Fixed frequency eigenfunction immersions and supremum
norms of random waves
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Hanin, Boris, and Yaiza Canzani. “Fixed Frequency
Eigenfunction Immersions and Supremum Norms of Random
Waves.” ERA-MS 22, no. 0 (January 2015): 76–86. ©
2015 American Institute of Mathematical Sciences
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Detailed Terms
ELECTRONIC RESEARCH ANNOUNCEMENTS
IN MATHEMATICAL SCIENCES
Volume 22, Pages 76–86 (August 31, 2015)
S 1935-9179 AIMS (2015)
doi:10.3934/era.2015.22.76
FIXED FREQUENCY EIGENFUNCTION IMMERSIONS AND
SUPREMUM NORMS OF RANDOM WAVES
YAIZA CANZANI AND BORIS HANIN
(Communicated by Christopher Sogge)
Abstract. A compact Riemannian manifold may be immersed into Euclidean
space by using high frequency Laplace eigenfunctions. We study the geometry
of the manifold viewed as a metric space endowed with the distance function
from the ambient Euclidean space. As an application we give a new proof
of a result of Burq-Lebeau and others on upper bounds for the sup-norms of
random linear combinations of high frequency eigenfunctions.
1. Introduction
Let (M, g) be an n-dimensional smooth compact Riemannian manifold without
boundary and write ∆g for the (non-negative) Laplace operator acting on L2 (M, g).
For each λ ≥ 0 we consider the space
M
Mλ =
ker(∆g − µ2 ).
(1)
µ∈(λ,λ+1]
λ
We set mλ := dim Mλ and fix an orthonormal basis {ϕj,λ }m
j=1 of Mλ consisting of
eigenfunctions for ∆g :
∆g ϕj,λ = µ2j,λ ϕj,λ ,
kϕj,λ kL2 = 1.
(2)
The purpose of this note is to prove a simple fact about the geometry of the immersions Φλ : M → Rmλ defined by
1
Φλ (x) :=
(ϕ1,λ (x), . . . , ϕmλ ,λ (x))
x ∈ M,
(3)
kλ
where
√ Γ mλ2+1
.
(4)
kλ := 2
Γ m2λ
Received by the editors January 25, 2015 and, in revised form, June 29, 2015.
1991 Mathematics Subject Classification. 35P05 (Pri), 53C42 (Sec).
Key words and phrases. immersions by eigenfunctions, supremum norms, random waves.
We would like to thank J. Toth and S. Zelditch for many useful comments on an earlier draft
of this article, and an anonymous referee for helping us improve the current version. We would
also like to thank N. Burq and S. Eswarathasan for pointing us to the work of N. Burq and
G. Lebeau [4], which gives the matching upper and lower bounds for the sup-norms of random
waves treated in Theorem 3. We would particularly like to thank N. Burq for explaining how to
extract an explicit constant from the proof in [4]. The second author would also like to thank
D. Baskin for several helpful conversations regarding the relationship between dλ and dgλ .
c
2015
American Institute of Mathematical Sciences
76
EIGENFUNCTION IMMERSIONS
77
That Φλ are immersions for λ sufficiently large follows from Theorem 2 below. The
constants kλ satisfy kλ2 = mλ + o(1) and, as we show in §2.1, ensure that
lim kΦλ (x)kl2 (Rmλ ) = p
λ→∞
1
.
volg (M )
(5)
Each immersion Φλ defines a (pseudo-)distance function dλ : M × M → R by
restricting the ambient Euclidean l2 distance:
dλ (x, y) := kΦλ (x) − Φλ (y)kl2 (Rmλ ) .
(6)
Our main result relates dλ to the Riemannian distance dg on M induced by g.
Recall that (M, g) is said to be an aperiodic manifold if for every x ∈ M, the set of
vectors ξ in Tx M for which the geodesic with initial conditions (x, ξ) returns to x
has Liouville measure 0. Any manifold with strictly negative sectional curvatures
is aperiodic. In contrast, Riemannian manifolds whose geodesic flow is periodic are
called Zoll manifolds. Examples of Zoll manifolds include the spheres S n endowed
with the round metric.
Theorem 1. Let (M, g) be a smooth compact Riemannian manifold without boundary that is either Zoll or aperiodic. There exists a constant C so that for all x, y ∈ M
and all λ ≥ 0
dλ (x, y) ≤ C · λ dg (x, y).
