# Improved fractal Weyl bounds for convex cocompact hyperbolic surfaces and
large resonance-free regions

Louis Soares<EMAIL_ADDRESS>
###### Abstract.

Let $X$ be a convex cocompact hyperbolic surface and let $\delta$ be the
dimension of its limit set. Let $N_{X}(\sigma,T)$ be the number of resonances
for $X$ inside the box $[\sigma,\delta]+i[0,T]$. We prove that for all
$\sigma>\delta/2$ we have

$N_{X}(\sigma,T)\ll_{\epsilon}T^{1+\delta-\frac{3\delta+2}{2\delta+2}(2\sigma-\delta)+\epsilon}.$

This improves upon previous “improved” fractal Weyl bounds of Naud [13] and
Dyatlov [4]. Moreover, this bound implies that there are resonance-free
rectangular boxes of arbitrary height in the strip

$\left\\{\delta-\frac{\delta^{2}}{6\delta+4}<\mathrm{Re}(s)<\delta\right\\}.$

We combine the approach of Naud [13] with the refined transfer operator
machinery introduced by Dyatlov–Zworski [5], and we use a new bound for
certain oscillatory integrals that arise naturally in our analysis.

###### Key words and phrases:

Resonances, hyperbolic surfaces, transfer operators

###### 2020 Mathematics Subject Classification:

Primary: 58J50, 81U24, Secondary: 11M36, 37C30

##  1\. Introduction and Statement of Results

In mathematical physics, resonances are the main spectral data in situations
where eigenvalues are missing, which is often the case when the underlying
geometry is not finite. In the last few decades, a great deal of research has
been devoted to the mathematical study of resonances. We refer to [20] for a
broad introduction to this subject.

In this paper, we are interested in “convex cocompact” hyperbolic surfaces,
that is, infinite-area surfaces of constant negative curvature $-1$ that can
be decomposed into a compact surface $N$ with geodesic boundary111The surface
$N$ is sometimes called the “Nielsen region” and a finite number of funnel
ends glued to $N$. Equivalently, a hyperbolic surface is said to be convex
cocompact if it is geometrically finite and has no cusps.

The resonances of a hyperbolic surface $X$ are defined as the poles of the
meromorphic continuation of the resolvent

(1)
$R_{X}(s)\stackrel{{\scriptstyle\text{def}}}{{=}}(\Delta_{X}-s(1-s))^{-1}\colon
C_{c}^{\infty}(X)\to C^{\infty}(X),$

where $\Delta_{X}$ is the positive Laplacian on $X$. A proof of the
meromorphic continuability of (1) to $s\in\mathbb{C}$ was given by [9]. We
also refer the reader to Borthwick’s book [1] which a good reference for the
spectral theory of infinite-area hyperbolic surfaces, a subject that has not
yet been fully explored.

Every hyperbolic surface $X$ can be realized as a quotient
$\Gamma\backslash\mathbb{H}^{2}$ where $\mathbb{H}^{2}$ is the hyperbolic
plane and $\Gamma\subset\mathrm{PSL}_{2}(\mathbb{R})$ is a Fuchsian group (a
discrete group of Möbius transformations). The limit set of $X$, denoted by
$\Lambda$, is defined as the set of accumulation points of all orbits of the
action of $\Gamma$ on $\mathbb{H}^{2}$. The limit set is a Cantor-like fractal
subset of $\partial\mathbb{H}^{2}\cong\mathbb{R}\cup\\{\infty\\}$ whose
Hausdorff dimension we denote by $\delta.$

By a result of Patterson [15], $X$ has a simple resonance at $s=\delta$ and
all the remaining resonances (of which there are infinitely many by [9]) lie
in half-plane $\mathrm{Re}(s)<\delta$. The multiplicity of a resonance $s_{0}$
is defined as the rank of the residual operator (1) at $s=s_{0}.$ From now on,
we assume that $X$ is convex cocompact and we denote by $\mathcal{R}_{X}$ the
set of resonances for $X$. We are interested in the resonance counting
functions

$M_{X}(\sigma,T)\stackrel{{\scriptstyle\text{def}}}{{=}}\\#\\{s\in\mathcal{R}_{X}:\text{$\mathrm{Re}(s)\geqslant\sigma$
and $|\mathrm{Im}(s)-1|\leqslant T$}\\}$

and

$N_{X}(\sigma,T)\stackrel{{\scriptstyle\text{def}}}{{=}}\\#\\{s\in\mathcal{R}_{X}:\text{$\mathrm{Re}(s)\geqslant\sigma$
and $0\leqslant\mathrm{Im}(s)\leqslant T$}\\},$

where resonances are counted with multiplicities. These functions have been
studied by a number of different authors. Using transfer operator techniques,
Guillopé–Lin–Zworski [8] proved that for all $\sigma<\delta$ we have, as
$T\to\infty$,

(2) $M_{X}(\sigma,T)\ll T^{\delta}$ and $N_{X}(\sigma,T)\ll T^{1+\delta}$.

(Note that the second bound follows by integrating the first one over $T$.)
This is consistent with the “fractal Weyl conjecture”. According to this
conjecture, there exists some $\sigma_{0}\in\mathbb{R}$ such that for all
$\sigma<\sigma_{0}$ we have

$N_{X}(\sigma,T)\asymp T^{\frac{1}{2}\dim\Omega(X)},$

where $\dim\Omega(X)$ is the dimension of the classical trapped set
$\Omega(X)$ of the geodesic flow on $X$, which in our setting is equal to
$\dim\Omega(X)=2(1+\delta).$ For more background on this conjecture we refer
to [12]. For numerical evidence in support of it we refer to [1, Chapter 16]
and [4].

Naud [13] improved the bound in (2) in the range $\sigma>\delta/2$ by showing
that there exists a function $\tau(\sigma)$, where $\tau(\sigma)$ is positive
for $\sigma\in(\delta/2,\delta)$, such that

(3) $M_{X}(\sigma,T)\ll T^{\delta-\tau(\sigma)}$ and $N_{X}(\sigma,T)\ll
T^{1+\delta-\tau(\sigma)}$.

This result goes in the direction of a conjecture of Jakobson–Naud [10], which
states that the “essential spectral gap” of a hyperbolic surface $X$ is equal
to $\delta/2$. This is equivalent to the statement that for all
$\sigma>\delta/2$ we have $N_{X}(\sigma,T)=O(1)$, which seems out of reach
given the current state of the art. The heuristic justification for this
conjecture is an estimate for the spectral radius of the transfer operator
associated the Schottky representation of $X$ (see §2.5).

Using the fractal uncertainty principle, Dyatlov [4] proved that

(4)
$M_{X}(\sigma,T)\ll_{\epsilon}T^{2(\delta-\sigma)+\epsilon}=T^{\delta-(2\sigma-\delta)+\epsilon}$

and consequently

(5)
$N_{X}(\sigma,T)\ll_{\epsilon}T^{1+2(\delta-\sigma)+\epsilon}=T^{1+\delta-(2\sigma-\delta)+\epsilon}$

Once again, this improves upon (2) when $\sigma>\delta/2,$ and makes Naud’s
bound (3) explicit. The purpose of this paper is to further improve upon (5)
by proving the following:

######  Theorem 1.1.

Let $X$ be a non-elementary, convex cocompact hyperbolic surface and let
$\delta$ be the dimension of its limit set. For every $\sigma>\delta/2$ and
every $\epsilon>0$ we have

(6)
$N_{X}(\sigma,T)\ll_{\epsilon}T^{1+\delta-\frac{3\delta+2}{2\delta+2}(2\sigma-\delta)+\epsilon}.$

We are not able to improve upon the bound for $M_{X}(\sigma,T)$ due to
technical reasons discussed in §1.1 below. Nevertheless, Theorem 1.1 shows
that there are resonance-free boxes with arbitrarily large height in the strip

$\left\\{\delta-\frac{\delta^{2}}{6\delta+4}<\mathrm{Re}(s)<\delta\right\\}.$

More precisely, we have the following:

######  Corollary 1.2.

Let $X$ be as in Theorem 1.1. For every
$\lambda\in\left(0,\frac{\delta^{2}}{2\delta+2}\right)$ and for every
$\epsilon>0$ sufficiently small there exists a density one subset
$\mathcal{S}\subseteq\mathbb{N}$ such that for all $n\in\mathcal{S}$ the box

$\left[\delta-\frac{\delta^{2}}{6\delta+4}+\frac{\delta+1}{3\delta+2}\lambda+\epsilon,\delta\right]+i[n,n+n^{\lambda}]$

contains no resonances for $X$.

###### Proof.

Fix $\sigma\in[\frac{\delta}{2},\delta]$ and $\epsilon>0$. By Theorem 1.1 the
number of resonances inside the box $[T,2T]+i[\sigma,\delta]$ is bounded by

$\\#\left([T,2T]+i[\sigma,\delta]\cap\mathcal{R}_{X}\right)\ll_{\epsilon}T^{1+\delta-\frac{3\delta+2}{2\delta+2}(2\sigma-\delta)+\epsilon}.$

The interval $[T,2T]$ contains roughly $T^{1-\lambda}$ mutually disjoint
intervals of the form $I_{n}=[n,n+n^{\lambda}]$ with $n\in\mathbb{N}$. Hence,
by the pigeonhole principle, for all but a tiny fraction of the intervals
$I_{n}$, the number of resonances inside $I_{n}+i[\sigma,\delta]$ is bounded
by

(7)
$\\#\left(I_{n}+i[\sigma,\delta]\cap\mathcal{R}_{X}\right)\ll_{\epsilon}T^{\delta-\frac{3\delta+2}{2\delta+2}(2\sigma-\delta)+\lambda+\epsilon}.$

A simple calculation shows that if we take

$\sigma=\delta-\frac{\delta^{2}}{6\delta+4}+\frac{\delta+1}{3\delta+2}\lambda+\epsilon,$

then the exponent in (7) drops below zero, and hence, for $T$ sufficiently
large, this implies that

$\\#\left(I_{n}+i[\sigma,\delta]\cap\mathcal{R}_{X}\right)=0,$

completing the proof. ∎

###  1.1. Outline of the proof

We now give an outline of the proof of Theorem 1.1. Fix a non-elementary,
convex cocompact hyperbolic surface $X$ and let $\Lambda$ be its limit set. We
exploit the fact that resonances for $X$ occur as zeros of the Fredholm
determinant

$f(s)=\det(1-A(s))$

for a suitable family of holomorphic trace class operators $A(s)$ with
$s\in\mathbb{C}$ acting on the functional space $H^{2}(\Omega)$, the Hilbert
space of holomorphic $L^{2}$-functions on some neighbourhood
$\Omega\subset\mathbb{C}$ of $\Lambda$. The operators $A(s)$ can be
constructed explicitly after having fixed a Schottky group $\Gamma$ such that
$X\cong\Gamma\backslash\mathbb{H}^{2}$, see §2. We could take $A(s)$ to be the
standard transfer operator $\mathcal{L}_{s}$ (see §2.5) or the “refined”
transfer operator $\mathcal{L}_{\tau,s}$ with some suitable resolution
parameter $\tau>0$ (see §2.6). In this paper we work with

$A(s)=\mathcal{L}_{\tau_{0},\tau_{1},s}^{2}$

for some suitable parameters $\tau_{0}>0$ and $\tau_{1}>0$, where
$\mathcal{L}_{\tau_{0},\tau_{1},s}$ is the composed operator

$\mathcal{L}_{\tau_{0},\tau_{1},s}\stackrel{{\scriptstyle\text{def}}}{{=}}\mathcal{L}_{\tau_{0},s}\circ\mathcal{L}_{\tau_{1},s}.$

The parameters $\tau_{0}$ and $\tau_{1}$ should be thought of as being small
and will be chosen depending on $T$ to optimize the estimates during the
proof.

Following an idea of Guillopé–Lin–Zworski [8], we introduce an additional
(small) parameter $h>0$ and we let $\mathcal{L}_{\tau_{0},\tau_{1},s}$ act on
the complex $h$-neighbourhood of the limit set

$\Omega(h)\stackrel{{\scriptstyle\text{def}}}{{=}}\Lambda+D_{\mathbb{C}}(0,h).$

If $h,\tau_{0},\tau_{1}$ are small enough, this gives a family of well-
defined, trace class operators

$\mathcal{L}_{\tau_{0},\tau_{1},s}\colon H^{2}(\Omega(h))\to
H^{2}(\Omega(h)).$

Finding an upper bound for $N_{X}(\sigma,T)$ amounts to counting the zeros of
the holomorphic function

$f(s)=\det(1-\mathcal{L}_{\tau_{0},\tau_{1},s}^{2})$

inside the rectangular box $[\sigma,\delta]+i[0,T]$. To that effect, we first
observe that the log-modulus of $f(s)$ is bounded by

$\log|f(s)|\leqslant\|\mathcal{L}_{\tau_{0},\tau_{1},s}\|_{\mathrm{HS},h}^{2},$

where $\|\cdot\|_{\mathrm{HS},h}$ denotes the Hilbert–Schmidt norm of
operators on $H^{2}(\Omega(h))$. Using a variant of Jensen’s formula we then
deduce that $N_{X}(\sigma,T)$ is bounded from above essentially by

$\frac{1}{T}\int_{0}^{T}\|\mathcal{L}_{\tau_{0},\tau_{1},\sigma+it}\|_{\mathrm{HS},h}^{2}dt,$

see Proposition 3.5 for a precise statement. A similar approach to count
resonances can be found in Naud’s work [13] who used suitable powers
$\mathcal{L}_{s}^{N}$ of the standard transfer operator instead of
$\mathcal{L}_{\tau_{0},\tau_{1},s}$.

To proceed, we note that elements of the free Schottky group $\Gamma$ can be
indexed by reduced words a in the alphabet $\mathcal{A}=\\{1,\dots,2m\\}$. We
can write down an explicit formula for the Hilbert–Schmidt norm of
$\mathcal{L}_{\tau_{0},\tau_{1},s}$ in terms of the Bergman kernel of
$H^{2}(\Omega(h))$. This formula is a sum over finitely many pairs of reduced
words $(\textbf{a},\textbf{b})$ in the alphabet $\mathcal{A}$. Using a
separation lemma and choosing $\tau_{0}\approx h$ this expression reduces to a
sum over pairs of words of the form
$(\textbf{a},\textbf{b})=(\textbf{c}\textbf{a}_{1},\textbf{c}\textbf{b}_{1})$
with a large common prefix c, that is, only “near-diagonal” terms have a non-
zero contribution. It turns out that by averaging these terms individually
over $|\mathrm{Im}(s)|=t\in[0,T]$, we can exhibit some significant
cancellation, allowing us to beat the estimate in (5). More precisely, we are
led to consider averaged oscillatory integrals of the form

(8)
$\frac{1}{T}\int_{0}^{T}\int_{\Omega_{b}(h)}e^{it\Phi_{\textbf{a},\textbf{b}}(z)}g_{\textbf{a},\textbf{b}}(z)\operatorname{dvol}(z)dt,$

where $\Phi_{\textbf{a},\textbf{b}}(z)$ is a so-called “phase function” and
$g_{\textbf{a},\textbf{b}}(z)$ is some function depending on the words a and
b. When $\textbf{a}\neq\textbf{b}$ we prove an upper bound for (8) that is
significantly better than what one would get with the triangle inequality, see
Proposition 3.4. In order to prove this bound, it is crucial to control the
derivatives of the phase $\Phi_{\textbf{a},\textbf{b}}(z)$.

We note that the same approach could, in principle, be used to estimate the
function $M_{X}(\sigma,T)$, which is bounded roughly by
$\|\mathcal{L}_{\tau_{0},\tau_{1},s}\|_{\mathrm{HS},h}^{2}$. Similarly as
above, this leads to oscillatory integrals of the form

(9)
$\int_{\Omega_{b}(h)}e^{it\Phi_{\textbf{a},\textbf{b}}(z)}g(z)\operatorname{dvol}(z).$

To the author’s knowledge, all the non-trivial estimates for (9) involve
derivatives of $g(z)$, which in our application are rather large as $h\to
0^{+}$, rendering these estimates unusable for our purposes. Fortunately, this
issue does not arise in the averaged version (8).

The cancellation in (8) should reflect the existence of large resonance-free
regions in the strip $\\{\delta/2<\mathrm{Re}(s)\leq\delta\\}$. Numerical
evidence for such regions can be found in the appendix of [4].

###  1.2. Structure of the paper

In §2 we review

  * •

§2.3: the construction of Schottky groups,

  * •

§2.4: the combinatorial notation for indexing words in Schottky groups

  * •

§2.5: the definition of the standard transfer operator and its connection to
the Selberg zeta function,

  * •

§2.6: the notion of partitions and refined transfer operators,

  * •

§2.7: basic estimates for the derivatives of elements of Schottky groups,

  * •

§2.8: the refined functional spaces on which the transfer operators act, and

  * •

§2.9: some useful properties of the Bergman kernel.

