#  n-th Root Optimal Rational Approximants
to Functions with Polar Singular Set

Laurent Baratchart  INRIA, Project FACTAS, 2004 route des Lucioles — BP 93,
06902 Sophia-Antipolis, France<EMAIL_ADDRESS>,  Herbert
Stahl  and  Maxim Yattselev  Department of Mathematical Sciences, Indiana
University Indianapolis, 402 North Blackford Street, Indianapolis, IN 46202
<EMAIL_ADDRESS>
###### Abstract.

Let $D$ be a bounded Jordan domain and $A$ be its complement on the Riemann
sphere. We investigate the $n$-th root asymptotic behavior in $D$ of best
rational approximants, in the uniform norm on $A$, to functions holomorphic on
$A$ having a multi-valued continuation to quasi every point of $D$ with
finitely many branches. More precisely, we study weak∗ convergence of the
normalized counting measures of the poles of such approximants as well as
their convergence in capacity. We place best rational approximants into a
larger class of $n$-th root optimal meromorphic approximants, whose behavior
we investigate using potential-theory on certain compact bordered Riemann
surfaces.

###### Key words and phrases:

Rational approximation, meromorphic approximation, AAK approximation, Nehari
approximation, weak distribution of poles, convergence in capacity

###### 2020 Mathematics Subject Classification:

30C15, 30E10, 31C40, 41A20, 41A25

The research of the last author was supported in part by a grant from the
Simons Foundation, CGM-706591.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries and Main Results
    1. 2.1 Class of Approximated Functions
    2. 2.2 Optimal Rational Approximants
    3. 2.3 Optimal Meromorphic Approximants
  3. 3 Proof of Theorem 2.5
    1. 3.1 Existence of Best Meromorphic Approximants
    2. 3.2 Reduction to the Unit Disk
    3. 3.3 Nehari Modifications
    4. 3.4 Notation
    5. 3.5 Stripped Error of Approximation
    6. 3.6 Asymptotic Distributions of Poles and Zeros
    7. 3.7 Logarithmic Error Function
    8. 3.8 Modified Logarithmic Error Function
    9. 3.9 Projected Logarithmic Error Function
    10. 3.10 Computation of the Logarithmic Error Function
    11. 3.11 Convergence in Capacity
  4. 4 Proof of Theorem 2.6
  5. A Potential Theory on a Riemann Surface
    1. A.1 Subharmonic Functions
    2. A.2 Green Functions
    3. A.3 Green Potentials
    4. A.4 Capacities
    5. A.5 Fine Topology
    6. A.6 Thinness
    7. A.7 Regularity
    8. A.8 Superlevel Sets of Green Potentials
    9. A.9 Balayage
    10. A.10 Green Potentials in $D$ and on ${\mathcal{R}}$
    11. A.11 Dirichlet problem

## List of Symbols

General Point Sets:
$\mathbb{T},\mathbb{D},\mathbb{C},\overline{\mathbb{C}}$ unit circle, open
unit disk, complex plane, and extended complex plane
$\mathbb{T}_{r},\mathbb{D}_{r}$ circle and open disk of radius $r$ centered at
the origin $T,D,A$ Jordan curve, its interior domain, and the closure of its
exterior domain in $\overline{\mathbb{C}}$

Riemann Surfaces:
${\mathcal{R}}_{*},p$ compact Riemann surface with natural projection
$p:{\mathcal{R}}_{*}\to\overline{\mathbb{C}}$ ${\mathcal{R}}$
${\mathcal{R}}:=\\{z\in{\mathcal{R}}_{*}:p(z)\in D\\}$ $\mathcal{T}$ a
connected component of $p^{-1}(T)$ homeomorphic to $T$ $\mathbf{rp}(\cdot)$
the set of ramification points of a given Riemann surface $m(z)$ ramification
order of a point $z$ on a Riemann surface $M$ total number of sheets of
${\mathcal{R}}$

Classes of Functions:
$C(E)$ continuous functions on a set $E$ $\mathcal{H}(A)$ functions analytic
in some neighborhood of $A$ $\mathcal{S}(A)$ subclass of $\mathcal{H}(A)$ of
functions multi-valued and quasi everywhere analytic off $A$ $\mathcal{E}(A)$
subclass of $\mathcal{H}(A)$ of functions single-valued and quasi everywhere
analytic off $A$ $\mathcal{F}({\mathcal{R}})$ class of functions quasi
everywhere analytic on $\overline{\mathcal{R}}$ $\mathcal{F}(A)$ subclass of
approximated functions analytic on $A$ $\mathcal{P}_{n}$ algebraic polynomials
of degree at most $n$ $\mathcal{M}_{n}(D)$ monic algebraic polynomials of
degree $n$ with all their zeros in $D$ ${\mathcal{R}}_{n}(D)$
$\mathcal{P}_{n}\mathcal{M}_{n}^{-1}(D)$ $H^{\infty}(D)$ space of bounded
holomorphic functions in $D$ $H_{n}^{\infty}(D)$
$H^{\infty}(D)\mathcal{M}_{n}^{-1}(D)$ $L^{2}(\mathbb{T})$ square integrable
functions on the unit circle $L^{\infty}(\mathbb{T})$ essentially bounded
functions on the unit circle $H^{2}$ Hardy space of functions in
$L^{2}(\mathbb{T})$ with vanishing Fourier coefficients of negative index
$H^{2}_{-}$ $L^{2}(\mathbb{T})\ominus H^{2}$

Operators:
$\mathbb{P}_{+},\mathbb{P}_{-}$ orthogonal projections from
$L^{2}(\mathbb{T})$ onto $H^{2}$, $H^{2}_{-}$ $\Gamma_{f}$ Hankel operator
from $H^{2}$ to $H^{2}_{-}$, $h\mapsto\mathbb{P}_{-}(fh)$ $s_{n}(\Gamma_{f})$
the $n$-th singular number of $\Gamma_{f}$

Potential Theory:
$\mathrm{cap}(K)$ logarithmic capacity of $K$ $\mathrm{cap}_{\Omega}(K)$
Greenian capacity of $K$ relative to $\Omega$ $\mathrm{cap}(E,K)$ capacity of
the condenser $(E,K)$ $g_{\Omega}(\cdot,w)$ Green function for $\Omega$ with
pole at $w$ $g(\sigma,\Omega;z)$ Green potential of the measure $\sigma$
relative to $\Omega$ $V^{\sigma}(z)$ logarithmic potential of the measure
$\sigma$ $\mu_{\Omega,K}$ Green equilibrium distribution on a set
$K\subset\Omega$ relative to $\Omega$ $\mathcal{B}_{v}^{E}$ balayage function
of a superharmonic function $v$ relative to a set $E$ $\sigma^{E}$ balayage
measure of a measure $\sigma$ onto a set $E$ $\widehat{\sigma}$ lift of a
measure $\sigma$ on $D$ to ${\mathcal{R}}$ $p_{*}(\sigma)$ projection
(pushforward) of a measure $\sigma$ on ${\mathcal{R}}$ to $D$
$\partial_{\mathsf{f}}A,\mathrm{clos}_{\mathsf{f}}(E)$ fine boundary and
closure of a set $E$ $b(E),i(E)$ base and the set of finely isolated points of
a set $E$

Various Symbols:
$\phi$ a conformal map from $\mathbb{D}$ onto $D$ $\mathcal{K}_{f}$ collection
of “branch cuts” for $f$ $K_{f}$ “branch cut” of minimal Greenian capacity for
$f\in\mathcal{S}(A)$ $\|\cdot\|_{K}$ essential supremum norm on $K$
$\rho_{n}(f,A)$ error of best rational approximation of $f$ analytic on $A$ by
functions in ${\mathcal{R}}_{n}(D)$

##  1\. Introduction

Rational approximation to holomorphic functions of one complex variable has
long been a requisite chapter of classical analysis with notable applications
to number theory [30, 56, 33], spectral theory [44, 26, 48] and numerical
analysis [25, 32, 24, 17]. Over the last decades it became a cornerstone of
modeling in Engineering [65, 66, 51, 1, 27], and it can also be viewed today
as a technique to regularize inverse potential problems [22, 31, 4]. Finding
best rational approximants of prescribed degree to a specific function, say in
the uniform norm on a given set, seems out of reach except in rare, particular
cases. Indeed, such approximants depend in a rather convoluted manner, both on
the approximated function and on the set where approximation should take
place. Accordingly, the constructive side of the theory has focused on
estimating optimal convergence rates as the degree grows large and devising
approximation schemes coming close to meet them [67, 7, 23, 62], or else
studying the behavior of natural, computationally appealing candidate
approximants like Padé interpolants and their variants [3, 61, 45, 38, 42].

When a function is holomorphic in some neighborhood of a continuum $A$, the
optimal speed of convergence for rational approximants on $A$ is at least
geometric in the degree. Then, a coarse but manageable estimate of this speed
proceeds via asymptotics of the $n$-th root of the error in approximation by
rational functions of degree $n$. For functions analytically continuable off
$A$ except over a polar set containing branchpoints (throughout _polar_ means
of logarithmic capacity zero), and provided that $A$ does not divide the
extended complex plane, Gonchar and Rakhmanov constructed, using multipoint
Padé interpolants and dwelling on work by the second author, a sequence of
rational approximants whose $n$-th root error on $A$ is asymptotically the
smallest possible. They further showed that these interpolants converge in
capacity on the complement of a compact set $K$ minimizing the capacity of the
condenser $(A,K)$ under the constraint that the function is analytic off $K$,
and proved that the normalized counting measures of their poles converge to
the condenser equilibrium distribution on $K$ [23]. It is remarkable that $K$
solves a geometric extremal problem from logarithmic potential theory, close
in spirit to the Lavrentiev type [34], and that it depends merely on the set
where approximation takes place, on the branchpoints of the approximated
function and its monodromy, but nothing else. Such a structure emerges because
only the $n$-th root of the error is considered, rather than the error itself.
Since then, it has been an open issue whether any $n$-th root optimal sequence
of approximants – in particular a sequence of best approximants – has the same
behavior. The present paper answers this question in the positive, at least
when the branchpoints are finite in number and order. In particular, our
results connect, apparently for the first time, the singularities of best
uniform approximants to those of the approximated function. We also address
the case of no branchpoints, when the approximated function is analytic except
over a polar set, in which the speed of best rational approximation on $A$ is
known to be faster than geometric with the degree. We prove that $n$-th root
optimal approximants converge in capacity outside the singular set of the
function, and that the “effective” poles converge in a sense to that set.

The gist of the paper becomes transparent upon observing that the behavior of
rational approximants can be surmised when the function to be approximated
extends analytically to a multiply-sheeted Riemann surface over the complex
plane. As rational functions are single-valued, this topological discrepancy
leads the approximants to mark out their domain of approximation by
accumulating poles so as to form a cut, thereby preventing single-valued
continuation in the limit. In the case of (diagonal) multipoint Padé
interpolants, this cut has been characterized as being of smallest weighted
capacity in a field that depends on the limiting distribution of the
interpolation points, and poles asymptotically distribute according to the
weighted equilibrium measure of that cut; moreover, the Padé interpolants
converge in capacity on the extremal domain thus demarcated. This was
established in [23], dwelling on the works [57, 58, 60, 61, 62] that deal with
classical Padé interpolants and correspond to the zero field and unit weight;
see also [46] for early developments along these lines, and [6] for
applications to $L^{2}$-best rational approximation on the circle.
Subsequently, by choosing interpolation points adequately and performing
surgery to eliminate spurious poles, the authors of [23] construct, on any
continuum $A$ not dividing the extended plane and contained in the analyticity
domain of a function indefinitely continuable except over a closed polar set
containing branchpoints, a sequence of rational approximants converging
uniformly to that function on $A$ as their degree $n$ grows large and whose
$n$-th root error has a liminf which is smallest possible, as well as a true
limit. For this weakly optimal choice of interpolation points (meaning that
the choice is optimal in the $n$-th root sense), the cut $K$ of minimum
weighted capacity is also the cut minimizing the condenser capacity of
$(A,K)$, as well as the cut of minimum Greenian capacity in the complement of
$A$. The smallest value for the limit of the $n$-th root error is a simple,
explicit function of this Greenian capacity, and the poles of the approximants
thus constructed distribute asymptotically according to the Green equilibrium
measure of that cut.

Now, assuming in addition that the branchpoints of the continuation off $A$ of
the function to be approximated are finite in number and of algebraic type, we
shall prove that _any_ sequence of rational (or meromorphic) approximants of
increasing degree $n$ whose $n$-th root error on $A$ converges to the smallest
possible limit – a fortiori every sequence of best approximants – has the same
asymptotic distribution of poles as the particular sequence constructed in
[23]. More precisely, if a function analytic in a simply connected
neighborhood of a continuum $A$ in the extended complex plane is indefinitely
continuable off that neighborhood except over a closed polar set containing
finitely many branchpoints, all of algebraic type, then _the normalized
counting measures of the poles of any sequence of rational approximants of
increasing degree $n$ with asymptotically optimal $n$-th root error on $A$ do
converge weak-star, as $n$ grows large, to the Green equilibrium distribution
of the compact set of minimum Greenian capacity outside of which the function
is single-valued; moreover, convergence holds in capacity everywhere off that
compact set._ Here, Green functions are understood with respect to the
complement of the continuum $A$ where approximation takes place. Finally, if
there are no branchpoints, that is, if the approximated function is single-
valued and analytic on the extended complex plane except possibly over a
closed polar set $E$, then there are rational approximants converging on $A$
faster than geometrically with the degree. We shall prove that such sequences
of approximants (as well as their meromorphic analogs) converge in capacity on
the extended plane deprived from $E$, and that retaining the singular part
that comes close to $E$ generates new sequences of approximants, still
converging faster than geometrically with the degree, while satisfying in
addition that any weak-star limit point of the normalized counting measures of
their poles is supported on $E$.

Previously cited references, which deal with Padé or multipoint Padé
interpolants, exploit the connection between denominators thereof and non-
Hermitian orthogonal polynomials on a system of arcs encompassing the singular
set of the interpolated function. Indeed, the core of the work in [60, 23] is
to derive asymptotics of such polynomials on extremal systems of arcs like
those constructed in [57, 58], so as to qualify the behavior of the poles of
the interpolants when the degree grows large and deduce from it the desired
convergence properties. Here, we proceed in the opposite direction: we assume
that the optimal rate is met in the $n$-th root sense and deduce from it the
behavior of the poles. For this we cannot make use of orthogonal polynomials,
and in fact it is not even known if interpolation takes place at all in the
case of best approximants. However, the construction from [57, 58] will still
be basic to our purposes.

_This work was initiated jointly by the three authors in 2009, but the
untimely passing away of the second one on April 22nd, 2013 prevented him from
seeing its completion. Still, some fundamental ideas are his._

##  2\. Preliminaries and Main Results

Given a function $f$ holomorphic in a neighborhood of a closed set
$A\subset\overline{\mathbb{C}}$, the error of approximation of $f$ on $A$ by
rational functions of degree $n$ is

(2.1) $\rho_{n}(f,A):=\inf_{r\in{\mathcal{R}}_{n}(\mathbb{C}\setminus
A)}\|f-r\|_{A},$

where $\|\cdot\|_{A}$ stands for the supremum norm on $A$ and, for
$\Omega\subset\overline{\mathbb{C}}$, we let ${\mathcal{R}}_{n}(\Omega)$ be
the class of rational functions of type $(n,n)$ with all their poles in
$\Omega$. That is, if $\mathcal{P}_{n}$ denotes the space of algebraic
polynomials of degree at most $n$ and $\mathcal{M}_{n}(\Omega)$ the monic
polynomials of degree $n$ with all zeros in $\Omega$, then
${\mathcal{R}}_{n}(\Omega):=\mathcal{P}_{n}\mathcal{M}_{n}^{-1}(\Omega)$. It
was shown by Walsh [67, 2], using interpolation techniques, that

(2.2)
$\limsup_{n\to\infty}\rho_{n}^{1/n}(f,A)\leq\exp\big{\\{}-1/\mathrm{cap}(A,K)\big{\\}},$

where $K$ is any closed set disjoint from $A$ in the complement of which $f$
is holomorphic and $\mathrm{cap}(A,K)$ denotes the capacity of the condenser
$(A,K)$. A definition of condenser capacity can be found in [55, Chapter VIII,
Section 3]; for our purposes, it is enough to know that if
$\overline{\mathbb{C}}\setminus A$ is connected, then $\mathrm{cap}(A,K)$
coincides with the Greenian capacity
$\mathrm{cap}_{\overline{\mathbb{C}}\setminus A}(K)$ defined in Section A.4,
see [55, Chapter VIII, Theorem 2.6 & Corollary 2.7] for this equivalence.

It is known that Walsh’s inequality (2.2) cannot be improved [37]. On the
other hand, it was conjectured by Gonchar [28] and proven by Parfënov when $A$
is a continuum with connected complement [47] (also later by Prokhorov for any
compact set $A$ [52]) that

(2.3)
$\liminf_{n\to\infty}\rho_{n}^{1/n}(f,A)\leq\exp\big{\\{}-2/\mathrm{cap}(A,K)\big{\\}}.$

Hence, $\rho_{n}(f,A)$ has no limit in general as $n\to\infty$, and when the
limit exists it cannot exceed the right-hand side of (2.3). For certain
classes of functions and certain loci of approximation $A$, it was
nevertheless shown that $\rho_{n}(f,A)$ does have a limit, which is equal to
the right-hand side of (2.3). More precisely, let $\mathcal{H}(A)$ denote the
space of functions holomorphic on a (variable) neighborhood of $A$, and
$\mathcal{S}(A)\subset\mathcal{H}(A)$ those functions continuable analytically
into the complement of $A$ along any path that avoids some compact polar111see
Section A.5 for a definition and basic properties of polar sets, that may be
defined as sets of zero logarithmic capacity. subset of
$\overline{\mathbb{C}}\setminus A$ (which may depend on the function); we
require in addition that this continuation is not single-valued, namely that
there are paths with the same endpoints leading to different analytic
branches. Now, when $A$ is a continuum that does not separate the plane and
$f\in\mathcal{S}(A)$, it follows from work by the second author in [57, 58,
59] and it was explicitly stated by Gonchar and Rakhmanov in [23, Theorem 1′]
that

(2.4)
$\lim_{n\to\infty}\rho_{n}^{1/n}(f,A)=\inf_{K}\exp\big{\\{}-2/\mathrm{cap}(A,K)\big{\\}},$

where the infimum is taken over all compact sets $K$ such that $f$ admits a
single-valued analytic continuation to $\overline{\mathbb{C}}\setminus K$.

Hereafter, we let $T\subset\mathbb{C}$ be a Jordan curve with interior domain
$D$, and we put $A:=\overline{\mathbb{C}}\setminus D$. In this setting
$\infty\in A$, which is no loss of generality for a preliminary Möbius
transform can ensure this; in contrast, our requirement that $D$ be a Jordan
domain is a regularity assumption on the set $A$ where approximation takes
place. Given $f\in\mathcal{H}(A)$, let $\mathcal{K}_{f}$ be the collection of
all compact sets $K\subset D$ such that $f$, initially defined on $A$, admits
a single-valued analytic continuation to $\mathbb{C}\setminus K$. It follows
from [57, 58] that there exists $K_{f}\in\mathcal{K}_{f}$, unique up to
addition and/or removal of a polar set, with

(2.5) $\mathrm{cap}_{D}(K_{f})\leq\mathrm{cap}_{D}(K),\quad
K\in\mathcal{K}_{f}.$

We can and will normalize $K_{f}$ to be the smallest possible, i.e., we make
it the intersection of all $K\in\mathcal{K}_{f}$ for which
$\mathrm{cap}_{D}(K)$ is minimal, see [58]. As
$\mathrm{cap}(K,A)=\mathrm{cap}_{D}(K)$, in light of equation (2.4), our main
goal is to investigate the asymptotic behavior of sequences $\\{r_{n}\\}$ of
rational functions of type $(n,n)$ meeting this optimal $n$-th root rate:

(2.6)
$\lim_{n\to\infty}\|f-r_{n}\|_{A}^{1/n}=\exp\big{\\{}-2/\mathrm{cap}_{D}(K_{f})\big{\\}}.$

We call any such sequence $\\{r_{n}\\}$ a sequence of _$n$ -th root optimal
rational approximants to $f$ on $A$_. In order to study $\\{r_{n}\\}$, we are
led to consider more general sequences of _meromorphic_ approximants of the
form $r_{n}+h_{n}$, where $h_{n}$ is holomorphic in $D$ and continuous on
$\overline{D}$, see Section 2.3 for the definitions. Even though best
meromorphic approximants may look less natural than rational ones, they make
contact with both the spectral theory of Hankel operators (through AAK theory)
and Green potentials (because they generate errors with constant modulus on
$T$), while remaining essentially equivalent to rational approximants as far
as $n$-th root error rates are concerned [47]. This is why $n$-th root optimal
meromorphic approximants (meeting (2.6) in place of $r_{n}$) are of principal
importance in our study. Yet, the potential-theoretic tools on Riemann
surfaces that we use only allow us to handle compact surfaces so far, and this
induces some finiteness conditions on the functions from the class
$\mathcal{S}(A)$ that we can deal with. These are set forth in the next
section.

