# Sectoriality of the Laplacian on Asymptotically Hyperbolic Spaces

Eric Bahuaud  Department of Mathematics, Seattle University, Seattle, WA,
98122, USA  bahuaude (at) seattleu.edu ,  Christine Guenther  Department of
Mathematics, Pacific University, Forest Grove, OR 97116, USA  guenther (at)
pacific.edu and  James Isenberg  Department of Mathematics, University of
Oregon, Eugene, OR 97403-1222 USA  isenberg (at) uoregon.edu

###### Abstract.

We prove that both the Laplacian on functions, and the Lichnerowicz Laplacian
on symmetric $2$-tensors with respect to asymptotically hyperbolic metrics,
are sectorial maps in weighted Hölder spaces. As an application, the machinery
of analytic semigroups then applies to yield well-posedness results for
parabolic evolution equations in these spaces.

###### Key words and phrases:

Asymptotically hyperbolic metrics; Sectorial Laplace operator; Resolvent
estimate

###### 2010 Mathematics Subject Classification:

58J35; 35J05, 35K, 35P

##  1\. Introduction

An initial-value problem for a parabolic partial differential equation is said
to be _well-posed_ if there is short-time existence, uniqueness, and
continuous dependence of the solution on the data in a given topology. Often
well-posedness is proved by setting up a contraction mapping argument in
appropriate function spaces. An elegant way to encode this argument involves
rewriting the equation as an ordinary differential equation with values in a
Banach space, and then applying semigroup methods to its linearization. This
point of view has been extensively developed in the literature and robust
results exist for linear, semilinear, quasilinear and fully nonlinear
equations; see for example the monographs [1, 8].

In this paper, with a view to applications in geometric analysis, we prove a
key estimate that allows us to prove well-posedness for heat-type equations on
a complete noncompact manifold $M^{n}$ with geometry modeled on an
asymptotically hyperbolic metric. We briefly describe the setting here, with
details given in §2.2. We assume that $M$ is a smooth manifold that is the
interior of a compact manifold with boundary $\overline{M}=M\cup\partial M$.
Given a smooth conformal structure $(\partial M,[\widehat{g}])$ on the
boundary, we assume there is a collar neighbourhood of $\partial M$ in
$\overline{M}$ and a diffeomorphism that identifies this collar with a product
$[0,1)_{\rho}\times\partial M$, and we consider the model metric in the
product neighbourhood given by

$g_{{\star}}=\frac{d\rho^{2}+\widehat{g}}{\rho^{2}}.$

This metric is smoothly conformally compact at $\rho=0$ with conformal
infinity $[\widehat{g}]$. A calculation shows that all sectional curvatures of
$g_{{\star}}$ tend to $-1$ as $\rho\to 0^{+}$. This condition does not depend
on the choice of representative in $[\widehat{g}]$ since changing
representatives leads to a new model of the same form up to diffeomorphisms
and error terms which vanish to higher order on the boundary.

Now use this model metric to define intrinsic weighted Hölder spaces of
$p$-tensors, $\rho^{\nu}\mathscr{C}^{\ell,\alpha}(M,T^{p}M)$, where
$\mathscr{C}^{\ell,\alpha}(M,T^{p}M)$ is the $(\ell,\alpha)$ little Hölder
space, defined in Section 2.2. We say that $g$ is an asymptotically hyperbolic
metric with conformal infinity $[\widehat{g}]$ if there exists $\nu>0$ and
some $k\in\rho^{\nu}\mathscr{C}^{\ell,\alpha}(M,T^{2}M)$, $\ell\geq 2$ so that
in the product neighborhood

$g=g_{{\star}}+k.$

Such a metric need not be smoothly conformally compact. These spaces have been
introduced by Biquard [3] and we use the analysis of the corresponding
geometric Laplace operators that appears in [4].

We motivate the analytic part of our result using the framework developed in
[8]. We suppose that $D\subset X$ is a dense inclusion of Banach spaces and we
suppose that an evolution equation for a quantity of interest $u$ is cast as
an ordinary differential equation with values in $X$,

$u^{\prime}(t)=F(u(t)),$

with $u:I\subset\mathbb{R}\longrightarrow X$, where $I$ is a subinterval of
the real line containing zero. Here $F:D\to X$ is a (possibly nonlinear)
Fréchet differentiable map that satisfies a number of structural assumptions,
the most important of which is that its linearization $L$ at a point $u_{0}\in
D$ be a sectorial operator $X\to X$. A linear operator $L:X\to X$ is sectorial
if its resolvent set (i.e., the set of complex numbers $\lambda$ for which
$\lambda I-L$ has bounded inverse) is contained in a certain sector
$S_{\omega}$ of the complex plane with vertex at $\omega$, and if there is a
constant $C>0$ so that for all $\lambda$ in this sector, the resolvent
estimate for the operator norm $\|\cdot\|_{\mathcal{L}(X,X)}$

$\left\|(\lambda
I-L)^{-1}\right\|_{\mathcal{L}(X,X)}\leq\frac{C}{|\lambda-\omega|}$

holds. This hypothesis allows one to define a heat semigroup $e^{tL}$ via the
functional calculus, upon which the subsequent theory of analytic semigroups
is based. This theory leads to a wealth of well-posedness results. Many
parabolic partial differential operators can be cast into this framework;
again see [8].

Given that many evolution equations in geometry have linearizations involving
geometric Laplacians, we need to be explicit about the function spaces in
which we have sectoriality. Motivated primarily by evolution equations for
functions and $2$-tensors, we show here that the Laplacian on functions and a
gauge-adjusted linearized Einstein operator on symmetric $2$-tensors are
sectorial in the little Hölder spaces defined above. Let $\Delta^{g}_{L}$
denote the Lichnerowicz Laplacian operator corresponding to a metric $g$
acting on symmetric $2$-tensors, $\mathcal{S}$.

In particular we prove

######  Theorem 1.

For an asymptotically hyperbolic metric $(M^{n},g)$, and any integer $\ell\geq
2$ and $\alpha\in(0,1)$, and $\mu\in(0,n-1)$, there exists a $\delta>0$ and
$\lambda_{0}>0$ such that

  1. (1)

on a sector of the form

$S=\left\\{\lambda=re^{i\theta}\in\mathbb{C}:r>0,\theta\in\left(-\frac{\pi}{2}-\delta,\frac{\pi}{2}+\delta\right)\right\\},$

$\Delta^{g}$ on functions is sectorial in
$\rho^{\mu}\mathscr{C}^{\ell,\alpha}(M,\mathbb{R})$;

  2. (2)

on a sector of the form

$S+\lambda_{0}=\left\\{\lambda+\lambda_{0}\;\Big{|}\;\lambda=re^{i\theta}\in\mathbb{C},r>0,\theta\in\left(-\frac{\pi}{2}-\delta,\frac{\pi}{2}+\delta\right)\right\\},$

$\Delta^{g}_{L}-2(n-1)$ on symmetric $2$-tensors is sectorial in
$\rho^{\mu}\mathscr{C}^{\ell,\alpha}(M,\mathcal{S})$.

There are two main parts to the proof of this Theorem. The first part is to
show that the resolvent set of the operator is confined to a suitable sector
of the complex plane. In our setting, this follows from a Fredholm theorem of
Biquard and Mazzeo and explains the range of weights $\mu$ for which our
result applies. The second part is to prove the resolvent estimate. Here we
proceed by contradiction and perform a blowup analysis. With this theorem, one
can then apply results in [8] to obtain well-posedness results for geometric
PDE.

In a forthcoming paper we discuss an application of these results to the
normalized Ricci flow.

#### Acknowledgments

We are happy to thank Jack Lee and Rafe Mazzeo for useful conversations during
the course of this work. This work was supported by collaboration grants from
the Simons Foundation (#426628, E. Bahuaud and #283083, C. Guenther). J.
Isenberg was supported by NSF grant PHY-1707427.

##  2\. Background

###  2.1. Analytic background

In this section we define a sectorial operator between Banach spaces. For
applications of such operators to evolution equations, see [1, 8]; for
applications to the Ricci flow, see [2, 6].

For complex Banach spaces $X,Y$ we let $\mathcal{L}(X,Y)$ denote the set of
bounded linear operators from $X$ to $Y$, with operator norm denoted by
$\|\cdot\|_{\mathcal{L}(X,Y)}$.

Consider a continuous inclusion of Banach spaces $D\hookrightarrow X$, and a
closed linear operator $L:D\to X$. We define the _spectrum_ of $L$,
$\sigma_{X}(L)$, to be the set of complex numbers $\lambda$ for which
$(\lambda I-L)$ does not have a bounded inverse, and the _resolvent_ set of
$L$ by $\rho_{X}(L):=\mathbb{C}\setminus\sigma_{X}(L)$.

######  Definition 2.1.

An operator $L:D\subset X\to X$ is sectorial in $X$ if:

  1. S1.

the resolvent set $\rho_{X}(L)$ contains a sector, i.e., if there exist
constants $\omega\in\mathbb{R}$ and $\theta\in(\pi/2,\pi)$ such that

$\rho_{X}(L)\supset
S_{\omega}:=\\{\lambda\in\mathbb{C}:\lambda\neq\omega,|\arg(\lambda-\omega)|<\theta\\},\;\;\;\mbox{and}$

  2. S2.

there exists a constant $C>0$ so that for all $\lambda$ in this sector, the
resolvent estimate holds, i.e., for $R_{\lambda}=R_{\lambda}(L)=(\lambda
I-L)^{-1}$,

(2.1) $\|R_{\lambda}\|_{\mathcal{L}(X,X)}=\|(\lambda
I-L)^{-1}\|_{\mathcal{L}(X,X)}\leq\frac{C}{|\lambda-\omega|}.$

In our proof of the resolvent estimate, we use the following lemma from [8].

######  Lemma 2.2 (Proposition 2.1.11 of [8]).

Let $X$ be a complex Banach space, and $L:D\subset X\to X$ a linear operator
such that the resolvent set $\rho_{X}(L)$ contains a half-plane
$\\{\lambda\in\mathbb{C}:\mathrm{Re}\;\lambda\geq\omega\\}$ for some
$\omega\in\mathbb{R}$. If there exists a constant $C>0$ such that for all
$\lambda$ in this half-plane,

$\|\lambda(\lambda-L)^{-1}\|_{\mathcal{L}(X,X)}\leq C,$

then $L$ is sectorial.

###  2.2. Geometric background

In this section we define the class of asymptotically hyperbolic metrics of
interest and refer the reader to [4] for further details. We begin by
discussing the model metrics.

Let $M^{n}$ be the interior of a smooth compact manifold with boundary
$\overline{M}=M\cup\partial M$. Recall that a smooth boundary defining
function $\rho$ for $\partial M$ is a function
$\rho\in\mathscr{C}^{\infty}(\overline{M})$ such that $\rho\geq 0$,
$\rho^{-1}(0)=\partial M$ and $d\rho\neq 0$ on $\partial M$. Given a conformal
class $[\widehat{g}]$ of metrics on $\partial M$ and a smooth representative
$\widehat{g}\in[\widehat{g}]$, it is always possible to find a neighbourhood
$\mathcal{U}$ of $\partial M$ in $\overline{M}$ and a diffeomorphism that
identifies $\mathcal{U}$ with $[0,1)_{\rho}\times\partial M$ where the metric
may be written

(2.2) $g_{{\star}}=\frac{d\rho^{2}+\widehat{g}}{\rho^{2}}.$

This definition is independent of the conformal representative chosen in the
sense that changing representatives changes the diffeomorphism and leads to a
metric of the same form up to higher order corrections in $\rho$. A
computation shows that all sectional curvatures of such metrics approach $-1$
near $\partial M$. We assume that $g_{{\star}}$ is extended into $M$ to remain
a smooth and complete metric. These model metrics are then smoothly
conformally compact since $\rho^{2}g_{{\star}}$ extends to a smooth metric on
$\overline{M}$.

By shrinking the collar neighborhood $\mathcal{U}$ if necessary, we may assume
that it is covered by finitely many smooth boundary coordinate charts
$\\{\mathcal{O}_{j},(\rho,\theta_{j}^{\alpha})\\}$ where $\rho$ is the
boundary defining function for $\partial M$ and $\theta_{j}^{\alpha}$,
$\alpha=1,\ldots,n-1,$ are local coordinates on $\partial M$. These are
extended into the interior of $\overline{M}$ by requiring them to be constant
along the integral curves of the gradient of $\rho$ with respect to
$\overline{g}_{{\star}}:=\rho^{2}g_{{\star}}$. Note that we often write
$\theta^{0}=\rho$ and use Latin indices for the range $0,1,\ldots,n-1$ and
Greek indices for the range $1,\ldots,n-1$.

