leq
B_{k}(\varrho_{1},\varrho_{2})\cdot\left\|\psi\right\|_{k}.$ (3.23)

We will need the following elementary estimate:

######  Proposition 3.4.

Let $\tau$ be the representation of $\mathbb{T}^{n}$ on
${\mathbb{C}}^{n}\setminus Z$ given by (3.20). For integers $m,k\geq 0$, and a
compact $K\subset{\mathbb{C}}^{n}\setminus Z$ there is a constant $C>0$ such
that for each $\psi\in\mathscr{D}_{K}$ and each $\alpha\in\mathbb{Z}^{n}$,
such that $\left|\alpha_{j}\right|\geq 2k$ for $1\leq j\leq n$, we have

$\left\|\bm{\pi}^{\tau}_{\alpha}\psi\right\|_{\mathcal{C}^{k}}\leq\frac{C}{\prod\limits_{j=1}^{n}\left|\alpha_{j}\right|^{m}}\cdot\left\|\psi\right\|_{\mathcal{C}^{2nk+nm}}.$

###### Proof.

For $r\in({\mathbb{R}}^{+})^{n},\alpha\in\mathbb{Z}^{n}$, define the
$\alpha$-th Fourier coefficient of $\psi$ by

$\widehat{\psi}(r,\alpha)=\frac{1}{(2\pi)^{n}}\int\limits_{[0,2\pi]^{n}}\psi(re^{i\theta})e^{-i\left\langle\alpha,\theta\right\rangle}d\theta,$
(3.24)

where
$re^{i\theta}=(r_{1}e^{i\theta_{1}},\dots,r_{n}e^{i\theta_{n}})\in{\mathbb{C}}^{n}\setminus
Z$,
$\left\langle\alpha,\theta\right\rangle=\sum_{j=1}^{n}\alpha_{j}\theta_{j}$
and $d\theta=d\theta_{1}\dots d\theta_{n}$ is the Lebesgue measure. We will
also write $(\psi)^{\wedge}$ for $\widehat{\psi}$ whenever convenient. We note
the following properties:

  1. (1)

We clearly have

$\sup_{r\in({\mathbb{R}}^{+})^{n}}\left|\widehat{\psi}(r,\alpha)\right|\leq\sup_{{\mathbb{C}}^{n}\setminus
Z}\left|\psi\right|.$ (3.25)

  2. (2)

If $\alpha\not=0$, by a standard integration by parts argument, for each
$\ell\in\mathbb{N}^{n}$,

$\widehat{\psi}(r,\alpha)=\frac{1}{(i\alpha)^{\ell}}\left(\frac{\partial^{\ell}\psi}{\partial\theta^{\ell}}\right)^{\wedge}(r,\alpha),$
(3.26)

where, as usual, we set for $\zeta\in{\mathbb{C}}^{n}$ and
$\beta\in\mathbb{N}^{n}$,
$\zeta^{\beta}=\zeta_{1}^{\beta_{1}}\dots\zeta_{n}^{\beta_{n}}$.

  3. (3)

Differentiating under the integral sign we have, for any
$\beta\in\mathbb{N}^{n}$,

$\left(\frac{\partial}{\partial
r}\right)^{\beta}\widehat{\psi}(r,\alpha)=\frac{1}{(2\pi)^{n}}\int\limits_{[0.2\pi]^{n}}\frac{\partial^{\beta}\psi}{\partial
r^{\beta}}(re^{i\theta})e^{i\left\langle\beta,\theta\right\rangle}e^{-i\left\langle\alpha,\theta\right\rangle}d\theta={\left(\frac{\partial^{\beta}\psi}{\partial
r^{\beta}}\right)}^{\wedge}(r,\alpha-\beta),$

and combining this with (3.26), we see that if $\beta\not=\alpha$, we have for
each integer $\ell\geq 0$ that

$\left(\frac{\partial}{\partial
r}\right)^{\beta}\widehat{\psi}(r,\alpha)=\frac{1}{\left(i(\alpha-\beta)\right)^{\ell}}\left(\frac{\partial^{\ell}}{\partial\theta^{\ell}}\frac{\partial^{\beta}}{\partial
r^{\beta}}\psi\right)^{\wedge}(r,\alpha-\beta).$ (3.27)

Now for $re^{i\theta}\in{\mathbb{C}}^{n}\setminus Z$ the evaluation
$\mathscr{D}({\mathbb{C}}^{n}\setminus Z)\to{\mathbb{C}}$,
$\psi\mapsto\psi(re^{i\theta})$ is continuous, so using (2.4) we see that for
each $\alpha\in\mathbb{Z}^{n}$ and each
$\psi\in\mathscr{D}({\mathbb{C}}^{n}\setminus Z)$, we have

$\displaystyle\bm{\pi}^{\tau}_{\alpha}\psi(re^{i\theta})$
$\displaystyle=\int_{\mathbb{T}}\lambda^{-\alpha}(\tau_{\lambda}\psi)(re^{i\theta})d\lambda=\frac{1}{(2\pi)^{n}}\int_{[0,2\pi]^{n}}e^{-i\left\langle\alpha,\phi\right\rangle}\psi(e^{i\phi}\cdot
re^{i\theta})d\phi=e^{i\left\langle\alpha,\theta\right\rangle}\cdot\widehat{\psi}(r,\alpha),$
(3.28)

where in the last step we make a change of variables in the integral from
$\phi$ to $\phi+\theta$. Therefore, if $\beta,\gamma\in\mathbb{N}^{n}$ with
$\beta_{j}\not=\alpha_{j}$ for $1\leq j\leq n$, then

$\frac{\partial^{\gamma}}{\partial\theta^{\gamma}}\frac{\partial^{\beta}}{\partial
r^{\beta}}(\bm{\pi}^{\tau}_{\alpha}\psi)(re^{i\theta})=e^{i\left\langle\alpha,\theta\right\rangle}\cdot\frac{(i\alpha)^{\gamma}}{\left(i(\alpha-\beta)\right)^{\ell}}{\left(\frac{\partial^{\ell}}{\partial\theta^{\ell}}\frac{\partial^{\beta}}{\partial
r^{\beta}}\psi\right)}^{\wedge}(r,\alpha-\beta).$

Now in the above formula, if we have
$\left|\beta\right|+\left|\gamma\right|\leq k$, and let

$\ell=(k+m,\dots,k+m)=(k+m)\bm{1},$

where $\bm{1}\in\mathbb{N}^{n}$ is the multi-index each of whose $n$ entries
is 1, we obtain, after taking absolute values of both sides

$\left|\frac{\partial^{\gamma}}{\partial\theta^{\gamma}}\frac{\partial^{\beta}}{\partial
r^{\beta}}(\bm{\pi}^{\tau}_{\alpha}\psi)(re^{i\theta})\right|=\frac{\prod\limits_{j=1}^{n}\left|\alpha_{j}\right|^{\gamma_{j}}}{\prod\limits_{j=1}^{n}\left|\alpha_{j}-\beta_{j}\right|^{m+k}}\cdot\left|\left(\frac{\partial^{(m+k)\bm{1}}}{\partial\theta^{(m+k)\bm{1}}}\frac{\partial^{\beta}}{\partial
r^{\beta}}\psi\right)^{\wedge}(r,\alpha-\beta)\right|.$ (3.29)

In the first factor on the right hand side, we have by hypothesis for each $j$
that $\left|\alpha_{j}\right|\geq 2k$ and $0\leq\beta_{j},\gamma_{j}\leq k$.
Therefore, we have
$\left|\alpha_{j}-\beta_{j}\right|\geq\frac{1}{2}\left|\alpha_{j}\right|$. The
first factor can be estimated as

$\frac{\prod\limits_{j=1}^{n}\left|\alpha_{j}\right|^{\gamma_{j}}}{\prod\limits_{j=1}^{n}\left|\alpha_{j}-\beta_{j}\right|^{m+k}}\leq\frac{\prod\limits_{j=1}^{n}\left|\alpha_{j}\right|^{k}}{\prod\limits_{j=1}^{n}\left(\frac{1}{2}\left|\alpha_{j}\right|^{m+k}\right)}=\frac{2^{m+k}}{\prod\limits_{j=1}^{n}\left|\alpha_{j}\right|^{m}}.$
(3.30)

Using (3.25), the second factor can be estimated as

$\left|\left(\frac{\partial^{(m+k)\bm{1}}}{\partial\theta^{(m+k)\bm{1}}}\frac{\partial^{\beta}}{\partial
r^{\beta}}\psi\right)^{\wedge}(r,\alpha-\beta)\right|\leq\sup_{{\mathbb{C}}^{n}\setminus
Z}\left|\frac{\partial^{(m+k)\bm{1}}}{\partial\theta^{(m+k)\bm{1}}}\frac{\partial^{\beta}}{\partial
r^{\beta}}\psi\right|\leq\left\|\psi\right\|_{2nk+nm},$ (3.31)

where in the last step we use the norm introduced in (3.21), and used the fact
that

$\left|(m+k)\bm{1}+\beta\right|=(m+k)n+\left|\beta\right|\leq(m+k)n+nk,$

since each $\beta_{j}\leq k$. Combining (3.29), (3.30) and (3.31) we see that

$\left|\frac{\partial^{\gamma}}{\partial\theta^{\gamma}}\frac{\partial^{\beta}}{\partial
r^{\beta}}(\bm{\pi}^{\tau}_{\alpha}\psi)(re^{i\theta})\right|\leq\frac{2^{m+k}}{\prod\limits_{j=1}^{n}\left|\alpha_{j}\right|^{m}}\cdot\left\|\psi\right\|_{2nk+nm},$

from which, taking a supremum on the left hand side over
$re^{i\theta}\in{\mathbb{C}}\setminus Z$, and remembering that
$\left|\beta\right|+\left|\gamma\right|\leq k$, we conclude that

$\left\|\bm{\pi}^{\tau}_{\alpha}\psi\right\|_{k}\leq\frac{2^{m+k}}{\prod\limits_{j=1}^{n}\left|\alpha_{j}\right|^{m}}\cdot\left\|\psi\right\|_{2nk+nm}.$

Recall that $K$ is a compact set in ${\mathbb{C}}^{n}\setminus Z$ such that
$\psi\in\mathscr{D}_{K}$, i.e. , the support of $\psi$ is contained in $K$.
Suppose that $\varrho_{1},\varrho_{2}$ are such that $K\subset A$, where $A$
is the product of annuli
$A=\\{z\in{\mathbb{C}}^{n}:\varrho_{1}<\left|z_{j}\right|<\varrho_{2},1\leq
j\leq n\\}$. Formulas (3.24) and (3.28) show that the compact support of
$\bm{\pi}^{\tau}_{\alpha}\psi$ is also contained in the set $A$. Therefore,
passing to the equivalent $\mathcal{C}^{k}$-norms using (3.23), we have

$\left\|\bm{\pi}^{\tau}_{\alpha}\psi\right\|_{\mathcal{C}^{k}}\leq
B_{k}(\varrho_{1},\varrho_{2})\cdot
B_{2nk+nm}(\varrho_{1},\varrho_{2})\cdot\frac{2^{m+k}}{\prod\limits_{j=1}^{n}\left|\alpha_{j}\right|^{m}}\cdot\left\|\psi\right\|_{\mathcal{C}^{2nk+nm}},$

which completes the proof of the result. ∎

###  3.6. The dual representation

Let $\Omega$ be a Reinhardt domain. Since for each $\lambda\in\mathbb{T}^{n}$,
the map $\mathrm{R}_{\lambda}$ maps $\Omega$ biholomorphically (and therefore
diffeomorphically) to itself. Consequently, we can define a representation of
$\mathbb{T}^{n}$ on the space of distributions $\mathscr{D}^{\prime}(\Omega)$
using the pullback operation

$\sigma_{\lambda}(T)=(\mathrm{R}_{\lambda})^{*}(T),\quad
T\in\mathscr{D}^{\prime}(\Omega).$ (3.32)

The representation $\sigma_{\lambda}$ is closely related to the representation
$\tau_{\lambda}$ introduced in (3.20). Clearly, $\mathscr{D}(\Omega)$ is an
invariant (dense) subspace of $\sigma$, on which $\sigma$ restricts to $\tau$.
So $\sigma$ is simply the extension of the representation $\tau$ of (3.20) by
continuity to the space of distributions.

