The trace formula for transversally elliptic operators on Riemannian foliations

18 Pages·0.22 MB·English

by  Yuri A. Kordyukov

The trace formula for transversally elliptic operators on
0
Riemannian foliations
0
0
2

n Yuri A. KORDYUKOV
a
J Department of Mathematics,

1
3 Ufa State Aviation Technical University,

K.Marx str. 12, Ufa 450000,Russia
]
G
e-mail: [email protected]
D
.
h September 1, 1999
t
a
m
[
1 Introduction

1
v
2 The main goal of the paper is to generalize the Duistermaat-Guillemin trace formula to the
8
case of transversally elliptic operators on a compact foliated manifold. First, let us recall
1
1 briefly the setting of the classical formula.
0
Let P be a positive self-adjoint elliptic pseudodifferential operator of order one on a
0
0 closed manifold M (for example, P = √∆, where ∆ is the Laplace-Beltrami operator of a
h/ Riemannian metric on M). For any function f ∈ Cc∞(R), the operator Uf = f(t)eitPdt
t can be shown to be of trace class, and the mapping θ : f trU is a continRuous linear
a f
m functional on C∞(R). Otherwise speaking, θ is a distributio7→n on R. The principal symbol
c
: p of the operator P is a smooth function on the symplectic manifold T∗M 0, and, by
v
\ ∗
i definition, the bicharacteristic flow f of the operator P is the Hamiltonian flow on T M 0
X t \
defined by the function p.
r
a By the theorem due to Colin de Verdi´ere and Chazarain [1, 2], the singularities of the
distribution θ are contained in the period set of closed trajectories of the bicharacteristic
flow f . Moreover, Duistermaat and Guillemin showed [3] that, under the assumption that
t
the bicharacteristic flow is clean, one can write down an asymptotic expansion for the dis-
tribution θ near a given period of closed bicharacteristic. A formula for the leading term of
this asymptotic expansion is the Duistermaat-Guillemin trace formula mentioned above. It
involves the geometry of the bicharacteristic flow in the form of Poincar´e map and Maslov
indices and provides a far-reaching generalization of the classical Poisson formula and the
Selberg trace formula on hyperbolic spaces.
In the paper, we prove a trace formula for an operator P = √A, where A is a positive
self-adjoint transversally elliptic pseudodifferential operator of order two with the positive,
holonomy invariant transverse principal symbol on a compact foliated manifold (M, ) (see
F
Theorem 6). One can consider such an operator as an elliptic operator on the singular
1
space M/ of leaves of the foliation (this statement can be made more precise, using
F F
the language of noncommutative geometry, see [4]) the trace formula stated in this paper
as an example of a trace formula for elliptic operators on singular spaces. We hope that
this formula will be useful in further study of a general trace formula in noncommutative
geometry (see, for instance, [5, 6] for discussion of a noncommutative trace formula).
It should be also noted that our trace formula can be viewed as a relative version of the
Duistermaat-Guillemin trace formula.
2 Preliminaries and main results
Let (M, ) be a compact, connected, oriented foliated manifold. We will use the following
F ∗
notation: T is the tangent bundle, H = TM/T is the normal bundle and N is the
F F F F
conormal bundle to . There is a short exact sequence
F
0 T TM H 0. (2.1)
−→ F −→ −→ F −→
We will consider linear operators, acting on half-densities. Recall that an α-density
(α R) on a real vector space V of dimension n is a map φ : ΛnV R such that φ(λv) =
∈ →
λ αφ(v),v ΛnV,λ R. For any real vector bundle E over a smooth manifold X, we will
| | ∈ ∈
denote by E α the α-density bundle of E.
| |
Given a pseudodifferential operator A Ψm(M, TM 1/2), the transversal principal
∈ | | ˜∗ ∗
symbol σ of Aisdefined to betherestriction of itsprincipal symbol a onN = N 0.