We need several function spaces defined in terms of $g_{{\star}}$. For any
tensor bundle $E$ over $M$, let $\mathscr{C}^{\infty}_{c}(M,E)$ denote the
space of compactly supported smooth tensors on $M$. For
$u,v\in\mathscr{C}^{\infty}_{c}(M,E)$, define the $L^{2}$ pairing by

$(u,v):=\int_{M}\left\langle
u,v\right\rangle_{g_{{\star}}}\operatorname{dvol}_{g_{{\star}}},$

where $\left\langle u,v\right\rangle_{g_{{\star}}}$ denotes the standard
tensor contraction induced by the metric $g_{{\star}}$, and define the
associated norm by $\|u\|_{L^{2}}^{2}=(u,u)$. The space $L^{2}(M,E)$ is
defined as the completion of $\mathscr{C}^{\infty}_{c}(M,E)$ with respect to
the $L^{2}$-norm. We take the Hermitian $L^{2}$ pairing in working with
complex vector bundles.

The $\mathscr{C}^{\ell}$ norm is defined by

$\|u\|_{\mathscr{C}^{\ell}(M,E)}:=\sum_{j=0}^{{\ell}}\sup_{p\in
M}\left|(\nabla^{g_{{\star}}})^{j}u\right|_{g_{{\star}}}(p).$

Before introducing the Hölder quotient, we note that by a path we mean any
piecewise $\mathscr{C}^{1}$ map $\gamma:[0,1]\to M$. The length of a path is
given by

$\mathrm{len(\gamma)}=\int_{0}^{1}|\gamma^{\prime}(s)|_{g_{{\star}}}\;ds.$

In what follows we let $\tau_{\gamma}:E|_{\gamma(0)}\longrightarrow
E|_{\gamma(1)}$ denote the parallel transport of sections of $E$ along
$\gamma$ with respect to the metric ${g_{{\star}}}$.

For any $\ell\in\mathbb{N}_{0}$ and $\alpha\in(0,1]$ we define the
$(\ell,\alpha)$-norm on $\mathscr{C}^{\ell}$ sections of $E$ by

$\|u\|_{\ell,\alpha}=\|u\|_{\mathscr{C}^{\ell,\alpha}(M,E)}:=[u]_{\ell,\alpha}+\|u\|_{\mathscr{C}^{\ell}(M,E)},$

where the Hölder seminorm is defined by

$[u]_{\ell,\alpha}:=\sup\left\\{\frac{\left|\tau_{\gamma}\left(\nabla_{g_{{\star}}}^{\ell}u(\gamma(0))\right)-\nabla_{g_{{\star}}}^{\ell}u(\gamma(1))\right|_{g_{{\star}}}}{\mathrm{len}(\gamma)^{\alpha}}:\gamma\;\mbox{is
a path with }\gamma(0)\neq\gamma(1)\right\\}.$

The space $\mathscr{C}^{\ell,\alpha}(M,E)$ is then defined as the closure of
the space of smooth sections111The Hölder spaces $\mathscr{C}^{\ell,\alpha}$
as defined here have desirable closure properties over merely requiring that
the Hölder space be taken over sections with finite norms. These are known as
_little Hölder spaces_ in the literature of analytic semigroups. of $E$ with
respect to $\|\cdot\|_{\ell,\alpha}$. We also need versions of these spaces
weighted by powers of the defining function $\rho$. Define the weighted space
$\mathscr{C}^{\ell,\alpha}_{\mu}(M,E)=\rho^{\mu}\mathscr{C}^{\ell,\alpha}(M,E)$.
Here the norm is given by

$\|u\|_{\ell,\alpha;\mu}=\|\rho^{-\mu}u\|_{\ell,\alpha}.$

We denote the bundle of symmetric $2$-tensors by $\mathcal{S}$, and are in
position to introduce the metrics of interest. We say that a metric $g$ on $M$
is _asymptotically hyperbolic_ if it can be written in the form

$g=g_{{\star}}+k,$

in the collar neighbourhood of the boundary, where $g_{{\star}}$ is the model
metric of equation (2.2), and
$k\in\rho^{\nu}\mathscr{C}^{\ell,\alpha}(M,\mathcal{S})$ for some $\nu>0$. The
rescaled tensor $\rho^{2}k$ need not be smooth up to the boundary, so $g$ need
not be smoothly conformally compact.

The operators of interest in this paper are geometric Laplace operators of the
form

(2.3) $P^{g}=\Delta^{g}+\mathcal{K}^{g}$

where $\Delta^{g}u=g^{ij}\nabla^{g}_{i}\nabla^{g}_{j}u$ is the covariant
Laplacian and $\mathcal{K}^{g}:E\to E$ is a real symmetric endomorphism
constructed from contractions of the curvature tensor of $g$ and the metric.
We require some theory of elliptic operators on these manifolds. In [9, 10],
Mazzeo constructs a pseudodifferential calculus, the $0$-calculus, that
contains the inverses of operators like $P^{g}$. We refer the reader to [4]
for a concise treatment of the details; see also [3, 7] for other approaches.

Condition S1 in Definition 2.1 is a statement of where the resolvent
$\lambda-P^{g}$ is invertible. In order to state the mapping properties of
$P^{g}$ and study the asymptotics of solutions of associated equations, we
need two simpler model operators. The first model operator is the normal
operator of $P^{g}$, which in our setting is the same operator but with
respect to the hyperbolic metric, so $P^{h}$ where $h$ is the hyperbolic
metric on the open unit ball. Geometric Laplace operators as in equation (2.3)
are said to be _fully elliptic_ if $P^{h}$ is invertible as an unbounded
operator in $L^{2}$ on hyperbolic space.

For the second model operator, given $s\in\mathbb{C}$ define the _indicial
operator_ $I_{s}(P^{g}):E|_{\partial M}\to E|_{\partial M}$ by

(2.4) $\displaystyle
I_{s}(P^{g})\overline{u}=\rho^{-s}P^{g}(\rho^{s}\overline{u})|_{\partial M},$

where $\overline{u}$ is a $\mathscr{C}^{2}$ section of $E|_{\partial M}$
extended arbitrarily into $M$. It can be shown that $I_{s}(P^{g})$ is a
polynomial in $s$ and is independent of the choice of extension. We call any
value of $s$ for which $I_{s}(P^{g})$ vanishes an _indicial root_. For any
operator $P^{g}$ of the form (2.3), the set of real parts of the indicial
roots are symmetric about $\frac{n-1}{2}$. For such $P^{g}$ there is a largest
positive constant $\Xi>0$ such that the interval
$\frac{n-1}{2}-\Xi<s<\frac{n-1}{2}+\Xi$ contains no other indicial roots;
$\Xi$ is called the indicial radius of $P^{g}$. Recall that $P^{g}$ is
Fredholm if $\dim\ker P^{g}$ and $\dim\ker(P^{g})^{*}$ are finite dimensional
and the range of $P^{g}$ is closed. The fundamental properties of $P^{g}$ are
encoded in the following result.

######  Theorem 2.3 (Theorem 13 of [4]).

Let $P^{g}$ be a fully elliptic Laplace operator as in (2.3). For any
$\ell\in\mathbb{N}_{0}$ and $\alpha\in(0,1)$, the natural extension

$P^{g}:\mathscr{C}^{\ell+2,\alpha}_{\mu}(M,E)\longrightarrow\mathscr{C}^{\ell,\alpha}_{\mu}(M,E)$

is Fredholm if and only if $|\mu-\frac{n-1}{2}|<\Xi$. Further, $P^{g}$ is an
isomorphism on these spaces if and only if $\ker_{L^{2}}(P^{g})=\\{0\\}$.

##  3\. Sectoriality of the Laplacian

In this section we prove Theorem 1. In §3.1 we explain how to obtain the
resolvent estimate on weighted spaces from an estimate for the operator
conjugated by powers of the defining function on unweighted spaces. We also
provide a key decomposition for the Laplace operator. In §3.2 we prove
sectoriality for the Laplacian acting on functions. In §3.3 we prove
sectoriality for the Lichnerowicz Laplacian on symmetric $2$-tensors.

###  3.1. The resolvent estimate

As in §2.2, let $(M,g)$ be an asymptotically hyperbolic metric of the form

$\displaystyle g$ $\displaystyle=g_{{\star}}+k$ (3.1)
$\displaystyle=\frac{d\rho^{2}+\widehat{g}}{\rho^{2}}+k,$

with $k\in\mathscr{C}_{\nu}^{\ell,\alpha}(M;\mathcal{S})$, $\nu>0$. Let
$\\{\mathcal{O}_{i},(\rho,\theta^{\alpha}_{i})\\}$ be a reference coordinate
chart in the collar neighbourhood of the boundary. From [10] we know that the
space of vector fields which vanish on the boundary, hereafter called
$0$-vector fields, are locally generated by

(3.2)
$\displaystyle\rho\frac{\partial}{\partial\rho},\;\;\rho\frac{\partial}{\partial\theta^{\alpha}}.$

In the calculations that follow we choose these vectors fields and their duals
$\frac{d\rho}{\rho}$, $\frac{d\theta^{\alpha}}{\rho}$ as the preferred bases
for tensor bundles.

It is convenient to use an equivalent norm for Hölder spaces, defined in terms
of the Möbius charts described in [7, Chapter 2]. Let $B_{r}$ denote the open
ball about $(1,0,\ldots,0)$ of (hyperbolic) radius $r$ in the half-space model
of hyperbolic space,
$\mathbb{U}^{n}=\\{(x,y^{1},\ldots,y^{n-1}):x>0,y^{\alpha}\in\mathbb{R}\\}$.
For $p\in\mathcal{U}$ in the domain of a reference chart $\mathcal{O}$ with
coordinates $(\rho,\theta^{\alpha})$, we denote
$p=(\rho_{0},\theta_{0}^{\alpha})$, and define a map
$\Phi_{p}:B_{r}\subset\mathbb{U}^{n}\to\mathcal{O}\subset M$ by

$\Phi_{p}(x,y^{\alpha})=(\rho_{0}x,\theta_{0}^{\alpha}+\rho_{0}y^{\alpha}).$

It is possible to cover $M$ by a locally uniformly finite covering of the
balls $\\{\Phi_{p}(B_{1})\\}$ near $\partial M$, and finitely many other
coordinate balls that cover $M\setminus\mathcal{U}$, which we label in the
same way for consistency. In sum we obtain a uniformly locally finite cover of
$M$ of the form $\\{\Phi_{p_{i}}(B_{1})\\}$. An equivalent norm for
$\mathscr{C}^{\ell,\alpha}_{\mu}(M,E)$ is given by

(3.3)
$\|u\|_{\ell,\alpha;\mu}=\sup_{i}\|\Phi_{p_{i}}^{*}(\rho^{-\mu}u)\|_{\mathscr{C}^{\ell,\alpha}_{\mathrm{Euc}}(B_{1})},$

where the norm on the right hand side is the Hölder norm of the components of
$u$ computed with respect to the background Euclidean metric on $B_{1}$. For
the remainder of this section, we use (3.3) as the norm on Hölder spaces.

We now set up the proof of the resolvent estimate. For the remainder of this
subsection we assume that $P=\Delta^{g}+\mathcal{K}$ is a geometric Laplace
operator on a tensor bundle $E$, and that for
$D=\mathscr{C}^{\ell+2,\alpha}_{\mu}(M;E)$,
$X=\mathscr{C}^{\ell,\alpha}_{\mu}(M;E)$ for some value of $\mu$, the
sectoriality property S1 of Definition 2.1 has been established on some sector
$S_{\omega}$. We drop the superscript $g$ on $P$ to lighten the notation. We
consider the case where $\ell=0$; the details for $\ell\in\mathbb{N}$ are
similar.

