Injectivity and stability for a generic class of generalized Radon transforms

0.17 MB·English

by  Andrew Homan

Injectivity and stability for a generic class of
5
1 generalized Radon transforms
0
2
y Andrew Homan and Hanming Zhou
a
M
Abstract
5
Let(M,g)beananalytic,compact,Riemannianmanifoldwithbound-

] ary, of dimension n ≥ 2. We study a class of generalized Radon trans-
G
forms, integrating over a family of hypersurfaces embedded in M, satis-
D fying the Bolker condition [23]. Using analytic microlocal analysis, we
. prove a microlocal regularity theorem for generalized Radon transforms
h
on analytic manifolds definedon an analyticfamily of hypersurfaces. We
t
a then show injectivity and stability for an open, dense subset of smooth
m generalized Radon transforms satisfying the Bolker condition, including
[ the analytic ones.

2
v 1 Introduction
0
1
Let (M,g) be an analytic, compact, Riemannian manifold with boundary, of
5
dimension n≥2, with volume form denoted by dVol. Let Σ be a family of em-
6
0 bedded hypersurfaces. A generalized Radon transform takes each f ∈ C∞(M)
. to the set of its integrals over the hypersurfaces of Σ, with respect to the sur-
2
0 face measure induced by the volume form. Often Σ is itself a smooth manifold;
5 for example, the Euclidean Radon transform is defined over the set of affine
1 hypersurfaces in Rn, which has R×Sn−1 as a double cover. Such transforms
:
v are found in applications to many other fields, including harmonic analysis,
i scattering theory, seismology and medical imaging.
X
The main questions regarding these transforms include determining condi-
r
a tions under which they are injective, finding when the transform has a stable
inversion, and characterizing the range. We concentrate on the first two ques-
tions here. These questions are also important in the partial data case, where
integrals are known only for a subset of Σ. In the case of the Euclidean Radon
transform, we refer the reader to [16, 22, 8, 17] and references therein for the
resolution of these problems, and their generalization to the Radon transform
over symmetric spaces and other related contexts.
The geometricdata ofa generalizedRadontransformcanbe encodedby an
incidencerelationbetweenpointsonM andthehypersurfacesinΣthatcontain
them. Let Λ⊂M ×Σ be this relation, i.e., the set of ordered pairs (x,σ) such
that x ∈ σ. One says Λ is a double fibration when it is a smooth, embedded
1
submanifold of M ×Σ such that both canonical projections are smooth and
their restrictions to Λ form a fiber bundle over M and Σ, respectively [10].
Guillemin and Sternberg [13, 12, 14] showed that given a Radon transform
R defined by a double fibration Λ, both R and its adjoint R∗ (often called
the generalizedbackprojectionby analogywith the Euclidean case) are Fourier
integraloperators,and the canonicalrelationof R is the conormalbundle N∗Λ
of the incidence relation. If in addition Λ satisfies the Bolker condition, which
says that the induced projection π∗ : N∗Λ → T∗Σ is an embedding, then the
normal operator R∗R is an elliptic pseudodifferential operator, which yields
invertibility up to smoothing error.
A stronger result than invertibility is a Helgason-type support theorem, by
analogy with that of the Euclidean Radon transform [15]. Such a support the-
orem implies that if f is a priori of compact support, and Rf = 0 for all
hypersurfaces intersecting the support, then f = 0. In the analytic category,
there has been much work in this area (for example, [3, 4, 23, 24]) using an-
alytic microlocal analysis and the Bolker condition to prove support theorems
for analytic generalized Radon transforms (i.e., with M,Σ analytic manifolds)
withnonvanishinganalyticweight. Ontheotherhand,inn=2thereisacoun-
terexampleinthe smoothcategorydue to Boman[2]ofafunctionsupportedin
the disk such that some weighted Radon transform over lines vanishes. For the
weighted X-ray transform over curves, there is an analogous Bolker condition
[11] and support theorems are known for a class of such transforms in n ≥ 3
[31, 32] including the geodesic ray transform on an analytic, simple manifold
over functions [20] and over symmetric tensor fields [21].