(7)
Relation (5) shows that the diameter of M with respect to dλ is bounded so
that (7) is a non-trivial statement only when dg (x, y) is on the order of λ−1 . In
this regime, much more can be said about dλ (x, y). Indeed, on manifolds without
conjugate points, the authors show in [5] that if λ dg (x, y) is bounded as λ → ∞,
then
!
n
n/2
n−2 (λ dg (x, y))
J
2
Γ
+
1
2
2
2
1 + o(λ2 d2g (x, y)) ,
1−
d2λ (x, y) =
n−2
vol(M, g)
n(2π)n
(λ dg (x, y)) 2
(8)
where Jν denotes the Bessel function of the first kind of order ν. In particular, if
λdg (x, y) → 0 as λ → ∞, we have
d2λ (x, y)
(λdg (x, y))
2
=
1
+ o(1).
n vol(M )(2π)n
The proof of the refined result (8) is significantly more delicate than the proof of
Theorem 1, and it can not be carried out on Zoll manifolds.
Note that all the even spherical harmonics take the same values at antipodal
points so that dλ is only a pseudo-distance function in general. In contrast, the
results in [5] prove that dλ is an honest distance function if, for example, (M, g) has
negative sectional curvatures. We deduce Theorem 1 from the following estimate
of Zelditch, which says that Φλ is an almost-isometric immersion for λ sufficiently
large. Let us write geucl for the flat metric on Rd for any d and introduce the
pullback metrics
mλ
X
−2
∗
gλ (x) := Φλ (geucl )(x) = kλ
dx φj,λ (x) ⊗ dy φj,λ (y)
(9)
j=1
for any x ∈ M.
x=y
78
YAIZA CANZANI AND BORIS HANIN
Theorem 2 (Zelditch [19]). Let (M, g) be a compact Riemannian manifold that is
either Zoll or aperiodic with dim M = n. Then, for any x ∈ M,
αn
2
gλ (x) =
(10)
n · λ g(x) (1 + o(1))
(2π)
as λ → ∞, where αn denotes the volume of the unit ball in Rn .
Equation (10) allows us to relate dg to the distance function dgλ of gλ . Theorem 1
then follows by observing that dλ ≤ dgλ (see §3 for details).
The maps Φλ are the Riemannian analogs of Kodaira-type projective embeddings
ΨN : M → PH 0 (L⊗N )∨ of a compact Kähler manifold M into the projectivization
of the dual of the space of global sections H 0 (L⊗N )∨ of the N th tensor power of an
ample holomorphic line bundle L M. Just as the choice of a Riemannian metric
g gives a weighted L2 space on M, a Hermitian metric h on L induces a weighted L2
inner product on H 0 (L⊗N ). This inner product gives rise to a Fubini Study metric
ωFNS on PH 0 (L⊗N )∨ , which plays the role of the Euclidean flat metric geucl used in
the present article.
The holmomorphic analog of Theorem 2 is that the pullback metrics Ψ∗N ωFNS
converge in the C ∞ topology to the curvature form of h. This statement is a result
of Zelditch in [21], with weaker versions going back to Tian [16]. Theorem 1 in this
holomorphic context follows easily from the holomorophic result of Zelditch and
was used to study the sup norms of random homomorphic sections by Feng and
Zelditch in [7].
1.1. Application to Sup Norms of Random Waves. Our main application of
Theorem 1 is to find upper bounds on L∞ -norms of random waves on (M, g).
Definition 1. A Gaussian random wave of frequency λ on (M, g) is a random
function φλ ∈ Mλ defined by
mλ
X
φλ :=
aj,λ ϕj,λ ,
j=1
N (0, kλ−2 )
where the aj,λ ∼
are independent and identically distributed standard
real Gaussian random variables and {ϕj,λ }j is an orthonormal basis for Mλ consisting of Laplace eigenfunctions as defined in (2).
The choice of kλ makes E [kφλ k2 ] = 1. Gaussian random waves were introduced
by Zelditch in [19], and, in addition to their Lp -norms, a number of subsequent
articles have studied their zero sets and critical points (cf., e.g., [9, 10, 13] and
references thererin).
The statistical features of Gaussian random waves of frequency λ are uniquely
determined by their so-called canonical distance, which is precisely dλ (cf. Definition 2). In particular, upper bounds on the expected value of their L∞ -norms
are related to the metric entropy of their canonical distance via Dudley’s entropy
method (see §2.2). We use Theorem 1 to prove in §4 the following result.
