Improvement of eigenfunction estimates on manifolds of nonpositive curvature

0.2 MB·English

by  Andrew Hassell

IMPROVEMENT OF EIGENFUNCTION ESTIMATES ON
MANIFOLDS OF NONPOSITIVE CURVATURE
3
1 ANDREWHASSELLANDMELISSATACY
0
2
Abstract. Let (M,g) be a compact, boundaryless manifold of dimension n

n withthepropertythateither(i)n=2and(M,g)hasnoconjugatepoints,or
a (ii) the sectional curvatures of (M,g) are nonpositive. Let ∆ be the positive
J LaplacianonM determinedbyg. WestudytheL2 Lp mappingproperties
0 of a spectral cluster of √∆ of width 1/logλ. Und→er the geometric assump-
3 tionsabove,[1]B´erardobtainedalogarithmicimprovementfortheremainder
term of the eigenvalue counting function which directly leads to a (logλ)1/2
] improvementforH¨ormander’sestimateontheL∞normsofeigenfunctions. In
P thispaperweextendthisimprovementtotheLp estimatesforallp> 2(n+1).
n−1
A
.
h
t
a
1. Introduction
m
[ We study the growth of the Lp norms of high eigenvalue eigenfunctions of the

Laplace-Beltramioperatoroncompactmanifolds. Thatis,weseekestimatesofthe
2
form
v
0 (1) u .f(λ,n,p) u
4 || ||Lp(M) || ||L2
5 where u is an eigenfunction, ∆u = λ2u, of the Laplace-Beltrami operator on a
2
n-dimensional, compact, boundaryless Riemannian manifold (M,g) (we adopt the
.
2 sign convention that the Laplacian ∆ is a positive operator). As is well known,
1 L2 L estimates for eigenfunctions follow directly from eigenvalue multiplicity
∞
2 →
bounds, and Ho¨rmander’s counting function remainder estimates [8] yield a bound
1
v: ||u||L∞ .λn−21 ||u||L2.
i
X For general manifolds without any additional geometric assumptions Sogge [10]
obtained Lp estimates of the form
r
a f(λ,n,p)=λδ(n,p)
(2) n 1 n 2(n+1) p
δ(n,p)= −2 − p n 1 ≤ ≤∞.
(n−41 − n2−p1 2≤−p≤ 2(nn+11)
−
These estimates are in fact for spectral clusters of window width one. While they
are sharp for such clusters the only known sharp eigenfunction examples are in
cases of high multiplicity of the spectrum. For the L2 L estimates more is
∞
→
known. Sogge and Zelditch [15] and Sogge, Zelditch and Toth [14] investigated
the conditions on M required for maximal L growth. They determined that to
∞
Key words and phrases. Eigenfunction estimates, nonpositive curvature, manifolds without
conjugatepoints,logarithmicimprovement,finitepropagationspeed.
A. H. was supported by Australian Research Council Future Fellowship FT0990895 and Dis-
coveryGrantsDP1095448 andDP120102019.
1
2 ANDREWHASSELLANDMELISSATACY
achievethe sharpL bound given by (2) it is necessary that at some point x M
∞
∈
both that the set of directions in S⋆M that loop back to x has positive measure
and that the first return map be recurrent. As L2 L estimates follow directly
∞
→
from multiplicity estimates an improvement in the remainder term of the counting
function automatically implies an improvement in L estimates. It is expected
∞
that systems whose classical dynamics exhibit chaotic behaviour will have lowered
multiplicity and therefore lowered L growth. In 1977 B´erard[1] obtained a logλ
∞
improvement for the remainder term in the counting function (and therefore an
improved multiplicity bound) for manifolds with nonpositive section curvature (in
two dimensions, the condition of no conjugate points suffices). His result directly
implies an improvement to the L bounds of
∞
λn−21
u . u .
|| ||L∞ (logλ)1/2 || ||L2
By interpolation with Sogge’s p= 2(n+1) estimate we can therefore easily obtain
n 1
−
n−1 n
λ 2 −p 2(n+1)
u . u p .