Our first result considers such analytic generalized Radon transforms and
shows their analytic microlocal regularity. This builds upon similar results for
theweightedX-raytransformoveragenericclassofcurves[7,9]andinparticu-
lar,geodesics[27,28]. Toavoidcomplicationsatthe boundaryofM,weembed
itisometricallyin a slightly larger,open analyticmanifold M . Insection2,we
1
show how to extend the definition of R to a transform on M in a stable way.
w 1
We show:
Theorem 1. Let R be an analytic generalized Radon transform satisfying the
w
Bolker condition, with w an analytic, nonvanishing weight. Let f ∈ E′(M ) be
1
suchthat R f(σ)=0 in a neighborhood of some hypersurface σ ∈Σ. Then the
w 0
analytic wavefront set WF (f) does not intersect the conormal bundle N∗σ .
A 0
The main tool is a complex stationary phase lemma of Sjo¨strand [26, Theo-
rem 2.8, 2.10 ff.] and related techniques, which suffice in lieu of a hypothetical
analytic calculus of Fourier integral operators. The proof of this theorem is
given is section 3.
Theorem 1 also implies a local support theorem [23, Prop 2.3] and in par-
ticular the injectivity of R : L2(M) → H(n−1)/2(Σ). In fact, the proof of
w
the theorem shows that it suffices for R f(σ) to be analytic in a neighborhood
w
of σ . One then obtains a unique continuation result of the following type: if
0
′
f ∈E (M ) is analytic on one side of σ , and R f(σ) is analytic in a neighbor-
1 0 w
hood of σ , then there is a neighborhood of σ on which f is analytic.
0 0
2
Our secondresult is a stability estimate for a generic class of smoothgener-
alized Radon transforms satisfying the Bolker condition. We restrict ourselves
to those generalized Radon transforms studied by Beylkin [1], which have Σ
parametrized globally by the level sets of a smooth defining function ϕ satisfy-
ing some conditions to be made explicit later.
Theorem 2. Let (M,g) be an analytic Riemannian manifold with boundary.
Take R : L2(M) → H(n−1)/2(Σ) to be an injective generalized Radon trans-
w
form defined by ϕ with weight w, satisfying the Bolker condition. Then there
exists K ≫ n and a neighborhood of (ϕ,w) ∈ CK such that the generalized
Radon transform R˜ defined on (M,g) by a defining function and weight in this
w˜
neighborhood is injective and for all f ∈L2(M) there exists C >0 such that
||f||L2(M) ≤C||R˜w∗˜R˜w˜f||Hn−1(M1).
ThisfollowsfromananalysisofthesymbolofthenormaloperatorR∗R . As
w w
mentioned above, under the Bolker condition it is an elliptic pseudodifferential
operator. We show that perturbing the defining function and weight slightly in
CK perturbs the operator slightly, preserving the stability estimate.
While we work entirely on an analytic Riemannian manifold (M,g) and
do not perturb the metric in this result, we use the metric only to provide a
convenientchoice of surface measure,andto ensure the existence ofa dense set
of injective, generalized Radon transforms. We may then conclude:
Corollary 1. On each analytic, compact Riemannian manifold with boundary,
there is a generic set of generalized Radon transforms satisfying the Bolker con-
dition that are both injective and stable.
We defer the proof of Theorem 2 to section 4.
Acknowledgements. ThefirstauthorispartlysupportedbyNSFGrantDMS–
1301646.
2 Generalized Radon Transforms
In this section we fix notation and establish some basic facts about the gen-
eralized Radon transform, including a statement of the Bolker condition. For
concreteness we consider the space of hypersurfaces Σ as parameterized by a
defining function, following Beylkin [1], though we only consider oriented hy-
persurfaces. To avoiddifficulties occuringat the boundaryof M,we assumeM
is isometrically embedded in a slightly larger open manifold M , whose metric
1
we also refer to by g. If we are considering the analytic category of Radon
transforms, we will also assume M is analytic. In the sequel, we will always
1
consider L2(M) to be functions on M , extended by zero.