The representation $\sigma$ on $\mathscr{D}^{\prime}(\Omega)$ is also “dual”
to the representation $\tau$ on $\mathscr{D}(\Omega)$:

$\left\langle\sigma_{\lambda}(T),\phi\right\rangle=\left\langle(\mathrm{R}_{\lambda})^{*}(T),\phi\right\rangle=\left\langle
T,\frac{1}{\det_{\mathbb{R}}(\mathrm{R}_{\lambda})}\phi\circ\mathrm{R}_{\lambda}^{-1}\right\rangle=\left\langle
T,\tau_{\lambda^{-1}}\phi\right\rangle,$ (3.33)

so that $\sigma_{\lambda}$ is the transpose of the map $\tau_{\lambda^{-1}}$.
The second equality in the chain may be proved by change of variables when $T$
is a test function, and then using density.

######  Proposition 3.5.

The representation $\sigma$ of $\mathbb{T}^{n}$ on
$\mathscr{D}^{\prime}(\Omega)$ defined in (3.32) is continuous.

###### Proof.

Let $\\{\lambda_{j}\\}$ be a sequence in $\mathbb{T}^{n}$ converging to the
identity element. Since pointwise convergence of a sequence in
$\mathscr{D}^{\prime}(\Omega)$ on each test function implies convergence in
the strong dual topology (see [Trè67]), to show that
$\sigma_{\lambda_{j}}(T)\to T$ in $\mathscr{D}^{\prime}(\Omega)$ we need to
show that for $\phi\in\mathscr{D}(\Omega)$ we have
$\left\langle\sigma_{\lambda_{j}}(T),\phi\right\rangle\to\left\langle
T,\phi\right\rangle$. But by (3.33),
$\left\langle\sigma_{\lambda_{j}}(T),\phi\right\rangle=\left\langle
T,\tau_{\lambda_{j}^{-1}}\phi\right\rangle$ and by the continuity of the
representation $\tau$, to follows that $\sigma_{\lambda_{j}}(T)\to T$ in
$\mathscr{D}^{\prime}(\Omega)$.

To complete the proof, by Proposition 2.3, we need to show that the topology
of $\Omega$ is generated by a collection of $\sigma$-invariant seminorms. Let
$B$ be a bounded subset of $\mathscr{D}(\Omega)$. Let

$\widetilde{B}=\\{\sigma_{\lambda}(\phi):\lambda\in\mathbb{T}^{n},\phi\in
B\\}.$

Then it is clear from (3.1) that $\widetilde{B}$ is also a bounded subset of
$\mathscr{D}(\Omega)$. With notation as it (3.3), it is clear that for each
$T\in\mathscr{D}^{\prime}(\Omega)$, we have $p_{B}(T)\leq
p_{\widetilde{B}}(T)$. Therefore the topology of
$\mathscr{D}^{\prime}(\Omega)$ is generated by the family of continuous
invariant seminorms $\\{p_{\widetilde{B}}\\}$. ∎

###  3.7. Fourier series of distributions on Reinhardt domains

Let $\Omega$ be a Reinhardt domain and let $T\in\mathscr{D}^{\prime}(\Omega)$
be a distribution. Then by the results of Section 2 we can expand $T$ in a
formal Fourier series with respect to the representation $\sigma$ of (3.32).
For simplicity of notation, whenever there is no possibility of confusion, we
denote the Fourier components of $T$ by

$T_{\alpha}=\bm{\pi}^{\sigma}_{\alpha}(T),\quad\alpha\in\mathbb{Z}^{n},$
(3.34)

so that the Fourier series of $T$ is written as
$T\sim\sum\limits_{\alpha\in\mathbb{Z}^{n}}T_{\alpha}$. We notice the
following properties of the Fourier components:

######  Proposition 3.6.

Let $\Omega$ and $T$ be as above and let $\alpha\in\mathbb{Z}^{n}$.

  1. (a)

If the distribution $T$ lies in one of the linear subspaces
$\mathscr{O}(\Omega),\mathcal{C}^{\infty}(\Omega),(\mathscr{O}\cap\mathcal{C}^{\infty})(\Omega)$
or $\mathscr{D}(\Omega)$ of $\mathscr{D}^{\prime}(\Omega)$, the Fourier
component $T_{\alpha}$ also lies in the same subspace.

  2. (b)

If $U$ is another Reinhardt domain such that $U\subset\Omega$, we have

$T_{\alpha}|_{U}=\left(T|_{U}\right)_{\alpha}.$ (3.35)

###### Proof.

All these are consequences of Part 2 of Proposition 2.4. To see Part (a), let
$X$ be the space $\mathscr{D}^{\prime}(\Omega)$ and let $Y$ be one of the
spaces $\mathscr{O}(\Omega),\mathcal{C}^{\infty}(\Omega)$ or
$\mathscr{D}(\Omega)$. Each of these has a natural structure of a complete
LCTVS (the space $\mathscr{O}(\Omega)$ as a closed subspace of
$\mathscr{D}^{\prime}(\Omega)$, $\mathcal{C}^{\infty}(\Omega)$ in its Fréchet
topology and $\mathscr{D}(\Omega)$ in its $LF$-topology), and for each one of
these topologies, the representation $\sigma$ restricts to a continuous
representation (in the stronger topology of the subspace), which we will call
$\tau$ (this coincides with the $\tau$ introduced in (3.20)). If $j$ is the
inclusion map of any one of these subspaces into
$\mathscr{D}^{\prime}(\Omega)$, then clearly it is continuous (where the
subspace has the natural topology). Therefore, by (2.7), we have
$j\left(\bm{\pi}^{\tau}_{\alpha}T\right)=\bm{\pi}^{\sigma}_{\alpha}(j(T))$. By
definition, the right hand side is precisely $T_{\alpha}$, i.e. the Fourier
component of $T$ as an element of $\mathscr{D}^{\prime}(\Omega)$. Notice that
$\bm{\pi}^{\tau}_{\alpha}T\in Y$, since it is the Fourier component of $T$ as
an element of $Y$. Since $j$ is the inclusion of $Y$ in $X$, it now follows
that $T_{\alpha}\in Y$ as well.

Since the result is true for $\mathscr{O}(\Omega)$ and
$\mathcal{C}^{\infty}(\Omega)$ it follows for
$(\mathscr{O}\cap\mathcal{C}^{\infty})(\Omega)$.

For part (b), in part 2 of Proposition 2.4, let $X=\mathscr{D}^{\prime}(U)$
with representation defined in (3.32), which we call $\sigma^{\prime}$ for
clarity, and let $Y=\mathscr{D}^{\prime}(\Omega)$ with the representation
$\sigma$ and let $j:Y\to X$ be the restriction map of distributions, which is
clearly continuous and intertwines the two representations. Then (2.7) takes
the form
$j\circ\bm{\pi}^{\sigma}_{\alpha}=\bm{\pi}^{\sigma^{\prime}}_{\alpha}\circ j$.
Applying this to the distribution $T$, we see that

$T_{\alpha}|_{U}=j\left(\bm{\pi}^{\sigma}_{\alpha}T\right)=\bm{\pi}^{\sigma^{\prime}}_{\alpha}\left(j(T)\right)=\left(T|_{U}\right)_{\alpha}.$

∎

We establish the following lemma:

######  Lemma 3.7.

Let $\Omega\subset{\mathbb{C}}^{n}$ be a Reinhardt domain, and let

$\mathscr{A}=\\{T\in\mathscr{O}(\Omega):\text{each Fourier component
}T_{\alpha}\text{ belongs to }\mathcal{C}^{\infty}(\Omega)\\}.$

Suppose that for each $T\in\mathscr{A}$ the Laurent series
$\sum\limits_{\alpha\in\mathbb{Z}^{n}}T_{\alpha}$ converges absolutely in the
Fréchet space $\mathcal{C}(\Omega)$ . Then the Laurent series of each element
of $\mathscr{A}$ converges absolutely in the space
$\mathcal{C}^{\infty}(\Omega)$.

###### Proof.

For a function $f\in\mathcal{C}(\Omega)$ and a compact set $K$, let $p_{K}$ be
the seminorm:

$p_{K}(f)=\sup_{K}\left|f\right|.$ (3.36)

Notice that the family $\\{p_{K}:K\subset\Omega\quad\text{compact}\\}$ is a
generating family of seminorms for $\mathcal{C}(\Omega)$, and the hypothesis
on $\mathscr{A}$ can be expressed as

$\sum\limits_{\alpha\in\mathbb{Z}^{n}}p_{K}(T_{\alpha})<\infty,$ (3.37)

for each compact $K\subset\Omega$.

The Fréchet topology of $\mathcal{C}^{\infty}(\Omega)$ is generated by the
seminorms $p_{K,L}$, where $K$ ranges over compact subsets of $\Omega$, $L$ is
a constant coefficient differential operator on ${\mathbb{C}}^{n}$, and

$p_{K,L}(f)=\sup_{K}\left|Lf\right|.$

We will continue to write $p_{K}$ for the seminorm (3.36) corresponding to
$L=1$. From the distributional chain rule (3.13), we have for each $j$ that

$\bm{D}_{j}\sigma_{\lambda}(T)=\bm{D}_{j}(\mathrm{R}_{\lambda}^{*}T)=(\bm{D}_{j}\mathrm{R}_{\lambda})\cdot\mathrm{R}_{\lambda}^{*}(\bm{D}_{j}T)=\lambda_{j}\cdot\sigma_{\lambda}(\bm{D}_{j}T).$

Let $\beta\in\mathbb{N}^{n}$, and denote
$\bm{D}^{\beta}=\bm{D}_{1}^{\beta_{1}}\dots\bm{D}_{n}^{\beta_{n}}$. Then, from
the above we have

$\displaystyle\bm{D}^{\beta}T_{\alpha}=\bm{D}^{\beta}\int_{\mathbb{T}^{n}}\lambda^{-\alpha}\sigma_{\lambda}(T)d\lambda=\int_{\mathbb{T}^{n}}\lambda^{-\alpha}\bm{D}^{\beta}\sigma_{\lambda}(T)d\lambda=\int_{\mathbb{T}^{n}}\lambda^{-(\alpha-\beta)}\sigma_{\lambda}(\bm{D}^{\beta}T)d\lambda=(\bm{D}^{\beta}T)_{\alpha-\beta},$

where
$(\bm{D}^{\beta}T)_{\alpha-\beta}=\bm{\pi}^{\sigma}_{{\alpha-\beta}}(\bm{D}^{\beta}T)$.
It follows that all Fourier coefficients of the distribution $\bm{D}^{\beta}T$
are in $\mathcal{C}^{\infty}(\Omega)$, so $\bm{D}^{\beta}T\in\mathscr{A}$.
Therefore,

$p_{K,\bm{D}^{\beta}}(T_{\alpha})=p_{K}\left(\bm{D}^{\beta}T_{\alpha}\right)=p_{K}\left((\bm{D}^{\beta}T)_{\alpha-\beta}\right).$

It follows therefore that

$\sum_{\alpha\in\mathbb{Z}^{n}}p_{K,\bm{D}^{\beta}}(T_{\alpha})=\sum_{\gamma\in\mathbb{Z}^{n}}p_{K}\left((\bm{D}^{\beta}T)_{\gamma}\right)<\infty,$

using assumption (3.37) on elements of $\mathscr{A}$. A partial derivative
$L=\frac{\partial^{\gamma}}{\partial
x^{\gamma}}\frac{\partial^{\delta}}{\partial y^{\delta}}$ on
${\mathbb{C}}^{n}$ can be rewritten as a polynomial in the commuting
differential operators
$\bm{D}_{1},\dots,\bm{D}_{n},\bm{\overline{D}}_{1},\dots,\bm{\overline{D}}_{n}$,
and by hypothesis the derivatives $\bm{\overline{D}}_{j}$ are zero. Therefore,
it follows that $\sum_{\alpha}p_{K,L}(T_{\alpha})<\infty$. Consequently the
series $\sum_{\alpha}T_{\alpha}$ is absolutely convergent in
$\mathcal{C}^{\infty}(\Omega)$.