To obtain the resolvent estimate between weighted spaces, it is convenient to
instead work in unweighted spaces. To achieve this, conjugate the operator $P$
by $\rho^{\mu}$ to obtain $\rho^{-\mu}P\rho^{\mu}$. For example, if $\lambda$
is in the resolvent set of $P$, then $\lambda-P$ is invertible and if
$u\in\mathscr{C}^{0,\alpha}_{\mu}$ we may set

$w=(\lambda-P)^{-1}u.$

This means that $u=(\lambda-P)w$, and since $u,w$ are both in
$\mathscr{C}^{0,\alpha}_{\mu}(M;E)$ we may write $u=\rho^{\mu}\overline{u}$
and $w=\rho^{\mu}\overline{w}$ for $\overline{u},\overline{w}$ in
$\mathscr{C}^{0,\alpha}(M;E)$. Thus we obtain

$\overline{u}=(\lambda-\rho^{-\mu}P\rho^{\mu})\overline{w}.$

######  Lemma 3.1.

If there exists $C>0$ so that for all $\lambda\in S_{\omega}$ and for any
$v\in\mathscr{C}^{0,\alpha}(M;E)$,

$\|(\lambda-\rho^{-\mu}P\rho^{\mu})^{-1}v\|_{0,\alpha}\leq\frac{C}{|\lambda-\omega|}\|v\|_{0,\alpha}.$

then for the same $C$ and $\lambda$ and any
$u\in\mathscr{C}^{0,\alpha}_{\mu}(M;E)$,

$\|(\lambda-P)^{-1}u\|_{0,\alpha;\mu}\leq\frac{C}{|\lambda-\omega|}\|u\|_{0,\alpha;\mu}.$

###### Proof.

Taking $u\in\mathscr{C}^{0,\alpha}_{\mu}(M,E)$ and $w=(\lambda-P)^{-1}u$, we
calculate

$\displaystyle\|(\lambda-P)^{-1}u\|_{0,\alpha;\mu}$
$\displaystyle=\|\rho^{-\mu}(\lambda-P)^{-1}u\|_{0,\alpha}$
$\displaystyle=\|\rho^{-\mu}(\lambda-P)^{-1}\rho^{\mu}\rho^{-\mu}u\|_{0,\alpha}$
$\displaystyle\leq\frac{C}{|\lambda-\omega|}\|\rho^{-\mu}u\|_{0,\alpha}=\frac{C}{|\lambda-\omega|}\|u\|_{0,\alpha;\mu}.$

∎

It is also convenient to have an expression for the conjugated operator
$\rho^{-\mu}P\rho^{\mu}$ near the boundary. Here this operator is well
approximated by a Laplacian plus terms that are under control.

######  Lemma 3.2.

For $\ell\geq 2$, $0<\alpha<1$, $\nu>0$, let $g=g_{{\star}}+k$, with
$k\in\mathscr{C}^{\ell,\alpha}_{\nu}$, be an asymptotically hyperbolic metric.
Let $P=\Delta^{g}+\mathcal{K}$ be a geometric Laplace operator on the tensor
bundle $E$. For a section $u$ of $E$,

(3.4)
$\displaystyle\rho^{-\mu}P[\rho^{\mu}u]=(\rho\partial_{\rho})^{2}u+\widehat{g}^{\alpha\beta}(\rho\partial_{\theta^{\alpha}})(\rho\partial_{\theta^{\beta}})u+{\mathcal{Z}_{\mathrm{van}}}u+{\mathcal{Z}_{\mathrm{bdd}}}u,$

where ${\mathcal{Z}_{\mathrm{van}}}$ is a second-order operator whose
coefficients lie in at least $\mathscr{C}^{\ell,\alpha}_{\nu}$ and
${\mathcal{Z}_{\mathrm{bdd}}}$ is a first-order operator with uniformly
bounded coefficients lying in at least $\mathscr{C}^{\ell-2,\alpha}_{\nu}$.

###### Proof.

We suppose that $u$ is a tensor of covariant rank $p$ and contravariant rank
$q$. In the components relative to the frame described at the beginning of
this section in 3.2, we write

$\displaystyle u$ $\displaystyle=u_{i_{1}\cdots i_{p}}^{j_{1}\cdots
j_{q}}(\rho\partial_{\theta^{j_{1}}})\otimes\cdots\otimes(\rho\partial_{\theta^{j_{q}}})\otimes\frac{d\theta^{i_{1}}}{\rho}\otimes\cdots\otimes\frac{d\theta^{i_{p}}}{\rho}.$

Thus

$\displaystyle\nabla^{g}u$ $\displaystyle=u_{i_{1}\cdots i_{p},a}^{j_{1}\cdots
j_{q}}(\rho\partial_{\theta^{j_{1}}})\otimes\cdots\otimes(\rho\partial_{\theta^{j_{q}}})\otimes\frac{d\theta^{i_{1}}}{\rho}\otimes\cdots\otimes\frac{d\theta^{i_{p}}}{\rho}\otimes\frac{d\theta^{a}}{\rho},$

where the components of $\nabla^{g}u$ are given by

$\displaystyle u_{i_{1}\cdots i_{p},a}^{j_{1}\cdots j_{q}}$
$\displaystyle=\rho\partial_{\theta^{a}}u_{i_{1}\cdots i_{p}}^{j_{1}\cdots
j_{q}}+\sum_{t=1}^{q}\Gamma_{sa}^{j_{t}}u_{i_{1}\cdots i_{p}}^{j_{1}\cdots
s\cdots j_{q}}-\sum_{t=1}^{p}\Gamma_{i_{t}a}^{s}u_{i_{1}\cdots s\cdots
i_{p}}^{j_{1}\cdots j_{q}}.$

Here, $\Gamma$ denotes the Christoffel symbols of the Levi-Civita connection
$\nabla^{g}$. As we do not need the precise form of this equation, we may
express it abstractly as

$\nabla_{a}u=\rho\partial_{\theta^{a}}u+\Gamma*u,$

where $*$ denotes linear combinations of tensor contractions of $\Gamma$ with
$u$ using $g$ and $g^{-1}$. Recall that
$\Delta^{g}u=g^{ab}\nabla_{b}\nabla_{a}u$; thus, applying a covariant
derivative to $\nabla_{a}u$ yields abstractly

$\displaystyle\nabla_{b}\nabla_{a}u$
$\displaystyle=\rho\partial_{\theta^{b}}(\rho\partial_{\theta^{a}}u+\Gamma*u)+\Gamma*(\rho\partial_{\theta^{a}}u+\Gamma*u)$
$\displaystyle=(\rho\partial_{\theta^{b}})(\rho\partial_{\theta^{a}})u+\Gamma*(\rho\partial
u)+(\rho\partial\Gamma)*u+\Gamma*\Gamma*u.$

We arrive at the expression

$\displaystyle g^{ab}\nabla_{b}\nabla_{a}u+\mathcal{K}u$
$\displaystyle=g_{{\star}}^{ab}(\rho\partial_{\theta^{b}})(\rho\partial_{\theta^{a}})u+\mathcal{Z}_{\mathrm{van}}u+\mathcal{Z}_{\mathrm{bdd}}u,$

where

$\mathcal{Z}_{\mathrm{van}}u=(g^{ab}-g_{{\star}}^{ab})(\rho\partial_{\theta^{b}})(\rho\partial_{\theta^{a}})u,$

and

$\mathcal{Z}_{\mathrm{bdd}}u=\Gamma*(\rho\partial_{\theta^{a}}u)+(\rho\partial_{\theta^{b}}\Gamma)*u+\Gamma*\Gamma*u+\mathcal{K}u.$

Regarding the second-order operator $\mathcal{Z}_{\mathrm{van}}$, since
$g=g_{{\star}}+k$, we may express $g^{-1}$ in terms of $g_{{\star}}^{-1}$ and
$k$ for $k$ with sufficiently small norm to ensure invertibility of $g$.
Writing $g^{ab}$ for the components of $g^{-1}$ and $g_{{\star}}^{ab}$ for the
components of $g_{{\star}}^{-1}$ relative to our choice of frame, one finds

$g^{ab}=(g_{{\star}}+k)^{ab}=g_{{\star}}^{ab}-g_{{\star}}^{al}g_{{\star}}^{bm}k_{ml}+(g_{{\star}}+k)^{bl}g_{{\star}}^{am}g_{{\star}}^{pq}k_{lp}k_{mq},$

which shows that $g^{ab}-g_{{\star}}^{ab}$ lies in
$\mathscr{C}^{\ell,\alpha}_{\nu}$.

Regarding the first order operator $\mathcal{Z}_{\mathrm{bdd}}$, one sees from
the formula for Christoffel symbols in terms of a local frame that $\Gamma$
involves one $0$-derivative of the metric, and the $0$th-order term
$\mathcal{K}$ depends on the curvature of $g$; thus,
$\mathcal{Z}_{\mathrm{bdd}}$ involves at most two $0$-derivatives of $g$. As
$\ell\geq 2$, all of the terms of $\mathcal{Z}_{\mathrm{bdd}}$ are uniformly
bounded on $M$.

In order to complete the proof we conjugate $P$ by $\rho^{\mu}$. Observe that
$\rho\partial_{\rho}[\rho^{\mu}u]=\rho^{\mu}\rho\partial_{\rho}u+\mu\rho^{\mu}u$
and
$(\rho\partial_{\rho})^{2}[\rho^{\mu}u]=\rho^{\mu}(\rho\partial_{\rho})^{2}u+2\mu\rho^{\mu}\rho\partial_{\rho}u+\mu^{2}\rho^{\mu}u$.
Thus the conjugated operator has the required form with the new terms absorbed
into $\mathcal{Z}_{\mathrm{bdd}}$. Since the application of $0$-derivatives
does not alter the power of vanishing in $\rho$, the
$\mathcal{Z}_{\mathrm{bdd}}$ remains uniformly bounded on $M$. ∎

We are now ready to establish the resolvent estimate in a half-plane,
presuming property S1 from Definition 2.1 holds. We present the argument for
general tensors but suppress indices to lighten the notation.

######  Proposition 3.3.

For $\ell\geq 2$, $0<\alpha<1$, $\nu>0$, let $g=g_{{\star}}+k$, with
$k\in\mathscr{C}^{\ell,\alpha}_{\nu}$, be an asymptotically hyperbolic metric.
Let $P=\Delta^{g}+\mathcal{K}$ be a geometric Laplace operator on the tensor
bundle $E$ whose resolvent set contains a sector $S_{\omega}$, as given in S1
in Definition 2.1 , with $\omega>0$. Then there exists $C>0$ so that for all
$\lambda\in\mathbb{C}$ such that $\mathrm{Re}\;\lambda\geq\omega$, we have
that for any $u\in\mathscr{C}^{0,\alpha}(M;E)$,

$\|\lambda(\lambda-\rho^{-\mu}P\rho^{\mu})^{-1}u\|_{0,\alpha}\leq
C\|u\|_{0,\alpha}.$

###### Proof.

We proceed by contradiction. If the estimate were false, then for every
$m\in\mathbb{N}$, there would exist $\lambda_{m}\in\mathbb{C}$ with
$\mathrm{Re}\;\lambda_{m}\geq\omega$ and $u_{m}\in\mathscr{C}^{0,\alpha}$
where

(3.5)
$\|\lambda_{m}(\lambda_{m}-\rho^{-\mu}P\rho^{\mu})^{-1}u_{m}\|_{0,\alpha}>m\|u_{m}\|_{0,\alpha}.$

We remind the reader that we sometimes write
$R_{\lambda}=R_{\lambda}(\rho^{-\mu}P\rho^{\mu})$ for
$(\lambda-\rho^{-\mu}P\rho^{\mu})^{-1}$.

Claim: We may assume $\lambda_{m}\to+\infty$ in the complex plane.

Indeed if this claim were false, then some subsequence of $\lambda_{m}$ would
converge to a finite point $\lambda_{*}$ with
$\mathrm{Re}\;\lambda_{*}\geq\omega$, which by assumption is in the resolvent
set. The resolvent operator $\lambda\mapsto
R_{\lambda}(\rho^{-\mu}P\rho^{\mu})$ is analytic in $\lambda$ and thus
continuous, so that $\|R_{\lambda_{m}}\|\to\|R_{\lambda_{*}}\|$ in the
operator norm; however, if a subsequence of $\lambda_{m}$ were to converge,
then estimate (3.5) would show that $\|R_{\lambda_{m}}\|\to\infty$, so that
$(\lambda_{*}-\rho^{-\mu}P\rho^{\mu})$ could not have a bounded inverse, and
so $\lambda_{*}$ could not be in the resolvent set. This contradiction
establishes the claim.