###  2.1. Class of Approximated Functions

We consider functions in $\mathcal{H}(A)$ such that

  * (i)

they can be continued into $D$ along any path originating on $T$ that stays in
$\overline{D}$ while avoiding a closed polar subset of $D$ (which may depend
on the function);

  * (ii)

they are not single-valued, meaning there are continuations along at least two
paths as in (i) with the same initial and terminal points that lead to
distinct function elements, but still they are finite-valued in that the
number of such function elements lying above a point of $D$ is uniformly
bounded (the bound may depend on the function);

  * (iii)

their number of _branchpoints_ (points in any neighborhood of which some
analytic continuation along a closed path in $D$ encircling that point while
avoiding the exceptional polar set leads to a different function element) is
finite.

In view of (i) and (ii), such functions lie in $\mathcal{S}(A)$. Note that
(iii) is not superfluous, for there are functions meeting (i) and (ii) with
infinitely many branchpoints. For instance, an open Riemann surface
$\mathcal{X}$ made of two copies of $\overline{\mathbb{C}}\setminus\\{0\\}$,
glued along a sequence of disjoint cuts in $\mathbb{D}$ shrinking to the point
$0$, has projection $p:\mathcal{X}\to\overline{\mathbb{C}}\setminus\\{0\\}$ a
two sheeted covering with infinitely many branchpoints of order $2$. As
$\mathcal{X}$ carries a holomorphic function $f$ assuming more than one value
on $p^{-1}(z)$ for $z$ not a critical value of $p$ [18, Theorem 26.7], we
deduce on putting $D=\mathbb{D}$ and
$A=\overline{\mathbb{C}}\setminus\mathbb{D}$ that each branch of $f\circ
p^{-1}$ is of the announced type.

We formalize (i), (ii) and (iii) as follows. Let ${\mathcal{R}}_{*}$ be an
auxiliary algebraic Riemann surface, whose set of ramification points
$\mathbf{rp}({\mathcal{R}}_{*})$ lies on top of $D$. That is, there exists an
irreducible polynomial in two variables $P(z,a)$, of degree at least 2 in $a$,
such that ${\mathcal{R}}_{*}=\\{(z,a):P(z,a)=0\\}$ and all branchpoints of the
algebraic function $a(z)$ lie in $D$. We denote by
$p:{\mathcal{R}}_{*}\to\overline{\mathbb{C}}$ the natural projection
$p((z,a))=z$, and we let ${\mathcal{R}}\subset{\mathcal{R}}_{*}$ be the (open)
Riemann surface defined as

${\mathcal{R}}:=\\{z\in{\mathcal{R}}_{*}:p(z)\in D\\};$

here and below, whenever it causes no confusion, we use letter $z$ to denote
both points in $\mathbb{C}$ and on ${\mathcal{R}}_{*}$. Clearly, the
ramification points of ${\mathcal{R}}_{*}$ and ${\mathcal{R}}$ are identical:
$\mathbf{rp}({\mathcal{R}}_{*})=\mathbf{rp}({\mathcal{R}})$. Let us denote by
$\overline{\mathcal{R}}$ the closure of ${\mathcal{R}}$ in
${\mathcal{R}}_{*}$, and define a class of functions
$\mathcal{F}({\mathcal{R}})$ by

(2.7) $\begin{array}[]{ll}\mathcal{F}({\mathcal{R}}):=\big{\\{}f:&f~{}\text{is
holomorphic and single-valued on}~{}\overline{\mathcal{R}}\setminus
E_{f},\vskip 3.0pt plus 1.0pt minus 1.0pt\\\ &E_{f}~{}\text{ is
closed},~{}p(E_{f})~{}\text{is polar and contained in}~{}D,\vskip 3.0pt plus
1.0pt minus 1.0pt\\\ &f(z_{1})\neq f(z_{2})~{}\text{for
some}~{}z_{1},z_{2}~{}\text{with}~{}p(z_{1})=p(z_{2})\big{\\}}.\end{array}$

In (2.7), we wrote $E_{f}$ for the singular set of $f$ on ${\mathcal{R}}$ but
it would have been more appropriate to write $E_{f}({\mathcal{R}})$, as the
complete Riemann surface of $f$ could be significantly larger than
${\mathcal{R}}$ and its singular set bigger than $E_{f}$. Since all
ramification points of ${\mathcal{R}}_{*}$ lie on top of $D$, the simple-
connectedness of $A$ implies that the preimage $p^{-1}(T)$ consists of
finitely many homeomorphic copies of $T$ under $p^{-1}$; we generically denote
by $\mathcal{T}$ such a copy, so that $p:\mathcal{T}\to T$ is a homeomorphism.
Then, denoting with a subscript $\mathcal{b}E$ the restriction to a set $E$,
the class of functions that we study is defined as

(2.8) $\mathcal{F}(A):=\bigg{\\{}f:~{}f~{}\text{is holomorphic
on}~{}A~{}\text{and}~{}\exists\,{\mathcal{R}}_{*},\,\mathcal{T},\,\widehat{f}\in\mathcal{F}({\mathcal{R}})~{}\text{with}~{}f_{\mathcal{b}T}=\widehat{f}\circ(p_{\mathcal{b}\mathcal{T}})^{-1}\bigg{\\}}.$

From (2.7) and (2.8), one sees that
$\mathcal{F}(A)\subset\mathcal{S}(A)\subset\mathcal{H}(A)$ and members of
$\mathcal{F}(A)$ meet (i), (ii), (iii). Conversely, when $f\in\mathcal{H}(A)$
satisfies (i), (ii) and (iii), one can check that $f\in\mathcal{F}(A)$.
Indeed, if $B$ is the closed polar subset of $D$ defined by (i), we get from
(ii) because $D\setminus B$ is connected, see Section A.5, that the number of
sheets of the Riemann surface of $f$ above $D\setminus B$ is a finite
constant, say $M$. Therefore, since the branchpoints are finitely many by
(iii), the algebraic surface $\mathcal{R}_{*}$ can be constructed by a
classical glueing process described in Section 3.2. The fine point, when
applying to the present case this familiar procedure based on glueing pairwise
in a certain order the banks of $M$ copies of a system of cuts joining the
branchpoints, is that any two points of $D$ can be joined by a smooth simple
arc entirely contained in $D\setminus B$, except for its endpoints if they lie
in $B$. It is so because $D\setminus B$ is a connected open set and each point
of $B$ is the center of a circle of arbitrary small radius contained in
$D\setminus B$, as well as the endpoint of a segment contained in $D\setminus
B$ (that may even be chosen to have quasi-any direction). These properties
hold because $B$ is polar, and therefore thin at each point of $\mathbb{C}$,
see Section A.6.

We also consider functions in $\mathcal{H}(A)$ meeting (i) but not (ii). These
are analytic and single-valued in $\overline{\mathbb{C}}\setminus E$, where
$E\subset D$ is closed and polar, i.e., there are no branchpoints. This case
complements the previous one on putting
$\mathcal{R}_{*}=\overline{\mathbb{C}}$ and omitting the last requirement in
(2.7); we denote the corresponding class of functions by $\mathcal{E}(A)$.
Since $\mathrm{cap}(A,E)=\mathrm{cap}_{D}(E)=0$ when $E\subset D$ is polar,
see Section A.4, we get from (2.3) and (2.2) that

(2.9) $\lim_{n\to\infty}\rho_{n}^{1/n}(f,A)=0,\qquad f\in\mathcal{E}(A).$

That is to say, some sequence $\\{r_{n}\\}_{n\in\mathbb{N}}$,
$r_{n}\in\mathcal{R}_{n}$, of rational functions, converges _faster than
geometrically_ towards $f$ on $A$ as the degree grows large, meaning that

(2.10) $\lim_{n\to\infty}\|f-r_{n}\|_{A}^{1/n}=0.$

We call any such sequence a sequence of _$n$ -th root optimal rational
approximants to $f\in\mathcal{E}(A)$ on $A$_.

###  2.2. Optimal Rational Approximants

Notions of potential theory in $D$ and ${\mathcal{R}}$ play a fundamental role
in what follows, and the reader might want to consult Appendix A for a
comprehensive account thereof. Let us here recall the definition of Green
potentials and Green equilibrium distributions.

The _Green function_ $g_{D}(\cdot,w)$ of the domain $D$ with pole at $w\in D$
is the unique non-negative harmonic function in $D\setminus\\{w\\}$, with
logarithmic singularity at $w$, whose largest harmonic minorant is zero. The
_Green potential_ of a positive Borel measure $\nu$ in $D$ is
$g(\nu,D;z):=\int g_{D}(z,w)d\nu(w)$. Putting $|\nu|$ for the total mass of
$\nu$, the _Greenian capacity_ (in $D$) of a Borel set $B\subset D$ is defined
by

(2.11) $\mathrm{cap}_{D}(B):=\bigg{(}\inf_{\nu\geq
0,|\nu|=1,\mathrm{supp}\,\nu\subseteq B}I_{D}(\nu)\bigg{)}^{-1},\quad
I_{D}(\nu):=\int\int g(z,w)d\nu(w)d\nu(z);$

the infimum above is taken over all probability Borel measures $\nu$ supported
on $B$. For any set $S\subset D$, the _outer Greenian capacity_ of $S$ in $D$
is defined as $\mathrm{cap}_{D}(S)=\inf\mathrm{cap}_{D}(U)$, where the infimum
is taken over all open sets $U\supset S$ in $D$ (using again the symbol
$\mathrm{cap}_{D}$ causes no confusion for the outer Greenian capacity is
known to coincide with the Greenian capacity on Borel sets). Polar subsets of
$D$ are those whose outer Greenian capacity is $0$. If $K$ is a non-polar
compact subset of $D$, then there exists a unique Borel probability measure
$\mu_{D,K}$ supported on $K$, called the _Green equilibrium distribution on
$K$ relative to $D$_, such that $\mathrm{cap}_{D}(K)=1/I_{D}(\mu_{D,K})$. It
is characterized by the property that its Green potential is bounded on $D$
and equal to its maximum (which is then necessarily $1/\mathrm{cap}_{D}(K)$)
_quasi everywhere_ (that is, up to a polar set) on $K$. To recap:

(2.12) $g(\mu_{D,K},D;z)\begin{cases}\leq 1/\mathrm{cap}_{D}(K),&z\in D,\\\
=1/\mathrm{cap}_{D}(K),&\text{for q.e. }\;z\in K,\\\
<1/\mathrm{cap}_{D}(K),&z\in D\setminus K,\end{cases}$

where the last inequality follows from the (generalized) maximum principle for
harmonic functions.

We are concerned with two types of asymptotics for sequences of rational
approximants: the weak∗ behavior of the normalized counting measures of their
poles, and the convergence in capacity of the functions themselves. More
precisely, given a rational function $r$ of type $(n,n)$, we define

(2.13) $\mu(r):=\frac{1}{n}\sum_{z:r(z)=\infty}\delta_{z},$

where each pole $z$ appears in the sum as many times as its order.
Equivalently $-2\pi\mu(r)$ is the distributional Laplacian
$\Delta(\log|q|^{1/n})$, with $q$ the denominator of an irreducible form of
$r$. One says that a sequence $\\{\nu_{n}\\}$ of finite Borel measures on $D$
converges _weak ∗_ to a measure $\nu$, denoted as

$\nu_{n}\overset{w*}{\to}\nu\quad\text{as}\quad n\to\infty,$

if $\int hd\nu_{n}\to\int hd\nu$ for every continuous compactly supported
function $h$ on $D$. We further say that a sequence of functions $h_{n}$
converges _in Greenian capacity_ to a function $h$ on a set $U\subset D$ if

(2.14) $\lim_{n\to\infty}\mathrm{cap}_{D}\big{(}\\{z\in
F:|h_{n}(z)-h(z)|>\epsilon\\}\big{)}=0$

for each $\epsilon>0$ and every compact $F\subset U$; we denote this claim by
$h_{n}\overset{\mathrm{cap}}{\to}h$ in $U$. The convergence (2.14) is said to
hold _at a geometric rate_ if we can replace $\epsilon$ in (2.14) by $a^{n}$
for some positive $a=a(F)<1$; the convergence rate is called _faster than
geometric_ if (2.14) holds with $\epsilon$ replaced by $a^{n}$ for any $a>0$.

######  Theorem 2.1.

Let $T\subset\mathbb{C}$ be a Jordan curve, $D$ its interior and $A$ the
complement of $D\cup T$ on the Riemann sphere. Given $f\in\mathcal{F}(A)$, let
$K_{f}\in\mathcal{K}_{f}$ be such that (2.5) holds and $\\{r_{n}\\}$ be a
sequence of $n$-th root optimal rational approximants to $f$ on $A$, as
defined in (2.6). Then,

(2.15) $\mu(r_{n})\overset{w*}{\to}\mu_{D,K_{f}},$

where $\mu(r_{n})$ is the normalized counting measure of the poles of $r_{n}$
and $\mu_{D,K_{f}}$ is the Green equilibrium distribution on $K_{f}$ relative
to $D$. Furthermore, it holds that

(2.16)
$\frac{1}{2n}\log|f-r_{n}|\overset{\mathrm{cap}}{\to}g(\mu_{D,K_{f}},D;\cdot)-\frac{1}{\mathrm{cap}_{D}(K_{f})}\quad\text{in}\quad
D\setminus K_{f}\quad\text{as}\quad n\to\infty,$

where $g(\mu_{D,K_{f}},D;\cdot)$ is the Green potential of $\mu_{D,K_{f}}$ in
$D$.

######  Remark 2.2.

_Theorem 2.1 and equation (2.12) imply that $n$-th root optimal rational
approximants converge to $f$ in capacity in $D\setminus K_{f}$, at a geometric
rate equal to $\exp\\{g(\mu_{D,K_{f}},D;\cdot)-1/\mathrm{cap}_{D}(K_{f})\\}$
pointwise. In fact, as shown in Section 3.11, convergence in Greenian capacity
in $D\setminus K_{f}$ and uniform convergence on $A$, along with the limiting
behavior (2.15) for the poles, together imply the seemingly stronger
convergence in logarithmic capacity on $\overline{D}\setminus K_{f}$, at a
geometric rate which is pointwise no slower than the previous one. Here,
convergence in logarithmic capacity is defined as in (2.14), except that the
logarithmic capacity replaces the Greenian one, see Section A.4 for a
definition of logarithmic capacity. This remark equally applies to the
forthcoming Theorem 2.5, as well as to Theorems 2.3 and 2.6 in which $K_{f}$
gets replaced by a polar set $E$ and convergence takes place at faster than
geometric rate. Moreover, if in Theorem 2.5 the error $f-M_{n}$ is
asymptotically circular on $T$, namely if
$\lim_{n}\inf_{T}|f-M_{n}|^{1/n}=\exp\\{-2/\mathrm{cap}_{D}(K_{f})\\}$ (in
particular when $M_{n}$ are their own Nehari modifications, see Section 3.3),
then one can draw the more precise conclusion that the exact pointwise rate of
geometric convergence of $M_{n}$ to $f$ in logarithmic capacity on
$\overline{D}\setminus K_{f}$ is the same as in Greenian capacity on
$D\setminus K_{f}$, namely
$\exp\\{g(\mu_{D,K_{f}},D;\cdot)-1/\mathrm{cap}_{D}(K_{f})\\}$._

➀➁➂➃$a_{2}$$a_{3}$$a_{2}$$a_{3}$$a_{4}$$a_{5}$$a_{4}$$a_{5}$$b_{1}$$b_{2}$$b_{3}$$\mathcal{T}$$\mathcal{U}_{f}$$\mathcal{U}_{f}$$\mathcal{U}_{f}$

Figure 1. Surface ${\mathcal{R}}$ with five ramification points
$\mathbf{rp}({\mathcal{R}})=\\{a_{1},a_{2},a_{3},a_{4},a_{5}\\}$ ($a_{1}$ is
not labeled) and four sheets. Ramification point $a_{1}$ has order $3$ and the
remaining points have order $2$. The sequence ➀-➁-➂-➃-➀ represents the
monodromy around $a_{1}$ while encircling it clockwise, where the transitions
happen across the dashed curves that stand for the cuts between different
sheets of ${\mathcal{R}}$. The domain $\mathcal{U}_{f}$ is depicted as a
shaded region. In this example $E_{0}=\varnothing$,
$E_{11}=\\{p(a_{1}),p(a_{2}),p(a_{3})\\}$ are the active branch points, and
$E_{10}=\\{b\\}$, where $b=p(b_{1})=p(b_{2})=p(b_{3})$. The set $K_{f}$ is a
threefold equal to the natural projection of the solid (purple) curves. The
solid curves also represent a different choice of the transition cuts between
different sheets of ${\mathcal{R}}$, with the latter the domain
$\mathcal{U}_{f}$ will lie entirely on one sheet.

In view of its importance, let us describe in greater detail the set $K_{f}$.
As mentioned before, the problem of finding elements of minimal capacity in
$\mathcal{K}_{f}$ was extensively studied by the second author [57, 58, 59].
The existence of such sets for $f\in\mathcal{H}(A)$ was proven in [57], and
one can choose $K_{f}$ so that $K_{f}\subseteq\tilde{K}$ for any
$\tilde{K}\in\mathcal{K}_{f}$ with
$\mathrm{cap}_{D}(\tilde{K})=\inf_{K}\mathrm{cap}_{D}(K)$, which makes $K_{f}$
unique [58]. The topological structure of $K_{f}$ was investigated in [59],
where it is shown that

(2.17) $K_{f}=E_{0}\cup E_{1}\cup\bigcup_{i}J_{i}$

where the $J_{i}$ are open analytic arcs, $E_{1}$ comprises the endpoints of
the arcs $J_{i}$, and $E_{0}$ is a subset of the singular set of $f$ in $D$
(the singular set consists of those points in $D$ to which some continuation
of $f$ from $T$ has a singularity). As soon as $f\in\mathcal{S}(A)$, the set
$E_{0}$ is polar by definition. To understand this decomposition better when
$f\in\mathcal{F}(A)$, let $J_{f}:=\cup_{i}\overline{J}_{i}$ so that
$E_{1}\subset J_{f}$. Then $f$ possesses a single-valued continuation into
$D\setminus J_{f}$ with singular set $E_{0}$, consisting of polar and
essential singularities (but no branching singularities). If $\mathcal{T}$,
$\widehat{f}$ are as in (2.8) and $\mathcal{U}_{f}$ is the connected component
of $p^{-1}(D\setminus J_{f})$ containing $\mathcal{T}$ in its boundary, then
we can further decompose:

$E_{0}=p(E_{\widehat{f}}\cap\mathcal{U}_{f}),\quad E_{1}=E_{10}\cup
E_{11},\quad
E_{11}:=p\big{(}\mathbf{rp}({\mathcal{R}})\cap\overline{\mathcal{U}}_{f}\big{)},\quad
E_{10}:=E_{1}\setminus E_{11}.$

That is, $E_{11}$ is the set of “active” branchpoints of $f$, i.e.,
branchpoints that can be reached by continuation of $f$ from $T$ into
$D\setminus J_{f}$, as opposed to those points in $\mathbf{rp}({\mathcal{R}})$
that cannot be so reached. Also, each $e\in E_{10}$ is an endpoint of at least
three arcs $J_{i}$, and generically $f$ possesses analytic continuations to
$e$ from any direction within $D\setminus J_{f}$ (unless by chance $e$ is a
singularity of $f$ as well), see Figure 1. As $\mathbf{rp}({\mathcal{R}})$ is
finite, so is the collection $\\{J_{i}\\}$ in cases that we consider.