A m
F F\
An operator A Ψm(M, TM 1/2) is said to be transversally elliptic, if σ (ν) = 0 for any
A
ν N˜∗ . ∈ | | 6
∈ F
For any smoothleafwise path γ fromx M to y M, sliding along leaves of thefoliation
∈ ∈
defines the holonomy map h , which associates to every germ of a local transversal to the
γ
foliation at the point x a germ of a local transversal to the foliation at the point y (this map
is a natural generalization of the Poincar´e first-return map for flows). The differential of this
map (the linear holonomy map) is well-defined as a linear map dh : H H and the
γ x y
∗ ∗ ∗ F → F
codifferential as a linear map dh : N N .
γ yF → xF
The transversal principal symbol σ of an operator A Ψm(M, TM 1/2) is said to be
A
∗ ∈ | |
holonomy invariant, if σ (dh (ν)) = σ (ν) for any smooth leafwise path γ from x to y
A γ A
and for any ν N˜∗ .
∈ yF
Throughout in the paper, we will assume that A is a linear operator in C∞(M, TM 1/2),
| |
satisfying the following conditions:
(A1) A Ψ2(M, TM 1/2) is a transversally elliptic operator with the positive, holonomy
∈ | |
invariant transversal principal symbol;
(A2) A is an essentially self-adjoint positive operator in L2 space of half-densities on M,
L2(M) (with the initial domain C∞(M, TM 1/2)).
| |
Example 1. Ageometricalexample ofanoperator,satisfying theconditions(A1) and(A2),
is given by the operator A = I+∆ , where ∆ is the transversal Laplacian of a bundle-like
H H
metric on a Riemannian foliation.
2
Recall that a foliation on a smooth Riemannian manifold (M,g ) is Riemannian if
M
F
it satisfies one of the following equivalent conditions (see, for instance, [7]):
1. (M, ) locally has the structure of Riemannian submersion;
F
⊥
2. the transverse part of the Riemannian metric g (that is, its restriction to H = T )
M
F
is holonomy invariant;
3. the horizontal distribution H is totally geodesic.
In this case, the metric g is called bundle-like.
M
∗
The Riemannian metric g defines a decomposition of the cotangent bundle T M into a
M
∗ ∗ ∗
direct sum T M = F H . With respect to this decomposition, the de Rham differential
∞ ∞ ⊕∗ ∞
d : C (M) C (M,T M) can be written as a sum d = d +d , where d : C (M)
F H F
∞ ∗ → ∞ ∞ ∗ →
C (M,F ) and d : C (M) C (M,H ).
H
→
The transversal Laplacian is a second order transversally elliptic differential operator in
∞
the space C (M) defined by the formula
∗
∆ = d d .
H − H H
Its principal symbol a is given by the formula
2
a (x,ξ) = g (ξ,ξ), (x,ξ) T˜∗M,
2 H
∈
and the holonomy invariance of the transverse principal symbol σ is equivalent to the
∆H
assumption on the Riemannian metric g to be bundle-like.
M
From now on, we will assume that A satisfies the assumptions (A1) and (A2). By
the spectral theorem, the operator P = √A generates a strongly continuous group eitP of
bounded operators in L2(M). To define a distributional trace of the operator eitP, one need
an additional regularization. First, let us introduce some notation.
Recall that the holonomy groupoid G = G of the foliation is the set of equivalence
F
F
classes of leafwise paths γ : [0,1] M with respect to an equivalence relation , setting
h
→ ∼
γ γ if γ and γ have the same initial and final points and the same holonomy maps.
1 h 2 1 2
∼
G is equipped with maps s,r : G M given by s(γ) = γ(0) and r(γ) = γ(1) and has a
→
composition law given by the composition of paths. For any γ ,γ G, the composition
1 2
γ γ makes sense iff r(γ ) = s(γ ). We will make use of standard n∈otation: Gx = r−1(x),
1 2 2 1
G ◦= s−1(x), Gx = s−1(x) r−1(x), x M. For any x M, Gx is the holonomy group of
x x ∩ ∈ ∈ x
the leaf L through the point x and the maps s : Gx L and r : G L are covering
x x x x
→ →
maps associated with Gx. We will identify a point x M with the element in G given by
x ∈
the constant path γ(t) = x,t [0,1].