|| ||Lp 1 n+1 || ||L2 ≥ n 1
(logλ)2−p(n−1) −
In this paper we show that under the same assumptions as in B´erard [1], the
improvement by a factor of (logλ)1/2 persists for p > 2(n+1). That is, our main
n 1
result is −
Theorem1.1. LetubeaneigenfunctionoftheLaplacian ∆u=λ2uonacompact
−
boundarylessRiemannianmanifold(M,g)ofdimensionn,suchthateither(i)n=2
and (M,g) has no conjugate points or (ii) the sectional curvatures of (M,g) are
nonpositive. Then we have the estimate
n−1 n
λ 2 −p
u C
|| ||Lp ≤ p(logλ)1/2
in the range
2(n+1)
(3) p> .
n 1
−
WeremarkthattheconstantC weobtainblowsupasptendstothe‘kinkpoint’
p
2(n+1). It remains unknown whether a logarithmic improvement, of any power, is
n 1
val−id at p = 2(n+1). For surfaces there are some similar results known below the
n 1
kink point, tha−t is p < 6. Bourgain [2] and Sogge [13] study the relationship
between u and u where γ is a geodesic. Sogge showsthat to improve
|| ||L4(M) || ||L2(γ)
the Lp(M) estimates for 2 < p < 6 it is necessary and sufficient to improve the
L2 restriction estimates. In related work Sogge and Zelditch [11] and Chen and
Sogge [6] study improvements to geodesic restriction theorems for low p in the
case of nonpositive curvature. In the high p range Chen [5] obtains logarithmic
improvements to the restriction theorems of Burq, G´erard and Tvetkov [4]. Also
in the setting of nonpositive curvature Bourgain, Shao, Sogge and Yao [3] obtain
logarithmic improvements for Lp norms of resolvent operators.
At this point we note the history of this paper. Theorem 1.1 was originally
presented by the second author at a conference at Dartmouth College in 2010 in
the special case of constant negative curvature. Following the general techniques
EIGENFUNCTION ESTIMATES ON MANIFOLDS OF NONPOSITIVE CURVATURE 3
outlined in that presentation Chen [5] in 2012 proved restriction estimates, in the
general nonpositive curvature case, which are logarithmic improvements on esti-
mates of Burq,G´erardand Tzvetkov [4]. This paper presents the originalresult in
this more general setting.
Toobtaineigenfunctionsestimateswestudyasmoothedspectralclusterχev (√∆)
λ,w
defined in the following section. Here the spectral cluster is centred at λ and has
effective width w. This spectral cluster operator is based on Sogge’s construction
[12, Section 5.1], with a slight variation as in [7, Section 8.1]. Sogge’s estimates
correspond to the case w = 1. B´erard obtained his logarithmic improvements by
shrinking the window width to 1/logλ. We will likewise aim to shrink window
widths by a logarithmic factor. B´erard’s method relied on expressing the solution
operator the the wave equation on M as a sum of shifted solutions operators on
the universal cover of M, denoted M. If M has no conjugate points, M has an
infinite injectivity radius. The logλ improvement is then obtained by estimating
the contribution from each copy offthe fundamental domain and estimfating the
number of copies that can be reached by finite speed propagation in logλ time.
2. Spectral cluster operator
Ourspectralclusteroperatorχev (√∆)isdefinedasfollows. LetχbeaSchwartz
λ,w
function such that χ(0)=1 and χˆ is supported in [ǫ,2ǫ]. Then that χ(0)>0, and
by taking ǫ sufficiently small, we can arrange that Reχ c>0 on [0,1]. We then
≥
define, following [7], and for 0<w 1,
≤
µ λ µ λ
χev (µ)=χ − +χ − − .
λ,w w w
(cid:18) (cid:19) (cid:18) (cid:19)
Thefirsttermontherighthandsideis,forw =1,preciselySogge’sspectralcluster
operator. Notice that χev (µ) is an even function of µ and for λ large enough we
λ,w
have
c
Reχev on [λ,λ+w].
