DYNAMICAL ZETA FUNCTIONS FOR ANOSOV …xiaolong/Han_Zeta.pdfDYNAMICAL ZETA FUNCTIONS VIA MICROLOCAL ANALYSIS 3 Conjecture 1.4 (Riemann Hypothesis, Riemann 1859). The non-trivial zeros

DYNAMICAL ZETA FUNCTIONS FOR ANOSOV SYSTEMS VIAMICROLOCAL ANALYSIS BY DYATLOV-ZWORSKI

XIAOLONG HAN

Contents

1. Introduction to the zeta functions 11.1. Riemann zeta function 21.2. Selberg zeta function 31.3. Ruelle zeta function 42. Preliminaries 72.1. Anosov systems 92.2. Poincare maps 92.3. Microlocal analysis 112.4. Flat trace 122.5. Atiyah-Bott-Guillemin trace formulae 152.6. Fredholm theory 182.7. Semiclassical analysis 192.8. Propagation of singularities I 223. The resolvent acting on anisotropic Sobolev spaces 283.1. Anosov systems: Extra facts 283.2. Propagation of singularities II 293.3. Anisotropic Sobolev spaces 333.4. Resolvent estimates on anisotropic Sobolev spaces 344. Meromorphic extension to the whole complex plane 384.1. Ruelle-Pollicott resonances 384.2. Ruelle zeta functions around resonances 41References 42

1. Introduction to the zeta functions

A zeta function ζ(z) is usually a function of a complex variable z ∈ C. There are three basicquestions which apply equally well to all such zeta functions:

(1). Where is the zeta function defined? How far can we extend it to an analytic or meromorphicfunction?

(2). Where are the zeros and poles of ζ(z)? What are the values of ζ(z) at particular values ofz in the domain?

(3). What does this tell us about certain counting quantities?

In the introduction, we mention three different types of zeta functions and their habitats,each of which serves as motivation to the next one in the development of the zeta functions.

(1). Riemann zeta function in number theory,

1

2 XIAOLONG HAN

(2). Selberg zeta function in hyperbolic geometry,(3). Ruelle zeta function in dynamical systems.

1.1. Riemann zeta function.

Definition (Riemann zeta function, Riemann 1859i). The Riemann zeta function is defined as

ζ(z) =∞∑n=1

1

nz.

Lemma 1.1 (Euler product, Euler 1730s).

ζ(z) =∞∑n=1

1

nz=

∏p prime

(1− 1

pz

)−1

.

Proof. Note that (∞∑n=1

1

nz

)(1− 1

2z

)=∞∑n=1

1

nz−∞∑n=1

1

(2n)z=

∞∑n=1,2-n

1

nz.

So (∞∑n=1

1

nz

) ∏p prime

(1− 1

pz

)=

∞∑n=1,p-n

1

nz= 1.

The Riemann zeta function ζ(z) is convergent if <(z) > 1. But it can be extended to theentire complex plane.

Theorem 1.2 (Riemann 1859).

(i). ζ(z) has a simple pole at z = 1.(ii). ζ(z) otherwise has an analytic extension to the entire complex plane C.

(iii). ζ(z) satisfies the functional equation

ζ(z) = 2zπz−1 sin(πz

2

)Γ(1− z)ζ(1− z).

(iv). ζ(z) has zeros −2k, k = 1, 2, ..., which are called the trivial zeros.

Theorem 1.3 (Prime Number Theorem, Hadamard 1896ii, de Valle Poussin 1896iii). The primenumber counting function π(x), i.e. the number of primes less than x, satisfies

π(x) ∼ x

log xas x→∞.

The standard proof uses the Riemann zeta function and its basic property: ζ(z) has a zero-free analytic extension to a neighbourhood of 1. See e.g. Stein-Shakarchi [SS, Chapter 7].Knowing more precisely about where the zeros of ζ(z) can improve the remainder term in thePrime Number Theorem. In particular,

iUeber die Anzahl der Primzahlen unter einer gegebenen Grosse. Monatsberichte der Berliner Akademie(1859).

iiSur la distribution des zeros de la fonction ζ(s) et ses consequences arithmetiques. Bull. Soc. Math.France 24 (1896), 199–220.

iiiRecherches analytiques sur la theorie des nombers premiers. Ann. Soc. Sci. Bruxelles 20 (1896), 183–256.

DYNAMICAL ZETA FUNCTIONS VIA MICROLOCAL ANALYSIS 3

Conjecture 1.4 (Riemann Hypothesis, Riemann 1859). The non-trivial zeros lie on the linez ∈ C : <(z) = 1/2.

A corollary of the Riemann Hypothesis is

Corollary 1.5.

π(x) = Li(x) +O(x

12 log x

)as x→∞,

in which Li is the offset logarithmic integral or Eulerian logarithmic integral

Li(x) =

∫ x

2

1

log udu.

Remark (Hilbert-Polya conjecture). Hilbert and Polya proposed the idea of understanding thelocation of the zeros of the Riemann zeta function in terms of eigenvalues of some undiscovered(as of yet) self-adjoint operator whose real eigenvalues are related to the zeros.

This idea has yet to reach fruition for the Riemann zeta function, (despite interesting workof [Berry-Keating 1999]i, [Connes 1999]ii, etc) but the approach works particularly well for theSelberg zeta function.

1.2. Selberg zeta function. Let M be a compact boundaryless Riemannian surface withconstant curvature −1.

Definition (Selberg zeta function, Selberg 1956iii). The Selberg zeta function is defined as

Z(z) =∏

γ prime

∞∏n=0

(1− e−(z+n)Tγ

).

We also write

ζ(z) =Z(z + 1)

Z(z)=

∏γ prime

(1− e−zTγ

)−1.

Here, the product runs over all the prime closed geodesics.

Remark (Closed geodesics). A closed geodesic is a closed curve that is geodesic at all of itspoints. (This is should be distinguished with a geodesic lasso, which is a closed curved that isgeodesic at all except one of its points, where it fails to be regular.) A geodesic γ : [0, T ]→Mis periodic with as a period T if γ(0) = γ(T ) and γ′(0) = γ′(T ), that is, (γ(t), γ′(t)), t ∈ [0, T ],is a closed orbit in the tangent bundle TM = (x, ξ) : x ∈ M, ξ ∈ TxM. A prime closedgeodesic of length T is a periodic geodesic whose least period is T .

The Selberg zeta function Z(z) is convergent if <(z) > 1. But it can be extended to theentire complex plane.

Theorem 1.6 (Selberg 1956).

(i). Z(z) has a simple zero at z = 1.(ii). Z(z) has an analytic extension to the entire complex plane C.

iThe Riemann zeros and eigenvalue asymptotics. SIAM Rev. 41 (1999), no. 2, 236–266.iiTrace formula in noncommutative geometry and the zeros of the Riemann zeta function. Selecta Math.

(N.S.) 5 (1999), no. 1, 29–106.iiiHarmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to

Dirichlet series. J. Indian Math. Soc. (N.S.) 20 (1956), 47–87.

4 XIAOLONG HAN

(iii). Z(z) satisfies the functional equation

Z(1− z) = Z(z)× exp

(−χ(M)

∫ z−1/2

0

πx tan(πx) dx

),

in which χ(M) is the Euler characteristic of M.(iv). Z(z) no further zero on the line z ∈ C : <(z) = 1.

As a corollary,

Corollary 1.7 (Prime Geodesic Theorem).

#γ : Tγ ≤ T ∼ eT

Tas T →∞.

Remark. The length T in the above theorem appear in the exponent since Tγ is in the exponentin the Selberg zeta function.

Now the Hilbert-Polya approach to the Riemann Hypothesis in the case of the Selberg zetafunction is successful. In particular, let

0 = λ20 < λ2

1 ≤ λ22 ≤ · · ·

be the eigenvalues of the (positive) Laplace-Beltrami operator on M. Then

Theorem 1.8 (Selberg 1956). The zeros of the Selberg zeta function Z(z) can be described by

(i). 1,(ii). trivial zerosi: −k, k ∈ N,

(iii). special zeros: 12±√

14− λ2

j , j = 1, 2....

This shows that the analogue of the Riemann Hypothesis holds in the case of the Selbergzeta function. It is proved by Selberg using his trace formula. One can then improve the PrimeGeodesic Theorem to

Corollary 1.9 (Improved Prime Geodesic Theorem).

#γ : Tγ ≤ T = Li(eT ) +O(e(1−ε)T ) as T →∞.

The value ε > 0 is related to the least distance of the zeros of Z(z) from the line <(z) = 1.This is determined by the smallest non-zero eigenvalue of the Laplace-Beltrami operator. BySchoen-Wolpert-Yau [SWY], it is comparable to the length of the shortest geodesic dividingthe surface into two pieces.

1.3. Ruelle zeta function. Let X be a compact Riemannian manifold and ϕt : X → X be asmooth flow, ϕt = etV , V ∈ C∞(X;TX). The flow is Anosov if the tangent space of X has acontinuous decomposition

TxX = Ec(x)⊕ Es(x)⊕ Eu(x),

in which Ec(x) = RV (x), dϕt(x)Es(x) = Es(ϕt(x)), and dϕt(x)Eu(x) = Eu(ϕt(x)). Here, Es(x)and Eu(x) are the stable and unstable subspaces in TxX, i.e. there exist C, θ > 0 such that

‖dϕt(x)v‖ϕ(x) ≤ Ce−θt‖v‖x, if v ∈ Es(x) and t > 0;

‖dϕt(x)v‖ϕ(x) ≤ Ce−θt‖v‖x, if v ∈ Eu(x) and t < 0.

iThey are also called the topological zeros, since the topological information of the Euler characteristic ofM is encoded in these zeros according to the functional equation of the Selberg zeta function.


Definition (Ruelle zeta function, Ruelle 1976i). The Ruelle zeta function is defined as

ζ(z) =∏

γ prime

(1− e−zTγ

)−1.

Here, the product runs over all the prime closed orbits of the flow ϕt.

Example (Anosov 1967ii). The most important example of Anosov flows is the geodesic flow onnegatively curved manifolds. That is, let M be a compact Riemannian manifold with negativesectional curvature. Then the geodesic flow on X = S∗M, the cosphere bundle of M, is Anosov.

In the case when M has constant curvature, the Ruelle zeta function is just ζ(z) appearedin the previous section. In particular, it readily reproduces the Selberg zeta function by theformula

Z(z) =∞∏n=1

ζ(z + n).

Theorem 1.10 (Topological entropy, Sinai 1966iii). The topological entropy h(ϕ) defined by

h(ϕ) = limT→∞

1

Tlog #γ : Tγ ≤ T

is finite.

Then one can argue as follows that ζ(z) converges when <(z) > h(ϕ): Firstly, notice that ifTγ 1, then ∣∣1− e−zTγ ∣∣ ∼ 1− e−<(z)Tγ < 1.

Hence,

|ζ(z)| =∏

γ prime

∣∣1− e−zTγ ∣∣−1

=∞∏n=1

∏Tγ∼n

∣∣1− e−zTγ ∣∣−1

.∞∏n=1

(1− e−<(z)n

)−ehn.

So

log |ζ(z)| . −∞∑n=1

ehn log(1− e−<(z)n

).

∞∑n=1

e(h−<(z))n <∞.

1.3.1. Meromorphic extension of Ruelle zeta functions. The (incomplete) list of results con-cerning meromorphic extension of the Ruelle zeta function ζ(z) is

iZeta-functions for expanding maps and Anosov flows. Invent. Math. 34 (1976), no. 3, 231–242.iiGeodesic flows on closed Riemannian manifolds of negative curvature. Trudy Mat. Inst. Steklov. 90

(1967) 209 pp.iiiAsymptotic behavior of closed geodesics on compact manifolds with negative curvature. Izv. Akad. Nauk

SSSR Ser. Mat. 30 (1966), 1275–1296.

6 XIAOLONG HAN

• [Ruelle 1976]: If Ec and Eu are analytici, then ζ(z) has a meromorphic extension to C. Inparticular, on Riemannian surfaces with constant curvature, it recovers the meromorphicextension of the Selberg zeta function (without using Selberg trace formula).• [Gallavotti 1976]ii: There is Cr Anosov flow ϕt for some r <∞ such that ζ(z) does not

have a meromorphic extension to C.• [Rugh 1992iii and 1996iv, Fried 1995v]: If the flow ϕt is analytic, then ζ(z) has a

meromorphic extension to C. In particular, this applies to the geodesic flow on analyticmanifolds with variable negative curvature.• [Policott 1985]vi: ζ(z) has a meromorphic extension to <(z) > h− ε for some ε > 0.• [Giulietti-Liverani-Pollicott [GLP] 2013 and Dyatlov-Zworski [DZ] 2016]:

ζ(z) has a meromorphic extension to C.

1.3.2. Location of zeros and poles of Ruelle zeta functions. The (incomplete) list of resultsconcerning the location of zeros and poles of the Ruelle zeta function is

• [Ruelle 1978vii, Parry-Pollicott 1983viii]: ζ(z) is zero-free and analytic on <(z) > h(ϕ),and has a simple pole at s = h(ϕ).• Let ϕt be the geodesic flow on the cosphere bundle X = S∗M of a Riemannian manifoldM with negative curvature. [Dolgopyat 1998]ix: If dimM = 2, then ζ(z) has an analytic zero-free extension to<(z) > h(ϕ)− ε, except for a simple pole at z = h(ϕ), for some ε > 0. [Giulietti-Liverani-Policott [GLP] 2013]: Let the sectional curvature K of M be

1/9-pinched, that is,

0 <1

9|Kmax| ≤ |K| ≤ |Kmin|.

Then ζ(z) has an analytic zero-free extension to <(z) > h(ϕ) − ε, except for asimple pole at z = h(ϕ), for some ε > 0.

In the last two cases, one has that

Theorem 1.11 (Improved Prime Geodesic Theorem).

#γ : Tγ ≤ T = Li(ehT)

+O(e(h(ϕ)−ε)T ) as T →∞.

The difference with the case of constant negative curvature is that there is no useful estimateon ε > 0.

iIt should be noted that the analyticity of the stable and unstable foliations holds in rather few examples.For example, if ϕt is the geodesic flow on X = S∗M, then the stable and unstable foliations are analytic whenM is locally symmetric (e.g. M has constant curvature). However, the foliations in general are only Holdercontinuous if M has variable negative curvature.

iiFunzioni zeta ed insiemi basilari. Atti. Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 61 (1976),no. 5, 309–317.

iiiThe correlation spectrum for hyperbolic analytic maps. Nonlinearity 5 (1992), no. 6, 1237–1263.ivGeneralized Fredholm determinants and Selberg zeta functions for Axiom A dynamical systems. Ergodic

Theory Dynam. Systems 16 (1996), no. 4, 805–819.vMeromorphic zeta functions for analytic flows. Comm. Math. Phys. 174 (1995), no. 1, 161–190.

viOn the rate of mixing of Axiom A flows. Invent. Math. 81 (1985), no. 3, 413–426.viiThermodynamic formalism. Addison-Wesley Publishing Co., Reading, MA, 1978.

viiiAn analogue of the prime number theorem for closed orbits of Axiom A flows. Ann. of Math. (2) 118(1983), no. 3, 573–591.

ixOn decay of correlations in Anosov flows. Ann. of Math. (2) 147 (1998), no. 2, 357–390.


