Annales Mathématiques Blaise Pascal

Pierre Fougères

Ivan Gentil

Boguslaw Zegarliński

Solution of a class of reaction-diffusion systems via logarithmic Sobolev inequality

Omar Anza Hafsa

Nicolas Clozeau

Jean-Philippe Mandallena

Homogenization of nonconvex unbounded singular integrals

Teodor Banica

Weingarten integration over noncommutative homogeneous spaces

Virginie Bonnaillie-Noël

Frédéric Hérau

Nicolas Raymond

ANNALES MATHÉMATIQUES
BLAISE PASCAL
Virginie Bonnaillie-Noël,
Frédéric Hérau & Nicolas Raymond
Holomorphic extension of the de Gennes function
Volume 24, no 2 (2017), p. 225-234.
<http://ambp.cedram.org/item?id=AMBP_2017__24_2_225_0>
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Annales mathématiques Blaise Pascal 24, 225-234 (2017)
Holomorphic extension of the de Gennes function
Virginie Bonnaillie-Noël
Frédéric Hérau
Nicolas Raymond
Abstract
This note is devoted to prove that the de Gennes function has a holomorphic
extension on a half strip containing R+.
1. Introduction
1.1. About the de Gennes operator
The de Gennes operator plays an important role in the investigation of
the magnetic Schrödinger operator. Consider the half-plane
R2+ = {(s, t) ∈ R2 : t > 0} ,
and a magnetic field B(s, t) = 1. We also introduce an associated vector
potential A(s, t) = (−t, 0) such that B(s, t) = ∇×A(s, t). Then, we define
LA the Neumann realization of the differential operator (−i∇+A)2 acting
on L2(R2+). By using the partial Fourier transform in s, we obtain the
direct integral:
FLAF−1 =
∫ ⊕
R
Lξ dξ ,
where, for ξ ∈ R, Lξ denotes the Neumann realization on R+ of the differ-
ential operator −∂2t + (ξ − t)2. The self-adjoint operator Lξ is called the
de Gennes operator with parameter ξ ∈ R. Let us give a precise definition
of this operator. For ξ ∈ R, we introduce the sesquilinear form
∀ ψ ∈ B1(R+) , qξ(ϕ,ψ) =
∫
R+
ϕ′ψ
′ + (t− ξ)2ϕψ dt ,
where
B1(R+) =
{
ψ ∈ H1(R+) : tψ ∈ L2(R+)
}
.
Keywords: de Gennes operator, holomorphic extension, holomorphic perturbation
theory.
2010 Mathematics Subject Classification: 81Q15, 32A10.
225
V. Bonnaillie-Noël, F. Hérau & N. Raymond
This form is continuous on the Hilbert space B1(R+) and Hermitian. Up
to the addition of a constant, qξ is coercive on B1(R+). Therefore, in
virtue of the Lax–Milgram representation theorem (see for instance [4,
Theorem 3.4]), we can consider the associated closed (and self-adjoint,
since the form is Hermitian) operator Lξ and its domain is
Dom(Lξ) =
{
ψ ∈ B2(R+) : ψ′(0) = 0
}
⊂ B1(R+) ,
where
B2(R+) =
{
ψ ∈ H2(R+) : t2ψ ∈ L2(R+)
}
.
Since B1(R+) is compactly embedded in L2(R+), we deduce that Lξ has
compact resolvent and we can consider the non-decreasing sequence of its
eigenvalues (µk(ξ))k∈N∗ . Each eigenspace is of dimension one (due to the
Neumann condition for instance).
Definition 1.1. We call the function R 3 ξ 7→ µ1(ξ) ∈ R the de Gennes
function. For shortness, we let µ = µ1.
In the terminology of Kato’s perturbation theory (see [6, Section VII.2]),
the family of self-adjoint operators (Lξ)ξ∈R is analytic of type (A). In other
words, for all ξ0 ∈ R, the family (Lξ)ξ∈R can be extended into a family of
closed operators (Lξ)ξ∈V with V a complex open ball containing ξ0 such
that
• the domain Dom(Lξ) does not depend on ξ ∈ V,