1
Definition. Let ϕ ∈ C∞(M × (Rn \0)). ϕ is a defining function when it
1
satisfies the following conditions:
3
1. ϕ(x,θ) is positive homogeneous of degree one in the fiber variable.
2. ϕ is non-degenerate in the sense that d ϕ(x,θ)6=0.
x
3. The mixed Hessian of ϕ is strictly positive, i.e.,
∂2ϕ
det >0.
(cid:18)∂xi∂θj(cid:19)
The level sets of ϕ will be denoted by
H ={x∈M :ϕ(x,θ)=s}.
s,θ 1
Note that by homogeneity, H = H for λ > 0. Therefore we can
s,θ λs,λθ
consider Σ as globally parameterized by (s,θ) ∈ R×Sn−1. Often we will also
implicitly consider ϕ as a function on M ×Sn−1.
1
ThethirdconditionimposedonadefiningfunctionisalocalformofBolker’s
condition. Thisallowsustolocallyidentify(x,θ)∈M ×Sn−1withthecovector
1
d ϕ(x,θ)/|d ϕ(x,θ)| ∈ S∗M . We will assume in addition a stronger, global
x x g x 1
Bolker condition.
Definition. A defining function ϕ satisfies the global Bolker condition if for
each θ ∈ Sn−1, the map x 7→ d ϕ(x,θ) is injective, and for each x ∈ M, the
θ
map θ 7→d ϕ(x,θ) is surjective.
x
The firstconditionis roughlyanalogousto the “noconjugate points” condi-
tionassumedby [9,20] for similar results regardingthe geodesicraytransform,
andthesecondensuresthateverysingularityisobservablefromsomehypersur-
faceinΣ. NotethatgeneralizedRadontransformsdefinedbyadoublefibration
satisfying the Bolker condition as stated by Guillemin et. al. also satisfy this
Bolker condition, see [23, Lemma 3.5].
∞
Consider f ∈ C (M). We extend it by zero to a function on M which
1
we also denote by f. Let the generalized Radon transform R determined by
w
(M,g,ϕ,w) be defined by
R f(s,θ)= w(x,θ)f(x)dµ ,
w s,θ
Z
Hs,θ
where w ∈ C∞(M ×Sn−1) is a smooth, nonvanishing weight and dµ is the
1 s,θ
volume form on H induced by dVol. There exists a smooth, nonvanishing
s,θ
function J(x,θ) such that
dµ (x)∧ds=J(x,θ)dVol.
s,θ
We calculate the adjoint of R in L2(M,dVol) to be
w
(R f)gdsdθ = w(x,θ)f(x)g(s,θ)dµ dsdθ
w s,θ
ZSn−1ZR ZSn−1ZRZHs,θ
= g(ϕ(x,θ),θ)w(x,θ)J(x,θ)f(x)dVoldθ
ZSn−1ZM1
4
Therefore
∗
R g(x)= w(x,θ)J(x,θ)g(ϕ(x,θ),θ)dθ.
w ZSn−1
This is simply a generalized backprojectionwith weight wJ.
3 Microlocal regularity
In this section, we take (M ,g) to be an analytic Riemannian manifold, ϕ to
1
be an analytic defining function, and w to be an analytic nowhere vanishing
weight. Given f ∈ E′(M ), we are interested in the microlocal analyticity of
1
′
f given that of R f. (We extend R to E (M ) by duality.) We will use the
w w 1
followingdefinitionoftheanalyticwavefrontset,followingSjo¨strand. Thereare
alternativeapproachestoanalyticwavefrontsetbySato,Kawai,Kashiwara[25]
and also Bros and Iagolnitzer [6], which were shown to be equivalent by Bony
[5].