1.3.3. Methods involved. The methods used to prove the above results on the Ruelle zeta func-tion can be roughly divided into

(1). Selberg trace formula on geodesic flows for which the manifold has constant negativecurvature, more generally, (generalized) Selberg trace formula for locally symmetric spaces,

(2). symbolic dynamics and Ruelle transfer operators for analytic Anosov flows,(3). anisotropic Sobolev spaces and Ruelle transfer operators for smooth Anosov flows.

See Giulietti-Liverani-Pollicott [GLP] for a complete review on the development on the studyof the Ruelle zeta function. The purpose of this note is to present Method (3) developed inDyatlov-Zworski [DZ].

2. Preliminaries

We fix the Anosov flow ϕt = etV on a compact Riemannian manifold X. Write P = −iV .Note that we do not require that dimEs = dimEu; we however require that the stable andunstable bundles are orientablei.

For a closed orbit of the flow ϕt, denote γ# its prime orbit. Furthermore, we use a sightlydifferent definition of Ruelle zeta function as

ζ(λ) =∏γ#

(1− eiλT

#γ

).

Notice the two differences with the previous Ruelle zeta function: λ→ iλ which accounts for arotation, and taking the reciprocal which accounts for changing poles to zeros and vice versa.Recall that

Theorem 2.1 (c.f. Theorem 3.4 in Stein-Shakarchi [SS]). Every meromorphic function in anopen set Ω is a quotient of two functions which are holomorphic in Ω.

So this new definition is harmless in proving its meromorphic extension; it will be convenientfor computation:

ζ(λ) =∏γ#

(1− eiλT

#γ

)

= exp

∑γ#

log(

1− eiλT#γ

)= exp

−∑γ#

∞∑m=1

eimλT#γ

m

= exp

(−∑γ

T#γ e

iλTγ

Tγ

)The first step is to reduce the above product to one involving the trace(s) of the linearisedPoincare map Pγ. To demonstrate the idea, let us look at a model first.

iThis assumption only comes in to play in the trace identity:

|det(Id− Pγ)| = (−1)q det(Id− Pγ), in which q is independent of γ.

This condition of orientablity can be removed; see Giulietti-Liverani-Policott [GLP, Appendix B].

8 XIAOLONG HAN

Example. Define

ζ0(λ) := exp

(−∑γ

T#γ e

iλTγ

Tγ| det(Id− Pγ)|

),

which differs with ζ(λ) only by a factor | det(Id−Pγ)| in the denominator of the exponent. (It,however, is an important factor!) The reason of adding this factor will become clear. Computethat

ζ ′0(λ)

ζ0(λ)=

d

dλlog ζ0(λ) =

1

i

∑γ

T#γ e

iλTγ

| det(Id− Pγ)|.

By Atiyah-Bott-Guillemin trace formula (see Theorem 2.11)

Trb(e−itP ) =∑γ

T#γ δ(t− Tγ)

| det(Id− Pγ)|for t > 0,

we deduce that

d

dλlog ξ0(λ) =

ζ ′0(λ)

ζ0(λ)=

1

i

∑γ

T#γ e

iλTγ

| det(Id− Pγ)|=

1

i

∫ ∞0

eitλTrb(e−itP ) dt.

To prove the meromorphic extension of ζ0, it now reduces to proving that the right-hand sideof the above equation has a meromorphic extension to C with simple poles and residues whichare integers. For example, if

1

i

∫ ∞0

eitλTrb(e−itP ) dt =∑j

kjλ− λj

for λj ∈ C and kj ∈ Z,

thenξ0(λ) =

∏j

(λ− λj)kj .

The distributionTrb(e−itP )

has a big singularity at t = 0 and also at t = Tγ > 0 for all closed orbit lengths Tγ. Seee.g. Duistermaat-Guillemin [DG], in which the (local) Weyl law is studied.i In particular, thesingularity at t = 0 corresponds to the leading term in the Weyl asymptotics while the othersingularities contribute to the reminder term.

However, in our case here, one has to deal the singularities corresponding to large times.Atiyah-Bott-Guillemin trace formula gives these singularities at t = Tγ. We further simplifythe argument as follows: Choose 0 < t0 < Tγ for all γ. Then noticing that Trb(e−itP ) = 0 when0 < t < t0, we have

1

i

∫ ∞t0

eitλTrb(e−itP ) dt

=1

ieit0λ

∫ ∞0

eitλTrb(ϕ∗−t0e−itP ) dt

=1

ieit0λTrb

(∫ ∞0

ϕ∗−t0eit(λ−P ) dt

)iIt should be noted that Duistermaat-Guillemin [DG] studied the half-wave propagator e−itP , in which

P =√

∆ is only a pseudodifferential operator, rather than a differential operator P = −iV treated in this note.


=1

ieit0λTrb

[ϕ∗−t0

(eit(λ−P )

i(λ− P )

∣∣∣∞0

)]when =(λ) 1

= −eit0λTrb(ϕ∗−t0(P − λ)−1

).

It now reduces to proving the meromorphic continuition of

Trb(ϕ∗−t0(P − λ)−1

).

In summary, one has to

• show that the flat trace Trb of e−itP (t > 0) and ϕ∗−t0(P −λ)−1 can be well-defined. Thisamounts to checking their wavefront sets do not intersect with N∗(∆(X)), the conormalbundle of the diagonal;• remove the factor | det(Id−Pγ)| in ζ0 to get the result for ζ. This amounts to using the

fact that

det(Id− Pγ) =n−1∑k=0

(−1)kTr(Pkγ ),

where by a modified Atiyah-Bott-Guillemin trace formula Tr(Pkγ ) is connected to

Trb[exp(−itP|C∞(X;Ek0 ))

],

in which P acts on vector bundles of differential forms in Ek0 .

2.1. Anosov systems. Recall that the Anosov flow ϕt = etV : X → X on a compact Rie-mannian manifold X has a continuous decomposition

TxX = Ec(x)⊕ Es(x)⊕ Eu(x),

where ϕt for t > 0 is length-preserving on Ec(x) → Ec(ϕ(x)), is contracting on Es(x) →Es(ϕ(x)), and is expanding on Eu(x)→ Eu(ϕ(x)).

2.2. Poincare maps. Let γ be a closed orbit with period t0 6= 0 (t0 needs not be the primitiveperiod) of ϕt. Let x0 ∈ γ. Then we define the linearised Poincare map Pγ as

Pγ = dϕ−t0|Es(x0)⊕Eu(x0).

Lemma 2.2. det(Id− Pγ) 6= 0 is independent of the base point on γ.

Proof. Let v = vs + vu ∈ Es(x0) ⊕ Eu(x0), where vs ∈ Es(x0) and vu ∈ Eu(x0). Assume thatPγv = v. Then PNγ v = v for all N ∈ Z. Hence,

‖v‖x0 = ‖PNγ v‖x0 = ‖PNγ vs + PNγ vu‖x0 ≥ ‖PNγ vs‖x0 − ‖PNγ vu‖x0≥ CeθNt0‖vs‖x0 − Ce−θNt0‖vu‖x0 ≥ C ′eθNt0‖vs‖x0

as N > 0 is large enough. So ‖vs‖x0 = 0. Letting N → −∞ similarly implies that ‖vu‖x0 = 0.Therefore, v = 0 and Id− Pγ is invertible.

To show that det(Id − Pγ) is independent of x0, we just need to notice that for each x =ϕs(x0) ∈ γ,

dϕ−t0(ϕs(x0)) = dϕs(x0) dϕ−t0(x0) dϕ−s(ϕs(x0)),

i.e. dϕ−t0(ϕs(x0)) is conjugate with dϕ−t0(x0) for all s ∈ R.

10 XIAOLONG HAN

Notice that we use negative times in Pγ and by Anosov properties the eigenvalues of Pγ|Eusatisfy |µ| < 1, therefore

det(Id− Pγ|Eu) > 0.

Similarly,det(Id− P−1

γ |Es) > 0.

From the assumption that Es is orientable, we have

det(Pγ|Es) = det(dϕ−Tγ |Es) > 0.

One also sees thatdet(Id− Pγ|Es) = det(−Pγ|Es) det(Id− P−1

γ |Es),so

| det(Id− Pγ|Es)| = (−1)dimEs det(Id− Pγ|Es),and we have proved that

Lemma 2.3. Let q = dimEs. Then

| det(Id− Pγ)| = (−1)q det(Id− Pγ).

We also need the elementary

Lemma 2.4. Let A be an n× n matrix. Then

det(Id− A) =n−1∑k=0

(−1)kTr(∧kA).

Now we can use the above two lemmas to remove the factor | det(Id − Pγ)| in ζ0(λ) to getζ(λ):

ζ(λ) = exp

(−∑γ

T#γ e

iλTγ

Tγ

)

= exp

(−∑γ

T#γ e

iλTγ (−1)q det(Id− Pγ)Tγ| det(Id− Pγ)|

)

= exp

(−

n−1∑k=0

∑γ

T#γ e

iλTγ (−1)q(−1)kTr(∧kPγ)Tγ| det(Id− Pγ)|

)

=n−1∏k=0

exp

((−1)k+q+1

∑γ

T#γ e

iλTγTr(∧kPγ)Tγ| det(Id− Pγ)|

)

:=n−1∏k=0

ζk(λ)(−1)k+q .

Here,

ζk(λ) = exp

(−∑γ

T#γ e


).

Write

fk(λ) :=1

i

∑γ

T#γ e

iλTγTr(∧kPγ)| det(Id− Pγ)|

=∂

∂λ

(−∑γ

T#γ e


). (2.1)


Then∂

∂λlog ζ(λ) =

n−1∑k=0

(−1)k+qfk(λ).

Thus to prove the meromorphic extension of ζ, it suffices to prove fk, k = 0, ..., n − 1, hasa meromorphic extension with simple poles. This will be subsequently treated using Atiyah-Bott-Guillemin trace formulae on differential forms.

2.3. Microlocal analysis. Let X be a manifold with a fixed volume form. We use the alge-bra of pseudodifferential operators Ψk(X), k ∈ R, with symbols lying in the class Sk(X) ⊂C∞(T ∗X):

a ∈ Sk(X)⇔ supx∈K|∂αx∂

βξ a(x, ξ)| ≤ Cα,β,K〈ξ〉k−|β| for each K b X.

Each properly supportedA ∈ Ψk(X) is bounded between Sobolev spacesHmcomp(X)→ Hm−k

loc (X),

or simply Hm(X) → Hm−k(X) if X is compact. The wavefront set WF(A) of A ∈ Ψk(X) isa closed conic subset of T ∗X \ 0, with 0 denoting the zero section; the complement of WF(A)consists of points in whose conic neighbourhoods the full symbol of A is O(〈ξ〉−∞).

The wavefront set WF(u) ⊂ T ∗X \ 0 of a distribution u ∈ D′(X) is defined as follows: Apoint (x, ξ) ∈ T ∗X \ 0 does not lie in WF(u) if there exists a conic neighbourhood U 3 (x, ξ)such that Au ∈ C∞(X) for each A ∈ Ψ0(X) with WF(A) ⊂ U . An equivalent definition isgiven in terms of the Fourier transform: (x, ξ) /∈WF(u) iff there exists χ ∈ C∞(X) with suppχcontained in some coordinate neighbourhood and χ(x) 6= 0 such that χu(η) = O(〈η〉−∞) for ηin a conic neighbourhood of ξ; here χu is considered a function on Rn using some coordinatesystem and ξ is accordingly considered as a vector in Rn.

The wavefront set WF(B) ⊂ T ∗(Y ×X) of an operator B : C∞0 (X)→ D′(Y ) is defined usingits Schwartz kernel KB(y, x) ∈ D′(Y ×X):

WF(B) := WF(KB),

we also writeWF′(B) = (y, η, x,−ξ) : (y, η, x, ξ) ∈WF(B).

If E is a smooth r-dim vector bundle over X, then we can consider distributions u ∈ D′(X; E)with values in E . The wavefront set WF(u), a closed conic subset of T ∗X \ 0, is defined asfollows: (x, ξ) /∈WF(u) iff for each local basis e1, ..., er ∈ C∞(U ; E) defined in a neighbourhoodU 3 x, and for

u|U =r∑j=1

ujej, where uj ∈ D′(U),

we have (x, ξ) /∈ WF(uj) for all j. Similarly, one can define WF(B) for an operator B withvalues in some smooth vector bundle over Y ×X.

An operator A : D′(X; E)→ D′(X; E) is said to be pseudodifferential in the class Ψk(X; Hom(E)),if WF(Au) ⊂ WF(u) for all u ∈ D′(X; E) and, for each local basis e1, ..., er ∈ C∞(U ; E) oversome open U ⊂ X, we have that

A(fel) =r∑j=1

(Ajlf)ej, for each f ∈ D′(X; E), supp f b U,

12 XIAOLONG HAN

where Ajl ∈ Ψk(U). As before, the wavefront set WF(A) on U is defined as the union ofWF(Ajl) over all j, l. The principal symbol

σ(A) ∈ Sk(X; Hom(E))/Sk−1(X; Hom(E))

is defined using the standard notion of the principal symbol σ(Ajl) ∈ Sk(X)/Sk−1(X) as follows:

σ(A)el =r∑j=1

σ(Ajl)ej, on U.

The operator A is called elliptic in the class Ψk(X; Hom(E)) at some point (x, ξ) ∈ T ∗X \ 0, if

〈η〉−kσ(A)(y, η)

is invertible as a homomorphism E → E uniformly as η →∞ for (y, η) in a conic neighbourhoodof (x, ξ); equivalently,

| det(〈η〉−kσ(A))| ≥ c > 0

in a conic neighbourhood of (x, ξ). The (open conic) set of all elliptic points of A is denotedEll(A).

2.4. Flat trace. Let A be a compact operator on a complex separable Hilbert space H. ThenA?A : H → H is a self-adjoint semidefinite compact operator, and hence it has discrete spectrum

‖A‖2 = s0(A)2 ≥ s1(A)2 ≥ · · · sj(A)2 → 0.

The singular values of A are defined as the nonnegative square roots of these eigenvalues sj(A),j = 0, 1, ...

Definition (Trace class operators). Let A be a compact operator on a complex separableHilbert space H. Then A is said to be of trace class, denoted as A ∈ L1(H), if

∞∑j=0

sj(A) <∞.

The trace class norm is defined as

‖A‖L1 =∞∑j=0

sj(A).

Theorem 2.5 (c.f. Theorem C.17 in Zworski [Z]). Let ej∞j=0 be an orthonormal basis of Hand A ∈ L1(H). Then

Tr(A) :=∞∑j=0

〈Aej, ej〉 <∞,

and is independent of the choice of ej∞j=0.

