Expected values of eigenfunction periods

0.22 MB·

by  Suresh Eswarathasan

EXPECTED VALUES OF EIGENFUNCTION PERIODS
SURESH ESWARATHASAN
Abstract. Let(M,g) be a compactRiemanniansurface. Considera family ofL2 normal-
4 ized Laplace-Beltrami eigenfunctions, written in the semiclassical form h2∆ ϕ = ϕ ,
01 whose eigenvalues satisfy hh−j1 ∈ (1,1+hD] for D > 0 a large enough−conjstagnth.j Let Phjh
2 be a uniform probability measure on the L2 unit-sphere S of this cluster of eigenfunctions
h

n and take u Sh. Given a closed curve γ M, there exists C1(γ,M),C2(γ,M) > 0 and
∈ ⊂
a h0 >0 such that for all h (0,h0],
J ∈
8 C1h1/2 Eh udσ C2h1/2.
≤ (cid:20) Zγ (cid:21)≤
] (cid:12) (cid:12)
P This result contrasts the deterministic (cid:12)(1) uppe(cid:12)rbounds obtained by Chen-Sogge [CS],
O
A Reznikov [Rez], and Zelditch [Zel]. Furthermore, we treat the higher dimensional cases
and compute large deviation estimates. Under a measure zero assumption on the periodic
.
h geodesics in S∗M, we can consider windows of small width D =1 and establish a (h1/2)
t O
a estimate. Lastly, we treat probabilistic Lq restriction bounds along curves.
m
[

1
v 1. Introduction and Main Results
0
1 1.1. Introduction. Let (M,g) be a smooth compact Riemmanian manifold without bound-
7
ary and ϕ be an L2 normalized eigenfunction of the Laplace-Beltrami operator, written in
1 h
. the semiclassical form h2∆ ϕ = ϕ . The quantities known as “periods” I , where
1 − g h h γ,h
0
4 I = ϕ dσ (1.1.1)
γ,h h
1
: Zγ
v (cid:12) (cid:12)
for γ a smoothclosed curve onM with arclen(cid:12)gth meas(cid:12)ure dσ, have garnered much interest as
i
X of late due to its connection with understanding nodal domains of eigenfunctions [JZ]. Such
r quantities are a complement to the study of restricted Lp norms [BGT, Hu] and quantum
a
ergodic restriction [TZ, DZ], and give us information on the fluctuations of eigenfunctions
along a curve.
For perspective, let us recall the global Lp estimates of Sogge [So] in dimension 2: There
exists C > 0 such that ϕ Ch δ(p) where δ(p) = 1 2 for p 6 and δ(p) = 1(1 1)
k hkLp(M) ≤ − 2−p ≥ 2 2−p
for p 6. The results for Lp restriction along curves, due in full generality to Burq, G´erard,
≤
and Tzvetkov [BGT], state: For any finite-length smooth curve γ, there exists C > 0 such
that ϕ Ch η(p) where η(p) = 1 for 2 p 4 and η(p) = 1 1 for p 4.
k hkLp(γ) ≤ − 4 ≤ ≤ 2 − p ≥
The question of asymptotics of (1.1.1), which do not seem to follow trivially from L2
restriction bounds even though an eigenfunction period is the zeroth coefficient of the eigen-
function’sFourierseriesexpansionalongthecurve, wasinitiallyposedoncompact hyperbolic
surfaces by Good and Hejhal [Go, Hej]. They showed, by using the Kuznecov trace formula,
that
I C (1.1.2)
γ,h γ
≤
1
2 SURESHESWARATHASAN
for some C > 0 and all h (0,1]. This demonstrates a stark constrast to the Lp results of
γ
∈
[So] and [BGT]. Subsequent work by Chen and Sogge [CS] generalized this bound to unit-
length geodesics γ on compact surfaces using the H¨ormander parametrix, while Reznikov
[Rez] proved the bound in the case of arithmetic surfaces using representation theory. How-
ever, before these results were proved, Zelditch [Zel] generalized the Kuznecov trace formula
to compact manifolds and obtained (1.1.2) for any closed hypersurface using the full power
of the global symbol calculus for homogeneous Lagrangian distributions. It is this trace
formula that Zelditch established and the consequential asymptotic
2
ϕ dσ = C h 1 + (1), (1.1.3)
hj γ − O
h−jX1<h−1(cid:12)(cid:12)Zγ (cid:12)(cid:12)
(cid:12) (cid:12)
along with an idea of Burq and Le(cid:12)beau [BL](cid:12)that play key roles in our main probabilistic
result. As the Weyl law tells us the number of terms in the sum of (1.1.3) is on the order of
h 2, it follows that “most” of the terms should be on the order of h.