Let s∗( T 1/2) and r∗( T ∈ 1/2) be the lifts of the vector bundle of leafwise half-densities
| F| | F|
T 1/2 to vector bundles on G via the mappings s and r respectively, and T 1/2 =
r|∗(FT| 1/2) s∗( T 1/2). The line bundle T 1/2 is the bundle of leafwise hal|f-dGe|nsities
| F| ⊗ | F| | G|
on G with respect to the natural foliation [8].
The space C∞(G, T 1/2) has the strucGture of involutive algebra (see, for instance, [8]).
There is a natucral -r|epGr|esentation R of the involutive algebra C∞(G, T 1/2) in L2(M).
∗ c | G|
3
For any k C∞(G, T 1/2), the operator R(k) in L2(M) is defined as follows. According to
∈ c | G|
the short exact sequence (2.1), the half-density vector bundle TM 1/2 can be decomposed
| |
as
TM 1/2 = T 1/2 H 1/2.
∼
| | | F| ⊗| F|
For any γ G,s(γ) = x,r(γ) = y, the corresponding linear holonomy map defines a map
∈
dh∗ : H 1/2 H 1/2.
γ | yF| → | xF|
Given u L2(M) of the form u = u u ,u L2(M, T 1/2),u L2(M, H 1/2),
1 2 1 2
∈ ⊗ ∈ | F| ∈ | F|
R(k)u L2(M) is defined by the formula
∈
∗ ∗
R(k)u(x) = k(γ)s u (γ) dh [u (s(γ))], x M.
Z 1 ⊗ γ−1 2 ∈
Gx
Proposition 2. Foranyk C∞(G, T 1/2) and f C∞(R), the operatorR(k) f(t)eitPdt
is of trace class. Moreover,∈for cany k| GC|∞(G, T ∈1/2),cthe formula R
∈ c | G|
θ ,f = trR(k) f(t)eitPdt, f C∞(R),
h k i Z ∈ c
defines a distribution θ on the real line R, θ ′(R).
k k
∈ D
Let be a foliation in N˜∗ , which is the horizontal foliation for the natural leafwise
N
flat connFection in N˜∗ (the BotFt connection). The leaf of the foliation through a point
N
ν N˜∗ is the set ofFall dh∗(ν) N˜∗ such that γ G,r(γ) = π(ν), wFhere π : N∗ M
∈ F γ ∈ F ∈ F →
is the natural projection.
∗ ∗
Denote by H the normal bundle to , H = T(N )/T . For any ν N ,
N N N N
∗ F F F ∗F F ∈ F
the space T N is a coisotropic subspace of the space T T M equipped with the canoni-
ν ν
F
cal symplectic structure, and T its skew-orthogonal complement, therefore, the normal
ν N
F
bundle H has a natural symplectic structure (see, for instance, [9]).
ν N
F
Given an operator A under the conditions (A1) and (A2) with the principal symbol a, let
p˜be a smooth function on T˜∗M homogeneous of degree one such that p˜(ξ) = 0 for ξ T˜∗M,
which is equal to p = a1/2 in some conical neighborhood of N∗ , and f˜6 the Ham∈iltonian
t
∗F 1/2
flow of the function p˜. Define σ to be the restriction of p on N : σ = σ . The function
P F P A
σ coincides with the transverse principal symbol of any operator P Ψ1(M, TM 1/2) such
P 1
that the principal symbols of P2 and A are equal on N∗ . ∈ | |
1 F
The holonomy invariance assumption on σ implies
A
dp˜(ν)(X) = 0, ν N˜∗ , X T . (2.2)
ν N
∈ F ∈ F
˜ ∗
Using (2.2) and the fact that f preserves the symplectic structure of T M, one can easily
t
˜ ∗
check that the Hamiltonian flow f can be restricted on N . The resulting flow will be
t
F
denoted by f . By definition, the flow f depends only on the 1-jet of the principal symbol
t t
∗
a on N , therefore, it doesn’t depend on a choice of p˜ and can be naturally called the
F
transverse bicharacteristic flow of the operator A.