Now define

(3.6)
$w_{m}=\frac{R_{\lambda_{m}}(\rho^{-\mu}P\rho^{\mu})u_{m}}{\|R_{\lambda_{m}}(\rho^{-\mu}P\rho^{\mu})u_{m}\|_{0,\alpha}}.$

Observe that by virtue of this definition, each $w_{m}$ satisfies an elliptic
partial differential equation

(3.7)
$\left(\lambda_{m}-\rho^{-\mu}P\rho^{\mu}\right)w_{m}=\frac{u_{m}}{\|R_{\lambda_{m}}(\rho^{-\mu}P\rho^{\mu})u_{m}\|_{0,\alpha}}=\frac{1}{\beta_{m}}u_{m},$

where
$\beta_{m}:=\|R_{\lambda_{m}}(\rho^{-\mu}P\rho^{\mu})u_{m}\|_{0,\alpha}$.

By elliptic regularity, $w_{m}$ is locally in $\mathscr{C}^{2,\alpha}$. By
construction, $\|w_{m}\|_{0,\alpha}\equiv 1$ for all $m$, so there must exist
a point $p_{m}\in M$ such that on the ball of $g$-radius 1,
$B_{m}:=B(p_{m},1)$, we have

(3.8) $\|w_{m}\|_{0,\alpha;B_{m}}\geq\frac{1}{2}.$

In the topology of the compact manifold with boundary $\overline{M}$, we may
now pass to a convergent subsequence $p_{m}\to p$. There are two model cases:
Case i) $p\in M$ (i.e., the subsequence accumulates in the interior of $M$,
and Case ii) $p\in\partial M$ (i.e., where the subsequence accumulates at a
particular point on the conformal boundary at infinity). In both cases we can
construct a solution to a limiting PDE and derive a contradiction. We detail
the case where $p\in\partial M$ and refer the reader to our published work [2]
for the interior case, which is similar to the case of a closed manifold.

Case ii): Assume $p\in\partial M$. It is helpful to recall the decomposition
of the metric $g$ near the boundary, as specified in equation (3.1). We may
without loss of generality assume that $p$ lies in a reference chart
$\left(\mathcal{O},(\rho,\theta^{\alpha})\right)$ on $\partial M$ which is
centered at $p$. In this chart, the metric takes the form

$g=\frac{d\rho^{2}+\widehat{g}_{\alpha\beta}(\theta)d\theta^{\alpha}d\theta^{\beta}}{\rho^{2}}+k,$

with $p=(\rho,\theta^{\alpha})=(0,0)$ in these coordinates. We advance and
relabel the subsequence so that that $p_{m}$ lies in this chart.

We now consider several simplifying transformations that enable us to “zoom
in” along $p_{m}$ and derive a solution to a PDE that results in a
contradiction. We begin by formulating the equation, and then we study the
estimates for the solutions.

The zooming sequence of equations. Begin with the sequence of equations (3.7).
Using Lemma 3.2 this sequence takes the form

(3.9)
$\displaystyle\left[\lambda_{m}-\left((\rho\partial_{\rho})^{2}+\widehat{g}^{\beta\nu}(\rho,\theta^{\alpha})\rho\partial_{\theta^{\beta}}\rho\partial_{\theta^{\nu}}\right)+\mathcal{Z}_{\mathrm{van}}+\mathcal{Z}_{\mathrm{bdd}}\right]{w_{m}}=\beta_{m}^{-1}{u_{m}}.$

For each $m$, we use the definition of Möbius charts to pull the equations
(3.9) back from $B_{m}$ to a fixed ball $B$. Under $\Phi_{p_{m}}:B\to B_{m}$
given by
$(\rho,\theta^{\alpha})=(\rho_{m}x,\theta^{\alpha}_{m}+\rho_{m}y^{\alpha})=\Phi_{p_{m}}(x,y^{\alpha})$,
we find that the derivatives
$\rho\partial_{\rho},\rho\partial_{\theta^{\alpha}}$ transform to
$x\partial_{x},x\partial_{y^{\alpha}}$ derivatives; thus we obtain

$\displaystyle\left[\lambda_{m}-\left((x\partial_{x})^{2}+\widehat{g}^{\beta\nu}(\rho_{m}x,\theta^{\alpha}_{m}+\rho_{m}y^{\alpha})x\partial_{y^{\beta}}x\partial_{y^{\nu}}\right)+\mathcal{Z}_{\mathrm{van}}+\mathcal{Z}_{\mathrm{bdd}}\right]{w_{m}}=\beta_{m}^{-1}{u_{m}}.$

It is helpful to divide both sides by $|\lambda_{m}|$,

(3.10)
$\left[\frac{\lambda_{m}}{|\lambda_{m}|}-\frac{1}{|\lambda_{m}|}\left((x\partial_{x})^{2}+\widehat{g}^{\beta\nu}(\rho_{m}x,\theta^{\alpha}_{m}+\rho_{m}y^{\alpha})x\partial_{y^{\beta}}x\partial_{y^{\nu}}\right)\right.\\\
\left.\phantom{\frac{\lambda_{m}}{|\lambda_{m}|}}+\mathcal{Z}_{\mathrm{van}}+\mathcal{Z}_{\mathrm{bdd}}\right]{w_{m}}=\frac{1}{|\lambda_{m}|\beta_{m}}{u_{m}}.$

Next, we apply a coordinate change in the $x$ variable given by $x=e^{-t}$.
This has the effect of mapping the center of $B$ to $(0,0)$ so that the image
of $B$ contains a ball in $(t,y^{\alpha})$ coordinates. Finally, for each
equation, we dilate the variables to effectuate the zoom. Let
$t=|\lambda_{m}|^{-1/2}r$, and let
$y^{\alpha}=|\lambda_{m}|^{-1/2}s^{\alpha}$. Thus the image of a ball of fixed
size in $(t,y^{\alpha})$ variables is mapped to a ball $\Omega_{m}$ in
$(r,s^{\alpha})$ coordinates of radius $|\lambda_{m}|^{1/2}$ which tends to
infinity in $m$. Thus the sequence of equations is defined on an exhaustion of
$\mathbb{R}^{n}$.

A chain rule computation shows that $x\partial_{x}$ transforms to
$-|\lambda_{m}|^{1/2}\partial_{r}$, and $x\partial_{y}^{\alpha}$ transforms to
$|\lambda_{m}|^{1/2}e^{-|\lambda_{m}|^{-1/2}r}\partial_{s^{\alpha}}$. We now
consider what happens if we convert equation (3.10) to these variables. Since
$\mathcal{Z}_{\mathrm{bdd}}$ contains at most first order derivatives with
bounded coefficients, there remains a prefactor of
$\frac{1}{|\lambda_{m}|^{1/2}}$ multiplying these terms after the change of
variables. Thus this operator tends to zero uniformly as $m\to\infty$. On the
other hand the second-order terms in equation (3.10) and in
$\mathcal{Z}_{\mathrm{van}}$ absorb the $|\lambda_{m}|$ upon conversion and we
obtain

(3.11)
$\left[\lambda_{m}-\left(\partial_{r}^{2}+\widehat{g}^{\beta\nu}\left(\rho_{m}e^{-\frac{r}{|\lambda_{m}|^{1/2}}},\theta_{m}^{\alpha}+\rho_{m}\frac{s^{\alpha}}{|\lambda_{m}|^{1/2}}\right)\partial_{s^{\beta}}\partial_{s^{\nu}}\right)\right.\\\
\left.+\mathcal{Z}_{\mathrm{van}}^{\prime}+\frac{1}{\sqrt{|\lambda_{m}|}}\mathcal{Z}_{\mathrm{bdd}}\right]\widetilde{w_{m}}=\widetilde{u_{m}}$

where we have written

$\displaystyle\widetilde{w_{m}}(r,s)$
$\displaystyle=w_{m}\left(\rho_{m}e^{-\frac{r}{|\lambda_{m}|^{1/2}}},\theta_{m}^{\alpha}+\rho_{m}\frac{s^{\alpha}}{|\lambda_{m}|^{1/2}}\right),$
$\displaystyle\widetilde{u_{m}}(r,s)$
$\displaystyle=\frac{1}{|\lambda_{m}|\beta_{m}}u_{m}\left(\rho_{m}e^{-\frac{r}{|\lambda_{m}|^{1/2}}},\theta_{m}^{\alpha}+\rho_{m}\frac{s^{\alpha}}{|\lambda_{m}|^{1/2}}\right),$

and

$\mathcal{Z}_{\mathrm{van}}^{\prime}=(e^{-2r/|\lambda_{m}|}-1)\partial_{s^{\beta}}\partial_{s^{\nu}}+(g^{ab}-g^{ab}_{+})\partial_{s^{a}}\partial_{s^{b}},$

where $\partial_{s^{a}}$ denotes either $\partial_{r}$ or
$e^{-r/|\lambda_{m}|}\partial_{s^{\alpha}}$. Regarding
$\mathcal{Z}_{\mathrm{van}}^{\prime}$, a MacLaurin expansion in $r$ for the
first term, and the fact that the asymptotically hyperbolic perturbation
tensor $k$ has the fall off rate $k=O(\rho^{\nu})$ in the second term, shows
that the coefficients of $\mathcal{Z}_{\mathrm{van}}^{\prime}$ tend to zero as
$m\to\infty$.

Estimates for the solutions. With the sequence of equations (3.11) in hand, we
now study estimates for the tensor fields $\widetilde{w_{m}}$ and
$\widetilde{u_{m}}$. Recall the definition of $w_{m}$ in (3.6) and the
estimate (3.8). From the form of the Hölder norm given in (3.3), we find that
$1/2\leq\|w_{m}\|_{0,\alpha;B,\textrm{Euc}}\leq 1$ in the Euclidean norm in
$(x,y^{\alpha})$ coordinates on $B$. Thus

(3.12) $\displaystyle\frac{1}{2}$ $\displaystyle\leq\sup_{(x,y)\in
B}|w_{m}(x,y)|+\sup_{(x,y)\neq(x^{\prime},y^{\prime})\in
B}\frac{|w_{m}(x,y)-w_{m}(x^{\prime},y^{\prime})|}{|x-x^{\prime}|^{\alpha}+|y-y^{\prime}|^{\alpha}}\leq
1.$

We now transfer the estimates for $w_{m}$ and $u_{m}$ to $\widetilde{w}_{m}$
and $\widetilde{u}_{m}$:

$\displaystyle|\widetilde{w}_{m}(r,s)$
$\displaystyle-\widetilde{w}_{m}(r^{\prime},s^{\prime})|$
$\displaystyle=\left|w_{m}\left(e^{-\frac{r}{\sqrt{|\lambda_{m}|}}},\frac{s}{\sqrt{|\lambda_{m}|}}\right)-w_{m}\left(e^{-\frac{r^{\prime}}{\sqrt{|\lambda_{m}|}}},\frac{s^{\prime}}{\sqrt{|\lambda_{m}|}}\right)\right|$
$\displaystyle\leq[w_{m}]_{0,\alpha;B}\left(\left|e^{-\frac{r}{\sqrt{|\lambda_{m}|}}}-e^{-\frac{r^{\prime}}{\sqrt{|\lambda_{m}|}}}\right|^{\alpha}+\frac{1}{|\lambda_{m}|^{\alpha/2}}\left|s-s^{\prime}\right|^{\alpha}\right)$
$\displaystyle\leq
C\frac{1}{|\lambda_{m}|^{\alpha/2}}\left(|r-r^{\prime}|^{\alpha}+\left|s-s^{\prime}\right|^{\alpha}\right),$

where we have used a mean-value estimate on $e^{-t}$ on the image of $B$ in
these coordinates and we have used the uniform bound on the
$\|w_{m}\|_{0,\alpha}$. Thus we conclude that the Hölder quotient of
$\widetilde{w}_{m}$ in $(r,s)$ coordinates satisfies

$[\widetilde{w}_{m}]_{0,\alpha}\leq\frac{C}{|\lambda_{m}|^{\alpha/2}}.$

Therefore as $m\to\infty$, the $[\widetilde{w}_{m}]_{0,\alpha}\to 0$ and the
Hölder norm is increasingly dominated by the sup-norm. In particular,
$\widetilde{w}_{m}$ are uniformly bounded in the Hölder norm and eventually
uniformly bounded from below by a positive constant.

Arguing analogously for the $\widetilde{u}_{m}$, observe that it follows from
estimate (3.5) that we have

$|\lambda_{m}|^{-1}\beta_{m}^{-1}\|u_{m}\|_{0,\alpha}\leq\frac{1}{m}.$

Thus, similar considerations as above imply that $\widetilde{u}_{m}\to 0$
uniformly in $\mathscr{C}^{0,\alpha}$ on compact subsets of $\mathbb{R}^{n}$.