In §3 we give the proof of Theorem 1.1, which is divided into the following
subsections:

  * •

§3.1: separation lemma,

  * •

§3.2: bounds for the derivatives of the phase function,

  * •

§3.3: key bound for averaged oscillatory integrals,

  * •

§3.4: adaptation of Jensen’s formula to count resonances,

  * •

§3.5: formula for Hilbert–Schmidt norm, and

  * •

§3.6: finishing the proof by putting everything together.

###  1.3. Notation

We write $f(x)\ll g(x)$ or $f(x)=O(g(x))$ interchangeably to mean that there
exists an implied constant $C>0$ such that $|f(x)|\leqslant C|g(x)|$. We write
$f(x)\ll_{y}g(x)$ or $f(x)=O_{y}(g(x))$ to mean that the implied constant
depends on $y$. We write $C=C(y)$ to emphasize that $C$ depends on $y$. In
this paper, all the implied constants are allowed to depend on the Schottky
data of the surface $X$, which we assume to be fixed throughout.

We write $s=\sigma+it\in\mathbb{C}$ to mean that $\sigma$ and $t$ are the real
and imaginary parts of $s$ respectively.

Given a finite set $S$, we denote its cardinality by $|S|.$

##  2\. Preliminaries

###  2.1. Hyperbolic geometry

We start by reviewing some basic facts about hyperbolic geometry. We refer the
reader to Borthwick’s book [1] for a comprehensive discussion. One of the
standard models for the hyperbolic plane is the Poincaré half-plane

$\mathbb{H}^{2}=\\{x+iy\in\mathbb{C}\ :\ y>0\\}$

endowed with its standard metric of constant curvature $-1$,

$ds^{2}=\frac{dx^{2}+dy^{2}}{y^{2}}.$

The group of orientation-preserving isometries of $(\mathbb{H}^{2},ds)$ is
isomorphic to $\mathrm{PSL}_{2}(\mathbb{R})$, which acts on the extended
complex plane $\overline{\mathbb{C}}=\mathbb{C}\cup\\{\infty\\}$ (and hence
also on $\mathbb{H}^{2}$) by Möbius transformations

$\gamma=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\mathrm{PSL}_{2}(\mathbb{R}),\quad
z\in\overline{\mathbb{C}}\Longrightarrow\gamma(z)=\frac{az+b}{cz+d}.$

  * •

“hyperbolic” if $|\operatorname{tr}(\gamma)|>2$, which implies that $\gamma$
has two distinct fixed points on the boundary $\partial\mathbb{H}^{2}$,

  * •

“parabolic” if $|\operatorname{tr}(\gamma)|<2$, which implies that $\gamma$
has precisely one fixed point on $\partial\mathbb{H}^{2}$, and

  * •

“elliptic” if $|\operatorname{tr}(\gamma)|=2$, which implies that $\gamma$ has
precisely one fixed point in the hyperbolic plane $\mathbb{H}^{2}$.

Discrete subgroups of $\mathrm{PSL}_{2}(\mathbb{R})$ are called “Fuchsian”
groups. A Fuchsian group is “torsion-free” if it contains no elliptic
elements. It is called “convex cocompact” if it is finitely generated and if
it contains neither elliptic nor parabolic elements. This is equivalent to the
“convex core” of $X=\Gamma\backslash\mathbb{H}^{2}$ being compact.

###  2.2. Selberg zeta function

Given a finitely generated Fuchsian group
$\Gamma<\mathrm{PSL}_{2}(\mathbb{R})$, the set of prime periodic geodesics on
$X=\Gamma\backslash\mathbb{H}^{2}$ is bijective to the set
$[\Gamma]_{\mathsf{prim}}$ of $\Gamma$-conjugacy classes of primitive
hyperbolic elements in $\Gamma$. We denote by $\ell(\gamma)$ the length of the
geodesic corresponding to the conjugacy class
$[\gamma]\in[\Gamma]_{\mathsf{prim}}$.

The “Selberg zeta function” is defined for $\mathrm{Re}(s)>\delta$ by the
infinite product

(10)
$Z_{\Gamma}(s)\stackrel{{\scriptstyle\text{def}}}{{=}}\prod_{k=0}^{\infty}\prod_{[\gamma]\in[\Gamma]_{\mathsf{prim}}}\left(1-e^{-(s+k)\ell(\gamma)}\right)$

The Selberg zeta function has a meromorphic continuation to $s\in\mathbb{C}.$
By Patterson–Perry [16] the zero set of $Z_{\Gamma}(s)$ consists of the so-
called “topological” zeros at $s=-k$ for $k\in\mathbb{N}_{0}$, and the set of
resonances, repeated according to multiplicity. Therefore, any problem about
resonances can be rephrased as a question about the distribution of the zeros
of the Selberg zeta function.

By the product representation (10) and by uniqueness of analytic
continuations, it follows that $Z_{\Gamma}(s)=Z_{\Gamma}(\overline{s})$ for
all $s\in\mathbb{C}$. Thus if $s_{0}$ is a resonance for $X$, then so is its
complex conjugate $\overline{s_{0}}$.

###  2.3. Schottky groups

By the result of Button [3], every convex cocompact hyperbolic surface $X$ can
be realized as the quotient of $\mathbb{H}^{2}$ by a so-called Schottky group
$\Gamma$, see also [1, Theorem 15.3]. Schottky groups stand out, among other
Fuchsian groups, by their simple geometric construction, which we recall now:

  * •

Define the alphabet $\mathcal{A}=\\{1,\dots,2m\\}$ and for each
$a\in\mathcal{A}$ define

$\overline{a}=\begin{cases}a+r&\text{ if }a\in\\{1,\dots,m\\}\\\ a-m&\text{ if
}a\in\\{m+1,\dots,2m\\}\end{cases}$

  * •

Fix open disks $D_{1},\dots,D_{2m}\subset\mathbb{C}$ centered on the real line
with mutually disjoint closures.

  * •

Fix isometries $\gamma_{1},\dots\gamma_{2m}\in\mathrm{PSL}_{2}(\mathbb{R})$
such that for all $a\in\mathcal{A}$

$\gamma_{a}(\overline{\mathbb{C}}\smallsetminus D_{\overline{a}})=D_{a}$ and
$\gamma_{\overline{a}}=\gamma_{a}^{-1}.$

(In the notation of [1, §15] we have $m=r$ and $\gamma_{a}=S_{a}^{-1}$.)

  * •

Let $\Gamma$ be the group generated by the elements
$\gamma_{1},\dots\gamma_{2m}$. This is a free group on $m$ generators, see for
instance [1, Lemma 15.2].

$\partial\mathbb{H}^{2}$$\mathbb{H}^{2}$$D_{1}$$D_{4}$$D_{2}$$D_{3}$$D_{5}$$D_{6}$$\gamma_{1}$$\gamma_{2}$$\gamma_{3}$
Figure 1. A configuration of Schottky disks and isometries with $m=3$

A Fuchsian group $\Gamma$ is called “elementary” if its limit set $\Lambda$ is
a finite set. If $\Gamma$ is a Schottky group as above, then it is elementary
precisely when $m=1$, in which case $\Gamma\backslash\mathbb{H}^{2}$ is a
hyperbolic cylinder.

Throughout the rest of this paper we assume that
$X=\Gamma\backslash\mathbb{H}^{2}$ is a convex cocompact quotient where
$\Gamma$ is a non-elementary Schottky group with Schottky data
$D_{1},\dots,D_{2m}$ and $\gamma_{1},\dots,\gamma_{2m}$ as above. This
assumption will not be repeated in the sequel.

###  2.4. Combinatorial notation for words

We will follow the combinatorial notation of Dyatlov–Zworski [5] for indexing
elements in the free group $\Gamma$.

  * •

A word a in the alphabet $\mathcal{A}=\\{1,\dots,2m\\}$ is a finite string
$\textbf{a}=a_{1}\dots a_{n}$ with $a_{1},\dots,a_{n}\in\mathcal{A}$. For
technical reasons, we also introduce the empty word $\emptyset$, a string of
length zero.

  * •

A word $\textbf{a}=a_{1}\dots a_{n}$ is said to be “reduced” if
$a_{j}\neq\overline{a_{j+1}}$ for all $j=1,\dots,n-1$. For all
$n\in\mathbb{N}$ denote by $\mathcal{W}_{n}$ the set of (reduced) words of
length $n$:

$\mathcal{W}_{n}=\left\\{a_{1}\cdots
a_{n}:\text{$a_{1},\dots,a_{n}\in\mathcal{A}$ s.t.
$a_{j}\neq\overline{a_{j+1}}$ for all $j=1,\dots,n-1$}\right\\}.$

Moreover, put $\mathcal{W}_{0}=\\{\emptyset\\}$ where $\emptyset$ is the empty
word.

  * •

Let $\mathcal{W}=\bigsqcup_{n\geqslant 0}\mathcal{W}_{n}$ be the set of all
reduced words and write $|\textbf{a}|=n$ if $\textbf{a}\in\mathcal{W}_{n}$. In
other words, $|\textbf{a}|$ is the reduced word length of a. Given
$m\in\mathbb{N}$ let $\mathcal{W}_{\geqslant m}=\bigsqcup_{n\geqslant
m}\mathcal{W}_{n}$ the set of all reduced words whose length is at least $m$,
and let $\mathcal{W}^{\circ}=\mathcal{W}_{\geqslant 1}$ be the set of all non-
empty reduced words.

  * •

Given a word $\textbf{a}=a_{1}\cdots a_{n}\in\mathcal{W}^{\circ}$ write
$\textbf{a}^{\prime}=a_{1}\cdots a_{n-1}\in\mathcal{W}$. Note that
$\mathcal{W}$ is a tree with root $\emptyset$ and $\textbf{a}^{\prime}$ is the
parent of $\textbf{a}.$

  * •

For $\textbf{a}=a_{1}\cdots a_{n}\in\mathcal{W}$ and $\textbf{b}=b_{1}\cdots
b_{m}\in\mathcal{W}$ write $\textbf{a}\to\textbf{b}$ if either
$\textbf{a}=\emptyset$, or $\textbf{b}=\emptyset$, or
$a_{n}\neq\overline{b_{1}}$. Note that in this case,
$\textbf{a}\textbf{b}\in\mathcal{W}$, that is, if $\textbf{a}\to\textbf{b}$,
then the concatenation ab is also a reduced word.

  * •

Given a word $\textbf{a}=a_{1}\cdots a_{n}$ let
$\overline{\textbf{a}}=\overline{a_{1}}\cdots\overline{a_{n}}$ be its “mirror”
word.

  * •

Write $\textbf{a}\prec\textbf{b}$ if a is a prefix b, that is, if
$\textbf{b}=\textbf{ac}$ for some $\textbf{c}\in\mathcal{W}$.

  * •

We have the one-to-one correspondence

$\textbf{a}=a_{1}\cdots
a_{n}\in\mathcal{W}\mapsto\gamma_{\textbf{a}}=\gamma_{a_{1}}\cdots\gamma_{a_{n}}\in\Gamma.$

Moreover, we have
$\gamma_{\textbf{ab}}=\gamma_{\textbf{a}}\gamma_{\textbf{b}}$,
$\gamma_{\textbf{a}}^{-1}=\gamma_{\overline{\textbf{a}}}$, and
$\gamma_{\textbf{a}}=\operatorname{id}$ if and only if $\textbf{a}=\emptyset.$

  * •

For $\textbf{a}=a_{1}\cdots a_{n}\in\mathcal{W}^{\circ}$ we define the disk

$D_{\textbf{a}}:=\gamma_{\textbf{a}^{\prime}}(D_{a_{n}}).$

If $\textbf{a}\prec\textbf{b}$ then $D_{\textbf{a}}\subset D_{\textbf{b}}$. On
the other hand, if $\textbf{a}\nprec\textbf{b}$ and
$\textbf{b}\nprec\textbf{a}$ then $D_{\textbf{a}}\cap
D_{\textbf{b}}=\emptyset$. We define the interval

(11) $I_{\textbf{a}}:=D_{\textbf{a}}\cap\mathbb{R}$

and we denote by $|I_{\textbf{a}}|$ its length which is equal to the diameter
of $D_{\textbf{a}}.$

  * •

Denote

$D=\bigsqcup_{a\in\mathcal{A}}D_{a}$ and $I=\bigsqcup_{a\in\mathcal{A}}I_{a}$.

  * •

In the above notation, the limit set of $\Gamma$ may be re-expressed as
follows:

$\Lambda=\bigcap_{n\geqslant
1}\bigsqcup_{\textbf{a}\in\mathcal{W}_{n}}I_{\textbf{a}}\subset\mathbb{R}.$

###  2.5. The standard transfer operator

Given any open set $\Omega\subset\mathbb{C}$ let $H^{2}(\Omega)$ be the
“Bergman space” over $\Omega$, which is defined as the Hilbert space of
square-integrable, holomorphic functions on $\Omega$:

(12)
$H^{2}(\Omega)\stackrel{{\scriptstyle\text{def}}}{{=}}\left\\{\text{$f\colon\Omega\to\mathbb{C}$
holomorphic}\left|\;\left\|f\right\|<\infty\right.\right\\},$

with $L^{2}$-norm given by

$\|f\|^{2}\stackrel{{\scriptstyle\text{def}}}{{=}}\int_{\Omega}\|f(z)\|_{V}^{2}\operatorname{dvol}(z).$

Here $\operatorname{vol}$ denotes the Lebesgue measure on the complex plane.

Now consider the Bergman space $H^{2}(D)$ over the union
$D=\bigsqcup_{a\in\mathcal{A}}D_{a}$ of the Schottky disks in §2.3. On this
space, we define the holomorphic family of operators, parametrized by
$s\in\mathbb{C}$,

(13) $\mathcal{L}_{s}\colon H^{2}(D)\to H^{2}(D)$

by the formula

(14)
$\mathcal{L}_{s}f(z)\stackrel{{\scriptstyle\text{def}}}{{=}}\sum_{\begin{subarray}{c}a\in\mathcal{A}\\\
a\to b\end{subarray}}^{2m}\gamma_{a}^{\prime}(z)^{s}f(\gamma_{a}(z))\
\mathrm{if}\ z\in D_{b}.$

The derivative satisfies $\gamma_{a}^{\prime}(z)>0$ for all $z\in I_{b}$, so
the complex power $\gamma_{a}^{\prime}(z)^{s}$ is uniquely defined and
holomorphic for $z\in D_{b}$ and $s\in\mathbb{C}.$ More concretely, we define

$\gamma_{a}^{\prime}(z)^{s}=\exp(s\mathbb{L}(\gamma_{a}^{\prime}(z))),$

where

(15) $\mathbb{L}(z)=\log|z|+\mathrm{arg}(z),$

with

$\mathrm{arg}\colon\mathbb{C}\smallsetminus(-\infty,0]\to(-\pi,\pi)$

being the principal value of the argument.

The operator in (14) is the well-known “transfer operator” associated to
$\Gamma$ which can be found for instance in Borthwick’s book [1, Chapter 15].
The following result relates the transfer operator to the Selberg zeta
function:

######  Proposition 2.1 (Fredholm determinant identity).

For every $s\in\mathbb{C}$ the operator (13) is trace class and we have the
identity

(16) $Z_{\Gamma}(s)=\det(1-\mathcal{L}_{s}).$

Proposition 2.1 characterizes resonances in terms of 1-eigenfunctions of the
transfer operator. Indeed, using standard properties of the Fredholm
determinant, this proposition says that $s\in\mathbb{C}$ is a resonance for
$X=\Gamma\backslash\mathbb{H}^{2}$ if and only if there exists a non-zero
$f\in H^{2}(D)$ such that $f=\mathcal{L}_{s}f.$

###  2.6. Partitions and refined transfer operators

In the present paper, we will work with refined transfer operators. These are
generalizations of the standard transfer operator $\mathcal{L}_{s}$ introduced
by Dyatlov–Zworski [5]. Let us now recall their definition.

Given a finite subset $Z\subset\mathcal{W}$, put
$Z^{\prime}=\\{\textbf{a}^{\prime}:\textbf{a}\in Z\\}$ and
$\overline{Z}=\\{\overline{\textbf{a}}:\textbf{a}\in Z\\},$ and for
$s\in\mathbb{C}$ define the family of operators

(17) $\mathcal{L}_{Z,s}\colon H^{2}(D)\to H^{2}(D)$

by the formula

(18)
$\mathcal{L}_{Z,s}f(z)\stackrel{{\scriptstyle\text{def}}}{{=}}\sum_{\begin{subarray}{c}\textbf{a}\in(\overline{Z})^{\prime}\\\
\textbf{a}\to b\end{subarray}}\gamma_{a}^{\prime}(z)^{s}f(\gamma_{a}(z))\
\mathrm{if}\ z\in D_{b}.$

Note that $\mathcal{L}_{Z,s}$ reduces to the standard transfer operator
$\mathcal{L}_{s}$ if $Z$ is taken to be $\mathcal{W}_{2}$, the set of reduced
words of length two.