Theorem 3. Let (M, g) be a smooth compact boundaryless Riemannian manifold
of dimension n. Assume that (M, g) is either aperiodic or Zoll, and let φλ be a
random wave of frequency λ. Then
s
E [kφλ k∞ ]
2n
≤ 16
.
(11)
lim sup √
volg (M )
log λ
λ→+∞
EIGENFUNCTION IMMERSIONS
79
Remark 1. If (M, g) is aperiodic, the more precise control of the distance function
described in (8) yields a slight
qimprovement of Theorem 3 by reducing the constant
on the right hand side to 16 volgn(M ) . We shall indicate how to use (8) to get the
improved upper bound in Remark 3.
√
The upper bounds of order log λ are not new. Indeed, a simple computation
shows that E [kφλ k∞ ] = E [kψλ k∞ ] if ψλ is chosen uniformly at random from the
unit sphere
SMλ := {f ∈ Mλ : kf k2 = 1}
endowed with the uniform probability measure. For the L2 -normalized random
waves ψλ , the expectation of the L∞ -norms has been studied on many occasions. On
round spheres (S n , ground ), VanderKam obtained in [18] that E [kφλ√k∞ ] = O(log2 λ).
Later, Neuheisel in [14] improved the bound to E [kφλ k∞ ] = O( log λ). On general smooth, compact, boundaryless Riemannian manifolds, Burq and Lebeau [4]
proved the existence of two positive constants C1 , C2 so that as λ → ∞,
p
p
C1 log λ ≤ E [kφλ k∞ ] ≤ C2 log λ.
Although an explicit value for C2 is not stated in [4], Burq relayed to the√authors
in a private communication how one may be extracted. Our constant 16 2 mul1
tiplying (n/volg (M )) 2 is larger (i.e., worse) than theirs, which is approximately
e−e + 1 + √12e . The reason for the discrepancy is that Dudley’s entropy method
makes no assumption on the structure of the probability space on which the Gaussian field in question is defined, while Burq and Lebeau use explicit concentration results on the spheres SMλ . There is a partial converse to Dudley’s entropy method,
called Sudakov minoration, which allows one to obtain lower bounds on the sup
norms of Gaussian fields. Even with the refined control (8) on the canonical
√ distance dλ from [5], the general lower bounds seem to be on the order of λ−1 log λ,
which are significantly worse than those given by Burq and Lebeau. Finally, we
mention that [4] studies more generally upper bounds on Lp norms, p ∈ [2, ∞),
of random waves combination of Laplace eigenfunctions. Results on Lp norms are
also contained in the work of Ayache-Tzvetkov [2] and Tzvetkov [17].
2. Preliminaries
The proof of Theorem 1 relies on three ingredients: the local Weyl law (§2.1),
Dudley’s entropy method (§2.2), and Zelditch’s almost-isometry result (Theorem 2
in the Introduction). Throughout this section (M, g) denotes a compact Riemannian manifold of dimension n.
2.1. Asymptotics for the Spectral Projector. For x, y ∈ M , we write
X
E(0,λ] (x, y) =
ϕj (x)ϕj (y)
λj ∈(0,λ]
for the Schwartz kernel of the orthogonal projection
M
E(0,λ] : L2 (M, g) ker ∆g − µ2 .
µ∈[0,λ)
80
YAIZA CANZANI AND BORIS HANIN
For λ > 0 we define
N (λ) := dim range E(0,λ]
to be the number of eigenvalues of ∆g smaller than λ2 counted with multiplicity
and set
ωn
αn :=
,
(2π)n
where ωn is the volume of the unit ball in Rn . We also write
X
E(λ,λ+1] (x, y) =
ϕj (x)ϕj (y)
λj ∈(λ,λ+1]
for the kernel of the orthogonal projection onto the span of the eigenfunctions of ∆g
2
whose eigenvalues lie in (λ2 , (λ + 1) ]. On several occasions, we use the following
result.
Proposition 1 (Local Weyl Law, [6] and [11]). Let (M, g) be a compact Riemannian manifold of dimension n that is either Zoll or aperiodic. Then
E(0,λ] (x, x) = αn λn + o(λn−1 )
as λ → ∞
E(λ,λ+1] (x, x) = nαn λn−1 + o(λn−1 )
as λ → ∞,
and thus
(12)
with the implied constants being uniform in λ and x ∈ M.