Limiting equation. Having completed these preliminaries, we now proceed to
construct a limiting equation from (3.11). Note that throughout this argument
we pass to subsequences and relabel the original sequence instead of using the
unwieldy subsequence notation $\lambda_{n_{k}}$. For the purposes of this
argument, we introduce the operator
$\square:=\partial_{r}^{2}+\widehat{g}^{\beta\nu}(0,0)\partial_{s^{\beta}}\partial_{s^{\nu}}$
on $\mathbb{R}^{n}$ in $(r,s)$ coordinates. Note that this is a constant
coefficient elliptic operator

Consider a subsequence $\lambda_{m}\in\mathbb{C}$ so that
$\lambda_{m}/|\lambda_{m}|$ converges to a unit modulus complex number
$\lambda_{*}$ with $\mathrm{Re}\;\lambda_{*}\geq\omega$. Using local elliptic
estimates and the Arzela-Ascoli lemma, on each ball of integer radius (with
respect to $(r,s)$ coordinates) in $\mathbb{R}^{n}$ we may obtain a
subsequence of $\widetilde{w}_{m}$ that converges to a $\mathscr{C}^{2}$
solution to $(\lambda_{*}-\square)\widetilde{w}=0$ on that ball. Then applying
a diagonal extraction argument, we conclude that there exists a bounded
$\mathscr{C}^{2}$ tensor field $\widetilde{w}$ on $\mathbb{R}^{n}$ whose
components each satisfy

(3.13) $\displaystyle(\lambda_{*}-\square)\widetilde{w}=0.$

Since the $\widetilde{w}_{m}$ are uniformly bounded from below, at least one
component of $\widetilde{w}$ is nonzero. But the only bounded solution to the
PDE (3.13) is $\widetilde{w}=0$, as can be seen by applying the Fourier
transform (in the sense of distributions) to equation (3.13), and observing
that the principal symbol is non-vanishing for
$\mathrm{Re}\;\lambda_{*}\geq\omega>0$. This contradiction concludes the proof
of Proposition 3.3. ∎

Proposition 3.3 and Lemma 2.2 establish the resolvent estimate in some sector,
given property S1 of Definition 2.1. In the following subsections we consider
specific cases in which we know that in fact property S1 holds.

###  3.2. The Laplacian on functions

We begin by considering the covariant Laplacian acting on functions,
$\Delta^{g}u=g^{ab}\nabla^{g}_{a}\nabla^{g}_{b}u$, thought of as an unbounded
operator on $L^{2}(M)$ with domain $D(\Delta^{g})=H^{2}(M)$.

This operator is a real, formally self-adjoint and closed operator. If $h$ is
the hyperbolic metric on the unit ball, a result of McKean [13] implies that
the whole $L^{2}$ spectrum of the Laplacian $\Delta^{h}$ is contained in the
ray $(-\infty,-\frac{(n-1)^{2}}{4}]$. If $g$ is asymptotically hyperbolic, a
result of Mazzeo [12] implies that the spectrum may additionally contain
finitely many eigenvalues in the interval
$\left(-\frac{(n-1)^{2}}{4},0\right)$. Thus $\lambda-\Delta^{g}$ is fully
elliptic away from $\lambda\in(-\infty,0)$.

For some weight $\mu$, to be specified below, let
$D=\mathscr{C}^{\ell+2,\alpha}_{\mu}(M)$ and
$X=\mathscr{C}^{\ell,\alpha}_{\mu}(M)$. We aim to to prove that $\Delta^{g}$
is sectorial on some sector with vertex $\omega$ = 0. We use Theorem 2.3 to
study the action of $\Delta^{g}$ on weighted little Hölder spaces, and thus we
require the indicial operator of equation (2.4). A computation using this
equation and equation (3.1) shows that for a $\mathscr{C}^{2}$ function
$\overline{u}$ on $\partial M$ extended arbitrarily into $M$,

$I_{s}(\Delta^{g})\overline{u}=(s^{2}-(n-1)s)\overline{u}.$

This shows that the indicial roots are $s=0$ and $s=n-1$ and that the indicial
radius is $\Xi=\frac{n-1}{2}$. Thus Theorem 2.3 implies that

$\Delta^{g}:\mathscr{C}^{\ell+2,\alpha}_{\mu}(M)\to\mathscr{C}^{\ell,\alpha}_{\mu}(M)$

is Fredholm for $\mu\in(0,n-1)$, with kernel equal to the $L^{2}$-kernel of
$\Delta^{g}$.

Now consider the effect of adding a spectral parameter $\lambda$. The indicial
operator becomes

$I_{s}(\lambda-\Delta^{g})\overline{u}=(-s^{2}+(n-1)s+\lambda)\overline{u},$

with corresponding indicial roots
$s=\frac{n-1}{2}\pm\frac{1}{2}\sqrt{\left(n-1\right)^{2}+4\lambda}$. While we
have a complex spectral parameter in mind, notice that if we momentarily think
of $\lambda$ as real-valued and let
$\lambda\searrow-\left(\frac{n-1}{2}\right)^{2}$, the range of weights for
which we have a Fredholm result shrinks to zero. In a similar way, certain
complex-valued $\lambda$ cause the Fredholm range to shrink. Thus given a
fixed $\mu\in(0,n-1)$ we must identify an appropriate set of $\lambda$ in the
complex plane on which we obtain a Fredholm result. Given our intended
application to sectorial operators, we then show that one such set is a
sector.

To assist our calculation, we use the following Lemma. Note that the proof is
a straightforward calculation using the polar representation of a complex
number.

######  Lemma 3.4.

For a real number $A>0$ and for any $\varepsilon>0$, there exists a $\delta>0$
so that if $\lambda$ belongs to the sector

$S=\left\\{\lambda=re^{i\theta}\in\mathbb{C}:r>0,\theta\in\left(-\frac{\pi}{2}-\delta,\frac{\pi}{2}+\delta\right)\right\\}$

then

$\mathrm{Re}\;\sqrt{A^{2}+4\lambda}>(1-\varepsilon)A.$

We then have the following proposition:

######  Proposition 3.5.

For an asymptotically hyperbolic metric $(M^{n},g)$, and any integer $\ell\geq
0$ and $\alpha\in(0,1)$, given $\mu\in(0,n-1)$ there exists a $\delta>0$ so
that for any $\lambda$ in the sector

$S=\left\\{\lambda=re^{i\theta}\in\mathbb{C}:r>0,\theta\in\left(-\frac{\pi}{2}-\delta,\frac{\pi}{2}+\delta\right)\right\\},$

the corresponding operator

$\lambda-\Delta^{g}:\mathscr{C}^{\ell+2,\alpha}_{\mu}(M;\mathbb{R})\to\mathscr{C}^{\ell,\alpha}_{\mu}(M;\mathbb{R})$

is a Fredholm map. Moreover the kernel of $\lambda-\Delta^{g}$ on
$\mathscr{C}^{\ell+2,\alpha}_{\mu}(M;\mathbb{R})$ coincides with its $L^{2}$
kernel.

###### Proof.

Given $\mu\in(0,n-1)$, choose $\varepsilon>0$ so small that
$\frac{n-1}{2}\varepsilon<\mu<n-1-\frac{n-1}{2}\varepsilon.$ By Lemma 3.4 with
$A=n-1$, there exists a $\delta>0$ so that for any $\lambda\in S$,
$\mathrm{Re}\;{\sqrt{(n-1)^{2}+4\lambda}}>(1-\varepsilon)(n-1)$. Thus $\mu$
lies in the Fredholm range of Theorem 2.3, and the result follows. ∎

Combining this proposition with our observations above confining the $L^{2}$
spectrum of the Laplacian to a certain ray of the negative real line yields
the following:

######  Corollary 3.6.

Given the hypotheses of Proposition 3.5, then for $\lambda\in S$,

$\lambda-\Delta^{g}:\mathscr{C}^{\ell+2,\alpha}_{\mu}(M;\mathbb{R})\to\mathscr{C}^{\ell,\alpha}_{\mu}(M;\mathbb{R})$

is an isomorphism.

###### Proof.

We check that the $L^{2}$ kernel of $\lambda-\Delta^{g}$ is empty for
$\lambda\in S$. But if $(\lambda-\Delta^{g})v=0$, then $v$ is an
$L^{2}$-eigenfunction of $\Delta^{g}$ with eigenvalue $\lambda$, which is
impossible since $S\cap(-\infty,0]=\varnothing$. ∎

We conclude that the resolvent set of $\Delta^{g}$ on little Hölder spaces
contains a sector; consequently, we have verified condition S1 of sectoriality
in Definition 2.1. In what follows we may always assume that the closure of
the sector $S$ still lies in the resolvent set; if not, we take slightly
smaller $\delta$ and use the fact that $0$ is not in the spectrum of
$\Delta^{g}$.

We now are ready to prove Theorem 1 for the Laplacian on functions.

Theorem 1 (part 1) _For an asymptotically hyperbolic metric $(M^{n},g)$, and
any integer $\ell\geq 0$ and $\alpha\in(0,1)$, given $\mu\in(0,n-1)$, there
exists a $\delta>0$ and a sector of the form_

$S=\left\\{\lambda=re^{i\theta}\in\mathbb{C}:r>0,\theta\in\left(-\frac{\pi}{2}-\delta,\frac{\pi}{2}+\delta\right)\right\\}$

_such that $\Delta^{g}$ is sectorial in $\mathscr{C}^{\ell,\alpha}_{\mu}$. _

###### Proof.

By Corollary 3.6, property S1 of sectoriality holds (see Definition 2.1).
Applying Lemmas 2.2 and 3.1, and Proposition 3.3 we obtain the resolvent
estimate S2 in this sector. Thus $\Delta^{g}$ is sectorial in these weighted
little Hölder spaces. ∎

###  3.3. The Lichnerowicz Laplacian on symmetric 2-tensors

We now consider the operator

$L^{g}=\Delta^{g}_{L}-2(n-1),$

where $\Delta^{g}_{L}$ is the Lichnerowicz Laplacian acting on symmetric
$2$-tensors given by

$\Delta_{L}^{g}u_{ij}=g^{kl}\nabla^{g}_{k}\nabla^{g}_{l}u_{ij}+2\mathrm{R}_{\phantom{k}ij\phantom{l}}^{k\phantom{ij}l}u_{kl}-{\operatorname{Rc}_{i}}^{k}u_{kj}-{\operatorname{Rc}_{j}}^{k}u_{ik}.$

Recall that $\mathcal{S}$ denotes the bundle of symmetric $2$-tensors. By
Lemma 4.9 of [7], $L^{g}$ defines an unbounded, closed, and self-adjoint
operator on $L^{2}(M;\mathcal{S})$ with domain
$D(L^{g})=H^{2}(M,\mathcal{S})$. Note that if $g=h$ is the model hyperbolic
metric, then
$L^{h}=h^{kl}\nabla^{h}_{k}\nabla^{h}_{l}u_{ij}+2\mathrm{R}_{\phantom{k}ij\phantom{l}}^{k\phantom{ij}l}u_{kl}$,
and by Lemma 14 of [4], $\lambda-L^{g}$ is fully elliptic if
$\lambda\not\in(-\infty,-\frac{(n-1)^{2}}{4}]$.

To understand the location of the essential spectrum, i.e., the set on which
$L^{g}$ is not Fredholm, one again needs to compute the indicial operator. We
only summarize this computation, which can be found in sources [5, Lemma 2.9]
and [11, Proposition 3]222There is an apparent discrepancy between these
references that traces back to the choice of basis for the bundle of symmetric
$2$-tensors; see the discussion on page 388 of [11] for a clear explanation.
Note that our convention from page 3.2 most closely resembles [11].
Additionally we take $\dim M=n$ instead of $n+1$.. The action of $L^{g}$ on
$\mathcal{S}$ respects the decomposition of the bundle of symmetric
$2$-tensors into their traces, $\mathcal{G}$, and their trace-free parts,
$\mathcal{S}_{0}$:

$\mathcal{S}=\mathcal{G}\oplus\mathcal{S}_{0}.$

Further, the bundle $\mathcal{S}_{0}$ of trace-free symmetric $2$-tensors
splits into three irreducible summands $\mathcal{V}_{i}$, corresponding to the
normal $\mathcal{V}_{1}$, mixed $\mathcal{V}_{2}$, and purely tangential
$\mathcal{V}_{3}$ components near the boundary relative to the decomposition
given in equation (3.1). For each of these sub-bundles one computes an
indicial operator. It turns out that the smallest interval centred about
$\frac{n-1}{2}$ between the roots of these polynomials is always given by the
indicial polynomial stemming from the action of $L^{g}$ on the bundle
$\mathcal{V}_{3}$ of purely tangential trace-free tensor fields. Indeed, the
indicial operator on $\mathcal{V}_{3}$ is

$I_{s}(L^{g}|_{\mathcal{V}_{3}})\overline{u}=(s^{2}-(n-1)s)\overline{u},$

where $\overline{u}$ is a $\mathscr{C}^{2}$ section of $\mathcal{S}|_{\partial
M}$ extended arbitrarily into $M$. Thus the indicial roots are again $s=0$ and
$s=n-1$ and the indicial radius is $\Xi=\frac{n-1}{2}$. Theorem 2.3 implies
that

$L^{g}:\mathscr{C}^{\ell+2,\alpha}_{\mu}(M;\mathcal{S})\to\mathscr{C}^{\ell,\alpha}_{\mu}(M;\mathcal{S})$

is Fredholm for $\mu\in(0,n-1)$, with kernel equal to its $L^{2}$-kernel.