A finite set $Z\subset\mathcal{W}^{\circ}$ is called a “partition” if there
exists $N\in\mathbb{N}$ such that for every reduced word
$\textbf{a}\in\mathcal{W}$ with $|\textbf{a}|\geqslant N$, there exists a
unique $\textbf{b}\in Z$ such that $\textbf{b}\prec\textbf{a}$. In terms of
the limit set, a finite set $Z\in\mathcal{W}^{\circ}$ is a partition if we
have the disjoint union

$\Lambda=\bigsqcup_{\textbf{b}\in Z}(I_{\textbf{b}}\cap\Lambda).$

Trivial examples of partitions are the sets $\mathcal{W}_{n}$ for
$n\in\mathbb{N}$, the set of reduced words of length $n$, in which case we
have $\mathcal{L}_{\mathcal{W}_{n},s}=\mathcal{L}_{s}^{n-1}.$

The fundamental fact about partitions is the following result of
Dyatlov–Zworski [5], which says that 1-eigenfunctions of $\mathcal{L}_{s}$
must also be 1-eigenfunctions of $\mathcal{L}_{Z,s}$, provided $Z$ is a
partition.

######  Lemma 2.2.

Let $Z$ be a finite subset of $\mathcal{W}_{\geqslant 2}=\bigsqcup_{n\geqslant
2}\mathcal{W}_{n}$. If $Z$ is a partition then for every $f\in H^{2}(D)$ the
following holds true:

$\mathcal{L}_{s}f=f\Longrightarrow\mathcal{L}_{Z,s}f=f.$

Combining this with Proposition 2.1 and the remark thereafter, we obtain that
for any given partition $Z\subset\mathcal{W}^{\circ}$, resonances for
$X=\Gamma\backslash\mathbb{H}^{2}$ are zeros of the Fredholm determinant
$\det(1-\mathcal{L}_{Z,s}).$ More generally, given a finite collection of
partitions $Z_{1},Z_{2},\dots,Z_{k}\subset\mathcal{W}_{\geqslant 2}$,
resonances for $X$ occur as zeros of the function

(19)
$f(s)=\det(1-\mathcal{L}_{Z_{1},s}\circ\cdots\circ\mathcal{L}_{Z_{k},s}).$

The most important family of partitions is defined as follows: given a
parameter $\tau>0$, called the “resolution parameter”, set

$Z(\tau)\stackrel{{\scriptstyle\text{def}}}{{=}}\\{\textbf{a}\in\mathcal{W}^{\circ}:|I_{\textbf{a}}|\leqslant\tau<|I_{\textbf{a}^{\prime}}|\\}.$

The set $Z(\tau)$ is a partition by virtue of the fact that the interval
length $|I_{\textbf{a}}|$ tends to zero as $|\textbf{a}|\to\infty$. This in
turn follows from the definition of the intervals $I_{\textbf{a}}$ in (11) and
from the uniform contraction property in Lemma 2.3 below.

Finally, we define the $\tau$-refined transfer operator by

(20)
$\mathcal{L}_{\tau,s}\stackrel{{\scriptstyle\text{def}}}{{=}}\mathcal{L}_{Z(\tau),s}.$

We can write down the following formula for $\mathcal{L}_{\tau,s}$ for every
$f\in H^{2}(D)$ and $b\in\mathcal{A}$:

(21) $\mathcal{L}_{\tau,s}f(z)=\sum_{\begin{subarray}{c}\textbf{a}\in
Y(\tau)\\\ \textbf{a}\to
b\end{subarray}}\gamma_{a}^{\prime}(z)^{s}f(\gamma_{a}(z))\ \mathrm{if}\ z\in
D_{b},$

where

(22)
$Y(\tau)\stackrel{{\scriptstyle\text{def}}}{{=}}\overline{Z(\tau)}^{\prime}.$

Note that the operator (20) is well-defined if and only if $Y(\tau)\subset
W^{\circ}$, or equivalently, $Z(\tau)\subset W_{\geqslant 2}$. This condition
is satisfied if the resolution parameter $\tau>0$ is small enough.

The main reason for using this special family of operators is that we can
control the size of $Y(\tau)$ size as well as the size of the derivatives
$\gamma_{\textbf{a}}$ with $\textbf{a}\in Y(\tau)$, see Lemma 2.4 below. This
is what enables us to obtain explicit exponents in Theorem 1.1.

###  2.7. Bounds for derivatives

Now we record some basic and well-known estimates for the derivatives of
elements of $\Gamma$. Following Magee–Naud [14], we use the following
notation: for every $a\in\mathcal{A}$ we pick a point $o_{a}\in D_{a}$ and for
any $\textbf{a}\in\mathcal{W}^{\circ}$ we set

$o_{\textbf{a}}\stackrel{{\scriptstyle\text{def}}}{{=}}o_{a}$

where $a\in\mathcal{A}$ is chosen such that $\textbf{a}\to a$.

Throughout the rest of this paper these points will remain fixed.

Moreover, put

$\Upsilon_{\textbf{a}}\stackrel{{\scriptstyle\text{def}}}{{=}}|\gamma_{\textbf{a}}^{\prime}(o_{\textbf{a}})|.$

The following basic estimates are due to Naud [13] and Magee–Naud [14]:

######  Lemma 2.3 (Basic distortion estimates).

The following estimates hold true with implied constants depending only on
$\Gamma$:

  1. (i)

Uniform contraction: There are constants $0<\theta_{1}<\theta_{2}<1$ and $C>0$
such that for all $b\in\mathcal{A}$ and for all $\textbf{a}\in\mathcal{W}$
with $\textbf{a}\to b$ and $z\in D_{b}$ we have

$C^{-1}\theta_{1}^{|\textbf{a}|}\leqslant|\gamma_{\textbf{a}}^{\prime}(z)|\leqslant
C\theta_{2}^{|\textbf{a}|}.$

  2. (ii)

Bounded distortion 1: For all $b\in\mathcal{A}$ and for all
$\textbf{a}\in\mathcal{W}$ with $\textbf{a}\to b$ and $z_{1},z_{2}\in D_{b}$
we have

$C^{-1}\leqslant\frac{|\gamma_{\textbf{a}}^{\prime}(z_{1})|}{|\gamma_{\textbf{a}}^{\prime}(z_{2})|}\leqslant
C.$

  3. (iii)

Bounded distortion 2: There exists a constant $C>0$ such that for all
$b_{1},b_{2}\in\mathcal{A}$, all $z_{1}\in D_{b_{1}}$ and all $z_{2}\in
D_{b_{2}}$, and all $\textbf{a}\in\mathcal{W}^{\circ}$ with $\textbf{a}\to
b_{1},b_{2}$ we have

$\left|\frac{\gamma^{\prime}_{\textbf{a}}(z_{1})}{\gamma^{\prime}_{\textbf{a}}(z_{2})}\right|\leqslant
C.$

  4. (iv)

For all $b\in\mathcal{A}$ and $z\in D_{b}$ with $\textbf{a}\to b$ we have
$|\gamma^{\prime}_{\textbf{a}}(z)|\asymp\Upsilon_{\textbf{a}}$.

  5. (v)

For all $\textbf{a}\in\mathcal{W}^{\circ}$ we have
$\Upsilon_{\textbf{a}}\asymp\Upsilon_{\textbf{a}}$.

  6. (vi)

For all $\textbf{a}\in\mathcal{W}^{\circ}$ we have
$\Upsilon_{\overline{\textbf{a}}}\asymp\Upsilon_{\textbf{a}}$.

  7. (vii)

For all $\textbf{a},\textbf{b}\in\mathcal{W}^{\circ}$ with
$\textbf{a}\to\textbf{b}$ we have
$\Upsilon_{\textbf{a}\textbf{b}}\asymp\Upsilon_{\textbf{a}}\Upsilon_{\textbf{b}}.$

The following following estimates concerning $Z(\tau)$ and $Y(\tau)$ are also
crucial:

######  Lemma 2.4 (Estimates for $Z(\tau)$ and $Y(\tau)$).

For all $\tau>0$ small enough the following estimates hold true with implied
constants depending only on $\Gamma$:

  1. (i)

For all $\textbf{a}\in Z(\tau)$ we have $\Upsilon_{\textbf{a}}\asymp\tau.$

  2. (ii)

For all $\textbf{a}\in Y(\tau)$ we have $\Upsilon_{\textbf{a}}\asymp\tau.$

  3. (iii)

$|Y(\tau)|\asymp|Z(\tau)|\asymp\tau^{-\delta}.$

The estimates for $Z(\tau)$ in Lemma 2.4 can be found in [2]. Lemma 2.4 can
also be deduced from the definitions of the sets $Z(\tau)$ and $Y(\tau)$ and
Lemma 2.3 above.

###  2.8. Refined function spaces

In this paper, we adopt the approach of Guillopé–Lin-Zworski [8], which we
present here. Roughly speaking, the idea is to let the transfer operators act
on “refined” function spaces. To define these spaces, let $h>0$ be a (small)
parameter and consider the complex $h$-neighbourhood of $\Lambda$:

$\Omega(h)\stackrel{{\scriptstyle\text{def}}}{{=}}\Lambda+D_{\mathbb{C}}(0,h),\quad
D_{\mathbb{C}}(0,h)=\\{z\in\mathbb{C}:|z|<h\\}.$

Note that for all $h>0$ small enough we have $\Omega(h)\subset D.$ For each
$b\in\mathcal{A}$ put

$\Omega_{b}(h)\stackrel{{\scriptstyle\text{def}}}{{=}}\Omega(h)\cap D_{b}.$

Given a partition $Z\subset\mathcal{W}_{\geqslant 2}$, we let the generalized
transfer operator $\mathcal{L}_{\tau,s}$ act on the functional space
$H^{2}(\Omega(h))$. This gives a family of operators

(23) $\mathcal{L}_{Z,s}\colon H^{2}(\Omega(h))\to H^{2}(\Omega(h)),$

defined for all functions $f\in H^{2}(\Omega(h))$ by the formula

(24)
$\mathcal{L}_{Z,s}f(z)=\sum_{\begin{subarray}{c}\textbf{a}\in\overline{Z}^{\prime}\\\
\textbf{a}\to
b\end{subarray}}\gamma^{\prime}_{\textbf{a}}(z)^{s}f(\gamma_{\textbf{a}}(z))\quad\text{if
$z\in\Omega_{b}(h)$}.$

By the definition of $\Omega(h)$ we have

$z\in\Omega(h)\Longrightarrow|\mathrm{Im}(z)|\leqslant h.$

Using the definition of the complex powers in (15), this implies that for all
$b\in\mathcal{A}$ and $\textbf{a}\in\mathcal{W}^{\circ}$ with $\textbf{a}\to
b$ we have for all $s=\sigma+it$:

(25)
$z\in\Omega_{b}(h)\Longrightarrow|\gamma_{\textbf{a}}^{\prime}(z)^{s}|\ll_{\sigma}\Upsilon_{\textbf{a}}^{\sigma}e^{Ch|t|}.$

Here the constant $C>0$ depends only on $\Gamma$ while the implied constant
depends only on $\Gamma$ and $\sigma.$ It is clear that in order to optimize
this bound one may choose $h\approx|t|^{-1}$ so that $e^{Ch|t|}=O(1)$. In
other words, $h$ can be chosen to eliminate exponential terms in
$|\mathrm{Im}(s)|$. This is the main reason for working with
$H^{2}(\Omega(h))$.

It remains to show that (23) is well-defined. The next lemma shows that the
operator (23) is well-defined and trace class, provided the parameter $h>0$ is
chosen small enough.

######  Lemma 2.5 (Contraction in $\Omega(h)$).

There exist constants $h_{0}>0$ and $\eta\in(0,1)$, depending only on
$\Gamma$, such that for all $h\in(0,h_{0})$, $b\in\mathcal{A}$ and
$\textbf{a}\in\mathcal{W}^{\circ}$ with $\mathbf{a}\to b$,

$\gamma_{\textbf{a}}(\Omega_{b}(h))\subset\Omega(\eta^{|\textbf{a}|}h).$

In particular,

$\mathrm{dist}(\gamma_{\textbf{a}}(z),\partial\Omega(h))>(1-\eta^{|\textbf{a}|})h\geqslant(1-\eta)h.$

###### Proof.

Note that $\Omega(h)$ and $\Omega_{b}(h)$ may be re-expressed as

$\Omega(h)=\\{z\in\mathbb{C}:\text{there exists some $p\in\Lambda$ such that
$|z-p|<h$}\\},$

and

$\Omega_{b}(h)=\\{z\in\mathbb{C}:\text{there exists some $p\in\Lambda\cap
D_{b}$ such that $|z-p|<h$}\\}.$

Let $z\in\Omega_{b}(h)$. Then, by definition, there exists some
$p\in\Lambda\cap D_{b}$ such that $|z-p|\leqslant h$. Now let
$a\in\mathcal{A}$ be such that $a\to b$. By the uniform contraction property
in Lemma 2.3 there exists $\eta\in(0,1)$ such that

$|\gamma_{a}(z)-\gamma_{a}(p)|\leqslant\eta|z-p|\leqslant\eta h.$

Applying this estimate repeatedly, it follows that for all reduced words
$\textbf{a}\in\mathcal{W}^{\circ}$ with $\textbf{a}\to b$ we have

(26)
$|\gamma_{\textbf{a}}(z)-\gamma_{\textbf{a}}(p)|\leqslant\eta^{|\textbf{a}|}h.$

But by $\Gamma$-invariance of $\Lambda$ it follows that
$\gamma_{\textbf{a}}(p)\in\Lambda$, so we obtain from (26) that
$\gamma_{\textbf{a}}(z)\in\Omega(\eta^{|\textbf{a}|}h),$ as claimed. ∎

We remark that for any finite collection of partitions
$Z_{1},Z_{2},\dots,Z_{k}\subset\mathcal{W}_{\geqslant 2}$, Lemma 2.5 shows
that for $h>0$ small enough, the operator

$\mathcal{L}_{Z_{1},s}\circ\cdots\circ\mathcal{L}_{Z_{k},s}\colon
H^{2}(\Omega(h))\to H^{2}(\Omega(h))$

is well-defined and trace-class. Moreover, the Fredholm determinant

(27) $\det(1-\mathcal{L}_{Z_{1},s}\circ\cdots\circ\mathcal{L}_{Z_{k},s})$

is independent of the choice of the parameter $h$. Indeed, carefully applying
the Lefschetz fixed point formula (see [1, Lemma 15.9]) shows that the traces
of powers of $\mathcal{L}_{Z_{1},s}\circ\cdots\circ\mathcal{L}_{Z_{k},s}$ (and
hence (27)) are independent of $h.$

Also crucial is the following upper bound for the Lebesgue measure of
$\Omega(h)$.

######  Lemma 2.6 (Volume bound).

There exists a constant $C>0$ such that for all $h>0$ sufficiently small we
have

$\mathrm{vol}(\Omega(h))\leqslant Ch^{2-\delta}$

###### Proof.

Consider the real $h$-neighbourhood of the limit set
$\Lambda(h)=\Lambda+[-h,h]$. Let $I_{l}(h)$ with $l=1,\dots,N(h)$ denote the
connected components of $\Lambda(h)$. It is well-known from [8] (see also [1,
Lemma 15.14]) that there exists some constant $C=C(\Gamma)>0$ such that for
all $h>0$ small enough the following hold true:

  * •

the diameter of each interval $I_{l}(h)$ is less than $Ch$, and

  * •

the number of connected components $N(h)$ is bounded by $N(h)\leqslant
Ch^{-\delta}.$

Now note that

$\Omega(h)\subseteq\bigsqcup_{l=1}^{N(h)}\left(I_{l}(h)+i[-h,h]\right),$

whence

$\mathrm{vol}(\Omega(h))\leqslant\sum_{l=1}^{N(h)}\mathrm{vol}(I_{l}(h)+i[-h,h])=h\sum_{l=1}^{N(h)}|I_{l}(h)|,$

where $|I_{l}(h)|$ is the diameter of $I_{l}(h)$. From the previously
mentioned facts it follows that the right hand side is at most
$O(h^{2-\delta})$, as claimed. ∎

###  2.9. Properties of Bergman kernels

We now recall some basic properties of the Bergman kernels which are crucial
in the proof of Theorem 1.1. We refer the reader to [11] for a more in-depth
account of the material given here.