∎

###  3.8.  Proof of Theorem 3.1: case of annular domains

We now prove a weaker version of Theorem 3.1. A Reinhardt domain
$\Omega\subset{\mathbb{C}}^{n}$ will be called _annular_ if it is disjoint
from the coordinate hyperplanes:

$\Omega\cap Z=\emptyset,$ (3.38)

where $Z$ is as in (3.7). We have the following:

######  Proposition 3.8.

Let $\Omega\subset{\mathbb{C}}^{n}\setminus Z$ be an annular Reinhardt domain
in ${\mathbb{C}}^{n}$. Then for each $\alpha\in\mathbb{Z}^{n}$ there is a
continuous linear functional $a_{\alpha}:\mathscr{O}(\Omega)\to{\mathbb{C}}$
such that for each $T\in\mathscr{O}(\Omega)$, the series of terms in
$\mathcal{C}^{\infty}(\Omega)$ given by
$\sum_{\alpha\in\mathbb{Z}^{n}}a_{\alpha}(T)e_{\alpha}$ converges absolutely
in $\mathcal{C}^{\infty}({\Omega})$ to a function $f$, and this
$\mathcal{C}^{\infty}$-smooth function generates the distribution $T$.

###### Proof.

Let $T\in\mathscr{O}(\Omega)$ and let $T_{\alpha}$ be its $\alpha$-th Fourier
component as in (3.34), and let $S=e_{-\alpha}\cdot T_{\alpha}$. The
distributional Leibniz rule

$\bm{\overline{D}}_{j}(fU)=\left(\bm{\overline{D}}_{j}f\right)\cdot
U+f\cdot\left(\bm{\overline{D}}_{j}U\right),\quad
f\in\mathcal{C}^{\infty}(\Omega)\text{ and }U\in\mathscr{D}^{\prime}(\Omega),$
(3.39)

shows that $S\in\mathscr{O}(\Omega)$, since $T_{\alpha}\in\mathscr{O}(\Omega)$
by Part (1) of Proposition 3.6. By Part (1) of Proposition 2.4, the Fourier
component $T_{\alpha}$ lies in the $\alpha$-th Fourier mode of
$\mathscr{O}(\Omega)$, i.e.,
$\sigma_{\lambda}(T_{\alpha})=\lambda^{\alpha}\cdot T_{\alpha}$, so

$\sigma_{\lambda}(S)=\lambda^{-\alpha}e_{-\alpha}\cdot\lambda^{\alpha}T_{\alpha}=S.$

Therefore, using polar coordinates $z_{j}=r_{j}e^{i\theta_{j}}$, with
$\epsilon_{j}(h)=(1,\dots,1,e^{ih},\dots,1)\in\mathbb{T}^{n}$ (with $e^{ih}$
in the $j$-th position) we have

$\frac{\partial}{\partial\theta_{j}}S=\lim_{h\to
0}\frac{1}{h}\left(\sigma_{\epsilon_{j}(h)}(S)-S\right)=0.$

Since $\bm{\overline{D}}_{j}S=0$, and in polar coordinates we have

$\bm{\overline{D}}_{j}=\frac{e^{i\theta_{j}}}{2}\left(\frac{\partial}{\partial
r_{j}}+\frac{i}{r_{j}}\frac{\partial}{\partial\theta_{j}}\right),$ (3.40)

we see that

$\dfrac{\partial}{\partial r_{j}}S=0$

also. Now from the representations (3.22) we have that
$\displaystyle{\frac{\partial}{\partial x_{j}}S=0}$ and
$\displaystyle{\frac{\partial}{\partial y_{j}}S=0}$ in
$\mathscr{D}^{\prime}(\Omega)$ for $1\leq j\leq n$. Therefore, by a classical
result in the theory of distributions (see [Sch66, Theorème VI, pp. 69ff.]) we
have that $S$ is locally constant on $\Omega$, or more precisely, $S$ is
generated by a locally constant function. Since $\Omega$ is connected, it
follows that there is a constant $a_{\alpha}(T)\in{\mathbb{C}}$ which
generates the distribution $S$, so that $T_{\alpha}=a_{\alpha}(T)e_{\alpha}.$
This can also be written as $a_{\alpha}(T)=e_{-\alpha}\cdot
T_{\alpha}=e_{-\alpha}\cdot\bm{\pi}^{\sigma}_{\alpha}(T)$. Since
multiplication by the fixed $\mathcal{C}^{\infty}$-function $e_{-\alpha}$ is a
continuous map on $\mathscr{D}^{\prime}(\Omega)$ (and therefore on the
subspace $\mathscr{O}(\Omega)$), and
$\bm{\pi}^{\sigma}_{\alpha}:\mathscr{O}(\Omega)\to\mathscr{O}(\Omega)$ is
continuous by part 1 of Proposition 2.4, it follows that
$a_{\alpha}:\mathscr{O}(\Omega)\to\mathscr{O}(\Omega)$ is continuous. But we
know that $a_{\alpha}$ takes values in the subspace of distributions generated
by constants. By a well-known theorem, on a finite-dimensional topological
vector space, there is only one topology. Therefore the topology induced from
$\mathscr{O}(\Omega)$ on the subspace of constants coincides with the natural
topology of ${\mathbb{C}}$. Therefore
$a_{\alpha}:\mathscr{O}(\Omega)\to{\mathbb{C}}$ is continuous.

By the above, the Fourier components $T_{\alpha}$ of the Fourier series of $T$
actually lie in $(\mathscr{O}\cap\mathcal{C}^{\infty})(\Omega)$, and thus any
partial sum (i.e. sum of finitely many terms) of this series also lies in
$(\mathscr{O}\cap\mathcal{C}^{\infty})(\Omega)$. We will now show that the
series (3.34) is absolutely convergent in the topology of
$\mathcal{C}^{\infty}(\Omega)$, in the sense of Proposition 2.2. Let
$K\subset\Omega$ be compact. Let
$\delta<\mathrm{dist}(K,{\mathbb{C}}^{n}\setminus\Omega)$, and let
$\Psi\in\mathscr{D}({\mathbb{C}}^{n})$ be a test function such that

$\mathrm{support}(\Psi)\subset
B(0,\delta),\quad\int_{\mathbb{C}}\Psi(\zeta)dV(\zeta)=1,\quad\text{ and
}\Psi(\zeta)=\Psi(\left|\zeta_{1}\right|,\dots,\left|\zeta_{n}\right|)\text{
for $\zeta\in{\mathbb{C}}^{n}$},$

i.e. $\Psi$ is radial around the origin in each complex coordinate. For $z\in
K$, define $\psi_{z}(\zeta)=\Psi(\zeta-z),$ so that $\psi_{z}$ is radially
symmetric around $z$ in each complex direction. Therefore, for $z\in K$, we
have, by Lemma 3.3 for each $\alpha$ that

$\left\langle T_{\alpha},\psi_{z}\right\rangle=\left\langle
a_{\alpha}(T)e_{\alpha},\psi_{z}\right\rangle=a_{\alpha}(T)z^{\alpha}.$

By the continuity of the mapping $T:\mathscr{D}({\Omega})\to{\mathbb{C}}$,
there is a constant $C>0$ and an integer $k\geq 0$ such that for all
$\psi\in\mathscr{D}({\Omega})$ with support in $K$, we have that

$\left|\left\langle T,\psi\right\rangle\right|\leq
C\cdot\left\|\psi\right\|_{\mathcal{C}^{k}({\Omega})}.$ (3.41)

Also notice that for any $\psi\in\mathscr{D}({\Omega})$ we have

$\displaystyle\left\langle\bm{\pi}^{\sigma}_{\alpha}(T),\psi\right\rangle$
$\displaystyle=\left\langle\int_{\mathbb{T}^{n}}\lambda^{-\alpha}\sigma_{\lambda}(T)d\lambda,\psi\right\rangle=\int_{\mathbb{T}^{n}}\lambda^{-\alpha}\left\langle\sigma_{\lambda}(T),\psi\right\rangle
d\lambda\qquad\text{using \eqref{eq-t}}$
$\displaystyle=\int_{\mathbb{T}^{n}}\lambda^{-\alpha}\left\langle
T,\tau_{\lambda^{-1}}\psi\right\rangle d\lambda=\left\langle
T,\int_{\mathbb{T}^{n}}\lambda^{-\alpha}\tau_{\lambda^{-1}}\psi
d\lambda\right\rangle\qquad\text{using \eqref{eq-dual} and \eqref{eq-t}}$
$\displaystyle=\left\langle
T,\int_{\mathbb{T}^{n}}\lambda^{\alpha}\tau_{\lambda}\psi
d\lambda\right\rangle$ using the invariance of Haar measure of a compact group
under inversion $\displaystyle=\left\langle
T,\bm{\pi}^{\tau}_{-\alpha}\psi\right\rangle.$ (3.42)

In the following estimates, let $C$ denote a constant that depends only on the
compact $K$ and the distribution $T$, and may have different values at
different occurrences. By combining (3.41) with (3.42), we see that for $z\in
K,\phi\in\mathscr{D}({\Omega})$ and each $\alpha$ with
$\left|\alpha\right|\geq 2k$ , using Lemma 3.3

$\displaystyle\left|a_{\alpha}(T)z^{\alpha}\right|$
$\displaystyle=\left|\left\langle
T_{\alpha},\psi_{z}\right\rangle\right|=\left|\left\langle\bm{\pi}^{\sigma}_{\alpha}T,\psi_{z}\right\rangle\right|=\left|\left\langle
T,\bm{\pi}^{\tau}_{-\alpha}\psi_{z}\right\rangle\right|$ $\displaystyle\leq
C\cdot\left\|\bm{\pi}^{\tau}_{-\alpha}\psi_{z}\right\|_{\mathcal{C}^{k}({\Omega})}\quad\text{using
\eqref{eq-distn1}}$
$\displaystyle\leq\frac{C}{\left|\alpha\right|^{2}}\cdot\left\|\psi_{z}\right\|_{\mathcal{C}^{2nk+2n}}\quad\text{by
Proposition~{}\ref{prop-est} with $m=2$,}$
$\displaystyle=\frac{C}{\left|\alpha\right|^{2}}\cdot\left\|\psi_{z}\right\|_{\mathcal{C}^{2nk+2n}},$

recalling that the local order $k$ depends only on the distribution $T$ and
the compact set $K$. Now, for each $z$ we have
$\left\|\psi_{z}\right\|_{\mathcal{C}^{2nk+2n}}=\left\|\Psi\right\|_{\mathcal{C}^{2nk+2n}}$
by translation invariance of the norm. Therefore, for each $\alpha\geq 2k$ we
obtain the estimate for the seminorm (with the same convention as above on the
constant $C$)

$p_{K}(T_{\alpha})=p_{K}(\bm{\pi}^{\sigma}_{\alpha}T)=\sup_{z\in
K}\left|a_{\alpha}(T)z^{\alpha}\right|\leq\frac{C}{\left|\alpha\right|^{2}}\cdot\left\|\Psi\right\|_{\mathcal{C}^{2nk+2n}}=\frac{C}{\left|\alpha\right|^{2}}.$
(3.43)

Clearly therefore
$\sum\limits_{\alpha\in\mathbb{Z}^{n}}p_{K}(T_{\alpha})<\infty$. By Lemma 3.7,
the series $\sum_{\alpha}T_{\alpha}$ converges absolutely in
$\mathcal{C}^{\infty}(\Omega)$. Let $f$ be its sum. Since the inclusion
$\mathcal{C}^{\infty}(\Omega)\subset\mathscr{D}^{\prime}(\Omega)$ is
continuous, we see easily that the Fourier series $\sum_{\alpha}T_{\alpha}$ of
$T\in\mathscr{D}^{\prime}(\Omega)$ converges absolutely in
$\mathscr{D}^{\prime}(\Omega)$. Now it follows from Corollary 2.5 that the sum
of the series is $T$. Thus $T$ is the distribution generated by $f$, and this
completes the proof of the proposition. ∎

###  3.9. Smoothness of holomorphic distributions

######  Corollary 3.9.

Let $\Omega\subset{\mathbb{C}}^{n}$ be an open set. Then each holomorphic
distribution on $\Omega$ is generated by a $\mathcal{C}^{\infty}$-smooth
function, i.e.
$\mathscr{O}(\Omega)=(\mathscr{O}\cap\mathcal{C}^{\infty})(\Omega)$.