λ,w ≥ 2
Hence we can write
c2
(Reχev )2 =F
λ,w − 8 λ,w
for some function F that is nonnegative on the interval [λ,λ+w]. Let u be an
λ,w
L2-normalizedeigenfunctionwitheigenvalueλ. ThentheLp normofuismajorized
bytheL2 Lpoperatornormof1l (√∆),orequivalentlytheLp′ L2norm
[λ,λ+w]
of the sam→e operator. However, for f Lp′, we compute as in [7, Sectio→n 8.1]
∈
c2 1l (√∆)f 2 = 1l (√∆)f,(Reχev (√∆))2 F (√∆) f
8 [λ,λ+w] L2 [λ,λ+w] λ,w − λ,w
(cid:13)(cid:13) = 1l[(cid:13)(cid:13)λ,λ+w](D√∆)Reχeλv,w(√∆)f,Reχeλv,w(√∆)f (cid:17) E
D E
F (√∆)1l (√∆)f,1l (√∆)f
λ,w [λ,λ+w] [λ,λ+w]
−
D Reχev (√∆)f 2 χev (√∆)f 2 .E
≤ λ,w L2 ≤ λ,w L2
So to obtain estimates(cid:13)for Laplacian e(cid:13)igenfun(cid:13)ctions it is(cid:13)enough to estimate the
operator norm of the o(cid:13)perator χev fro(cid:13)m Lp′(cid:13)to L2 for a(cid:13)ny w, 0 < w 1. The
λ,w ≤
4 ANDREWHASSELLANDMELISSATACY
parameter w controls the effective size of the spectral window; we aim to make w
as small as possible. We have
χ (√∆)u= eit√∆/w+e it√∆/w e itλ/wχˆ(t)udt
λ,w − −
(4) Z (cid:16) (cid:17)
=2w cos(t√ ∆)e itλχˆ(wt)udt.
−
−
Z
Remark 2.1. The point of considering χev (√∆) rather than χ (√∆/w) is that
λ,w λ
the former operator can be expressed in terms of the cosine kernel, i.e. the kernel
of cost√∆, instead of the half-wave operator eit√∆. This allows us to exploit the
finite propagationspeed of the cosine kernel (see Proposition 3.6).
3. Proof of Theorem 1.1
We work with a smoothed spectral cluster of window width 1/A, we will allow
the parameter A to remain free at the moment and later set it as required. With
χ defined as above, we need L2 Lp mapping estimates for
→
(5) χev (√∆)=χ A(√∆ λ) +χ A( √∆ λ) .
λ,1/A − − −
(cid:16) (cid:17) (cid:16) (cid:17)
The proof of Theorem 1.1 reduces to showing that for any p in the range (3) there
exists an α and a C such that for A=α logλ
p p p
λδ(n,p)
χev (√∆)u C u .
λ,1/A Lp ≤ p(logλ)1/2 || ||L2
(cid:12)(cid:12) (cid:12)(cid:12)
We approach this p(cid:12)(cid:12)roblem via a(cid:12)T(cid:12) T⋆ method. That is we seek estimates of the
(cid:12)(cid:12) (cid:12)(cid:12)
form
λ2δ(n,p)
χeλv,1/A(√∆)χeλv,1/A(√∆)⋆u Lp ≤Cp A ||u||Lp′
(cid:12)(cid:12) (cid:12)(cid:12)
We have that (cid:12)(cid:12) (cid:12)(cid:12)
(cid:12)(cid:12) (cid:12)(cid:12)
χev (√∆)u=2 e itλAcos tA√∆ χˆ(t)udt
λ,1/A −
Z
so (cid:0) (cid:1)
χev (√∆)⋆u=2 eisλAcos sA√∆ χˆ(s)uds
λ,1/A
Z
therefore it suffices to estimate (cid:0) (cid:1)
1
(6) T u= χev(1/A)χev(1/A)⋆u
λ,A 2 λ λ
=2 e iλA(t s)cos tA√∆ cos sA√∆ χˆ(t)χˆ(s)udtds
− −
ZZ
(cid:0) (cid:1) (cid:0) (cid:1)
= e−iλA(t−s) cos (t+s)A√∆ +cos (t s)A√∆ χˆ(t)χˆ(s)udtds.