We again add a complex eigenvalue parameter $\lambda$. The next proposition
documents the indicial roots of $\lambda-L^{g}$. Note that this proposition
and proof is essentially a transcription of Proposition 3 of [11] to our
setting; however, the operator $L^{g}$ differs by a sign according to our
convention and we take $\dim M=n$ instead of $n+1$.

######  Proposition 3.7.

For an asymptotically hyperbolic manifold $(M^{n},g)$, the set of indicial
roots of $\lambda-L^{g}$ corresponds to the pairs

$\zeta_{1}^{\pm}=\frac{1}{2}\left(n-1\pm\sqrt{(n-1)^{2}+8(n-1)+4\lambda}\right),$
$\zeta_{2}^{\pm}=\frac{1}{2}\left(n-1\pm\sqrt{(n-1)^{2}+4(n-1)+4+4\lambda}\right),$
$\zeta_{3}^{\pm}=\frac{1}{2}\left(n-1\pm\sqrt{(n-1)^{2}+4\lambda}\right),$

where $\zeta_{i}^{\pm}$ are the indicial roots of the operator on the sub-
bundle $\mathcal{V}_{i}\subset\mathcal{S}_{0}$ defined above, and the indicial
roots of $\lambda-L^{g}$ on $\mathcal{G}$ are also given by $\zeta_{1}^{\pm}$.

Combining Lemma 3.4, Proposition 3.7, and Theorem 2.3 yields

######  Proposition 3.8.

For an asymptotically hyperbolic metric $(M^{n},g)$, and any integer $\ell\geq
0$, given $\mu\in(0,n-1)$, there exists a $\delta>0$ so that for any $\lambda$
in the sector

$S=\left\\{\lambda=re^{i\theta}\in\mathbb{C}:r>0,\theta\in\left(-\frac{\pi}{2}-\delta,\frac{\pi}{2}+\delta\right)\right\\}$

the operator

$\lambda-L^{g}:\mathscr{C}^{\ell+2,\alpha}_{\mu}(M;\mathcal{S})\to\mathscr{C}^{\ell,\alpha}_{\mu}(M;\mathcal{S})$

is a Fredholm map. Moreover the kernel of $\lambda-L^{g}$ on
$\mathscr{C}^{\ell+2,\alpha}_{\mu}(M;\mathcal{S})$ coincides with its $L^{2}$
kernel.

Consequently, in order to study the location of the spectrum of $L^{g}$ on
weighted little Hölder spaces, we need to better understand the $L^{2}$
kernel. In general as a real and self-adjoint operator, $L^{g}$ has its
spectrum confined to the real line, with the possibility of finitely many
eigenvalues outside of $(-\infty,-\frac{(n-1)^{2}}{4}]$. So $\lambda-L^{g}$
has no $L^{2}$ kernel if $\lambda$ has nonzero imaginary part. Define

(3.14)
$\lambda_{0}=\sup_{v\in\mathscr{C}^{\infty}_{c}(M,\mathcal{S})}\frac{(v,L^{g}v)}{\|v\|_{2}^{2}}.$

It follows that $\lambda-L^{g}$ has no $L^{2}$ kernel on the translated sector
$S+\lambda_{0}$, and so condition $S1$ of sectoriality Definition 2.1 holds on
$S+\lambda_{0}$.

We are ready to finish the proof of the second part of Theorem 1, restated
here:

Theorem 1 (part 2) _For an asymptotically hyperbolic metric $(M^{n},g)$, and
any integer $\ell\geq 0$ and $\alpha\in(0,1)$, given $\mu\in(0,n-1)$, there
exists a $\delta>0$ and a sector of the form $S+\lambda_{0}$ so that
$\Delta^{g}_{L}-2(n-1)$ on symmetric $2$-tensors is sectorial in
$\mathscr{C}^{\ell,\alpha}_{\mu}$._

###### Proof.

The remarks immediately above show that property S1 for sectoriality holds on
$S+\lambda_{0}$. We may assume the vertex $\omega$ is strictly positive by
further translation of the sector, if necessary. Applying Lemma 3.1 and
Proposition 3.3 we obtain the resolvent estimate S2 in this sector. Thus
$L^{g}$ is sectorial in these weighted little Hölder spaces. ∎

## References

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  * [7] J. M. Lee, _Fredholm operators and Einstein metrics on conformally compact manifolds_ , Memoirs AMS (2006) No. 864
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# Multi-modal Retrieval of Tables and Texts Using Tri-encoder Models

Bogdan Kostić  Julian Risch  Timo Möller
deepset
{bogdan.kostic, julian.risch<EMAIL_ADDRESS>
###### Abstract

Open-domain extractive question answering works well on textual data by first
retrieving candidate texts and then extracting the answer from those
candidates. However, some questions cannot be answered by text alone but
require information stored in tables. In this paper, we present an approach
for retrieving both texts and tables relevant to a question by jointly
encoding texts, tables and questions into a single vector space. To this end,
we create a new multi-modal dataset based on text and table datasets from
related work and compare the retrieval performance of different encoding
schemata. We find that dense vector embeddings of transformer models
outperform sparse embeddings on four out of six evaluation datasets. Comparing
different dense embedding models, tri-encoders with one encoder for each
question, text and table increase retrieval performance compared to bi-
encoders with one encoder for the question and one for both text and tables.
We release the newly created multi-modal dataset to the community so that it
can be used for training and evaluation.

##  1 Introduction

Finding the answer to a factual question in a large collection of documents is
a tedious task that many people from a broad range of domains have to complete
on a daily basis. In order to address this task with machine learning
approaches, it has been formalized as open-domain extractive question-
answering (QA). More specifically, given a natural language question and a
database of text documents as a knowledge base, open-domain extractive QA aims
to extract a substring that answers the given question out of one of the
documents.

The standard approach for this task is a pipeline architecture consisting of
two components: a retriever selecting a small subset of relevant documents
from the database and a reader extracting granular answers out of each of
these retrieved documents Voorhees and Tice (2000). In this paper, we focus on
the retriever and present a transformer-based tri-encoder model as an
implementation of this component. Retrievers can also be implemented with bag-
of-words retrieval methods, such as TF-IDF or BM25, that transform all
documents and the question to sparse vector representations. However, as these
methods rely on a lexical overlap of the question and the documents, they fail
to capture synonymy and other semantic relationships. This limitation
motivates the use of dense vector representations and we compare dense
retrieval models to a BM25 baseline in our experiments. A survey on term-
based, early semantic, and neural semantic models for document retrieval as a
first step before document reranking and down-stream tasks, such as question
answering, has been published by Cai et al. (2021).

To date, most of the research centered around question-answering focuses on
using free-form text as the single source for answering questions. However,
valuable information can obviously be found in other modalities as well. For
instance, a lot of information is stored in semi-structured tables; according
to Cafarella et al. (2008), more than 14.1 billion tables can be found on the
World Wide Web. Given that a user typically does not know in advance in which
modality the answer to their question resides, a QA system capable of jointly
handling text and tables is needed. One major challenge in building such a
system is to represent texts and tables in a way that allows capturing
semantic similarity and retrieving texts and tables that are semantically
related to a given question.

#### Contributions

The contributions of this paper can be summarized as follows: (1) we present
bi-encoder and tri-encoder models that are capable of joint retrieval of
tables and texts; (2) we create and release a multi-modal dataset for training
and evaluating models on this task;111https://multimodalretrieval.s3.eu-
central-1.amazonaws.com/data.zip (3) we compare sparse retrieval models with
dense retrieval models using bi-encoders and tri-encoders on our new multi-
modal dataset and on five uni-modal datasets from related work.

#### Outline

The remainder of this paper is structured as follows: Section 2 summarizes
existing methods for uni-modal retrieval of texts on the one hand and of
tables on the other hand. Further, it discusses the only two, recently
published approaches for joint retrieval of tables and texts. Section 3
briefly describes the existing uni-modal datasets and our new multi-modal
dataset, which we use to train the retrieval models presented in Section 4 and
to evaluate these models in Section 5. Section 6 concludes the paper and gives
directions for future work.

##  2 Related Work

To the best of our knowledge the only existing work that addresses joint
retrieval of tables and texts is by Chen et al. (2021), Talmor et al. (2021)
and Li et al. (2021). To address the challenge of limited context available
for tables, Chen et al. (2021) fuse a table segment and text passages into one
block if they mention the same named entities. Each block is represented with
a single dense embedding so that a table and relevant passages are jointly
retrieved as a group if the embedding is similar to that of the question. This
grouping makes sense because Chen et al. (2021) address the task of multi-hop
QA, where information needs to be aggregated from multiple tables and texts to
answer a question. In contrast to that, we address the slightly different task
of single-hop QA, where either only one table or one text is needed to answer
a question. Therefore, our approach represents tables and texts with separate
embeddings, which are in the same embedding space. The advantage here is that
the model learns to estimate relevance on the more fine-grained level of
individual tables or texts and can decide whether a particular table or text
is more relevant to the question.

Li et al. (2021) also address the task of multi-hop QA on texts and tables but
retrieve them individually. They make use of the sparse retrieval method BM25
and a transformer-based reranker to generate a set of candidate texts and
tables. With two separate BM25 indices for texts and tables, they retrieve a
set of documents for each modality. In a second step, they apply a joint BERT-
based reranker to reduce the size of candidate texts and tables. Talmor et al.
(2021) create a MultiModalQA dataset containing questions that require joint
reasoning over tables, texts, and images.

Otherwise closely related to our approach are TaBERT by Yin et al. (2020) and
TaPas by Herzig et al. (2020), who use tables and texts for language model
pre-training but do not consider joint retrieval of tables and texts. TURL
focuses on representation learning for tables but also does not consider the
retrieval task Deng et al. (2020). Due to this limited amount of prior
research, we discuss related work on the separate tasks of uni-modal text
retrieval and table retrieval in the following.

###  2.1 Text Retrieval

Dense Passage Retrieval (DPR) by Karpukhin et al. (2020) relies on a bi-
encoder model comprising two separate BERT models Devlin et al. (2019).
Similar to TF-IDF, DPR is a vector space model that represents both queries
and documents in the same vector space. However, while TF-IDF represents text
documents as very high dimensional sparse vectors, DPR relies on relatively
low dimensional dense embeddings. While one of the models, the passage encoder
(BERTp), is used to encode text passages at indexing time, the second model,
the question encoder (BERTq), is used to encode questions at query time. Since
BERT’s [CLS]-token is particularly designated to capture the meaning of the
whole input sequence, its embedding is used as a representation vector for
both the text passages and the questions.

The training aims to increase the dot product or cosine similarity of
semantically similar passages and questions. In order to achieve this,
Karpukhin et al. (2020) use, besides the question’s positive passage, hard
negative passages as well as in-batch negative passages as training signal.
Hard negatives are sampled utilizing a BM25-based retriever on the whole
English Wikipedia dump. For each question, they use the highest ranked passage
not containing the question’s answer string. DPR drastically outperforms BM25
by almost 20 percentage points with regard to recall@20 on the Natural
Questions (NQ) dataset by Kwiatkowski et al. (2019).

###  2.2 Table Retrieval

Most existing table retrieval approaches rely on supervised learning-to-rank
approaches. Zhang and Balog (2018) combine a set of hand-crafted query
features, table features and query-table features with semantic similarity of
table and query as additional feature. To get table and query representations,
they use the average of pre-trained word2vec Mikolov et al. (2013) and RDF2vec
embeddings Ristoski and Paulheim (2016). These features are then used to train
a random forest regressor to get relevance scores of the tables with regard to
the query. Interestingly, training word embeddings on a corpus of tables
instead of texts does not improve performance Zhang et al. (2019).