−
Using a simple application of Chebyshev’s inequality and the asymptotic (1.1.3), Jung and
Zelditch were able to obtain the following upper bound:
Theorem. [JZ] For any compact surface M and closed curve γ M, and any g(t) such
⊂
that lim g(t) = , there exists a subsequence of eigenfunctions ϕ of density one such
t→∞ ∞ hj
that
ϕ dσ = (g(h 1)h )1/2 . (1.1.4)
hj O −j j
(cid:12)Zγ (cid:12)
(cid:12) (cid:12) (cid:0) (cid:1)
We would like to mention th(cid:12)at withou(cid:12)t using the Kuznecov trace formula, we can replace
(cid:12) (cid:12)
the estimate (1.1.4) with the little-o estimate o(1) just by assuming a generic condition on γ
and using the “quantum ergodic restriction” result of Toth and Zelditch [TZ, DZ]. Chen and
Sogge [CS] were also able to prove that the constant in (1.1.2) can be replaced with o(1) in
the strictly negative curvature setting via a refined analysis with the Hadamard parametrix.
Both of these estimates can be related to the “Random wave conjecture” of Berry [Berr].
Please see our remarks section and the discussion below.
1.2. Main Results. Let S denote the L2 unit sphere of the space E = u L2(M) =
h h
z ϕ (x),z C endowed with a uniform probability measu{re∈P , defined
hh−j1∈[1,1+hDM) j j j ∈ } h
in Section 2, for some large enough constant D > 0 that is independent of h.
P
Also, let “f(h) g(h)” be the notationthat there exists constants C ,C > 0 independent
1 2
≃
of h and an h > 0 such that for h (0,h ], we have C g(h) f(h) C g(h).
0 0 1 2
∈ ≤ ≤
In this note, we prove the following theorem:
Theorem 1.2.1. Let (M,g) be a compact Riemannian manifold of dimension n and S be a
closed submanifold of dimension d with the induced measure dσ. Then for a positive integer
p and u S , there exists h > 0 such that
h 0
∈
p dp
Eh udσ h 2
≃
ZS
(cid:2)(cid:12) (cid:12) (cid:3)
for h (0,h ], where E is taken with (cid:12)respect to(cid:12) P .
0 h h
∈
PERIODS OF RANDOM EIGENFUNCTION CLUSTERS 3
Furthermore, for M being the median value of the random variable u ,
1,h | S |
P udσ M > r Cexp h dr2 , R
h 1,h −
| |− ≤ −
(cid:20) ZS (cid:21)
where C > 0 is independen(cid:12)t of h. (cid:12) (cid:0) (cid:1)
(cid:12) (cid:12)
That is, we are able to prove for random linear combinations of eigenfunctions u in the
spectral window [1,1+hD), (1.1.4) is improved to a h1/2 upperbound and lowerbound when
p = d = 1 and n = 2, and that the random variable concentrates around the median with
h. Moreover, we can set D = ε for any fixed ε > 0 under a measure zero assumption for
the set of periodic geodesics in S M, but must settle for a h1/2 upperbound only; please see
∗
Section 2.2 for the precise measure 0 assumption and Corollary 4.0.19. We also establish a
deviation estimate for renormalized random variables when p > 1. However, due to the fact
that our pth moments are just the pth powers of our random variables, the most meaningful
deviation estimate occurs when p = 1.