−
ZZ (cid:16) (cid:0) (cid:1) (cid:0) (cid:1)(cid:17)
Wewillseparatetheoperatorintotwopieces,onecapturingshorttimepropagation
and the other long time propagation. We wish to set the problem up so that the
short time piece captures behaviour limited to one fundamental domain while the
long time pieces captures the behaviour across multiple domains. The scaling A
canbe seenasatime scalingsoshorttime inthis settingis At s 2ǫwhereǫ is
| − |≤
somesmallparameter. Accordinglyletζ be asmoothcutofffunctionsupportedin
EIGENFUNCTION ESTIMATES ON MANIFOLDS OF NONPOSITIVE CURVATURE 5
[ 2ǫ,2ǫ] and equal to one on [ ǫ,ǫ]. We decompose T into the operators T1 ,
− − λ,A λ,A
T2 and T3 given by
λ,A λ,A
(7) T1 = e iλA(t s)cos A√∆(t s) χˆ(t)χˆ(s)ζ(A(t s))dtds
λ,A − − − −
ZZ
(cid:0) (cid:1)
(8) T2 = e iλA(t s)cos A√∆(t s) χˆ(t)χˆ(s)(1 ζ(A(t s)))dtds
λ,A − − − − −
ZZ
and (cid:0) (cid:1)
(9) T3 = e iλA(t s)cos A√∆(t+s) χˆ(t)χˆ(s)dtds.
λ,A − −
ZZ
We treat T1 first. (cid:0) (cid:1)
λ,A
Proposition 3.1. Let T1 be given by (7). Then for any A 1
λ,A ≥
λ2δ(n,p)
(10) Tλ1,Au Lp ≤C A ||u||Lp′ .
We give a self-containe(cid:12)d(cid:12) proof(cid:12)(cid:12)of Proposition 3.1. A sketch of a shorter proof
(cid:12)(cid:12) (cid:12)(cid:12)
which uses Sogge’s estimate for unit length spectral windows as an input is given
in Remark 3.4. This is similar to the argument used in [3].
Proof. We have set the cut off function ζ such that T1 captures the contribution
λ,A
to T arising from one fundamental domain; indeed, using the finite speed of
λ,A
propagation of the cosine kernel and the support properties of ζ, we see that its
kernel is supported where d(x,y) 2ǫ. Therefore we do not need to sum over
≤
shifted solutions on M (the terms T2 and T3 will require such a sum). Hence,
λ,A λ,A
for this term, we choose to express cos √∆(t) in terms of eit√∆ and e it√∆. In
−
the short time settinfg this use of the half wave kernel is advantageous.
(cid:0) (cid:1)
Applying a change of variable t tA and s sA we have
→ →
(11)
1 t s d(x,y)
T1 = e iλ(t s) e i√∆(t s) χˆ χˆ ζ((t s))ζ dtds.
λ,A 2A2 − − ± − A A − 2
ZZ (cid:16)X± (cid:17) (cid:18) (cid:19) (cid:16) (cid:17) (cid:18) (cid:19)
Letus write Tλ1,,A± forthe expressionabovewhere wereplacethe sum by either
the + or the − term. Thus Tλ1,A = Tλ1,,A+ +Tλ1,,A−. We only treat thPe±term Tλ1,,A+
below (Tλ1,,A− is trivial to estimate as the integral corresponding to (12) below has
no stationary points).
We thereforeneedanexpressionforei(t s)√∆ where t s 2ǫ. Inthis casewe
−
| − |≤
may use the short time parametrix of Ho¨rmander [8] (see also Sogge [12, Section
4.1])
ei(t−s)√∆u= ei(φ(x,y,ξ)−(t−s)|ξ|g(y))a(t,x,y,ξ)u(y)dydξ
Z
whereinlocalcoordinates ξ = gij(x)ξ ξ andφsatisfiestheHamilton-Jacobi
g(x) i j
| |
equation
p
φ = ξ
x g(x) g(y)
|∇ | | |
with initial condition
φ(x,y,ξ)=0when x y,ξ =0,
h − i
6 ANDREWHASSELLANDMELISSATACY
while a satisfies
a(0,x,x,ξ) 1 S−1.
− ∈
Rescaling the time variables s As, t At we can write the Schwartz kernel of
T1,+ as → →
λ,A
(12)
t s d(x,y) dtdsdξ
e−iλ(t−s)ei(φ(x,y,ξ)−(t−s)|ξ|)a(t s,x,y,ξ)χˆ χˆ ζ(t s)ζ
− A A − 2 A2
Z (cid:18) (cid:19) (cid:16) (cid:17) (cid:18) (cid:19)
where ξ = ξ . Changing variables s =t s and ξ λξ we can write this in
g(y) ′
| | | | − →
the form
1 2ǫA t
T1,+ = χˆ T1,+(t)dt,
λ,A A2 A λ,A
(13) ZǫA (cid:18) (cid:19)
Tλ1,,A+(t):=λn e−iλ(s′+φ(x,y,ξ)−s′|ξ|)a˜(s′,x,y,ξ)χˆ t−As′ ζ(s′)ds′dξ.