Integrating the above expressions, one has that on compact aperiodic (or Zoll)
manifolds,
N (λ) = αn volg (M )λn + o(λn−1 )
as λ → ∞.
Continuing to write mλ = dim Mλ , we see that
mλ = N (λ + 1) − N (λ) = nαn volg (M )λn−1 + o(λn−1 )
as λ → ∞.
(13)
We also need Hörmander’s off-diagonal pointwise Weyl law:
Lemma 2 ([8]). Let (M, g) be a compact Riemannian manifold of dimension n.
Fix x, y ∈ M. Then
E(0,λ] (x, y) = O(λn−1 ).
(14)
Using the definition (4) of kλ , equation (13), and the fact that
√
√
Γ x + 21 / Γ (x) = x + O(1/ x)
as
x → ∞,
we have from (13) that
kλ2 = mλ + O(λ−
n−1
2
= n αn volg (M )λ
)
n−1
+ o(λn−1 ).
Finally, by combining (12) and (15), we see that for all x ∈ M ,
1q
1
kΦλ (x)kl2 (Rmλ ) =
E(λ,λ+1] (x, x) = p
+ o(1)
kλ
volg (M )
as λ → ∞, which confirms (5).
(15)
(16)
EIGENFUNCTION IMMERSIONS
81
2.2. Dudley’s Entropy Method. Let (Ω, A, P) be a complete probability space.
A measurable mapping φ : Ω → RM is called a random field on M . If for every
N
finite collection {xj }N
j=1 ∈ M the random vector {φ(xj )}j=1 is Gaussian, then φ is
said to be a Gaussian field on M . In addition, if E [φ(x)] = 0 for all x ∈ M , then
φ is said to be centered. The Gaussian random waves of Definition 1 on (M, g) are
examples of centered Gaussian random fields on M .
Definition 2. Let φ be a centered Gaussian field on M. The canonical distance on
M induced by φ is
1
dφ (x, y) := E (φ(x) − φ(y))2 2
for x, y ∈ M.
The law of of any centered Gaussian field is determined completely by its canonical distance function. We note that in general dφ turns M into a pseudo-metric
space. Let us define for each ε > 0 the ε-covering number of (M, dφ ) as
n
o
Ndφ (ε) := inf ` ≥ 1 : ∃ x1 , . . . , x` ∈ M such that ∪`j=1 Bε (xj ) = M ,
(17)
where Bε (x) is the ball of radius ε centered at x. Dudley’s entropy method says
that
√ Z Dφ q
log Ndφ (ε) dε,
(18)
E sup φ(x) ≤ 8 2
x∈M
0
where Dφ = diam(M, dφ )/2. We refer the reader to [1, Theorem 1.3.3] for the
statement √
with an unspecified constant and to the notes [3] of Bartlett where the
constant 8 2 appears.
√ Unfortunately, we are unaware of a refereed article in which
the explicit value 8 2 is given.
We observe that if (M, g) is a smooth compact Riemannian manifold and φλ is
a random wave on M with frequency λ, then it follows from Definition 1 that dλ
and dφλ coincide. Indeed, for x, y ∈ M
d2λ (x, y) = kΦλ (x) − Φλ (y)k2
1
= 2 E(λ,λ+1] (x, x) + E(λ,λ+1] (y, y) − 2E(λ,λ+1] (x, y)
kλ
= E (φλ (x) − φλ (y))2 .
(19)
(20)
3. Proof of Theorem 1
The starting point for our proof is the following estimate on diam(M, dλ ), the
diameter of M with respect to dλ .
Proposition 3. Let (M, g) be a smooth compact boundaryless Riemannian manifold of dimension n. Assume that (M, g) is either aperiodic or Zoll. There exists
δ > 0 so that as λ → ∞,
2
δ ≤ diam(M, dλ ) ≤ p
+ o(1).
volg (M )
(21)
Remark 2. The authors’ results in a forthcoming paper [5] allow one to prove
matching upper and lower bounds in (21) and give the precise asymptotic:
√
2
diam(M, dλ ) = p
+ o(1).
volg (M )
82
YAIZA CANZANI AND BORIS HANIN
The corresponding statement for Zoll manifolds is simpler and follows from the fact
that E(0,λ] has a complete asymptotic expansion (cf. [20]).