As a simple justification for this estimate, consider the Gaussian random plane waves on
the 2-torus of the form 1 Nh a exp(ih 1 x) for h h 1 [1,1+Dh]. Here, N is equal to
kh j=1 j −j · | −j | ∈ h
the dimension of the eigenspace associated to the window [1,1+Dh], D > 0 is large enough
P
such that N h 1 by Weyl’s Law [Ho], k a normalizing factor that is asymptotic to N1/2,
h ∼ − h h
and the coefficients a are i.i.d. Gaussian random variables. Simply calculating upperbounds
j
on the period of these quantities demonstrates our asymptotic.
For the purpose of comparison, let us observe some deterministic cases. Take S2 and let
Z be a zonal harmonic associated to the eigenvalue h 2 and γ be a small segment going
h −
through the north pole that is also part of a great circle; then Z dσ = (h1/2). In a
| γ h | O
large contrast however, letting γ be the equator saturates the Chen-Sogge bound (1.1.2).
R
Now, on the 2-torus T2 with eigenfunctions of the form einx2, taking γ to be a small segment
of a vertical geodesic gives einx2dσ = (n 1). And similarly to the spherical case, for a
| γ | O −
given closed geodesic γ on T2, there exist a sequence of eigenfunctions ϕ such that ϕ = 1
R j jγ
|
and therefore saturates the corresponding period bound (1.1.2) as well. It is clear that
among these two manifolds, which have diametrically opposite spectral theoretic settings,
there exists a large range of decay rates.
Our probabilistic results can be interpreted as saying a “typical” eigenfunction cluster
should oscillate enough in order to induce a decaying period on a closed curve, along which
the deterministic bounds seemed to be saturated, and that decay better/worse than h1/2 is
“atypical”. According to the “Random wave conjecture” of M. Berry [Berr], which states
the Gaussian random waves model the quantum ergodic case, our estimates appear to be
consistent after applying a simple observation using results in [TZ]; please see our remarks
section.
We spend a majority of our note proving Theorem 1.2.1, but also give the following related
result onrestricted Lq norms alongfinite-length smoothcurves. The result essentially follows
from Theorem 4 in [BL] after making a simple observation.
Theorem 1.2.2. Given the same setup as Theorem 1.2.1 in the 2-dimensional case with
L being the median value of the random variable u for q [2, ), we have that for
q,h Lq(γ)
k k ∈ ∞
all h (0,1]
∈
P u L > r Ce cqh−δ(q)r2
h Lq(γ) q,h −
k k − ≤
(cid:0)(cid:12) (cid:12) (cid:1)
(cid:12) (cid:12)
4 SURESHESWARATHASAN
where C,c > 0 and δ(q) > 0. Moreover, given q [2, ) there exists h such that for all
q 0
∈ ∞
h (0,h ], we have that
0
∈
L 1.
q,h
≃
We would like to mention that [BL] also treats the case of probabilistic L bounds on
∞
M, from which the restricted L bounds on γ follow immediately. Independently and using
∞
different methods, CanzaniandHanin[CH]wereabletoretrievethesesameL upperbounds
∞
over aperiodic manifolds M with more precise constants.
Acknowledgements. TheauthorwouldliketothankJohnTothforsuggesting theproblem
of eigenfunction periods and numerous helpful discussions. Furthermore, the author would
like to thank Nicolas Burq for the suggestion to look at the paper [BL], from which this
article was inspired, and St´ephane Nonnenmacher for his valuable comments on previous
versions of this paper, as well his useful perspective on quantum ergodic restriction. Finally,
the author would like to thank the IHE´S, where the writing of this article took place.
2. Probability and measure concentration on spheres
The probabilistic setup, and following exposition in the first two sections, we use is pat-
terned after that in [BL].
2.1. Measure concentration on spheres. In this section, we define the probability mea-
sures used in our note.
Even though we are dealing with CN, we are only using its properties as a vector space.