4
˜ ∗
Since f preserves the symplectic structure of T M and T is the skew-adjoint comple-
t N
∗ F
ment to TN , f maps leaves of the foliation to leaves. In particular, the differential
t N
F F
df defines a map T T and a symplectic map H H .
t νFN → ft(ν)FN νFN → ft(ν)FN
We say that a point ν N˜∗ is a relative fixed point of the diffeomorphism f :
t
N˜∗ N˜∗ (with respect∈to theFfoliation ), if there exist γ G such that r(γ) = π(ν)
N
F → ∗F F ∈
and f dh (ν) = ν.
−t γ
For any t R, denote by Z the set of relative fixed points of f . We also introduce
t t
∈ ∗ ∗
the corresponding set in the cospherical bundle SN = ν N : σ (ν) = 1 : SZ =
P t
Z SN∗ .Thissetmightbenotclosed, but, foranykF C∞{(G∈, T F1/2)),thecorre}sponding
t∩ F ∈ c | G|
part
∗ ∗
SZ = ν SN : ( γ suppk,r(γ) = π(ν))f dh (ν) = ν
t,k { ∈ F ∃ ∈ −t γ }
is closed. By the tranversal ellipticity of σ , the flow f is transverse to , therefore, the
A t N
F
relative period set = t R : SZ = ∅ is a discrete subset of R.
k t,k
T { ∈ 6 }
The following theorem was proved in [10], but we will give an independent proof.
Theorem 3. Given anoperatorAundertheconditions(A1) and(A2) and k C∞(G, T 1/2),
∈ c | G|
the distribution θ is smooth outside of the relative period set of the transverse bicharac-
k k
T
terictic flow f .
t
Let G denote the holonomy groupoid of the foliation . G consists of all pairs
(γ,ν) GFN N˜∗ such that r(γ) = π(ν) with the source mapFsN : GFN N˜∗ ,s (γ,ν) =
dh∗(ν)∈, anFd×the Ftarget map r : G N˜∗ ,r (γ,ν) = ν.NTherFeNis→a prFojecNtion π :
γ N FN → F N G
GFN → GF given by the formula πG(γ,ν) = γ. Put also GSN∗F = GFN ∩rN−1(SN∗F).
For any (γ,ν) G , denote by dH the associated linear holonomy map:
∈ FN (γ,ν)
dH(γ,ν) : Hdh∗(ν) N Hν N.
γ F → F
It is easy to see that dH preserves the symplectic structure of H .
(γ,ν) N
∗ F
Denote by Q : TN H the projection map. The differential of the map (r ,s ) :
N N N
G N˜∗ N˜∗ dFefi→nes aFn isomorphism of the tangent space T G with the set
FN → F × F∗ ∗ (γ,ν) FN
of all (v1,v2) TνN Tdh∗(ν)N such that the normal components of v1 and v2 are
∈ F ⊕ γ F
connected by the holonomy map: Q(v ) = dH (Q(v )).
1 (γ,ν) 2
The holonomy groupoid G has the natural foliation such that the tangent bundle
F F
N G N
T(γ,ν)GFN correspondstoTνFN⊕Tdh∗γ(ν)FN under theisomorphism described above. The nor-
mal space H(γ,ν)GFN to GFN is isomorphic to the set of all (v1,v2) ∈ HνFN ⊕Hdh∗γ(ν)FN such
that v = dH (v ), and therefore the maps dr (ds ) define isomorphisms of H
1 (γ,ν) 2 N N (γ,ν)GFN
with Hν N (Hdh∗(ν) N) respectively.
F γ F
∗
Lemma 4. The set Z is a saturated subset of N , that is, it is a union of leaves of the
t
F
foliation .