Let $\Omega\subset\mathbb{C}$ be a non-empty bounded (not necessarily
connected) open set and let $H^{2}(\Omega)$ be the Bergman space over
$\Omega$. Since $H^{2}(\Omega)$ is a closed subspace of $L^{2}(\Omega)$, it is
a separable Hilbert space. Let $(\varphi_{n})$ be a orthonormal basis of
$H^{2}$. Given any function $f\in H^{2}(\Omega)$, we can expand it as

$f=\sum_{n}c_{n}(f)\varphi_{n},\quad
c_{n}(f)=\int_{\Omega}f(w)\overline{\varphi_{n}(w)}\operatorname{dvol}(w),$

where the sum converges absolutely and uniformly on compact sets with respect
to the $L^{2}$-norm. By the Riesz representation theorem there exists some
unique $\overline{B_{\Omega}(z,\cdot)}\in H^{2}(\Omega)$, called the “Bergman
reproducing kernel”, such that

(28) $f(z)=\int_{\Omega}B(z,w)f(w)\operatorname{dvol}(w).$

For any fixed $z\in\Omega$, we can expand $\overline{B(z,w)}$ as follows:

(29) $\overline{B(z,w)}=\sum_{n}c_{n}\varphi_{n}(w),$

where

(30)
$c_{n}=\int_{\Omega}\overline{B(z,w)}\,\overline{\varphi_{n}(w)}\operatorname{dvol}(w).$

Inserting (30) and (29) into (28) and using the uniqueness of the Bergman
kernel reveals that

(31) $B(z,w)=\sum_{n}\varphi_{n}(z)\overline{\varphi_{n}(w)},$

where the series is uniformly convergent on compact subsets of
$\Omega\times\Omega.$

######  Lemma 2.7 (Basic properties of the Bergman kernel).

The following holds true:

  1. (i)

If $z$ and $w$ lie in different components of $\Omega$ then
$B_{\Omega}(z,w)=0.$

  2. (ii)

For all $z,w$ we have $|B_{\Omega}(z,w)|^{2}\leqslant
B_{\Omega}(z,z)B_{\Omega}(w,w).$

  3. (iii)

Variational Formula: $B_{\Omega}(z,z)=\sup\\{|f(z)|^{2}:f\in
H^{2}(\Omega),\|f\|=1\\}.$

  4. (iv)

Comparison Formula: If $z\in\Omega_{1}\subset\Omega_{2}$ then
$B_{\Omega_{2}}(z,z)\leqslant B_{\Omega_{1}}(z,z).$

  5. (v)

For the disk $D(z_{0},r)=\\{|z-z_{0}|<r\\}$ we have the explicit formula

$B_{D(z_{0},r)}(z,w)=\frac{2r^{2}}{\pi\left(r^{2}-(z-z_{0})(\overline{w}-\overline{w}_{0})\right)^{2}}.$

###### Proof.

Properties (i)–(iv) are easy to deduce from the definition of the Bergman
kernel, from the formula (31), and the Cauchy–Schwarz inequality. To prove
(v), we use (31) with the following explicit orthonormal basis for
$H^{2}(D(z_{0},r))$:

$\varphi_{n}(z)=\sqrt{\frac{n+1}{\pi
r^{2}}}\left(\frac{z-z_{0}}{r}\right)^{n},\quad n\in\mathbb{N}_{0}.$

∎

######  Lemma 2.8 (Upper bound for the Bergman kernel).

For all $z\in\Omega$ set

$\mathrm{dist}(z,\partial\Omega)\stackrel{{\scriptstyle\text{def}}}{{=}}\inf_{z^{\prime}\in\partial\Omega}|z-z^{\prime}|.$

Then for all $z,w\in\Omega$ we have

$|B_{\Omega}(z,w)|\leqslant\frac{1}{\pi\mathrm{dist}(z,\partial\Omega)\mathrm{dist}(w,\partial\Omega)}.$

###### Proof.

Note that $D(z,r_{z})\subset\Omega$ and $D(w,r_{w})\subset\Omega$ for any
$r_{z}<\mathrm{dist}(z,\partial\Omega)$ and
$r_{w}<\mathrm{dist}(w,\partial\Omega)$, so we can use Parts (ii), (iv), and
(v) of Lemma 2.7 to obtain

$|B_{\Omega}(z,w)|^{2}\leqslant B_{\Omega}(z,z)B_{\Omega}(w,w)\leqslant
B_{D(z,r_{z})}(z,z)B_{D(w,r_{w})}(w,w)\leqslant\frac{1}{\pi
r_{z}^{2}}\cdot\frac{1}{\pi r_{w}^{2}}.$

This gives

$|B_{\Omega}(z,w)|\leqslant\frac{1}{\pi r_{z}r_{w}}.$

Letting $r_{z}\nearrow\mathrm{dist}(z,\partial\Omega)$ and
$r_{w}\nearrow\mathrm{dist}(w,\partial\Omega)$ finishes the proof. ∎

##  3\. Proof of Main Theorem

This chapter is devoted to the proof of Theorem 1.1 and is divided into
several sections. Recall that $X=\Gamma\backslash\mathbb{H}^{2}$ is a convex
co-compact hyperbolic quotient, that we fix a Schottky representation for
$\Gamma$ as in §2.3, and that we use the notations introduced in §2.

###  3.1. A separation lemma

In this section, we prove the following “separation” lemma which is an
adaptation of a result of Jakobson–Naud [10, Lemma 4.4]. This result is a
consequence of the total discontinuity of the limit set $\Lambda.$

######  Lemma 3.1 (Separation Lemma).

There exist constants $c>0$ and $h_{0}>0$, depending only on $\Gamma$, such
that for all $h\in(0,h_{0})$ and $\tau\in(0,ch)$ the following holds true: for
all $b\in\mathcal{A}$, $\textbf{a},\textbf{b}\in Y(\tau)$ with
$\textbf{a},\textbf{b}\to b$, and $z_{1},z_{2}\in D_{b}$ we have

$\gamma_{\textbf{a}}(z_{1})$ and $\gamma_{\textbf{b}}(z_{2})$ belong to the
same connected component of $\Omega(h)\Longrightarrow\textbf{a}=\textbf{b}$.

###### Proof.

Fix $b\in\mathcal{A}$, $\textbf{a},\textbf{b}\in Y(\tau)$ with
$\textbf{a},\textbf{b}\to b$, and $z_{1},z_{2}\in D_{b}$. Suppose that
$\textbf{a}\neq\textbf{b}$ and that $\gamma_{\textbf{a}}(z_{1})$ and
$\gamma_{\textbf{b}}(z_{2})$ lie in the same component of $\Omega(h).$ Suppose
further that $\tau<ch$. We seek to obtain a contradiction for some
sufficiently small $c=c(\Gamma)>0$. By [1, Lemma 15.14] the diameter of each
connected component of $\Omega(h)$ is less that $Ch$, so we get

(32) $|\gamma_{\textbf{a}}(z_{1})-\gamma_{\textbf{b}}(z_{2})|\leqslant Ch.$

If the first letter of a and b are different, then
$\gamma_{\textbf{a}}(z_{1})$ and $\gamma_{\textbf{b}}(z_{2})$ lie in different
Schottky disks. This implies that

$|\gamma_{\textbf{a}}(z_{1})-\gamma_{\textbf{b}}(z_{2})|\geqslant K,$

where $K>0$ denotes the minimal distance between two Schottky disks, so we get
a contradiction for $h$ small enough. Thus we may assume that a and b have a
common prefix, so we write $\textbf{a}=\textbf{c}\textbf{a}_{1}$ and
$\textbf{b}=\textbf{c}\textbf{b}_{1}$ for some non-empty word
$\textbf{c}\in\mathcal{W}^{\circ}$. We assume that $|\textbf{c}|$ is maximal
with this property, that is, we assume that the first letters of
$\textbf{a}_{1}$ and $\textbf{b}_{1}$ are different, in which case

(33) $|\gamma_{\textbf{a}_{1}}(z_{1})-\gamma_{\textbf{b}_{1}}(z_{2})|\geqslant
K.$

Now we use the following identity which in valid for all
$\gamma\in\mathrm{PSL}_{2}(\mathbb{R})$ and all $w_{1},w_{2}\in\mathbb{C}$:

$|\gamma(w_{1})-\gamma(w_{2})|^{2}=|\gamma^{\prime}(w_{1})||\gamma^{\prime}(w_{2})||w_{1}-w_{2}|^{2}.$

Applying this to $\gamma=\gamma_{\textbf{c}}$,
$w_{1}=\gamma_{\textbf{a}_{1}}(z)$, and $w_{2}=\gamma_{\textbf{a}_{1}}(z)$
yields

$\displaystyle|\gamma_{\textbf{a}}(z_{1})-\gamma_{\textbf{b}}(z_{2})|^{2}$
$\displaystyle=|\gamma_{\textbf{c}}(\gamma_{\textbf{a}_{1}}(z))-\gamma_{\textbf{c}}(\gamma_{\textbf{b}_{1}}(z))|^{2}$
$\displaystyle=|\gamma_{\textbf{c}}^{\prime}(\gamma_{\textbf{a}_{1}}(z))||\gamma_{\textbf{c}}^{\prime}(\gamma_{\textbf{b}_{1}}(z))||\gamma_{\textbf{a}_{1}}(z)-\gamma_{\textbf{b}_{1}}(z)|^{2},$

or equivalently,

(34)
$|\gamma_{\textbf{a}}(z_{1})-\gamma_{\textbf{b}}(z_{2})|=|\gamma_{\textbf{c}}^{\prime}(\gamma_{\textbf{a}_{1}}(z))|^{1/2}|\gamma_{\textbf{c}}^{\prime}(\gamma_{\textbf{b}_{1}}(z))|^{1/2}|\gamma_{\textbf{a}_{1}}(z)-\gamma_{\textbf{b}_{1}}(z)|.$

Note that $\gamma_{\textbf{a}_{1}}^{\prime}(z_{1})$ is bounded from above by
some constant that depends solely on $\Gamma.$ Hence, using the chain rule and
the fact that $\textbf{a}\in Y(\tau)$, it follows from Lemma 2.4 that

$\gamma_{\textbf{c}}^{\prime}(\gamma_{\textbf{a}_{1}}(z_{1}))=\frac{\gamma_{\textbf{a}}^{\prime}(z_{1})}{\gamma_{\textbf{a}_{1}}^{\prime}(z_{1})}\geqslant
c_{0}\tau$

for some constant $c_{0}=c_{0}(\Gamma)>0$. Similarly, we obtain

$\gamma_{\textbf{c}}^{\prime}(\gamma_{\textbf{b}_{1}}(z_{2}))\geqslant
c_{0}\tau.$

Inserting this into (34) and using (33) yields

$|\gamma_{\textbf{a}}(z_{1})-\gamma_{\textbf{b}}(z_{2})|\geqslant
c_{0}\tau|\gamma_{\textbf{a}_{1}}(z_{1})-\gamma_{\textbf{b}_{1}}(z_{2})|\geqslant
c_{0}\tau K.$

When combined with (32), this forces $c_{0}K\tau\leqslant Ch,$ or equivalently
$h\geqslant\frac{c_{0}K}{C}\tau.$ We obtain the desired contradiction by
taking $c=\frac{c_{0}K}{2C}$. ∎

###  3.2. The phase and its derivative

Given words $\textbf{a},\textbf{b}\in\mathcal{W}^{\circ}$ and $z\in D$, define
the “phase” by

(35)
$\Phi_{\textbf{a},\textbf{b}}(z)\stackrel{{\scriptstyle\text{def}}}{{=}}\mathbb{L}(\gamma_{\textbf{a}}^{\prime}(z))-\mathbb{L}(\gamma_{\textbf{b}}^{\prime}(z)),$

where $\mathbb{L}$ is the complex logarithm defined in (15), so that for all
$s=\sigma+it$,

$\gamma_{\textbf{a}}^{\prime}(z)^{s}\overline{\gamma_{\textbf{b}}^{\prime}(z)^{s}}=\gamma_{\textbf{a}}^{\prime}(z)^{\sigma}\overline{\gamma_{\textbf{b}}^{\prime}(z)}^{\sigma}e^{it\Phi_{\textbf{a},\textbf{b}}(z)}.$

Moreover, define the quantity

(36)
$\mathcal{D}_{\textbf{a},\textbf{b}}\stackrel{{\scriptstyle\text{def}}}{{=}}\left(\frac{\Upsilon_{\textbf{a}}\Upsilon_{\textbf{b}}}{\Upsilon_{\textbf{a}\overline{\textbf{b}}}}\right)^{1/2}.$

It turns out that the derivatives of $\Phi_{\textbf{a},\textbf{b}}(z)$ are
controlled by $\mathcal{D}_{\textbf{a},\textbf{b}}$.

######  Proposition 3.2 (Phase derivatives).

For all $b\in\mathcal{A}$ and $\textbf{a},\textbf{b}\in\mathcal{W}^{\circ}$
with $\textbf{a},\textbf{b}\to b$ and $\textbf{a}\neq\textbf{b}$ the following
estimates hold true with implied constants depending only on $\Gamma$:

  1. (i)

For all $z\in D_{b}$ we have

$|\Phi_{\textbf{a},\textbf{b}}^{\prime}(z)|\asymp\mathcal{D}_{\textbf{a},\textbf{b}}.$

  2. (ii)

There exists some $C=C(\Gamma)>0$ such that for all $n\in\mathbb{N}$ the
$n$-th derivative satisfies

$|\Phi_{\textbf{a},\textbf{b}}^{(n)}(z)|\leqslant
n!C^{n}\mathcal{D}_{\textbf{a},\textbf{b}}.$

  3. (iii)

There exists some $r_{0}=r_{0}(\Gamma)>0$ such that for any two points
$z,z^{\prime}\in D_{b}$ with $|z^{\prime}-z|<r_{0}$ we have

$|\Phi_{\textbf{a},\textbf{b}}(z)-\Phi_{\textbf{a},\textbf{b}}(z^{\prime})|\gg\mathcal{D}_{\textbf{a},\textbf{b}}|z^{\prime}-z|.$

We need the following lemma:

######  Lemma 3.3.

For every $\textbf{a}\in\mathcal{W}^{\circ}$ write
$\gamma_{\textbf{a}}=\begin{pmatrix}a_{\textbf{a}}&b_{\textbf{a}}\\\
c_{\textbf{a}}&d_{\textbf{a}}\end{pmatrix}\in\mathrm{SL}_{2}(\mathbb{R})$.
Then we have

$|c_{\textbf{a}}|\asymp\Upsilon_{\textbf{a}}^{-1/2}$

with implied constants depending only on $\Gamma.$

###### Proof.

For all $b\in\mathcal{A}$, $\textbf{a}\in\mathcal{W}^{\circ}$ with
$\mathbf{a}\to b$, and $z\in D_{b}$ we have

$\gamma^{\prime}_{\textbf{a}}(z)=\frac{1}{(c_{\textbf{a}}z+d_{\textbf{a}})^{2}}=\frac{1}{c_{\textbf{a}}^{2}(z-x_{\textbf{a}})^{2}},$

where $x_{\textbf{a}}=-d_{\textbf{a}}/c_{\textbf{a}}.$ Write
$\textbf{a}=a_{1}\cdots a_{n}$ and note that $\mathbf{a}\to b$ is equivalent
to $\overline{a}_{n}\neq b$. Moreover, observe that

$x_{\textbf{a}}=\gamma_{\textbf{a}}^{-1}(\infty)=\gamma_{\overline{\textbf{a}}}(\infty)\in
D_{\overline{a}_{n}},$

which implies that for all $z\in D_{b}$ the quantity $|z-x_{\textbf{a}}|$ is
bounded from below and from above by some positive constants depending only on
$\Gamma$. Therefore, we have

$|\gamma^{\prime}_{\textbf{a}}(z)|\asymp\frac{1}{c_{\textbf{a}}^{2}}.$

Combining this with Part (iv) of Lemma 2.3, which asserts that
$\Upsilon_{\textbf{a}}\asymp|\gamma^{\prime}_{\textbf{a}}(z)|$, proves the
claim. ∎

###### Proof of Proposition 3.2.

Throughout this proof fix $b\in\mathcal{A}$, $z\in D_{b}$, and
$\textbf{a},\textbf{b}\in\mathcal{W}^{\circ}$ such that
$\textbf{a},\textbf{b}\to b$ and $\textbf{a}\neq\textbf{b}$. A simple
calculation shows that

(37)
$\Phi_{\textbf{a},\textbf{b}}^{\prime}(z)=2\left(\frac{c_{\textbf{a}}}{c_{\textbf{a}}z+d_{\textbf{a}}}-\frac{c_{\textbf{b}}}{c_{\textbf{b}}z+d_{\textbf{b}}}\right)=2\frac{c_{\textbf{a}}d_{\textbf{b}}-c_{\textbf{b}}d_{\textbf{a}}}{(c_{\textbf{a}}z+d_{\textbf{a}})(c_{\textbf{b}}z+d_{\textbf{b}})},$

which implies

$|\Phi_{\textbf{a},\textbf{b}}^{\prime}(z)|=2|c_{\textbf{a}}d_{\textbf{b}}-c_{\textbf{b}}d_{\textbf{a}}|\cdot|\gamma_{\textbf{a}}^{\prime}(z)|^{1/2}|\gamma_{\textbf{b}}^{\prime}(z)|^{1/2}.$

Combining this with Part (iv) of Lemma 2.3 gives

(38)
$|\Phi_{\textbf{a},\textbf{b}}^{\prime}(z)|\asymp|c_{\textbf{a}}d_{\textbf{b}}-c_{\textbf{b}}d_{\textbf{a}}|\cdot\Upsilon_{\textbf{a}}^{1/2}\Upsilon_{\textbf{b}}^{1/2}$

with implied constants depending only on $\Gamma.$ Now note that

$\gamma_{\textbf{a}\overline{\textbf{b}}}=\gamma_{\textbf{a}}\gamma_{\textbf{b}}^{-1}=\begin{pmatrix}a_{\textbf{a}}&b_{\textbf{a}}\\\
c_{\textbf{a}}&d_{\textbf{a}}\end{pmatrix}\begin{pmatrix}d_{\textbf{b}}&-b_{\textbf{b}}\\\
-c_{\textbf{b}}&a_{\textbf{b}}\end{pmatrix}=\begin{pmatrix}\ast&\ast\\\
c_{\textbf{a}}d_{\textbf{b}}-c_{\textbf{b}}d_{\textbf{a}}&\ast\end{pmatrix},$

so we can apply Lemma 3.3 to obtain

$|c_{\textbf{a}}d_{\textbf{b}}-c_{\textbf{b}}d_{\textbf{a}}|=|c_{\textbf{a}\overline{\textbf{b}}}|\asymp\frac{1}{\Upsilon_{\textbf{a}\overline{\textbf{b}}}^{1/2}},$

which when inserted into (38) leads to

$|\Phi_{\textbf{a},\textbf{b}}^{\prime}(z)|\asymp\frac{\Upsilon_{\textbf{a}}^{1/2}\Upsilon_{\textbf{b}}^{1/2}}{\Upsilon_{\textbf{a}\overline{\textbf{b}}}^{1/2}}=\mathcal{D}_{\textbf{a},\textbf{b}},$

proving Part (i).