Definition ([26, Def. 6.1]). Let (x ,ξ ) ∈ T∗Rn \ 0 and let ψ(x,y,ξ) be an
0 0
analytic function defined in a neighborhood U of (x ,x ,ξ )∈C3n such that
0 0 0
1. For all (x,x,ξ)∈U (i.e., x=y), we have
ψ(x,x,ξ)=0 and ∂ ψ(x,x,ξ)=ξ.
x
2. There exists C >0 such that for all (x,y,ξ)∈U, we have
Imψ(x,y,ξ)≥C|x−y|2.
Let a(x,y,ξ) be an elliptic classical analytic symbol defined on U, see, e.g., [26,
Theorem 1.5].
We say u ∈ D′(Rn) is analytic microlocally near (x ,ξ ) if there exists a
0 0
cut-off function χ∈C∞(Rn) with χ(x )=1 such that
c 0
eiλψ(x,y,ξ)a(x,y,ξ)χ(y)u(y)dy =O(e−λ/C),
Z
for some C >0, uniformly in a conic neighborhood of (x ,ξ ).
0 0
The analytic wavefront set is theclosed conic setWF (u)⊂T∗Rn\0,which
A
is the complement of the set of covectors near which u is microlocally analytic.
Wenotethatthisdefinitionismicrolocalandinvariantlydefined,andthere-
fore can be extended to distributions on analytic manifolds (see [19, Theorem
8.5.1] and the remarks following). In this case, for u ∈ D′(M ), WF (u) is a
1 A
closed conic subset of T∗M \0.
1
Recall that since the mixed Hessian of ϕ is strictly positive, we may locally
identify (x,θ) ∈ M × Sn−1 with the unit covector d ϕ(x,θ)/|d ϕ(x,θ)| ∈
1 x x g
∗ ∗
S M . Fix acovector(x ,θ )∈T M \0with s =ϕ(x ,θ ). Fromnow onwe
1 0 0 1 0 0 0
will work in a small conic neighborhood of this covector.
5
Proposition 1. If R f(s,θ) = 0 for (s,θ) in a neighborhood of (s ,θ ), then
w 0 0
(x ,d ϕ(x ,θ ))6∈WF (f).
0 x 0 0 A
Proof. Let us fix a coordinate system. We already have local coordinates (x,θ)
on T∗M \0. Without loss of generality we can take s = 0 and |θ | = 1. To
0 0
simplify the coordinates on Σ, we perform a stereographic projection onto the
tangentplaneofthe sphereatθ ,whichis ananalyticdiffeomorphismmapping
0
a neighborhood of θ ∈ Sn−1 to a neighborhood of the origin in Rn−1. We
0
refer to the coordinates on this tangent plane by ξ, and pass to a perhaps
smaller neighborhood of Σ with |s| < 2ǫ and |ξ| < δ, with ǫ,δ > 0 being small
parameters.
Much of the complexity of analytic microlocal calculus is due to the dif-
ficulty of localizing in the analytic category, as there are no suitable cut-off
functions. Instead one often uses a sequence of quasianalytic cut-off functions
χ ∈ C∞(R), depending on ǫ, for whose construction we refer to [18, 30]. We
N c
will only use the following properties of this sequence:
1. suppχ ⊂(−2ǫ,2ǫ) and χ (−ǫ,ǫ)=1.
N N
2. For all N ∈N and k ≤N, the estimate
∂(k)χ (s) ≤(CN)k
s N
(cid:12) (cid:12)
(cid:12) (cid:12)
holds for a constant C >0(cid:12)independe(cid:12)nt of N.