For $f\in\mathcal{F}(A)$, Theorem 2.1 asserts two things: (i) the weak∗
convergence of $\mu(r_{n})$ to $\mu_{D,K_{f}}$ whenever $\\{r_{n}\\}$ is an
$n$-th root optimal sequence of rational approximants to $f$ on $A$, and (ii)
the convergence in capacity of $r_{n}$ to $f$ at a geometric rate on
$D\setminus K_{f}$. If now $f\in\mathcal{E}(A)$ and $\\{r_{n}\\}$ is a
sequence of $n$-th root optimal rational approximants to $f$ on $A$, i.e., a
sequence meeting (2.10) and thus converging faster than geometrically to $f$
on $A$, then we shall see that $r_{n}$ converges in capacity to $f$ at faster
than geometric rate in $D$. However, one can no longer expect a specific
behavior of $\mu(r_{n})$ in this case, for if $R_{n}$ is a sequence in
$\mathcal{R}_{n}(D)$ that converges to zero faster than geometrically on $T$,
then $r_{n/2}+R_{n/2}$ is again a sequence of $n$-th root optimal rational
approximants to $f$ on $A$, whereas the weak∗ limit points of $\mu(R_{n})$ can
be arbitrary amongst positive measures of mass at most $1$ on $D$. Thus,
faster than geometric convergence does not qualify rational approximants
enough to imply much on the behavior of their poles. Still, those poles of
$r_{n}$ that stay away from the singular set of $f$ cannot account for the
rate of convergence. This is made precise in the following result, which
complements Theorem 2.1 in the case of no branchpoints.

######  Theorem 2.3.

Let $T$, $A$ and $D$ be as in Theorem 2.1. Given $f\in\mathcal{E}(A)$, let
$\\{r_{n}\\}$ be a sequence of rational functions of type $(n,n)$ meeting
(2.10). Then, it holds that

$r_{n}\overset{\mathrm{cap}}{\to}f\quad\text{in}\quad D\setminus E$

at faster than geometric rate, where $E\subset D$ a closed polar set outside
of which $f$ is analytic and single-valued. Moreover, for any neighborhood $V$
of $E$ there is a sequence $\\{R_{k_{n}}\\}$,
$R_{k_{n}}\in\mathcal{R}_{k_{n}}(V)$, $k_{n}\leq n$, such that the poles of
$R_{k_{n}}$ are among the poles of $r_{n}$ lying in $V$ and

(2.18) $\lim_{n\to\infty}\|f-R_{k_{n}}\|_{A}^{1/n}=0.$

Using neighborhoods $V$ shrinking to $E$ and a diagonal argument, Theorem 2.3
yields a corollary of independent interest.

######  Corollary 2.4.

If $f\in\mathcal{E}(A)$ and $E\subset D$ a closed polar set outside of which
$f$ is analytic and single-valued, then there is a sequence of rational
functions $\\{r_{n}\\}$, $r_{n}\in\mathcal{R}_{n}(D)$, converging faster than
geometrically to $f$ on $A$, and such that every weak∗ cluster point of the
sequence $\\{\mu(r_{n})\\}$ of normalized counting measures of the poles is
supported on $E$.

###  2.3. Optimal Meromorphic Approximants

Let $H^{\infty}(D)$ denote the space of bounded analytic functions on $D$ and
$\mathcal{A}(D)$ the subspace of those extending continuously to
$\overline{D}$. When $T$ is rectifiable each $h\in H^{\infty}(D)$ has a non-
tangential limit almost everywhere on $T$ with respect to arclength, that we
still call $h$, and putting $\|\cdot\|_{T}$ for the essential supremum norm on
$T$ (with respect to arclength) it holds that $\|h\|_{T}=\|h\|_{D}$ [12,
Theorems 10.3 & 10.5]. When $T$ is a non-rectifiable Jordan curve, however,
limiting values of $H^{\infty}(D)$-functions on $T$ generally exist at
sectorially accessible points only, and such points may reduce to a set of
zero linear measure [41]. This will force us into a somewhat careful
discussion of meromorphic approximants. Remember the set $\mathcal{M}_{n}(D)$
of monic polynomials of degree $n$ whose zeros belong to $D$, and put

$\begin{cases}H_{n}^{\infty}(D)&:=\big{\\{}h/q:~{}h\in
H^{\infty}(D),\;q\in\mathcal{M}_{n}(D)\big{\\}},\vskip 3.0pt plus 1.0pt minus
1.0pt\\\
\mathcal{A}_{n}(D)&:=\big{\\{}h/q:~{}h\in\mathcal{A}(D),\;q\in\mathcal{M}_{n}(D)\big{\\}}.\end{cases}$

That is, $H_{n}^{\infty}(D)$ is the set of meromorphic function with at most
$n$ poles in $D$ that are bounded near $T$, and $\mathcal{A}_{n}(D)$ is the
subset of those extending continuously to $T$. We shall say that $\\{M_{n}\\}$
is a sequence of _$n$ -th root optimal meromorphic approximants to
$f\in\mathcal{S}(A)$ on $T$_ if $M_{n}\in\mathcal{A}_{n}(D)$ and

(2.19)
$\limsup_{n\to\infty}\|f-M_{n}\|_{T}^{1/n}\leq\exp\big{\\{}-2/\mathrm{cap}_{D}(K_{f})\big{\\}}.$

Any sequence of $n$-th root optimal rational approximants is a particular
sequence of $n$-th root optimal meromorphic ones, by (2.6). When $T$ is
rectifiable, Corollary 2.7 to come will entail that in (2.19) the condition
$M_{n}\in\mathcal{A}_{n}(D)$ can be traded for the seemingly weaker
requirement $M_{n}\in H^{\infty}_{n}(D)$. However, it is important to require
that $M_{n}\in\mathcal{A}_{n}(D)$ when $T$ is not rectifiable, for otherwise
the left-hand side of (2.19) may no longer make sense for the essential
supremum norm (with respect to arclength) and the considerations below would
not apply. In his proof of (2.3), Parfënov [47] has shown that the limit
inferior of $\rho_{n}^{1/n}(f,A)$ is the same as the one of the $n$-th root of
the error in best meromorphic approximation, see (2.22) for a definition of
best (not just $n$-th root optimal) meromorphic approximants. Hence, in light
of (2.6), replacing rational approximants with meromorphic ones does not
improve the rate of decay of the $n$-th root error, and $n$-th root optimal
meromorphic approximants to functions in $\mathcal{S}(A)$ could be defined
with the limit superior and the inequality sign replaced by a full limit and
the equality sign in (2.19).

######  Theorem 2.5.

With the notation of Theorem 2.1, let $\\{M_{n}\\}$ be a sequence of $n$-th
root optimal meromorphic approximants to$f\in\mathcal{F}(A)$ on $T$, see
(2.19). Then, the conclusions of Theorem 2.1 hold with $r_{n}$ replaced by
$M_{n}$.

Let us reiterate that $n$-th root optimal rational approximants are just an
instance of $n$-th root optimal meromorphic approximants, and therefore
Theorem 2.1 is a special case of Theorem 2.5.

When $f\in\mathcal{E}(A)$, we define meromorphic approximants in a manner
similar to (2.19) but this time with an eye on (2.9). Namely, we say that
$\\{M_{n}\\}$ is a sequence of _$n$ -th root optimal meromorphic approximants
to $f\in\mathcal{E}(A)$ on $T$_ if $M_{n}\in\mathcal{A}_{n}(D)$ and

(2.20) $\lim_{n\to\infty}\|f-M_{n}\|_{T}^{1/n}=0.$

The following complements Theorem 2.5 in the case of no branchpoints, and
subsumes Theorem 2.3.

######  Theorem 2.6.

With the notation of Theorem 2.3, let $\\{M_{n}\\}$ be a sequence of $n$-th
root optimal meromorphic approximants on $T$ to $f\in\mathcal{E}(A)$, see
(2.20). Then, the conclusions of Theorem 2.3 hold with $r_{n}$ replaced by
$M_{n}$.

Besides best rational approximants, a noteworthy instance of $n$-th root
optimal meromorphic approximants are the AAK (short for Adamyan-Arov-Krein)
approximants. These are best meromorphic approximants with at most $n$ poles.
Specifically, consider the following (Nehari-Takagi) problem: given $f\in
L^{\infty}(\mathbb{T})$, find $M_{n}^{\infty}\in H_{n}^{\infty}(\mathbb{D})$
such that

(2.21) $\|f-M_{n}^{\infty}\|_{\mathbb{T}}=\inf_{M\in
H_{n}^{\infty}(\mathbb{D})}\|f-M\|_{\mathbb{T}}.$

If $n=0$, then (2.21) reduces to the question of best analytic approximation
of bounded functions on the unit circle by elements of
$H^{\infty}(\mathbb{D})$, which is the so-called _Nehari problem_ named after
[43] (that deals with an equivalent issue). It is known that $M_{n}^{\infty}$
always exists and that it is unique when $f$ lies in
$C(\mathbb{T})+H^{\infty}(\mathbb{D})$, see [48, Chapter 4]. Moreover, if $f$
is Dini-continuous on $\mathbb{T}$, then $M_{n}^{\infty}$ is continuous on
$\mathbb{T}$. Indeed, if we write $M_{n}^{\infty}=r_{n}+g$ where
$r_{n}\in\mathcal{R}_{n}(\mathbb{D})$ and $g\in H^{\infty}(\mathbb{D})$, we
see that $g$ must be the best Nehari approximant to $f-r_{n}$, which is Dini-
continuous on $\mathbb{T}$, and the claim follows from [9]. When $T$ is a
rectifiable Jordan curve, one can readily replace $\mathbb{D}$ by $D$ in
(2.21) and carry over to $T$ all the properties of best meromorphic
approximants on $\mathbb{T}$ by conformal mapping. When $T$ is non-
rectifiable, the very existence of approximants depends on the analyticity of
$f$ on $T$, and follows from the proof of the next corollary to Theorem 2.5.

######  Corollary 2.7.

Let $T$, $A$ and $D$ be as in Theorem 2.1. Given $f\in\mathcal{F}(A)$ or
$\mathcal{E}(A)$, to each integer $n$ there exists a unique
$M_{n}^{\infty}\in\mathcal{A}_{n}(D)$ such that

(2.22) $\|f-M_{n}^{\infty}\|_{T}=\inf_{M\in\mathcal{A}_{n}(D)}\|f-M\|_{T},$

and if $T$ is rectifiable then $\mathcal{A}_{n}(D)$ can be replaced by
$H_{n}^{\infty}(D)$ in (2.22) without changing $M_{n}^{\infty}$. Of necessity,
the conclusions of Theorem 2.5 and 2.6 hold with $M_{n}$ replaced by
$M_{n}^{\infty}$.

Much interest in best meromorphic approximants stems from their striking
connection to operator theory. Denote by $L^{2}(\mathbb{T})$ the space of
square integrable functions on $\mathbb{T}$, and let $H^{2}\subset
L^{2}(\mathbb{T})$ be the Hardy space of functions whose Fourier coefficients
with negative index do vanish. It is known that $H^{2}$ can be identified with
(non-tangential limits on $\mathbb{T}$ of) analytic functions in $\mathbb{D}$
whose $L^{2}$-means on circles centered at the origin are uniformly bounded,
see [12, Theorem 3.4]. Set $H^{2}_{-}:=L^{2}(\mathbb{T})\ominus H^{2}$ to be
the orthogonal complement of $H^{2}$, which is the Hardy space of
$L^{2}$-functions whose Fourier coefficients with nonnegative index are equal
to zero. The latter can be identified with functions analytic in
$\overline{\mathbb{C}}\setminus\overline{\mathbb{D}}$ that vanish at infinity,
and whose $L^{2}$-means with respect to normalized arclength on circles
centered at the origin are uniformly bounded. Let
$\mathbb{P}_{-}:L^{2}(\mathbb{T})\to H^{2}_{-}$ be the orthogonal projection.
Given $f\in L^{\infty}(\mathbb{T})$, one defines the _Hankel operator with
symbol_ $f$ to be

(2.23) $\Gamma_{f}:~{}H^{2}\to
H^{2}_{-},\quad\quad\Gamma_{f}(g):=\mathbb{P}_{-}(gf).$

For $n$ a non-negative integer, let $s_{n}(\Gamma_{f})$ be the $(n+1)$-th
singular number of the operator $\Gamma_{f}$, that is
$s_{n}(\Gamma_{f}):=\inf_{\mathrm{rank}\,R\leq n}\|\Gamma_{f}-R\|$, where the
infimum is taken over all operators $R:H^{2}\to H^{2}_{-}$ of rank at most $n$
and $\|\cdot\|$ stands for the operator norm. Then, one has that

(2.24) $\|f-M_{n}^{\infty}\|_{\mathbb{T}}=s_{n}(\Gamma_{f}).$

If, in addition, $f\in C(\mathbb{T})+H^{\infty}(\mathbb{D})$, then
$\Gamma_{f}$ is compact so that $s_{n}^{2}(\Gamma_{f})$ is the $(n+1)$-st
eigenvalue of $\Gamma_{f}^{*}\Gamma_{f}$ when these are arranged in non-
increasing order, and (2.24) becomes a pointwise equality:

(2.25) $|(f-M_{n}^{\infty})(z)|=s_{n}(\Gamma_{f})\qquad\text{a.e.\ on}\
\mathbb{T}.$

Moreover, if $v_{n}$ is an $(n+1)$-st singular vector of $\Gamma_{f}$, i.e.,
an eigenvector of $\Gamma_{f}^{*}\Gamma_{f}$ with eigenvalue
$s_{n}^{2}(\Gamma_{f})$, then $M_{n}^{\infty}$ is explicitly given in terms of
$f$ and $v_{n}$ by the formula

(2.26) $f-M_{n}^{\infty}=\frac{\Gamma_{f}(v_{n})}{v_{n}}.$

Though not obvious at first glance, the right-hand side of (2.26) is
independent of which $n$-th singular vector $v_{n}$ is chosen and it has
constant modulus on $\mathbb{T}$, in accordance with (2.25). The next
corollary, of independent interest, follows from (2.25), (2.19), (2.9), and
Theorems 2.5–2.6.

######  Corollary 2.8.

Let $\Gamma_{f}$ be the Hankel operator with symbol
$f\in\mathcal{F}\big{(}\overline{\mathbb{C}}\setminus\mathbb{D}\big{)}$. Then,
it holds that

$\lim_{n\to\infty}s_{n}^{1/n}(\Gamma_{f})=\exp\big{\\{}-2/\mathrm{cap}_{\mathbb{D}}(K_{f})\big{\\}}.$

Moreover, if $f\in\mathcal{E}(\overline{\mathbb{C}}\setminus\mathbb{D})$ ,
then the above limit is equal to $0$.

##  3\. Proof of Theorem 2.5

###  3.1. Existence of Best Meromorphic Approximants

Below, we establish the existence and uniqueness part of Corollary 2.7, along
with the assertion that $H_{n}^{\infty}(D)$ may replace $\mathcal{A}_{n}(D)$
when $T$ is rectifiable. The rest of the corollary will follow from Theorem
2.5 upon conclusion of its proof.

Let $\phi:\mathbb{D}\to D$ be a conformal map. Since $D$ is a Jordan domain,
$\phi$ extends to a homeomorphism from $\overline{\mathbb{D}}$ to
$\overline{D}$ by Carathéodory theorem [50, Section 2]. Let
$L^{2}(\mathbb{T})$, $H^{2}$, and $H^{2}_{-}$ be as defined after Corollary
2.7 and $\mathbb{P}_{+}:L^{2}(\mathbb{T})\to H^{2}$,
$\mathbb{P}_{-}:L^{2}(\mathbb{T})\to H^{2}_{-}$ be the orthogonal projections.
If we pick a continuous function $f$ on $T$, then $f\circ\phi$ is continuous
on $\mathbb{T}$ and, in particular, it lies in $L^{2}(\mathbb{T})$. Set
$F:=\mathbb{P}_{-}(f\circ\phi)$ and $G:=\mathbb{P}_{+}(f\circ\phi)$, so that
$F\in H^{2}_{-}$ and $G\in H^{2}$. Since $\mathbb{P}_{+}+\mathbb{P}_{-}$ is
the identity operator, it holds that

(3.1) $F(z)=(f\circ\phi)(z)-G(z),\quad\mathrm{a.e.\ }z\in\mathbb{T}.$

Now, if $f\in\mathcal{H}(A)$, then $f\circ\phi(z)$ is holomorphic in $r<|z|<1$
and continuous in $r\leq|z|\leq 1$, for $r$ close enough to $1$. Hence, the
right-hand side of (3.1) is holomorphic in $r<|z|<1$ with uniformly bounded
$L^{2}$-means on circles centered at the origin, while the left-hand side lies
in $H^{2}_{-}$ and both sides have the same non-tangential limit on
$\mathbb{T}$. By an easy variant of Morera’s theorem [21, Chapter II, Exercise
12], the function equal to $F(z)$ for $|z|>1$ and to $(f\circ\phi)(z)-G(z)$
for $r<|z|<1$ is holomorphic across $\mathbb{T}$, in particular $F$ extends
analytically across $\mathbb{T}$ and $G$ extends continuously to
$\overline{\mathbb{D}}$. Then, as described after (2.21), the best meromorphic
approximant $M_{n}^{\infty}\in H_{n}^{\infty}(\mathbb{D})$ to $f\circ\phi$
exists and is unique, moreover it is readily checked that $M_{n}^{\infty}$ is
equal to the sum of $G$ (a member of $\mathcal{A}(\mathbb{D})$) and of the
best approximant to $F$ from $H_{n}^{\infty}(\mathbb{D})$, which lies in
$\mathcal{A}_{n}(\mathbb{D})$ because $F$ is analytic across $\mathbb{T}$ and
so is a fortiori Dini-continuous on $\mathbb{T}$; hence, we get that
$M_{n}^{\infty}\in\mathcal{A}_{n}(\mathbb{D})$. If now
$M\in\mathcal{A}_{n}(D)$, then $M\circ\phi\in\mathcal{A}_{n}(\mathbb{D})$ and

$\|f-M\|_{T}=\|f\circ\phi-M\circ\phi\|_{\mathbb{T}}\geq\|f\circ\phi-
M_{n}^{\infty}\|_{\mathbb{T}}$

by definition of $M_{n}^{\infty}$. As
$M_{n}^{\infty}\circ\phi^{-1}\in\mathcal{A}_{n}(D)$, it is the unique best
meromorphic approximant to $f$ we are looking for. The previous argument also
shows that, when $f\in\mathcal{H}(A)$, the best meromorphic approximant to
$f\circ\phi$ necessarily belongs to $\mathcal{A}_{n}(\mathbb{D})$. Because
composition with $\phi$ is an isometric isomorphism $L^{\infty}(T)\to
L^{\infty}(\mathbb{T})$ (understood with respect to arclength measure) when
$T$ is rectifiable, one can equivalently use $H_{n}^{\infty}(D)$ instead of
$\mathcal{A}_{n}(D)$ in definition (2.22).

###  3.2. Reduction to the Unit Disk

Let $K$ be a compact subset of $\mathbb{D}$. Since Green potentials are non-
negative superharmonic functions whose largest harmonic minorant is zero,
while a characteristic property of Green equilibrium potentials is to be
constant quasi everywhere on the support of their defining measure while being
no greater than this constant everywhere in the domain, it holds that

$g(\mu_{\mathbb{D},K},\mathbb{D};z)=g(\mu_{D,\phi(K)},D;\phi(z))\quad\text{and}\quad\mathrm{cap}_{\mathbb{D}}(K)=\mathrm{cap}_{D}(\phi(K))$

where $\phi:\mathbb{D}\to D$ is a conformal map, see (2.12) as well as
Sections A.3 and A.4. One has in this case that
$\mu_{D,\phi(K)}=\phi_{*}(\mu_{\mathbb{D},K})$, the pushforward of
$\mu_{\mathbb{D},K}$ under $\phi$. Hence, we get that

$\frac{1}{n}\sum_{i=1}^{n}\delta_{z_{i}}\stackrel{{\scriptstyle*}}{{\to}}\mu_{\mathbb{D},K}\quad\text{if
and only
if}\quad\frac{1}{n}\sum_{i=1}^{n}\delta_{\phi(z_{i})}\stackrel{{\scriptstyle*}}{{\to}}\mu_{D,\phi(K)},$

where the weak∗ convergence is understood for $n\to\infty$. Furthermore, by
conformal invariance, a sequence of functions $h_{n}$ converges in Greenian
capacity to a function $h$ in $D$ if and only if the functions
$h_{n}\circ\phi$ converge in Greenian capacity to the function $h\circ\phi$ in
$\mathbb{D}$.