Z (cid:18) (cid:19)
Here and in what follows we will abuse notation somewhat and refer to both an
operatorand its Schwartzkernelwith the same notation. To establish(10), itthus
suffices to obtain a bound of the form
(14) T1,+(t) Cλ2δ(n,p).
λ,A Lp′ Lp ≤
→
We will do this, followin(cid:13)g the str(cid:13)ategy in [12], by applying the stationary phase
(cid:13) (cid:13)
lemma to the integrals in (13). We first note that if we localize the integral in
the definition of T1,+(t) away from ξ = 1 then we see that the phase is non-
λ,A | |g(y)
stationary in s. It follows (by integrating by parts in the s variable) that the
′ ′
integralis O(λ N) for every N. Therefore we may assume that the symbol a˜(x,ξ)
−
is supportedwhere 1 ǫ ξ 1+ǫ. We mayalsoassumethat a˜ is supported
g(y)
− ≤| | ≤
where d(x,y) 2ǫ.
≤
To prepare for stationary phase calculations, for each fixed y we may choose a
coordinate system centered at y such that ξ g(y) = ξ Rn = ξ . To estimate the
| | | | | |
kernelofT1,+(t)wefirstperformanangularintegrationinξ. Thatiswedecompose
λ,A
ξ into polar coordinates ξ =(r,ω). In these coordinates
φ(x,y,ξ)= x y,ξ +O(ξ x y 2)
h − i | || − |
= x y rcos(θ(ω))+O(rx y 2)
| − | | − |
for θ(ω) the angle between (x y) and ξ. Therefore the critical points occur when
−
x y rsin(θ(ω))θ =O(rx y 2)
| − | ωi | − |
and the Hessian of φ in ω has lower bound
x y r
∂2 φ> | − | Id.
ωω 2
We now obtain by stationary phase an expression for T1,+(t) of the form
λ,A
λn eiλ(s′+φ(x,y,r,ω(r,x,y))−s′r)a˜˜(s′,x,y,r,λ)χˆ t−s′ ζ(s)ds′dr,
A
Z (cid:18) (cid:19)
where ω(r,x,y) is the critical point of φ in the ω variable for fixed (r,x,y) and
β+γ
∂sα′∂xβ∂yγ∂rκa˜˜(s′x,y,r,λ) ≤Cα,β,γ,κ(1+λd(x,y))−n−21 1+λλd(x,y) | | | |
(cid:18) (cid:19)
(cid:12) (cid:12)
(cid:12) (cid:12)
EIGENFUNCTION ESTIMATES ON MANIFOLDS OF NONPOSITIVE CURVATURE 7
as follows from [12, Section 1.1] for example. We have φ(x,y,r,ω(r,x,y)) = x
| −
y r + O(rx y 2), so ∂ φ(x,y,r,ω(r,x,y)) = O(d(x,y)). It follows that there
r
| | − |
is a nondegenerate stationary point of the phase in the (s,r) variables, for s =
′
x y +O(x y 2), r =1 . Another application of stationary phase gives
| − | | − |
λn 1eiλψ(x,y)b(x,y,t,λ)
−
where
ψ(x,y)= x y +O(x y 2)
| − | | − |
is the Riemannian distance between x and y, and
β+γ
∂xβ∂yγb(x,y,t,λ) ≤Cβ,γ(1+λd(x,y))−n−21 1+λλd(x,y) | | | |.
(cid:18) (cid:19)
(cid:12) (cid:12)
Such a k(cid:12)ernel is similar i(cid:12)n form to those operators analyzed by Sogge [12, Section
2.2]by calculatingχ (1)χ⋆(1), andtherefore wecanexpectthem to havethe same
λ λ
Lp′ Lp mapping normof λ2δ(n,p). This can be easilyshownby following Sogge’s
→
“freezing” argument, found in [12, Section 2.2]. This argument is often used to
proveeigenfunctionsestimates(seeforexample[17],[9],[16]). Wefirstexcisethose
points within λ 1 of the diagonal. That is
−
T1,+(t)=W +R , with
λ,A λ λ
W =λn 1eiλψ(x,y)b(x,y,t,λ)(1 ζ(λd(x,y))) and
λ −
−
R =λn 1eiλψ(x,y)b(x,y,t,λ)ζ(λd(x,y)).