Hence, let ustakeRN withtheLebesguemeasure dx = Π dx , andconsider theuniform
1 j N j
probability measure on the unit sphere SN 1 RN of d≤im≤ension N 1. This probability
−
⊂ −
measure P on S(N) := SN 1 is exactly the pushforward of the Gaussian probability mea-
N −
sure
1
Π1 j N e−|xj|2/2dxj
≤ ≤ √2π
on RN via the projection map
x
π : x ,
→ x
| |
where x is the standard Euclidean norm. The main idea is to consider large N and use
| |
these Gaussian probability measures on the corresponding spheres.
For our purposes, we want to compute the corresponding distribution functions explicitly
andutilize theconcentration ofmeasure phenomenon (tobedescribed moreprecisely below).
In order to do this, let us fix M 1 and group the variables on a large sphere S(N) into
≥
M parts. For N 1, we consider N = M N . Using this decomposition of N, we can
j ≥ j=1 j
rewrite the coordinates on RN as (x ,...,x ), with each component x having the polar
1 PM j
decomposition xj = ρjωj RNj for ρj > 0 and ωj SNj−1.
∈ ∈
As we need a particular representation of the distribution function P , we consider the
N
decomposition of N = 2+(N 2) where M = 2. An explicit calculation shows that
−
P ( x > t) = 1 (1 t2)N/2 1, (2.1.1)
N 1 t [0,1) −
| | ∈ −
where (x ,x ) S(N) with N = 2 and N = N 2.
1 2 1 2
∈ −
Furthermore, we have the following deviation formula.
PERIODS OF RANDOM EIGENFUNCTION CLUSTERS 5
Proposition2.1.2. SupposeF is a Lipschitz functionon the sphere SN 1 = S(N), endowed
−
with the uniform probability measure P given above, with respect to the natural geodesic
N
distance. For the median value M(F), we have that for all r > 0
(N 2) r2
PN( F M(F) > r) 2e− − 2kFk2Lip. (2.1.3)
| − | ≤
This large deviation estimate is more commonly referred to as the “concentration of measure
phenomenon”.
The factors in the exponential essentially say that the width of the distribution for F is on
the order of kFkLip for N large. Please see [Led] for further results on measure concentration.
√N
2.2. Probabilistic decomposition of L2(M). Although the results we present were in-
spired by the eigenfunction treatment in [BL], these results were mainly intended for prob-
abilistic applications towards the damped and non-linear wave equations. The probabilistic
treatment in [BL] uses the dyadic Littlewood-Paley decomposition in harmonic analysis
through eigenfunction clusters. For more on the relationship between Littlewood-Paley the-
ory and probabilistic methods, please see [BL].
General manifolds. Take two h dependent sequences a and b (with limits a and b, respec-
h h
tively) with the relation that a < b for all h (0,1] and
h h
∈
0 lima limb .
h h
≤ h 0 ≤ h
→
Furthermore, in the case that a = lima = b = limb exist, we impose a > 0 and that the
h h
rate of decay cannot be too fast, i.e.
Dh b a
h h
≤ −
for some large constant D > 0 that we will specified shortly.
As (M,g) is a compact manifold with the semiclassical relation h2∆ ϕ = ϕ and L2
− j g j j
orthonormal basis of eigenfunctions ϕ , we can consider the set
{ j}∞j=1
I := k N : hh 1 (a ,b ]
h { ∈ −k ∈ h h }
and the spectral cluster
E = u = z ϕ (x), z C ,
h k k k
{ ∈ }
kX∈Ih
which is independent of the choice of eigenbasis. Lastly, set Nh = dimCEh. The Weyl
formula, in its semiclassical form (please see [Zw]) where λ 1 = h, states that there exists a
−
constant C > 0 independent of h such that
Vol(M)
N c (h 1b )n (h 1a )n Ch (n 1). (2.2.1)
h − n (2π)n − h − − h ≤ − −
(cid:12) (cid:0) (cid:1)(cid:12)
A further calculatio(cid:12)n shows that N is asymptotic to eit(cid:12)her c Vol(M)h n(bn an) when
h n (2π)n − h − h
0 a < b or c nan 1Vol(M)h n (b a ) + (h + (b a )2 + (b a ) a a ) when
≤ n − (2π)n − h − h O h − h h − h | − h|
0 < a = b.