Let now ${\mathcal{R}}_{*}$ and ${\mathcal{R}}$ be as in Section 2.1. Denote
by $M$ the number of sheets of ${\mathcal{R}}$, so that a generic point in $D$
has $M$ preimages under the natural projection $p:{\mathcal{R}}\to D$. Let
$J\subset D$ be a smooth oriented Jordan arc joining two points of $p({\bf
rp}({\mathcal{R}}))$ while passing through all others exactly once.
Constructing $J$ is tantamount to enumerating the points of $p({\bf
rp}({\mathcal{R}}))$. Write $J=\cup_{l}J_{l}$ where each $J_{l}$ connects
exactly two points in $p({\bf rp}({\mathcal{R}}))$. The surface
${\mathcal{R}}$ can be realized as $M$ copies $U_{1},\ldots,U_{M}$ of
$D\setminus J$, suitably glued to each other along the banks of the cuts
$J_{l}$ in each copy $U_{i}$ (the glueing rule can be encoded, for example, as
a collection of 4-tuples $(l,i,j,k)$ telling one that the left and right banks
of the cut $J_{l}$ in $U_{i}$ need to be glued to the right bank of the cut
$J_{l}$ in $U_{j}$ and the left bank of the cut $J_{l}$ in $U_{k}$
respectively, see [40] for a discussion of Hurwitz’s theorem). Using the same
gluing rule, we can construct another $M$-sheeted surface $\mathcal{S}$ out of
the domains $\phi^{-1}(U_{1}),\ldots,\phi^{-1}(U_{M})$. The map $\phi$ can
then be lifted to a conformal map $\Phi:\mathcal{S}\to{\mathcal{R}}$ for which
$\phi(\pi(z))=p(\Phi(z))$, $z\in\mathcal{S}$, where
$\pi:\mathcal{S}\to\mathbb{D}$ is the natural projection.

Let $f\in\mathcal{F}(A)$ and $\widehat{f}\in\mathcal{F}({\mathcal{R}})$ be as
in (2.8) and (2.7). Clearly $\widehat{f}\circ\Phi\in\mathcal{F}(\mathcal{S})$,
and the surface $\mathcal{S}_{*}$ can be constructed by gluing $M$ copies of
$\overline{\mathbb{C}}\setminus\mathbb{D}$ to $\mathcal{S}$ along the $M$
homeomorphic copies of $\mathbb{T}$ that comprise the boundary of
$\mathcal{S}$. Argueing as we did after (3.1), we find that
$F:=\mathbb{P}_{-}(f\circ\phi)$ lies in
$\mathcal{F}(\overline{\mathbb{C}}\setminus\mathbb{D})$ with the corresponding
$\widehat{F}\in\mathcal{F}(\mathcal{S})$ given by
$\widehat{f}\circ\Phi-G\circ\pi$, where $G:=\mathbb{P}_{+}(f\circ\phi)$. Note
that the conformal equivalence of Greenian capacities implies that
$K_{F}=\phi^{-1}(K_{f})$. Hence, if $\\{M_{n}\\}$ is an $n$-th root optimal
sequence of meromorphic approximants of $f$ in $D$ as defined in (2.19), then
$\\{\widetilde{M}_{n}:=M_{n}\circ\phi-G\\}$ is an $n$-th root optimal sequence
of meromorphic approximants to $F$ on $\mathbb{T}$ and

(3.2) $(f-M_{n})(\phi(z))=(F-\widetilde{M}_{n})(z),\quad
z\in\overline{\mathbb{D}}\setminus K_{F}.$

Therefore, it is sufficient to study the asymptotic behavior of
$F-\widetilde{M}_{n}$ as well as the limit distribution of poles of
$\widetilde{M}_{n}$. That is, it is enough to prove Theorem 2.5 on the unit
disk.

###  3.3. Nehari Modifications

For $f\in\mathcal{H}(\overline{\mathbb{C}}\setminus\mathbb{D})$ and
$M_{n}\in\mathcal{A}_{n}(\mathbb{D})$, let $h_{f-M_{n}}$ be the best
holomorphic (Nehari) approximant of $f-M_{n}$ in $H^{\infty}(\mathbb{D})$.
That is, $\|f-M_{n}-h_{f-M_{n}}\|_{\mathbb{T}}=\inf_{h\in
H^{\infty}(\mathbb{D})}\|f-M_{n}-h\|_{T}$, and $h_{f-M_{n}}\in
H^{\infty}(\mathbb{D})$. Let us set

(3.3) $N(M_{n})(z):=(M_{n}+h_{f-M_{n}})(z),$

and call $N(M_{n})$ the _Nehari modification_ of $M_{n}$. The discussion after
(2.21) shows that $N(M_{n})$ lies in $\mathcal{A}_{n}(\mathbb{D})$. Indeed, we
may write $M_{n}=g_{n}+r_{n}$ with $r_{n}\in\mathcal{R}_{n}(\mathbb{D})$ and
$g_{n}\in\mathcal{A}(\mathbb{D})$. Then, one can readily check that
$h_{f-M_{n}}=g_{n}+h_{f-r_{n}}$. As $f-r_{n}$ is analytic across $\mathbb{T}$
and in particular Dini-continuous there, it follows that $h_{f-r_{n}}$ belongs
to $\mathcal{A}(\mathbb{D})$ and so does $h_{f-M_{n}}$.

Since $\|f-N(M_{n})\|_{\mathbb{T}}\leq\|f-M_{n}\|_{\mathbb{T}}$ and $N(M_{n})$
lies in $\mathcal{A}_{n}(\mathbb{D})$, the sequence $\\{N(M_{n})\\}$ is one of
$n$-th root optimal meromorphic approximants to $f$, whenever $\\{M_{n}\\}$ is
such a sequence. It is beneficial for us to consider Nehari modifications
because they enjoy the additional property that the error they generate has
constant modulus on $\mathbb{T}$, i.e., it follows from (2.24) that

(3.4) $|(f-N(M_{n}))(z)|=\|\Gamma_{f-M_{n}}\|\quad\text{ for a.e.}\quad
z\in\mathbb{T}.$

_We claim_ that it is enough to prove Theorem 2.5 for Nehari modifications
only, as we now show.

Assume that Theorem 2.5 holds for $\\{N(M_{n})\\}$. As the poles of $M_{n}$
and $N(M_{n})$ are the same, this automatically yields the statement about
weak∗ convergence of the counting measures of the poles. Moreover, given
$\epsilon>0$ and $K\subset\mathbb{D}\setminus K_{f}$ a compact set, let us put

$E(K,\epsilon,N(M_{n})):=\left\\{z\in
K:\left|\frac{1}{2n}\log|(f-N(M_{n}))(z)|-g(\mu_{\mathbb{D},K_{f}},\mathbb{D};z)+\frac{1}{\mathrm{cap}_{\mathbb{D}}(K_{f})}\right|>\epsilon\right\\}.$

Define $E(K,\epsilon,M_{n})$ analogously. According to our assumption it holds
that

(3.5)
$\lim_{n\to\infty}\mathrm{cap}_{\mathbb{D}}\big{(}E(K,\epsilon,N(M_{n}))\big{)}=0,$

and we need to show that (3.5) holds with $N(M_{n})$ replaced by $M_{n}$.
Given $\varepsilon\in(0,1)$, define

$F_{n,\varepsilon}:=\left\\{z\in
K:\,\,\frac{1}{2n}\log|(f-N(M_{n}))(z)|>g\big{(}\mu_{\mathbb{D},K_{f}},\mathbb{D};\cdot\big{)}-\frac{1}{\mathrm{cap}_{\mathbb{D}}(K_{f})}-\varepsilon
m_{K}\right\\},$

where
$m_{K}:=\min_{K}g\big{(}\mu_{\mathbb{D},K_{f}},\mathbb{D};\cdot\big{)}>0$.
Since $N(M_{n})-M_{n}$ is analytic in $\mathbb{D}$, we get from the maximum
modulus principle and the triangle inequality that $|N(M_{n})(z)-M_{n}(z)|\leq
2\|f-M_{n}\|_{\mathbb{T}}$ for $z\in\mathbb{D}$. Since
$\exp\\{(1-\varepsilon)m_{K}\\}>1$, relation (2.19) implies that

$2\|f-M_{n}\|_{\mathbb{T}}<\frac{1}{2}\exp\left\\{(1-\varepsilon)2n\,m_{K}-\frac{2n}{\mathrm{cap}_{\mathbb{D}}(K_{f})}\right\\}$

for all $n$ large enough. Hence, by the previous estimates, it holds for all
such $n$ and $z\in F_{n,\varepsilon}$ that

$\left|\frac{N(M_{n})(z)-M_{n}(z)}{f(z)-N(M_{n})(z)}\right|<\frac{1}{2}e^{2n(m_{K}-g(\mu_{\mathbb{D},K_{f}},\mathbb{D};z))}\leq\frac{1}{2}.$

In particular, for any $0<\varepsilon<\varepsilon^{\prime}$, there exists
$n_{0}$ depending on $K$ and $\varepsilon^{\prime}-\varepsilon$ such that

(3.6)
$\frac{1}{2n}\left|\log\left|\frac{f(z)-M_{n}(z)}{f(z)-N(M_{n})(z)}\right|\right|<(\varepsilon^{\prime}-\varepsilon)m_{K}$

for all $z\in F_{n,\varepsilon}$ and $n\geq n_{0}$. Therefore, we get from the
triangle inequality that

$E(K,\varepsilon^{\prime}m_{K},M_{n})\subseteq(K\setminus
F_{n,\varepsilon})\cup E(K,\varepsilon m_{K},N(M_{n}))=E(K,\varepsilon
m_{K},N(M_{n}))$

for all $n\geq n_{0}$. The above inclusion clearly yields that (3.5) holds
with $N(M_{n})$ replaced by $M_{n}$ for $\epsilon=\varepsilon^{\prime}m_{K}$.
As $\varepsilon$, $\varepsilon^{\prime}$, and $K$ were arbitrary, _the claim
follows_.

###  3.4. Notation

We fix $f\in\mathcal{F}(\overline{\mathbb{C}}\setminus\mathbb{D})$ and, with a
slight abuse of notation, we keep denoting by $f$ the corresponding function
in $\mathcal{F}({\mathcal{R}})$ (that was denoted by $\widehat{f}$ in (2.8)).
Take $\\{M_{n}\\}$ to be a sequence of $n$-th root optimal meromorphic
approximants of $f$ and let $\\{N(M_{n})\\}$ be the sequence of corresponding
Nehari modifications. Since $\\{N(M_{n})\\}$ is also $n$-th root optimal, it
holds that

(3.7)
$\lim_{n\to\infty}\frac{1}{n}\log\|f-N(M_{n})\|_{\mathbb{T}}=-\frac{2}{\mathrm{cap}_{\mathbb{D}}(K_{f})},$

see the discussion after (2.19). We shall need an exhaustion of
${\mathcal{R}}\setminus E_{f}$ by open sets with “nice” boundaries. That is,
we consider a sequence $\\{\Omega_{m}\\}_{m\geq 1}$ of open sets such that

(3.8) $\Omega_{m}\subset{\mathcal{R}}\setminus
E_{f},\quad\overline{\Omega}_{m}\subset\Omega_{m+1},\quad\partial{\mathcal{R}}\subset\partial\Omega_{m},\quad{\mathcal{R}}\setminus
E_{f}=\cup_{m}\Omega_{m},$

and each $\Omega_{m}$, when viewed as an open subset with compact closure of
${\mathcal{R}}_{*}$, is regular for the Dirichlet problem, see Section A.7. We
will require in addition that
$\lambda(\partial\Omega_{m}\setminus\partial{\mathcal{R}})=0$ for some Radon
measure $\lambda$ on ${\mathcal{R}}$ that will be specified at the beginning
of Section 3.5 (a Radon measure is a positive Borel measure which is finite on
compact sets). To design regular $\Omega_{m}$ that meet (3.8) is possible
because $E_{f}$, being compact in ${\mathcal{R}}$, is a countable intersection
of compact sets $K_{l}\subset{\mathcal{R}}$ with smooth boundary. Indeed,
there is a smooth function $h\geq 0$ on ${\mathcal{R}}$ such that $E_{f}$ is
the zero set of $h$ and $h\geq c>0$ outside a compact neighborhood of $E_{f}$,
hence we can pick $K_{l}$ to be the sublevel set $\\{z:\,h(z)\leq t_{l}\\}$,
where $\\{t_{l}\\}$ a sequence of regular values of $h$ tending to 0 (almost
every positive number is a regular value by Sard’s theorem). In fact, using
smooth partitions of unity and local coordinates, existence of such $h$
quickly reduces to the corresponding issue in Euclidean space, where it
follows easily from a combination of [63, Chapter VI, Theorem 2] and [68,
Theorem I] (this result is named after H. Whitney). Furthermore, since the
sets $C_{a,b}:=\\{z:\,a\leq h(z)\leq b\\}$ are compact for $0<a<b<\varepsilon$
and $\varepsilon>0$ small enough, $\lambda(C_{a,b})<\infty$. So,
$\lambda(\\{z:\,h(z)=t\\})\neq 0$ for at most countably many positive
$t<\varepsilon$, and we can assume that $t_{l}$ chosen above are not such
values. The domains $\Omega_{m}$ thus constructed satisfy all our
requirements.

Let $\Omega$ be a subdomain of $\mathbb{D}$ or ${\mathcal{R}}$. Given a Borel
measure $\sigma$ on $\Omega$, we denote the Green potential of $\sigma$
relative to $\Omega$ by $g(\sigma,\Omega;\cdot)$, see Section A.2. Hereafter,
every measure is Borel unless otherwise stated. If $\sigma$ is a measure on a
Borel set containing $\Omega$, we write for simplicity
$g(\sigma,\Omega;\cdot)$ to mean $g(\sigma_{\mathcal{b}\Omega},\Omega;\cdot)$.
Conversely, for a measure $\sigma$ on a Borel set $B_{0}\subset\Omega$, we
still denote by $\sigma$ the measure on $\Omega$ mapping a Borel set $B$ to
$\sigma(B\cap B_{0})$, and write $g(\sigma,\Omega;\cdot)$ for its potential.
When $\sigma=\sum_{j}\delta_{z_{j}}$ is a (possibly infinite) sum of Dirac
delta measures, we put

(3.9) $b(\sigma,\Omega;z)=\exp\big{\\{}-g^{*}(\sigma,\Omega;z)\big{\\}}$

to stand for the corresponding generalized Blaschke product, where
$g^{\ast}(\sigma,\Omega;\cdot)$ is a complexified Green potential, i.e., it is
locally holomorphic in $\Omega\setminus\mathrm{supp}(\sigma)$ and
$\operatorname{Re}g^{\ast}(\sigma,\Omega;\cdot)=g(\sigma,\Omega;\cdot)$. If
$g(\sigma,\Omega;\cdot)\equiv+\infty$, which can happen when the points
$z_{j}$ accumulate in $\Omega$ or to the boundary $\partial\Omega$
sufficiently slowly, then $b(\sigma,\Omega;\cdot)$ is identically zero.
Otherwise, $b(\sigma,\Omega;\cdot)$ is well defined up to a unimodular
constant (because the periods of a conjugate function of
$g(\sigma,\Omega;\cdot)$ are integral multiples of $2\pi$ by Gauss’ theorem),
holomorphic in $\Omega$, unimodular quasi everywhere on $\partial\Omega$
(everywhere if the latter is regular and the points $z_{j}$ are finite in
number), and it vanishes only at the points $z_{j}$ (with multiplicities
represented by repetition).

###  3.5. Stripped Error of Approximation

We shall study the asymptotics of the error functions $|f-N(M_{n})\circ p|$.
In this section, we strip off their poles and zeros to take logarithms and
obtain harmonic functions whose limiting behavior we then investigate. To this
end, we set

(3.10)
$\widetilde{\mu}_{n}:=\sum\delta_{v_{n,j}},\;\;\mu_{n}:=\widetilde{\mu}_{n}/n,\quad\text{and}\quad\widetilde{\nu}_{n}:=\sum\delta_{u_{n,j}},\;\;\nu_{n}:=\widetilde{\nu}_{n}/n,$

where $\\{v_{n,j}\\}\subset\mathbb{D}$ are the poles of $N(M_{n})$ and
$\\{u_{n,j}\\}\subset{\mathcal{R}}\setminus E_{f}$ the zeros of
$f-N(M_{n})\circ p$, with multiplicities counted by repetition. Before we
proceed, let us specify the measure $\lambda$ appearing in the definition of
the exhaustion $\\{\Omega_{m}\\}$ in the previous subsection. To this end,
recall that on a locally compact space $X$, a sequence of Radon measures
$\\{\sigma_{n}\\}$ converges _vaguely_ to a Radon measure $\sigma$ if $\int
gd\sigma_{n}\to\int gd\sigma$ for every $g$ in $C_{c}(X)$, the space of
continuous functions with compact support on $X$ endowed with the $\sup$-norm.
Moreover, if the measures $\sigma_{n}$ are locally bounded on $X$, then they
do contain a vaguely convergent subsequence, see for example [14, Theorem
1.41] for an argument on $\mathbb{R}^{n}$ which is applicable to any
$\sigma$-compact locally convex space222Vague convergence is called weak
convergence in [14].. Hence, since the measures $\mu_{n}$ have mass at most 1,
we get if $\nu_{n}$ is locally bounded along some sequence of integers that
there exists a subsequence $\mathcal{N}\subseteq\mathbb{N}$, a measure
$\nu^{*}$ on ${\mathcal{R}}\setminus E_{f}$ and a measure $\mu$ on
$\mathbb{D}$ such that $\nu_{n}$ and $\mu_{n}$ converge vaguely to $\nu^{*}$
and $\mu$, respectively, along $\mathcal{N}$; if the measures $\nu_{n}$ have
no locally bounded subsequence, i.e., if there exists a compact set
$K\subset{\mathcal{R}}\setminus E_{f}$ such that $\nu_{n}(K)\to\infty$ as
$\mathbb{N}\ni n\to\infty$, then we put $\nu^{*}=0$ and we only require the
vague convergence $\mu_{n}\to\mu$ along $\mathcal{N}$. In any case we take
$\lambda$ to be $\nu^{*}+\widehat{\mu}$, where $\widehat{\mu}$ is the lift of
$\mu$ to ${\mathcal{R}}$ defined via (A.33).

Using (3.9), we define Blaschke products vanishing at the poles of $N(M_{n})$
and the zeros of $f-N(M_{n})\circ p$. Namely, we put

$b_{n}^{pole}(z):=b(\widetilde{\mu}_{n},\mathbb{D};z)\quad\mbox{and}\quad
b_{n,m}^{zero}(z):=b(\widetilde{\nu}_{n},\Omega_{m};z).$

These functions are not identically zero, as the number of poles of $N(M_{n})$
is at most $n$ while the number of zeros of $f-N(M_{n})\circ p$ in each
$\Omega_{m}$ is finite. To see the latter point, recall from (3.4) that
$|f-N(M_{n})|$ is constant on $\mathbb{T}$, hence the error $f-N(M_{n})$ can
be meromorphically continued across $\mathbb{T}$ by reflection. It implies
that $N(M_{n})$ can be meromorphically continued across $\mathbb{T}$ and this
continuation is necessarily analytic in some neighborhood of $\mathbb{T}$.
Thus, $f-N(M_{n})\circ p$ is analytic in a neighborhood of
$\partial{\mathcal{R}}$ by the analyticity of $f$ there, so the zeros of
$f-N(M_{n})\circ p$ can only accumulate on $E_{f}$ and not on
$\partial{\mathcal{R}}$. Thus, there are at most finitely many of them in each
$\Omega_{m}$.

Using these Blaschke products, we define

(3.11) $h_{n,m}(z):=\frac{1}{n}\log\left|\left(f-N(M_{n})\circ
p\right)(z)(b_{n}^{pole}\circ p)(z)/b_{n,m}^{zero}(z)\right|,\quad
z\in\Omega_{m},$

which is harmonic in $\Omega_{m}$. Recall that superharmonic functions on a
hyperbolic Riemann surface are either identically $+\infty$ or finite quasi
everywhere, and any two of them that coincide almost everywhere (with respect
to Lebesgue measure in local coordinates) are in fact equal (the weak identity
principle), see Section A.1.

######  Lemma 3.1.

There exist a subsequence $\mathcal{N}^{\prime}\subseteq\mathcal{N}$ and a
non-negative superharmonic function $u^{\prime}(z)$ on ${\mathcal{R}}$ such
that

(3.12) $-h_{n_{m},m}(z)\to u^{\prime}(z)\quad\text{as}\quad m\to\infty,$

locally uniformly in $z\in{\mathcal{R}}\setminus E_{f}$, for any subsequence
$\\{n_{m}\\}_{m=1}^{\infty}\subseteq\mathcal{N}^{\prime}$. If $u^{\prime}$ is
finite quasi everywhere, then one has a decomposition

(3.13) $u^{\prime}=g(\nu^{\prime},{\mathcal{R}};\cdot)+h^{\prime},$

where $\nu^{\prime}$ is a finite positive Borel measure supported on $E_{f}$
and $h^{\prime}$ is a non-negative harmonic function on ${\mathcal{R}}$.