Using (37) we calculate the $n$-th derivative of
$\Phi_{\textbf{a},\textbf{b}}$:

(39)
$\Phi_{\textbf{a},\textbf{b}}^{(n)}(z)=2(n-1)!\left(\frac{c_{\textbf{a}}^{n}}{(c_{\textbf{a}}z+d_{\textbf{a}})^{n}}-\frac{c_{\textbf{b}}^{n}}{(c_{\textbf{b}}z+d_{\textbf{b}})^{n}}\right).$

Writing

(40)
$\frac{c_{\textbf{a}}}{c_{\textbf{a}}z+d_{\textbf{a}}}=\frac{1}{z-x_{\textbf{a}}}$

with
$x_{\textbf{a}}=\gamma_{\textbf{a}}^{-1}(\infty)=\gamma_{\overline{\textbf{a}}}(\infty)$
and arguing as in the proof of Lemma 3.3, we deduce that (40) is bounded from
above by some constant $C>0$ depending only on $\Gamma$. Clearly, the same
holds true for $\textbf{b}.$ Now note that for all $u,v\in\mathbb{C}$ with
$|u|,|v|\leqslant C$ we have the bound $|u^{n}-v^{n}|\leqslant nC^{n}|u-v|.$
Applying this to (39), we obtain

$\displaystyle|\Phi_{\textbf{a},\textbf{b}}^{(n)}(z)|$ $\displaystyle\leqslant
2(n-1)!\left|\frac{c_{\textbf{a}}^{n}}{(c_{\textbf{a}}z+d_{\textbf{a}})^{n}}-\frac{c_{\textbf{b}}^{n}}{(c_{\textbf{b}}z+d_{\textbf{b}})^{n}}\right|$
$\displaystyle\leqslant
2n!C^{n}\left|\frac{c_{\textbf{a}}}{c_{\textbf{a}}z+d_{\textbf{a}}}-\frac{c_{\textbf{b}}}{c_{\textbf{b}}z+d_{\textbf{b}}}\right|$
$\displaystyle=n!C^{n}|\Phi_{\textbf{a},\textbf{b}}^{\prime}(z)|.$

Thus we obtain from Part (i) (possibly with a larger constant $C$)

$|\Phi_{\textbf{a},\textbf{b}}^{(n)}(z)|\leqslant
C^{n}n!\mathcal{D}_{\textbf{a},\textbf{b}},$

proving Part (ii).

Now we prove Part (iii). Fix points $z,z^{\prime}\in D_{b}$ and Taylor-expand
$\Phi_{\textbf{a},\textbf{b}}(z^{\prime})$ around $z$ to obtain

(41)
$\Phi_{\textbf{a},\textbf{b}}(z^{\prime})=\Phi_{\textbf{a},\textbf{b}}(z)+\Phi_{\textbf{a},\textbf{b}}^{\prime}(z)(z^{\prime}-z)+\sum_{n=2}^{\infty}\frac{\Phi_{\textbf{a},\textbf{b}}^{(n)}(z)}{n!}(z^{\prime}-z)^{n}.$

Using the triangle inequality as well as Parts (i) and (ii) gives

$\displaystyle|\Phi_{\textbf{a},\textbf{b}}(z^{\prime})-\Phi_{\textbf{a},\textbf{b}}(z)|$
$\displaystyle\geqslant|\Phi_{\textbf{a},\textbf{b}}^{\prime}(z)||z^{\prime}-z|-\sum_{n=2}^{\infty}\frac{|\Phi_{\textbf{a},\textbf{b}}^{(n)}(z)|}{n!}|z^{\prime}-z|^{n}$
$\displaystyle\geqslant
C_{0}\mathcal{D}_{\textbf{a},\textbf{b}}|z^{\prime}-z|-\sum_{n=2}^{\infty}C^{n}\mathcal{D}_{\textbf{a},\textbf{b}}|z^{\prime}-z|^{n}.$

Thus, if $|z^{\prime}-z|<r_{0}$ with some $r_{0}=r_{0}(C,C_{0})>0$ small
enough, we obtain

$\displaystyle|\Phi_{\textbf{a},\textbf{b}}(z^{\prime})-\Phi_{\textbf{a},\textbf{b}}(z)|$
$\displaystyle\geqslant
C_{0}\mathcal{D}_{\textbf{a},\textbf{b}}|z^{\prime}-z|-\frac{Cr_{0}}{1-Cr_{0}}\mathcal{D}_{\textbf{a},\textbf{b}}|z^{\prime}-z|$
$\displaystyle\geqslant\left(C_{0}-\frac{Cr_{0}}{1-Cr_{0}}\right)\mathcal{D}_{\textbf{a},\textbf{b}}|z^{\prime}-z|$
$\displaystyle\geqslant\frac{C_{0}}{2}\mathcal{D}_{\textbf{a},\textbf{b}}|z^{\prime}-z|,$

as desired. ∎

###  3.3. Averaged oscillatory integrals

Recall from §2 the following notations:

  * •

$D=\bigsqcup_{a\in\mathcal{A}}D_{a}$ is the union of all the Schottky disks
$D_{a}$, and

  * •

for $h>0$ and $a\in\mathcal{A}$ we define
$\Omega(h)=\Lambda+D_{\mathbb{C}}(0,h)$ and $\Omega_{a}(h)=\Omega(h)\cap
D_{a}$.

In our proof of Theorem 1.1 we are led to consider oscillatory integrals of
the form

$\int_{\Omega_{b}(h)}e^{it\Phi_{\textbf{a},\textbf{b}}(z)}f_{\textbf{a},\textbf{b}}(z)\operatorname{dvol}(z),$

where $\Phi_{\textbf{a},\textbf{b}}$ is the phase given by (35). The aim of
this section is to prove the following key estimate:

######  Proposition 3.4 (Key bound for averaged oscillatory integrals).

For all $b\in\mathcal{A}$, $\textbf{a},\textbf{b}\in\mathcal{W}^{\circ}$ with
$\textbf{a},\textbf{b}\to b$ and $\textbf{a}\neq\textbf{b}$, for all $T\gg 1$
large, for all $h>0$ small enough, for all measurable functions $f\colon
D\to\mathbb{C}$, and for all $\epsilon>0$ we have

$\frac{1}{T}\int_{0}^{T}\left|\int_{\Omega_{b}(h)}e^{it\Phi_{\textbf{a},\textbf{b}}(z)}f(z)\operatorname{dvol}(z)\right|dt\ll_{\epsilon}h^{1-\frac{\delta}{2}}\mathcal{D}_{\textbf{a},\textbf{b}}^{-1}T^{-1+\epsilon}e^{ChT}\|f\|_{\infty,D_{b}},$

where $\|f\|_{\infty,D_{b}}\stackrel{{\scriptstyle\text{def}}}{{=}}\sup_{z\in
D_{b}}|f(z)|$ and $\mathcal{D}_{\textbf{a},\textbf{b}}$ is the quantity in
(36). The implied constant and $C>0$ depend only on $\Gamma.$

###### Proof.

For the rest of this proof fix $h>0$ small enough so that $\Omega(h)\subset
D$, fix $b\in\mathcal{A}$ and $\textbf{a},\textbf{b}\in\mathcal{W}^{\circ}$
with $\textbf{a},\textbf{b}\to b$ and $\textbf{a}\neq\textbf{b}$, and write

$I_{T}\stackrel{{\scriptstyle\text{def}}}{{=}}\frac{1}{T}\int_{0}^{T}\left|\int_{\Omega_{b}(h)}e^{it\Phi_{\textbf{a},\textbf{b}}(z)}f(z)\operatorname{dvol}(z)\right|dt.$

Let $r_{0}>0$ be as in Part (iii) of Proposition 3.2. Clearly, $D$ can be
covered by finitely many euclidean disks of radius $r_{1}=r_{0}/2$. Let
$B_{1},\dots B_{m}$ be the sub-collection of those disks covering
$\Omega_{b}(h)=\Omega(h)\cap D_{b}$. Set $\Omega_{b}^{i}(h)=\Omega_{b}(h)\cap
B_{i}$ for all $i=1,\dots,m$. By construction, for any pair of points
$z,z^{\prime}\in\Omega_{b}^{i}(h)$, we have $|z^{\prime}-z|<r_{0}.$ Hence, by
Part (iii) of Proposition 3.2, we have for all
$z,z^{\prime}\in\Omega_{b}^{i}(h)$:

(42)
$|\Phi_{\textbf{a},\textbf{b}}(z)-\Phi_{\textbf{a},\textbf{b}}(z^{\prime})|\gg\mathcal{D}_{\textbf{a},\textbf{b}}|z^{\prime}-z|.$

We will use this bound further below. Here and throughout this proof, implied
constants are allowed to depend solely on $\Gamma$.

For all $i=1,\dots,m$ write

$I_{T,i}\stackrel{{\scriptstyle\text{def}}}{{=}}\frac{1}{T}\int_{0}^{T}\left|\int_{\Omega_{b}^{i}(h)}e^{it\Phi_{\textbf{a},\textbf{b}}(z)}f(z)\operatorname{dvol}(z)\right|dt.$

Clearly, by the triangle inequality we have

$I_{T}\leqslant I_{T,i}+\cdots+I_{T,m},$

so it suffices to prove the bound

$I_{T,i}\ll_{\epsilon}h^{1-\frac{\delta}{2}}\mathcal{D}_{\textbf{a},\textbf{b}}^{-1}T^{-1+\epsilon}e^{ChT}\|f\|_{\infty,D_{b}}$

for each $I_{T,i}$ individually. For the rest of this proof fix some index
$i\in\\{1,\dots,m\\}.$

Next, let $\psi\in C_{c}(\mathbb{R})$ be a non-negative, smooth function,
supported in $[-2,2]$ with $\psi=1$ in $[-1,1]$, and define

$\psi_{T}(x)\stackrel{{\scriptstyle\text{def}}}{{=}}\psi\left(\frac{x}{T}\right).$

Clearly, we have

$I_{T,i}\leqslant\frac{1}{T}\int_{-\infty}^{\infty}\psi_{T}(t)\left|\int_{\Omega_{b}^{i}(h)}e^{it\Phi_{\textbf{a},\textbf{b}}(z)}f(z)\operatorname{dvol}(z)\right|dt.$

Applying the Cauchy–Schwarz inequality gives

$\displaystyle I_{T}^{2}$
$\displaystyle\leqslant\frac{1}{T^{2}}\left(\int_{-\infty}^{\infty}\psi_{T}(t)dt\right)\left(\int_{-\infty}^{\infty}\psi_{T}(t)\left|\int_{\Omega_{b}^{i}(h)}e^{it\Phi_{\textbf{a},\textbf{b}}(z)}f(z)\operatorname{dvol}(z)\right|^{2}dt\right)$
$\displaystyle\ll\frac{1}{T}\int_{-\infty}^{\infty}\psi_{T}(t)\left|\int_{\Omega_{b}^{i}(h)}e^{it\Phi_{\textbf{a},\textbf{b}}(z)}f(z)\operatorname{dvol}(z)\right|^{2}dt$
$\displaystyle=\frac{1}{T}\int_{-\infty}^{\infty}\psi_{T}(t)\int_{\Omega_{b}^{i}(h)}\int_{\Omega_{b}^{i}(h)}e^{it(\Phi_{\textbf{a},\textbf{b}}(z_{1})-\Phi_{\textbf{a},\textbf{b}}(\overline{z_{2}}))}f(z_{1})\overline{f(z_{2})}\operatorname{dvol}(z_{1})\operatorname{dvol}(z_{2}).$

For last equality we opened the squared brackets and we used that

$\overline{\Phi_{\textbf{a},\textbf{b}}(z)}=\Phi_{\textbf{a},\textbf{b}}(\overline{z}).$

Using Fubini’s theorem (which is justified because the support of $\psi_{T}$
is compact) we can interchange integrals to obtain

$I_{T,i}^{2}\ll\frac{1}{T}\int_{\Omega_{b}^{i}(h)}\int_{\Omega_{b}^{i}(h)}\widehat{\psi}_{T}\left(\Phi_{\textbf{a},\textbf{b}}(\overline{z_{2}})-\Phi_{\textbf{a},\textbf{b}}(z_{1})\right)f(z_{1})\overline{f(z_{2})}\operatorname{dvol}(z_{1})\operatorname{dvol}(z_{2}),$

where the Fourier transform is defined for all $\varphi\in
C_{c}^{\infty}(\mathbb{R})$ as usual by

$\widehat{\varphi}(z)\stackrel{{\scriptstyle\text{def}}}{{=}}\int_{-\infty}^{\infty}\varphi(t)e^{-itz}dt.$

Using basic properties of the Fourier transform we find that
$\widehat{\psi}_{T}(z)=T\widehat{\psi}(Tz),$ whence

$I_{T,i}^{2}\ll\int_{\Omega_{b}(h)}\int_{\Omega_{b}(h)}\widehat{\psi}\left(T\left(\Phi_{\textbf{a},\textbf{b}}(\overline{z_{2}})-\Phi_{\textbf{a},\textbf{b}}(z_{1})\right)\right)f(z_{1})\overline{f(z_{2})}\operatorname{dvol}(z_{1})\operatorname{dvol}(z_{2}).$

Since $\psi$ is smooth and supported in $[-2,2]$ we have that for all $A>0$,

$|\widehat{\psi}(z)|\ll_{A}\frac{e^{2|\mathrm{Im}(z)|}}{(1+|z|)^{A}}.$

Therefore,

$\displaystyle I_{T,i}^{2}$
$\displaystyle\ll_{A}\left\|f\right\|_{\infty,D_{b}}^{2}\cdot\int_{\Omega_{b}^{i}(h)}\int_{\Omega_{b}^{i}(h)}\frac{e^{2T|\mathrm{Im}\left(\Phi_{\textbf{a},\textbf{b}}(\overline{z_{2}})-\Phi_{\textbf{a},\textbf{b}}(z_{1})\right)|}}{\left(1+T\left|\Phi_{\textbf{a},\textbf{b}}(\overline{z_{2}})-\Phi_{\textbf{a},\textbf{b}}(z_{1})\right|\right)^{A}}\operatorname{dvol}(z_{1})\operatorname{dvol}(z_{2})$
$\displaystyle\ll_{A}e^{2ChT}\left\|f\right\|_{\infty,D_{b}}^{2}\cdot\int_{\Omega_{b}^{i}(h)}\int_{\Omega_{b}^{i}(h)}\frac{1}{\left(1+T\left|\Phi_{\textbf{a},\textbf{b}}(\overline{z_{2}})-\Phi_{\textbf{a},\textbf{b}}(z_{1})\right|\right)^{A}}\operatorname{dvol}(z_{1})\operatorname{dvol}(z_{2}).$

For the second inequality we used that for all $z\in\Omega_{b}(h)$,

$|e^{it\Phi_{\textbf{a},\textbf{b}}(z)}|=|\gamma_{\textbf{a}}^{\prime}(z)^{it}\gamma_{\textbf{b}}^{\prime}(z)^{-it}|\ll
e^{Ch|t|}.$

To proceed, divide the double integral above as follows:

$\int_{\Omega_{b}^{i}(h)}\int_{\Omega_{b}^{i}(h)}=\int_{A^{r}}+\int_{B^{r}},$

where

$A^{r}=\\{(z_{1},z_{2})\in\Omega_{b}^{i}(h)\times\Omega_{b}^{i}(h):|z_{1}-\overline{z_{2}}|>r\\}$

and

$B^{r}=\\{(z_{1},z_{2})\in\Omega_{b}^{i}(h)\times\Omega_{b}^{i}(h):|z_{1}-\overline{z_{2}}|\leqslant
r\\}.$

Now take $r=\mathcal{D}_{\textbf{a},\textbf{b}}^{-1}T^{-1+\epsilon}$ for some
$\epsilon>0$. Invoking the estimate in (42), we obtain that for all
$(z_{1},z_{2})\in A^{r}$:

$\left|\Phi_{\textbf{a},\textbf{b}}(\overline{z_{2}})-\Phi_{\textbf{a},\textbf{b}}(z_{1})\right|\gg\mathcal{D}_{\textbf{a},\textbf{b}}r\gg
T^{-1+\epsilon}.$

It follows that

$\displaystyle\int_{A^{r}}\frac{1}{\left(1+T\left|\Phi_{\textbf{a},\textbf{b}}(\overline{z_{2}})-\Phi_{\textbf{a},\textbf{b}}(z_{1})\right|\right)^{A}}\operatorname{dvol}(z_{1})\operatorname{dvol}(z_{2})$
$\displaystyle\ll\int_{A^{r}}\frac{1}{\left(1+T^{\epsilon}\right)^{A}}\operatorname{dvol}(z_{1})\operatorname{dvol}(z_{2})$
$\displaystyle\ll T^{-\epsilon A}\operatorname{vol}(\Omega(h))^{2},$

where in the last inequality we simply used the fact that
$A^{r}\subseteq\Omega(h)\times\Omega(h).$ Using the volume bound from Lemma
2.6 we conclude that the integral over $A^{r}$ is bounded from above by

(43) $\ll_{A}T^{-\epsilon A}h^{4-2\delta}.$

The integral over $B^{r}$ can be bounded as follows:

$\displaystyle\int_{B^{r}}\frac{1}{\left(1+T\left|\phi(\overline{z_{2}})-\phi(z_{1})\right|\right)^{A}}\operatorname{dvol}(z_{1})\operatorname{dvol}(z_{2})$
$\displaystyle\leqslant\int_{B^{r}}1\operatorname{dvol}(z_{1})\operatorname{dvol}(z_{2})$
$\displaystyle\leqslant\int_{\Omega_{b}(h)}\int_{D(z_{1},r)}1\operatorname{dvol}(z_{2})\operatorname{dvol}(z_{1})$
$\displaystyle=\int_{\Omega_{b}(h)}\operatorname{vol}(D(z_{1},r))\operatorname{dvol}(z_{1})$
$\displaystyle\ll\int_{\Omega_{b}(h)}r^{2}\operatorname{dvol}(z_{2}).$

By the volume estimate and by our choice of $r$, the integral over $B_{r}$ is
less than

(44) $\operatorname{vol}(\Omega(h))r^{2}\ll
h^{2-\delta}\mathcal{D}_{\textbf{a},\textbf{b}}^{-2}T^{-2+2\epsilon}.$

Putting together the estimates (43) and (44), we obtain

$I_{T,i}^{2}\ll_{A}\left(h^{4-\delta}T^{-\epsilon
A}+h^{2-\delta}\mathcal{D}_{\textbf{a},\textbf{b}}^{-2}T^{-2+2\epsilon}\right)e^{2ChT}\left\|f\right\|_{\infty,D_{b}}^{2}.$

Finally, for any fixed $\epsilon>0$ we can choose $A>0$ so large that the
first term on the right gets absorbed by the second one, giving

$I_{T,i}^{2}\ll_{\epsilon}h^{2-\delta}\mathcal{D}_{\textbf{a},\textbf{b}}^{-2}T^{-2+2\epsilon}e^{2ChT}\left\|f\right\|_{\infty,D_{b}}^{2},$

as desired. ∎

###  3.4. Applying Jensen’s Formula

Given $h>0$ and resolution parameters $\tau_{0}>0$ and $\tau_{1}>0$, define
the composite operator

(45)
$\mathcal{L}_{\tau_{0},\tau_{1},s}\stackrel{{\scriptstyle\text{def}}}{{=}}\mathcal{L}_{\tau_{0},s}\circ\mathcal{L}_{\tau_{1},s}\colon
H^{2}(\Omega(h))\to H^{2}(\Omega(h)),$

where $\mathcal{L}_{\tau,s}$ is the $\tau$-refined transfer operator from
§2.6. Recall that (45) is well-defined, provided $h,\tau_{0},\tau_{1}$ are
small enough.

Given a trace-class operator $A\colon H\to H$ on a separable Hilbert space
$H$, the Hilbert–Schmidt norm is defined by

$\|A\|_{\mathrm{HS}}^{2}=\mathrm{tr}\left(A^{\ast}A\right),$

where $A^{\ast}$ denotes the adjoint of $A$. The next result relates the
resonance counting function $N_{X}(\sigma,T)$ to the Hilbert–Schmidt norm of
(45):

######  Proposition 3.5 (Key resonance counting bound).

There exist constants $K_{0}=K_{0}(\Gamma)>0$, $T_{0}=T_{0}(\Gamma)>0$, and
$\epsilon_{0}=\epsilon_{0}(\Gamma)>0$ such that for all $T>T_{0}$, $K>K_{0}$,
and $h,\tau_{0},\tau_{1}\in(0,\epsilon_{0})$ the following holds true:

$N_{X}(\sigma,T)\ll
K\max_{\begin{subarray}{c}\sigma-\frac{\alpha}{K}\leqslant\mathrm{Re}(s)\leqslant\beta
K\\\ |\mathrm{Im}(s)|\leqslant\beta
K\end{subarray}}\left(\int_{0}^{T}\|\mathcal{L}_{\tau_{0},\tau_{1},s+it}\|_{\mathrm{HS},h}^{2}dt\right)+(C\tau_{0}\tau_{1})^{K}T,$

where $\|\cdot\|_{\mathrm{HS},h}$ denotes the Hilbert–Schmidt norm of
operators $H^{2}(\Omega(h))\to H^{2}(\Omega(h)).$ The implied constants as
well as the constants $\alpha,\beta,C>0$ depend solely on $\Gamma.$

Proposition 3.5 is a consequence of the following variant of Jensen’s formula
that we specifically tailored to our purposes:

######  Lemma 3.6 (Adaptation of Jensen’s Formula).

Let $f$ be an entire function and let

$N_{f}(\sigma,T)\stackrel{{\scriptstyle\text{def}}}{{=}}\\#\\{s\in\mathbb{C}:\text{$f(s)=0$,
$\sigma\leqslant\mathrm{Re}(s)\leqslant\delta$,
$0\leqslant\mathrm{Im}(s)\leqslant T$}\\}.$

Then, for all $K>0$ sufficiently large and for all $\sigma\in\mathbb{R}$ with
$\sigma<\delta$, we have

$N_{f}(\sigma,T)\ll
K\max_{\begin{subarray}{c}\sigma-\frac{\alpha}{K}\leqslant\mathrm{Re}(s)\leqslant\beta
K\\\ |\mathrm{Im}(s)|\leqslant\beta
K\end{subarray}}\left(\int_{0}^{T}\log|f(s+it)|dt\right)-\int_{0}^{T}\log|f(\delta+K+it)|dt.$

The implied constant as well as $\alpha>0$ and $\beta>0$ are independent of
$f$, $\sigma$ and $K$.

###### Proof.

Fix some $t\in\mathbb{R}$ and consider the pair of concentric disks
$D_{i}=D_{\mathbb{C}}(s_{0},r_{i})$ with $i\in\\{1,2\\}$ centered at the point
$s_{0}=\sigma_{0}+it$ and with radii $r_{2}>r_{1}>0$. Assume that
$\sigma_{0},r_{1},r_{2}$ are chosen in such a way that

(46) $\\{s\in\mathbb{C}:\text{$\sigma\leqslant\mathrm{Re}(s)\leqslant\delta$,
$|\mathrm{Im}(s)-t|\leqslant 1$}\\}\subset\overline{D_{1}}\subset D_{2}.$

Define

$M_{f}(\sigma,t)\stackrel{{\scriptstyle\text{def}}}{{=}}\\#\\{s\in\mathbb{C}:\text{$f(s)=0$,
$\sigma\leqslant\mathrm{Re}(s)\leqslant\delta$, $|\mathrm{Im}(s)-t|\leqslant
1$}\\}$

Now we apply Jensen’s formula from complex analysis. We refer to Titchmarsh’s
classical book [19] for a proof of Jensen’s formula and its relatives. We
obtain

(47)
$M_{f}(\sigma,t)\leqslant\frac{1}{\log(r_{2}/r_{1})}\int_{0}^{1}\log|f(\sigma_{0}+r_{2}e^{2\pi
i\theta}+it)|d\theta-\log|f(\sigma_{0}+it)|.$

By integrating this estimate, we obtain

$\displaystyle N_{f}(\sigma,T)$
$\displaystyle\leqslant\int_{0}^{T}M_{f}(\sigma,t)dt$
$\displaystyle\leqslant\frac{1}{\log(r_{2}/r_{1})}\int_{0}^{1}\left(\int_{0}^{T}\log|f(\sigma_{0}+r_{2}e^{2\pi
i\theta}+it)|dt\right)d\theta-\int_{0}^{T}\log|f(\sigma_{0}+it)|dt.$

Let us now choose appropriate parameters $\sigma_{0},r_{1},r_{2}.$ For $K>1$,
we put

$\text{$\sigma_{0}=\delta+K$, $r_{1}=\sqrt{(\sigma_{0}-\sigma)^{2}+1}$ and
$r_{2}=r_{1}+1/K$}.$

One can verify that this choice guarantees that the inclusions in (46) hold
true. Furthermore, for $K$ large, the following estimates hold true with
absolute implied constants:

  1. (i)

$r_{1}\asymp r_{2}\asymp\sigma_{0}-\sigma\asymp K,$

  2. (ii)

$\sqrt{1+\frac{1}{(\sigma_{0}-\sigma)^{2}}}=1+O(\frac{1}{K^{2}}),$

  3. (iii)

$r_{1}=\sqrt{(\sigma_{0}-\sigma)^{2}+1}=(\sigma_{0}-\sigma)\sqrt{1+\frac{1}{(\sigma_{0}-\sigma)^{2}}}=(\sigma_{0}-\sigma)+O(\frac{1}{K}),$
and

  4. (iv)

$r_{2}=\sigma_{0}-\sigma+O(\frac{1}{K}).$

These estimates imply that for all $s=\sigma_{0}+r_{2}e^{2\pi i\theta}$ with
$\theta\in[0,1]$ we have

$\mathrm{Re}(s)\geqslant\sigma_{0}-r_{2}\geqslant\sigma-O(\frac{1}{K})$ and
$|\mathrm{Im}(s)|\leqslant\sigma_{0}+r_{2}=O(K)$.

Thus, we have

$\displaystyle\frac{1}{\log(r_{2}/r_{1})}\int_{0}^{1}\left(\int_{0}^{T}\log|f(\sigma_{0}+r_{2}e^{2\pi
i\theta}+it)|dt\right)d\theta$ $\displaystyle\qquad\qquad\qquad\ll
K\max_{\begin{subarray}{c}\sigma-O(\frac{1}{K})\leqslant\mathrm{Re}(s)\leqslant
O(K)\\\ |\mathrm{Im}(s)|\leqslant
O(K)\end{subarray}}\left(\int_{0}^{T}\log|f(s+it)|dt\right),$

where all implied constants are absolute. The proof of complete. ∎

We also need the following pointwise estimate:

######  Lemma 3.7 (Pointwise estimate).

For all sufficiently small resolution parameters $\tau_{0}>0$ and
$\tau_{1}>0$, and for all $s=\sigma+it\in\mathbb{C}$ with $\sigma>\delta$ the
Fredholm determinant of $\mathcal{L}_{\tau_{0},\tau_{1},s}^{2}$ satisfies

$-\log|\det\left(1-\mathcal{L}_{\tau_{0},\tau_{1},s}^{2}\right)|\leqslant\frac{(C\tau_{0}\tau_{1})^{2(\sigma-\delta)}}{1-(C\tau_{0}\tau_{1})^{2(\sigma-\delta)}},$

where $C>0$ depends only on $\Gamma.$

###### Proof.

We will adapt the argument of Magee–Naud [14]. Given a trace class operator
$A\colon H\to H$ acting on a separable Hilbert space $H$ with $\|A\|_{H}<1$,
we have the absolutely convergent series expansion

(48)
$\det(1-A)=\exp\left(-\sum_{n=1}^{\infty}\frac{1}{n}\mathrm{tr}(A^{n})\right),$

see for instance [6]. In particular,

(49)
$-\log|\det(1-A)|\leqslant\sum_{n=1}^{\infty}\frac{1}{k}|\mathrm{tr}(A^{n})|.$

Specializing this to $A=\mathcal{L}_{\tau_{0},\tau_{1},s}^{2}$ with
$\sigma=\mathrm{Re}(s)>\delta,$ we get

(50)
$-\log|\det(1-\mathcal{L}_{\tau_{0},\tau_{1},s}^{2})|\leqslant\sum_{n=1}^{\infty}\frac{1}{k}|\mathrm{tr}(\mathcal{L}_{\tau_{0},\tau_{1},s}^{2n})|.$

Therefore, proving the lemma is reduced to finding a suitable upper bound for
the trace of $\mathcal{L}_{\tau_{0},\tau_{1},s}^{2n}$. Using the formula (21)
we may write

(51) $\mathcal{L}_{\tau_{0},\tau_{1},s}f(z)=\sum_{\textbf{b}\in
S_{b}}\gamma^{\prime}_{\textbf{b}}(z)^{s}f(\gamma_{\textbf{b}}(z))\quad\text{for}\quad
z\in\Omega_{b}(h),$

where for each $b\in\mathcal{A}$ we put

$S_{b}=\\{\textbf{a}_{0}\textbf{a}_{1}\in Y(\tau_{0})\times
Y(\tau_{1}):\textbf{a}_{0}\to\textbf{a}_{1}\to b\\}.$

Here $\textbf{a}_{0}\textbf{a}_{1}\in Y(\tau_{0})\times Y(\tau_{1})$ means
that for $i\in\\{0,1\\}$ the sub-word $\textbf{a}_{i}$ belongs to
$Y(\tau_{i})$. Defining $S=S_{1}\sqcup\cdots\sqcup S_{b}$ and carefully
applying the Lefschetz fixed point formula (see [1, Lemma 15.9]), we get for
every $n\in\mathbb{N}$:

$\mathrm{tr}(\mathcal{L}_{\tau_{0},\tau_{1},s}^{n})=\sum_{\begin{subarray}{c}\textbf{b}_{n}\to\textbf{b}_{1}\to\dots\to\textbf{b}_{n}\\\
\textbf{b}_{1},\dots,\textbf{b}_{n}\in
S\end{subarray}}\frac{\gamma^{\prime}_{\textbf{b}_{1}\textbf{b}_{2}\cdots\textbf{b}_{n}}(x_{\textbf{b}_{1}\textbf{b}_{2}\cdots\textbf{b}_{n}})^{s}}{1-\gamma^{\prime}_{\textbf{b}_{1}\textbf{b}_{2}\cdots\textbf{b}_{n}}(x_{\textbf{b}_{1}\textbf{b}_{2}\cdots\textbf{b}_{n}})},$

where $x_{\textbf{b}_{1}\textbf{b}_{2}\cdots\textbf{b}_{n}}$ is the unique
attracting fixed point of
$\gamma_{\textbf{b}_{1}\textbf{b}_{2}\cdots\textbf{b}_{n}}$. Note that
$x_{\textbf{b}_{1}\textbf{b}_{2}\cdots\textbf{b}_{n}}\in D_{b}$ where $b$ is
the first letter of $\textbf{b}_{1}$ and the last letter of $\textbf{b}_{n}$.
Thus, applying Part (vii) of Lemma 2.3 $(n-1)$ times, we obtain for some
$C=C(\Gamma)>0$,

$\gamma^{\prime}_{\textbf{b}_{1}\textbf{b}_{2}\cdots\textbf{b}_{n}}(x_{\textbf{b}_{1}\textbf{b}_{2}\cdots\textbf{b}_{n}})\ll\Upsilon_{\textbf{b}_{1}\textbf{b}_{2}\cdots\textbf{b}_{n}}\leqslant
C^{n}\Upsilon_{\textbf{b}_{1}}\cdots\Upsilon_{\textbf{b}_{n}}.$

By the definition of $S$ it follows that for all $\textbf{b}\in S$,

$\Upsilon_{\textbf{b}}\ll\tau_{0}\tau_{1}.$

Thus (increasing $C$ if necessary) we have

$\gamma^{\prime}_{\textbf{b}_{1}\textbf{b}_{2}\cdots\textbf{b}_{n}}(x_{\textbf{b}_{1}\textbf{b}_{2}\cdots\textbf{b}_{n}})\leqslant(C\tau_{0}\tau_{1})^{n}.$

Now if $C\tau_{0}\tau_{1}<\frac{1}{2}$ then

$\frac{\gamma^{\prime}_{\textbf{b}_{1}\textbf{b}_{2}\cdots\textbf{b}_{n}}(x_{\textbf{b}_{1}\textbf{b}_{2}\cdots\textbf{b}_{n}})^{\sigma}}{1-\gamma^{\prime}_{\textbf{b}_{1}\textbf{b}_{2}\cdots\textbf{b}_{n}}(x_{\textbf{b}_{1}\textbf{b}_{2}\cdots\textbf{b}_{n}})}\ll\gamma^{\prime}_{\textbf{b}_{1}\textbf{b}_{2}\cdots\textbf{b}_{n}}(x_{\textbf{b}_{1}\textbf{b}_{2}\cdots\textbf{b}_{n}})^{\sigma}\ll(C\tau_{0}\tau_{1})^{\sigma
n},$

and therefore,

$\mathrm{tr}(\mathcal{L}_{\tau_{0},\tau_{1},s}^{n})\ll|S|^{n}(C\tau_{0}\tau_{1})^{\sigma
n}.$

By Lemma 2.4 the size of $S$ is bounded from above by

(52) $|S|\ll|Y(\tau_{0})||Y(\tau_{1})|\ll(\tau_{0}\tau_{1})^{-\delta},$

which when inserted above yields (possibly with a larger constant $C$)

$\mathrm{tr}(\mathcal{L}_{\tau_{0},\tau_{1},s}^{n})\ll(C\tau_{0}\tau_{1})^{n(\sigma-\delta)}.$

Returning to (50), taking $\tau_{0}$ and $\tau_{1}$ small, and using a
geometric series summation, we obtain

$\displaystyle-\log|\det(1-\mathcal{L}_{\tau_{0},\tau_{1},s}^{2})|$
$\displaystyle\ll\sum_{n=1}^{\infty}|\mathrm{tr}(\mathcal{L}_{\tau_{0},\tau_{1},s}^{2n})|$
$\displaystyle\ll\sum_{n=1}^{\infty}(C\tau_{0}\tau_{1})^{2n(\sigma-\delta)}$
$\displaystyle=\frac{(C\tau_{0}\tau_{1})^{2(\sigma-\delta)}}{1-(C\tau_{0}\tau_{1})^{2(\sigma-\delta)}},$

completing the proof. ∎

We are now ready to finish the proof of Proposition 3.5.