Theorem 2.6 (c.f. Theorem C.18 in Zworski [Z]). Suppose that B is an operator of trace class

on L2(X; Ω12 (X)), given by the integral kernel

KB ∈ C∞(X ×X,Ω12 (X ×X)).

Then KB|∆(X), the restriction to the diagonal

∆(X) = (x, x) : x ∈ X,


has a well-defined density; and

Tr(B) =

∫∆(X)

KB|∆(X).

Remark (c.f. Theorem C.16 in Zworski [Z]).

Ψ−n−(Rn) ⊂ L1(L2(Rn)).

However, if

Bf(x) =

∫X

KB(x, y)f(y) dy

in the distribution sense, then generally KB|∆(X) is not well-defined density. One can in factdefine KB|∆(X) in the distributional sense, if WF(B) has some additional property.

Theorem 2.7 (c.f. Theorem 2.5.11 in Hormander [H]). Suppose that WF(KB)∩N∗∆(X) = ∅,where N∗∆(X) is the conormal bundle of ∆(X) . Then KB|∆(X) can be defined in a uniqueway as a distribution on ∆(X). More precisely, let ι : x→ (x, x). Then

KB|∆(X) = ι∗KB,

and we define the flat trace of B as

Trb(B) :=

∫X

ι∗KB(x) dx,

which is independent of the density dx.

One can also approximate Trb by the regular trace as follows. Let d(x, y) be the geodesicdistance of x and y in a neighbourhood of ∆(X) with respect to some fixed Riemannian metric.Let ψ ∈ C∞0 (R), 0 ≤ ψ ≤ 1, and is equal to 1 near 0. Write

Eε(x, y) =1

Fε(x)ψ

(d(x, y)

ε

),

where Fε(x) is chosen so that∫XEε = 1 and satisfies

εn

C≤ Fε(x) ≤ Cεn.

Define Eε : D′(X)→ C∞(X) as

Eεu(x) =

∫X

Eε(x, y)u(y) dy for u ∈ C∞(X).

Then

Lemma 2.8 (Approximation of identity).

Eε → Id in Ψ0+(X).

Proof. In a local coordinate patch where the density coincides with the Riemannian volume,we can write

Eεu(x) =

∫Rnφε(x− y)u(y) dy,

where φ ∈ C∞0 (Rn) and φε(x) = ε−nφ(x/ε). So

Eεu(x) =1

(2π)n

∫∫R2n

ei(x−y)·ξφε(ξ)u(y) dy =1

(2π)n

∫∫R2n

ei(x−y)·ξφ(εξ)u(y) dy,

14 XIAOLONG HAN

in whichφ(εξ) ∈ S−∞(Rn) uniformly for ε→ 0

since φ ∈ S(Rn). To see Eε → Id in Ψ0+(X), notice that Eε1 = 1 implies that∫Rnφε(x) dx = φ(0) = 1.

So since φ ∈ S(Rn) again

φ(εξ)− 1→ 0 in S0+(Rn) uniformly for ε→ 0.

The next lemma shows that the flat trace is well-approximated by regular traces.

Lemma 2.9. Suppose that WF(KB) ∩N∗∆(X) = ∅. Then

Trb(B) = limε→0

Tr(EεBEε),

where the trace on the right-hand side is well-defined since EεBEε is smoothing and in L1(L2(X)).

Proof. Since WF(KB) ∩N∗∆(X) = ∅, we can find a closed conic set Γ ⊂ T ∗(X ×X) \ 0 suchthat WF(KB) ⊂ Γ and Γ does not intersect a conic neighbourhood of N∗∆(X).

Notice that

Trb(B) :=

∫X

ι∗KB(x) dx = ι∗KB1.

We only need to show that

ι∗KB → ι∗KEεBEε as ε→ 0 in D′(X),

(in fact, D′0(X) is sufficient.) Now we use Hormander [H, Theorem 2.5.11’],

ι∗ : D′Γ(X ×X)→ D′(X)

is continuous, where u ∈ D′Γ(X × X) means that u ∈ D′(X × X) and WF(u) ⊂ Γ. Thus itsuffices to show that

KB → KEεBEε as ε→ 0 in D′Γ(X ×X).

It further reduces to proving

(1).KB → KEεBEε as ε→ 0 in D′(X ×X).

(2).AKB → AKEεBEε as ε→ 0 in C∞(X ×X)

if A ∈ Ψ0(X ×X) and WF(A) ∩ Γ = ∅.(1). According to Schwartz kernel theorem, we only need to show that if f, g ∈ C∞(X), then

KB(f ⊗ g)→ KEεBEε(f ⊗ g) as ε→ 0.

But this is obvious by the previous lemma: Eε → Id as ε → 0; in particular, Eεf → f andEεg → g in C∞(X) as ε→ 0.

(2). We use a local version and prove that for each ψ ∈ C∞0 (R2n) and each closed coneV ⊂ R2n such that

Γ ∩ (suppψ × V ) = ∅,we have that for N = 1, 2,...

supV〈ξ〉N

∣∣∣ψKB(ξ)− ψKEεBEε(ξ)∣∣∣→ 0 as ε→ 0.


Note that it follows from (1) that

supV bV〈ξ〉N

∣∣∣ψKB(ξ)− ψKEεBEε(ξ)∣∣∣→ 0 as ε→ 0.

So we only need to show that

supV

∣∣∣ ψKEεBEε(ξ)∣∣∣ <∞ uniformly for all ε→ 0.

Recall that

Eεu(x) =

∫Rnφε(x− y)u(y) dy.

Compute that

KEεBEε(x, y) =

∫∫R2n

φε(x− z)KB(z, w)φε(w − y) dzdw.

So

ψKEεBEε(ξ) =

∫∫R2n

ei(x,y)·ξψ(x, y)φε(x− z)KB(z, w)φε(w − y) dzdwdxdy

→∫∫

R2n

ei(z,w)·ξψ(z, w)KB(z, w) dzdw

is uniformly bounded C∞(V ) as ε→ 0. This is because WF(KB) ⊂ Γ and Γ∩ (suppψ× V ) =∅.

We now generalise the flat trace to operators acting on vector bundles. Let

B : C∞(X; E)→ D′(X; E),

where E is the vector bundle of differential forms of all orders onX. Let e1, ..., er be a local frameof E . Suppose that B is supported in the domain of this local frame and WF(KB)∩N∗∆(X) = ∅.Then we define

Trb(B) = Trb

(r∑j=1

Bjj

),

where

B(fel) =r∑j=1

(Bjlf)ej for f ∈ C∞(X).

The general case can be handled by a partition of unity and the independent of the choice ofthe frame can be verified.

2.5. Atiyah-Bott-Guillemin trace formulae. Now we want define the flat trace of

ϕ∗−t = e−tV = e−itP

as a distribution of t ∈ R \ 0, where ϕ∗−t is the pullback of ϕ−t. To do this, we chooseχ ∈ C∞0 (R \ 0) and write

Tχ :=

∫Rχ(t)ϕ∗−t dt.

By Theorem 2.7, we need to check WF(Tχ) does not intersect with N∗∆(X) so we can takethe flat trace. Firstly,

ϕ∗−tf(y) = f(ϕ−t(y)) =

∫X

K(t, y, x)f(x) dx,

16 XIAOLONG HAN

where K(t, y, x) = δ(x − ϕ−t(y)) formally. The singular support of K is y = ϕt(x) ⊂R×X ×X. The wavefront set is contained in the conormal bundle of this surface. Since thetangent vectors at (t, y = ϕt(x), x) are of the form

(a, aV (ϕt(x)) + dϕt(x)b, b), where a ∈ R, b ∈ Rn,

the conormal vectors are of the form

(−V (ϕ(x)) · η, η,−Tdϕt(x)η), where η ∈ Rn.

Therefore,

WF(K) ⊂ (t,−V (x) · η, ϕ(x), η, x,−Tdϕt(x) · η) : t ∈ R, x ∈ X, η ∈ T ∗ϕt(x)X \ 0.Next by Hormander [H, Theorem 2.5.13],

WF(Tχ) ⊂ (y, η, x, ξ) : ∃t ∈ suppχ, (t, 0, y, η, x, ξ) ∈WF(K).So

(t, τ, y, η, x, ξ) ∈WF(Tχ) ∩N∗∆(X)

ifft ∈ suppχ, τ = −V (x) · η = 0, y = ϕ(x) = x.

This implies that η ∈ Es(x)⊕ Eu(x) and

η =T dϕt(x) · η.Hence, η = 0 since Id− Pγ is nonsingular.

Lemma 2.10. Let x0 ∈ γ and t0 6= 0 such that φt0(x0) = x0. Then there exists ε > 0and a neighbourhood U ⊂ X of x0 such that ϕ(x0) ∈ U for |s| < ε and for each χ(t, x) ∈C∞0 ((t0 − ε, t0 + ε)× U), we have∫

R×Xχ(t, x)K(t, x, x) dtdx =

1

| det(Id− Pγ)|

∫ ε

−εχ(t0, ϕs(x0)) ds.

Proof. We choose a local coordinate system such that

x0 = 0, V = ∂w1 , Eu(x0)⊕ Es(x0) = dw1 = 0.Define the maps A : BRn−1(0, ε)→ BRn−1(0, ε) and F : BRn−1(0, ε)→ (−ε1, ε1) such that

ϕ−t0(0, w′) = (F (w′), A(w′)), where w′ ∈ Rn−1 and |w′| < ε.

So we can easily see thatF (0) = 0 and A(0) = 0.

Moreover, for |t− t0| < ε and (w1, w′) ∈ B(0, ε), we have

ϕ−t(w1, w′) = (−t+ t0 + w1 + F (w′), A(w′)).

Hence,K(t, (z1, z

′), (w1, w′)) = δ(w′ − A(w′))δ(w1 + t− t0 − z1 − F (z′)).

Then ∫R×B(0,ε)

χ(t, (w1, w′))δ(w′ − A(w′))δ(t− t0 − F (z′)) dtdw1dw

′

=

∫B(0,ε)

χ(t0 + F (w′), (w1, w′))δ(w′ − A(w′)) dw1dw

′

=1

| det(Id− dA(0))|

∫ ε

−εχ(t0, (w1, 0)) dw1


=1

| det(Id− Pγ)|

∫ ε


Here, dA(0) = Pγ and the equation w′ = A(w′) has exactly one solution w′ = 0 if |w′| < ε.

By a partition of unity, we have that for each χ(t, x) ∈ C∞0 ((R \ 0)×X),∫R×X

χ(t, x)K(t, x, x) dtdx =∑γ

1

| det(Id− Pγ)|

∫γ#χ(Tγ, x) dl(x).

Here, the sum is over all closed orbits γ with period Tγ and dl refers to the measure dt onγ(t) = ϕt(x0). Furthermore, taking χ(t, x) = χ(t) reduces to∫

R×Xχ(t)K(t, x, x) dtdx =

∑γ

T#γ χ(Tγ)

| det(Id− Pγ)|.

This is indeed

Theorem 2.11 (Atiyah-Bott-Guillemin trace formula). Let t > 0.

Trb(e−itP ) =∑γ

T#γ δ(t− Tγ)

| det(Id− Pγ)|.

In fact, the above trace formula can be generalised to

e−itP, where P : C∞(X; E)→ C∞(X; E).

Here, P acting on vector bundle of differential forms is defined by P = LV /i. Write Ek0 as thesmooth invariant subbundle of E of all differential k-forms u satisfying V yu = 0, where y is thecontraction operator by a vector field. Then

Theorem 2.12 (Atiyah-Bott-Guillemin trace formula on differential forms). Let t > 0 andk = 0, ..., n− 1.

Trb(e−itP|C∞(X;Ek0 )

)=∑γ

T#γ Tr(∧kPγ)δ(t− Tγ)| det(Id− Pγ)|

.

Proof. It suffices to prove a local version∫R×X

χ(t, x)Kk(t, x, x) dtdx =Tr(∧kPγ)

| det(Id− Pγ)|

∫ ε


Here, e1, ..., er form a local frame of Ek0 on a small neighbourhood U near x0, where r = dim Ek0 ,

ϕ∗−t(fel) =r∑j=1

(Bjlf)ej for f ∈ C∞(X),

and Kk as the kernel of the operatorr∑j=1

Bjj.

Define the functions bjl on (t0 − ε, t0 + ε)× U by

ϕ∗−tel =r∑j=1

bjl(t)ej.

18 XIAOLONG HAN

ThenBjl(t)f = bjl(t)(ϕ

∗−tf),

which means that

Kk(t, x, y) =r∑j=1

bjj(t, y)K(t, x, y).

Then by Lemma 2.10,∫R×X

χ(t, x)Kk(t, x, x) dtdx =1

| det(Id− Pγ)|

∫ ε

−εχ(t0, ϕs(x0))

r∑j=1

bjj(t0, ϕs(x0)) ds.

It remains to observe thatr∑j=1

bjj(t0, ϕs(x0)) = Tr(∧k Tdϕ−t0(x0)|E∗s (x0)⊕E∗u(x0)) = Tr(∧kPγ).

Let Pk = P|C∞(X;Ek0 ). From the above theorem, we immediately derive that fk defined in

(2.1) can be written as

fk(λ) =1

i

∑γ

T#γ e

iλTγTr(∧kPγ)| det(Id− Pγ)|

=1

iTrb∫ ∞

0

eit(λ−Pk) dt

=1

iTrb(ϕ∗−t0

∫ ∞0

eit(λ−Pk) dt

)= −Trb

(ϕ∗−t0(Pk − λ)−1

)= −eiλt0Trb

(e−it0Pk(Pk − λ)−1

)(2.2)

for =(λ) 1 and 0 < t0 < Tγ for all γ.

2.6. Fredholm theory. This section follows Zworski [Z, Appendix D]. Let T : B1 → B2 be alinear transformation between two finite dimensional vector spaces B1 and B2. Then

rankT := dimB1 − dim kerT,

which equals the dimension of the range TB1. So we have that

rankT = dimB2 − dim cokerT,

in which cokerT := B2/TB1. If we define the index of T as

indT := dim kerT − dim cokerT,

then indT = dimB1 − dimB2 is independent of T . This is the basic reason for the stabilityproperties of the left-hand side in the infinite dimensional cases.

If B1 and B2 are two Banach spaces and T ∈ L(B1, B2), then kerT ⊂ B1 is closed, but neednot be finite dimensional; the range TB1 ⊂ B2 need not be closed. However,

Lemma 2.13. If TB1 has finite codimension in B2, then TB1 is closed in B2.


Definition (Fredholm operators). Let B1 and B2 be two Banach spaces and T ∈ L(B1, B2).If dim kerT <∞ and TB1 (is closed and) has finite codimension, then we say T is a Fredholmoperator and define

indT = dim kerT − dim cokerT.

Example. Let K : B → B be compact. Then Id +K is a Fredholm operator with index 0.

In fact, the index does not change under continuous deformations of Fredholm operators(with respect to operator norm topology).