###### Proof.

The regularity of $\partial\Omega_{m}$ implies that $|b_{n,m}^{zero}(z)|\equiv
1$ for $z\in\partial\Omega_{m}$, and likewise $|b_{n}^{pole}(z)|=1$ for $z$ on
$\mathbb{T}$. In particular we get that

$\big{|}N(M_{n})(z)b_{n}^{pole}(z)\big{|}\leq\|N(M_{n})\|_{\mathbb{T}},\quad
z\in\mathbb{D},$

by the maximum modulus principle for $H^{\infty}$-functions. Set
$\Gamma_{n}:=\Gamma_{f-M_{n}}$ be the Hankel operator with symbol $f-M_{n}$,
see (2.23). It follows from the definition of the singular values, (2.24), and
(3.3) that
$\|\Gamma_{n}\|=s_{0}(\Gamma_{n})=\|f-N(M_{n})\|_{\mathbb{T}}\leq\|f-M_{n}\|_{\mathbb{T}}$.
Since the norms $\|f-M_{n}\|_{\mathbb{T}}$ tend to zero by assumption, we get
that

(3.14) $\|N(M_{n})\|_{\mathbb{T}}\leq\|f\|_{\mathbb{T}}+\|\Gamma_{n}\|\leq
C_{f}$

for some constant $C_{f}$ that depends only on $f$ and the sequence
$\\{M_{n}\\}$. Thus, we get from the maximum principle for harmonic functions
that

(3.15)
$h_{n,m}(z)\leq\frac{1}{n}\log\big{(}\|f\|_{\partial\Omega_{m}}+C_{f}\big{)},\qquad
z\in\Omega_{m}.$

Set $\mathcal{N}_{0}:=\mathcal{N}$. Proceeding inductively on $m\geq 1$, we
deduce from (3.15) that the sequence $\\{h_{n,m}\\}_{n\in\mathcal{N}_{m-1}}$
is uniformly bounded above in $\Omega_{m}$ and therefore, by Harnack’s
theorem, see Section A.1, there exists a subsequence
$\mathcal{N}_{m}\subseteq\mathcal{N}_{m-1}$ such that

(3.16) $h_{n,m}(z)\to h_{m}(z)\quad\text{as}\quad\mathcal{N}_{m}\ni
n\to\infty$

locally uniformly in $\Omega_{m}$, where $h_{m}(z)$ is either identically
$-\infty$ or a non-positive harmonic function. Define $\mathcal{N}_{*}$ to be
the diagonal of the table $\\{\mathcal{N}_{m}\\}_{m=1}^{\infty}$; that is, the
$m$-th element of $\mathcal{N}_{*}$ is the $m$-th element of
$\mathcal{N}_{m}$. Then, (3.16) holds along $n\in\mathcal{N}_{*}$ for each
$m$.

Additionally, it follows from the maximum modulus principle for holomorphic
functions that

$|b_{n,m_{1}}^{zero}(z)|>|b_{n,m_{2}}^{zero}(z)|,\quad
z\in\Omega_{m_{1}},\quad m_{1}<m_{2}.$

Therefore, we get from (3.11) and (3.15) that $h_{m_{1}}(z)\leq
h_{m_{2}}(z)\leq 0$ for $z\in\Omega_{m_{1}}$ when $m_{1}<m_{2}$. Thus, if a
finite limit $h_{m_{*}}(z)$ exists for some index $m_{*}$, then it exists for
all $m>m_{*}$. Hence, either the functions $h_{n,m}$ converge to $-\infty$ as
$\mathcal{N}_{*}\ni n\to\infty$ locally uniformly in each $\Omega_{m}$, in
which case $h_{m}\equiv-\infty$ for all $m$, or else the functions $h_{m}$ are
finite and harmonic for all $m$ large enough.

If $h_{m}\equiv-\infty$ for all $m$, select for each $m$ some
$n_{m}\in\mathcal{N}_{*}$ such that $h_{n,m}(z)<-m$ for
$z\in\overline{\Omega}_{m-1}$ and all $n\geq n_{m}$; we may require in
addition that $n_{m}>n_{m-1}$ for $m\geq 1$. Then (3.12) holds with
$\mathcal{N}^{\prime}:=\\{n_{m}\\}_{m=1}^{\infty}$ and
$u^{\prime}\equiv+\infty$.

If on the contrary the functions $h_{m}$ are finite, they form an increasing
sequence on $\Omega_{\ell}$ for $m\geq\ell$ and fixed $\ell$. As they are non-
positive, they converge locally uniformly in ${\mathcal{R}}\setminus E_{f}$ to
a non-positive harmonic function, say $-u^{\prime}$, again by Harnack’s
theorem. Since $E_{f}$ is a closed polar set and $u^{\prime}$ is non-negative,
it follows from the Removability theorem, see Section A.5, that $u^{\prime}$
extends to a superharmonic function on ${\mathcal{R}}$ that we keep denoting
by $u^{\prime}$. Because $u^{\prime}$ is superharmonic on ${\mathcal{R}}$ and
harmonic on ${\mathcal{R}}\setminus E_{f}$, its Laplacian is a negative Radon
measure $-\nu^{\prime}$ supported on $E_{f}$, which is necessarily finite
since $E_{f}$ is compact. Thus, by the Riesz representation theorem, see
Section A.3, equation (3.13) takes place with $h^{\prime}$ the largest
harmonic minorant of $u^{\prime}$.

Given $m$, choose $\tilde{n}_{m}\in\mathcal{N}_{*}$ such that
$|h_{n,m}(z)-h_{m}(z)|\leq 1/m$ for $z\in\overline{\Omega}_{m-1}$ and all
$n\geq\tilde{n}_{m}$. Define
$\mathcal{N}^{\prime}:=\\{\tilde{n}_{m}\\}_{m=1}^{\infty}$, where we again
additionally require that $\tilde{n}_{m}>\tilde{n}_{m-1}$. Given a compact set
$K\subset{\mathcal{R}}\setminus E_{f}$ and $\epsilon>0$, we can pick $m$ large
enough that

$K\subseteq\overline{\Omega}_{m-1},\quad
1/m\leq\epsilon/2,\quad\text{and}\quad|u^{\prime}(z)+h_{m}(z)|\leq\epsilon/2,\quad
z\in K.$

Then, it follows from the last two inequalities that
$|h_{n,m}(z)+u^{\prime}(z)|\leq\epsilon$ for $z\in K$ and any
$\mathcal{N}^{\prime}\ni n\geq\tilde{n}_{m}$. Since $\epsilon$ and $K$ were
arbitrary, this finishes the proof of (3.12). ∎

######  Lemma 3.2.

If $u^{\prime}\not\equiv+\infty$ in Lemma 3.1, then $h^{\prime}$ in (3.13)
continuously extends to $\partial{\mathcal{R}}$ and

(3.17)
$h^{\prime}(z)=\begin{cases}\displaystyle\frac{2}{\mathrm{cap}_{\mathbb{D}}(K_{f})},&z\in\mathcal{T},\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
0,&z\in\partial{\mathcal{R}}\setminus\mathcal{T}.\end{cases}$

###### Proof.

Let $\Omega_{*}:=p^{-1}(\\{z:r<|z|<1\\})$, with $r>0$ close enough to $1$ that
$\overline{\Omega}_{*}\setminus\partial{\mathcal{R}}\subset\Omega_{m}$ for
each $m$. It follows from the proof of Lemma 3.1 that $h_{n,m}(z)\to h_{m}(z)$
as $\mathcal{N}^{\prime}\ni n\to\infty$, locally uniformly in
$\overline{\Omega}_{*}\setminus\partial{\mathcal{R}}$. Given a connected
component $\Omega$ of $\Omega_{*}$, let
$\delta_{z}^{{\mathcal{R}}\setminus\Omega}$ be its harmonic measure, see
Section A.9. Then

(3.18)
$\log|p(z)-(1+\epsilon)p(\xi)|=\int\log|p(\zeta)-(1+\epsilon)p(\xi)|d\delta_{z}^{{\mathcal{R}}\setminus\Omega}(\zeta),\quad
z\in\Omega,$

for any $\xi\in\partial\Omega\cap\partial{\mathcal{R}}$ and $\epsilon>0$, see
(A.28). It then follows from the monotone convergence theorem that we can take
$\epsilon=0$ in (3.18). Recall that $f-N(M_{n})\circ p$ is analytic across
$\partial{\mathcal{R}}\cap\partial\Omega$ and hence is non-vanishing there
except perhaps for finitely many zeros counting multiplicities. Since
$h_{n,m}$ is harmonic in $\Omega$, is continuous on
$\partial\Omega\setminus\partial{\mathcal{R}}$, and is equal to
$\frac{1}{n}\log|f-N(M_{n})\circ p|$ on
$\partial{\mathcal{R}}\cap\partial\Omega$ (i.e., it is continuous on
$\partial\Omega\setminus\partial{\mathcal{R}}$ except perhaps for finitely
many logarithmic singularities), we get from (3.18) and (A.28) that

$h_{n,m}(z)=\int_{\partial\Omega\setminus\partial{\mathcal{R}}}h_{n,m}d\delta_{z}^{{\mathcal{R}}\setminus\Omega}+\int_{\partial{\mathcal{R}}\cap\partial\Omega}\frac{1}{n}\log|f-N(M_{n})\circ
p|d\delta_{z}^{{\mathcal{R}}\setminus\Omega},\quad z\in\Omega.$

By (3.7) the pointwise limit of $\frac{1}{n}\log|f-N(M_{n})\circ p|$ on
$\partial{\mathcal{R}}$ is minus the right-hand side of (3.17) except perhaps
for a finite subset of $\partial\mathcal{R}\setminus\mathcal{T}$, where
$|f-N(M_{n})\circ p|$ may go to zero, contained in

$\big{\\{}\zeta|~{}\exists\,\eta\neq\zeta:~{}p(\zeta)=p(\eta),\;f(\zeta)=f(\eta)\big{\\}}\setminus\mathcal{T}.$

As $\delta_{z}^{{\mathcal{R}}\setminus\Omega}$ does not charge polar, thus
finite sets, the convergence in fact holds almost everywhere with respect to
$\delta_{z}^{{\mathcal{R}}\setminus\Omega}$ for each fixed $z$. So, if we can
justify the second equality in the following relation:

(3.19) $h_{m}(z)=\lim_{\mathcal{N}^{\prime}\ni
n\rightarrow\infty}\left(\int_{\partial\Omega\setminus\partial{\mathcal{R}}}h_{n,m}d\delta_{z}^{{\mathcal{R}}\setminus\Omega}+\int_{\partial{\mathcal{R}}\cap\partial\Omega}\frac{1}{n}\log|f-N(M_{n})\circ
p|d\delta_{z}^{{\mathcal{R}}\setminus\Omega}\right)\\\
=\int_{\partial\Omega\setminus\partial{\mathcal{R}}}h_{m}d\delta_{z}^{{\mathcal{R}}\setminus\Omega}+\int_{\partial{\mathcal{R}}\cap\partial\Omega}\lim_{\mathcal{N}\ni
n\rightarrow\infty}\frac{1}{n}\log|f-N(M_{n})\circ
p|d\delta_{z}^{{\mathcal{R}}\setminus\Omega},$

we shall get that $h_{m}(z)$ solves a Dirichlet problem on $\Omega$ with
boundary data equal to
$h_{m\mathcal{b}\partial\Omega\setminus\partial{\mathcal{R}}}$ on
$\partial\Omega\setminus\partial{\mathcal{R}}$, and to the negative of the
right-hand side of (3.17) on $\partial{\mathcal{R}}\cap\partial\Omega$. As
such, $h_{m}$ must extend continuously to $\partial\Omega$ where it is equal
to the boundary data, since $\partial\Omega$ is non-thin at any of its points.
Subsequently, as $h_{m}$ converges to $-u^{\prime}$ locally uniformly on
${\mathcal{R}}\setminus E_{f}$ (see the proof of Lemma 3.1), passing to the
limit in the leftmost and rightmost sides of (3.19) when $m\to\infty$ yields
that $u^{\prime}$ extends continuously to
$\partial\Omega\cap\partial{\mathcal{R}}$ with values given there by the
right-hand side of (3.17). This will give us the desired conclusion, because
$\nu^{\prime}$ is compactly supported in ${\mathcal{R}}$ and therefore
$g(\nu^{\prime},{\mathcal{R}};\cdot)$ continuously extends by zero to
$\partial{\mathcal{R}}$ since ${\mathcal{R}}$ is regular.

Altogether, it only remains to justify the swapping of the limit and
integration signs in (3.19). On
$\partial\Omega\setminus\partial{\mathcal{R}}$, one can invoke the dominated
convergence theorem. Thus, we only need to consider the integral over
$\partial{\mathcal{R}}\cap\partial\Omega$. According to Vitali’s convergence
theorem, it is enough to show that for every $\epsilon>0$ there exists
$\delta>0$ and $n_{\varepsilon}\in\mathbb{N}$ for which

(3.20) $\int_{E}\left|\frac{1}{n}\log|f-N(M_{n})\circ
p|\right|d\delta_{z}^{{\mathcal{R}}\setminus\Omega}<\epsilon\quad\mathrm{as\
soon\
as}\quad|\delta_{z}^{{\mathcal{R}}\setminus\Omega}(E)|<\delta\quad\mathrm{and}\quad
n\geq n_{\varepsilon}.$

For this, we first deduce from (3.14) that

(3.21) $\int_{\partial{\mathcal{R}}}\frac{1}{n}\log|f-N(M_{n})\circ
p|d\delta_{z}^{{\mathcal{R}}\setminus\Omega}\leq\frac{1}{n}\log\big{(}\|f\|_{\partial{\mathcal{R}}}+C_{f}\big{)}$

as $\delta_{z}^{{\mathcal{R}}\setminus\Omega}$ is a probability measure. Now,
if $\partial{\mathcal{R}}\cap\partial\Omega=\mathcal{T}$, then it follows from
(3.7) that

(3.22) $\int_{E}\frac{1}{n}\log|f-N(M_{n})\circ
p|d\delta_{z}^{{\mathcal{R}}\setminus\Omega}>-c\delta_{z}^{{\mathcal{R}}\setminus\Omega}(E),$

for any $E\subseteq\mathcal{T}$ and some positive constant $c$. If
$\partial{\mathcal{R}}\cap\partial\Omega=\mathcal{T}^{\prime}\neq\mathcal{T}$,
set $d(\eta):=f(\eta)-f(\zeta)$ where $\eta\in\mathcal{T}^{\prime}$,
$\zeta\in\mathcal{T}$, and $p(\eta)=p(\zeta)$. Then, we get from (3.14) that

(3.23) $\log|(f-N(M_{n})\circ
p)(\eta)|\geq\log\big{|}|d(\eta)|-\|\Gamma_{n}\|\big{|}=\log\left|\frac{|d(\eta)|^{2}-\|\Gamma_{n}\|^{2}}{|d(\eta)|+\|\Gamma_{n}\|}\right|\\\
\geq\log\big{|}|d(\eta)|^{2}-\|\Gamma_{n}\|^{2}\big{|}-\log(\|f\|_{\mathcal{T}^{\prime}}+C_{f}).$

Since $E_{f}$, the singular set of $f$, and ${\bf rp}({\mathcal{R}})$, the
ramification set of ${\mathcal{R}}$, are closed and lie on top of
$\mathbb{D}$, $d(\eta)$ extends to a holomorphic function, non-identically
zero in a neighborhood of $\mathcal{T}^{\prime}$. Then,

$\mathcal{D}(\eta):=d(\eta)\overline{d(p^{-1}(1/\overline{p(\eta)})}\quad\text{satisfies}\quad\mathcal{D}(\eta)=|d(\eta)|^{2},\quad\eta\in\mathcal{T}^{\prime},$

and is holomorphic about $\mathcal{T}^{\prime}$. Pick an open set
$W\supset\mathcal{T}^{\prime}$ such that $\overline{W}\cap{\bf
rp}({\mathcal{R}})=\varnothing$ and $\mathcal{D}$ is holomorphic in
$\overline{W}$ with no zero on $\partial W$; then, so is
$\mathcal{D}-\|\Gamma_{n}\|^{2}$ for $n$ large as $\|\Gamma_{n}\|^{2}\to 0$.
Let $\ell$ and $\ell_{n}$ be minimal degree polynomials, normalized by
imposing $\|\ell\|_{\mathbb{T}}=\|\ell_{n}\|_{\mathbb{T}}=1$, such that

$\frac{\mathcal{D}(\eta)}{\ell(p(\eta))}\quad\text{and}\quad\frac{\mathcal{D}(\eta)-\|\Gamma_{n}\|^{2}}{\ell_{n}(p(\eta))}$

are holomorphic and non-vanishing in $W$. Since $\|\Gamma_{n}\|^{2}\to 0$, the
zeros of $\mathcal{D}-\|\Gamma_{n}\|^{2}$ in $W$ tend to those of
$\mathcal{D}$ by Rouché’s theorem, and so our normalization implies that
$\ell_{n}\to\ell$ uniformly in $\overline{W}$ as $n\to\infty$. Hence,
$(\mathcal{D}-\|\Gamma_{n}\|^{2})/(\ell_{n}\circ p)$ converges to
$\mathcal{D}/(\ell_{n}\circ p)$ uniformly in $\overline{W}$, in particular it
is uniformly bounded away from zero there. Consequently, it follows from
(3.23) that

(3.24) $\log|(f-N(M_{n})\circ p)(\eta)|\geq
C+\log|\ell_{n}(p(\eta))|,\quad\eta\in\mathcal{T}^{\prime},$

for some finite constant $C$. Note that $\log|\ell_{n}|\leq 0$ in $\mathbb{D}$
according to our normalization. Let us write
$\ell_{n}(x)=a_{n}\prod_{i}(x-x_{i,n})$ and define the reciprocal polynomial
$\tilde{\ell}_{n}$ of $\ell_{n}$ by

$\tilde{\ell}_{n}(x):=a_{n}\prod_{i}\left\\{\begin{array}[]{ll}x-x_{i,n}&\mathrm{if\
}|x_{i,n}|\geq 1,\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
1-x\bar{x}_{i,n}&\mathrm{if\ }|x_{i,n}|<1.\end{array}\right.$

Clearly, $|\ell_{n}(\xi)|=|\tilde{\ell}_{n}(\xi)|$ for $|\xi|=1$, and the
maximum principle for harmonic functions implies that

(3.25)
$\int_{\mathcal{T}^{\prime}}\log|\tilde{\ell}_{n}(p(\eta))|d\delta_{z}^{{\mathcal{R}}\setminus\Omega}\geq\log\big{|}\tilde{\ell}_{n}(p(z))\big{|},\quad
z\in\Omega,$

since both sides of (3.25) are harmonic in $\Omega$ and have the trace
$\log|\ell_{n}(p(\eta))|$ on $\mathcal{T}^{\prime}$ while the left-hand side
has zero trace on $\partial\Omega\setminus\mathcal{T}^{\prime}$ and the right-
hand side satisfies $\log|\ell_{n}\circ p|\leq 0$ there. Thus, we get from
(3.24) and (3.25) that

(3.26)
$\displaystyle\int_{\mathcal{T}^{\prime}}\frac{1}{n}\log|f-N(M_{n})\circ
p|d\delta_{z}^{{\mathcal{R}}\setminus\Omega}$ $\displaystyle\geq$
$\displaystyle\frac{C}{n}+\frac{1}{n}\int_{\mathcal{T}^{\prime}}\log|\tilde{\ell_{n}}(p(\eta))|d\delta_{z}^{{\mathcal{R}}\setminus\Omega}$
$\displaystyle\geq$
$\displaystyle\frac{C}{n}+\frac{1}{n}\log\big{|}\tilde{\ell}_{n}(p(z))\big{|}.$

As the last term goes to zero uniformly on $\Omega$ and since
$\tilde{\ell}_{n}\to\tilde{\ell}$, where $\tilde{\ell}$ is the reciprocal
polynomial of $\ell$ defined similarly to $\tilde{\ell}_{n}$, the estimate
(3.20) now follows from (3.21), (3.22), and (3.26). ∎

###  3.6. Asymptotic Distributions of Poles and Zeros

Recall the measures $\mu_{n}$ introduced in (3.10). Since these measures have
mass at most $1$, it follows from Helly’s selection theorem [55, Theorem
0.1.3] that there exists a Borel measure $\mu^{\prime}$, supported in
$\overline{\mathbb{D}}$ with mass at most $1$, such that

(3.27)
$\mu_{n}\overset{w*}{\to}\mu^{\prime}\quad\text{as}\quad\mathcal{N}^{\prime}\ni
n\to\infty,$

perhaps at a cost of further restricting $\mathcal{N}^{\prime}$, where
$\overset{w*}{\to}$ stands for weak∗ convergence of finite (signed) measures
(a sequence of Borel measures $\\{\sigma_{n}\\}$ on a locally compact space
$X$ converges weak∗ to a measure $\sigma$ if $\int gd\sigma_{n}\to\int
gd\sigma$ for every continuous function $g$ in $C_{0}(X)$, the completion of
$C_{c}(X)$ in the supremum norm). Observe that
$\mu^{\prime}_{\mathcal{b}\mathbb{D}}=\mu$, where $\mu$ was defined as the
vague limit $\mu_{n}$ in $\mathbb{D}$ along
$\mathcal{N}\supseteq\mathcal{N}^{\prime}$. In particular,
$\mu_{n}\overset{w*}{\to}\mu$ in $\mathbb{D}$.