λ −
By interpolating between L2 L2 and L1 L , we see that R has Lp′ Lp
∞ λ
mapping norm of at most λ(n→−1)−2pn = λ2δ(n→,p) and therefore we may restric→t our
attention to W . Further we place cut off functions to define a principal direc-
λ
tion. That is for any given direction η we pick a coordinate system such that η
corresponds with the x axis. Then writing x=(x ,x) we have
1 1 ′
x y
Wλ =λn−1eiλψ(x,y)b(x,y,t,λ)(1 ζ(λd(x,y)))ζ | ′− ′| .
− d(x,y)
(cid:18) (cid:19)
Notethatofnthesupportoftheintegrand x1 y1 d(x,y). Sincewemayexpress
| − |∼
W as a sum of a finite number of such operators we need only estimate W . We
λ λ
freeze the variables x and y and consider the operator
1 1
f
(15) Wλ(x1,y1)=λn−21eiλψ(x1,x′,y1,y′)b(x1,y1,x′,y′,t,λ)
x y
′ ′
f 1 ζ(λx1 y1 ) ζ | − | .
× − | − | x y
(cid:18)| 1− 1|(cid:19)
(cid:0) (cid:1)
We may read off a L1 L estimate from the pointwise kernel bounds:
∞
→
(16) Wλ(x1,y1)u L∞ .λn−21(λ−1+|x1−y1|)−n−21 ||u(y1,·)||L1y′ .
(cid:12)(cid:12) (cid:12)(cid:12)
FollowingS(cid:12)(cid:12)o(cid:12)(cid:12)fgge[12, Sect(cid:12)(cid:12)(cid:12)(cid:12)ion2]wemay obtainthe requiredLp′ →Lp estimates via
interpolation and Hardy-Littlewood-Sobolev if we also have
W (x ,y )u . u(y , ) .
(cid:12)(cid:12) λ 1 1 (cid:12)(cid:12)L2x′ || 1 · ||L2y′
Thus the proof of Pro(cid:12)(cid:12)p(cid:12)(cid:12)fosition 3.1 i(cid:12)(cid:12)s(cid:12)(cid:12)completed by the following lemma. (cid:3)
8 ANDREWHASSELLANDMELISSATACY
Lemma 3.2. Suppose x y >Kλ 1 and W (x ,y ) be given by (15). Then
1 1 − λ 1 1
| − |
W (x ,y )u C u(y , )
(cid:12)(cid:12) λ 1 1 (cid:12)(cid:12)L2x′ ≤ f|| 1 · ||L2y′
where the constant C is(cid:12)(cid:12)u(cid:12)(cid:12)fniform in x(cid:12)(cid:12)1(cid:12)(cid:12), y1.
Proof. We have
2
(17) W (x ,y )u = K(t,x ,y ,y ,z )u(y ,y )u¯(z ,z )dy dz
λ 1 1 1 1 ′ ′ 1 ′ 1 ′ ′ ′
(cid:12)(cid:12) (cid:12)(cid:12)L2x′ Z
where (cid:12)(cid:12)(cid:12)(cid:12)f (cid:12)(cid:12)(cid:12)(cid:12) e
(18) K(t,x1,y1,y′,z′)=λn−1 x1 y1 −(n−1) eiλ(ψ(x1,x′,y1,y′)−ψ(x1,x′,y1,z′)
| − |
Z
x y x z
e ˜b(t,x1,y1,x′,y′,z′,λ)ζ | ′− ′| ζ | ′− ′| dx′
× x y x y
(cid:18)| 1− 1|(cid:19) (cid:18)| 1− 1|(cid:19)
where
α
λ | | 1
∂αb(t,x ,y ,x,y ,z ,λ) C . .
| x′ 1 1 ′ ′ ′ |≤ α 1+λd(x,y) x y α
(cid:18) (cid:19) | 1− 1|| |
Weseektoestimate K(t,x ,y ,y ,z ) viaintegrationbypartsinx. Thereforewe
1 1 ′ ′ ′
| |
expand the phase function
e
ψ(x1,x′,y1,y′) ψ(x1,x′,y1,z′)
−
as a Taylor series about z =y . That is
′ ′
ψ(x1,x′,y1,y′) ψ(x1,x′,y1,z′)= y′ψ(x1,x′,y1,y′) (y′ z′)+O(y′ z′ 2).