###### Proof.

Let $T\in\mathscr{O}(\Omega)$. It suffices to show that $T$ is a smooth
function in a neighborhood of each point $p\in\Omega$. Without loss of
generality, thanks to the invariance of the space of holomorphic functions
under translations (Proposition 3.2 and comments after it) , we can assume
that $p=0$. Let $r>0$ be such that the polydisc $P(r)$ given by

$P(r)=\\{\left|z_{j}\right|<r,1\leq j\leq n\\}\subset{\mathbb{C}}^{n}$

is contained in $\Omega$. We will show that $T$ is a smooth function on
$P^{\prime}=P(\frac{r}{6})$, i.e.,
$T\in(\mathscr{O}\cap\mathcal{C}^{\infty})(P^{\prime})$.

By the previous section, holomorphic distributions on an annular Reinhardt
domain are smooth. By translation invariance, the same is true if the annular
domain is centered at a point $a\in{\mathbb{C}}^{n}$ different from the
origin, i.e. for a domain of the form $\mathrm{M}_{a}(A)$ where $A$ is an
annular Reinhardt domain centered at the origin, with $\mathrm{M}_{a}$ as in
(3.15) is the translation by $a$.

Indeed, let $a=\left(\frac{r}{3},\dots,\frac{r}{3}\right)$, and set
$Q=\mathrm{M}_{a}\left(P(\frac{r}{2})\setminus Z\right)$, so that $Q$ is the
annular Reinhardt domain centered at $a$ given by

$Q=\left\\{z\in{\mathbb{C}}^{n}:0<\left|z_{j}-\frac{r}{3}\right|<\frac{r}{2},1\leq
j\leq n\right\\},$

for which it is easy to verify that $P^{\prime}\subset Q\subset P(r)$. Now
since $T|_{Q}\in(\mathscr{O}\cap\mathcal{C}^{\infty})(Q)$ is a holomorphic
function, the result follows. ∎

###  3.10. Extension to the log-convex hull

In this section, we prove the following special case of Theorem 3.1 for
annular Reinhardt domains:

######  Proposition 3.10.

Let $\Omega$ be an annular Reinhardt domain in ${\mathbb{C}}^{n}$. Then the
Laurent series of a distribution $T\in\mathscr{O}(\Omega)$ converges
absolutely in $\mathcal{C}^{\infty}(\widehat{\Omega})$ where
$\widehat{\Omega}$ is the smallest log-convex annular Reinhardt domain
containing $\Omega$.

Introduce the following notation: for a compact subset
$K\subset{\mathbb{C}}^{n}\setminus Z=({\mathbb{C}}^{*})^{n}$, we let

$\widehat{K}=\left\\{z\in{\mathbb{C}}^{n}\setminus
Z:\left|e_{\alpha}(z)\right|\leq\sup_{K}\left|e_{\alpha}\right|,\text{for all
$\alpha\in\mathbb{Z}^{n}$}\right\\}.$ (3.44)

######  Lemma 3.11.

Let $\Omega$ be an annular Reinhardt domain in ${\mathbb{C}}^{n}$ and let
$\widehat{\Omega}$ be the log-convex annular Reinhardt domain containing
$\Omega$. Then for each point $p\in\widehat{\Omega}$ there is a compact
neighborhood $M$ of $p$ in $\widehat{\Omega}$ and a compact subset
$K\subset\Omega$ such that $M\subset\widehat{K}$.

###### Proof.

Let $\Lambda:{\mathbb{C}}^{n}\setminus Z\to{\mathbb{R}}^{n}$ be the map

$\Lambda(z_{1},\dots,z_{n})=(\log\left|z_{1}\right|,\dots,\log\left|z_{n}\right|).$
(3.45)

From (3.8), it follows that an annular Reinhardt domain
$V\subset{\mathbb{C}}^{n}$ is log-convex if and only if the subset
$\Lambda(V)$ of ${\mathbb{R}}^{n}$ (called the _logarithmic shadow_ of $V$) is
convex. The set $\widehat{\Omega}$ is characterized by the fact that it is the
convex hull of $\Lambda(\Omega)$. Therefore, if $p\in\widehat{\Omega}$, there
exist $m$ points $q^{1},\dots,q^{m}\in V\setminus Z$ and positive numbers
$t_{1},\dots,t_{m}$, with $\sum\limits_{k=1}^{m}t_{k}=1$ such that for $1\leq
j\leq n$, we have $\Lambda(p)=\sum_{k=1}^{m}t_{k}\Lambda(q^{k})$, i.e.,

$\log\left|p_{j}\right|=\sum\limits_{k-1}^{m}t_{k}\log\left|q_{j}^{k}\right|.$

In other words $\Lambda(p)$ lies in the convex hull of the $m$-element set
$\\{\Lambda(q^{1}),\dots,\Lambda(q^{m})\\}$. (By a theorem of Carathéodory,
$m\leq n+1$, but we do not need this fact.) We will show that we can “fatten”
one the points in this set so that the convex hull of the fattened set
contains a neighborhood of the point $\Lambda(p)$.

Without loss of generality, we can assume that $t_{m}\not=0$. Consider the
affine self-map $F$ of ${\mathbb{R}}^{n}$ given by

$F(x)=t_{m}x+\sum_{k=1}^{m-1}t_{k}\Lambda(q_{j}^{k}),$

which is clearly an affine automorphism of ${\mathbb{R}}^{n}$. Therefore if
$L$ is a compact neighborhood of $\Lambda(p)$ in ${\mathbb{R}}^{n}$, then
$F^{-1}(L)$ is a compact neighborhood of $\Lambda(q^{m})$, which after
shrinking can be taken to be contained in $\Lambda(\Omega).$ Let
$K_{0}=\Lambda^{-1}(F^{-1}(L))$, and set

$K=\\{q^{1},\dots,q^{m-1}\\}\cup K_{0}.$

We claim that $\widehat{K}$ contains the compact neighborhood
$\Lambda^{-1}(L)$ of the point $p$. Let $\zeta\in\Lambda^{-1}(L)$. Then there
is a point $z\in K_{0}$ such that
$\Lambda(\zeta)=\sum_{k=1}^{m-1}t_{k}\Lambda(q^{k})+t_{m}\Lambda(z)$, i.e. for
each $1\leq j\leq n$ we have

$\log\left|\zeta_{j}\right|=\sum_{k=1}^{m-1}t_{k}\log\left|q^{k}_{j}\right|+t_{m}\log\left|z_{j}\right|.$

For simplicity of writing, introduce new notation as follows: we let
$z^{k}=q^{k}$ for $1\leq k\leq m-1$ and $z^{m}=z$. Then for a multi-index
$\alpha\in\mathbb{Z}^{n}$ and the point $\zeta\in\Lambda^{-1}(L)$ we have

$\displaystyle\left|e_{\alpha}(\zeta)\right|$
$\displaystyle=\prod_{j=1}^{n}\left|\zeta_{j}\right|^{\alpha_{j}}=\exp\left(\sum_{j=1}^{n}\alpha_{j}\log\left|\zeta_{j}\right|\right)$
$\displaystyle=\exp\left(\sum_{j=1}^{n}\alpha_{j}\sum_{k=1}^{m}t_{k}\log\left|z^{k}_{j}\right|\right)=\exp\left(\sum_{k=1}^{m}t_{k}\sum_{j=1}^{n}\alpha_{j}\log\left|z^{k}_{j}\right|\right)$
$\displaystyle\leq\exp\left(\max_{1\leq k\leq
m}\sum_{j=1}^{n}\alpha_{j}\log\left|z^{k}_{j}\right|\right)$ (3.46)
$\displaystyle=\max_{1\leq k\leq
m}\exp\left(\sum_{j=1}^{n}\alpha_{j}\log\left|z^{k}_{j}\right|\right)$
$\displaystyle=\max_{1\leq k\leq
m}\left|e_{\alpha}(z^{k})\right|=\max\left\\{\max_{1\leq k\leq
m-1}\left|e_{\alpha}(q^{k})\right|,\left|e_{\alpha}(z)\right|\right\\}$
$\displaystyle\leq\max_{K}\left|e_{\alpha}\right|,$ (3.47)

this shows that $\zeta\in\widehat{K}$, and completes the proof. ∎

###### Proof of Proposition 3.10.

By Lemma 3.7 it is sufficient to show that the Laurent series
$\sum_{\alpha}a_{\alpha}(T)e_{\alpha}$ (whose existence was proved in
Proposition 3.8, and whose terms lie in
$(\mathscr{O}\cap\mathcal{C}^{\infty})(\Omega)$) converges absolutely in
$\mathcal{C}(\widehat{\Omega})$. It is sufficient to show that each point $p$
of $\widehat{\Omega}$ has a compact neighborhood $M$ such that we have

$\sum_{\alpha}p_{M}(a_{\alpha}(T)e_{\alpha})<\infty.$

Now by Lemma 3.11, there is a compact subset $K\subset\Omega$ such that
$M\subset\widehat{K}$. We observe, using the definition (3.44) that

$\displaystyle p_{M}(a_{\alpha}(T)e_{\alpha})$ $\displaystyle\leq
p_{\widehat{K}}(a_{\alpha}(T)e_{\alpha})=\sup_{\widehat{K}}\left|a_{\alpha}(T)e_{\alpha}\right|=\left|a_{\alpha}(T)\right|\cdot\sup_{\widehat{K}}\left|e_{\alpha}\right|$
$\displaystyle=\left|a_{\alpha}(T)\right|\cdot\sup_{{K}}\left|e_{\alpha}\right|=\sup_{{K}}\left|a_{\alpha}(T)e_{\alpha}\right|=p_{K}(a_{\alpha}(T)e_{\alpha}),$

so that

$\sum_{\alpha}p_{M}(a_{\alpha}(T)e_{\alpha})\leq\sum_{\alpha}p_{K}(a_{\alpha}(T)e_{\alpha})<\infty,$

using the estimate (3.43) which holds since $K\subset\Omega$. ∎

###  3.11. Laurent series on non-annular Reinhardt domains

We now consider the case of a general (i.e. possibly non-annular) Reinhardt
domain $\Omega$. We begin by showing that the only monomials that occur in the
Laurent series are the ones smooth on $\Omega:$

######  Proposition 3.12.

Let $\Omega$ be a (possibly non-annular) Reinhardt domain in
${\mathbb{C}}^{n}$. Then for each $\alpha\in\mathcal{S}(\Omega)$ (where
$\mathcal{S}(\Omega)$ is as in (3.11)), there is a continuous linear
functional $a_{\alpha}:\mathscr{O}(\Omega)\rightarrow{\mathbb{C}}$ such that
the Fourier series of a $T\in\mathscr{O}(\Omega)$ is of the form

$T\sim\sum_{\alpha\in\mathcal{S}(\Omega)}a_{\alpha}(T)e_{\alpha}.$ (3.48)

###### Proof.

Since $T\in\mathscr{O}(\Omega)$ we have $T|_{\Omega\setminus
Z}\in\mathscr{O}(\Omega\setminus Z)$. Notice that $\Omega\setminus Z$ is an
annular Reinhardt domain and therefore the proof of Proposition 3.8 shows that
the Fourier components are given for $\alpha\in\mathbb{Z}^{n}$ by

$\left(T|_{\Omega\setminus
Z}\right)_{\alpha}=\widehat{a}_{\alpha}(T|_{\Omega\setminus Z})e_{\alpha},$

where $\widehat{a}_{\alpha}:\mathscr{O}(\Omega\setminus Z)\to{\mathbb{C}}$ is
the $\alpha$-th coefficient functional associated to the domain
$\Omega\setminus Z$ (see Proposition 3.8). Thanks to (3.35), we know that
$\left(T|_{\Omega\setminus Z}\right)_{\alpha}=(T_{\alpha})|_{\Omega\setminus
Z}$, so we have