(cid:0) (cid:1)
In either case, there exists D > 0 and constants 0 < α < β such that for D h b a
M M h h
≤ −
and h (0,1],
∈
αh n(b a ) N βh n(b a ) (2.2.2)
− h h h − h h
− ≤ ≤ −
6 SURESHESWARATHASAN
which implies N 2. We choose the constant D > 0 in this particular way in order to
h M
≥
guarantee that N properly goes to infinity with h, which is important for our asymptotic
h
results. Otherwise, we may be in a geometrical setting where this is not the case. For
instance, choosing D = 1 in the case of the S2 with the round metric would give us N = 0
M 2 h
for infinitely many h.
Althoughwe canconsider windows ofgrowingsize, wewill beprimarily concerned withthe
case b = 1+D h and a = 1, with the spaces E L2(M) corresponding to the spectral
h M h h
⊂
window (1,1 + hD ]. Moreover, it follows that the quantity N h (n 1), suggesting to
M h − −
≃
us that it should be thought of as the remainder term in the semiclassical form of the Weyl
law.
Aperiodic manifolds. As the case of small-length spectral windows (1,1 + hD], for D > 0
arbitrarily small but fixed, is also of interest, we must make a further dynamical assumption
on M in order to obtain (2.2.2). First, we will review some important results. For the
boundaryless situation, Duistermaat and Guillemin [DG] proved that if the set of periodic
geodesics forms a set of measure zero in S M, then for any ε > 0
∗
Vol(M)
# k : hλ (a ,b ] c (h 1b )n (h 1a )n εh (n 1),
{ k ∈ h h }− n (2π)n − h − − h ≤ − −
(cid:12) (cid:0) (cid:1)(cid:12)
forallh h (ε). Thisisequivalent tosayingthattheremaindertermR(λ)intheWeyllawis
(cid:12)0 (cid:12)
≤
o(λn 1). Compact manifoldswiththisassumption willbereferredtoasbeing“aperiodic”. In
−
thecase ofmanifoldswithboundary, Weyl conjectured that thereexist two-termasymptotics
with R(λ) = o(λn 1); this was proved in the 1980s by Ivrii [Ivr] under the same measure
−
zero assumption for the billiard flow (this result was also proved independently by Melrose
[Mel] using some stronger assumptions involving convexity) .
Hence, choosing ε > 0 small enough implies there exists β > α > 0 such that
0
αh (n 1) N βh (n 1) (2.2.3)
− − h − −
≤ ≤
for h [0,h (ε )) in the case of small-length D spectral windows.
0 0
∈
Probability measure. We are able to endow the unit sphere S E , with respect to the
h h
⊂
L2 norm, with the uniform probability measure P defined in the previous section. An
Nh
important set of random variables we will consider, for u S , are defined in the following
h
∈
way:
b
x,h
ev (u) = u(x) = z ϕ (x) = z, b (2.2.4)
x k k x,h
h b i| |
x,h
hλkX∈(ah,bh] | |
for b = (ϕ (x))Nh . Clearly, the above representation only holds for x such that b = 0,
x,h k k=1 x,h 6
and is the only situation of relevance to our results as we will be computing probabilites of
positive quantities. Notice that b 2 = Nh + (h (n 1)) by the pointwise Weyl law [Ho].
| x,h| Vol(M) O − −
For the simplicity of notation, we will abbreviate P by using the symbol P in the later
Nh h
sections.
3. Generalized Kuznecov trace formulas
The tool of trace formulas has a rich history and has deeply influenced the spectral theory
of automorphic forms, as well as spectral geometry. Even to today, microlocal analysts are
finding new ways to use trace formulas to derive spectral data on manifolds via dynamical
PERIODS OF RANDOM EIGENFUNCTION CLUSTERS 7
information. For an extensive survey on the far-reaching effects of semiclassical trace for-
mulae and their applications towards the spectral theory of the Laplace-Beltrami operator,
please see the survey [CdV] of Colin de Verdiere.