− ∇ · − | − |
Now differentiating in x we have that
′
∇x′[ψ(x1,x′,y1,y′)−ψ(x1,x′,y1,z′)]=[∂x2′y′ψ](y′−z′)+O(|y−z′|2)
As
ψ(x1,x′,y1,y′)= x y +O(x y 2)
| − | | − |
and x y ǫx y , we have
′ ′
| − |≤ | − |
1
∂x2′y′ = x− y [Id+O(ǫ)]
| − |
and therefore
y z
′ ′
x′[ψ(x1,x′,y1,y′) ψ(x1,x′,y1,z′)] >c | − |.
|∇ − | x y
1 1
| − |
Therefore integrating (18) by parts we gain a factor of cλ 1 y z 1 x y for
− ′ ′ − 1 1
| − | | − |
eachiterationandatworstdifferentiatingproducesgrowthof x y 1. Therefore
1 1 −
| − |
we have
(19) K(t,x1,y1,y′,z′) CNλn−1 x1 y1 −(n−1)(1+λy′ z′ )−N
| |≤ | − | | − |
x z
e ˜b(t,x1,y1,y′,z′,λ)ζ | ′− ′| dx′
× x y
Z (cid:18)| 1− 1|(cid:19)
CNλn−1(1+λy′ z′ )−N
≤ | − |
EIGENFUNCTION ESTIMATES ON MANIFOLDS OF NONPOSITIVE CURVATURE 9
astheintegrandin(19)isonlysupportedonaregion x y ǫx y . Therefore
′ ′ 1 1
| − |≤ | − |
we apply Ho¨lder and Young to (17) to obtain
2
2
W (x ,y )u . u(y ,y )
(cid:12)(cid:12) λ 1 1 (cid:12)(cid:12)L2x′ || 1 ′ ||Ly′
as required. (cid:12)(cid:12)(cid:12)(cid:12)f (cid:12)(cid:12)(cid:12)(cid:12)
(cid:3)
Remark 3.3. For this small t s term there is no restriction on how large the
−
parameter A can be. It is from the large t s term that will will arrive at the
−
restriction that A must grow logarithmically in λ.
Remark 3.4. One can give an alternative proof of Proposition3.1 by showing that
T1,+(t) is a Schwartz function φ (√∆ λ) of √∆ λ, where φ is uniformly
λ,A A − − A
bounded in A 1 in the sense that every seminorm is uniformly bounded for
≥
A 1. Then one can show that any Schwartz function φ of √∆ λ is bounded
fro≥m Lp′ to Lp with norm estimate Cλ2δ(n,p), where C depends o−nly on a finite
numberofseminormsofφ. ThisisessentiallyequivalenttotheargumentinLemma
2.3 of [3].
Remark 3.5. One can avoid the use of Lemma 3.2 with the observation that (13)
is a semiclassical Fourier Integral operator. In fact, if we localize the symbol in
(13) in the variable ω = ξ/ξ near some direction, which (by rotating in the ξ
| |
coordinate) we may assume to be the e direction, and then fix the variables x ,
1 1
y , the resulting frozen operator, acting in the remaining n 1 variables, is an
1
−
FIO oforderzeroassociatedto a canonicalgraph,and thereforeL2 bounded. This
localized and frozen operator is analogous to W above, with the only difference
λ
beingthatthelocalizationisdoneintheξvariablesbeforestationaryphase,instead
of in the spatial variables directly. g
We now treat T2 .