By assumption R f(s,ξ)=0 for |s|<2ǫ and |ξ|<δ. Let λ≫1 be a large
w
parameter, to be fixed later. This implies that
0= eiλsχ (s) w(x,ξ)f(x)dµ ds. (1)
N s,ξ
Z Z
Hs,ξ
Recall that ξ are analytic coordinates for the neighborhood of θ in Sn−1 that
0
we are concerned with, and so here and in the sequel we write for brevity, e.g.,
w(x,ξ)=w(x,θ(ξ)) and dµ =dµ .
s,ξ s,θ(ξ)
It follows from Beylkin’s construction that
dµ ∧ds=J(x,ξ)dVol
s,ξ
whereJ(x,ξ)isananalytic,nonvanishingJacobiananddVolisthevolumeform
on M associated to the metric. Hence (1) reduces to the oscillating integral
1
eiλϕ(x,ξ)a (x,ξ)f(x)dVol=0. (2)
N
Z
Here a (x,ξ) is a sequence of classicalanalytic symbols on the same neighbor-
N
hood of (x ,0) ∈ M ×Rn−1. The coordinates on x and ξ are real-analytic,
0 1
and so we may extend their domain of definition slightly by analytic continu-
ation to a Grauert tube of a small neighborhood of H ⊂ M (for x) and a
0,0 1
small neighborhood of the origin in Cn−1 for ξ. This continuation in principle
6
depends on the choice of analytic coordinates, but as the analytic wavefront
set is invariantly defined the final result does not depend on this choice. We
choosea perhaps smallerδ suchthat {ξ ∈Cn−1 :|ξ|<δ/2}is containedin this
neighborhood. We denote the local complex coordinate patch of x as U ⊂Cn.
0
Let y ∈ U and η ∈ Cn−1, with |η| < δ/2. Let ρ(ξ) = 1 when |ξ| ≤ δ and
zero otherwise. Then we multiply (2) by
λ
ρ(ξ−η)exp − |ξ−η|2−iλϕ(y,ξ) ,
(cid:18) 2 (cid:19)
and integrate with respect to ξ. The resulting integral is of the form
eiλΦ(x,y,ξ,η)b (x,ξ,η)f(x)dVol(x)dξ =0. (3)
N
ZZ
Here b is a sequence of classicalanalytic symbols defined on a complex neigh-
N
borhood of H ×{0}×{0} and Φ is the augmented phase function given by
0,0
i
Φ(x,y,ξ,η)= |ξ−η|2+ϕ(x,ξ)−ϕ(y,ξ).
2
To estimate the left-hand side of (3), we intend to use the method of com-
plex stationary phase. Therefore, we are interested in the critical points of the
function ξ 7→Φ(x,y,ξ,η). Note that
Φ (x,y,ξ,η)=i(ξ−η)+∂ ϕ(x,ξ)−∂ ϕ(y,ξ).
ξ ξ ξ
There are clearly real critical points ξ when ξ = η and x = y. These crit-
ical points are non-degenerate, and therefore induce complex critical points
ξ (x,y,η)=η+i(y−x)+O(δ).
c
Consider the situation when y = 0. Then for x 6= 0, the only real critical
points are where ∂ ϕ(x,ξ) = ∂ ϕ(y,ξ). However, this cannot happen by the
ξ ξ
global Bolker condition that we imposed on the defining function. By non-
degeneracy again we see there are no real or complex critical points other than
ξ (x,y,η) for (x,y,ξ,η) where |y|<δ and |ξ−η|<δ.
c
Now we apply the complex stationary phase lemma [26, Theorem 2.8, 2.10]
to (3). As a preparatory step divide the integral into two regions; one over the
region
I ={(x,y,ξ,η):|x−y|≤δ/C ,|ξ−η|<δ}
+ 0
and one over the region
I− ={(x,y,ξ,η):|x−y|>δ/C0,|ξ−η|<δ}.
Here C >0 is a constantchosenso that the criticalpoints ξ (x,y,η) lie within
0 c
I+ and none lie in I−.
In I−, we may define the usual operator L such that LeiλΦ =eiλΦ via
∂ Φ·∂
ξ ξ
L= .
iλ|∂ Φ|2
ξ
7
This is well-defined as there are no critical points in I−, so we may repeatedly
integrate by parts:
eiλΦb fdVoldξ = (LNeiλΦ)b fdVoldξ
(cid:12) N (cid:12) (cid:12) N (cid:12)
(cid:12)ZI− (cid:12) (cid:12)ZI− (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) N
(cid:12) (cid:12) (cid:12) (cid:12)
≤ eiλΦ(L∗)N[b f]dVoldξ + |B |.