Forourpurposes, wewill beconcentrating onlyontheKuznecov traceformula [Ku], whose
original formulation was
U(t,x,y)dµ (x)dµ (y) = I ([σ]),
1 2 t
ZY1ZY2 [σ]∈ΓYX1\(Γ/ΓY2)
where U(t,x,y) is the full half-wave kernel e it√∆ on a compact quotient M = Γ/D of a
−
non-compact symmetric space D, Y areclosed geodesics for i = 1,2 onM, Γ isthe isotropy
i Yi
group in Γ of Y , and I ([σ]) is a distribution on R that is invariantly associated to [σ]. This
i t
notation was generalized to compact Riemannian manifolds by Zelditch [Zel].
We now provide the spectral asymptotics that were computed in [Zel] from the Kuznevoc
trace formula. The following is a key estimate in the proof of Theorem 1.2.1.
Proposition 3.0.5. For a closed submanifold S of dimension d with surface measure dσ,
we have
ϕ (s)dσ(s) 2 c Vol(S(N S))h (n d) c h (n d)+1
j n ∗ − − S − −
− ≤
(cid:12)(cid:12){j:h−jX1≤h−1}(cid:12)ZS (cid:12) (cid:12)(cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
for N S de(cid:12)noting the normal bundle over S and S(N S) denotin(cid:12)g its unit sphere bundle.
∗ ∗
Now, let us set
E(h,S) := ϕ (s)dσ(s) 2,
j
| |
j:h−jX1≤h−1 ZS
which is also independent of the choice of eigenbasis. We can consider an analogous quantity
to N , which we will label as N(S) and set it as
h h
N(S) := E(h 1b ,S) E(h 1a ,S).
h − h − h
−
Moreover,
N(S) c Vol(S(N S)) (h 1b )n d (h 1a )n d C h ((n d) 1) (3.0.6)
h n ∗ − h − − h − S − − −
− − ≤
for h (cid:12)(0,1]. Furthermore, we(cid:0)set b := ( ϕ (s)dσ(s(cid:1))(cid:12)) CNh. Similarly to the
pointw∈is(cid:12)e Weyl law after (2.2.4), N(S)S,h= b 2S. j (cid:12) j∈Ih ∈
h S,h
| |R
˜
Following the calculations after (2.2.1), we see that there exists D > 0 and constants
S
0 < α < β such that for D˜ h b a and h (0,1],
S h h
≤ − ∈
αh (n d)(b a ) N(S) βh (n d)(b a ). (3.0.7)
− − h h h − − h h
− ≤ ≤ −
4. Proof of main theorem
The main idea in the article [BL] is that by having spectral asymptotics associated to a
summation formula (in our case, the eigenvalue counting function and the Kuznecov sum
8 SURESHESWARATHASAN
formula), we can obtain explicit expected values and deviation estimates. Therefore, let us
consider the random variable defined by
b
S,h
u(s)dσ(s) = z e (s)dσ(s) = z,b = z, b , (4.0.8)
k k S,h S,h
h i h b i| |
ZS ZSkX∈Ih | S,h|
where dσ is the surface measure on S. Once again, we note that the above representation
only holds for b = 0, which is exactly our case since we will compute P [ u > λ] for
| S,h| 6 h | S |
λ 0.
≥ R
We note again that it is useful to view N(S) = b 2 as sort of remainder term for the
h S,h
| |
asymptotics of E(h 1b ,S). For instance, if we again consider the spectral window of large
− h
size D, i.e. b = 1 + hD and a = 1 where D is the supremum of the constants in (2.2.2)
h h
and (3.0.7), then it is clear that N(S) h ((n d) 1). Notice that for n = 2 and d = 1,
h − − −
≃
N(S) 1.
h
≃
Lemma 4.0.9. For an integral p 1 and u S , there exists C > 0 such that for h (0,1],
h p
≥ ∈ ∈
we have
F (u) C (N(S) )p/2
p Lip p h
k k ≤
where F (u) = u(s)dσ(s) p = F (u)p.
p | S | 1
Proof. We have,Rafter (4.0.8) with u = z e and v = w e , that
k∈Ih k k k∈Ih k k
p p P bS,h p P bS,h p
u v = z, b w, b .