$(T_{\alpha})|_{\Omega\setminus Z}=\widehat{a}_{\alpha}(T|_{\Omega\setminus
Z})e_{\alpha}.$

By Proposition 3.6, the Fourier components of a holomorphic distribution are
holomorphic distributions, so we have $T_{\alpha}\in\mathscr{O}(\Omega)$ since
$T\in\mathscr{O}(\Omega)$. Therefore, since by Corollary 3.9, each holomorphic
distribution is a smooth function, we know that
$T_{\alpha}\in\mathcal{C}^{\infty}(\Omega)$. Therefore, the function
$T_{\alpha}|_{\Omega\setminus Z}=\widehat{a}_{\alpha}(T|_{\Omega\setminus
Z})e_{\alpha}$ admits a $\mathcal{C}^{\infty}$ extension through $Z$. If
$\widehat{a}_{\alpha}(T|_{\Omega\setminus Z})\not=0$ for some
$T\in\mathscr{O}(\Omega)$, this means that $e_{\alpha}$ itself admits a
$\mathcal{C}^{\infty}$ extension to $\Omega$, i.e.,
$\alpha\in\mathcal{S}(\Omega)$, where $\mathcal{S}(\Omega)$ is as in (3.11).
Therefore if $\alpha\not\in\mathcal{S}(\Omega)$, the corresponding term in the
Laurent series of $T|_{\Omega\setminus Z}$ vanishes, and the series takes the
form

$T|_{\Omega\setminus
Z}=\sum_{\alpha\in\mathcal{S}(\Omega)}\widehat{a}_{\alpha}(T|_{\Omega\setminus
Z})e_{\alpha}.$

Now for each $\alpha\in\mathcal{S}(\Omega)$ define the map
$a_{\alpha}:\mathscr{O}(\Omega)\to{\mathbb{C}}$ by
$a_{\alpha}(T)=\widehat{a}_{\alpha}(T|_{\Omega\setminus Z}).$ Since both the
restriction map and the coefficient functional $\widehat{a}_{\alpha}$ are
continuous, it follows that $a_{\alpha}$ is continuous. The extension of
$e_{\alpha}$ from $\Omega\setminus Z$ to $\Omega$ is still the monomial
$e_{\alpha}$, so the Fourier series of $T$ in $\mathscr{O}(\Omega)$ is of the
form (3.48). ∎

Notice that each term of (3.48) is in $\mathcal{C}^{\infty}(\Omega)$ and by
Proposition 3.8, the series converges absolutely in
$\mathcal{C}^{\infty}(\Omega\setminus Z)$ when its terms are restricted to
$\Omega\setminus Z$, i.e., it converges uniformly along with all derivatives
on those compact sets in $\Omega$ which do not intersect $Z$.

###  3.12. Extension to the relative completion

Given a Reinhardt domain $D\subset{\mathbb{C}}^{n}$, its _relative completion_
is the smallest relatively complete domain containing $D$ (see above before
the statement of Theorem 3.1 for the definition of relative completeness of a
domain.) Notice that the relative completion of $D$ coincides with the unions
of the sets $D^{(j)}$ of (3.10), where the union is taken over those $j$ for
which $D\cap\\{z_{j}=0\\}$ is nonempty. The following general proposition,
which encompasses classical examples of the Hartogs phenomenon, e.g. in the
“Hartogs figure”, will be needed to complete the proof of Theorem 3.1.

######  Proposition 3.13.

Let $D$ be a Reinhardt domain. Then each holomorphic function on $D$ extends
holomorphically to the relative completion of $D$.

###### Proof.

We may assume that $n\geq 2$ since each Reinhardt domain in the plane is
automatically relatively complete. If for each $j$, the intersection
$D\cap\\{z_{j}=0\\}=\emptyset$, then the domain $D$ is annular, and its
relative completion is itself so there is nothing to prove. Suppose therefore
that there is $1\leq j\leq n$ such that $D\cap\\{z_{j}=0\\}\not=\emptyset$. We
need to prove that each function in $\mathscr{O}(D)$ extends holomorphically
to $D^{(j)}$.

Without loss of generality we can assume that $j=1$. Write the coordinates of
a point $z\in{\mathbb{C}}^{n}$ as $z=(z_{1},\widetilde{z})$ where
$\widetilde{z}\in{\mathbb{C}}^{n-1}$. Let $f\in\mathscr{O}(D)$. By Proposition
3.12 $f$ admits a Laurent series representation

$f=\sum_{\alpha\in\mathcal{S}(D)}a_{\alpha}(f)e_{\alpha}$

with $S(D)$ as in (3.11), and the series converges absolutely in
$\mathcal{C}^{\infty}(D\setminus Z)$. To prove the proposition it suffices to
show that the series in fact converges absolutely in
$\mathcal{C}^{\infty}(D^{(1)})$, which by Lemma 3.7 is equivalent to the
following: for each point $p\in D^{(1)}$ there is a compact neighborhood $M$
of $p$ in $D^{(1)}$ such that

$\sum_{\alpha\in\mathcal{S}(D)}p_{M}(a_{\alpha}(f)e_{\alpha})<\infty.$ (3.49)

We claim the following: for each $p\in D^{(1)}$ there is a compact
neighborhood $M$ of $p$ and a compact subset $K\subset D$ such that
$p_{M}(e_{\alpha})\leq p_{K}(e_{\alpha})$ for each $\alpha\in\mathcal{S}(D)$.
Assuming the claim for a moment, we see that we have

$p_{M}(a_{\alpha}(f)e_{\alpha})=\left|a_{\alpha}(f)\right|p_{M}(e_{\alpha})\leq\left|a_{\alpha}(f)\right|p_{K}(e_{\alpha})=p_{K}(a_{\alpha}(f)e_{\alpha}),$

so that (3.49) follows since by (3.43) we do have
$\sum_{\alpha\in\mathcal{S}(D)}p_{K}(a_{\alpha}(f)e_{\alpha})<\infty$ for a
compact subset $K$ of $D$. Since each point $p\in D^{(1)}$ has such a
neighborhood this completes the proof, modulo the claim above.

To establish the claim, we may assume that $p\not\in D$, since otherwise there
is nothing to prove. Therefore $p\in D^{(1)}\setminus D$, and consequently,
there is a $z\in D$ and a $\lambda\in\mathbb{D}$ such that $p=(\lambda
z_{1},\widetilde{z})$, where $\widetilde{z}=(z_{2},\dots,z_{n})$. Let $K$ be a
compact neighborhood of $z$ in $D$ of the form $K=K_{1}\times\widetilde{K}$,
where $\widetilde{K}\subset{\mathbb{C}}^{n-1}$ and
$K_{1}=\\{\zeta\in{\mathbb{C}}:\left|\zeta-z_{1}\right|\leq\epsilon\\}$ is a
closed disk. Let $L_{1}$ be the disk
$\\{\zeta\in{\mathbb{C}}:\left|\zeta\right|\leq\left|z_{1}\right|+\epsilon\\}$,
so that $K_{1}\subset L_{1}$, and

$\gamma={z_{1}}+\frac{z_{1}}{\left|z_{1}\right|}\epsilon$

is a point of maximum modulus (i.e. maximum distance from the origin) in both
sets $K_{1}$ and $L_{1}$. Note that

$\left|\gamma\right|=\left|\frac{z_{1}}{\left|z_{1}\right|}\cdot\left|z_{1}\right|+\frac{z_{1}}{\left|z_{1}\right|}\epsilon\right|=\left|z_{1}\right|+\epsilon.$

We set $M=L_{1}\times\widetilde{K}$. We will show that these sets $K,M$
satisfy the conditions of the claim.

Now let $\alpha\in\mathcal{S}(D)$. Since $D\cap\\{z_{1}=0\\}\not=\emptyset$,
it follows that $\alpha_{1}\geq 0$. Let
$\widetilde{\alpha}=(\alpha_{2},\dots,\alpha_{n})\in\mathbb{Z}^{n-1}$, and set

$B=\sup_{\widetilde{w}\in\widetilde{K}}\left|\widetilde{w}^{\widetilde{\alpha}}\right|\quad\text{where
$\widetilde{w}=(w_{2},\dots,w_{n})\in{\mathbb{C}}^{n-1}$ and
$\left|\widetilde{w}^{\widetilde{\alpha}}\right|=\left|w_{2}^{\alpha_{2}}\right|\cdots\left|w_{n}^{\alpha_{n}}\right|$},$

so that we have

$p_{M}(e_{\alpha})=\sup_{w\in M}\left|e_{\alpha}(w)\right|=\sup_{w_{1}\in
L_{1}}\left|w_{1}\right|^{\alpha_{1}}\cdot
B=(\left|z_{1}\right|+\epsilon)^{\alpha_{1}}\cdot B.$

On the other hand

$p_{K}(e_{\alpha})=\sup_{w\in K}\left|e_{\alpha}(w)\right|=\sup_{w_{1}\in
K_{1}}\left|w_{1}\right|^{\alpha_{1}}\cdot
B=(\left|z_{1}\right|+\epsilon)^{\alpha_{1}}\cdot B.$

Consequently, in fact we have $p_{M}(e_{\alpha})=p_{K}(e_{\alpha})$, and the
claim is proved, thus completing the proof. ∎

###  3.13. End of proof of Theorem 3.1

It only remains to put together the various pieces to note that all parts of
Theorem 3.1 have been established. If $\Omega$ is annular, i.e. $\Omega$ has
empty intersection with the set $Z$ of (3.7), then Proposition 3.8 takes care
of the complete proof. When $\Omega$ is allowed to be non-annular, we see from
Proposition 3.12 that the Laurent series representation has only monomials
which are smooth functions on $\Omega$. Now it is not difficult to see that
the smallest log-convex relatively complete Reinhardt domain
$\widehat{\Omega}$ containing $\Omega$ can be constructed from $\Omega$ in two
steps. First, we construct the log-convex hull $\Omega_{1}$ of the set
$\Omega\setminus Z$. Notice that $\Omega\setminus Z$ and $\Omega_{1}$ are both
annular. The second step consists of constructing the relative completion of
the domain $\Omega_{1}$, thus obtaining the domain $\widehat{\Omega}$. Now by
Proposition 3.10, the Laurent series of a holomorphic distribution on $\Omega$
converges absolutely in $\mathcal{C}^{\infty}(\Omega_{1})$. Applying
Proposition 3.13 (with $D=\Omega_{1}$), we see that the Laurent series
actually converges absolutely in the space
$\mathcal{C}^{\infty}(\widehat{\Omega})$. The sum of this series is the
required holomorphic extension of a given holomorphic distribution on
$\Omega$. The result has been completely established.

##  4\. Missing monomials

In Theorem 3.1, we considered the natural representation of the torus
$\mathbb{T}^{n}$ on the space $\mathscr{O}(\Omega)$ of holomorphic functions
on a Reinhardt domain $\Omega$. In applications, one often deals with a
subspace of functions $Y\subset\mathscr{O}(\Omega)$ such that

  1. (1)

the subspace $Y$ is invariant under the natural representation $\sigma$, i.e.,
if $f\in Y$ then $f\circ\mathrm{R}_{\lambda}\in Y$ for each
$\lambda\in\mathbb{T}^{n}$, where $\mathrm{R}_{\lambda}$ is as in (3.6),

  2. (2)

there is a locally convex topology on $Y$ in which it is complete, and such
that the inclusion map

$j:Y\hookrightarrow\mathscr{O}(\Omega)$

is continuous, and

  3. (3)

when $Y$ is given this topology, the representation $\sigma$ restricts to a
continuous representation on $Y$.