• for all ψ ∈ Dom(Lξ0), the map V 3 ξ 7→ Lξψ is holomorphic.
By using Kato’s theory and since µ is a simple eigenvalue, µ is real analytic,
or equivalently, for all given ξ0 ∈ R, µ has a local holomorphic extension
in a neighborhood of ξ0. We refer to [8, Section 2.4] for a direct proof. We
also recall that µ has a minimum (see [2] or [3, Proposition 3.2.2]). We let
Θ0 = min
ξ∈R
µ(ξ).
This note answers the following question: Can the de Gennes function
µ be holomorphically extended on a strip about R+?
1.2. Motivations and result
The aim of this note is to prove the following theorem.
226
Holomorphic extension
Theorem 1.2. There exist ε > 0 and F a holomorphic function on the
half strip
Sε := {ξ ∈ C : Re ξ > 0 , |Im ξ| < ε} ,
such that, for all ξ ∈ R, µ(ξ) = F (ξ). Moreover, for all ξ ∈ Sε, we have
ReF (ξ) ≥ µ(Re ξ)− (Im ξ)2 ,
and F (ξ) belongs to the discrete spectrum of Lξ.
Remark 1.3. As we can see from the proof, for all k ∈ N∗, µk has a
holomorphic extension on a half strip about R+. Of course, the size of this
strip depends on k and is expected to shrink when k becomes large.
Remark 1.4. The method used in this note can be applied to the family
of Montgomery operators
−∂2t +
(
ξ − t
n+1
n+ 1
)2
,
acting on L2(R), with n ∈ N∗ and ξ ∈ R. Their eigenvalues have holo-
morphic extensions on a half strip about R+. The Montgomery operators
appear in the case of vanishing magnetic fields.
This theorem is motivated by various spectral questions. Firstly, it plays
an important role in the investigation of the resonances induced by local
perturbations of LA. Indeed, these resonances can often be defined by
analytic dilations (see for instance [5, Chapter 16]) and the existence of a
holomorphic extension of the band functions µk could be useful to reveal
new magnetic spectral phenomena. Secondly, it is also strongly related
to complex WKB analysis (as we have shown in [1]). In particular, the
WKB constructions in [1, Section 1.2.2] are a priori local and an accurate
knowledge of the holomorphy strip would allow to extend the domain of
validity of these constructions (see [1, Section 4.2] where the size of the
holomorphy strip is involved). Moreover, this holomorphic extension is
crucial in the study of the semiclassical magnetic tunneling effect when
there are symmetries. Actually, this effect can only be fully understood in
the complexified phase space.
227
V. Bonnaillie-Noël, F. Hérau & N. Raymond
2. Proof of the theorem
2.1. Preliminary considerations
Let us prove the following separation lemma.
Lemma 2.1. For all k ∈ N∗, there exists ck > 0 such that
∀ ξ ∈ R , µk+1(ξ)− µk(ξ) ≥ ck .
Proof. We recall that the functions µk are real analytic and that, for all
integer k, we have µk < µk+1. As a consequence of the harmonic approxi-
mation (see for instance [3, Section 7.1] or [8, Section 3.2]), we have
∀ k ∈ N∗ , lim
ξ→+∞
µk(ξ) = 2k − 1 . (2.1)
Therefore, for all k ∈ N∗, there exist Ξk, dk > 0 such that, for all ξ ≥ Ξk,
µk+1(ξ)− µk(ξ) ≥ dk .
For α > 0, let us consider L−α. By dilation, this operator is unitarily
equivalent to the Neumann realization on R+ of the differential operator
α2(α−4D2τ + (τ + 1)2) ,
where we denote Dt = −i∂t. The potential R+ 3 τ 7→ (τ + 1)2 is minimal
at τ = 0 and thus a variant of the harmonic approximation near τ = 0