S,h S,h
− h b i| | − h b i| |
(cid:12) ZS ZS (cid:12) (cid:12) | S,h| | S,h| (cid:12)
(cid:12)(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)(cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
Using the fact t(cid:12)h(cid:12)at u,(cid:12)v S(cid:12)h giv(cid:12)e(cid:12)s us(cid:12)(cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) ∈ (cid:12) (cid:12) (cid:12)
p 1
b p z, bS,h w, bS,h − z, bS,h p−1−j w, bS,h j ,
S,h
| | · h b i − h b i · h b i · h b i
(cid:12) | S,h| | S,h| (cid:12) (cid:18) j=0 | S,h| | S,h| (cid:19)
(cid:12)(cid:12) (cid:12) (cid:12) (cid:12)(cid:12) X(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12)(cid:12) (cid:12) (cid:12) (cid:12)(cid:12) (cid:12) (cid:12) b (cid:12) (cid:12)
(cid:12) (cid:12) p b p z w, S,h .
S,h
≤ | | · h − b i
S,h
| |
(cid:12) (cid:12)
(cid:12) (cid:12)
Also b p = N(S)p/2, and applying the Cauchy-Schwarz inequality once more shows
| S,h| h
p p p/2
u v p N(S) u v .
− ≤ · h k − kL2(M)
(cid:12) ZS ZS (cid:12)
(cid:12)(cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
Finally, as ku−vkL2(M)(cid:12)(cid:12)(cid:12)= |z−(cid:12) w|(cid:12)≤ C ·(cid:12)d(cid:12)(cid:12)istSNh−1(z,w) = C ·distS(Nh)(u,v), our propositio(cid:3)n
is proved.
Theorem 4.0.10. (Average value A )
p,h
p
A = E u p = p N(S)p/2β( ,N ),
p,h h | | · h 2 h
ZS
(cid:2) (cid:3)
which as h 0 gives us the asymptotic
→
A = C (N(S) /N )p/2 (1+o(1))
p,h p h h
· ·
for some C > 0.
p
PERIODS OF RANDOM EIGENFUNCTION CLUSTERS 9
Proof. It is known that
E u p = ∞pλp 1P u > λ dλ.
h − h
| |
(cid:20) ZS (cid:21) Z0 (cid:20) ZS (cid:21)
(cid:12) (cid:12)
Furthermore, after using (4.0.8) we obtain (cid:12) (cid:12)
b λ
P u > λ = P z, S,h > .
h h
h b i b
(cid:20) ZS (cid:21) (cid:20) | S,h| | S,h|(cid:21)
(cid:12) (cid:12) (cid:12) (cid:12)
Therefore, after using (2.1.1(cid:12)), we(cid:12)see that (cid:12) (cid:12)
λ2
Eh(cid:20)|ZS u|p(cid:21) = pZ0∞λp−1 ·1|bSλ,h|∈[0,1] ·(1− |bS,h|2)Nh−1dλ
1
= p bS,h p ηp−1 (1 η2)Nh−1dλ
·| | · −
Z0
p
= p b p β( ,N ).
S,h h
·| | · 2
Recalling that the beta function has the closed form β(p,N ) = Γ(p/2)Γ(Nh), and as N
2 h Γ((p/2)+Nh) h → ∞
we have the following well-known asymptotic for fixed p:
p
p/2
β(2,Nh) = Γ(p/2)Nh− (1+oh(1)).
Using N(S) = b 2 completes our proof. (cid:3)
h S,h
| |
Theorem 4.0.11. (Estimation of the median M and large deviation)
p,h
For all p Z+ and h [0,1),
∈ ∈
0 M 2p/2(A )p/2 (4.0.12)
p,h 2,h
≤ ≤
and the following deviation estimates hold in the range p < n and h (0,h ]:
n d ∈ 0
−
π 2N(S)p 1
A M h 2 (4.0.13)
p,h p,h
− ≤ 2 · N 2
(cid:12) (cid:12) h −
(cid:12) (cid:12) (cid:0) (cid:1)
and (cid:12) (cid:12)
(cid:12) (cid:12) N 2
P F (u) M > r 2exp h − r2 . (4.0.14)
h | p − p,h| ≤ − p(N(S) )p
h
(cid:0) (cid:1) (cid:0) (cid:1)
Proof. Equation (4.0.12) follows immediately fromproperties of themedian andChebyshev’s
inequality. Then, equation (4.0.13) is just another direct calculation after applying (2.1.3).