Proof. For x, y ∈ M , we have
2
d2λ (x, y) =
kΦλ (x) − Φλ (y)k
.
kλ2
(22)
2
From (15) and the fact that kΦλ (x)k = Π(λ,λ+1] (x, x) for all x ∈ M , we conclude
2
d2λ (x, y) =
kΦλ (x) − Φλ (y)k
4
≤
+ o(1).
mλ + O(1)
volg (M )
Taking the supremum over x, y ∈ M proves the upper bound in (21). To prove the
lower bound in (21), we proceed by contradiction. That is, suppose that d2λ (x, y)
is not bounded below. In virtue of (22) and (15), this means that
kΦλ (x) − Φλ (y)k = o(λn−1 )
(23)
for all x, y ∈ M. Consider the map Ψλ : M → RN (λ) given by
Ψλ (x) = ϕ1 (x), . . . , ϕN (λ) (x)
for x ∈ M,
where we continue to write N (λ) for the number of eigenvalues of ∆g in the interval
(0, λ2 ]. Note that the difference between Ψλ and Φλ is that Ψλ includes all the
eigenfunctions up to eigenvalue λ2 . The local Weyl law (14) shows that for x, y ∈ M
with x 6= y,
2
kΨλ (x) − Ψλ (y)k = E(0,λ] (x, x) + E(0,λ] (y, y) − 2E(0,λ] (x, y) = Ω (λn ) .
(24)
On the other hand, for any positive integer k,
2
kΨk (x) − Ψk (y)k =
k−1
X
2
kΦj (x) − Φj (y)k .
j=0
The assumption (23) shows that
2
kΨk (x) − Ψk (y)k = o(k n ),
which contradicts (24).
Let us denote by dgλ the distance function for the metric gλ defined in (9).
We continue to write dg for the distance function of the background metric g.
Combining
p (21) with (15), we see that Theorem 1 reduces to showing that for any
K > 2/ volg (M ) there exist positive constants C, λ0 depending on (M, g) such
(x,y)
that ddλg (x,y)
≤ Cλ for all λ ≥ λ0 and x, y ∈ M with dg (x, y) ≤ K/λ. We write
dλ (x, y)
dλ (x, y) dgλ (x, y)
=
·
.
dg (x, y)
dgλ (x, y) dg (x, y)
(25)
Note that ddgλ (x,y)
(x,y) ≤ 1. Indeed, let γ : [0, 1] → M be any length minimizing
λ
geodesic between two given points x and y with respect to the metric gλ = Φ∗λ (geucl )
and consider the curve Φ ◦ γ : [0, 1] → Rmλ joining Φλ (x) and Φλ (y). Then
Z 1
d
Φλ (γ(t))
dλ (x, y) = kΦλ (x) − Φλ (y)kgeucl ≤
dt = dgλ (x, y).
(26)
dt
g
0
eucl
EIGENFUNCTION IMMERSIONS
d
83
(x,y)
We turn to show that there exists C > 0 such that dggλ(x,y) ≤ Cλ. Fix x, y ∈ M
and a unit speed geodesic γ from x to y with respect to the metric g. Theorem 2
n+1
states that there exists C > 0 so that as λ → ∞, one has gλ = C · λmλ g (1 + o(1)).
Therefore, after suitably adjusting C,
Z dg (x,y)
q
d
γ(t) dt ≤ C λn+1 dg (x, y).
(27)
dg (x, y) ≤
λ
0
dt
gλ
mλ
Substituting (26) and (27) into (25) and using the asymptotics (15) for mλ
completes the proof of Theorem 1.
4. Proof of Theorem 3
Let φλ be a Gaussian random wave of (M, g) with frequency λ. As explained
in (20), the distance dλ induced by the immersions and the distance dφλ induced
by φλ coincide. Theorem 1 therefore allows us to control the metric entropy Ndλ
of dφλ (see (17)).
Proposition 4. Let (M, g) be a compact aperiodic or Zoll Riemannian manifold.
There exist positive constants C and λ0 such that if φλ is a Gaussian random wave
of frequency λ, then
λn
Ndλ (ε) ≤ C n
ε
for all ε > 0 and λ ≥ λ0 .