(cid:12) N (cid:12) k
(cid:12)(cid:12)ZI− (cid:12)(cid:12) Xk=1
(cid:12) (cid:12)
(cid:12) (cid:12)
ThetermsB areboundarytermsthatdecayexponentially,duetothefactthat
k
ImΦ>0for|ξ−η|=O(δ). Asforthe integralontheright-handside,werecall
that b is defined by
N
b (x,ξ,η)=ρ(ξ−η)χ (ϕ(x,ξ))w(x,ξ)J(x,ξ). (4)
N N
Theworstpossiblegrowthof(L∗)Nb intermsofN occurswhenallderivatives
N
are applied to χ (ϕ(x,ξ)), and in this case we may apply the estimate
N
∂(N)χ (s) ≤(CN)N,
s N
(cid:12) (cid:12)
(cid:12) (cid:12)
which follows from the con(cid:12)struction o(cid:12)f the sequence of quasianalytic cut-off
functions. Therefore,
eiλΦb fdx =O (CN/λ)N +CNe−λ/C . (5)
(cid:12) N (cid:12)
(cid:12)ZI− (cid:12) (cid:16) (cid:17)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
As for the integral over I , the cut-off functions χ (ϕ(x,ξ)) are all equal
+ N
to one. Therefore the amplitude on I does not depend on N; we remove this
+
dependence and refer to the amplitude restricted to this region as b. We know
allofthecriticalpointsofξ 7→Φandcanthereforeapplythecomplexstationary
phase lemma. This yields an estimate of the form
eiλΦbfdVoldξ =Cλ−n/2 eiλψBfdVol+O (CN/λ)N +Ne−λ/C .
Z Z (cid:16) (cid:17)
Here ψ(x,y,η) = Φ(x,y,ξ (x,y,η),η) and B(x,y,η) = b(x,ξ (x,y,η),η). We
c c
may now fix N such that N ≤(λ/Ce)≤N +1 to ensure the error is exponen-
tially small.
eiλψBfdVol=O(e−λ/C).
Z
Now B(x,y,η) is an elliptic analytic symbol near (x,y,η) = 0 and ψ(x,y,η)
is a non-degenerate phase function. To show this implies (x ,θ ) 6∈ WF (f),
0 0 A
we check the details of the characterizationof the analytic wave front set given
above. Recall
i
ψ(x,y,η)= |ξ (x,y,η)−η|2+ϕ(x,ξ (x,y,η))−ϕ(y,ξ (x,y,η)).
c c c
2
8
Note that ξ (x,x,η)=η for x real, and therefore ψ(x,x,η)=0. In addition
c
∂ ψ(x,x,η)=∂ ϕ(x,η)=−∂ ψ(x,x,η).
x x y
By the global Bolker condition we can make a change of variables η′ so that
η′ = d ϕ(x,η). Finally, it is clear that Imψ(x,y,ξ) ≥ C|x−y|2 for x,y real.
x
Therefore, (x ,d ϕ(x ,θ ))6∈WF (f).
0 x 0 0 A
Theorem 1 follows from applying the proposition to all conormals of a fixed
hypersurface σ .
0
Remark. Fromthe proof we see that it suffices for R f(σ) to be analytic in a
w
neighborhoodof(s ,θ ). Aftermicrolocalization,the right-handsideof (2)will
0 0
be O(e−λ/C) instead of zero, but this poses no problem.
4 Stability
We now return to generalizedRadon transforms with smooth defining function
ϕ:M ×Sn−1 and smooth, nonvanishing weight w :M ×Sn−1. The object of
1 1
interest in this section is the normal operator N = R∗R . It is known that
w w w
theglobalBolkerconditionimpliesN isapseudodifferentialoperator[13,Prop
w
8.2]. However,we require more detailed knowledge of the symbol of N for the
w
kind of stability estimates we prove later.