######  Lemma 3.3.

For any subsequence $\\{n_{m}\\}_{m=1}^{\infty}\subseteq\mathcal{N}^{\prime}$,
it holds that

(3.28)
$\liminf_{m\to\infty}\frac{1}{n_{m}}\log\big{|}b_{n_{m}}^{pole}(z)\big{|}^{-1}\left\\{\begin{array}[]{c}\geq\\\
=\end{array}\right\\}g(\mu,\mathbb{D};z),$

where the inequality holds for every $z\in\mathbb{D}$ and equality holds for
quasi every $z\in\mathbb{D}$.

###### Proof.

Observe that
$\frac{1}{n}\log\big{|}b_{n}^{pole}(z)\big{|}^{-1}=g(\mu_{n},\mathbb{D};z)$,
see (3.9). Since $\mu_{n}\overset{w*}{\to}\mu$ on $\mathbb{D}$, the conclusion
follows from the Principle of Descent and the Lower Envelope Theorem, see
Section A.7. ∎

We cannot immediately get an analog of the previous lemma for the measures
$\nu_{n}=\widetilde{\nu}_{n}/n$, because we do not know if these Radon
measures on ${\mathcal{R}}\setminus E_{f}$ have uniformly bounded masses.
Instead, we shall study the asymptotic behavior of their Green potentials in
the style of Lemmas 3.1 and 3.2.

######  Lemma 3.4.

If the Radon measures $\sigma_{n}$ converge vaguely to $\sigma$, $K\subset E$
is compact, and $\sigma(\partial K)=0$, then the restrictions
$\sigma_{n\mathcal{b}K}$ have uniformly bounded masses,
$\sigma_{n\mathcal{b}K}\overset{w*}{\to}\sigma_{\mathcal{b}K}$ on $K$, and
$\sigma_{n}(\partial K)\to 0$.

###### Proof.

For each $\varepsilon>0$ there is an open set $V\supset\partial K$ such that
$\sigma(V)<\varepsilon$ (by outer regularity of $\sigma$), and an open set $W$
with compact closure satisfying $V\supset\overline{W}\supset W\supset\partial
K$ together with a continuous function $\varphi\geq 0$, supported in $V$,
which is $1$ on $\overline{W}$ (by Urysohn’s lemma). Thus,
$K_{\varepsilon}:=K\setminus W$ is a compact subset of $\mathrm{int}\,K$ (the
interior of $K$) such that, for $n$ large enough that $|\int\varphi
d\sigma_{n}-\int\varphi d\sigma|<\varepsilon$,

$\sigma_{n}(\partial K)\leq\sigma_{n}(K\setminus
K_{\varepsilon})\leq\sigma_{n}(W)\leq\int\varphi
d\sigma_{n}<\varepsilon+\int\varphi d\sigma\leq 2\varepsilon.$

Therefore $\int gd\sigma_{n}\to\int gd\sigma$ as $n\to\infty$ for any bounded
continuous function $g$ on $\mathrm{int}\,K$ [11, Proposition 6.18], and since
$\sigma_{n}(\partial K)\to 0$ while $\sigma(\partial K)=0$ it implies the
weak∗ convergence of $\sigma_{n\mathcal{b}K}$ to $\sigma_{\mathcal{b}K}$. The
uniform boundedness of the masses $\sigma_{n}(K)=\int_{K}1d\sigma_{n}$ now
follows. ∎

From now on we employ standard notation $\mathbb{D}_{r}:=\\{z:|z|<r\\}$ and
$\mathbb{T}_{r}:=\partial\mathbb{D}_{r}$.

######  Lemma 3.5.

There exist a subsequence
$\mathcal{N}^{\prime\prime}\subseteq\mathcal{N}^{\prime}$ and a non-negative
superharmonic function $u^{\prime\prime}(z)$ on ${\mathcal{R}}$ such that

(3.29)
$\liminf_{m\to\infty}\frac{1}{n}\log\left|b_{n_{m},m}^{zero}(z)\right|^{-1}\left\\{\begin{array}[]{c}\geq\\\
=\end{array}\right\\}u^{\prime\prime}(z)$

for any $\\{n_{m}\\}_{m=1}^{\infty}\subseteq\mathcal{N}^{\prime\prime}$, where
the inequality in (3.29) holds everywhere on ${\mathcal{R}}\setminus E_{f}$
while the equality needs only to hold quasi everywhere. When
$u^{\prime\prime}\not\equiv+\infty$, it holds that

(3.30)
$u^{\prime\prime}=g(\nu^{\prime\prime},{\mathcal{R}};\cdot)+h^{\prime\prime}$

for some Radon measure $\nu^{\prime\prime}$ carried by ${\mathcal{R}}$ and
some non-negative function $h^{\prime\prime}$ harmonic on ${\mathcal{R}}$.

###### Proof.

Observe that
$\frac{1}{n}\log\big{|}b_{n,m}^{zero}(z)\big{|}^{-1}=g(\nu_{n},\Omega_{m};z)$
according to (3.9). Below, we distinguish two cases: (i) when the measures
$\nu_{n}$ possess a subsequence which is locally bounded in
${\mathcal{R}}\setminus E_{f}$, i.e., having uniformly bounded masses on each
compact subset of ${\mathcal{R}}\setminus E_{f}$, and (ii) when there exists a
compact set $K\subset{\mathcal{R}}\setminus E_{f}$ such that
$\nu_{n}(K)\to\infty$ as $\mathcal{N}^{\prime}\ni n\to\infty$.

In case (ii) relation (3.29) holds with $u^{\prime\prime}\equiv\infty$ and
$\mathcal{N}^{\prime\prime}=\mathcal{N}^{\prime}$ because $\min_{w\in
K}g_{\Omega_{\ell}}(z,w)>0$ for $z\in\Omega_{\ell}$ and every $\ell$ such that
$K\subset\Omega_{\ell}$, and therefore

$g(\nu_{n_{m}^{\prime}},\Omega_{m};z)\geq
g(\nu_{n^{\prime}_{m}},\Omega_{\ell};z)\geq\nu_{n_{m}^{\prime}}(K)\min_{w\in
K}g_{\Omega_{\ell}}(z,w)\to\infty$

as $m\to\infty$ and
$\\{n_{m}^{\prime}\\}_{m=1}^{\infty}\subseteq\mathcal{N}^{\prime}$, where the
first inequality holds for $m\geq\ell$.

In case (i), the measures $\nu_{n}$ converge vaguely to $\nu^{*}$ in
${\mathcal{R}}\setminus E_{f}$ along $\mathcal{N}^{\prime}$. Let
$\\{r_{l}\\}_{l=1}^{\infty}$ be a positive real sequence increasing to $1$
with $r_{1}$ large enough that
$p(\partial\Omega_{m}\setminus\partial{\mathcal{R}})\subset\mathbb{D}_{r_{1}}$
and $\nu^{*}(p^{-1}(\mathbb{T}_{r_{l}}))=0$ for each $l$. This is possible,
because for $0<a<b<1$ the set $\cup_{a\leq r\leq b}p^{-1}(\mathbb{T}_{r})$ is
compact, so there are at most countably many $r\in[a,1)$ with
$\nu^{*}(p^{-1}(\mathbb{T}_{r}))\neq 0$.

We now argue by double induction over $m$ and $l$: the reasoning below should
be applied inductively in $m\geq 1$, so as to define a sequence of integers
$\mathcal{N}_{m}$ for each $m$. Let $\Omega_{m,l}:=\Omega_{m}\cap
p^{-1}(\mathbb{D}_{r_{l}})$ and proceed inductively in $l\geq 1$, starting
with $\mathcal{N}_{m,0}:=\mathcal{N}_{m-1}$ where
$\mathcal{N}_{0}=\mathcal{N}^{\prime}$. Since
$\nu_{n}(\overline{\Omega}_{m})<\infty$ for each $n$, $m$ by definition of
$\nu_{n}$, we can define

(3.31)
$h_{n,m,l}(z):=g(\nu_{n},\Omega_{m};z)-g\big{(}\nu_{n\mathcal{b}\overline{\Omega}_{m,l}},\Omega_{m};z\big{)},\quad
z\in\Omega_{m,l},$

which is a non-negative harmonic function in $\Omega_{m,l}$. By Harnack’s
theorem, either there is a subsequence
$\mathcal{N}_{m,l}\subseteq\mathcal{N}_{m,l-1}$ of indices $n$ along which
$h_{n,m,l}\to h_{m,l}$, locally uniformly in $\Omega_{m,l}$, for some non-
negative harmonic function $h_{m,l}$, or else $h_{n,m,l}$ tends to infinity
with $n\in\mathcal{N}_{m,l-1}$, locally uniformly in $\Omega_{m,l}$. In the
latter case, we set $\mathcal{N}_{m,l}:=\mathcal{N}_{m,l-1}$ and
$h_{m,l}\equiv+\infty$. Clearly, $h_{n,m,l}\geq h_{n,m,l+1}$ and so, for fixed
$m$, either $h_{m,l}\equiv+\infty$ for all $l$ or the $h_{m,l}$ are finite for
$l$ large enough. Let $\mathcal{N}_{m}$ be the diagonal of the table
$\\{\mathcal{N}_{m,l}\\}_{l=1}^{\infty}$. Since $\mathcal{N}_{m}$ is
eventually a subsequence of every $\mathcal{N}_{m,l}$, it holds that
$h_{n,m,l}\to h_{m,l}$ as $\mathcal{N}_{m}\ni n\to\infty$ for every $l\geq 1$,
locally uniformly in $\Omega_{m,l}$.

In another connection, since $\nu^{*}(\partial\Omega_{m,l})=0$ by construction
and the $\nu_{n\mathcal{b}\overline{\Omega}_{m,l}}$ have uniformly bounded
mass over $n$ by the assumptions of the considered case, we deduce from Lemma
3.4 that

$\nu_{n\mathcal{b}\overline{\Omega}_{m,l}}\overset{w*}{\to}\nu^{*}_{\mathcal{b}\overline{\Omega}_{m,l}}\quad\text{on}\quad\overline{\Omega}_{m,l}\quad\text{as}\quad\mathcal{N}_{0}\ni
n\to\infty.$

Now, $\nu_{n\mathcal{b}\overline{\Omega}_{m,l}}$ defines a measure on
$\Omega_{m}$ in a natural way, and the weak∗ convergence above implies weak∗
convergence on $\Omega_{m}$, because every continuous function with compact
support in $\Omega_{m}$ restricts to a continuous function on
$\Omega_{m}\cap\overline{p^{-1}(\mathbb{D}_{r_{l}})}$ which itself extends to
$\overline{\Omega}_{m,l}=\overline{\Omega}_{m}\cap\overline{p^{-1}(\mathbb{D}_{r_{l}})}$
continuously by zero. As $\Omega_{m}$ is a regular open set with compact
closure on the surface ${\mathcal{R}}_{*}$, the Principle of Descent and the
Lower Envelope Theorem yield for any subsequence
$\mathcal{N}^{*}\subseteq\mathcal{N}_{0}$ that

$\liminf_{\mathcal{N}^{*}\ni
n\to\infty}g\big{(}\nu_{n\mathcal{b}\overline{\Omega}_{m,l}},\Omega_{m};z\big{)}\left\\{\begin{array}[]{c}\geq\\\
=\end{array}\right\\}g\big{(}\nu^{*}_{\mathcal{b}\overline{\Omega}_{m,l}},\Omega_{m};z\big{)},$

where the inequality holds everywhere in $\Omega_{m}$ and the equality may
only hold quasi everywhere.

In view of (3.31), the above inequality and the very definition of
$\mathcal{N}_{m}$ imply that

(3.32) $\liminf_{\mathcal{N}^{*}\ni
n\to\infty}g\big{(}\nu_{n},\Omega_{m};z\big{)}\left\\{\begin{array}[]{c}\geq\\\
=\end{array}\right\\}g\big{(}\nu^{*}_{\mathcal{b}\overline{\Omega}_{m,l}},\Omega_{m};z\big{)}+h_{m,l}(z),\qquad
z\in\Omega_{m,l},$

along any subsequence $\mathcal{N}^{*}\subseteq\mathcal{N}_{m}$, where the
inequality holds everywhere in $\Omega_{m,l}$ while the equality needs only to
hold quasi everywhere. As the left-hand side of (3.32) does not depend on $l$
and the right-hand side is superharmonic, we get from the weak identity
principle that successive right-hand sides are superharmonic continuations of
each other when $l$ increases. Let $u_{m}(z)$ be the superharmonic function in
$\Omega_{m}$ given on each $\Omega_{m,l}$ by the right-hand side of (3.32).
Since a smooth function with compact support in $\Omega_{m}$ is eventually
supported in $\Omega_{m,l}$ for large $l$, we get from the definition that
either $u_{m}\equiv+\infty$ or $\Delta u_{m}=-\nu^{*}$. In the latter case,
the Riesz representation theorem yields that

(3.33) $u_{m}(z)=g(\nu^{*},\Omega_{m};z)+h_{m}(z),\qquad z\in\Omega_{m},$

for some non-negative harmonic function $h_{m}$, which is the largest harmonic
minorant of $u_{m}$.

Let $\widetilde{\mathcal{N}}$ be the diagonal of the table
$\\{\mathcal{N}_{m}\\}_{m=1}^{\infty}$. As $\widetilde{\mathcal{N}}$ is
eventually a subsequence of each $\mathcal{N}_{m}$, we get from (3.32) that
for each $m$ and any subsequence
$\mathcal{N}^{*}\subseteq\widetilde{\mathcal{N}}$ it holds that

(3.34) $\liminf_{\mathcal{N}^{*}\ni
n\to\infty}g(\nu_{n},\Omega_{m};z)\left\\{\begin{array}[]{c}\geq\\\
=\end{array}\right\\}u_{m}(z),$

where the inequality takes place everywhere in $\Omega_{m}$ and equality at
least quasi everywhere. Because the left-hand side of (3.34) increases with
$m$, we have that $u_{m}(z)\leq u_{m+1}(z)$ for quasi every $z\in\Omega_{m}$.
Thus, either $u_{m}\equiv+\infty$ for all $m$ large enough or else $u_{m}$ is
finite quasi everywhere on $\Omega_{m}$ for all $m$. In the latter case, since
$\Delta u_{m}=\Delta u_{m+1\mathcal{b}\Omega_{m}}$
($=-\nu^{*}_{\mathcal{b}\Omega_{m}}$), we get that $u_{m+1}-u_{m}$ is harmonic
on $\Omega_{m}$. Hence, $0\leq u_{m}\leq u_{m+1}$ everywhere on $\Omega_{m}$,
and so $u^{\prime\prime}:=\lim_{m}u_{m}$ is positive and superharmonic on
${\mathcal{R}}\setminus E_{f}$. If $u^{\prime\prime}\equiv+\infty$ we are
done, for we get (3.29) from (3.34) with
$\mathcal{N}^{\prime\prime}=\widetilde{\mathcal{N}}$. Otherwise
$u^{\prime\prime}$ is locally integrable and therefore (3.33), together with
the Riesz representation theorem, imply that

(3.35) $u^{\prime\prime}(z)=g(\nu^{*},{\mathcal{R}}\setminus
E_{f};z)+\tilde{h}(z),\quad z\in{\mathcal{R}}\setminus E_{f},$

where $\tilde{h}(z)$ is a non-negative function that is the largest harmonic
minorant of $u^{\prime\prime}$ on ${\mathcal{R}}\setminus E_{f}$.

As $E_{f}$ is polar and compact in ${\mathcal{R}}$, we deduce from the
Removability theorem and the Riesz representation theorem that

(3.36) $\tilde{h}(z)=h^{\prime\prime}(z)+g(\tilde{\nu},{\mathcal{R}};z),\quad
z\in{\mathcal{R}}\setminus E_{f},$

where $h^{\prime\prime}$ is a non-negative harmonic function on
${\mathcal{R}}$ and $\tilde{\nu}$ a finite positive measure supported on
$E_{f}$. Moreover, since for $z\in{\mathcal{R}}\setminus E_{f}$ the function
$g_{\mathcal{R}}(z,\cdot)-g_{{\mathcal{R}}\setminus E_{f}}(z,\cdot)$ is non-
negative harmonic on ${\mathcal{R}}\setminus E_{f}$ and bounded above near
$E_{f}$, the Removability theorem for harmonic functions yields that
$g_{\mathcal{R}}(z,\cdot)-g_{{\mathcal{R}}\setminus E_{f}}(z,\cdot)\equiv 0$
as it extends to a non-negative harmonic minorant of
$g_{\mathcal{R}}(z,\cdot)$ on ${\mathcal{R}}$. Hence,

(3.37) $g(\nu^{*},{\mathcal{R}}\setminus
E_{f};z)=g(\nu^{*},{\mathcal{R}};z),\quad z\in{\mathcal{R}}\setminus E_{f},$

and equations (3.35)–(3.37) imply that $u^{\prime\prime}(z)$ extends
superharmonically to the entire surface ${\mathcal{R}}$ by

(3.38)
$u^{\prime\prime}(z)=g(\nu^{\prime\prime},{\mathcal{R}};z)+h^{\prime\prime}(z),\quad\nu^{\prime\prime}=\nu^{*}+\tilde{\nu}.$

Now, for any subsequence
$\\{\tilde{n}_{m}\\}_{m=1}^{\infty}\subseteq\widetilde{\mathcal{N}}$, it holds
in view of (3.34) that for each $m_{0}\in\mathbb{N}$ and $z\in\Omega_{m_{0}}$

(3.39)
$\liminf_{m\to\infty}g\big{(}\nu_{\tilde{n}_{m}},\Omega_{m};z\big{)}\geq\liminf_{m\to\infty}g\big{(}\nu_{\tilde{n}_{m}},\Omega_{m_{0}};z\big{)}\geq
u_{m_{0}}(z),$

and we obtain the inequality in (3.29) by letting $m_{0}$ tend to infinity.