Observe that
A M = F (u) M
p,h − p,h k p kL1(S(Nh) −k p,hkL1(S(Nh))
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
F (u) M
(cid:12)(cid:12) (cid:12)(cid:12) ≤ (cid:12)(cid:12) p − p,h L1(S(Nh)) (cid:12)(cid:12)
(cid:13) ∞ (cid:13)
= (cid:13) P F (u)(cid:13) M > λ dλ
h p p,h
−
Z0
2 ∞e−(cid:2)(cid:12)(cid:12)NN(hS−)p2λ2dλ. (cid:12)(cid:12) (cid:3) (4.0.15)
h
≤
Z0
10 SURESHESWARATHASAN
Hence, we obtain that (4.0.15) is equal to
π 2N(S)p 1
h 2.
2 N 2
h
−
(cid:0) (cid:1)
For the purpose of obtaining a nontrivial deviation estimate, we must have that lim
h 0
2N(S)ph = 0, which is only possible if p < n after using (2.2.2) and (3.0.7). Lastly, (4.0.1→4)
cNomh−e2s from the Lipschitz estimate on F (nu−)dand (2.1.3). (cid:3)
p
Corollary 4.0.16. Let M˜ be the median value for the renormalized random variable
p,h
F˜ (u) := Fp(u) . Then for all p Z+, there exists C > 0 such that for all h [0,1),
p pN(S)p/2 ∈ ∈
h
P F˜ (u) M˜ > r Cexp N r2 . (4.0.17)
h p p,h h
| − | ≤ −
Remark 4.0.18. Notice the(cid:0)deviation estimate (cid:1)(4.0.14) is(cid:0)meaningf(cid:1)ul only in the case p = 1,
as the deviation is larger than the average for p > 1 because F (u) = F (u)p. This is our
p 1
reasoning for renormalizing in Corollary 4.0.16.
Using the asymptotics in (2.2.2) and (3.0.7), along with Theorems 4.0.10 and 4.0.11,
immediately gives us
Corollary 4.0.19. For a spectral window of large enough constant size D,
A hdp/2.
p,h
≃
Furthermore, the random variable F (u) concentrates around the median M .
1 1,h
Remark 4.0.20. Under the aperiodicty assumption on M, we can consider windows of small
constant length D > 0, but the asymptotic in Corollary 4.0.19 becomes only a (hpd/2)
O
bound as we are unable to refine the estimate (3.0.6) further at this time. However, the
deviation estimate continues to hold.
Remark 4.0.21. Based on the analysis made in the proofs of Theorems 4.0.10 and 4.0.11, we
see that obtaining actual asymptotics with error terms for the eigenfunction periods comes
down to deducing more precise asymptotics for the remainders of both the Weyl law and
Kuznecov sum formula. Given the various works that improve on the remainder estimate
under different assumptions (for instance [DG, Ivr] and the references therein), it is clear
that this problem is difficult.
5. Proof of restricted versions of the Burq-Lebeau statistics
We now present probabilistic restricted Lq estimates in the spectral window [1,1 + Dh)
with D > 0 large in the general case and with D > 0 small in the aperiodic case, for which
the even periods u2ldσ for l Z+ follow immediately. Before beginning, we want to make
γ ∈
note that the proofs for the global Lq bounds in Section 2.3 of [BL] go through with minor
R
changes, such as establishing the new Lipschitz estimate and the new deviation estimate.
For the sake of simplicity, we write our theorems in the case of finite-length smooth curves
in surfaces γ M. Using the general theorems in Burq, G´erard, and Tzvetkov [BGT] that
⊂
give the corresponding Lipschitz estimates for lower dimensional submanifolds, the higher
dimensional formulations of our theorems follow similarly.