The _locus classicus_ here is the theory of Hardy spaces on the disc. We can
make the following elementary observation:

######  Proposition 4.1.

Let $Y$ and $\sigma$ be as above, and set

$\mathcal{S}(Y)=\\{\alpha\in\mathbb{Z}^{n}:e_{\alpha}\in Y\\}.$

Then the Laurent series of a function $f\in Y$ is of the form

$f=\sum_{\alpha\in\mathcal{S}(Y)}a_{\alpha}(f)e_{\alpha}.$

###### Proof.

It suffices to show that if for $\alpha\in\mathbb{Z}^{n}$, if the monomial
$e_{\alpha}$ does not belong to $Y$, we have that $a_{\alpha}(f)=0$, where
$a_{\alpha}(f)$ is the Laurent coefficient of $f$ as in (3.12). By part (1) of
Proposition 2.4, we see that $\bm{\pi}^{\sigma}_{\alpha}(f)\in Y$. By part (2)
of the same proposition, taking $X$ to be the space $\mathscr{O}(\Omega)$, for
$f\in Y$ we have that
$\bm{\pi}^{\sigma}_{\alpha}(j(f))=j(\bm{\pi}^{\sigma}_{\alpha}(f))$, and from
the description of the Fourier components of a holomorphic function in the
proof of Theorem 3.1, we see that
$\bm{\pi}^{\sigma}_{\alpha}(j(f))=a_{\alpha}(j(f))e_{\alpha}=a_{\alpha}(f)e_{\alpha}$.
Therefore $j(\bm{\pi}^{\sigma}_{\alpha}(f))=a_{\alpha}(f)e_{\alpha}$, which
contradicts the fact that $\bm{\pi}^{\sigma}_{\alpha}(f)\in Y$ unless
$a_{\alpha}(f)=0$. ∎

This simple observation can be called the “principle of missing monomials”,
since it says that certain monomials cannot occur in the Laurent series of the
function $f$. It can be thought to be the reason behind several phenomena
associated to holomorphic functions. We consider two examples:

  1. (1)

Bergman spaces in Reinhardt domains: Let $\Omega$ be a Reinhardt domain in
${\mathbb{C}}^{n}$ and let $\lambda>0$ be a radial weight on $\Omega$, i.e.,
for $z\in\Omega$ we have

$\lambda(z_{1},\dots,z_{n})=\lambda(\left|z_{1}\right|,\dots,\left|z_{n}\right|).$

The _ $L^{p}$-Bergman space_ $A^{p}(\Omega,\lambda)$ is defined to be the
subspace of the weighted $L^{p}$-space $L^{p}(\Omega,\lambda)$ consisting of
holomorphic functions, where the norm on the weighted $L^{p}$-space is given
by

$\left\|f\right\|_{L^{p}(\Omega,\lambda)}^{p}=\int_{\Omega}\left|f\right|^{p}\lambda
dV,$

where $dV$ is the Lebesgue measure. It is well-known that
$A^{p}(\Omega,\lambda)$ is a closed subspace of the Banach space
$L^{p}(\Omega,\lambda)$ and therefore a Banach space ([DS04]). It is also easy
to see (using standard facts about $L^{p}$-spaces) that the natural
representation $\sigma$ of $\mathbb{T}^{n}$ on $L^{p}(\Omega,\lambda)$ is
continuous for $1\leq p<\infty$, so it follows that the representation on
$A^{p}(\Omega,\lambda)$ is also continuous. It now follows from Proposition
4.1 that the Laurent series expansion of a function $f\in
A^{p}(\Omega,\lambda)$ consists only of terms with monomials $e_{\alpha}$ such
that $e_{\alpha}\in L^{p}(\Omega,\lambda)$. The case $\lambda\equiv 1$ of this
fact was deduced by a different argument in [CEM18].

  2. (2)

Extension of holomorphic functions smooth up to the boundary: Let
$\Omega\subset{\mathbb{C}}^{n}$ be a Reinhardt domain such that the origin
(which is the center of symmetry) is on the boundary of $\Omega$. A classic
example of this is the _Hartogs triangle_
$\\{\left|z_{1}\right|<\left|z_{2}\right|<1\\}$ in ${\mathbb{C}}^{2}$. In
[Cha19], the following extension theorem was proved: there is a complete
Reinhardt domain $V\subset{\mathbb{C}}^{n}$ such that $\Omega\subset V$ and
each function in the space
$\mathcal{O}(\Omega)\cap\mathcal{C}^{\infty}(\overline{\Omega})$ of
holomorphic functions on $\Omega$ smooth up to the boundary extends
holomorphically to the domain $V$. This was noted for the Hartogs triangle in
[Sib75], where $V$ is the unit bidisk.

To deduce this from the principle of missing monomials, it suffices to
consider the case when $\Omega$ is bounded. We notice that
$\mathcal{O}(\Omega)\cap\mathcal{C}^{\infty}(\overline{\Omega})$ is a Fréchet
space in its usual topology of uniform convergence on $\overline{\Omega}$ with
all partial derivatives. A generating family of seminorms for this topology is
given by the norms $\\{p_{k}\\}$ where
$p_{k}(f)=\left\|f\right\|_{C^{k}(\overline{\Omega})}$. The natural
representation $\sigma$ on $\mathscr{O}(\Omega)$ restricts to a continuous
representation on
$\mathcal{O}(\Omega)\cap\mathcal{C}^{\infty}(\overline{\Omega})$. Therefore,
the principle of missing monomials applies and the only monomials $e_{\alpha}$
that occur in the Laurent expansion of a function
$f\in\mathcal{O}(\Omega)\cap\mathcal{C}^{\infty}(\overline{\Omega})$ are such
that
$e_{\alpha}\in\mathcal{O}(\Omega)\cap\mathcal{C}^{\infty}(\overline{\Omega})$.
For such a multi-index $\alpha$, write the multiindex $\alpha=\beta-\gamma$,
where $\beta_{j}=\max(\alpha_{j},0)$ and $\gamma_{j}=(-\alpha_{j},0)$. Then
$\beta,\gamma\in\mathbb{N}^{n}$ , and we can apply the differential operator
$\left(\frac{\partial}{\partial z}\right)^{\beta}$ to obtain

$\left(\frac{\partial}{\partial
z}\right)^{\beta}e_{\alpha}(z)=\left(\frac{\partial}{\partial
z}\right)^{\beta}\frac{e_{\beta}(z)}{e_{\alpha}(z)}=\frac{\beta!}{z^{\gamma}}.$

Since $e_{\alpha}\in\mathcal{C}^{\infty}(\overline{\Omega})$ this means that
$e_{\gamma}\in\mathcal{C}^{\infty}(\overline{\Omega})$, which is possible only
if $\gamma=0$. Thus $\alpha\in\mathbb{N}^{n}$, and the Laurent series of each
function $f\in\mathcal{O}(\Omega)\cap\mathcal{C}^{\infty}(\overline{\Omega})$
is a Taylor series which converges in some complete (log-convex) Reinhardt
domain $V$, and this $V$ must strictly contain $\Omega$, since $\Omega$ is not
complete.

##  5\. Classical characterizations of holomorphic functions

In this section we show how one can avoid the machinery of generalized
functions and weak derivatives altogether, and still use Fourier methods to
prove the basic facts of function theory. We confine ourselves to one complex
variable and the simple geometry of the disk.

###  5.1. Goursat’s characterization

The starting point of a traditional account of holomorphic functions of a
single variable is typically Goursat’s definition ( [Gou00]): a function
$f:\Omega\to{\mathbb{C}}$ on an open subset $\Omega\subset{\mathbb{C}}$ is
holomorphic, if it is _complex differentiable_ , i.e., for each point
$w\in\Omega$, the limit

${\lim_{z\to w}\frac{f(z)-f(w)}{z-w}}$ (5.1)

exists. The result that a holomorphic function in this sense is infinitely
many times complex differentiable and even admits a convergent power series
representation near each point is rightly celebrated as one of the most
elegant and surprising in all of mathematics. Unfortunately, we cannot use it
as a definition, if we want to apply the theory of abstract Fourier expansions
as developed in Section 2. Denoting by $G(\Omega)$ the collection of
holomorphic functions in the sense of Goursat in an open set
$\Omega\subset{\mathbb{C}}$, we notice that the space $G(\Omega)$ does not
have a nice a priori linear locally convex topology in which it is complete
and such that when $\Omega$ is a disc or an annulus, the natural action of the
group $\mathbb{T}$ on the space $G(\Omega)$ is a continuous representation.
Though Goursat’s definition carries the weight of a century of academic
tradition, we will start from an alternative definition which lends itself
better to the application of the methods of Section 2. We also note that the
characterization of holomorphic functions by complex-differentiability cannot
be used for natural generalizations of complex analysis, e.g. quaternionic
analysis, analysis on Clifford algebras etc. (see [GM91, pp. 87–93]).

###  5.2. Morera’s definition

Let $\Omega$ be an open subset of the complex plane ${\mathbb{C}}$, and let
$f:\Omega\to{\mathbb{C}}$ be a continuous function. In honor of [Mor02], let
us say that the function $f$ is _holomorphic in the sense of Morera_ (_Morera-
holomorphic_ for short) if for each triangle $T$ contained (with its interior)
in $\Omega$, we have the vanishing of the complex line integral of $f$ around
the boundary of $T$:

$\int_{\partial T}f(z)dz=0,$ (5.2)

where $\partial T$ denotes the boundary of $T$, oriented counterclockwise. For
an open set $\Omega\subset{\mathbb{C}}$, let us denote by
$\mathscr{O}_{M}(\Omega)$ the collection of Morera-holomorphic functions on
$\Omega$. It is known by Morera’s theorem that a Morera-holomorphic function
is Goursat-holomorphic, and one can develop function theory starting from
Morera’s definition (see [Hef55, MW67]). Notice that the a priori regularity
of Morera-holomorphic functions (assumed to be only continuous) is even less
than that assumed for Goursat-holomorphic functions (assumed also to admit the
limit (5.1) at each $w$.)

It is immediate from the definition that $\mathscr{O}_{M}(\Omega)$ is a closed
linear subspace of the Fréchet space $\mathcal{C}(\Omega)$ of continuous
functions. “Closed” means that the limit of a sequence of Morera-holomorphic
functions converging uniformly on compact subsets of $\Omega$ is itself
Morera-holomorphic, a fact that was already noted in [Mor86]. The proof of
this crucial fact starting from the Goursat definition must pass through a
lengthy development of integral representations, so this is definitely a
pedagogical advantage of Morera’s definition over Goursat’s.

The notion of Morera-holomorphicity is local: i.e.,
$f\in\mathscr{O}_{M}(\Omega)$ if and only if there is an open cover
$\\{\Omega_{j}\\}_{j\in J}$ of $\Omega$ such that
$f|_{\Omega_{j}}\in\mathscr{O}_{M}(\Omega_{j})$ for each $j$. One half of this
claim is trivial, and for the other half, for a triangle $T$ in $\Omega$, we
can perform repeated barycentric subdivisions till the triangles so formed are
each contained in some element of the open cover $\\{\Omega_{j}\\}_{j\in J}$.
We therefore conclude that $\mathscr{O}_{M}$ is a _sheaf_ of Fréchet spaces on
${\mathbb{C}}$.

The following local description of Morera-holomorphic functions is well-known,
and a proof can be found in e.g. [Rem91, pp. 186-189].

######  Proposition 5.1.

Let $\Omega\subset{\mathbb{C}}$ be convex. Then the following statements about
a continuous function $f\in\mathcal{C}(\Omega)$ are equivalent:

  1. (A)

$f\in\mathscr{O}_{M}(\Omega)$.

  2. (B)

$f$ has a holomorphic primitive, i.e., there is an $F$ which is complex-
differentiable on $\Omega$ and $F^{\prime}=f$.

  3. (C)

for each piecewise $\mathcal{C}^{1}$ closed path $\gamma$ in $\Omega$ we have

$\int_{\gamma}f(z)dz=0.$ (5.3)

Recall that assuming (A), the primitive $F$ in (B) is constructed by fixing
$z_{0}\in\Omega$, and setting
$\displaystyle{F(z)=\int_{[z_{0},z]}f(\zeta)d\zeta}$ where $[z_{0},z]$ denotes
the line segment from $z_{0}$ to $z$. Proposition 5.1 allows us to give
examples of Morera-holomorphic functions. Recall from (3.9) that for an
integer $n$, we use the notation $e_{n}(z)=z^{n}$ for the holomorphic
monomials.