Theorem 2.14 (Invariance of the index under deformations). The set of Fredholm operatorsis open in L(B1, B2), and the index is constant in each component.

The following theorem is used in the main proof.

Theorem 2.15 (Analytic Fredholm theory). Suppose Ω ⊂ C is a connected open set andA(λ)λ∈Ω is a family of Fredholm operators depending holomorphically on λ.

Then if A(λ0)−1 exists at some point λ0 ∈ Ω, then the mapping λ→ A(λ)−1 is a meromorphicfamily of operators on Ω.

2.7. Semiclassical analysis. Let X be a manifold with a fixed volume form. We use thealgebra of semiclassical pseudodifferential operators Ψk

h(X), k ∈ R, with symbols lying in theclass Skh(X):

a(x, ξ;h) ∈ Skh(X)⇔ a ∈ Sk(X) uniformly as h→ 0.

Each properly supported A ∈ Ψkh(X) is bounded between Sobolev spaces Hm

h,comp(X) →Hm−1h,loc (X), or simply Hm

h (X) → Hm−kh (X) if X is compact. The semiclassical wavefront set

WFh(A) of A ∈ Ψkh(X) is a closed (not necessarily conic) subset of the fiber-radially compact-

ified cotangent bundle T∗X; a point (x, ξ) ∈ T ∗X does not lie in WFh(A) iff the full symbol a

of A satisfies a(x′, ξ′) = O(h∞〈ξ′〉−∞) for h small enough and (x′, ξ′) ∈ T ∗X in a neighborhood

of (x, ξ) ∈ T ∗X.

The wavefront set WFh(u) ⊂ T∗X of an h-tempered family of distributions u = u(h) ∈

D′(X) is defined as follows: A point (x, ξ) ∈ T ∗X does not lie in WFh(u) if there exists aneighbourhood U 3 (x, ξ) such that ‖Au‖L2 ∈ O(h∞) for each A ∈ Ψ0(X) with WFh(A) ⊂ U .An equivalent definition is given in terms of the Fourier transform: (x, ξ) /∈ WFh(u) iff thereexists χ ∈ C∞(X) with suppχ contained in some coordinate neighbourhood and χ(x) 6= 0 such

that Fh(χu)(η) = O(h∞〈η〉−∞) for η in a neighbourhood of ξ ∈ T ∗X. Here, χu is considereda function on Rn using some coordinate system and ξ is accordingly considered as a vector inRn, and Fh is the semiclassical Fourier transform:

Fhu(ξ) =

∫Rne−ix·ξ/hu(x) dx = u

(ξ

h

).

This characterisation immediately implies that if u is independent of h, then

WFh(u) = WF(u) ∪ (suppu× 0) and WF(u) = WFh(u) ∩ (T ∗X \ 0). (2.3)

The wavefront set WFh(B) ⊂ T∗(Y ×X) of an operator B : C∞0 (X)→ D′(Y ) is defined using

its Schwartz kernel KB(y, x) ∈ D′(Y ×X):

WFh(B) := WFh(KB),

we also writeWF′h(B) = (y, η, x,−ξ) : (y, η, x, ξ) ∈WFh(B).

It can be characterised as

20 XIAOLONG HAN

Lemma 2.16. Let B : C∞0 (X) → D′(Y ) be an h-tempered family of operators. A point(y, η, x, ξ) ∈ T ∗(Y ×X) does not lie in WF′h(B) iff there exists neighbourhoods U of (x, ξ) andV of (y, η) such that

WFh(f) ⊂ U ⇒WFh(Bf) ∩ V = ∅for each h-tempered family of functions f(h) ∈ C∞0 (X).

Proof. Using local coordinates, we reduce to the case X = Rn and Y = Rm.⇐: Assume that there exists neighbourhoods U of (x, ξ) and V of (y, η) such that

WFh(f) ⊂ U ⇒WFh(Bf) ∩ V = ∅for each h-tempered family of functions f(h) ∈ C∞0 (X). Take χx ∈ C∞0 (Rn) and χy ∈ C∞0 (Rm)such that χx(x) 6= 0 and χy(y) 6= 0; choose Uξ and Vη as two neighbourhoods of ξ and η suchthat

suppχx × Uξ ⊂ U and suppχy × Vη ∈ V.Let

K ′B(y′, x′) = χy(y′)KB(y′, x′)χx(x

′).

Take ξ′ ∈ Uξ and η′ ∈ Vη. Then

Fh(K ′B)(η′,−ξ′) =

∫∫Rm+n

e−i(y′·η′−x′·ξ′)/hK ′B(y′, x′) dy′dx′ = Fh(χyBf)(η′),

wheref(x′) = χx(x

′)eix′·ξ′/h.

We see thatWFh(f) ⊂ suppχx × ξ′ ⊂ U.

So WFh(Bf) ∩ V = ∅ by assumption. Therefore,

WFh(χyBf) ∩ (Rm × Vη) = ∅,which implies that

Fh(K ′B)(η′,−ξ′) = Fh(χyBf) = O(h∞).

By the characterisation of WFh, we have that

(y, η, x, ξ) /∈WF′h(B).

⇒: Assume that (y, η, x, ξ) /∈ WF′h(B). Then take χx ∈ C∞0 (Rn) and χy ∈ C∞0 (Rm) suchthat χx(x) = 1 in a neighbourhood Ux 3 x and χy(y) = 1 in a neighbourhood Vy 3 y; chooseUξ and Vη as two neighbourhoods of ξ and η such that

(suppχx × Uξ × suppχy × Vη) ∩WF′h(B) = ∅.Now write

U = Ux × Uξ and V = Vy × Vη.Let f be an h-tempered family of distribution on X with WFh(f) ⊂ U . Then

f(x′) = χx(x′)

1

(2πh)n

∫Uξ

eix′·ξ′/hFh(f)(ξ′) dξ′ + (1− χx(x′))f(x′)

+χx(x′)

1

(2πh)n

∫Rn\Uξ

eix′·ξ′/hFh(f)(ξ′) dξ′

= χx(x′)

1

(2πh)n

∫Uξ

eix′·ξ′/hFh(f)(ξ′) dξ′ +O(h∞)


by the characterisation of WFh via semiclassical Fourier transform. TakeK ′B(y′, x′) = χy(y′)KB(y′, x′)χx(x

′).Then

Fh(χyBf)(η′) =

∫Rm

e−iy′·η′/hχy(y

′)

∫RnKB(y′, x′)f(x′) dx′dy′

=

∫Rm

e−iy′·η′/hχy(y

′)

∫RnKB(y′, x′)χx(x

′)1

(2πh)n

∫Uξ

eix′·ξ′/hFh(f)(ξ′) dξ′dx′dy′

+O(h∞)

=1

(2πh)n

∫Uξ

∫Rm+n

e−i(y′·η′−x′·ξ′)/hχy(y

′)KB(y′, x′)χx(x′) dx′dy′Fh(f)(ξ′) dξ′

+O(h∞)

=1

(2πh)n

∫Uξ

Fh(K ′B)(η′,−ξ′)Fh(f)(ξ′) dξ′ +O(h∞).

We have thatFh(K ′B)(η′,−ξ′) = O(h∞)

for all (η′, ξ′) ∈ Vη × Uξ. So Fh(χyBf)(η′) = O(h∞) for all η′ ∈ Vη, implying that WFh(Bf) ∩V = ∅.

To be able to work with differential forms, we consider a semiclassical pseudodifferentialoperator P ∈ Ψk

h(X; Hom(E)) acting on h-tempered family of distributions u(h) ∈ D(X; E).

Proposition 2.17 (Elliptic estimate). Let u(h) ∈ D(X; E) be h-tempered. Then

(i). If A ∈ Ψ0h(X) (acting on D′(X, E) diagonally) and P ∈ Ψk

h(X; Hom(E)) is elliptic onWFh(A), then for each m,

‖Au‖Hmh (X;E) ≤ C‖Pu‖Hm−k

h (X;E) +O(h∞).

(ii). If Ellh(P) denotes the elliptic set of P, then

WFh(u) ∩ Ellh(P) ⊂WFh(Pu).

Proof. (i). Reducing to a local frame of E , we see that σ(P)−1 (as a matrix inverse) of σ(P) iswell-defined in C∞(X; Hom(E)); moreover,

σ(P)−1 ⊂ S−kh (X; Hom(E)) near WFh(A).

One can then construct Q0 ∈ Ψ−kh (X; Hom(E)) with principal symbol σ(P)−1 such that

Q0P = Id + hR,

where R ∈ Ψ−1h (X; Hom(E)). Repeating this procedure and using the completeness of symbol

classes, one constructs Q ∈ Ψ−kh (X; Hom(E)) such that

QP = Id +O(h∞)Ψ−∞ .

SoAu = AQPu +O(h∞)C∞ ,

and‖Au‖Hm

h (X;E) ≤ ‖AQPu‖Hmh (X;E) +O(h∞) ≤ C‖Pu‖Hm−k

h (X;E) +O(h∞).

(ii). We only need to show that if (x, ξ) ∈ Ellh(P) \WFh(Pu), then (x, ξ) /∈ WFh(u). (Infact, if there exists (x, ξ) ∈WFh(u)∩Ellh(P) but 6∈WFh(Pu), then (x, ξ) ∈ Ellh(P)\WFh(Pu)but ∈ WFh(u).) Indeed, take a neighbourhood U 3 (x, ξ) such that U b Ellh(P) \WFh(Pu)

22 XIAOLONG HAN

and choose B ∈ Ψ0h(X) such that U ⊂ Ellh(B) and WFh(B) ∩WFh(Pu) = ∅. Then BP is

elliptic on U and ‖BPu‖Hm−kh

= O(h∞) for all m. By (i) applied to BP in place of P, we have

that for all m and all A ∈ Ψ0h(X) such that WFh(A) ⊂ U ,

‖Au‖Hmh (X;E) ≤ C‖BPu‖Hm−k

h (X;E) +O(h∞) = O(h∞).

So U ∩WFh(u) = ∅ and (x, ξ) /∈WFh(u).

Notice that the correspondence Oph : a ∈ Skh → Ψkh(X) is not generally not positive, i.e.

a ≥ 0⇒ Oph(a) ≥ 0. Such property concerning positivity is in the following

Theorem 2.18 (Garding inequality). Suppose that a ∈ Skh(X) and a ≥ 0. Then for someC > 0,

〈Oph(a)u, u〉 ≥ −Ch‖u‖2

Hk−12

h (X)

for all u ∈ S(X).

See Zworski [Z, Theorems 4.30 and 9.11] for proofs.

Remark (Fefferman-Phong inequality). One can get a stronger Gading inequality for Weylquantisation: Suppose that a ∈ Skh(X) and a ≥ 0. Then for some C > 0,

〈Opwh (a)u, u〉 ≥ −Ch2‖u‖2

Hk−22

h (X)

for all u ∈ S(X).

2.8. Propagation of singularities I.

Proposition 2.19 (Propagation of singularities). Assume that P ∈ Ψ1h(X, ; E) and the semi-

classical principal symbol

σh(P) ∈ S1h(X; Hom(E))/hS0

h(X; Hom(E))

is diagonal with entries p ∈ S1(X;R) independent of h. Let Q ∈ Ψ0h(X; Hom(E)) with principal

symbol q ≥ 0 everywhere. Assume also that p is homogeneous of degree 1 in ξ for |ξ| largeenough. Let γ(t) = etHp be the Hamiltonian flow of p on T ∗X and u(h) ∈ D′(X; E) be anh-tempered family of distributions. Then

(i). Assume that A,B,B1 ∈ Ψ0h(X) and for each γ(0) ∈ WFh(A), there exists T ≥ 0 with

γ(T ) ∈ Ellh(B) and γ(t) ∈ Ellh(B1) for t ∈ [−T, 0]. Then for each m,

‖Au‖Hmh (X;E) ≤ C‖Bu‖Hm

h (X;E) + Ch−1‖B1(P− iQ)u‖Hmh (X;E) +O(h∞)‖u‖H−∞h (X;E).

(ii). For each T > 0,

γ(T ) /∈WFh(u), γ([−T, 0]) ∩WFh((P− iQ)u) = ∅ ⇒ γ(0) /∈WFh(u).

Propagation of singularities states in particular that if (P − iQ)u = O(h∞)C∞ and u =O(1)Hm

hmicrolocally near some (x, ξ) ∈ T ∗X, then u = O(1)Hm

hmicrolocally near etHp(x, ξ) for

t ≥ 0; in other words, regularity can be propagated forward along the Hamiltonian flow lines.(If q ≤ 0 instead, then regularity could be propagated backward.)

Remark (Complex absorbing operators). The operator Q = Oph(q) is called the complexabsorbing operator. It fixes the direction of propagation. To see how the sign of q affects thedirection of propagation, one can consider the one-dimensional example with P = hDx and


Q = q(x) ≥ 0 everywhere, a(x) is supported near x = 1 and b(x) is supported near x = 0.Then the solutions to

(P − iQ)u = 0, i.e. ∂xu(x) = −q(x)

hu(x)

are multiples of

u(x) = exp

(−1

h

∫ x

0

q(y) dy

)u(0).

We see that‖a(x)u(x)‖ . ‖b(x)u(x)‖,

but not the other way around.

2.8.1. First reduction: Hormander’s positive commutator argument. For simplicity, we adaptWeyl quantisation so P is self-adjoint. (Other quantisations differ by a lower order term.)Compute that

〈[P, F ?F ]u, u〉 = 〈PF ?Fu, u〉 − 〈F ?FPu, u〉= 〈F ?Fu, Pu〉 − 〈Pu, F ?Fu〉= 2i=〈F ?Fu, Pu〉,

where F = Oph(f) with f ∈ C∞0 and f ≥ 0. We know that

[P, F ?F ] =h

iOph(Hp(f

2)) +O(h2).

So

−2

h=〈F ?Fu, Pu〉 = 〈Oph(Hp(f

2))u, u〉+O(h)‖u‖2L2 .

If Hpf ≤ −C0f for some C0 > 0, then

〈Oph(Hp(f2))u, u〉 = 〈Oph(2fHpf)u, u〉 ≤ −2C0〈F ?Fu, u〉 −O(h)‖u‖2

L2

by Garding inequality Zworski [Z, Theorem 9.11]. Hence,

−2C0〈F ?Fu, u〉 ≥ 〈Oph(Hp(f2))u, u〉+O(h)‖u‖2

L2

≥ −2

h=〈F ?Fu, Pu〉+O(h)‖u‖2

L2

≥ −2

h‖F ?Fu‖L2‖Pu‖L2 +O(h)‖u‖2

L2 ,

which implies that‖Fu‖2

L2 . h−1‖Fu‖L2‖Pu‖L2 +O(h)‖u‖2L2 .

So‖Fu‖L2 ≤ Ch−1‖Pu‖L2 + Ch1/2‖u‖L2 .

However, a nonnegative function f ∈ C∞0 in general can not satisfy Hpf ≤ −C0f everywhere;its existence depends on the dynamical properties of Hp, (e.g. if Hp = ∂θ on a circle, then it isimpossible for f ∈ C∞0 such that Hpf ≤ −C0f everywhere.)