Shraga et al. (2020a) combine intrinsic and extrinsic table similarity scores.
For the intrinsic table similarity, they concatenate the table’s title,
caption, header and data rows and use a sliding window over this concatenated
string to get a set of candidate passages for each table. Each candidate
passage is scored using BM25 and the maximum of all passages of a table is the
intrinsic score with regard to a query. The extrinsic score is based on table-
table similarities to ensure that similar tables get a similar final score.

Bagheri and Al-Obeidat (2020) focus on hard queries that contain terms that do
not occur in the relevant tables, which means the query and the relevant
tables have a low lexical overlap. To this end, they learn low dimensional
latent factor matrices to represent tables as well as queries, i.e., they
learn term co-occurrences to be able to get tables that address the same topic
but only partially overlap on a lexical level. Based on this result, we
compare results on datasets with high or low lexical overlap in our
experiments.

There are four deep learning approaches for table retrieval Shraga et al.
(2020b); Pan et al. (2021); Chen et al. (2020b); Herzig et al. (2021). Shraga
et al. (2020b) treat tables and queries as multi-modal objects that consist of
query, table caption, schema, rows and columns. Each component is encoded
using its own neural network encoder that accounts for its special
characteristics. Subsequently, these uni-modal encodings are joined into a
single representation, which is passed on to fully connected layers that
predict whether the input table is relevant with regard to the input query.

Pan et al. (2021) stack two retrieval components. First, they use BM25 to
produce a large subset of possibly relevant tables. These tables are then
passed to a row-column intersection model that generates a probability
distribution over the table cells whether they contain the answer to the
user’s query. The maximum cell-level score for each table represents its
retrieval score.

Chen et al. (2020b) apply the transformer-based language model BERT Devlin et
al. (2019) to the table retrieval task by combining BERT embeddings with
other, hand-curated table and query features. Their approach consists of
several components, including a step where the concatenation of a query, a
table’s context fields and selected relevant table rows are processed by a
BERT model. A significant downside of this approach is that it becomes
inefficient with increasing number of tables: for each query, all tables need
to be passed through a BERT network.

Herzig et al. (2021) solve this efficiency problem by adapting Karpukhin et
al.’s (2020) dense passage retrieval approach to dense table retrieval (DTR).
To this end, they make use of TaPas Herzig et al. (2020), a transformer-based
language model that has been pre-trained on millions of tables. TaPas extends
BERT by adding three different types of positional embeddings to encode the
two-dimensional tabular structure: row, column and rank embeddings. This
allows to flatten the table by concatenating the rows to a one-dimensional
sequence of tokens. Similar to Karpukhin et al. (2020), Herzig et al. (2021)
make use of a bi-encoder approach. However, they use two TaPas instances
instead of BERT instances to encode the queries and the tables, respectively.
The goal of training this bi-encoder is to build an embedding model that
generates similar embeddings for questions and their relevant tables. As in
Karpukhin et al.’s (2020) approach, this goal is achieved using hard-negatives
retrieved from all the tables from the English Wikipedia dump as well as in-
batch negatives. DTR outperforms BM25 by more than 40 percentage points on the
NQ-Tables dataset Herzig et al. (2021). However, the experiments also show
that TaPas requires additional pre-training on the task of table retrieval on
millions of tables scraped from Wikipedia. As a further research direction for
future work, Herzig et al. (2021) propose to combine tables and texts for
multi-modal open-domain QA. We contribute towards this goal in our paper by
providing a multi-modal retriever as one component of a multi-modal open-
domain QA pipeline on tables and texts.

##  3 Datasets

The training and evaluation of models examined in this paper on the task of
multi-modal retrieval makes use of five datasets from related work: NQ
Kwiatkowski et al. (2019), NQ-Tables Herzig et al. (2021), WikiSQL Zhong et
al. (2017), a subset of WikiSQL, which we call WikiSQL${}_{\text{ctx-
independent}}$, and OTT-QA Chen et al. (2021). This section briefly explains
the characteristics of these datasets and of our newly created multi-modal
retrieval dataset comprising tables and texts, which we call
MultiModalRetrieval. Table 1 gives an overview of the modality and the number
of samples in each dataset.

Dataset | Modality | Train | Test | Ctx-ind.
---|---|---|---|---
NQ | text | 58,880 | 3,610 | ✔
NQ-Tables | table | 9,594 | 966 | ✔
WikiSQL | table | 56,355 | 15,878 | ✖
WikiSQL${}_{\text{ctx-independent}}$ | table | 7,336 | 2,101 | ✔
OTT-QA | table | 41,469 | 2,158 | ✔
MultiModalRetrieval | text & table | 120,239 | 4,937 | ✔
Table 1: Modality and number of train and test samples for multi-modal
retrieval models. Only WikiSQL is not context-independent (ctx-ind.), which is
why we create the subset WikiSQL${}_{\text{ctx-independent}}$.

#### NQ

Natural Questions (NQ) Kwiatkowski et al. (2019) is an open-domain QA dataset
on Wikipedia articles. It consists of questions, their answers and the text
passages the answers reside in. The questions are _natural_ because they
consist of real user queries issued to the Google search engine instead of
questions posed by annotators after reading a text passage, which was done to
create other popular QA datasets, such as the Stanford Question Answering
Dataset (SQuAD) Rajpurkar et al. (2018). Having _natural_ questions does not
only ensure that the questions correspond to information needs by real users
but also makes the dataset open-domain, i.e., the questions are context-
independent and can be understood without their accompanying text passage that
contains the answer. For the purpose of retrieval, we utilize Karpukhin et
al.’s (2020) preprocessed variant of NQ.

#### NQ-Tables

The answers to most of the questions inside the NQ dataset can be found in
plain unstructured text. However, a subset of the questions are answered by
tables. As a consequence, Herzig et al. (2021) construct NQ-Tables, a table-
specific QA dataset based on NQ. To achieve this objective, they extract all
the tables and all the questions whose answers reside inside a table. They
come up with a dataset consisting of 9,594 questions in the training set,
1,068 questions in the development set, 966 questions in the test set and
169,898 tables in total. Given that this dataset is a subset of NQ, the
questions share the characteristic of being context-independent.

#### WikiSQL

WikiSQL Zhong et al. (2017) is a closed-domain QA dataset consisting of 24,241
tables and 80,654 natural language questions together with their corresponding
SQL query. To build this dataset, Zhong et al. (2017) generate a number of
random SQL queries for each table. These SQL queries are then transformed into
crude questions using templates. Finally, Amazon Mechanical Turk crowd workers
paraphrase these crude questions into natural language questions, which are
checked by two additional crowd workers.

#### WikiSQL${}_{\text{ctx-independent}}$

Since WikiSQL is a closed-domain QA dataset, the majority of its questions is
context-dependent, i.e., they do not provide enough context to be answered
without their accompanying table and are therefore not suitable for training
or evaluation on the retrieval task. One example for such insufficient
questions inside the dataset is: _“Who is the player that wears number 42?”_ ,
which cannot be answered without additional context given in the table, such
as the name of a sports team and a year. As a consequence, to make use of
WikiSQL, all questions that do not provide enough context for retrieval need
to be filtered out. For automating this filtering, we labeled a subset of
WikiSQL’s questions with regard to whether they are either context-independent
or under-specified resulting in 4,553 labels as training set and 612 labels as
test set. These labels are then used to train a classifier that predicts
whether a question provides enough context. We fine-tune a RoBERTa-base Liu et
al. (2019) language model achieving an accuracy of 0.8134 and a macro-averaged
F1-score of 0.7748 on the test set. Next, we apply this classifier to the
whole WikiSQL dataset to filter out all the questions that are predicted as
under-specified.

#### OTT-QA

OTT-QA Chen et al. (2021) is an open-domain multi-hop QA dataset of texts and
tables from Wikipedia built upon HybridQA Chen et al. (2020a), a closed-domain
multi-hop QA dataset. Multi-hop QA refers to the fact that most questions
inside the dataset require a combination of different texts and/or tables
instead of a single document in order to be answered. To generate an open-
domain version of HybridQA, Chen et al. (2021) let crowd workers
decontextualize all the questions. Furthermore, they add additional question-
answer pairs on newly crawled tables. Since the published annotations contain
only the gold tables but not the gold texts, we use OTT-QA only for generating
table retrieval training samples and evaluating the uni-modal retrieval of
tables.

#### MultiModalRetrieval

Given that no multi-modal dataset of tables and texts is readily available,
this paper newly introduces such a dataset based on datasets from related
work. For this purpose, we combine the question-passage pairs from NQ as
questions requiring a text passage to be answered with the question-table
pairs from NQ-Tables, WikiSQL${}_{\text{ctx-independent}}$, and OTT-QA as
questions requiring a table to be answered. Both Karpukhin et al. (2020) for
text retrieval and Herzig et al. (2021) for table retrieval show that adding
hard negatives as training signal boosts the retrieval performance of the
model significantly. Therefore, we index 21 million Wikipedia passages (from
Karpukhin et al. (2020)) and 7 million Wikipedia tables (used by Eisenschlos
et al. (2020) to pre-train TaPas) with Elasticsearch to sample hard negatives
using BM25. For each question, the highest ranked passages or tables that do
no contain the answer string were chosen, i.e., a question originating from a
tabular question-answering dataset can also have a text passage as hard
negative, and vice versa.

##  4 Approach

Our two approaches for the joint retrieval of tables and texts comprise bi-
encoder and tri-encoder models based on uni-modal dense retrieval methods by
Karpukhin et al. (2020) and Herzig et al. (2021).

In our first approach, the bi-encoder uses one language model to encode the
questions and a second model to encode both the tables and the text passages.
Our second approach adds a third encoder, such that there is one separate
encoder for questions, text passages and tables. In contrast to the bi-encoder
where tables and texts are encoded by the same model, the tri-encoder routes
tables to the table encoder model and text passages to the text encoder model.

We train three different bi-encoders and three different tri-encoders, which
differ in the underlying language models as specified in Table 2. The first
multi-modal bi-encoder consists of two different BERT-small instances that
serve as question encoder and table and text encoder, respectively. Given that
BERT models only allow one-dimensional strings of text as input, the two-
dimensional tables are transformed into one dimension by concatenating the
titles of the page and the section the table occurs in, the caption of the
table, and each row of the table. In order to analyze whether the table-
specific language model TaPas Herzig et al. (2020); Eisenschlos et al. (2020)
gives a performance boost compared to a plain BERT model, the remaining two
bi-encoders make use of a TaPas model that is pre-trained for the task of
table retrieval Herzig et al. (2021) for at least one of their encoders. Thus,
the second bi-encoder uses a BERT-small instance as question encoder and a
TaPas-small instance as table and text encoder. The third bi-encoder utilizes
two TaPas-small models, one to encode the questions and the second to encode
both text and tables. Using BERT-small instead of BERT-base or BERT-large
models drastically reduces the number of parameters and allows to fit more
training samples into one batch. In contrast to a BERT-large model with 24
transformer layers, hidden representations of size 1024, 16 attention heads,
and a total number of 335M parameters, BERT-small consists of only 4
transformer layers, hidden representations of size 512, 8 attention heads, and
a total number of 29.1M parameters.

The first tri-encoder model uses BERT-small instances for all of its three
encoders. Also for the tri-encoder approach, we analyze the impact of using
TaPas for at least one of the encoders. Therefore, the second architecture
makes use of a BERT-small model for both the question encoder and the text
encoder and utilizes a TaPas-small instance to encode the tables. The third
tri-encoder model uses two TaPas-small instances to encode questions and
tables and a BERT-small instance to encode text passages.

Herzig et al. (2021) use an additional down-projection layer to reduce the
dimensionality of the question and table embeddings. We evaluated models with
and without such a down-projection layer and found that their results do not
differ significantly. Given that the models using an additional down-
projection layer are more complex than models that directly utilize the
embedding of the [CLS]-token, we consider only TaPas-models without a down-
projection layer throughout the remainder of this paper, including the
experiments.

| Encoders
---|---
| Question | Text | Table
Bi-encoder | BERT-small | —— BERT-small ——
BERT-small | —— TaPas-small ——
TaPas-small | —— TaPas-small ——
Tri-encoder | BERT-small | BERT-small | BERT-small
BERT-small | BERT-small |  TaPas-small
TaPas-small | BERT-small |  TaPas-small
Table 2: Examined multi-modal bi-encoder and tri-encoder models. | Bi-encoders | Tri-encoders
---|---|---
Learning rate | 1e-5 | 1e-5
LR schedule | linear | linear
Warm-up steps | 10% | 10%
Batch size | 38 | 28
Epochs | 10 | 10
Optimizer | Adam | Adam
Table 3: Hyperparameters used to train the bi-encoder and tri-encoder multi-
modal retrieval models.