###### Proof of Proposition 3.5.

By Lemma 2.2 and the remarks thereafter, resonances for $X$ occur as zeros of

$f(s)=\det(1-\mathcal{L}_{\tau_{0},\tau_{1},s}^{2}).$

By Lemma 3.6 and Lemma 3.7, we have for all $K$ sufficiently large and for all
$\tau_{0}>0$ and $\tau_{1}>0$ sufficiently small:

(53) $N_{X}(\sigma,T)\ll
K\max_{\begin{subarray}{c}\sigma-\frac{\alpha}{K}\leqslant\mathrm{Re}(s)\leqslant\beta
K\\\ |\mathrm{Im}(s)|\leqslant\beta
K\end{subarray}}\left(\int_{0}^{T}|\det(1-\mathcal{L}_{\tau_{0},\tau_{1},s+it}^{2})|dt\right)+(C\tau_{0}\tau_{1})^{2K}T.$

It remains to show that

$|\det(1-\mathcal{L}_{\tau_{0},\tau_{1},s}^{2})|\leqslant\|\mathcal{L}_{\tau_{0},\tau_{1},s}\|_{\mathrm{HS},h}^{2},$

where $\|\cdot\|_{\mathrm{HS},h}$ denotes the Hilbert–Schmidt norm of
operators $H^{2}(\Omega(h))\to H^{2}(\Omega(h)).$ To that effect, let us
recall some basic facts on trace class operators and Fredholm determinants,
referring the reader to [6, 7, 18] for details. Weyl’s estimate on Fredholm
determinants states that for every trace-class operator $A\colon H\to H$ on a
separable Hilbert space $H$ we have

(54) $\log|\det\left(1-A\right)|\leqslant\|A\|_{\mathrm{tr}},$

where $\|\cdot\|_{\mathrm{tr}}$ denotes the trace norm. Moreover, for any two
Hilbert–Schmidt operators $A_{1},A_{2}\colon H\to H$ we have the inequality

(55)
$\|A_{1}A_{2}\|_{\mathrm{tr}}\leqslant\|A_{1}\|_{\mathrm{HS}}\|A_{2}\|_{\mathrm{HS}}.$

Now recall that if $h,\tau_{0},\tau_{1}$ are small enough, then

$\mathcal{L}_{\tau_{0},\tau_{1},s}\colon H^{2}(\Omega(h))\to H^{2}(\Omega(h))$

is a well-defined defined family of trace class operators, in which case we
can apply the above facts to $A=A_{1}=A_{2}=\mathcal{L}_{\tau_{0},\tau_{1},s}$
and $H=H^{2}(\Omega(h))$ to obtain

$|\det(1-\mathcal{L}_{\tau_{0},\tau_{1},s}^{2})|\leqslant\|\mathcal{L}_{\tau_{0},\tau_{1},s}^{2}\|_{\mathrm{tr}}\leqslant\|\mathcal{L}_{\tau_{0},\tau_{1},s}\|_{\mathrm{HS},h}^{2}$

as desired. ∎

###  3.5. Hilbert–Schmidt norm

The goal of this section is to prove the following explicit formula for the
Hilbert–Schmidt norm of $\mathcal{L}_{\tau_{0},\tau_{1},s}$:

######  Proposition 3.8 (Hilbert–Schmidt norm of
$\mathcal{L}_{\tau_{0},\tau_{1},s}$).

For all sufficiently small parameters $h,\tau_{0},\tau_{1}>0$ the
Hilbert–Schmidt norm of

$\mathcal{L}_{\tau_{0},\tau_{1},s}=\mathcal{L}_{\tau_{0},s}\circ\mathcal{L}_{\tau_{1},s}\colon
H^{2}(\Omega(h))\to H^{2}(\Omega(h))$

is given by

$\|\mathcal{L}_{\tau_{0},\tau_{1},s}\|_{\mathrm{HS},h}^{2}=\sum_{b\in\mathcal{A}}\sum_{\begin{subarray}{c}\textbf{a}=\textbf{a}_{0}\textbf{a}_{1}\in
Y(\tau_{0})\times Y(\tau_{1})\\\ \textbf{b}=\textbf{b}_{0}\textbf{b}_{1}\in
Y(\tau_{0})\times Y(\tau_{1})\\\ \textbf{a}_{0}\to\textbf{a}_{1}\to b\\\
\textbf{b}_{0}\to\textbf{b}_{1}\to
b\end{subarray}}\int_{\Omega_{b}(h)}\gamma^{\prime}_{\textbf{a}}(z)^{s}\overline{\gamma^{\prime}_{\textbf{b}}(z)^{s}}B_{\Omega(h)}(\gamma_{\textbf{a}}(z),\gamma_{\textbf{b}}(z))\operatorname{dvol}(z).$

Here, we write $\textbf{a}=\textbf{a}_{0}\textbf{a}_{1}\in S_{0}\times S_{1}$
to mean that for $i\in\\{0,1\\}$ the sub-word $\textbf{a}_{i}$ belongs to
$S_{i}$. Moreover, $B_{\Omega(h)}(\cdot,\cdot)$ is the Bergman kernel of
$H^{2}(\Omega(h))$.

###### Proof.

Proofs of formulas for the Hilbert–Schmidt norm for similar operators can be
found in [14, Lemma 4.7] and in [17, Proposition 5.5]. We will give an
alternative but essentially equivalent argument. First, we write down the
formula for $\mathcal{L}_{\tau_{0},\tau_{1},s}$ using (21):

(56)
$\mathcal{L}_{\tau_{0},\tau_{1},s}f(z)=\sum_{\begin{subarray}{c}\textbf{a}=\textbf{a}_{0}\textbf{a}_{1}\in
Y(\tau_{0})\times Y(\tau_{1})\\\ \textbf{a}_{0}\to\textbf{a}_{1}\to
b\end{subarray}}\gamma^{\prime}_{\textbf{a}}(z)^{s}f(\gamma_{\textbf{a}}(z))\quad\text{for}\quad
z\in\Omega_{b}(h).$

For notational convenience we rewrite this as

(57) $\mathcal{L}_{\tau_{0},\tau_{1},s}f(z)=\sum_{\textbf{a}\in
S_{b}}\gamma^{\prime}_{\textbf{a}}(z)^{s}f(\gamma_{\textbf{a}}(z))\quad\text{for}\quad
z\in\Omega_{b}(h),$

where for each $b\in\mathcal{A}$ we put

$S_{b}\stackrel{{\scriptstyle\text{def}}}{{=}}\\{\textbf{a}=\textbf{a}_{0}\textbf{a}_{1}\in
Y(\tau_{0})\times Y(\tau_{1}):\textbf{a}_{0}\to\textbf{a}_{1}\to b\\}.$

Recall from §2.9 that the Bergman kernel $B_{h}(z,w)$ is defined by the
property

$\int_{\Omega(h)}B_{\Omega(h)}(z,w)f(w)\operatorname{dvol}(w)=f(z)$

for all $f\in H^{2}(D)$ and $z\in D$. Thus the operator
$\mathcal{L}_{\tau_{0},\tau_{1},s}$ is a kernel integral operator

$\mathcal{L}_{\tau_{0},\tau_{1},s}f(z)=\int_{\Omega(h)}K(z,w)f(w)\operatorname{dvol}(w)$

whose kernel is given by the formula

$K(z,w)=\sum_{\textbf{a}\in
S_{b}}\gamma^{\prime}_{\textbf{a}}(z)^{s}B_{\Omega(h)}(\gamma_{\textbf{a}}(z),w)\quad\text{for}\quad
z\in\Omega_{b}(h).$

The Hilbert–Schmidt norm can now be computed as follows:

$\displaystyle\|$
$\displaystyle\mathcal{L}_{\tau_{0},\tau_{1},s}\|_{\mathrm{HS}}^{2}=\int_{\Omega(h)}\int_{\Omega(h)}|K(z,w)|^{2}\operatorname{dvol}(w)\operatorname{dvol}(z)$
$\displaystyle=\sum_{b\in\mathcal{A}}\int_{\Omega_{b}(h)}\int_{\Omega(h)}|K(z,w)|^{2}\operatorname{dvol}(w)\operatorname{dvol}(z)$
$\displaystyle=\sum_{b\in\mathcal{A}}\int_{\Omega_{b}(h)}\int_{\Omega(h)}\sum_{(\textbf{a},\textbf{b})\in
S_{b}\times
S_{b}}\gamma^{\prime}_{\textbf{a}}(z)^{s}\overline{\gamma^{\prime}_{\textbf{b}}(z)^{s}}B_{\Omega(h)}(\gamma_{\textbf{a}}(z),w)\overline{B_{\Omega(h)}(\gamma_{\textbf{b}}(z),w)}\operatorname{dvol}(w)\operatorname{dvol}(z)$

Using the defining property of the Bergman kernel we obtain the relation

$\displaystyle\int_{\Omega(h)}B_{\Omega(h)}(\gamma_{\textbf{a}}(z),w)\overline{B_{\Omega(h)}(\gamma_{\textbf{b}}(z),w)}\operatorname{dvol}(w)$
$\displaystyle=\int_{\Omega(h)}B_{\Omega(h)}(\gamma_{\textbf{a}}(z),w)B_{\Omega(h)}(w,\gamma_{\textbf{b}}(z))\operatorname{dvol}(w)$
$\displaystyle=B_{\Omega(h)}(\gamma_{\textbf{a}}(z),\gamma_{\textbf{b}}(z)),$

which when inserted into the last line above completes the proof. ∎

###  3.6. Finishing the proof of the main theorem

Now we finish the proof of Theorem 1.1. It follows directly from Proposition
3.5 and the next estimate:

######  Proposition 3.9 (Main technical estimate).

For all $T\gg 1$ large, $\sigma\in[0,\delta]$ and $s\in\mathbb{C}$ with
$\mathrm{Re}(s)\geqslant\sigma$ and $|\mathrm{Im}(s)|\leqslant T$ there exist
positive parameters $h,\tau_{0},\tau_{1}$ such that for all $\epsilon>0$:

$\frac{1}{T}\int_{0}^{T}\|\mathcal{L}_{\tau_{0},\tau_{1},s+it}\|_{\mathrm{HS},h}^{2}dt\ll_{\epsilon}T^{\delta-\frac{3\delta+2}{2\delta+2}(2\sigma-\delta)+\epsilon}.$

Furthermore, this is achieved by choosing $h=T^{-1}$ and resolution parameters
$\tau_{0}$ and $\tau_{1}$ satisfying $\tau_{0}\asymp T^{-1}$ and
$\tau_{1}\asymp T^{-\frac{\delta}{2\delta+2}}$ with implied constants
depending only on $\Gamma$.

We now deduce Theorem 1.1 from this estimate. By Proposition 3.5, we have for
all $K$ large enough

$N_{X}(\sigma,T)\ll
K\max_{\begin{subarray}{c}\sigma-\frac{\alpha}{K}\leqslant\mathrm{Re}(s)\leqslant\beta
K\\\ |\mathrm{Im}(s)|\leqslant\beta
K\end{subarray}}\left(\int_{0}^{T}\|\mathcal{L}_{\tau_{0},\tau_{1},s+it}\|_{\mathrm{HS},h}^{2}dt\right)+(C\tau_{0}\tau_{1})^{K}T.$

Combining this with Proposition 3.9 with parameters $h=T^{-1}$,
$\tau_{0}\asymp T^{-1},\tau_{1}\asymp T^{-\frac{\delta}{2\delta+2}}$ as in the
statement, we obtain

$\displaystyle N_{X}(\sigma,T)$
$\displaystyle\ll_{\epsilon}KT^{1+\delta-\frac{3\delta+2}{2\delta+2}(2\sigma-\delta)+O(\frac{1}{K})+\epsilon}+T^{1-AK},$

where $A>0$ is independent of $T.$ It remains to choose $K$ suitably, which
may be done by taking $K=\log T$. Clearly, with this choice, the second term
$T^{1-AK}$ gets absorbed by the first one, and
$KT^{O(\frac{1}{K})}\ll_{\epsilon}T^{\epsilon}$. This concludes the proof of
Theorem 1.1 assuming Proposition 3.9. The proof of the latter occupies the
rest of this section.

###### Proof of Proposition 3.9.

Let $h,\tau_{0},\tau_{1}$ be sufficiently small positive parameters such that
the operator

(58) $\mathcal{L}_{\tau_{0},\tau_{1},s}\colon H^{2}(\Omega(h))\to
H^{2}(\Omega(h))$

is well-defined. Furthermore, assume $T\gg 1$ to be large and take $h=T^{-1}$.
The parameters $\tau_{0}$ and $\tau_{1}$ will be chosen in the course of the
proof.

Using Proposition 3.8 we can write down the following formula for the
Hilbert–Schmidt norm of (58):

$\|\mathcal{L}_{\tau_{0},\tau_{1},s}\|_{\mathrm{HS},h}^{2}=\sum_{b\in\mathcal{A}}\sum_{\begin{subarray}{c}\textbf{a}=\textbf{a}_{0}\textbf{a}_{1}\in
Y(\tau_{0})\times Y(\tau_{1})\\\ \textbf{b}=\textbf{b}_{0}\textbf{b}_{1}\in
Y(\tau_{0})\times Y(\tau_{1})\\\ \textbf{a}_{0}\to\textbf{a}_{1}\to b\\\
\textbf{b}_{0}\to\textbf{b}_{1}\to
b\end{subarray}}\int_{\Omega_{b}(h)}\gamma^{\prime}_{\textbf{a}}(z)^{s}\overline{\gamma^{\prime}_{\textbf{b}}(z)^{s}}B_{\Omega(h)}(\gamma_{\textbf{a}}(z),\gamma_{\textbf{b}}(z))\operatorname{dvol}(z).$

Now we invoke the separation lemma (Lemma 3.1). Fix $b\in\mathcal{A}$, a point
$z\in D_{b}$, and words
$\textbf{a}=\textbf{a}_{0}\textbf{a}_{1},\textbf{b}=\textbf{b}_{0}\textbf{b}_{1}\in
Y(\tau_{0})\times Y(\tau_{1})$ such that $\textbf{a}_{0}\to\textbf{a}_{1}\to
b$ and $\textbf{b}_{0}\to\textbf{b}_{1}\to b.$ If we take $\tau_{0}=ch$ with
some sufficiently small $c>0$, then $\textbf{a}_{0}\neq\textbf{b}_{0}$ implies
that $\gamma_{\textbf{a}}(z)$ and $\gamma_{\textbf{b}}(z)$ lie in different
connected components of $\Omega(h)$. Thus, taking $\tau_{0}=ch$ and recalling
that $B_{\Omega(h)}(z,w)=0$ if $z$ and $w$ are in different connected
components, the above sum reduces to summands with
$\textbf{a}_{0}=\textbf{b}_{0}$:

$\|\mathcal{L}_{\tau_{0},\tau_{1},s}\|_{\mathrm{HS},h}^{2}=\sum_{b\in\mathcal{A}}\sum_{\begin{subarray}{c}\textbf{a}=\textbf{a}_{0}\textbf{a}_{1}\in
Y(\tau_{0})\times Y(\tau_{1})\\\ \textbf{b}=\textbf{b}_{0}\textbf{b}_{1}\in
Y(\tau_{0})\times Y(\tau_{1})\\\ \textbf{a}_{0}\to\textbf{a}_{1}\to b\\\
\textbf{b}_{0}\to\textbf{b}_{1}\to b\\\
\textbf{a}_{0}=\textbf{b}_{0}\end{subarray}}\int_{\Omega_{b}(h)}\gamma^{\prime}_{\textbf{a}}(z)^{s}\overline{\gamma^{\prime}_{\textbf{b}}(z)^{s}}B_{\Omega(h)}(\gamma_{\textbf{a}}(z),\gamma_{\textbf{b}}(z))\operatorname{dvol}(z).$

For ease of notation, we re-express this as

(59)
$\|\mathcal{L}_{\tau_{0},\tau_{1},s}\|_{\mathrm{HS},h}^{2}=\sum_{b\in\mathcal{A}}\sum_{(\textbf{a},\textbf{b})\in
Q_{b}}\int_{\Omega_{b}(h)}\gamma^{\prime}_{\textbf{a}}(z)^{s}\overline{\gamma^{\prime}_{\textbf{b}}(z)^{s}}B_{\Omega(h)}(\gamma_{\textbf{a}}(z),\gamma_{\textbf{b}}(z))\operatorname{dvol}(z),$

where for each $b\in\mathcal{A}$ the set $Q_{b}$ consists of all pairs of
words $(\textbf{a},\textbf{b})$ satisfying

  * •

$\textbf{a}=\textbf{c}\textbf{a}_{1}$ and
$\textbf{b}=\textbf{c}\textbf{b}_{1}$ where $\textbf{c}\in Y(\tau_{0})$ and
$\textbf{a}_{1},\textbf{b}_{1}\in Y(\tau_{1})$, and

  * •

$c\to\textbf{a}_{1},\textbf{b}_{1}\to b.$

Next we decompose each $Q_{b}$ as

$Q_{b}=Q_{b}^{(1)}\sqcup Q_{b}^{(2)}$

where

$Q_{b}^{(1)}\stackrel{{\scriptstyle\text{def}}}{{=}}\\{(\textbf{a},\textbf{a})\in
Q_{b}\\},$

is the diagonal subset in $Q_{b}$, and $Q_{b}^{(2)}$ is the set of pairs
$(\textbf{a},\textbf{b})$ where $\textbf{a}=\textbf{c}\textbf{a}_{1}$ and
$\textbf{b}=\textbf{c}\textbf{b}_{1}$ satisfy the above conditions and
$\textbf{a}_{1}\neq\textbf{b}_{1}$.