N
F
Proof. By the holonomy invariance of p, the Hamiltonian vector field Ξ with the Hamilto-
p
nian p satisfies the identity
∗ ˜∗
dH (Q(Ξ (dh (ν)))) = Q(Ξ (ν)), ν N , γ G, r(γ) = π(ν),
(γ,ν) p γ p ∈ F ∈
5
therefore, there exists a vector field Ξˆ on G such that
p FN
ds (Ξˆ (γ,ν)) = Ξ (dh∗(ν)), dr (Ξˆ (γ,ν)) = Ξ (ν), (γ,ν) G . (2.3)
N p p γ N p p ∈ FN
Let Fˆ be a flow on G generated by Ξˆ . By (2.3), we have
t FN p
f r = r Fˆ, f s = s Fˆ,
t N N t t N N t
◦ ◦ ◦ ◦
or, if we write Fˆ : G G as Fˆ(γ,ν) = (F (γ,ν),f (ν)),
t FN → FN t t t
∗ ∗
f (dh (ν)) = dh (f (ν)). (2.4)
t γ Ft(γ,ν) t
∗
Take any ν Z with the corresponding γ G such r(γ) = π(ν),f dh (ν) = ν. Let
∈ t ∈ −t γ
(γ ,ν) G . Then we have
1 ∈ FN
∗ ∗
f (dh (ν)) = dh (f (ν)) by (2.4)
t γ1 Ft(γ1,ν) t
∗ ∗
= dh (dh (ν))
Ft(γ1,ν) γ
∗ ∗ ∗ ∗
= (dh dh dh )(dh (ν))
Ft(γ1,ν) ◦ γ ◦ γ1−1 γ1
∗ ∗
= dh (dh (ν)),
γ′ γ1
where γ′ = γ−1 γ F (γ ,ν) that implies dh∗ (ν) Z .
1 ◦ ◦ t 1 γ1 ∈ t
The relative fixed point sets Z can be naturally lifted to the holonomy groupoid G :
t FN
∗
Zt = {(γ,ν) ∈ GFN : f−t dhγ(ν) = ν}, SZt = Zt ∩GSN∗F,
By Lemma 4, Z = r ( ) = s ( ).
t N t N t
Z Z
Let us assume that is a smooth submanifold of G . By Lemma 4, the tangent space
Zt FN
to at a point (γ,ν) contains a subspace F , which is the graph of the linear
t t (γ,ν) t
Z ∈ Z Z
map dft(ν) : TνFN → Tdh∗γ(ν)FN = Tft(ν)FN:
F(γ,ν) t = (v1,v2) Tν N Tdh∗(ν) N : v2 = dft(ν)(v1) .
Z { ∈ F × γ F }
Let
H = T /F , H S = T S /F .
(γ,ν) t (γ,ν) t (γ,ν) t (γ,ν) t (γ,ν) t (γ,ν) t
Z Z Z Z Z Z
Definition 5. Let t R be a relative period of the flow f . We say that the flow f is clean
t t
∈
on , if:
t
Z
(1) is a smooth submanifold of G ;
Zt FN
(2) thenormalspace H at anypoint (γ,ν) coincides withthe set ofall(v ,v )
(γ,ν) t t 1 2
Z ∈ Z ∈
H such that v = df (ν)(v ).
(γ,ν)GFN 2 t 1
6
Let T 1/2 be the vector bundle of leafwise half-densities on N∗ , and s∗ ( T 1/2)
and r∗ (|TFN|1/2) are the lifts of this vector bundle to vector bundles oFn G vNia|thFeNm|ap-
N | FN| FN
pings s and r respectively. Let T 1/2 be the vector bundle of leafwise half-densities
N N | GFN|
on G :
F
N
T 1/2 = r∗ ( T 1/2) s∗ ( T 1/2).
| GFN| N | FN| ⊗ N | FN|
The projection π : G G defines a local diffeomorphism π : , that induces a
G FN → G GFN → G
map
π∗ : C∞(G, T 1/2) C∞(G , T 1/2).
G c | G| → FN | GFN|
Define a restriction map
R : C∞(G , T 1/2) C∞( , T 1)
Z c FN | GFN| → c Zt | FN|
as follows. If ρ = fr∗ ρ s∗ ρ , f C∞(G ), ρ ,ρ C∞(M, T 1/2), then
N 1 ⊗ N 2 ∈ c FN 1 2 ∈ c | FN|
∗ ∗
R ρ(γ,ν) = f(γ,ν)ρ (γ,ν)df (ν)[ρ (dh (ν))], (γ,ν) ,
Z 1 t 2 γ ∈ Zt
where the map df∗(ν) : T 1/2 T 1/2 is induced by the linear map df (ν) :
t | ft(ν)FN| → | νFN| t
T T .