Table 3 specifies the hyperparameters used to train the bi-encoder and tri-
encoder models on the training split of the MultiModalRetrieval dataset
described in Section 3. The learning objective is to create similar embeddings
for relevant texts and/or tables with regard to a question. To train the
models more efficiently, we make use of in-batch negatives besides each
question’s hard negative text or table as suggested by Karpukhin et al. (2020)
in the context of text retrieval. Given that the training samples inside a
batch are randomly selected from all training examples, questions comprising a
text passage as gold-label might have tables as negative labels, and vice
versa.

##  5 Experiments

We compare the presented dense retrieval models to the sparse retrieval method
BM25 and evaluate them based on recall@$k$ with $k\in\\{10,20,100\\}$. The
search space to evaluate the models needs to consist of both texts and tables.
For this purpose, 500,000 text passages are randomly sampled from Karpukhin et
al. (2020)’s preprocessed Wikipedia passages making sure that the gold
passages are among these passages. Furthermore, besides the text passages, all
the tables from WikiSQL, OTT-QA, and NQ-Tables are used, resulting in 656,166
tables and therefore approximately 1.2 million documents in total. The models
are evaluated on a random sample of 1,000 questions of each dataset’s test
split as listed in Table 1 and the full test split of the MultiModalRetrieval
dataset.

Following Karpukhin et al. (2020), a document retrieved for a question
originating from NQ or NQ-Tables is considered a correct match if the document
contains the answer string of the granular answer. Given that the derivation
of the granular answer for questions originating from WikiSQL and OTT-QA might
need further aggregation, such as summation or counting, and, therefore, the
answer string does not need to be present in a relevant document, a retrieved
document is only considered a correct match if it is the gold annotated table.
This evaluation procedure might have the effect of incorrectly judging non-
gold tables that contain the answer to a query as irrelevant. However, since
we apply the same evaluation procedure for all models, the numbers should be
comparable. Table 4 specifies the evaluation results for BM25, all bi-encoder
and all tri-encoder models.

Encoders | NQ | WikiSQL | WikiSQL${}_{\text{ctx-independent}}$
---|---|---|---
Question | Text | Table | R@10 | R@20 | R@100 | R@10 | R@20 | R@100 | R@10 | R@20 | R@100
——— BM25 ——— | 53.3 | 59.8 | 73.9 | 42.1 | 47.1 | 59.5 | 61.2 | 67.2 | 81.0
BERT | — BERT — | 70.1 | 76.0 | 84.2 | 17.5 | 22.8 | 39.6 | 30.7 | 40.9 | 59.1
BERT | — TaPas — | 50.0 | 57.1 | 72.7 | 6.4 | 8.9 | 20.3 | 12.3 | 17.1 | 36.7
TaPas | — TaPas — | 59.6 | 67.4 | 78.2 | 9.5 | 13.5 | 26.3 | 16.6 | 23.6 | 45.9
BERT | BERT | BERT | 69.1 | 75.0 | 83.5 | 20.2 | 26.8 | 43.3 | 30.8 | 38.2 | 60.7
BERT | BERT | TaPas | 59.0 | 67.1 | 77.6 | 3.5 | 5.1 | 15.2 | 7.5 | 12.0 | 30.9
TaPas | BERT | TaPas | 46.2 | 53.5 | 68.0 | 10.7 | 14.0 | 28.8 | 17.1 | 23.3 | 45.0
|  |  |  |  |  |  |  |  |  |  |
Encoders | OTT-QA | NQ-Tables | MultiModalRetrieval
Question | Text | Table | R@10 | R@20 | R@100 | R@10 | R@20 | R@100 | R@10 | R@20 | R@100
——— BM25 ——— | 40.2 | 45.6 | 58.2 | 56.6 | 65.1 | 82.8 | 50.7 | 57.0 | 71.1
BERT | — BERT — | 72.9 | 78.0 | 89.4 | 84.9 | 91.2 | 96.7 | 55.2 | 61.8 | 73.8
BERT | — TaPas — | 30.6 | 40.0 | 63.1 | 71.6 | 72.6 | 90.0 | 34.2 | 39.1 | 56.6
TaPas | — TaPas — | 49.9 | 60.4 | 82.8 | 78.9 | 86.0 | 94.0 | 42.9 | 50.2 | 65.4
BERT | BERT | BERT | 73.8 | 79.7 | 90.1 | 86.4 | 91.6 | 96.7 | 56.1 | 62.3 | 74.9
BERT | BERT | TaPas | 25.8 | 34.3 | 59.4 | 52.3 | 62.1 | 80.7 | 29.6 | 36.1 | 52.8
TaPas | BERT | TaPas | 50.8 | 60.9 | 79.8 | 76.9 | 82.8 | 92.9 | 40.3 | 46.9 | 62.9
Table 4: Evaluation results of BM25 and bi-encoder and tri-encoder retrieval
models on 1000 random samples of the test splits of NQ, WikiSQL, context-
independent questions of WikiSQL, OTT-QA, and NQ-Tables and the full test set
of our new MultiModalRetrieval dataset with regard to recall@10, recall@20,
and recall@100.

The evaluation shows that BM25 outperforms all dense methods on both the full
WikiSQL dataset and WikiSQL’s context-independent questions. It outperforms
the best dense retrieval model on this dataset, the tri-encoder consisting of
three BERT models, by 21.9 percentage points on all WikiSQL questions and 30.4
percentage points on context-independent WikiSQL questions with regard to
recall@10.

Analysing the WikiSQL dataset in more detail shows a very high lexical overlap
of questions with their accompanying table. A combination of the Jaccard
coefficient and Gestalt-Pattern-Matching222We use an implementation from:
https://github.com/seatgeek/fuzzywuzzy#token-set-ratio allows to quantify the
lexical overlap of the questions with their corresponding tables without
incorporating neither duplicate occurrences of the same word nor the order of
the words inside the questions and the tables. This word order independence is
particularly important for the analysis, given that the sparse retrieval
method BM25 is order-agnostic. Even after lower-casing questions and tables
and removing stopwords in the WikiSQL dataset, 40.68% of the questions
lexically overlap completely with the relevant table, according to the
combination of the Jaccard coefficient and Gestalt-Pattern-Matching. This
large lexical overlap explains BM25’s strong performance on that dataset in
Table 4. In contrast, only 0.27% of the questions in OTT-QA overlap completely
with their accompanying table. For the other datasets, 15.73% of the questions
in NQ-Tables, 14.47% of the questions in NQ, and 21.96% of the questions in
MULTIMODALRETRIEVAL overlap completely with their accompanying table or text.

To better understand how the lexical overlap of questions and accompanying
tables influences BM25’s performance, we split the test sets of WikiSQL and
WikiSQL${}_{\text{ctx-independent}}$ into subsets with different ranges of
lexical overlap. As can be observed in Figure 1, the recall of both BM25 and
dense retrieval highly correlates with lexical overlap. Furthermore, while
BM25 outperforms dense retrieval for questions with high lexical overlap, it
is the other way round for questions with low lexical overlap.

(a) WikiSQL

(b) WikiSQL${}_{\text{ctx-independent}}$

Figure 1: Recall@20 of the BM25 model and the BERT-BERT-BERT tri-encoder
across different percentage of lexical overlap of question and table.
Performance of BM25 drops drastically if the lexical overlap is low.

On the sample of the NQ-Tables test set, all dense retrieval models outperform
BM25, except for the tri-encoder that consists of BERT instances as question
and text passage encoder and a TaPas instance as table encoder. The best
performing model on this dataset is the tri-encoder consisting of three BERT
encoders. This model outperforms the sparse retrieval method BM25 by 29.8
percentage points with regard to recall@10.

For the sampled questions of the OTT-QA development set, four out of the six
dense retrieval models outperform BM25. The best performing model is again the
tri-encoder model that is composed of three BERT-small encoders. This model
outperforms BM25 by 33.6 percentage points with regard to recall@10. BM25
outperforms the bi-encoder consisting of a BERT model as question encoder and
a TaPas model as text and table encoder as well as the tri-encoder consisting
of two BERT models serving as question encoder and text encoder, respectively,
and a TaPas model serving as table encoder.

When it comes to the performance on the text modality, i.e., questions
deriving from NQ whose gold-label answer resides in a text passage, four out
of the six dense retrieval models outperform BM25. For this case, the best
performing model is not a tri-encoder but the bi-encoder comprising two BERT
models. This model outperforms BM25 by 16.8 percentage points with regard to
recall@10. However, this bi-encoder exceeds the tri-encoder consisting of
three BERT encoders only slightly by one percentage point. The sparse
retrieval method BM25 beats the bi-encoder consisting of a BERT model as
question encoder and a TaPas model as text and table encoder as well as the
tri-encoder consisting of two TaPas models serving as question encoder and
table encoder, respectively, and a BERT model serving as text encoder.

In summary, the best performance on the WikiSQL test set is achieved by the
sparse retrieval method BM25. The tri-encoder consisting of three BERT
encoders shows the best performance on the remaining two tabular datasets,
OTT-QA and NQ-Tables. On the NQ dataset, i.e., questions whose answers reside
in the textual modality, the bi-encoder consisting of two BERT encoders
performs best but is almost on par with the tri-encoder consisting of three
BERT models.

We can conclude that, under the limited experimental conditions, in
particular, on the datasets used in our study, models involving TaPas as
question, text, and/or table encoder perform worse than models that rely only
on BERT language models. Herzig et al. (2021) show that to be able to use
TaPas for the retrieval of tables, TaPas needs to be additionally pre-trained
on the table retrieval task. These pre-trained table retrieval models, which
are used in the bi-encoders and tri-encoders that involve one or more TaPas
instances as encoder, are, however, pre-trained solely on the task of table
retrieval and not text retrieval. Given the fact that the plain TaPas model
cannot be adapted to retrieval from scratch but needs this special pre-
training, it might be the case that, to use TaPas efficiently for the
retrieval of both texts and tables, it needs to be pre-trained in a multi-
modal setting on the retrieval of both texts and tables. Furthermore, batch
size is significant for training retrieval models, as higher batch sizes make
the training harder by adding more in-batch negatives. While the training of a
bi-encoder does not allow a batch size higher than 38 and the training of a
tri-encoder does not allow a batch size higher than 28 on a Tesla V100 GPU
with 16 GB of memory, Herzig et al. (2021) make use of a batch size of 256 for
training their TaPas-based table retrieval models. Accordingly, it might be
the case that TaPas is more unstable to train and requires, therefore, larger
batch sizes.

##  6 Conclusion and Future Work

This paper presented a transformer-based approach using bi-encoder and tri-
encoder models for multi-modal retrieval of tables and texts. With experiments
on five datasets from related work and one newly created dataset, we show that
the presented dense retrieval models outperform the sparse retrieval model
BM25 if there is a low lexical overlap of questions and relevant tables and
texts. More specifically, the tri-encoder architecture performs better on OTT-
QA and NQ-Tables, which represent the tabular modality, while the bi-encoder
architecture performs slightly better on the NQ dataset representing the
textual modality. We observe that the best retrieval models are those that
rely only on BERT models as encoder and do not make use of TaPas.

From an application point of view, future work could integrate the presented
retrieval models as one component in a multi-modal open-domain QA pipeline and
evaluate it on a real-world use case. Such a pipeline would facilitate
information access immensely by combining valuable information from both
sources rather than relying only on either texts or tables. Another promising
path for future work is to extend our approach to more modalities with
transformer-based models for images, videos, or speech. These models could
serve as encoders for documents of different modalities to jointly train an
$n$-encoder architecture, where one encoder is tailored to the queries and the
remaining $n-1$ encoders are tailored to each of the modalities that the user
would like to search on. Last but not least, the research community would
surely benefit from the creation of more multi-modal datasets to improve
training and evaluation of multi-modal retrieval models and we are only making
a first step in this direction with creating and releasing a dataset of tables
and texts.

## Acknowledgements

We would like to thank Jonathan Herzig and Julian Eisenschlos for taking the
time to discuss ideas with us and to give early feedback on experiment
results.

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