Moreover, we do not have the control of ‖Au‖ by ‖Bu‖ and ‖Pu‖ for which WFh(A) andWFh(B) supported in different regions along one bicharacteristic line. In order to remedy this,we need to design the escape function.

24 XIAOLONG HAN

2.8.2. Second reduction: Escape functions. Assume that a, b ∈ C∞0 (R2n) such that for eachγ(0) ∈ supp a, there exists T ≥ 0 with γ(−T ) ∈ b 6= 0. We design f ∈ C∞0 (R2n) such that

(1). f ≥ 0 everywhere;(2). f > 0 near supp a;(3). Hpf ≤ −C0f outside of supp b.

The construction of this escape function is indeed one-dimensional: We identify a tubularneighbourhood of γ([−T, 0]) with

|θ| < δ × (−T − δ, δ) ∈ R2n−1θ × Rt,

where Hp = ∂t, supp a ⊂ |θ| < δ×(−δ/2, δ/2), and supp b ⊂ |θ| < δ×(−T−δ/2,−T+δ/2).Let ψ0(t) ∈ C∞0 (−T − δ, δ) such that ψ0 ≥ 0, ψ0(0) = 1, and ψ′0 ≤ 0 on (−T + δ, δ). Then

ψ(t) = e−C0tψ0(t)

satisfies that ψ ≥ 0, ψ(0) = 1, and ψ′ ≤ −C0ψ on (−T + δ, δ). Now f(θ, t) = χ(θ)ψ(t) forχ ∈ C∞0 with suppχ ∈ |θ| < δ satisfies the conditions of the escape function.

Now write A = Oph(a) and B = Oph(b). With the help of the escape function, we canconnect Au with Bu and Pu. Note that Hpf ≤ −C0f outside of supp b. We can then chooseC1 large enough such that

fHpf ≤ −C0f2 + C1b

2 everywhere.

Then by Garding inequality,

〈Oph(fHpf)u, u〉 ≤ −C0‖Fu‖2L2 + C1‖Bu‖2

L2 +O(h)‖u‖2L2 .

Hence,

−1

h‖F ?Fu‖L2‖Pu‖L2 ≤ −1

h=〈F ?Fu, Pu〉

≤ 1

2〈Oph(Hp(f

2))u, u〉

≤ −C0‖Fu‖2L2 + C1‖Bu‖2

L2 +O(h)‖u‖2L2 .

So for 0 < ε 1

‖Fu‖2L2 . ‖Bu‖2

L2 + h−1‖F ?Fu‖L2‖Pu‖L2 +O(h)‖u‖2L2

≤(‖Bu‖L2 + ε−1h−1‖Pu‖L2 + ε‖Fu‖L2 + Ch1/2‖u‖L2

)2,

which implies that

‖Fu‖L2 ≤ C‖Bu‖L2 + Ch−1‖Pu‖L2 + Ch1/2‖u‖L2 .

We notice that f > 0 on supp a so by elliptic estimate,

‖Au‖L2 ≤ ‖Fu‖L2 +O(h∞)‖u‖L2 ≤ C‖Bu‖L2 + Ch−1‖Pu‖L2 + Ch1/2‖u‖L2 .

2.8.3. Third reduction: Presence of complex absorbing operator. We also want to remove theterm h1/2‖u‖L2 in the estimate. In order to do this, we assume that all of our operations areon supp b1 where b1 ∈ C∞0 (R2n). That is, a, b, b1 ∈ C∞0 (R2n) such that for each γ(0) ∈ supp a,there exists T ≥ 0 with γ(−T ) ∈ b1 6= 0 and γ([−T, 0]) ⊂ supp b1. Adopt the escape functionf such that supp f ⊂ supp b1 with C0 to be fixed later. Compute that

=〈(P − iQ)u, F ?Fu〉

=1

2i

[〈(P − iQ)u, F ?Fu〉 − 〈F ?Fu, (P − iQ)u〉

]


=1

2i

[〈Pu, F ?Fu〉 − 〈F ?Fu, Pu〉

]+

1

2i

[〈−iQu, F ?Fu〉 − 〈F ?Fu,−iQu〉

]=

1

2i〈[F ?F, P ]u, u〉 − 1

2

[〈Qu, F ?Fu〉+ 〈F ?Fu,Qu〉

]=

i

2〈[P, F ?F ]u, u〉 − <〈Qu, F ?Fu〉.

The first term in the above equation can be estimated as before:

i

2〈[P, F ?F ]u, u〉

=h

2〈Oph(Hp(f

2))u, u〉+O(h2)‖B1u‖2L2 +O(h∞)‖u‖2

L2

= h〈Oph(fHpf)u, u〉+O(h2)‖B1u‖2L2 +O(h∞)‖u‖2

L2

≤ −C0h‖Fu‖2L2 + C1h‖Bu‖2

L2 +O(h2)‖B1u‖2L2 +O(h∞)‖u‖2

L2 ;

while the second term can be estimated asi

<〈Qu, F ?Fu〉= <〈FQu, Fu〉= <〈QFu, Fu〉 − <〈F ?[Q,F ]u, u〉≥ −C2h‖Fu‖2

L2 +O(h2)‖B1u‖2L2 +O(h∞)‖u‖2

L2 ;

by Garding inequality and the fact that

F ?[Q,F ] =h

iOph(fHqf) +O(h2),

where fHqf is real-valued, so F ?[Q,F ] = O(h2). Here, C2 depends only on Q.Combine these two estimates together, we arrive at

−|〈(P − iQ)u, F ?Fu〉|≤ =〈(P − iQ)u, F ?Fu〉≤ −C0h‖Fu‖2

L2 + C1h‖Bu‖2L2 +O(h2)‖B1u‖2

L2 + C2h‖Fu‖2L2 +O(h∞)‖u‖2

L2 .

So choosing C0 C2, we have that for 0 < ε 1,

C‖Fu‖2L2

≤ h−1|〈B1(P − iQ)u, F ?Fu〉|+ C1‖Bu‖2L2 +O(h)‖B1u‖2

L2 +O(h∞)‖u‖2L2

≤[ε−1h−1‖B1(P − iQ)u‖L2 + ε‖Fu‖L2 + C1‖Bu‖L2 + Ch1/2‖B1u‖L2 +O(h∞)‖u‖L2

]2,

which implies that

‖Fu‖L2 ≤ C‖Bu‖L2 + Ch−1‖B1(P − iQ)u‖L2 + Ch1/2‖B1u‖L2 +O(h∞)‖u‖L2 .

Since f > 0 on supp a,

‖Au‖L2 ≤ C‖Bu‖L2 + Ch−1‖B1(P − iQ)u‖L2 + Ch1/2‖B1u‖L2 +O(h∞)‖u‖L2

by elliptic estimate.To remove the term h1/2‖B1u‖L2 , we can assume without loss of generality that for each

(x, ξ) ∈ supp b1, there exist T ≥ 0 and t ∈ [−T, 0] such that γ(t) = etHp(x, ξ) ∈ b 6= 0 (by

iNote that we do not want to use Garding inequality directly because we want a bound of ‖Fu‖ instead of‖u‖.

26 XIAOLONG HAN

enlarging the support of b1 if necessary). Then choose b2 such that γ([−T, 0]) ⊂ supp b2. Sosupp b1 ⊂ supp b2.

Applying the above inequality for (b1, b, b2), we have

‖B1u‖L2 ≤ C‖Bu‖L2 + Ch−1‖B1(P − iQ)u‖L2 + Ch1/2‖B2u‖L2 +O(h∞)‖u‖L2 ,

which leads to

‖Au‖L2 ≤ C ′‖Bu‖L2 + C ′h−1‖B1(P − iQ)u‖L2 + C ′h‖B2u‖L2 +O(h∞)‖u‖L2 .

Repeated process would give us that

‖Au‖L2 ≤ CN‖Bu‖L2 + CNh−1‖B1(P − iQ)u‖L2 + CNh

N‖BNu‖L2 +O(h∞)‖u‖L2 ,

for some B1, BN ∈ Ψ0h(X). Therefore,

‖Au‖L2 ≤ C‖Bu‖L2 + Ch−1‖B1(P − iQ)u‖L2 +O(h∞)‖u‖L2 .

2.8.4. Fourth reduction: Regularity in Hmh for arbitrary m. We still use the same escape func-

tion f . It now suffices to show that

‖F u‖L2 ≤ C‖Bu‖Hmh

+ Ch−1‖(P − iQ)u‖Hmh

+ Ch1/2‖B1u‖Hm−1/2h

+O(h∞)‖u‖H−∞h ,

for some F ∈ Ψmh (X) with symbol f = 〈ξ〉mf ; one can then change the left-hand side by

‖Au‖Hmh

since f > 0 on supp a. Recall that

=〈(P − iQ)u, F ?F u〉 =i

2〈[P, F ?F ]u, u〉 − <〈Qu, F ?F u〉.

To estimate the first term in the right-hand side, notice that

i

h[P, F ?F ] = Oph(Hp(f

2)) +O(h)Ψ2m−1h (X) ∈ Ψ2m

h (X)

with principal symbol

Hp(f2) = Hp(〈ξ〉2mf 2) = 2m〈ξ〉2m−1Hp(〈ξ〉)f 2 + 2〈ξ〉2mfHpf ;

since Hp(〈ξ〉) = O(〈ξ〉) (because p is homogeneous of degree 1) and fHpf ≤ −C0f2 + C1b

2,

Hp(〈ξ〉2mf 2) ≤ 2(m− C0)〈ξ〉2mf 2 + 2C1〈ξ〉2mb2 = 2(m− C0)f 2 + 2C1(〈ξ〉mb)2.

Notice also that if A ∈ Ψkh(X), then

|〈Au, u〉| ≤ ‖Au‖H−k/2h‖u‖

Hk/2h≤ C‖u‖2

Hk/2h

.

Hence, by Garding inequality

i

2〈[P, F ?F ]u, u〉

=h

2〈Oph(Hp(f

2)u, u〉+O(h2)‖B1u‖2

Hm−1/2h

+O(h∞)‖u‖2H−∞h

≤ (m− C0)h‖F u‖2L2 + C1h‖Bu‖2

Hmh

+O(h2)‖B1u‖2

Hm−1/2h

+O(h∞)‖u‖2H−∞h

≤ −C0h‖F u‖2L2 + C1h‖Bu‖2

Hmh

+O(h2)‖B1u‖2

Hm−1/2h

+O(h∞)‖u‖2H−∞h

by choosing larger C0 and C1 depending on m.To estimate the second term, compute that

<〈Qu, F ?F u〉= <〈QFu, Fu〉 − <〈F ?[Q, F ]u, u〉


≥ −C2h‖F u‖2L2 +O(h2)‖B1u‖2

Hm−1h

+O(h∞)‖u‖2H−∞h

;

by Garding inequality and the fact that

F ?[Q, F ] =h

iOph(fHqf) +O(h2)Ψ2m−2

h,

where fHqf is real-valued, so <F ?[Q, F ] = O(h2)Ψ2m−2h

. Here, C2 depends only on Q.

Combine these two estimates together, we arrive at

−|〈(P − iQ)u, F ?F u〉|≤ =〈(P − iQ)u, F ?F u〉≤ −C0h‖F u‖2

L2 + C1h‖Bu‖2Hmh

+O(h2)‖B1u‖2

Hm−1/2h

+ C2h‖F u‖2L2 +O(h∞)‖u‖2

H−∞h.

This implies that (by choosing larger C0)


+ Ch−1‖B1(P − iQ)u‖Hmh

+ Ch1/2‖B1u‖Hm−1/2h

+O(h∞)‖u‖H−∞h .

Remark. There is an issue in the above argument. That is, we assume that u ∈ Hm−1/2h so

(P − iQ)u ∈ Hm−3/2h and F ?F u ∈ H−m−1/2

h

since P − iQ ∈ Ψ1h and F ∈ Ψm

h ; hence we can not define

〈(P − iQ)u, F ?F u〉in apriori. To fix this problem, we introduce Fε ∈ Ψm−1

h with symbol 〈ξ〉m〈εξ〉−1f , whereε ∈ (0, 1). Then

F ?ε Fεu ∈ H

−m+3/2h

and 〈(P − iQ)u, F ?F u〉 can be well-defined. One can proceed the same argument as above andget

‖Fεu‖L2 ≤ C‖Bu‖Hmh

+ Ch−1‖(P − iQ)u‖Hmh

+ Ch1/2‖B1u‖Hm−1/2h

+O(h∞)‖u‖H−∞huniformly for ε. Notice that

Fε → F in Ψm+1/2h ,

soFεu→ F u in H−1

h .

This means that〈Fεu, v〉 → 〈F u, v〉 for all v ∈ H1

h,

in particular, for all v ∈ L2 ∩H1h

〈Fεu, v〉 → 〈F u, v〉,which would imply that F u ∈ L2 and


+ Ch−1‖B1(P − iQ)u‖Hmh

+ Ch1/2‖B1u‖Hm−1/2h

+O(h∞)‖u‖H−∞h .

28 XIAOLONG HAN

3. The resolvent acting on anisotropic Sobolev spaces

3.1. Anosov systems: Extra facts. The principal symbol of P = −iV , p = ξ(V (x)), ishomogeneous of degree 1. The Hamiltonian flow etHp is a natural lift of etV to T ∗X, that is,

etHp(x, ξ) =(ϕ(x),

(Tdϕt(x)

)−1ξ).

The above definition can be easily derived since etHp preserves the canonical symplectic formdξ ∧ dx. There is also decomposition

T ∗xX = E∗c (x)⊕ E∗s (x)⊕ E∗u(x),

where

E∗c (x)(Es(x)⊕ Eu(x)) = 0, E∗s (x)(Ec(x)⊕ Es(x)) = 0, E∗u(x)(Ec(x)⊕ Eu(x)) = 0.

So(Tdϕt(x)

)−1for t > 0 is length-preserving on E∗c (x)→ E∗c (ϕ(x)), is contracting on E∗s (x)→

E∗s (ϕ(x)) and is expanding on E∗u(x)→ E∗u(ϕ(x)).Denote κ : T ∗X \ 0→ S∗X the natural projection map.

Definition (Radial source and radial sink). Let L ⊂ T ∗X \ 0 be a closed conic set which isinvariant under the flow etHp .

(i). L is called a radial source if there exists an open conic neighbourhood U of L such thatfor some C, θ > 0, as t→ +∞,• d(κ(e−tHp(U)), κ(L))→ 0;• (x, ξ) ∈ U ⇒ |e−tHp(x, ξ)| ≥ C−1eθ|t||ξ| for any norm on the fibers.

(ii). L is called a radial sink if there exists an open conic neighbourhood U of L such that forsome C, θ > 0, as t→ −∞,• d(κ(e−tHp(U), κ(L))→ 0;• (x, ξ) ∈ U ⇒ |e−tHp(x, ξ)| ≥ C−1eθ|t||ξ| for any norm on the fibers.