We decompose the right hand side of (59) accordingly:

$\|\mathcal{L}_{\tau_{0},\tau_{1},s}\|_{\mathrm{HS},h}^{2}=\Sigma^{(1)}(s)+\Sigma^{(2)}(s),$

where

$\displaystyle\Sigma^{(1)}(s)$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\sum_{b\in\mathcal{A}}\sum_{(\textbf{a},\textbf{a})\in
Q_{b}^{(1)}}\int_{\Omega_{b}(h)}\gamma^{\prime}_{\textbf{a}}(z)^{s}\overline{\gamma^{\prime}_{\textbf{a}}(z)^{s}}B_{\Omega(h)}(\gamma_{\textbf{a}}(z),\gamma_{\textbf{a}}(z))\operatorname{dvol}(z)$
$\displaystyle=\sum_{b\in\mathcal{A}}\sum_{\begin{subarray}{c}\textbf{a}=\textbf{c}\textbf{a}_{1}\in
Y_{0}\times Y_{1}\\\ \textbf{c}\to\textbf{a}_{1}\to
b\end{subarray}}\int_{\Omega_{b}(h)}\gamma^{\prime}_{\textbf{a}}(z)^{s}\overline{\gamma^{\prime}_{\textbf{a}}(z)^{s}}B_{\Omega(h)}(\gamma_{\textbf{a}}(z),\gamma_{\textbf{a}}(z))\operatorname{dvol}(z)$

and

$\Sigma^{(2)}(s)\stackrel{{\scriptstyle\text{def}}}{{=}}\sum_{b\in\mathcal{A}}\sum_{(\textbf{a},\textbf{b})\in
Q_{b}^{(2)}}\int_{\Omega_{b}(h)}\gamma^{\prime}_{\textbf{a}}(z)^{s}\overline{\gamma^{\prime}_{\textbf{b}}(z)^{s}}B_{\Omega(h)}(\gamma_{\textbf{a}}(z),\gamma_{\textbf{b}}(z))\operatorname{dvol}(z).$

To proceed we need to recall a few estimates for the terms appearing above. By
the contraction property (Lemma 2.5) we have for all
$(\textbf{a},\textbf{b})\in Q_{b}$ and $z\in D_{b}$:

$\mathrm{dist}(\gamma_{\textbf{a}}(z),\partial\Omega(h))\gg h$

and

$\mathrm{dist}(\gamma_{\textbf{b}}(z),\partial\Omega(h))\gg h.$

Combining this with Lemma 2.8 gives

$B_{\Omega(h)}(\gamma_{\textbf{a}}(z),\gamma_{\textbf{b}}(z))\leqslant\frac{1}{\pi\mathrm{dist}(\gamma_{\textbf{a}}(z),\partial\Omega(h))\mathrm{dist}(\gamma_{\textbf{b}}(z),\partial\Omega(h))}\ll
h^{-2}.$

Let $\sigma\in[0,\delta]$ and let $s\in\mathbb{C}$ with
$\mathrm{Re}(s)\geqslant\sigma$ and $|\mathrm{Im}(s)|\leqslant K$. By (25)
there exists $C=C(\Gamma)>0$ such that for all $t\in(0,T)$, and for all
$b\in\mathcal{A}$ and $(\textbf{a},\textbf{b})\in Q_{b}$:

$|\gamma^{\prime}_{\textbf{a}}(z)^{s+it}|\ll\Upsilon_{\textbf{a}}^{\sigma}e^{Ch(K+t)}\ll\tau_{0}^{\sigma}\tau_{1}^{\sigma}e^{Ch(K+T)}.$

Recalling further that $K=O(T)$ and $h=T^{-1}$, this gives

$|\gamma^{\prime}_{\textbf{a}}(z)^{s+it}|\ll\tau_{0}^{\sigma}\tau_{1}^{\sigma}.$

Similarly, we have

$|\gamma^{\prime}_{\textbf{b}}(z)^{s+it}|\ll\tau_{0}^{\sigma}\tau_{1}^{\sigma}.$

Furthermore, by Lemma 2.4 the cardinality of the set $Y(\tau)$ is bounded by

$|Y(\tau)|\asymp\tau^{-\delta},$

and by Lemma 2.6 the volume of $\Omega(h)$ is bounded by

$\operatorname{vol}(\Omega(h))\ll h^{2-\delta}.$

Using the triangle inequality and the above estimates, the diagonal term
$\Sigma^{(1)}$ may be bounded for all $t\in[0,T]$ as follows:

$\displaystyle\Sigma^{(1)}(s+it)$
$\displaystyle\ll\sum_{b\in\mathcal{A}}\sum_{\begin{subarray}{c}\textbf{a}=\textbf{c}\textbf{a}_{1}\in
Y_{0}\times Y_{1}\\\ \textbf{c}\to\textbf{a}_{1}\to
b\end{subarray}}\int_{\Omega_{b}(h)}|\gamma^{\prime}_{\textbf{a}}(z)^{s}|^{2}|B_{\Omega(h)}(\gamma_{\textbf{a}}(z),\gamma_{\textbf{a}}(z))|\operatorname{dvol}(z)$
$\displaystyle\ll\sum_{b\in\mathcal{A}}\sum_{\begin{subarray}{c}\textbf{a}=\textbf{c}\textbf{a}_{1}\in
Y_{0}\times Y_{1}\\\ \textbf{c}\to\textbf{a}_{1}\to
b\end{subarray}}\tau_{0}^{2\sigma}\tau_{1}^{2\sigma}\operatorname{vol}(\Omega(h))h^{-2}$
$\displaystyle\ll|Y(\tau_{0})||Y(\tau_{1})|\tau_{0}^{2\sigma}\tau_{1}^{2\sigma}\operatorname{vol}(\Omega(h))h^{-2}$
$\displaystyle\ll\tau_{0}^{-\delta+2\sigma}\tau_{1}^{-\delta+2\sigma}\operatorname{vol}(\Omega(h))h^{-2}$
$\displaystyle\ll\tau_{0}^{-\delta+2\sigma}\tau_{1}^{-\delta+2\sigma}h^{-\delta}.$

Recalling that $\tau_{0}\asymp h$ and $h=T^{-1}$ gives

$\Sigma^{(1)}(s+it)\ll T^{2\delta-2\sigma}\tau_{1}^{-\delta+2\sigma}$

for all $t\in[0,T]$, and consequently

(60) $\frac{1}{T}\int_{0}^{T}\Sigma^{(1)}(s+it)dt\ll
T^{2\delta-2\sigma}\tau_{1}^{-\delta+2\sigma}.$

In order to bound the non-diagonal term $\Sigma^{(2)}(s+it)$, we appeal to the
oscillatory integral estimate in Proposition 3.4. First observe that

$\gamma^{\prime}_{\textbf{a}}(z)^{s+it}\overline{\gamma^{\prime}_{\textbf{b}}(z)^{s+it}}=\gamma^{\prime}_{\textbf{a}}(z)^{s}\overline{\gamma^{\prime}_{\textbf{b}}(z)^{s}}e^{it\Phi_{\textbf{a},\textbf{b}}(z)},$

where $\Phi_{\textbf{a},\textbf{b}}$ is the phase defined in (35), so

$\Sigma^{(2)}(s+it)=\sum_{b\in\mathcal{A}}\sum_{(\textbf{a},\textbf{b})\in
Q_{b}^{(2)}}\int_{\Omega_{b}(h)}\gamma^{\prime}_{\textbf{a}}(z)^{s}\overline{\gamma^{\prime}_{\textbf{b}}(z)^{s}}e^{it\Phi_{\textbf{a},\textbf{b}}(z)}B_{\Omega(h)}(\gamma_{\textbf{a}}(z),\gamma_{\textbf{b}}(z))\operatorname{dvol}(z).$

Recall that $Q_{b}^{(2)}$ contains all pairs $(\textbf{a},\textbf{b})$ of
words $\textbf{a}=\textbf{c}\textbf{a}_{1}$ and
$\textbf{b}=\textbf{c}\textbf{b}_{1}$ such that

  * •

$\textbf{c}\in Z(\tau_{0})$ and $\textbf{a}_{1},\textbf{b}_{1}\in
Y(\tau_{1})$,

  * •

$\textbf{a}_{1}\neq\textbf{b}_{1}$, and

  * •

$\textbf{c}\to\textbf{a}_{1},\textbf{b}_{1}\to b$.

In particular, we have
$\textbf{a}\overline{\textbf{b}}=\textbf{c}\textbf{a}_{1}\overline{\textbf{b}}_{1}\overline{\textbf{c}}$
and

$\textbf{c}\to\textbf{a}_{1}\overline{\textbf{b}}_{1}\to\overline{\textbf{c}}.$

Therefore, applying Parts (vi) and (vii) of Lemma 2.3 gives

  * •

$\Upsilon_{\textbf{a}\overline{\textbf{b}}}=\Upsilon_{\textbf{c}\textbf{a}_{1}\overline{\textbf{b}}_{1}\overline{\textbf{c}}}\asymp\Upsilon_{\textbf{c}}\Upsilon_{\textbf{a}_{1}\overline{\textbf{b}}_{1}}\Upsilon_{\overline{\textbf{c}}}\ll\Upsilon_{\textbf{c}}^{2},$

  * •

$\Upsilon_{\textbf{a}}\asymp\Upsilon_{\textbf{c}}\Upsilon_{\textbf{a}_{1}},$
and

  * •

$\Upsilon_{\textbf{b}}\asymp\Upsilon_{\textbf{c}}\Upsilon_{\textbf{b}_{1}}.$

This implies that

$\mathcal{D}_{\textbf{a},\textbf{b}}=\left(\frac{\Upsilon_{\textbf{a}}\Upsilon_{\textbf{b}}}{\Upsilon_{\textbf{a}\overline{\textbf{b}}}}\right)^{1/2}\gg\left(\frac{\Upsilon_{\textbf{c}}^{2}\Upsilon_{\textbf{a}_{1}}\Upsilon_{\textbf{b}_{1}}}{\Upsilon_{\textbf{c}}^{2}}\right)^{1/2}=\Upsilon_{\textbf{a}_{1}}^{1/2}\Upsilon_{\textbf{b}_{1}}^{1/2}\gg\tau_{1}.$

Thus, by Proposition 3.4 and the previous estimates, we obtain

$\displaystyle\frac{1}{T}\int_{0}^{T}$ $\displaystyle\Sigma^{(2)}(s+it)dt$
$\displaystyle\ll\sum_{b\in\mathcal{A}}\sum_{(\textbf{a},\textbf{b})\in
Q_{b}^{(2)}}\frac{1}{T}\int_{0}^{T}\left|\int_{\Omega_{b}(h)}\gamma^{\prime}_{\textbf{a}}(z)^{s}\overline{\gamma^{\prime}_{\textbf{b}}(z)^{s}}e^{it\Phi_{\textbf{a},\textbf{b}}(z)}B_{\Omega(h)}(\gamma_{\textbf{a}}(z),\gamma_{\textbf{b}}(z))\operatorname{dvol}(z)\right|dt$
$\displaystyle\ll_{\epsilon}\sum_{b\in\mathcal{A}}\sum_{(\textbf{a},\textbf{b})\in
Q_{b}^{(2)}}\mathcal{D}_{\textbf{a},\textbf{b}}^{-1}T^{-1+\epsilon}h^{1-\delta/2}\left\|\gamma^{\prime}_{\textbf{a}}(\cdot)^{s}\overline{\gamma^{\prime}_{\textbf{b}}(\cdot)^{s}}B_{\Omega(h)}(\gamma_{\textbf{a}}(\cdot),\gamma_{\textbf{b}}(\cdot))\right\|_{\infty,D_{b}}$
$\displaystyle\ll\sum_{(\textbf{a},\textbf{b})\in
Q_{b}^{(2)}}\tau_{0}^{2\sigma}\tau_{1}^{2\sigma}\mathcal{D}_{\textbf{a},\textbf{b}}^{-1}T^{-1+\epsilon}h^{-1-\delta/2}$
$\displaystyle\ll|Q_{b}^{(2)}|\cdot\tau_{0}^{2\sigma}\tau_{1}^{2\sigma}\tau_{1}^{-1}T^{-1+\epsilon}h^{-1-\delta/2}.$

The size of $Q_{b}^{(2)}$ is bounded by

$|Q_{b}^{(2)}|\ll|Y(\tau_{0})|\cdot|Y(\tau_{1})|^{2}\ll\tau_{0}^{-\delta}\tau_{1}^{-2\delta}.$

Inserting this into the previous line and using $\tau_{0}\asymp h$ and
$h=T^{-1}$ gives

(61) $\displaystyle\frac{1}{T}\int_{0}^{T}\Sigma^{(2)}(s+it)dt$
$\displaystyle\ll_{\epsilon}\tau_{0}^{-\delta+2\sigma}\tau_{1}^{-2\delta+2\sigma}\tau_{1}^{-1}T^{-1+\epsilon}h^{-1-\delta/2}$
(62) $\displaystyle\ll
T^{2\delta-2\sigma+\epsilon}\tau_{1}^{-2\delta+2\sigma}\cdot\tau_{1}^{-1}T^{-\delta/2}.$

Putting everything together yields

$\displaystyle\frac{1}{T}\int_{0}^{T}\|\mathcal{L}_{\tau_{0},\tau_{1},s+it}\|_{\mathrm{HS},h}^{2}dt$
$\displaystyle=\frac{1}{T}\int_{0}^{T}\Sigma^{(1)}(s+it)dt+\frac{1}{T}\int_{0}^{T}\Sigma^{(2)}(s+it)dt$
$\displaystyle\ll\eqref{final1}+\eqref{final2}$
$\displaystyle\ll_{\epsilon}T^{2\delta-2\sigma}\tau_{1}^{-\delta+2\sigma}+T^{2\delta-2\sigma+\epsilon}\tau_{1}^{-2\delta+2\sigma}\cdot\tau_{1}^{-1}T^{-\delta/2}$
$\displaystyle=T^{2\delta-2\sigma+\epsilon}\tau_{1}^{-2\delta+2\sigma}\cdot\left(\tau_{1}^{\delta}+\tau_{1}^{-1}T^{-\delta/2}\right).$

It remains to choose $\tau_{1}$ optimally. This may be done by taking
$\tau_{1}=T^{-\frac{\delta}{2\delta+2}}$, whence

$\frac{1}{T}\int_{0}^{T}\|\mathcal{L}_{\tau_{0},\tau_{1},s+it}\|_{\mathrm{HS},h}^{2}dt\ll
T^{2\delta-2\sigma-\frac{\delta}{2\delta+2}(2\sigma-\delta)}=T^{\delta-\frac{3\delta+2}{2\delta+2}(2\sigma-\delta)}.$

This concludes the proof of Proposition 3.9. ∎

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