λ,A
Proposition 3.6. Suppose T2 is given by (8) then for any p there exists an α
λ,A p
and a C such that with A=α logλ we have
p p
λ2δ(n,p)
T2 u C
λ,A Lp ≤ p logλ
(cid:12)(cid:12) (cid:12)(cid:12)
Proof. We have (cid:12)(cid:12) (cid:12)(cid:12)
T2 u= e iλA(t s)cos A(t s)√∆ χˆ(t)χˆ(s)(1 ζ(A(t s)))udtds
λ,A − − − − −
ZZ
(cid:0) (cid:1)
After scaling t At and s As and setting s =t s we obtain
′
→ → −
Tλ2,Au= A12 e−iλs′cos s′√∆ χˆ At χˆ t−As′ (1−ζ(s′))udtds′.
ZZ (cid:18) (cid:19) (cid:18) (cid:19)
(cid:0) (cid:1)
We estimate this term by interpolating between L2 L2 estimates and L1
→ →
L estimates. Using the bound cost 1 we get a trivial L2 L2 estimate for
∞
| | ≤ →
T2 :
λ,A
(20) T2 C,
λ,A L2 L2 ≤
→
uniformly in λ and A. (cid:13) (cid:13)
(cid:13) (cid:13)
10 ANDREWHASSELLANDMELISSATACY
An L1 L estimate on T2 is equivalent to a pointwise kernel estimate. To
→ ∞ λ,A
obtain such an estimate we express the cosine kernel on M in terms of the cosine
kernel on the universal cover M, following B´erard. This exploits the fact that the
universal cover M has no conjugate points. B´erard [1, Section D] expresses the
cosine kernel e˜(t,x,y) on M infterms of a finite series plus an error term:
f
N
(21) e˜(t,x,y)=C0 (−1f)k4−kuk(x,y)|t|χ+k−(n+1)/2 d(x,y)2−t2 +˜ǫN(t,x,y),
Xk=0 (cid:16) (cid:17)
where d(x,y) is the Riemannian distance between x and y on M, and χa(r) is the
+
analyticcontinuationofra/Γ(a+1)intheparameterafromtheregionRea> 1.
+ −
It is shown in [1] that under the assumptions of Theorem 1f.1, u satisfies the
k
estimate
(22) u (x,y) C eckd(x,y),
k k
≤
(cid:12) (cid:12)
and that provided that N >(cid:12) d + 1,(cid:12)ǫ˜N is a continuous function satisfying the
(cid:12) (cid:12)
estimate
(23) ǫ˜ (t,x,y) C ecNt.
N N
≤
(cid:12) (cid:12)
The cosine kernel e(t,x,y) o(cid:12)n M is re(cid:12)lated to that on M by
(cid:12) (cid:12)
(24) e(t,x,y)= e˜(t,x,γy)
f
γ Γ
X∈
where Γ is the fundamental group of M, acting on M by deck transformations.
Here we make crucial use of the fact that the cosine kernel is supported where
d(x,y) t: this means that for fixed x,y M the sumfin (24) is finite for each t.
≤ ∈
In fact, since x,y may be viewed as points in M in the same fundamental domain,
i.e. distance O(1) apart in M, the number of terms in the sum (24) is O(eCt)
| |
for some C, since M has bounded sectional cfurvatures and therefore the volume
growth of balls is at most expfonential.
f
Lemma3.7. Letη >0begiven. SupposethatA=αlogλ. Thenifαissufficiently
small and λ sufficiently large, we have
(25) 1 e−iλs′e(s′,x,y)χˆ t−s′ (1 ζ(s′))ds′ Cλ(n−1)/2+η
A A − ≤
(cid:12)Z (cid:18) (cid:19) (cid:12)
uniformly fo(cid:12)(cid:12)r t [0,2ǫA] and (x,y) M2. (cid:12)(cid:12)
(cid:12) ∈ ∈ (cid:12)
Proof of Lemma 3.7. Weexpressthecosinekernele(t,x,y)via(24)anddecompose
e˜as in (21), where we choose N =n/2+1 if n is even and N =n/2+1/2 if N is
odd. Let
ǫN(s′,x,y)= ǫ˜N(s′,x,γy).
γ Γ
X∈
Then since χˆ is supported in [ǫ,2ǫ], the integrand in (25) is zero unless s 2Aǫ.
′
| |≤
In that case, due to the estimate (23) and the exponential estimate of the number
of nonzero terms in this sum, we find that
ǫN(s′,x,y) Ce(CN+C′)αA,
| |≤
showing that for A = αlogλ and α sufficiently small, the contribution of the ǫ
N
term to (25) is O(λη) for any η >0.