ε
Proof. From Theorem 1, it follows that if x, y ∈ M are such that dg (x, y) ≤ Cλ
and
λ exceeds some fixed λ0 , then dλ (x, y) ≤ ε. Writing Bd (x, r) for a ball of radius r
ε
⊆ Bdλ (x, ε) for each x ∈ M. Hence,
in the distance d, we see that Bdg x, Cλ
ε (28)
Ndλ (ε) ≤ Ndg
Cλ
for every λ ≥ λ0 . Define αg := inf{kg (x) : x ∈ M }, where kg denotes the sectional
√
curvature function for (M, g), and set π/ αg = ∞ whenever αg ≤ 0. For λ large
enough,
n
o
π
ε
< min injg (M ), √ , 2π ,
Cλ
αg
and so we may apply [12, Lemma 4.1] to obtain
ε 2n n−1 ε −n
Ndg
≤ volg (M )
π
,
Cλ
sn−1
Cλ
(29)
where sn−1 is the volume of the (n − 1)-dimensional unit sphere in Rn . The claim
follows from combining (28) and (29).
To complete the proof of Theorem 3, we input the upper bound on Ndλ of
Proposition 4 into Dudley’s entropy estimate (18) to conclude that for a constant
C depending (M, g) and all λ exceeding some λ0 ,
r
C 1/n λ √ Z Dλ
E sup φλ (x) ≤ 8 2n
log
dt
t
x∈M
0
r √ Z 1
αλ
= 8Dλ 2n
log
dt,
t
0
84
YAIZA CANZANI AND BORIS HANIN
where
Dλ :=
diam(M, dλ )
1
≤p
+ o(1),
2
volg (M )
and
αλ :=
C 1/n
λ.
Dλ
(30)
Setting aλ := 1/ log αλ , we get
Z 1
√
√ p
(1 − aλ log t)1/2 dt.
E sup φλ (x) ≤ 8 2Dλ n log αλ
x∈M
(31)
0
We now use the estimate
Z 1
aλ
1/2
(1 − aλ · log t) dt − 1 ≤
2
0
as aλ → 0,
(32)
whose proof we give in Claim 5 below. Hence, combining (32) with (31) and the
definition (30) of αλ , we have
√
√ q
aλ .
(33)
E sup φλ (x) ≤ 8 2Dλ n log λ + log C 1/n /Dλ 1 +
2
x∈M
Since Proposition 3 guarantees that log(C 1/n /Dλ ) is uniformly bounded in λ, and
aλ → 0, we conclude from (33) that
√ r
E [ supx∈M φλ (x)]
n
√
≤8 2
+ o(1).
volg (M )
log λ
Since φλ is symmetric,
s
E [ kφλ k∞ ]
E [ supx∈M φλ (x)]
2n
√
√
+ o(1).
≤2
≤ 16
volg (M )
log λ
log λ
(34)
Taking the lim sup as λ → ∞ in (34) completes the proof.
Remark 3. The proof of the result stated in Remark 1 for the aperiodic case follows
from using the diameter asymptotics in Remark 2, which give Dλ = √ 1
+ o(1)
2 volg (M )
instead of Dλ ≤ √
1
volg (M )
+ o(1).
Claim 5 (Proof of (32)). As a → 0,
Z 1
a
1/2
(1 − a · log x) dx − 1 ≤ .
2
0
1/2
Proof. Making the change of variables u = (1 − a log x) , we get
Z
Z 1
2 1 ∞ 2 − u2
1/2
u e a du.
(1 − a · log x) dx = e a
a
1
0
2
a ∂
−u /a
Observe that − 2u
and integrate by parts in (35) to get
∂u preserves e
Z 1
Z ∞
1
u2
1/2
a
(1 − a · log x) dx = 1 + e
e− a du
0
r1 Z ∞
1
t2
a
= 1 + ea
e− 2 dt
√
2
2
a
(35)
EIGENFUNCTION IMMERSIONS
85
Using the classical estimate
Z
∞
2
e
x
− y2
x2
e− 2
dy ≤
x
shows that, as a → 0,
Z
0
1
1/2
(1 − a · log x)
a
dx − 1 ≤ ,
2
as desired.
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YAIZA CANZANI AND BORIS HANIN
Department of Mathematics, Harvard University, Cambridge, United States.
E-mail address: canzani@math.harvard.edu
Department of Mathematics, Massachusetts Institute of Technology, Cambridge,
United States.
E-mail address: bhanin@mit.edu