Theorem 3.1. The closed conic sets E∗s and E∗u are a radial source and a radial sink, respec-tively.

Proof.

(i). Let ξ be a covector near E∗s (x) in T ∗xX. Write ξ = ξc + ξs + ξu ∈ T ∗xX, where ξc ∈ E∗c (x),0 6= ξs ∈ E∗s (x), and ξu ∈ E∗u(x). Since (Tdϕt(x))−1 for t > 0 is length-preserving onE∗c (x) → E∗c (ϕ(x)), is contracting on E∗s (x) → E∗s (ϕ(x)) and is expanding on E∗u(x) →E∗u(ϕ(x)), we have that for t→ −∞

‖(Tdϕt(x)

)−1ξc‖ϕ(x) = ‖ξc‖x,

‖(Tdϕt(x)

)−1ξs‖ϕ(x) ≥ Ceθt‖ξs‖x →∞,

and‖(Tdϕt(x)

)−1ξu‖ϕ(x) ≤ Ce−θt‖ξu‖x → 0.

Hence, as t→ −∞,(Tdϕt(x)

)−1ξ

‖ (Tdϕt(x))−1 ξ‖ϕ(x)

=

(Tdϕt(x)

)−1ξc


+

(Tdϕt(x)

)−1ξs


+

(Tdϕt(x)

)−1ξu



→(Tdϕt(x)

)−1ξs

‖ (Tdϕt(x))−1 ξs‖ϕ(x)

∈ S∗ϕ(x)X.

This shows that d(κ(e−tHp(U), κ(E∗s ))→ 0 as t→ +∞; indeed,

d(κ(e−tHp(x, ξ), κ(E∗s )) .

∥∥∥(Tdϕt(x))−1

(ξc + ξu)∥∥∥ϕ(x)


→ 0.

The second condition is obvious since the stable covector is expanding as t→ −∞.(ii). Reverse the time and one can get the conclusions in a similar fashion.

3.2. Propagation of singularities II. In this subsection, we prove two results on propagationof singularities adapted for radial sources and radial sinks, respectively. Just as what we didin Propagation of singularities I, we need to construct escape functions. We still assume thatp is homogeneous of degree 1 in ξ for |ξ| large enough. So V = κ∗Hp is a natural projection asa smooth vector field on the cosphere bundle S∗X.

Lemma 3.2 (Escape functions adapted for radial sources). Suppose L is a radial source. Thenthere exist

(i). f0 ∈ C∞(T ∗X \0; [0, 1]), homogeneous of degree 0 and such that f0 = 1 near L, supp f0 ⊂U , and Hpf0 ≤ 0.

(ii). f1 ∈ C∞(T ∗X \ 0; [0,∞)), homogeneous of degree 1 and such that 0 < c|ξ| ≤ f1(x, ξ) ≤c−1|ξ| everywhere and Hpf1 ≤ −cf1 on U for some c > 0.

Proof. (i). Our construction is on S∗X. Since L is invariant under the flow etHp , κ(L) isinvariant under the flow etV . We will construct F ∈ C∞(S∗X; [0, 1]) such that F = 1 on aneighbourhood of κ(L), suppF ⊂ κ(U), and V (F ) ≤ 0. Then f0 = κ∗F will be a functionsatisfying the conditions in (i).

To obtain F , fix F0 ∈ C∞(S∗X; [0, 1]) such that F0 = 1 near κ(L) and suppF0 ⊂ κ(U). Bythe first assumption of radial sources, we have for T > 0 large enough,

e−tV (suppF0) ⊂ F0 = 1 for t ≥ T.

Also since U is invariant under the flow, supp (F etV ) ⊂ κ(U) for all t ≥ T .Now we see that F0(w) ≥ F0(eTV (w)) for all w ∈ S∗X. Indeed, if eTV (w) ∈ suppF0, then

there exists w′ ∈ suppF0 such that e−TV (w′) = w and hence F0(w) = F0(e−TV (w′)) = 1;otherwise F0(eTV (w)) = 0 but 0 ≤ F0 ≤ 1 everywhere. Then write

F (w) =1

T

∫ 2T

T

F0(etV (w)) dt.

If w is closed enough (depending on T ) to κ(L) such that e2T (w) ∈ F0 = 1. Therefore,F (w) = 1 near κ(L). F (w) obviously bounded between 0 and 1. Furthermore,

V (F )(w) =1

T

[F0(e2TV (w))− F0(eTV (w))

]≤ 0.

(ii). To find f1, fix a smooth norm | · | of the fibers of T ∗X. By the second assumption ofradial sources, we have for T1 > 0 large enough,

e−tHp(x, ξ) ≥ 2|ξ| for (x, ξ) ∈ U, t ≥ T1.

30 XIAOLONG HAN

Then the function

f1(x, ξ) =

∫ T1

0

|e−tHp(x, ξ)| dt

is homogenous of degree 1 and 0 < c|ξ| ≤ f1(x, ξ) ≤ c−1|ξ| everywhere. Furthermore,

Hpf1(x, ξ) = |ξ| −∣∣e−T1Hp(x, ξ)∣∣ ≤ −|ξ| ≤ −cf1(x, ξ) for (x, ξ) ∈ U.

Remark. We see that the construction of f1 (resp. f2) uses only the first (resp. second)assumption of the radial sources. Moreover, notice that

f(x, ξ) = f0(x, ξ)fm1 (x, ξ) ∈ C∞(T ∗X \ 0; [0,∞))

is homogeneous of degree m, is supported in U , f ≥ c〈ξ〉m near κ(L), and since Hpf1 ≤ −cf1

on U ,Hpf = fm1 Hpf0 +Hp(f

m1 )f0 ≤ −cmf0f

m1 = −cmf

everywhere bacause f = 0 outside of U . Such escape function would give us the regularitynear κ(L). However, we should begin with some apriori regularity assumption since the aboveescape function depends on m.

The following theorem shows that for sufficiently high regularity the wavefront set at theradial sources is controlled.

Proposition 3.3 (Propagation of singularities adapted for radial sources). Assume that P ∈Ψ1h(X, ; E) and the semiclassical principal symbol


h(X; Hom(E))


symbol q ≥ 0 everywhere. Let L ⊂ T ∗X \ 0 be a radial source. Then there exists m0 > 0 suchthat

(i). For each B1 ∈ Ψ0h(X) elliptic on κ(L) ⊂ S∗X, there exists A ∈ Ψ0

h(X) elliptic on κ(L)such that if u(h) ∈ D′(X; E) is h-tempered, then for each m ≥ m0,

Au ∈ Hm0h (X; E)⇒ ‖Au‖Hm

h (X;E) ≤ Ch−1‖B1(P− iQ)u‖Hmh (X;E) +O(h∞)‖u‖H−∞h (X;E).

(ii). If u ∈ D′(X; E) is h-tempered and B1 ∈ Ψ0h(X) elliptic on κ(L), then

B1u ∈ Hm0h (X; E),WFh((P− iQ)u) ∩ κ(L) = ∅ ⇒WFh(u) ∩ κ(L) = ∅.

Remark. In particular, if u solves Pu = 0, then u ∈ Hm0h near a radial source L would imply

that u ∈ Hmh near L for all m ≥ m0.

Proof. As before, we only need to prove (ii). It suffices to prove

‖Au‖Hmh≤ Ch−1‖B1(P − iQ)u‖Hm

h+ Ch1/2‖B1u‖Hm−1/2

h+O(h∞)‖u‖H−∞h .

Indeed, without loss of generality we may assume that WFh(B1) ⊂ U . Then by the assumptionof the radial source, each backward flow line of Hp starting on WFh(B1) reaches Ellh(A).Combining the above inequality and Propagation of singularities I, we have that for someB2 ∈ Ψ0

h(X) elliptic on κ(L),


h+ Ch1/2‖B1u‖Hm−1/2

h+O(h∞)‖u‖H−∞h

≤ Ch−1‖B2(P − iQ)u‖Hmh

+ Ch1/2‖Au‖Hm−1/2h

+O(h∞)‖u‖H−∞h .


Iterating this estimate, we arrive at


h+ Ch∞‖Au‖Hm0

h+O(h∞)‖u‖H−∞h ,

where the term h∞‖Au‖Hm0h

can be trivially removed provided that Au ∈ Hm0h .

Let F ∈ Ψmh (X) with symbol f = f0f

m1 , m > 0. Then

=〈(P − iQ)u, F ?Fu〉 =i

2〈[P, F ?F ]u, u〉 − <〈Qu, F ?Fu〉.


i


2)) +O(h)Ψ2m−2h (X) ∈ Ψ2m−1

h (X)


Hp(f2) = Hp(f

20 f

2m1 ) = 2f0f

2m1 Hpf0 + 2mf 2

0 f2m−11 Hpf1;

since Hpf1 ≤ −cf1 on U and f = 0 outside of U ,

Hp(f2) ≤ −2cmf 2 everywhere.

Hence, by Garding inequality

i

2〈[P, F ?F ]u, u〉

=h

2〈Oph(Hp(f

2)u, u〉+O(h2)‖B1u‖2Hm−2h

+O(h∞)‖u‖2H−∞h

≤ −cmh‖Fu‖2L2 +O(h2)‖B1u‖2

Hm−1/2h

+O(h∞)‖u‖2H−∞h

≤ −C0h‖Fu‖2L2 +O(h2)‖B1u‖2

Hm−1/2h

+O(h∞)‖u‖2H−∞h

if m ≥ m0 ≥ C0/c and C0 to be chosen later (and thus m0).The second term inherits

<〈Qu, F ?Fu〉 ≤ −C2h‖Fu‖2L2 +O(h2)‖B1u‖2

Hm−1h

+O(h∞)‖u‖2H−∞h

,

where C2 depends only on Q.Combine these two estimates together, we arrive at


L2 +O(h2)‖B1u‖2

Hm−1/2h

+ C2h‖Fu‖2L2 +O(h∞)‖u‖2

H−∞h.

This implies that (by choosing large C0)

‖Fu‖L2 ≤ Ch−1‖B1(P − iQ)u‖Hmh

+ Ch1/2‖B1u‖Hm−1/2h

+O(h∞)‖u‖H−∞h .

Notice that 0 < c|ξ| ≤ f1 ≤ c−1|ξ| and f0 = 1 near L. One can then find A ∈ Ψ0h(X) elliptic

on κ(L) such that F is elliptic on WFh(A) so


h+ Ch1/2‖B1u‖Hm−1/2


for all m ≥ m0.

The escape functions needed to prove the propagation of singularities adapted for radial sinksare in

32 XIAOLONG HAN

Lemma 3.4 (Escape functions adapted for radial sinks). Suppose L is a radial sink. Thenthere exist

(i). f0 ∈ C∞(T ∗X \0; [0, 1]), homogeneous of degree 0 and such that f0 = 1 near L, supp f0 ⊂U , and Hpf0 ≥ 0.

(ii). f1 ∈ C∞(T ∗X \ 0; [0,∞)), homogeneous of degree 1 and such that 0 < c|ξ| ≤ f1(x, ξ) ≤c−1|ξ| everywhere and Hpf1 ≥ cf1 on U for some c > 0.

The proof follows easily by noticing that a radial sink for Hp is the radial source for H−p =−Hp.

The following theorem shows that for sufficiently low regularity the wavefront set at theradial sinks is controlled.

Proposition 3.5 (Propagation of singularities adapted for radial sinks). Assume that P ∈Ψ1h(X, ; E) and the semiclassical principal symbol


h(X; Hom(E))


symbol q ≥ 0 everywhere. Let L ⊂ T ∗X \ 0 be a radial sink. Then there exists m0 > 0 suchthat for each B1 ∈ Ψ0

h(X) elliptic on κ(L) ⊂ S∗X, there exists A ∈ Ψ0h(X) elliptic on κ(L) and

B ∈ Ψ0h(X) with WFh(B) ⊂ Ellh(B1) \ κ(L) such that if u(h) ∈ D′(X; E) is h-tempered, then

for each m ≤ −m0,

‖Au‖Hmh (X;E) ≤ C‖Bu‖Hm

h (X;E) + Ch−1‖B1(P− iQ)u‖Hmh (X;E) +O(h∞)‖u‖H−∞h (X;E).

Proof. As before, it suffices to prove

‖Au‖Hmh≤ C‖Bu‖Hm

h+ Ch−1‖B1(P − iQ)u‖Hm

h+ Ch1/2‖B1u‖Hm−1/2

h+O(h∞)‖u‖H−∞h .

Let F ∈ Ψmh (X) with symbol f = f0f

m1 , m > 0. Then

=〈(P − iQ)u, F ?Fu〉 =i

2〈[P, F ?F ]u, u〉 − <〈Qu, F ?Fu〉.


i


2)) +O(h)Ψ2m−2h (X) ∈ Ψ2m−1

h (X)


Hp(f2) = Hp(f

20 f

2m1 ) = 2f0f

2m1 Hpf0 + 2mf 2

0 f2m−11 Hpf1;

since Hpf1 ≥ cf1 on U and f = 0 outside of U , (note that m < 0)

Hp(f2) ≤ 2cmf 2 everywhere.

In particular, for each C0 > 0, by choosing m ≤ −m0 for m0 large enough (depending on C0),

Hp(f2) ≤ 2cmf 2 ≤ −2cm0f

2 ≤ −2C0f2 + 2〈ξ〉2mb2 everywhere.

for some b ∈ C∞0 (T ∗X) with supp b ∈ Ellh(B1) \ κ(L).Hence, by Garding inequality

i

2〈[P, F ?F ]u, u〉

=h

2〈Oph(Hp(f

2)u, u〉+O(h2)‖B1u‖2Hm−2h

+O(h∞)‖u‖2H−∞h

≤ −C0h‖Fu‖2L2 +O(h)‖Bu‖Hm

h+O(h2)‖B1u‖2

Hm−1/2h

+O(h∞)‖u‖2H−∞h

.


The second term inherits

<〈Qu, F ?Fu〉 ≤ −C2h‖Fu‖2L2 +O(h2)‖B1u‖2

Hm−1h

+O(h∞)‖u‖2H−∞h

,

where C2 depends only on Q.Combine these two estimates together, we arrive at


L2 +O(h)‖Bu‖Hmh

+O(h2)‖B1u‖2

Hm−1/2h

+ C2h‖Fu‖2L2 +O(h∞)‖u‖2

H−∞h.

This implies that (by choosing large C0)

‖Fu‖L2 ≤ Ch−1‖B1(P − iQ)u‖Hmh

+ C‖Bu‖Hmh

+ Ch1/2‖B1u‖Hm−1/2h

+O(h∞)‖u‖H−∞h .

Notice that 0 < c|ξ| ≤ f1 ≤ c−1|ξ| and f0 = 1 near L. One can then find A ∈ Ψ0h(X) elliptic

on κ(L) such that F is elliptic on WFh(A) so

‖Au‖Hmh≤ C‖Bu‖Hm

h+ Ch−1‖B1(P − iQ)u‖Hm

h+ Ch1/2‖B1u‖Hm−1/2


for all m ≤ −m0.