Thus, it only remains to prove the equality quasi everywhere in (3.29) when
$u^{\prime\prime}(z)$ is quasi everywhere finite. As before, the argument
should be applied inductively on $m$ with
$\widetilde{\mathcal{N}}_{0}:=\widetilde{\mathcal{N}}$. The functions
$g_{n,l,m}:=g(\nu_{n},\Omega_{l};\cdot)-g(\nu_{n},\Omega_{m};\cdot)$ are non-
negative and harmonic in $\Omega_{m}$ for $l>m$. Therefore, by Harnack’s
theorem, there are subsequences
$\widetilde{\mathcal{N}}_{m,l}\subseteq\widetilde{\mathcal{N}}_{m,l-1}$,
$\widetilde{\mathcal{N}}_{m,0}:=\widetilde{\mathcal{N}}_{m-1}$ such that
$g_{n,l,m}$ converges locally uniformly to some function $H_{l,m}$ harmonic in
$\Omega_{m}$ as $\widetilde{\mathcal{N}}_{m,l}\ni n\to\infty$ (note that
$g_{n,l,m}$ cannot go to $+\infty$ otherwise so would
$g(\nu_{n},\Omega_{l};\cdot)$, and in view of (3.34) $u_{l}(z)$ would be
infinite for quasi every $z$, contradicting that $u_{l}\leq
u^{\prime\prime}<\infty$). Of necessity, $H_{l,m}=u_{l}-u_{m}$ by (3.34), and
a diagonal argument gives us a single subsequence
$\widetilde{\mathcal{N}}_{m}^{*}\subseteq\widetilde{\mathcal{N}}$ along which
the convergence $g_{n,l,m}\to H_{l,m}$ takes place for any $l>0$. Now, for
fixed $m$ and each $l>m$, select
$\tilde{n}_{l}\in\widetilde{\mathcal{N}}_{m}^{*}$ such that

(3.40) $\big{|}g_{n,l,m}(z)-\big{(}u_{l}(z)-u_{m}(z)\big{)}\big{|}\leq
1/l,\quad z\in\overline{\Omega}_{m-1},\quad\tilde{n}_{l}\leq
n\in\widetilde{\mathcal{N}}_{m}^{*}.$

Since the functions $u_{l}-u_{m}$ are harmonic in $\Omega_{m}$ and increase
with $l$, they converge locally uniformly to $u^{\prime\prime}-u_{m}$ there by
Harnack’s theorem and the definition of $u^{\prime\prime}$. Thus, taking
(3.40) into account, for any $\epsilon>0$ there exists $L>0$ such that

(3.41)
$\big{|}g_{n,l,m}(z)-\big{(}u^{\prime\prime}(z)-u_{m}(z)\big{)}\big{|}\leq\epsilon,\quad
z\in\overline{\Omega}_{m-1},\quad l\geq L,\quad\tilde{n}_{l}\leq
n\in\widetilde{\mathcal{N}}_{m}^{*}.$

Define $\widetilde{\mathcal{N}}_{m}:=\\{\tilde{n}_{l}\\}_{l=1}^{\infty}$. Then
it follows from (3.41) and (3.34) that

(3.42)
$\liminf_{l\to\infty}g\big{(}\nu_{n_{l}},\Omega_{l};z\big{)}=\liminf_{l\to\infty}g\big{(}\nu_{n_{l}},\Omega_{m};z\big{)}+\lim_{l\to\infty}g_{n_{l},l,m}(z)=u^{\prime\prime}(z)$

for quasi every $z\in\overline{\Omega}_{m-1}$, whenever
$\\{n_{l}\\}_{l=1}^{\infty}\subseteq\widetilde{\mathcal{N}}_{m}$. Finally, let
$\mathcal{N}^{\prime\prime}$ be the diagonal sequence of the table
$\big{\\{}\widetilde{\mathcal{N}}_{m}\big{\\}}_{m=1}^{\infty}$. Since
$\mathcal{N}^{\prime\prime}$ is eventually a subsequence of every
$\widetilde{\mathcal{N}}_{m}$ it follows from (3.42) that

(3.43)
$\liminf_{m\to\infty}g\big{(}\nu_{n_{m}},\Omega_{m};z\big{)}=u^{\prime\prime}(z),\quad\\{n_{m}\\}_{m=1}^{\infty}\subseteq\widetilde{\mathcal{N}}^{\prime\prime},\quad\text{for
q.e.}\quad z\in{\mathcal{R}}\setminus E_{f},$

which is the equality case in (3.29). ∎

###  3.7. Logarithmic Error Function

Hereafter, we redefine $\mathcal{N}$ to be $\mathcal{N}^{\prime\prime}$
constructed in Lemma 3.5. By Lemmas 3.1, 3.3 and 3.5, the limits (3.7),
(3.12), (3.28), and (3.29) hold along this new sequence.

Since $\mu$ is finite, there is a $G_{\delta}$ polar set
$\widetilde{N}_{0}\subset\mathbb{D}$ such that $g(\mu,\mathbb{D};x)<+\infty$
for $x\in\mathbb{D}\setminus\widetilde{N}_{0}$, see Sections A.3 and A.5. Let
us put $N_{0}:=p^{-1}(\widetilde{N}_{0})$, which is a $G_{\delta}$ polar
subset of ${\mathcal{R}}$, see Section A.5. We now introduce the function
$ler:{\mathcal{R}}\setminus N_{0}\to[-\infty,+\infty)$ (“ler” for “logarithmic
error”), by putting

(3.44)
$ler(z):=g(\mu,\mathbb{D};p(z))-u^{\prime}(z)-u^{\prime\prime}(z),\qquad
z\in{\mathcal{R}}\setminus N_{0},$

where $u^{\prime}$ and $u^{\prime\prime}$ are as in Lemmas 3.1 and 3.5.
Clearly, $ler(z)$ is a $\delta$-subharmonic function (the difference of two
subharmonic functions), and it is well defined for $z\notin N_{0}$ since
$g(\mu,\mathbb{D};p(z))$ is finite there. As introduced, $ler$ depends on the
choice of the subsequence $\mathcal{N}$, but later we shall see that it is in
fact unique.

######  Lemma 3.6.

There exists a polar set $A_{0}\subset{\mathcal{R}}\setminus N_{0}$ such that,
whenever $z_{1},z_{2}$ are distinct points in ${\mathcal{R}}\setminus N_{0}$
with $p(z_{1})=p(z_{2})$ and $ler(z_{i})<0$ for $i=1,2$, then $z_{1},z_{2}\in
A_{0}$.

###### Proof.

By the equality quasi everywhere in (3.28), there exists a polar set
$A_{1}\subset\mathbb{D}$ such that, for every $x\in\mathbb{D}\setminus A_{1}$,
one can find a sequence $\mathcal{N}_{x}\subseteq\mathcal{N}$ along which

$\lim_{\mathcal{N}_{x}\ni
n\rightarrow\infty}\frac{1}{n}\log\big{|}b_{n}^{pole}(x)\big{|}^{-1}=g(\mu,\mathbb{D};x).$

Together with (3.11), (3.12), the inequality in (3.29) and (3.44), this gives
us

(3.45) $\limsup_{\mathcal{N}_{x}\ni
n\rightarrow\infty}\frac{1}{n}\log\left|\left(f-N(M_{n})\circ
p\right)(z)\right|\,\leq\,ler(z)$

for every $z\notin N_{0}$ such that $z\in p^{-1}(x)$ with $x\not\in A_{1}\cup
p(E_{f})$. Assume now that $z_{1},z_{2}\in{\mathcal{R}}\setminus N_{0}$
satisfy $z_{1}\neq z_{2}$ and $p(z_{1})=p(z_{2})$, as well as $ler(z_{i})<0$
for $i=1,2$. Let $x\in\mathbb{D}$ be such that $z_{1},z_{2}\in p^{-1}(x)$. If
$x\not\in A_{1}\cup p(E_{f})$, it follows from (3.45) that

$f(z_{1})=\lim_{\mathcal{N}_{x}\ni
n\to\infty}N(M_{n})(p(z_{1}))=\lim_{\mathcal{N}_{x}\ni
n\to\infty}N(M_{n})(p(z_{2}))=f(z_{2})$

and necessarily $z_{1},z_{2}\in
A_{2}:=\big{\\{}\zeta|~{}\exists\,\eta\neq\zeta:~{}p(\zeta)=p(\eta),\;f(\zeta)=f(\eta)\big{\\}}$.
Now, the conditions placed on the class $\mathcal{F}({\mathcal{R}})$ imply
that the set $A_{2}$ is finite, and therefore the lemma holds with
$A_{0}:=p^{-1}\big{(}A_{1}\cup p(E_{f})\cup p(A_{2})\big{)}$ which is polar,
as inverse image under $p$ of a polar set. ∎

######  Lemma 3.7.

The inequality $ler(z)>-\infty$ holds for quasi every
$z\in{\mathcal{R}}\setminus N_{0}$. In particular
$u^{\prime}\not\equiv+\infty$ and $u^{\prime\prime}\not\equiv+\infty$ in
Lemmas 3.1 and 3.5, so $h^{\prime}$ and $h^{\prime\prime}$ are finite non-
negative harmonic functions on ${\mathcal{R}}$.

###### Proof.

Since $g(\mu;\mathbb{D};p(\cdot))$ is finite on ${\mathcal{R}}\setminus N_{0}$
while $u^{\prime},u^{\prime\prime}$ are non-negative and either identically
$+\infty$ or finite quasi everywhere, $ler$ is in turn either identically
$-\infty$ or finite quasi everywhere on ${\mathcal{R}}\setminus N_{0}$. The
former possibility contradicts Lemma 3.6, therefore the latter prevails so
that $u^{\prime}\not\equiv+\infty$ and $u^{\prime\prime}\not\equiv+\infty$.
Hence, Lemmas 3.1 and 3.5 imply that $h^{\prime}$ and $h^{\prime\prime}$ are
finite on ${\mathcal{R}}$. ∎

Due to the previous lemma, we can rewrite (3.44) as

(3.46)
$ler(z)=g(\mu,\mathbb{D};p(z))-g(\nu,{\mathcal{R}};z)-h_{\mathcal{R}}(z),\quad
z\in{\mathcal{R}}\setminus N_{0},$

where we have set $\nu:=\nu^{\prime}+\nu^{\prime\prime}$, which is a locally
finite measure on ${\mathcal{R}}$ with quasi everywhere finite potential, and
$h_{\mathcal{R}}:=h^{\prime}+h^{\prime\prime}$ which is a positive harmonic
function on ${\mathcal{R}}$.

######  Lemma 3.8.

It holds that $\lim_{z\to\zeta}h_{\mathcal{R}}(z)=0$ for every
$\zeta\in\partial{\mathcal{R}}\setminus\mathcal{T}$.

###### Proof.

Fix $\zeta\in\partial{\mathcal{R}}\setminus\mathcal{T}$, and let
$\xi\in\mathcal{T}$ be such that $p(\zeta)=p(\xi)$. _We claim_ that

(3.47) $\liminf_{z\to\zeta}h_{\mathcal{R}}(z)=0.$

Indeed, if $\liminf_{z\to\zeta}h_{\mathcal{R}}(z)=l>0$, take
$0<2\epsilon:=\min\\{l,2/\mathrm{cap}_{\mathbb{D}}(K_{f})\\}$. Let
$S\subset\mathbb{D}$ be the radial segment
$\\{z:z=rp(\zeta),~{}r\in[1-\delta,1)\\}$ and $S_{\zeta}$ (resp. $S_{\xi}$) be
the connected component of $p^{-1}(S)$ accumulating on $\partial{\mathcal{R}}$
to $\zeta$ (resp. $\xi$). If $\delta>0$ is small enough, then

(3.48) $h_{\mathcal{R}}(z)\geq\epsilon,\quad z\in S_{\zeta}\cup S_{\xi},$

by Lemma 3.2. Furthermore, Lemma 3.6 yields that either $ler(z_{1})\geq 0$ or
$ler(z_{2})\geq 0$ if $p(z_{1})=p(z_{2})\in S\setminus p(A_{0}\cup N_{0})$. In
particular, we get from (3.46) and (3.48) that

(3.49) $g(\mu,\mathbb{D};z)\geq\epsilon,\quad z\in S\setminus p(A_{0}\cup
N_{0}),$

where we notice that $A_{0}\cup N_{0}$ as well as $p(A_{0}\cup N_{0})$ are
polar. This contradicts Lemma A.5, applied with
$g(\sigma,D;\cdot)=g(\mu,\mathbb{D};\cdot)$ and $\xi$ being $p(\zeta)$, since
$R_{\epsilon/2}$ from that lemma would necessarily be a subset of $p(A_{0}\cup
N_{0})$. _This proves our claim_ (3.47).

Next, assume for a contradiction that
$\limsup_{z\to\zeta}h_{\mathcal{R}}(z)=l^{\prime}>0$, and pick
$0<2\epsilon\leq\min\\{l^{\prime},2/\mathrm{cap}_{\mathbb{D}}(K_{f})\\}$ such
that the level line $L_{\epsilon}:=\\{z:h_{{\mathcal{R}}}(z)=\epsilon\\}$ is a
smooth 1-dimensional submanifold of ${\mathcal{R}}$ (this can be achieved
according to Sard’s theorem). Notice that $\zeta$ must be a limit point of
$L_{\epsilon}$ because any neighborhood of $\zeta$ in ${\mathcal{R}}_{*}$
contains a connected open set $U\ni\zeta$ with $U\cap{\mathcal{R}}$ connected
($p$ is a local homeomorphism at $\zeta$ and we may take $p(U)$ to be a disk)
in which $h_{{\mathcal{R}}}$ assumes values arbitrary close to $0$ and
$l^{\prime}$ by definition of $\liminf$ and $\limsup$; hence, as
$h_{{\mathcal{R}}}(U\cap{\mathcal{R}})$ is connected, it contains the value
$\epsilon$.

Let $D_{0}$ be a disk centered at $p(\zeta)$ of small enough radius so that
$D_{\zeta}$ and $D_{\xi}$, the components of $p^{-1}(D_{0})$ in
${\mathcal{R}}_{*}$ that contain respectively $\zeta$ and $\xi$, are in one-
to-one correspondence with $D_{0}$ under $p$. Decreasing the radius of $D_{0}$
if necessary, we can assume that $h_{{\mathcal{R}}}(z)\geq\epsilon/2$ for
$z\in D_{\xi}$ by Lemma 3.2. Let us redefine

(3.50) $S_{\zeta}:=L_{\epsilon}\cap\overline{D}_{\zeta},\quad
S:=p(S_{\zeta}),\quad\text{and}\quad
S_{\xi}:=p^{-1}(S)\cap\overline{D}_{\xi}.$

Observe that $h_{{\mathcal{R}}}$ cannot be constant in view of (3.47) and
(3.17). Therefore, no connected component of $S$ can be a closed curve in
$D_{0}$ by the maximum principle for harmonic functions. In addition, if $S$
has a connected component, say $S_{*}$, accumulating at $z_{*}\in\mathbb{T}$,
then

$g(\mu,\mathbb{D};z)\geq\epsilon/2,\quad z\in S_{*}\setminus p(A_{0}\cup
N_{0}),$

exactly as in (3.49). Let $R_{\epsilon/3}$ be as in Lemma A.5, applied with
$g(\sigma,D;\cdot)=g(\mu,\mathbb{D};\cdot)$ and $\xi$ being $z_{*}$. Then
$S_{*}$ must intersect every circle $\\{z\in\mathbb{D}:|z-z_{*}|=1-r\\}$,
$r\in R_{\epsilon/3}$, by connectedness. Necessarily, the intersection must be
a subset of $p(A_{0}\cup N_{0})$ and therefore polar. Since contractive maps
do not increase the logarithmic capacity [53, Theorem 5.3.1], this means that
$R_{\epsilon/3}$ is polar, which contradicts Lemma A.5 (as we explain in
Section A.4, polar subsets of $\mathbb{D}$ have Greenian and logarithmic outer
capacity zero). Thus, $S$ is a system of smooth curves, each connected
component of which has at least one limit point on $\partial
D_{0}\cap\mathbb{D}$. Consequently, if $T_{0}\subset D_{0}$ is a circle
centered at $p(\zeta)$, then any connected component of $S$ intersecting the
interior of $T_{0}$ must intersect $T_{0}$ as well since it accumulates at a
point of $\partial D_{0}$. That is, $S$ must intersect any such circle and we
arrive at a contradiction exactly as above. ∎

The exceptional set where inequality is strict in Lemmas 3.3 and 3.5 a priori
depends on the subsequence $\\{n_{m}\\}$ under consideration. The next lemma
shows that there exists a polar set outside of which equality holds, both in
(3.28) and (3.29), for one and the same subsequence, see (3.51).

######  Lemma 3.9.

For quasi every $z\in{\mathcal{R}}$, there is a sequence
$\mathcal{N}_{z}=\\{n_{m}^{z}\\}_{m=1}^{\infty}\subseteq\mathcal{N}$ such that

$\left\\{\begin{array}[]{lcl}\displaystyle\lim_{m\to\infty}g(\nu_{n_{m}^{z}},\Omega_{m};z)&=&u^{\prime\prime}(z),\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\displaystyle\lim_{m\to\infty}g(\mu_{n_{m}^{z}},\mathbb{D};p(z))&=&g(\mu,\mathbb{D};p(z)).\end{array}\right.$

###### Proof.

Our goal is to show that there exists a subsequence
$\\{n_{m}^{*}\\}\subseteq\mathcal{N}$ such that

(3.51)
$\liminf_{m\to\infty}\big{(}g(\nu_{n_{m}^{*}},\Omega_{m};z)+g(\mu_{n_{m}^{*}},\mathbb{D};p(z))\big{)}=g(\mu,\mathbb{D};p(z))+u^{\prime\prime}(z)$

for quasi every $z\in{\mathcal{R}}$. Since the inequalities in (3.28) and
(3.29) hold for every $z\in{\mathcal{R}}$, this will indeed imply the claim of
the lemma.

To prove (3.51), we shall rewrite the sum of two potentials in the left-hand
side as a single potential on ${\mathcal{R}}$. To this end, we lift $\mu_{n}$
and $\mu$ to ${\mathcal{R}}$ via the construction described in (A.33).
Specifically, with the notation introduced there, it follows from (A.34) that

(3.52)
$g({\widehat{\mu}}_{n},\mathcal{R};z)=g(\mu_{n},\mathbb{D};p(z))\quad\text{and}\quad
g({\widehat{\mu}},{\mathcal{R}};z)=g(\mu,\mathbb{D};p(z)),\quad
z\in{\mathcal{R}}.$

Now, we can write $g(\nu_{n},\Omega_{m};z)+g(\mu_{n},\mathbb{D};p(z))$ as a
sum of three terms:

(3.53)
$g(\nu_{n}+{\widehat{\mu}}_{n},\Omega_{m};z)+\big{(}g({\widehat{\mu}}_{n\mathcal{b}\Omega_{m}},{\mathcal{R}},z)-g({\widehat{\mu}}_{n},\Omega_{m};z)\big{)}+g({\widehat{\mu}}_{n\mathcal{b}{\mathcal{R}}\setminus\Omega_{m}},{\mathcal{R}};z),$

and we shall study their behavior separately.

To start, recall that $\mu=\mu_{\mathcal{b}\mathbb{D}}^{\prime}$, where
$\mu^{\prime}$ is the weak∗ limit of $\mu_{n}$ along $\mathcal{N}$ in
$\overline{\mathbb{D}}$. Thus, by the discussion after (A.34), an analogous
relation holds between ${\widehat{\mu}}$ and ${\widehat{\mu}}_{n}$. Namely,
since $\mu_{n}$ has total mass $1$, the total mass of ${\widehat{\mu}}_{n}$ is
equal to $M$, the number of sheets of $\overline{\mathcal{R}}$. In particular,
the sequence ${\widehat{\mu}}_{n}$ converges weak∗ on $\overline{\mathcal{R}}$
to ${\widehat{\mu}}^{\prime}$, and on $\mathcal{R}$ to
${\widehat{\mu}}^{\prime}_{\mathcal{b}{\mathcal{R}}}={\widehat{\mu}}$.

Since the sets $\Omega_{m}$ exhaust ${\mathcal{R}}\setminus E_{f}$, it holds
that
${\widehat{\mu}}({\mathcal{R}}\setminus\Omega_{m})-{\widehat{\mu}}(E_{f})\to
0$ as $m\to\infty$. Moreover, as each ${\mathcal{R}}\setminus\Omega_{m}$ is
compact with boundary of ${\widehat{\mu}}$-measure zero, it follows from Lemma
3.4 that the measures
${\widehat{\mu}}_{n\mathcal{b}{\mathcal{R}}\setminus\Omega_{m}}$ converge
weak∗ to ${\widehat{\mu}}_{\mathcal{b}{\mathcal{R}}\setminus\Omega_{m}}$ along
$\mathcal{N}$. In particular,
${\widehat{\mu}}_{n}({\mathcal{R}}\setminus\Omega_{m})-{\widehat{\mu}}({\mathcal{R}}\setminus\Omega_{m})\to
0$ as $n\to\infty$. Hence, for each $m$ there exists
$n_{m}^{\prime}\in\mathcal{N}$ such that

$|{\widehat{\mu}}_{n}(\mathcal{R}\setminus\Omega_{m})-{\widehat{\mu}}(E_{f})|\leq
2|{\widehat{\mu}}({\mathcal{R}}\setminus\Omega_{m})-{\widehat{\mu}}(E_{f})|\quad\text{as
soon as}\quad n\geq n_{m}^{\prime}.$

Let $\sigma$ be a weak∗ limit point on ${\mathcal{R}}$ of the family
$\\{{\widehat{\mu}}_{n_{m}^{\prime}\mathcal{b}{\mathcal{R}}\setminus\Omega_{m}}\\}_{m\in\mathbb{N}}$.
Clearly $\mathrm{supp}(\sigma)\subseteq E_{f}$ and $\sigma$ is also a weak∗
limit point of this family on the fixed compact set
${\mathcal{R}}\setminus\Omega_{1}$. Thus, integrating against any function
which is identically $1$ on ${\mathcal{R}}\setminus\Omega_{1}$ and passing to
the limit gives us $\sigma(E_{f})={\widehat{\mu}}(E_{f})$ by our very choice
of $n^{\prime}_{m}$. In another connection, if $\varphi:{\mathcal{R}}\to[0,1]$
is a continuous function with compact support which is $1$ on a compact set
$K\subseteq E_{f}$, then

$\sigma(K)\leq\int\varphi d\sigma\leq\limsup_{m}\int\varphi
d{\widehat{\mu}}_{n_{m}^{\prime}\mathcal{b}{\mathcal{R}}\setminus\Omega_{m}}\leq\lim_{m}\int\varphi
d{\widehat{\mu}}_{n_{m}^{\prime}}=\int\varphi d{\widehat{\mu}}.$

As the infimum over $\varphi$ of the rightmost term above is
${\widehat{\mu}}(K)$ by Urysohn’s lemma and the outer regularity of Radon
measures, we get that $\sigma\leq{\widehat{\mu}}_{\mathcal{b}E_{f}}$ by the
inner regularity of Radon measures on ${\mathcal{R}}$, see [54, Theorem 2.18].
Altogether, we deduce that $\sigma={\widehat{\mu}}_{\mathcal{b}E_{f}}$, and
consequently the measures
${\widehat{\mu}}_{n_{m}^{\prime}\mathcal{b}{\mathcal{R}}\setminus\Omega_{m}}$
converge weak∗ to ${\widehat{\mu}}_{\mathcal{b}E_{f}}$ along
$\mathcal{N}^{\prime}:=\\{n_{m}^{\prime}\\}$. In particular,

(3.54)
$\lim_{m\to\infty}g({\widehat{\mu}}_{n_{m}^{\prime}\mathcal{b}{\mathcal{R}}\setminus\Omega_{m}},{\mathcal{R}};z)=g({\widehat{\mu}}_{\mathcal{b}E_{f}},{\mathcal{R}};z),\quad
z\in{\mathcal{R}}\setminus E_{f},$

which settles the asymptotic behavior of the last term in (3.53) along the
sequence of indices $\\{n_{m}^{\prime}\\}_{m}$. Notice that by the definition
of $n^{\prime}_{m}$, if $\mathcal{N}^{\prime}$ is replaced by an arbitrary
subsequence $\\{n^{\prime\prime}_{m}\\}_{m}$ thereof, then (3.54) continues to
hold along this new subsequence of indices. This will be used in the
forthcoming steps.