νFN → ft(ν)FN
If the flow f is clean, there is defined a natural density dµ on H , being the fixed
t Z (γ,ν) t
Z
point set of the symplectic linear map dH df (ν) of the symplectic space H (see, for
(γ,ν) t ν N
◦ F
instance, [3, Lemma 4.3]). Dividing dµ by dσ , we get a density dµ on H S .
Z P SZ (γ,ν) t
Z
Using the natural isomorphism
TS = F HS .
t ∼ t
| Z | | Z|⊗| Z |
∗ ∞ ∞
one can combine the densities R π k C (S , FS ) and dµ C (S , HS ) to
∗ Z G ∈ c Zt | Zt| SZ ∈ c Zt | Zt|
get a smooth density R π kdµ on S .
Let σsub(A) denote tZheGsubprSiZncipal sZytmbol of A. Define σsub(P) = 21a−21σsub(A) in some
∗ ∗
conic neighborhood of N . The restriction of σ (P) on N is equal to the restriction
sub
on N∗ of the subprincipFal symbol of any operator P Ψ1F(M, TM 1/2) such that the
1
compleFte symbols of P2 and A are equal mod S−∞ in som∈e neighb|orho|od of N∗ .
1 F
Theorem 6. Let t R be a relative period of the flow f . Assume that the relative fixed
t
point set is clean.∈Then, for any k C∞(G, T 1/2) and for any τ in some neighborhood
Zt ∈ c | G|
of t, we have
+∞
θ (τ) = α (s,k)eis(τ−t)ds, (2.5)
k Z j
X −∞
Z
j
where:
1. j are all connected components of the set S t in GSN∗F of dimensions dj = dim j;
Z Z Z
7
2. α has an asymptotic expansion
j
α (s,k) s (dj−p−1)/2i−σj +∞ α (k)s−r, s + (2.6)
j j,r
∼ (cid:16)2πi(cid:17) X → ∞
r=0
with α given by the formula
j,0
αj,0(k) = Z eiR0tσsub(P)(f−τdh∗γ(ν))dτRZπG∗k(γ,ν)dµSZj(γ,ν), (2.7)
Z
j
where σ denotes the Maslov index associated with the connected component (see
j j
Z
below for the definition).
3 Reduction to the case when A is elliptic
In this section, we will assume that A is an operator under the assumptions (A1) and (A2).
We will use the classes Ψm,−∞(M, , TM 1/2) of transversal pseudodifferential operators
F | |
(see [4] for the definition) and the Sobolev spaces Hs(M) of half-densities on M. Put also
Ψ∗,−∞(M, , TM 1/2) = Ψm,−∞(M, , TM 1/2).
F | | m F | |
By [4], the operator PS= A1/2 satisfies the following conditions:
(H1) P has the form
P = P +R ,
1 1
where:
(a) P Ψ1(M, TM 1/2) is a transversally elliptic operator with the positive, holonomy
1
∈ | |
invariant transversal principal symbol such that the complete symbols of P2 and A are equal
1
−∞ ∗
mod S in some neighborhood of N ;
F
(b)R isaboundedoperatorfromL2(M)toH−1(M)andforanyK Ψ∗,−∞(M, , TM 1/2)
1
∈ F | |
the operator KR is a smoothing operator in L2(M), that is, it defines a bounded operator
1
from L2(M) to C∞(M, TM 1/2).
| |
(H2) P is essentially self-adjoint in L2(M) (with the initial domain C∞(M, TM 1/2)).
| |
Lemma 7 ([10]). Any operator P, satisfying the conditions (H1) and (H2), can be repre-
sented in the form
P = P +R , (3.1)
2 2
where:
(a) P Ψ1(M, TM 1/2) is an essentially self-adjoint, elliptic operator with the posi-
2
∈ | |
tive principal symbol and the holonomy invariant transversal principal symbol such that the
−∞ ∗
complete symbols of P and P are equal mod S in some neighborhood of N ;
1 2
F
(b) R is a boundedoperatorfrom L2(M) to H−1(M) and, foranyK Ψ∗,−∞(M, , TM 1/2),
2
∈ F | |
the operator KR is a smoothing operator in L2(M).