3.3. Anisotropic Sobolev spaces. Recall that the Sobolev space

Hs(X) =(Op(〈ξ〉s)

)−1L2(X) and ‖u‖Hs = ‖Op(〈ξ〉s)u‖L2 .

An ansisotropic Sobolev space is one that we modify the symbol 〈ξ〉s in the above equation.Precisely, let

Hsm(X) =(Op(〈ξ〉sm)

)−1L2(X),

where m(x, ξ) ∈ C∞(T ∗X \ 0; [−1, 1]). In particular, we want

(1). m = 1 near E∗s ;(2). m = −1 near E∗u;(3). Hpm ≤ 0 everywhere.

The existence of such function is guaranteed by Lemmas 3.2 and 3.4, particularly, the f0

components therein. The reason why we want m to be such will be made clear in the proofof Proposition 3.6; it is in fact natural, given the dynamics. See also Faure-Tsujii [FT, §3.1]for a simple motivation of the above conditions in the investigation of the transfer operatoru(x)→ u(x/θ), θ > 1, on anisotropic Sobolev spaces.

For computational simplicity and consistence, let m := mG and define G ∈ Ψ0+(X) withsymbol mG(x, ξ) log |ξ|. Define the anisotropic Sobolev spaces

HsG = e−sGL2 and ‖u‖HsG = ‖esGu‖L2 ,

where L2 functions can take values on E . We immediately see that

Hs ⊂ HsG ⊂ H−s.

Recall that P acts on vector bundle of differential forms E is defined by P = LV /i. Wewant to establish the resolvent estimates of P on the spaces HsG. This allows us to accessTrb((P− λ)−1|C∞(X;Ek0 )), and subsequently ζ(λ). Note that

(P− λ)−1 = i

∫ ∞0

eiϕ∗−t dt

34 XIAOLONG HAN

is well-defined when =(λ) 1. We want to extend it to the region

=(λ) ≥ −C0 for C0 > 0 arbitarily large.

To achieve this, we perform a semiclassical rescaling hP ∈ Ψ1h(X; E) and study the action of

hP on semiclassical anisotropic Sobolev spaces.

3.4. Resolvent estimates on anisotropic Sobolev spaces. Define G(h) ∈ Ψ0+h (X) with

symbolσh(G(h))(x, ξ) = (1− χ(x, ξ))mG(x, ξ) log |ξ|, (3.1)

where χ ∈ C∞0 (T ∗X) is equal to 1 near the zero section ξ| = 0. Define the anisotropicSobolev spaces

HsG(h) = e−sG(h)L2 and ‖u‖HsG(h)= ‖esG(h)u‖L2 .

Define the domain, DsG(h), of hP as the set of u ∈ HsG(h) such that hPu ∈ HsG(h). The Hilbertspace norm on DsG(h) is given by

‖u‖2DsG(h)

= ‖u‖2HsG(h)

+ ‖hPu‖2HsG(h)

.

We also induce the complex absorbing potential iQδ ∈ Ψ0h(X), which provides a localization to

a neighbourhood of the zero section:

WFh(Qδ) ⊂ |ξ| < δ, qδ := σh(Qδ) on |ξ| ≤ δ/2, qδ ≥ 0 everywhere.

Here, | · | is a fixed norm on the fibers of T ∗X. Define for z = hλ that

Pδ(z) := hP− iQδ − z : DsG(h) → HsG(h).

The following proposition is the key element in establishing the meromorphic extension of ζ(λ).

Proposition 3.6 (Resolvent estimates on anisotropic Sobolev spaces). Fix a constant C0 > 0and ε > 0. Then for s > 0 large enough depending on C0 and h small enough, the operator

Pδ(z) : DsG(h) → HsG(h), −C0h ≤ =(z) ≤ 1, |<(z)| ≤ hε,

is invertible, and the inverse, Rδ(z), satisfies

‖Rδ(z)‖HsG(h)→HsG(h)≤ Ch−1, WF′h(Rδ(z)) ∩ T ∗(X ×X) ⊂ ∆(T ∗X) ∪ Ω+,

where ∆(T ∗X) is the diagonal and Ω+ is the positive flow-out of etHp on p = 0:Ω+ = (etHp(x, ξ), x, ξ) : t ≥ 0, p(x, ξ) = 0.

We use the propagation of semiclassical singularities and the elimination of trapping due tothe complex absorbing potential to establish existence and properties of the inverse of Pδ(z).The idea is simple and natural given the dynamics of the flow: Given bounds on ‖Pδ(z)u‖HsG(h)

,

we first establish bounds on u microlocally near the sources κ(E∗s ) by Proposition 3.3. Byellipticity (Proposition 2.17) we can also estimate u on p 6= 0 and in |ξ| < δ/2, wherethe latter is made possible by the potential Qδ. The resulting estimates can be propagatedforward along the flow etHp , using Proposition 2.19, to the whole T ∗X \κ(E∗u); finally, to boundu microlocally near κ(E∗u), we use Proposition 3.5. The spaces HsG(h) provide the correctregularity for Propositions 3.3 and 3.5.

Note that the action

Pδ(z) = hP − iQδ − z : HsG(h) → HsG(h)

is equivalent to the action on L2 of the conjugated operator

Pδ,s(z) := esG(h)Pδ(z)e−sG(h)


=[esG(h)Pδ(z)− Pδ(z)esG(h) + Pδ(z)esG(h)

]e−sG(h)

= Pδ(z)− [Pδ(z), esG(h)]e−sG(h),

where

[Pδ(z), esG(h)] =h

iOph

(Hσh(Pδ(z))(e

sG(h)))

+O(h2)Ψ−1+h

.

Hence, the principal symbol

σh

(i

h[Pδ(z), esG(h)]e−sG(h)

)= Hσh(Pδ(z))(e

sG(h))e−sG(h)

= sHσh(Pδ(z))G(h)

= σh

(i

h[Pδ(z), sG(h)]

);

thus,[Pδ(z), esG(h)]e−sG(h) = [Pδ(z), sG(h)] = −s[G(h), hP ]

because [G(h), iQδ] = O(h∞)Ψ−∞hsince WFh(G(h)) ∩WFh(Qδ) = ∅. Therefore,

Pδ,s(z) = Pδ(z) + s[G(h), hP ] +O(h2)Ψ−1+h

.

In general, for all A ∈ Ψmh (X), we have that

σh(A) = σh(esG(h)Ae−sG(h)

)and WFh(A) = WFh

(esG(h)Ae−sG(h)

).

3.4.1. Boundedness. Write f = Pδ(z)u for u ∈ DsG(h). We first prove

‖u‖HsG(h)≤ Ch−1‖f‖HsG(h)

. (3.2)

By a microlocal partition of unity, it suffices to obtain the bounds on Au, where A ∈ Ψ0h(X)

falls into one of the following five cases.

• Case 1. WFh(A) ∩ p = 0 ∩ |ξ| ≥ δ/2 = ∅;• Case 2. WFh(A) is near κ(E∗s );• Case 3. WFh(A) is near some point (x0, ξ0) ∈ p = 0 \ E∗u;• Case 4. WFh(A) is near some point (x0, ξ0) ∈ E∗u;• Case 5. WFh(A) is near κ(E∗u).

• Case 1. WFh(A) ∩ p = 0 ∩ |ξ| ≥ δ/2 = ∅. Therefore, Pδ,s(z) is elliptic on WFh(A).We have

‖Au‖HsG(h)= ‖esG(h)Au‖L2 = ‖esG(h)Ae−sG(h)esG(h)u‖L2 = ‖AsesG(h)u‖L2 ,

whereAs = esG(h)Ae−sG(h) ∈ Ψ0

h(X) and WFh(As) ⊂WFh(A).

Note that Pδ,s(z) is elliptic on WFh(A) ⊃ WFh(As). By the elliptic estimate in Proposition

2.17,‖AsesG(h)u‖L2 ≤ C‖Bs

1Pδ,s(z)esG(h)u‖L2 +O(h∞),

where Bs1 ∈ Ψ0

h(X) is microlocalized in a neighbourhood of WFh(A). Putting

B1 := e−sG(h)Bs1esG(h),

we derive that

‖Au‖HsG(h)= ‖AsesG(h)u‖L2

36 XIAOLONG HAN

≤ C‖esG(h)B1e−sG(h)Pδ,s(z)esG(h)u‖L2 +O(h∞)

= C‖esG(h)B1Pδ(z)u‖L2 +O(h∞)

= C‖esG(h)B1f‖L2 +O(h∞)

= C‖B1f‖HsG(h)+O(h∞),

where B1 ∈ Ψ0h(X) is microlocalized near WFh(A).

• Case 2. WFh(A) is contained in a small neighbourhood of κ(E∗s ). From the constructionof the anisotropic Sobolev space, since σh(e

sG(h)) = |ξ|s near κ(E∗s ), HsG(h) is microlocallyequivalent to the space Hs

h near κ(E∗s ).By =(z) ≥ −C0h, we have

=σh(Pδ(z)) = −qδ −=(z) ≤ 0 in S1h(X)/hS0

h(X).

Because E∗s is a radial source, we apply Proposition 3.3 and obtain, for sufficiently large s,

‖Au‖HsG(h)≤ Ch−1‖B1f‖HsG(h)

+O(h∞),

where B1 ∈ Ψ0h(X) is microlocalized near κ(E∗s ).

• Case 3. WFh(A) is contained in a small neighbourhood of some point (x0, ξ0) ∈ p =

0 \ E∗u, where E∗u = E∗u ∪ κ(E∗u) is the closure of E∗u in T∗X. Since E∗s is a radial source, for

any fixed neighbourhood U of κ(E∗s ), there exists B ∈ Ψ0h(X) with WFh(B) ⊂ U and T > 0

such that e−THp(WFh(A)) ⊂ Ellh(B). In other words, the backward flow will take WFh(A) toa neighbourhood of κ(E∗s ).

Notice that

[G(h), hP ] = −hi

Oph(Hp(σ(G(h))) +O(h2)Ψ−1+h

= ihOph(Hp(σ(G(h))) +O(h2)Ψ−1+h

,

in which

Hp(σ(G(h)) = Hp((1− χ(x, ξ))mG(x, ξ) log |ξ|)= Hp(1− χ(x, ξ))mG(x, ξ) log |ξ|+ (1− χ(x, ξ))Hp(mG(x, ξ)) log |ξ|

+(1− χ(x, ξ))mG(x, ξ)Hp(log |ξ|)= Hp(mG(x, ξ)) log |ξ|+O(1)S0

h.

From the construction of mG, we have that Hp(mG) ≤ 0. We see that =(z) ≥ −C0h so

=σh(Pδ,s(z)) = −qδ −=(z) + s=σ([G(h), hP ])

= −qδ −=(z) + shHp(σ(G(h)))

≤ −qδ + C0h+ shHp(σ(G(h)))

≤ 0 in S1h(X)/hS0

h(X).

Similar to Case 1, we have

‖Au‖HsG(h)= ‖esG(h)Au‖L2 = ‖esG(h)Ae−sG(h)esG(h)u‖L2 = ‖AsesG(h)u‖L2 ,

whereAs = esG(h)Ae−sG(h) ∈ Ψ0

h(X) and WFh(As) ⊂WFh(A).

By Propagation of singularities I in Proposition 2.19,

‖AsesG(h)u‖L2 ≤ C‖BesG(h)u‖L2 + C‖Bs2Pδ,s(z)esG(h)u‖L2 +O(h∞),

where Bs2 ∈ Ψ0

h(X) is microlocalized in a neighbourhood of ∪t∈[−T,0]etHp(WFh(A)). Putting

Bs := e−sG(h)BesG(h) and B2 := e−sG(h)Bs1esG(h),


so WFh(Bs) ⊂WFh(B) is near κ(E∗s ) and we can use Case 2 to get

‖Bsu‖HsG(h)≤ Ch−1‖B1f‖HsG(h)

+O(h∞).

Hence, we derive that

‖Au‖HsG(h)= ‖AsesG(h)u‖L2

≤ C‖BesG(h)u‖L2 + Ch−1‖esG(h)B2e−sG(h)Pδ,s(z)esG(h)u‖L2 +O(h∞)

= C‖esG(h)Bsu‖L2 + Ch−1‖esG(h)B2Pδ(z)u‖L2 +O(h∞)

= C‖Bsu‖HsG(h)+ Ch−1‖B2f‖HsG(h)

+O(h∞)

= Ch−1(‖B1f‖HsG(h)+ ‖B2f‖HsG(h)

) +O(h∞),

where B1 ∈ Ψ0h(X) is microlocalized near κ(E∗s ) and B2 ∈ Ψ0

h(X) is microlocalized in a neigh-bourhood of ∪t∈[−T,0]e

tHp(WFh(A)).• Case 4. WFh(A) is contained in a small neighbourhood of some (x0, ξ0) ∈ E∗u. Since E∗u

is a radial sink, there exists T > 0 such that e−THp(WFh(A)) ⊂ |ξ| < δ/2. Similar to Case3, we have

‖Au‖HsG(h)≤ C‖Bu‖HsG(h)

+ Ch−1‖B2f‖HsG(h)+O(h∞),

where WFh(B) ⊂ |ξ| < δ/2 and B2 ∈ Ψ0h(X) is microlocalized in a neighbourhood of

∪t∈[−T,0]etHp(WFh(A)). So ‖Bu‖HsG(h)

can be estimated by Case 1 and we have

‖Au‖HsG(h)≤ Ch−1(‖B1f‖HsG(h)

+ ‖B2f‖HsG(h)) +O(h∞),

• Case 5. WFh(A) is contained in a small neighbourhood of κ(E∗u). From the constructionof the anisotropic Sobolev space, since σh(e

sG(h)) = |ξ|−s near κ(E∗u), HsG(h) is microlocallyequivalent to the space H−sh near κ(E∗u).

Since E∗u is a radial sink we can apply Proposition 3.5 and obtain, for sufficiently large s,

‖Au‖HsG(h)≤ C‖Bu‖HsG(h)

+ Ch−1‖B1f‖HsG(h)+O(h∞),

where B,B1 ∈ Ψ0h(X) is microlocalized near κ(E∗u) and WFh(B)∩κ(E∗u) = ∅. Then ‖Bu‖HsG(h)

can be estimated by a combination of the preceding cases; this gives

‖Au‖HsG(h)≤ Ch−1‖Bf‖HsG(h)

+O(h∞).

Combining Cases 1-5, we have

‖u‖HsG(h)≤ Ch−1‖f‖HsG(h)

.