######  Proposition 5.2.

If $n\geq 0$ then $e_{n}\in\mathscr{O}_{M}({\mathbb{C}})$ and if $n<0$ then
$e_{n}\in\mathscr{O}_{M}({\mathbb{C}}\setminus\\{0\\})$.

###### Proof.

First note that $e_{n}$ is continuous, on all of ${\mathbb{C}}$ if $n\geq 0$
and on ${\mathbb{C}}\setminus\\{0\\}$ if $n<0$. If
$g_{n}(z)=\frac{z^{n}}{n+1}$ for $n\not=-1$, we can verify from the definition
(5.1) that $g_{n}$ is complex-differentiable and $g_{n}^{\prime}=e_{n}$ so
that by part (B) of Proposition 5.1 the result follows for $n\not=-1$. For
$n=-1$, we can construct for each $p\in{\mathbb{C}}\setminus\\{0\\}$, a local
primitive of $e_{-1}$ near $p$ by setting

$g_{-1}(z)=\ln\left|z\right|+i\arg z,$

where $\arg$ denotes a branch of the argument defined near the point $p$. A
direct computation shows that $g_{-1}^{\prime}=e_{-1}$ near $p$, so that again
we see that $e_{-1}\in\mathscr{O}_{M}({\mathbb{C}}\setminus\\{0\\}).$∎

###  5.3. Products of Morera-holomorphic functions

In the proof of Theorem 3.1, an important role is played by the fact that if
$U$ is a holomorphic distribution (in the sense of (3.5))and $f$ is a
holomorphic function (i.e. a holomorphic distribution which is
$\mathcal{C}^{\infty}$, Section 3.3 above), then the product distribution $fU$
is also a holomorphic distribution. This is an immediate consequence of the
distributional Lebniz formula (3.39). A similar result, proved in [MW67], will
be needed in order to develop the properties of holomorphic functions starting
from Morera’s definition.

######  Proposition 5.3.

Let $f,g\in\mathscr{O}_{M}(\Omega)$, and assume that $g$ is locally Lipschitz
at each point, i.e. for each $w\in\Omega$ and each compact $K\subset\Omega$
such that $w\in K$, there is an $M>0$ such that for $z\in K$ we have

$\left|g(z)-g(w)\right|\leq M\left|z-w\right|.$ (5.4)

Then the product $fg$ also belongs to $\mathscr{O}_{M}(\Omega)$.

The proof is based on a version of the classical Goursat lemma ([Gou00,
Pri01]). This is of course the main ingredient in the standard textbook proof
of the Cauchy theorem for triangles for Goursat-holomorphic functions. Recall
that two triangles are _similar_ if they have the same angles.

######  Lemma 5.4.

Let $\Omega$ be an open subset of ${\mathbb{C}}$ and let $\lambda$ be a
complex valued function defined on the set of triangles contained in $\Omega$
such that the following two conditions are satisfied:

  1. (1)

$\lambda$ is additive in the following sense: if a triangle $\Delta$ is
represented as a union of smaller triangles
$\Delta=\bigcup_{k=1}^{n}\Delta_{k}$ with pairwise disjoint interiors then

$\lambda(\Delta)=\sum_{k=1}^{n}\lambda(\Delta_{k}).$ (5.5)

  2. (2)

For each $w\in\Omega$ and each triangle $\Delta_{0}$, we have

$\lim_{\begin{subarray}{c}\Delta\downarrow w\\\
\Delta\sim\Delta_{0}\end{subarray}}\frac{\lambda(\Delta)}{\left|\Delta\right|}=0,$
(5.6)

where $\left|\Delta\right|$ denotes the area of the triangle $\Delta$, and the
limit is taken along the family of triangles similar to $\Delta_{0}$ and
containing the point $w$, as these triangles shrink to the point $w$.

Then $\lambda\equiv 0$.

###### Proof.

For completeness, we recall the classic argument. Let $T$ be a triangle
contained in $\Omega$. We construct a sequence of triangles
$\\{T_{k}\\}_{k=0}^{\infty}$ with $T_{0}=T$ using the following recursive
procedure. Assuming that $T_{k}$ has been constructed, we divide $T_{k}$ into
four similar triangles with half the diameter of $T_{k}$ by three line
segments each parallel to a side of $T_{k}$ and passing through the midpoints
of the other two sides. Denote the four triangles so obtained by
$\Delta_{j},1\leq j\leq 4$. Then, by (5.5), we have

$\lambda(T_{k})=\sum_{j=1}^{4}\lambda(\Delta_{j}).$

Choose $T_{k+1}$ to be one of $\Delta_{j},1\leq j\leq 4$ such that the value
of $\left|\lambda(T_{k+1})\right|$ is the largest. Then, by the triangle
inequality we have $\left|\lambda(T_{k})\right|\leq
4\left|\lambda(T_{k+1})\right|,$ and by induction it follows that

$\left|\lambda(T)\right|\leq
4^{k}\left|\lambda(T_{k})\right|=4^{k}\left|T_{k}\right|\cdot\frac{\left|\lambda(T_{k})\right|}{\left|T_{k}\right|}=\left|T\right|\cdot\frac{\left|\lambda(T_{k})\right|}{\left|T_{k}\right|},$
(5.7)

where in the last step we have used the fact that $T_{k+1}$ has one-fourth the
area of $T_{k}$, so
$\left|T_{k}\right|=4^{-k}\left|T_{0}\right|=4^{-k}\left|T\right|$. Since the
diameters of the $T_{k}$ go to zero, by compactness, there is a unique point
$w$ in the intersection $\bigcap_{k=0}^{\infty}T_{k}$. Since the family
$\\{T_{k}\\}$ is a subfamily of all the triangles containing $w$, and each
$T_{k}$ is similar to $T_{0}$, therefore by letting $k\to\infty$ in (5.7) and
using (5.6) the result follows. ∎

###### Proof of Proposition 5.3.

For a triangle $\Delta\subset\Omega$, define

$\lambda(\Delta)=\int_{\partial\Delta}f(z)g(z)dz.$

To prove the result, we need to show that $\lambda\equiv 0$. Since condition
(5.5) of Lemma 5.4 is obvious, we need to show (5.6) to complete the proof.
Let $w\in\Omega$, let $K$ be a compact neighborhood of $w$ in $\Omega$, and
denote by $M$ the Lipschitz constant corresponding to this $w$ and this $K$ in
(5.4). Now, let $\Delta_{0}$ be a triangle and let $\Delta$ be a triangle
similar to $\Delta_{0}$ such that $w\in\Delta\subset K$. Then observe that, by
the hypothesis of Morera-holomorphicity of $f$ and $g$ we have

$\lambda(\Delta)=\int_{\partial\Delta}\left(f(z)-f(w)\right)\left(g(z)-g(w)\right)dz.$

Therefore, denoting the perimeter of the triangle $\Delta$ by
$\left|\partial\Delta\right|$,

$\displaystyle\left|\lambda(\Delta)\right|$
$\displaystyle\leq\sup_{z\in\partial\Delta}\left|\left(f(z)-f(w)\right)\left(g(z)-g(w)\right)\right|\left|\partial\Delta\right|$
$\displaystyle\leq
M\cdot\left|\partial\Delta\right|\cdot\sup_{z\in\partial\Delta}\left(\left|f(z)-f(w)\right|\left|z-w\right|\right)$
$\displaystyle\leq
M\cdot\left|\partial\Delta\right|\cdot\mathrm{diam}(\Delta)\cdot\sup_{z\in\partial\Delta}\left|f(z)-f(w)\right|$
$\displaystyle=M\cdot\left|\Delta\right|\cdot\left(\frac{\left|\partial\Delta\right|\cdot\mathrm{diam}(\Delta)}{\left|\Delta\right|}\right)\cdot\sup_{z\in\partial\Delta}\left|f(z)-f(w)\right|$
$\displaystyle=M\cdot\left|\Delta\right|\cdot\left(\frac{\left|\partial\Delta_{0}\right|\cdot\mathrm{diam}(\Delta_{0})}{\left|\Delta_{0}\right|}\right)\cdot\sup_{z\in\partial\Delta}\left|f(z)-f(w)\right|,$

where in the last step we use the fact that $\Delta$ and $\Delta_{0}$ are
similar, so the quantity in parentheses (which is clearly invariant under
dilations) is the same. So we have

$\frac{\left|\lambda(\Delta)\right|}{\left|\Delta\right|}\leq
C\cdot\sup_{z\in\partial\Delta}\left|f(z)-f(w)\right|,$

for a constant $C$ independent of the triangle $\Delta$ as long as $\Delta$ is
similar to $\Delta_{0}$ and $w\in\Delta\subset K$. Letting $\Delta$ shrink to
$w$, we have (5.6) and the proof is complete. ∎

###  5.4. Fourier expansion of a Morera-holomorphic function

We will now prove the following analog of Theorem 3.1. In particular, it shows
that holomorphic functions in the sense of Morera are identical to the
holomorphic distributions considered in Section 3.

######  Theorem 5.1.

Let $f\in\mathscr{O}_{M}(\mathbb{D})$ where
$\mathbb{D}=\\{\left|z\right|<1\\}$ is the open unit disc. Then there is a
sequence $\\{a_{n}\\}_{n=0}^{\infty}$ of complex numbers, such that

$f=\sum_{n=0}^{\infty}a_{n}e_{n},$ (5.8)

where $e_{n}$ is as in (3.9), and the series on the right converges absolutely
in $\mathcal{C}(\mathbb{D})$ to the function $f$.

Let $\sigma$ be the natural representation of $\mathbb{T}$ on
$\mathcal{C}(\mathbb{D})$ given by

$\sigma_{\lambda}(f)(z)=f(\lambda
z),\quad\lambda\in\mathbb{T},z\in\mathbb{D}.$ (5.9)

######  Proposition 5.5.

The space $\mathscr{O}_{M}(\mathbb{D})$ is invariant under $\sigma$ and the
resulting representation of $\mathbb{T}$ on $\mathscr{O}_{M}(\mathbb{D})$ is
continuous.

###### Proof.

Let $\lambda\in\mathbb{T}$, let $T$ be a triangle in $\mathbb{D}$. Notice that

$\lambda^{-1}T=\\{\lambda^{-1}z:z\in T\\}$

is itself a triangle, and we have

$\int_{\partial T}(\sigma_{\lambda}f)(z)dz=\int_{\partial T}f(\lambda
z)dz=\lambda^{-1}\int_{\partial(\lambda^{-1}T)}f(w)dw=0.$

It follows that $\mathscr{O}_{M}(\mathbb{D})$ is invariant under $\sigma$.

It suffices to show that the representation $\sigma$ is continuous on
$\mathcal{C}(\mathbb{D})$. For $0<r<1$, let $p_{r}$ be the seminorm on
$\mathcal{C}(\mathbb{D})$ given by

$p_{r}(f)=\sup_{\left|z\right|\leq r}\left|f(z)\right|.$ (5.10)

It is clear that $p_{r}(\sigma_{\lambda}(f))=p_{r}(f)$ for each
$0<r<1,\lambda\in\mathbb{T}$ and $f\in\mathcal{C}(\mathbb{D})$. Also $f_{j}\to
f$ in $\mathcal{C}(\mathbb{D})$ if and only if for each $r$, we have
$p_{r}(f-f_{j})\to 0$, so the family $\\{p_{r}\\}$ is a $\sigma$-invariant
family of seminorms that generates the topology of $\mathcal{C}(\mathbb{D})$.
Further, given $f\in\mathcal{C}(\mathbb{D})$, by uniform continuity, for each
$0<r<1$, we have

$p_{r}(\sigma_{\lambda}(f)-f)=\sup_{\left|z\right|\leq r}\left|f(\lambda
z)-f(z)\right|\to 0\quad\text{ as }\lambda\to 1,$

so that $\lim_{\lambda\to 1}\sigma_{\lambda}(f)=f$ in the space
$\mathcal{C}(\mathbb{D})$. Therefore both conditions of Proposition 2.3 are
satisfied, and the representation $\sigma$ is continuous. ∎

In view of the above, the machinery of Section 2 applies. We now compute the
Fourier components (2.5).