Next, let $f_{m}(z,w):=g_{\mathcal{R}}(z,w)-g_{\Omega_{m}}(z,w)$ for
$z,w\in\Omega_{m}$. Clearly, $f_{m}(z,\cdot)$ is harmonic in $\Omega_{m}$ and
continuous on $\overline{\Omega}_{m}$, by the regularity of $\Omega_{m}$. Its
boundary values are equal to $g_{\mathcal{R}}(z,\cdot)$ on
$\partial\Omega_{m}$, in particular they are identically zero on
$\partial{\mathcal{R}}$. Fix $z\in{\mathcal{R}}\setminus E_{f}$, and let $k$
be an integer such that $z\in\Omega_{k}$. Then, we get for all $m>k$ that

(3.55) $0\leq
f_{m}(z,w)\leq\max_{w\in\partial\Omega_{m}\setminus\partial{\mathcal{R}}}g_{\mathcal{R}}(z,w)\leq\max_{w\in\partial\Omega_{k+1}}g_{\mathcal{R}}(z,w)=:C_{k},\quad
w\in\overline{\Omega}_{m},$

where the constant $C_{k}$ is finite, independent of $m$, and we used the
maximum principle for harmonic functions twice (once for $f_{m}(z,\cdot)$ on
$\Omega_{m}$ and once for $g_{\mathcal{R}}(z,\cdot)$ on
${\mathcal{R}}\setminus\Omega_{k+1}$). Observe further that the functions
$f_{m}(z,\cdot)$, $m>k$, are not only positive harmonic in each $\Omega_{l}$
for fixed $l$ satisfying $k<l\leq m$, but form a decreasing sequence there.
Therefore they converge in $\Omega_{l}$ to a non-negative harmonic function,
say $f^{\\{l\\}}(z,\cdot)$, by Harnack’s theorem. As this claim is true for
all large $l$, the $f^{\\{l\\}}(z,\cdot)$ inductively define a harmonic
function $f(z,\cdot)$ on ${\mathcal{R}}\setminus E_{f}$ that satisfies $0\leq
f(z,\cdot)\leq C_{k}$ when $z\in\Omega_{k}$. Thus, it extends harmonically to
the entire surface ${\mathcal{R}}$ by the Removability theorem for harmonic
functions, and as its trace on $\partial{\mathcal{R}}$ is zero we conclude
that $f(z,\cdot)\equiv 0$.

Observe now that $f_{m}(z,w)=f_{m}(w,z)$ for $m$ large enough so that
$z,w\in\Omega_{m}$. Thus, it is jointly harmonic in both variables [35, p.
561]. Hence, by Harnack’s theorem and the diagonal argument, any subsequence
of $\\{f_{m}(\cdot,\cdot)\\}$ has a further subsequence converging locally
uniformly in ${\mathcal{R}}\setminus E_{f}\times{\mathcal{R}}\setminus E_{f}$,
and we know from what precedes that the limit function can only be zero. In
particular, it holds that

(3.56) $\eta_{l,m}:=\max_{z,w\in\overline{\Omega}_{l}}f_{m}(z,w)\to
0\quad\text{as}\quad m\to\infty,$

where we used that each $f_{m}(\cdot,\cdot)$ extends continuously by zero to
$\partial{\mathcal{R}}\times\partial{\mathcal{R}}$, and therefore the maximum
principle for harmonic functions can be applied to show that the convergence
is indeed uniform on $\overline{\Omega}_{l}$. Given $m$, define

$l_{m}:=\max\big{\\{}l\,:\,1\leq l\leq m,\,\eta_{l,m}\leq 1/l\big{\\}}.$

It follows from (3.56) that $l_{m}\to\infty$ and $\eta_{l_{m},m}\to 0$ when
$m\to\infty$, whence for each fixed $z\in\Omega_{k}$ and all $m>k$ it holds
that

$\displaystyle 0\leq\int_{\Omega_{m}}f_{m}(z,w)d{\widehat{\mu}}_{n}(w)$
$\displaystyle=$
$\displaystyle\left(\int_{\Omega_{m}\setminus\Omega_{l_{m}}}+\int_{\Omega_{l_{m}}}\right)f_{m}(z,w)d{\widehat{\mu}}_{n}(w)$
(3.57) $\displaystyle\leq$ $\displaystyle
C_{k}{\widehat{\mu}}_{n}(\overline{\Omega}_{m}\setminus\Omega_{l_{m}})+M\eta_{l_{m},m},$

where we used (3.55) while recalling that $M$ is the number of sheets of
${\mathcal{R}}$, which is a bound for the total mass of each
${\widehat{\mu}}_{n}$. Now, we obtain from Lemma 3.4 and our choice of
$\Omega_{m}$ that the measures ${\widehat{\mu}}_{n}$ converge weak∗ to
${\widehat{\mu}}$ along $\mathcal{N}$, not only on ${\mathcal{R}}$, but also
on every compact set of the form
$\overline{\Omega}_{m}\setminus\Omega_{l_{m}}$. Hence, taking into account
that ${\widehat{\mu}}\big{(}({\mathcal{R}}\setminus
E_{f})\setminus\Omega_{l_{m}}\big{)}\to 0$ as $m\to\infty$, we can associate
to each $m$ an integer $n_{m}^{\prime\prime}\in\mathcal{N}^{\prime}$ such that
$n_{m}^{\prime\prime}\geq n_{m}^{\prime}$ and

${\widehat{\mu}}_{n_{m}^{\prime\prime}}\big{(}\overline{\Omega}_{m}\setminus\Omega_{l_{m}}\big{)}\to
0\quad\text{as}\quad m\to\infty.$

Clearly, the choice of
$\mathcal{N}^{\prime\prime}:=\\{n_{m}^{\prime\prime}\\}$ is independent of
$z$. Thus, we get from (3.57) together with the choices of $\\{l_{m}\\}$ and
$\mathcal{N}^{\prime\prime}$ that for every $z\in{\mathcal{R}}\setminus E_{f}$
it holds

(3.58)
$\lim_{m\to\infty}g({\widehat{\mu}}_{n_{m}^{\prime\prime}\mathcal{b}\Omega_{m}},{\mathcal{R}};z)-g({\widehat{\mu}}_{n_{m}^{\prime\prime}},\Omega_{m};z)=\lim_{m\to\infty}\int_{\Omega_{m}}f_{m}(z,w)d{\widehat{\mu}}_{n_{m}^{\prime\prime}}(w)=0,$

which settles the asymptotic behavior of the middle term in (3.53) along
$\mathcal{N}^{\prime\prime}$.

Lastly, to describe asymptotics of the first term in (3.53), we need to repeat
some steps of the proof of Lemma 3.5 with $\nu_{n}$ replaced by
$\nu_{n}+{\widehat{\mu}}_{n}$. Recall the definition of the sets
$\Omega_{m,l}$ given just before (3.31). We may adjust it so that
$(\nu^{*}+{\widehat{\mu}})(\partial\Omega_{m,l})=0$. Indeed,
$\partial\Omega_{m,l}=(\partial\Omega_{m}\setminus\partial{\mathcal{R}})\cup
p^{-1}(\mathbb{T}_{r_{l}})$. We already know that
$(\nu^{*}+{\widehat{\mu}})(\partial\Omega_{m}\setminus\partial{\mathcal{R}})=0$,
so we only need to ensure that
$(\nu^{*}+{\widehat{\mu}})(p^{-1}(\mathbb{T}_{r_{l}}))=0$ for each $l$. This
can be achieved as before since ${\widehat{\mu}}$ is finite and therefore
$\nu^{*}+{\widehat{\mu}}$ is still a Radon measure. Since
${\widehat{\mu}}^{\prime}(\partial\Omega_{m,l})={\widehat{\mu}}(\partial\Omega_{m,l})=0$
by construction, it follows from Lemma 3.4 that

${\widehat{\mu}}_{n\mathcal{b}\overline{\Omega}_{m}\setminus\Omega_{m,l}}\overset{w^{*}}{\to}{\widehat{\mu}}^{\prime}_{\mathcal{b}\overline{\Omega}_{m}\setminus\Omega_{m,l}}\quad\text{and}\quad{\widehat{\mu}}_{n}(p^{-1}(\mathbb{T}_{r_{l}}))\to
0$

for all $m,l$. So, when $z\in\Omega_{m,l}$, we get that

(3.59) $\lim_{\mathcal{N}^{\prime\prime}\ni
n\to\infty}g\big{(}{\widehat{\mu}}_{n\mathcal{b}\Omega_{m}\setminus\overline{\Omega}_{m,l}},\Omega_{m};z\big{)}=\lim_{\mathcal{N}^{\prime\prime}\ni
n\to\infty}g\big{(}{\widehat{\mu}}_{n\mathcal{b}\overline{\Omega}_{m}\setminus\Omega_{m,l}},\Omega_{m};z\big{)}\\\
=g\big{(}{\widehat{\mu}}^{\prime}_{\mathcal{b}\overline{\Omega}_{m}\setminus\Omega_{m,l}},\Omega_{m};z\big{)}=g\big{(}{\widehat{\mu}}_{\mathcal{b}\Omega_{m}\setminus\overline{\Omega}_{m,l}},\Omega_{m};z\big{)}$

since $g_{\Omega_{m}}(\cdot,z)$ is continuous on
$\overline{\Omega}_{m}\setminus\Omega_{m,l}$ and vanishes on
$\partial{\mathcal{R}}$ by regularity of $\Omega_{m}$. Moreover, the
convergence is locally uniform in $z\in\Omega_{m,l}$ by the continuity of
Green functions with respect to both variables off the diagonal. From (3.59)
and the reasoning used after (3.31), it follows that

$\lim_{\mathcal{N}^{\prime\prime}\ni
n\to\infty}g\big{(}(\nu_{n}+{\widehat{\mu}}_{n})_{\mathcal{b}\Omega_{m}\setminus\overline{\Omega}_{m,l}},\Omega_{m};z\big{)}=h_{m,l}(z)+g\big{(}{\widehat{\mu}}_{\mathcal{b}\Omega_{m}\setminus\overline{\Omega}_{m,l}},\Omega_{m};z\big{)},$

locally uniformly in $\Omega_{m,l}$ for each $l$ and some subsequence
$\mathcal{N}^{\prime\prime}_{m}$ of $\mathcal{N}^{\prime\prime}$. Arguing as
we did to obtain (3.32), but applying this time the Lower Envelope theorem to
$(\nu_{n}+{\widehat{\mu}}_{n})_{\mathcal{b}\overline{\Omega}_{m,l}}$, we get
in view of the above limit that

$\displaystyle\liminf_{\widetilde{\mathcal{N}}^{\prime\prime}\ni
n\to\infty}g(\nu_{n}+{\widehat{\mu}}_{n},\Omega_{m};z)$
$\displaystyle=g((\nu^{*}+{\widehat{\mu}})_{\mathcal{b}\overline{\Omega}_{m,l}},\Omega_{m};z)+h_{m,l}(z)+g\big{(}{\widehat{\mu}}_{\mathcal{b}\Omega_{m}\setminus\overline{\Omega}_{m,l}},\Omega_{m};z\big{)}$
$\displaystyle=g(\nu^{*}+{\widehat{\mu}},\Omega_{m};z)+h_{m}(z)$

for quasi every $z\in\Omega_{m}$, where
$\widetilde{\mathcal{N}}^{\prime\prime}$ is the diagonal of the table
$\\{{\mathcal{N}}_{m}^{\prime\prime}\\}_{m=1}^{\infty}$ and to get the second
equality we used (3.33) along with the explanation preceding it on the
inductive definition of $u_{m}$ by the right-hand side of (3.32). The previous
equation stands analog to (3.34), and continues to hold if
$\widetilde{\mathcal{N}}^{\prime\prime}$ is replaced by any subsequence
thereof. Now, the last part of the proof in Lemma 3.5 was predicated on the
limit

$u_{m}(z)=g(\nu^{*},\Omega_{m};z)+h_{m}(z)\to
u^{\prime\prime}(z)\quad\mathrm{as}\quad m\to\infty,\quad
z\in{\mathcal{R}}\setminus E_{f},$

where $u_{m}$ were initially defined inductively by the right-hand side of
(3.32) and their limit $u^{\prime\prime}$ assumed the form (3.35). In the
present case, $u_{m}$ is replaced by
$u_{m}+g({\widehat{\mu}},\Omega_{m};\cdot)$ and the monotone convergence
theorem together with the polar character of $E_{f}$ imply that

$\lim_{m}g({\widehat{\mu}},\Omega_{m};z)=g({\widehat{\mu}},{\mathcal{R}}\setminus
E_{f};z)=g({\widehat{\mu}}_{\mathcal{b}{\mathcal{R}}\setminus
E_{f}},{\mathcal{R}};z),\quad z\in{\mathcal{R}}\setminus E_{f}.$

Hence, arguing as we did after (3.39), we obtain similarly to (3.43) that

(3.60)
$\liminf_{m\to\infty}g(\nu_{n_{m}^{*}}+{\widehat{\mu}}_{n_{m}^{*}},\Omega_{m};z)=u^{\prime\prime}(z)+g({\widehat{\mu}}_{\mathcal{b}{\mathcal{R}}\setminus
E_{f}},{\mathcal{R}};z),$

for quasi every $z\in{\mathcal{R}}\setminus E_{f}$ and some subsequence
$\\{n_{m}^{*}\\}\subseteq\widetilde{\mathcal{N}}^{\prime\prime}$. The desired
limit (3.51) now follows from (3.52), (3.53), (3.54), (3.58), and (3.60). ∎

Hereafter we shall deal with the _fine topology_ , which is the coarsest
topology for which superharmonic and therefore $\delta$-subharmonic functions
are continuous. In this connection, the reader may want to consult the
definitions and properties collected in Section A.5.

Recall the definition of $N_{0}\subset{\mathcal{R}}$ before (3.44) as being
the $+\infty$-set of $g(\mu,\mathbb{D};p(\cdot))$, i.e.,
$N_{0}=p^{-1}(\widetilde{N}_{0})$ where $\widetilde{N}_{0}\subset\mathbb{D}$
is the $+\infty$-set of $g(\mu,\mathbb{D};\cdot)$. Clearly, $N_{0}$ is a
finely closed and polar $G_{\delta}$-set. Let us define

(3.61) $G_{\pm}:=\big{\\{}z\in{\mathcal{R}}\setminus N_{0}:\pm
ler(z)>0\big{\\}}.$

It is easily seen from (3.44) that $ler:\mathcal{R}\setminus
N_{0}\to[-\infty,+\infty)$ is finely continuous and that $G_{-}$ and $G_{+}$
are finely open in $\mathcal{R}\setminus N_{0}$. Since $N_{0}$ is polar, if
the complement of either $G_{+}$ or $G_{-}$ in ${\mathcal{R}}\setminus N_{0}$
is thin at a point $z$, then the respective complement in ${\mathcal{R}}$ is
also thin at $z$. Hence, $G_{+}$ and $G_{-}$ are in fact finely open in
${\mathcal{R}}$. Hereafter, we put $D_{\pm}:=p(G_{\pm})$.

######  Lemma 3.10.

For $z\in G_{+}$ and any $\zeta\in{\mathcal{R}}$ with $p(\zeta)=p(z)$, it
holds that $\zeta\in G_{+}$ and $ler(z)=ler(\zeta)$. In particular
$G_{+}=p^{-1}(D_{+})$. Moreover, $D_{+}$ is finely open.

###### Proof.

By the definition of $ler$ given in (3.46), we get from equation (3.11)
together with Lemmas 3.1, 3.7, and 3.9 (recall $\mathcal{N}^{\prime\prime}$
was renamed as $\mathcal{N}$ at the top of Section 3.7) that there exists a
polar set $B\subset{\mathcal{R}}$ with the following property: for each $z\in
G_{+}\setminus B$ there is a subsequence $\mathcal{N}_{z}\subset\mathcal{N}$
such that

(3.62) $\left\\{\begin{array}[]{lcl}\displaystyle\lim_{\mathcal{N}_{z}\ni
n\rightarrow\infty}\frac{1}{n}\log\left|\left(f-N(M_{n})\circ
p\right)(z)\right|&=&ler(z),\\\ \displaystyle\lim_{\mathcal{N}_{z}\ni
n\rightarrow\infty}g(\mu_{n},\mathbb{D};p(z))&=&g(\mu,\mathbb{D};p(z)),\end{array}\right.$

where we note that $g(\mu,\mathbb{D};p(z))<+\infty$ since $z\notin N_{0}$.
Without loss of generality, we may assume that $B$ contains $p^{-1}(p(E_{f}))$
and $A_{0}$, since both sets are polar, see Lemma 3.6 for the definition of
$A_{0}$. As $ler(z)>0$ and $f(z)$ is finite on $G_{+}\setminus B$, the above
limit implies that

$\lim_{\mathcal{N}_{z}\ni
n\rightarrow\infty}\frac{1}{n}\log\big{|}N(M_{n})(p(z))\big{|}=ler(z).$

Now, if $\zeta\in{\mathcal{R}}\setminus B$ is such that $p(\zeta)=p(z)$,
obviously $\zeta\notin N_{0}$ and in addition $\zeta\notin E_{f}$, by
definition of $B$. Thus, on account of the finiteness of $f(\zeta)$, we get
that

(3.63) $\lim_{\mathcal{N}_{z}\ni
n\rightarrow\infty}\frac{1}{n}\log\big{|}\left(f-N(M_{n})\circ
p\right)(\zeta)\big{|}=\lim_{\mathcal{N}_{z}\ni
n\rightarrow\infty}\frac{1}{n}\log\big{|}N(M_{n})(p(z))\big{|}=ler(z).$

On the other hand, from the second equation in (3.62), we deduce as in (3.45)
that the first limit in (3.63) is at most $ler(\zeta)$, whence $0<ler(z)\leq
ler(\zeta)$. In particular, $\zeta\in G_{+}$, and reversing the roles of $z$
and $\zeta$ gives $ler(z)=ler(\zeta)$. This proves the first assertion of the
lemma when $z\in G_{+}\setminus B$.

To prove it on all of $G_{+}$, pick $z,\zeta\in{\mathcal{R}}$ such that
$p(z)=p(\zeta)$ and let $D_{0}\subset\mathbb{D}$ be a disk centered at $p(z)$.
Denote by $D_{z}$, $D_{\zeta}$ the connected components of $p^{-1}(D_{0})$
that contain $z$, $\zeta$ respectively, and make $D_{0}$ small enough so that