For the dynamics of −Hp, E∗s is a radial sink and E∗u is a radial source. Hence the proof of

(3.2) applies to −Pδ(z)∗ = −(hP − iQδ − z)∗, and we obtain the adjoint bound

‖v‖H−sG(h)≤ Ch−1‖Pδ(z)∗v‖H−sG(h)

. (3.3)

3.4.2. Invertibility. Injectivity of

Pδ(z) : DsG(h) → HsG(h), −C0h ≤ =(z) ≤ 1, |<(z)| ≤ hε,

follows the boundedness of the resolvent in (3.2) directly. To show the surjectivity, we first notethat (3.2) implies that if uj ∈ HsG(h) and Pδ(z)uj is a Cauchy sequence in HsG(h), then uj is aCauchy sequence in HsG(h) as well; since the operator Pδ(z) is closed on HsG(h) with domainDsG(h), we see that the image of Pδ(z) is a closed subspace of HsG(h). We need to show thatthis subspace is actually the whole HsG(h).

38 XIAOLONG HAN

Now, H−sG(h) is the dual to HsG(h) with respect to an inner product in L2. Let v ∈ H−sG(h)

be orthogonal to the range of Pδ(z). Then

〈Pδ(z)u, v〉 = 〈u, Pδ(z)∗v〉 = 0 for all u ∈ HsG(h),

But this means that Pδ(z)∗v = 0, and therefore v = 0 in the view of the boundedness of theadjoint operator in (3.3). Hence, Pδ(z) is surjective and invertible.

3.4.3. Wavefront set. The statement that

WFh(Rδ(z)) ∩ T ∗(X ×X) ⊂ ∆(T ∗X) ∪ Ω+,

where Ω+ is the positive flow-out of etHp on p = 0 is a rather standard result from Duistermaat-Hormander [DH]. The only conceptional difference is that here the semiclassical wavefront setis defined to include the fiber infinity S∗X, so the intersection with T ∗(X ×X) eliminates suchregions.

4. Meromorphic extension to the whole complex plane

Our plan of attack to prove the meromorphic extension of ζ(λ) to the whole complex planeis through

Trb(ϕ∗−t0(P− λ)−1

).

So we need to study the spectral properties of P on differential forms.

4.1. Ruelle-Pollicott resonances. First we need to transfer our knowledge on

Pδ(z) = hP− iQδ − z : DsG(h) → HsG(h), −C0h ≤ =(z) ≤ 1, |<(z)| ≤ h12 ,

toP− λ : DsG → HsG, λ = z/h.

We assume that λ varies in some compact subset of =(λ) ≥ −C0. Then

Pδ(z) = h(P− λ)− iQδ : DsG(h) → HsG(h)

is invertible and has bounded inverse by Proposition 3.6. Note also that for fixed h > 0, thetopology of HsG(h) is equivalent with the one of HsG. Hence,

P− λ =1

h(Pδ(z) + iQδ)

is a Fredholm operator of index 0, since Qδ is smoothing and therefore is compact. This isexactly

Proposition 4.1 (Fredholm property). Let C0 and s be the constants in Proposition 3.6. Then

P− λ : DsG → HsG

is a Fredholm operator of index 0 in the region =(λ) > −C0.

To apply the analytic Fredholm theory in Theorem 2.15, we also need to estimate P − λwhen =(λ) 1.

Proposition 4.2. Let C0 and s be the constants in Proposition 3.6. Then there exists C1

depending on s such that for =(λ) > C1,

P− λ : DsG → HsG

is invertible and

(P− λ)−1 = i

∫ ∞0

eiλtϕ∗−tf dt,


where ϕ∗−t : C∞(X; E) → C∞(X; E) is the pullback operator by φ−t on differential forms andthe integral on the right-hand side converges in the operator norm Hs → Hs and H−s → H−s.

Proof. The proof is easy considering that once =(λ) 1, then

ϕ∗t = O(eC1t) : H±s → H±s,

where C1 depends on s. Given u ∈ HsG ⊂ H−s and (P − λ)u = f ∈ HsG, recalling thatP = V/i, we have that

∂t(eiλtϕ∗−tu) = iλeiλtϕ∗−tu− eiλtV (ϕ∗−tu) = −ieiλtϕ∗−t((P − λ)u).

Hence,

u = −∫ ∞

0

∂t(eiλtϕ∗−tu) dt = i

∫ ∞0

eiλtϕ∗−t((P − λ)u) dt = i

∫ ∞0

eiλtϕ∗−tf dt,

where the integrals converge in H−s. This implies hat P − λ is injective from DsG to HsG andthus is invertible.

Now we denote R(λ) = (P − λ)−1 on HsG. The invertibility of P − λ for =(λ) 1 impliesthe following theorem from the analytic Fredholm theory in Theorem 2.15, since P − λ is anFredholm operator in =(λ) ≥ −C0.

Proposition 4.3. Let C0 and s be the constants in Proposition 3.6. Assume that =(λ0) > −C0.Then for λ near λ0,

R(λ) = RH(λ)−J(λ0)∑j=1

(P− λ0)j−1Π

(λ− λ0)j,

where RH(λ) is holomorphic near λ0, Π(λ0) : HsG → HsG is the commuting projection ontothe kernel of (P− λ0)J(λ0), and

WF′(RH(λ)) ⊂ ∆(T ∗X) ∪ Ω+ ∪ (E∗u × E∗s ), WF′(Π(λ0)) ⊂ (E∗u × E∗s ).

Proof. From Propositions 4.1 and 4.2, R(λ) is a family of meromorphic operators in =(λ) ≥−C0, using the analytic Fredholm theory in Theorem 2.15, we have that near λ0 with =(λ0) ≥−C0,

1

P − λ= R(λ) = RH(λ) +

J(λ0)∑j=1

Aj(λ0)

(λ− λ0)j,

where RH(λ) is holomorphic and Aj(λ0) are operators of finite rank, moreover,

A1(λ0) =1

2πi

∮λ0

1

P − λdλ and Π(λ0) := −A1(λ0) =

1

2πi

∮λ0

1

λ− Pdλ.

So [Π, P ] = 0 and using Cauchy’s theorem, Π2 = Π. We expand (P − λ)R(λ) = IdHsG and get

Id = (P − λ)R(λ)

= [(P − λ0)− (λ− λ0)]

RH(λ) +

J(λ0)∑j=1

Aj(λ0)

(λ− λ0)j

= (P − λ)RH(λ) +

J(λ0)∑j=1

(P − λ0)Aj(λ0)

(λ− λ0)j−

J(λ0)∑j=1

Aj(λ0)

(λ− λ0)j−1.

40 XIAOLONG HAN

Equating the powers of λ− λ0 we have

Aj(λ0) = (P − λ0)Aj−1(λ0) = −(P − λ0)j−1Π(λ0) for j = 2, ..., J(λ0),

and(P − λ0)AJ(λ0)(λ0) = −(P − λ0)J(λ0)Π(λ0) = 0,

that is, Π is the projection onto the kernel of (P − λ0)J(λ0).Recall that R(λ) = (P − λ)−1 = h(hP − z)−1 and Rδ(z) = (hP − iQδ − z)−1, where z = hλ.

Compute

h(Rδ(z)− iRδ(z)QδRδ(z))−Rδ(z)QδR(λ)QδRδ(z)

= hRδ(z)− ihRδQδ

(1− iQδ(hP − z)−1

)Rδ(z)

= hRδ(z)− ihRδQδ(hP − z)−1

= hRδ(z)(1− iQδ(hP − z)−1

)= h(hP − z)−1

= R(λ). (4.1)

Applying Proposition 3.6 that

WF′h(Rδ(z)) ∩ T ∗(X ×X) ⊂ ∆(T ∗X) ∪ Ω+,

we seeWF′h(Rδ(z)− iRδ(z)QδRδ(z)) ∩ T ∗(X ×X) ⊂ ∆(T ∗X) ∪ Ω+.

Here, we use the theorem on the wavefront set of composition of two operators in Hormander[H, Theorem 2.5.15] and the fact that Qδ ∈ Ψ0

h(X) is pseudodifferential.To handle the term Rδ(z)QδR(λ)QδRδ(z), note that if

(ρ′, ρ) ∈WF′h(Rδ(z)QδR(λ)QδRδ(z)) ∩ T ∗(X ×X),

thenρ′ ∈WF′h(Rδ(z)Qδ) and ρ ∈WF′h(QδRδ(z)).

But this just means that there exist t, s ≥ 0 such that

e−sHp(ρ′) ∈WFh(Qδ) and etHp(ρ) ∈WFh(Qδ).

Therefore,WF′h(Rδ(z)QδR(λ)QδRδ(z)) ∩ T ∗(X ×X) ⊂ Υδ,

whereΥδ := (ρ′, ρ) : ∃t, s ≥ 0 : etHp(ρ) ∈WFh(Qδ), e

−sHp(ρ′) ∈WFh(Qδ).Hence,

WF′h(R(λ)) ∩ T ∗(X ×X) ⊂ ∆(T ∗X) ∪ Ω+ ∪Υδ.

Since R(λ) is independent of δ,

WF′h(R(λ)) ∩ T ∗(X ×X) ⊂ ∆(T ∗X) ∪ Ω+ ∪⋂δ>0

Υδ = ∆(T ∗X) ∪ Ω+ ∪ (E∗u × E∗s ).

Here, we use the fact that if for all δ > 0, there exists s ≥ 0 such that e−sHp(ρ′) ∈WFh(Qδ) ⊂|ξ| ≤ δ, then ρ′ ∈ E∗u; similarly, if for all δ > 0, there exists t ≥ 0 such that etHp(ρ) ∈WFh(Qδ), then ρ ∈ E∗s .

Now R(λ) is independent of h, using the identity (2.3) that

WF(R(λ)) = WFh(R(λ)) ∩ (T ∗(X ×X) \ 0),


we derive thatWF′(R(λ)) ⊂ ∆(T ∗X) ∪ Ω+ ∪ (E∗u × E∗s ).

• λ is not a pole of R. So R(λ) = RH(λ) and Π(λ0) = 0.• λ is in a neighbourhood of a pole λ0 of R. So

R(λ) = RH(λ) +

J(λ0)∑j=1

Aj(λ0)

(λ− λ0)j,

and thus

(λ− λ0)J(λ0)R(λ) = (λ− λ0)J(λ0)RH(λ) +

J(λ0)∑j=1

Aj(λ0)(λ− λ0)J(λ0)−j,

From (4.1) we have

(λ− λ0)J(λ0)R(λ)

= (λ− λ0)J(λ0)RH(λ) +

J(λ0)∑j=1

Aj(λ0)(λ− λ0)J(λ0)−j

= h(λ− λ0)J(λ0)(Rδ(z)− iRδ(z)QδRδ(z))− (λ− λ0)J(λ0)Rδ(z)QδR(λ)QδRδ(z).

= h(λ− λ0)J(λ0)(Rδ(z)− iRδ(z)QδRδ(z))− (λ− λ0)J(λ0)Rδ(z)QδRH(λ)QδRδ(z)

−J(λ0)∑j=1

(λ− λ0)J(λ0)Rδ(z)QδAj(λ0)QδRδ(z). (4.2)

So similar to the previous case, one can derive that

WF′((λ− λ0)J(λ0)R(λ)) ⊂ ∆(T ∗X) ∪ Ω+ ∪ (E∗u × E∗s ).uniformly in λ near λ0.

Taking J(λ0) derivatives at λ = λ0 to (4.2) we have

WF′(RH(λ)) ⊂ ∆(T ∗X) ∪ Ω+ ∪ (E∗u × E∗s ).Taking J(λ0)− 1 derivatives at λ = λ0 to (4.2) we have

A1(λ0) = −Rδ(z)QδA1(λ0)QδRδ(z).

soWF′(Π(λ0)) = WF′(A1(λ0)) ⊂ E∗u × E∗s .

Definition (Ruelle-Pollicott resonances). The poles of R(λ), which are independent of s andG, are called Ruelle-Pollicott resonances.

4.2. Ruelle zeta functions around resonances. Recall that from (2.2)

fk(λ) = −eiλt0Trb(e−it0Pk(Pk − λ)−1

)= −eiλt0Trb

(e−it0PkRk(λ)

).

for =(λ) 1.For all λ ∈ C, the right-hand side of the above equation is well defined because WF(e−it0PkRk(λ))∩

∆(X ×X) = ∅ in Proposition 4.3 and Theorem 2.7. Then by an approximation argument in

42 XIAOLONG HAN

Lemma 2.9, the above equation holds for all λ ∈ C. Now near any resonance λ0, i.e. a pole ofRk,

Rk(λ) = RH,k(λ)−J(λ0)∑j=1

(Pk − λ0)j−1Πk

(λ− λ0)j,

where RH,k(λ) is holomorphic near λ0, Πk : HsG → HsG is the commuting projection onto thekernel of (P− λ0)J(λ0). So

Trb(e−it0PkRk(λ)

)= Trb

(e−it0PkRH,k(λ)

)− Trb

e−it0Pk J(λ0)∑j=1

(Pk − λ0)j−1Πk

(λ− λ0)j

.

The first term is holomorphic; we are therefore done if the second term only contributes asimple pole. This is because

Lemma 4.4. Suppose that a linear map A : Cm → Cm satisfies (A−λ0)J = 0 for some λ0 ∈ C.Then for ϕ holomorphic near λ0 we have

limλ→λ0

(λ− λ0)Tr

(ϕ(A)

J∑j=1

(A− λ0)j−1

(λ− λ0)j

)= mϕ(λ0).

Proof. Note that the traces of nilpotent operators are zero, so

Tr((A− λ0)j−1

)= 0 for j ≥ 2.

Hence,

limλ→λ0

(λ− λ0)Tr

(ϕ(A)

J∑j=1

(A− λ0)j−1

(λ− λ0)j

)

= limλ→λ0

(λ− λ0)Tr

(ϕ(A)

IdCm

λ− λ0

)= mϕ(λ0).

References

[DG] J. J. Duistermaat and V. Guillemin, The spectrum of positive elliptic operators and periodic bicharacter-istics. Invent. Math. 29 (1975), 39–79.

[DH] J. J. Duistermaat and L. Hormander, Fourier integral operators. II. Acta Math. 128 (1972), no. 3-4,183–269.

[DZ] S. Dyatlov and M. Zworski, Dynamical zeta functions for Anosov flows via microlocal analysis. Ann. Sci.

Ec. Norm. Super. (4) 49 (2016), no. 3, 543–577.[FT] Frederic Faure and Masato Tsujii, Semiclassial approach for the Ruelle-Pollicott spectrum of hyperbolic

dynamics. Online lecture notes.[GLP] P. Giulietti, C. Liverani, and M. Pollicott, Anosov flows and dynamical zeta functions. Ann. of Math.

(2) 178 (2013), no. 2, 687–773.[H] L. Hormander, Fourier integral operators. I. Acta Math. 127 (1971), no. 1-2, 79–183.[SS] E. S. Stein and R. Shakarchi, Complex analysis. Princeton Lectures in Analysis, II. Princeton University

Press, Princeton, NJ, 2003.[SWY] R. Schoen, S. Wolpert, and S. T. Yau, Geometric bounds on the low eigenvalues of a compact surface.

Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979), pp.279–285, Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, RI, 1980.

[Z] M. Zworski, Semiclassical analysis. American Mathematical Society, Providence, RI, 2012.


Department of Mathematics, The Australian National University, Canberra, ACT 2601,Australia

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