2
8
Proof. Take a foliated coordinate chart Ω on M with coordinates (x,y) Ip Iq (I is the
∈ ×
open interval (0,1)) such that the restriction of on U is given by the sets y = const. Let
F
p S1(In Rn) be the complete symbol of the operator P in this chart. Assume that
1 1
∈ ×
p (x,y,ξ,η) is invertible for any (x,y,ξ,η) U, ξ 2 + η 2 > R2, where R > 0, U is a conic
1
neighborhood of the set η = 0. Take any f∈uncti|on| φ | |C∞(In Rn),φ = φ(x,y,ξ,η),x
∈ × ∈
Ip,y Iq,ξ Rp,η Rq, homogeneous of degree 0 in (ξ,η) for ξ 2 + η 2 > 1, which is
∈ ∈ ∈ | | | |
supported in some conic neighborhood of η = 0 and is equal to 1 in U, and put
p (x,y,ξ,η) = φp (x,y,ξ,η)+(1 φ)(1+ ξ 2 + η 2)1/2.
2 1
− | | | |
Take P to be the operator p (x,y,D ,D ) with the complete symbol p (or, more precisely,
2 2 x y 2
∗
p (x,y,D ,D ) + p (x,y,D ,D ) to provide self-adjointness) and put R = P P . The
2 x y 2 x y 2 2
∗ −
operator P P has order in some conic neighbourhood of N , therefore, for any K
1 2
Ψ∗,−∞(M, −, TM 1/2) the−op∞erator K(P P ) is a smoothing opeFrator [4], that complete∈s
1 2
F | | −
immediately the proof.
Denote by W(t) = eitP2 the wave group generated by the elliptic operator P . It is
2
well-known that W(t) is a Fourier integral operator (see below for more details). Put also
R(t) = eitP W(t).
−
Proposition 8. For any K Ψ∗,−∞(M, , TM 1/2), the family KR(t),t R, is a smooth
family of bounded operators f∈rom L2(M) tFo C| ∞(M| , TM 1/2). ∈
| |
Proof. Since P2 = A Ψ2(M, TM 1/2), by interpolation and duality, P defines a bounded
operator from H1(M)∈to L2(M|) an|d from L2(M) to H−1(M) and, for any natural N, PN
defines a bounded operator from HN(M) to L2(M) and from L2(M) to H−N(M). Since
R = P P and P Ψ1(M, TM 1/2), R also defines a bounded operator from H1(M) to
2 2 2 2
L2(M) a−nd from L2(M∈ ) to H−|1(M|).
By assumption, for any K Ψ∗,−∞(M, , TM 1/2), the operator KR is a smooth-
2
∈ F | |
ing operator, therefore, the operator KR PN is defined as an operator from HN(M) to
2
C∞(M, TM 1/2).
| |
Lemma 9. For any K Ψ∗,−∞(M, , TM 1/2) and for any N N, the operator KR PN
2
extends to a bounded ope∈rator from LF2(M| ) to| C∞(M, TM 1/2). ∈
| |
Proof. We will prove the lemma by induction on N. For N = 0, the statement is true by as-
sumption. LetusassumethatitistrueforsomeN,thatis,foranyK Ψ∗,−∞(M, , TM 1/2),
the operator KR PN extends to a bounded operator from L2(M) to∈C∞(M, TMF1/|2). |
2
We have the equality P2 P2 = R P + P R as operators from H1(M|) to|H−1(M),
therefore, for any K Ψ∗,−∞(−M,2 , TM2 1/2), 2 2
∈ F | |
KR PN+1 = K(R P)PN = K(P2 P2)PN KP R PN.