First we obtain a representation of the Schwartz kernel of R .
w
Lemma 1. The Schwartz kernel K ∈D′(R×Sn−1×M ) of R is
Rw 1 w
K (s,θ,y)=(2π)−1δ(s−ϕ(y,θ))w(y,θ)J(y,θ)
Rw
where J(y,θ) is the smooth, nonvanishing function such that
dµ (y)∧ds=J(y,θ)dVol(y).
s,θ
Proof. We perform a partial Fourier transform of R f(s,θ) in the s variable,
w
taking s′ to be the dual variable of s. The change of variables then yields
F R f(s′,θ)= e−iss′ w(y,θ)f(y)dµ ds
s w s,θ
ZR ZHs,θ
= e−is′ϕ(y,θ)w(y,θ)J(y,θ)f(y)dVol(y).
Z
M1
Therefore
R f(s,θ)=(2π)−1 ei(s−ϕ(y,θ))s′w(y,θ)J(y,θ)f(y)dVol(y)ds′
w
ZRZM1
= K (s,θ,y)f(y)dVol(y).
Z Rw
M1
9
Similarly, the kernel of the generalized backprojectionR′ is
w
KR∗ =(2π)−1δ(ϕ(x,θ)−s)w(x,θ)J(x,θ).
w
From this we see that the kernel of N is
w
K =(2π)−1 eis′(ϕ(x,θ)−ϕ(y,θ))w(x,θ)J(x,θ)w(y,θ)J(y,θ)ds′dθ. (6)
Nw ZZ
We can now use this representation to find the principal symbol of the normal
operator N .
w
Lemma 2. The principal symbol of N is
w
W(x,x,ξ/|ξ|)+W(x,x,−ξ/|ξ|)
p(x,ξ)=(2π)1−n ,
|ξ|n−1
where W is the auxillary function
W(x,y,θ)=w(x,θ)J(x,θ)w(y,θ)J(y,θ).
Proof. Beginning from (6), we split the integration over R into {s′ > 0} and
{s′ < 0}. Using the positive homogeneity of the defining function, we rewrite
the integral as
∞
K = ei(ϕ(x,s′θ)−ϕ(y,s′θ))W(x,y,θ)ds′dθ
Nw ZSn−1Z0
∞
+ e−i(ϕ(x,s′θ)−ϕ(y,s′θ))W(x,y,θ)ds′dθ.
ZSn−1Z0
=K+ +K− .
Nw Nw
Here K+ and K− are the Schwartz kernels of the operators N+ and N−
respectiNvewly, so thNatw N = N+ + N−. We work with each termwseparatelwy.
w w w
Let ξ = s′θ be polar coordinates for Rn. This change of variables is justified
whenthe kernelis appliedto a testfunctionin C∞(M ); usingthe proofof[19,
c 1
Theorem7.8.2]it canbe shownthat it is justified for the kernelitself. Thenwe
obtain
ξ
K+ = ei(ϕ(x,ξ)−ϕ(y,ξ))W x,y, |ξ|1−ndξ
Nw ZRn (cid:18) |ξ|(cid:19)
By the global Bolker condition, ∂ ϕ(x,ξ) = ∂ ϕ(y,ξ) implies x = y. A sta-
ξ ξ
tionary phase argument implies that K+ is a smooth function away from the
Nw
diagonal of M ×M .
1 1
Fixx ∈M . ThereexistsaneighborhoodU ofx onwhichwehavenormal
0 1 0
coordinates, which we refer to again with (xi), such that x(x ) = 0. We then
0
use (xi,yi) as coordinates on U ×U, with xi = yi. We consider the localized
kernel
ξ
χK+ χ= ei(ϕ(x,ξ)−ϕ(y,ξ))W x,y, χ(x)χ(y)|ξ|1−ndξ.
Nw ZRn (cid:18) |ξ|(cid:19)
10