######  Proposition 5.6.

For $f\in\mathscr{O}_{M}(\mathbb{D})$, and with $\sigma$ the natural
representation (5.9), the Fourier components of $f$ are of the form:

$\bm{\pi}^{\sigma}_{n}(f)=\begin{cases}a_{n}e_{n}&\text{ if }n\geq 0\\\
0&\text{ if }n<0,\end{cases}$

where $a_{n}\in{\mathbb{C}}$ and $e_{n}(z)=z^{n}$.

The proof will use the following lemma:

######  Lemma 5.7.

A radial function in $\mathscr{O}_{M}(\mathbb{D}\setminus\\{0\\})$ is
constant.

###### Proof.

Let $f\in\mathscr{O}_{M}(\mathbb{D}\setminus\\{0\\})$ be radial, and define
the complex valued continuous function $u$ on the interval $(0,1)$ by
restriction, $u(r)=f(r)$, so that we have $f(z)=u(\left|z\right|)$ by the
radiality of $f$. To prove the theorem, it suffices to show that $u$ is a
constant .

Fix $0\leq\alpha<\beta<\pi$ and $0<\rho<1$. For $R$ in the interval $(\rho,1)$
consider the curvilinear quadrilateral defined by

$S(R)=\\{re^{i\theta}:\rho\leq r\leq R,\alpha\leq\theta\leq\beta\\},$ (5.11)

and notice that $S(R)$ lies in the upper half disc, which is convex. The
region $S(R)$ is bounded by the two circular arcs

$AB=\\{\rho e^{i\theta}:\alpha\leq\theta\leq\beta\\},\quad
CD=\\{Re^{i\theta}:\alpha\leq\theta\leq\beta\\},$

along with the two radial line segments

$AD=\\{re^{i\alpha}:\rho\leq r\leq R\\},\quad BC=\\{re^{i\beta}:\rho\leq r\leq
R\\}.$ $A$$D$$C$$B$$S(R)$

Orient the boundary $\partial S(R)$ counterclockwise. We can write

$\int_{\partial S(R)}f(z)dz=\int_{AD}+\int_{DC}+\int_{CB}+\int_{BA}=0,$

where the vanishing of the integral follows from part (c) of Proposition 5.1
above. Parametrizing $AD$ by $z=re^{i\alpha}$ where $\rho\leq r\leq R$ , and
using $dz=e^{i\alpha}dr$ we get

$\int_{AD}f(z)dz=\int_{\rho}^{R}f(re^{i\alpha})e^{i\alpha}dr=e^{i\alpha}\int_{\rho}^{R}u(r)dr.$

Similarly,

$\int_{CB}f(z)dz=-e^{i\beta}\int_{\rho}^{R}u(r)dr.$

Now, parametrizing $DC$ by $z=Re^{i\theta},\alpha\leq\theta\leq\beta$, we have
$dz=Rie^{i\theta}d\theta$, so

$\int_{DC}f(z)dz=\int_{\theta=\alpha}^{\beta}f(Re^{i\theta})\cdot
Rie^{i\theta}d\theta=Ru({R})\int_{\alpha}^{\beta}ie^{i\theta}d\theta=Ru(R)(e^{i\beta}-e^{i\alpha}).$

Similarly,

$\int_{BA}f(z)dz=\rho u(\rho)(e^{i\beta}-e^{i\alpha}).$

Therefore, adding the four integrals we have

$\int_{\partial
S}f(z)dz=\left({\rho}u({\rho})-Ru(R)+\int_{\rho}^{R}u(r)dr\right)(e^{i\alpha}-e^{i\beta})=0,$

so that we have the relation

$\int_{\rho}^{R}u(r)dr=Ru(R)-{\rho}u({\rho}),$ (5.12)

which can be written as

$u(R)=\frac{1}{R}\left(\int_{\rho}^{R}u(r)dr+{\rho}u({\rho})\right).$

Since $u$ is continuous, the right hand side is a function of $R$ which is
continuously differentiable on $(\rho,1)$. Thus $u\in\mathcal{C}^{1}(\rho,1)$.
Differentiating both sides of (5.12) with respect to $R$ we have
$u(R)=Ru^{\prime}(R)+u(R)$ so that we have $u^{\prime}(R)=0$ for
$R\in(\rho,1)$. But $0<\rho<1$ can be chosen arbitrarily, so it follows that
$u^{\prime}\equiv 0$ on $(0,1)$ and the result follows. ∎

###### Proof of Proposition 5.6.

Let $n\in\mathbb{Z}$ and for simplicity of notation, let
$f_{n}=\bm{\pi}^{\sigma}_{n}(f)$. Then by Proposition 2.4, we see that $f_{n}$
lies in the Fourier mode $[\mathscr{O}_{M}(\mathbb{D})]_{n}^{\sigma}$ so that
$f_{n}\in\mathscr{O}_{M}(\mathbb{D})$ and for $\lambda\in T,z\in\mathbb{D}$ we
have $f_{n}(\lambda z)=\lambda^{n}f_{n}(z)$. Since clearly
$f_{n}|_{\mathbb{D}\setminus\\{0\\}}\in\mathscr{O}_{M}(\mathbb{D}\setminus\\{0\\})$,
it follows by Proposition 5.3 that the function $h_{n}$ on
$\mathbb{D}\setminus\\{0\\}$ defined by

$h_{n}=e_{-n}\cdot f_{n}$

lies in $\mathscr{O}_{M}(\mathbb{D}\setminus\\{0\\})$. Further for $\lambda\in
T,z\in\mathbb{D}$ we have

$h_{n}(\lambda z)=e_{n}(\lambda z)f_{n}(\lambda
z)=\lambda^{-n}e_{-n}(z)\cdot\lambda^{n}f_{n}(z)=h_{n}(z),$

so $h_{n}$ is radial, and hence by Lemma 5.7, $h_{n}$ is a constant, which we
call $a_{n}$. Therefore, on $\mathbb{D}\setminus\\{0\\}$, we have
$f_{n}=a_{n}e_{n}$, so the product $a_{n}e_{n}$ extends for each $n$ to a
continuous function on $\mathbb{D}$. If $n<0$, this is possible only if
$a_{n}=0$, and this completes the proof. ∎

###  5.5. Conclusion of the proof of Theorem 5.1

We will first show that the series $\sum_{n=0}^{\infty}a_{n}e_{n}$ converges
absolutely in the space $\mathcal{C}(\mathbb{D})$. It suffices to show that
for each $0<r<1$, we have

$\sum_{n=0}^{\infty}p_{r}(a_{n}e_{n})<\infty,$

where $p_{r}$ is as in (5.10). Fix an $r$ with $0<r<1$ and let $\rho$ be such
that $r<\rho<1$. Applying the Cauchy estimate (2.9) to the seminorm $p_{\rho}$
we see that for each $n\in\mathbb{Z}$:

$p_{\rho}(\bm{\pi}^{\sigma}_{n}f)\leq p_{\rho}(f)$

Thanks to Proposition 5.6 we have for each $n\geq 0$,

$p_{r}(\bm{\pi}^{\sigma}_{n}f)=\sup_{\left|z\right|\leq
r}\left|a_{n}z^{n}\right|=\left|a_{n}\right|r^{n}\leq\left(\frac{r}{\rho}\right)^{n}\cdot\left|a_{n}\right|\rho^{n}\leq\left(\frac{r}{\rho}\right)^{n}\cdot
p_{\rho}(f).$

Therefore we have

$\sum_{n=-\infty}^{\infty}{p_{r}(\bm{\pi}^{\sigma}_{n}f)}=\sum_{n=0}^{\infty}{p_{r}(\bm{\pi}^{\sigma}_{n}f)}\leq\left(\sum_{n=0}^{\infty}\left(\frac{r}{\rho}\right)^{n}\right)\cdot
p_{\rho}(f)=\frac{1}{1-\frac{r}{\rho}}\cdot p_{\rho}(f)<\infty.$

This proves that the series $\sum_{n=0}^{\infty}a_{n}e_{n}$ converges
absolutely in $\mathcal{C}(\mathbb{D})$. Let $g$ be its sum. Since the partial
sums are all in $\mathscr{O}_{M}(\mathbb{D})$ by Proposition 5.2, we see that
$g\in\mathscr{O}_{M}(\mathbb{D})$. However, it follows from Corollary 2.5 that
the sum of the series is $f$, therefore $f=g$, and the proof of the theorem is
complete.

###  5.6. Pompeiu’s characterization of holomorphic functions

Goursat’s definition is non-quantitative, since it is framed in terms of the
_existence_ of the limit (5.1), and does not provide a way to measure the
degree of non-holomorphicity of a function. For example, if $\epsilon$ is
small and nonzero, all it says about the functions
$f(z)=z+\epsilon\overline{z}$ and $g(z)=\overline{z}+\epsilon{z}$ is that both
are non-holomorphic. Morera’s definition of a holomorphic function does not
have this shortcoming, as the integral $\int_{\partial T}f(z)dz$ gives a
measure of the amount of non-holomorphicity of $f$ on the triangle $T$.

It is possible to normalize and localize this measure of non-holomorphicity,
as was realized by Pompeiu (see [Pom12, MŞ98]). The quantity

$\frac{1}{\left|T\right|}\int_{\partial T}f(z)dz$ (5.13)

is a numerical measure of the “average density of non-holomorphicity” of a
continuous function $f$ on a triangle (or other region with piecewise smooth
boundary) $T$, where $\left|T\right|$ denotes the area of $T$. To localize
this, we can consider, for a point $w\in\Omega$, and a continuous function
$f:\Omega\to{\mathbb{C}}$, the following limit (called the “areolar
derivative” by Pompieu) as a measure of the degree of non-holomorphicity of
$f$ at the point $w$:

$\lim_{T\downarrow w}\frac{1}{\left|T\right|}\int_{\partial T}f(z)dz$ (5.14)

where the limit is taken over the family of triangles containing the point $w$
and contained in $\Omega$, as these triangles shrink to the point $w$. Let us
say that a function is _Pompeiu-holomorphic_ if the limit (5.14) exists at
each $w\in\Omega$, and is equal to zero. The following simple and well-known
observations clarify the meaning of this notion:

######  Proposition 5.8.

  1. (a)

A continuous function $f:\Omega\to{\mathbb{C}}$ is Pompieu-holomorphic if and
only if it is Morera-holomorphic.

  2. (b)

If the function $f$ is in $\mathcal{C}^{1}(\Omega)$, then the limit (5.14)
exists, and is equal to $\displaystyle{2i\frac{\partial
f}{\partial\overline{z}}(w)}$.

###### Proof.

For part (a), if $f$ is Morera-holomorphic, then the quantity (5.13) vanishes
for each triangle $T$, so the limit (5.14) vanishes. Conversely, suppose that
the limit (5.14) vanishes at each point $w\in\Omega$, and let $T$ be a
triangle in $\Omega$. In Lemma 5.4, if we take
$\lambda(\Delta)=\int_{\partial\Delta}f(z)dz$, then the additivity condition
(5.5) is clear, and the limit condition (5.6) holds by hypothesis. The result
follows.

For part (b), using Stoke’s theorem

$\displaystyle\frac{1}{\left|T\right|}\int_{\partial T}f(z)dz$
$\displaystyle=\frac{1}{\left|T\right|}\int_{\partial
T}f(z)dx+if(z)dy=\frac{1}{\left|T\right|}\int_{T}\left(i\frac{\partial
f}{\partial x}(z)-\frac{\partial f}{\partial y}(z)\right)dx\wedge dy$
$\displaystyle=2i\int_{T}\frac{\partial f}{\partial\overline{z}}dx\wedge dy.$

Since the last integral has a continuous integrand, we may take the limit as
the triangle $T$ shrinks to the point $w$ to obtain:

$\lim_{T\downarrow w}\frac{1}{\left|T\right|}\int_{\partial
T}f(z)dz=2i\frac{\partial f}{\partial\overline{z}}(w).$

∎

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