2 2 − 2 − 2 2
The operator P2 P2 Ψ2(M, TM 1/2) has order in some conic neighbourhood of
N∗ , therefore, fo−r an2y∈K Ψ∗,−|∞(M|, , TM 1/2) th−e∞operator K(P2 P2) extends to a
bouFnded operator from Hs(∈M) to C∞(MF,|TM|1/2) for any s and K(P2− P2 2)PN extends
to a bounded operator from L2(M) to C∞(|M, T| M 1/2). − 2
| |
9
Since P Ψ1(M, TM 1/2) and K Ψ∗,−∞(M, , TM 1/2), by the composition theorem
2
[4], KP ∈Ψ∗,−∞(M|, , |TM 1/2) and∈, by inductFion| hyp|othesis, KP R PN extends to a
2 2 2
bounded o∈perator fromFL2|(M)|to C∞(M, TM 1/2).
| |
By the Duhamel formula, we have
t
R(t)u = i eiτP2R ei(t−τ)Pudτ, u H1(M) D(P),
Z 2 ∈ ⊂
0
therefore, for any K Ψ∗,−∞(M, , TM 1/2),
∈ F | |
t t
KR(t) = i KeiτP2 R ei(t−τ)P dτ = i eiτP2e−iτP2KeiτP2 R ei(t−τ)P dτ.
Z 2 Z 2
0 0
Any operator K Ψ∗,−∞(M, , TM 1/2) is a Fourier integral operator (see below for more
∈ F | |
details) and,using thecompositiontheoremforFourierintegraloperators,onecancheck that
e−iτP2KeiτP2 Ψ∗,−∞(M, , TM 1/2). Therefore, the operator e−iτP2KeiτP2R extends to
2
a bounded op∈erator from LF2(|M) t|o C∞(M, TM 1/2). Since eiτP2 maps C∞(M, TM 1/2) to
C∞(M, TM 1/2) and, by the spectral theor|em,|ei(t−τ)P is a bounded operator| in L|2(M),
for any|K | Ψ∗,−∞(M, , TM 1/2), the operator KR(t) extends to a bounded opera-
tor from L2(∈M) to C∞(MF, |TM |1/2). Moreover, one can be easily seen from above argu-
| |
ments that the function KR(t) is continuous as a function on R with values in the space
(L2(M),C∞(M, TM 1/2)) of bounded operators from L2(M) to C∞(M, TM 1/2).
L | | | |
For any u H1(M), the function R H : t KR(t)u is differentiable, and
∈ → 7→
d
KR(t)u = iK(PeitPu P eitP2u) = i(KP R(t)+KR eitP)u.
2 2 2
dt −
TheoperatorKP R(t)+KR eitP extendstoaboundedoperatorfromL2(M)toC∞(M, TM 1/2),
2 2
| |
and, moreover, the function t KP R(t) + KR eitP is a continuous function on R with
2 2
values in (L2(M),C∞(M, TM7→1/2)). Using this, one can be easily seen that the function
t KR(Lt) is differentiable|as a|function on R with values in (L2(M),C∞(M, TM 1/2))
7→ L | |
and
d
KR(t) = i(KP R(t)+KR eitP).
2 2
dt
Let us proceed by induction. Assume that, for any K Ψ∗,−∞(M, , TM 1/2) and for
∈ F | |
some natural n, the function KR(t) is n-times differentiable as a function on R with values
in (L2(M),C∞(M, TM 1/2)) and the derivative KR(n)(t),t R satisfies the equation
L | | ∈
KR(n)(t) = iKP R(n−1)(t)+inKR Pn−1eitP. (3.2)
2 2
To prove that thefunction t KR(n)(t) isdifferentiable asa functiononRwith values in
(L2(M),C∞(M, TM 1/2)),as7→above,itsufficestoprovethatthederivative(d/dt)KR(n)(t)u
L | |
exists for any u from a dense subspace of L2(M), it extends to a bounded operator from
L2(M) to C∞(M, TM 1/2), and its extension is continuous as a function on R with values
in (L2(M),C∞(M| , T|M 1/2)).
L | |
10