shows that (see for instance [7, Théorème 1.7])
∀ k ∈ N∗ , µk(−α) =
α→+∞
α2 + νkα
2
3 + o(α
2
3 ) , (2.2)
where (νk)k∈N∗ is the increasing sequence of the eigenvalues of the Neu-
mann realization on R+ of D2τ + 2τ . Let us briefly recall the main steps of
the proof of (2.2). The regime α→ +∞ is equivalent to the semiclassical
regime h = α−2 → 0 and one knows that the eigenfunctions associated
with the low lying eigenvalues are concentrated near the minimum of the
potential τ = 0. Near this point, we can perform a Taylor expansion so
that the operator asymptotically becomes
α2(α−4D2τ + 1 + 2τ) .
Then, we homogenize this operator with the rescaling τ = h
2
3 τ̃ and the
conclusion follows.
We deduce that
lim
α→+∞
(µk+1(−α)− µk(−α)) = +∞ . (2.3)
228
Holomorphic extension
Combining (2.1), (2.3), the simplicity of the µk and their continuity, the
result follows. 
2.2. The family (Lξ)ξ∈Sε
Let us fix ε > 0 and explain how the operator Lξ is defined for ξ ∈ Sε. We
let, for all ξ ∈ Sε,
∀ ϕ,ψ ∈ B1(R+) , qξ(ϕ,ψ) =
∫
R+
ϕ′ψ
′ + (t− ξ)2ϕψ dt .
The sesquilinear form qξ is well defined and continuous on B1(R+). In
order to apply the Lax–Milgram representation theorem, let us consider
the quadratic form B1(R+) 3 ψ 7→ Re qξ(ψ,ψ). We have
Re qξ(ψ,ψ) = ‖ψ′‖2 +
∫
R+
Re (t− ξ)2|ψ|2 dt ,
and
Re (t− ξ)2 = (t− Re ξ)2 − (Im ξ)2 .
In particular, we have
Re qξ(ψ,ψ) ≥ qRe ξ(ψ,ψ)− (Im ξ)2‖ψ‖2 ≥ (µ(Re ξ)− (Im ξ)2)‖ψ‖2
≥ (Θ0 − (Im ξ)2)‖ψ‖2 . (2.4)
Thus, up to the addition of constant, Re qξ is coercive on B1(R+). We
deduce that, for all ε > 0 and all ξ ∈ Sε, there exist C, c > 0 such that,
for all ψ ∈ B1(R+), ∣∣∣qξ(ψ,ψ) + C‖ψ‖2∣∣∣ ≥ c‖ψ‖2B1(R+) .
Therefore, we can apply the Lax–Milgram theorem and define a closed
operator Lξ. It satisfies
∀ ψ ∈ B1(R+) , ∀ ϕ ∈ Dom(Lξ) , qξ(ϕ,ψ) = 〈Lξϕ,ψ〉 .
We can easily show that
Dom(Lξ) =
{
ψ ∈ B2(R+) : ψ′(0) = 0
}
,
so that Lξ has compact resolvent and (Lξ)ξ∈Sε is a holomorphic family of
type (A). If ε ∈ (0,
√
Θ0), as a consequence of (2.4), we get that, for all
ξ ∈ Sε, Lξ is bijective. Indeed, we have, for all ψ ∈ Dom(Lξ),
(Θ0 − ε2)‖ψ‖ ≤ ‖Lξψ‖ .
229
V. Bonnaillie-Noël, F. Hérau & N. Raymond
From this, the operator Lξ is injective with closed range. Since L∗ξ = Lξ,
we deduce that L∗ξ is also injective and thus Lξ has a dense image. We get
that, for all ξ ∈ Sε,
‖L−1ξ ‖ ≤ (Θ0 − (Im ξ)
2)−1 .
2.3. Resolvent and projection estimates
Let us now prove the main result of this note.
2.3.1. Difference of resolvents
For 0 < r1 < r2, and z0 ∈ C, we define the annulus
Ar1,r2(z0) = {z ∈ C : r1 < |z − z0| < r2} .
We consider, for all r > 0 and all ξ ∈ Sε,
Ar,ξ := Ar,2r(µ(Re ξ)) .
Using Lemma 2.1 with k = 1, we know that there exists r0 > 0 such that
• for all ξ ∈ Sε,
Ar0,ξ ⊂ ρ(LRe ξ) , dist(Ar0,ξ, sp(LRe ξ)) ≥ r0 > 0 ,
where ρ(LRe ξ) denotes the resolvent set of LRe ξ,
• the disk of center µ(Re ξ) with radius r0, denoted by D(µ(Re ξ), r0),
contains only one eigenvalue of LRe ξ.
The following proposition states an approximation of the resolvent of Lξ
by the one of LRe ξ when ε goes to 0.
Proposition 2.2. There exist C, ε0 > 0 such that for all ε ∈ (0, ε0) and
all ξ ∈ Sε, we have Ar0,ξ ⊂ ρ(Lξ). In addition, we have, for all ξ ∈ Sε
and z ∈ Ar0,ξ, ∥∥∥(Lξ − z)−1 − (LRe ξ − z)−1∥∥∥ ≤ Cε. (2.5)
Proof. For all z ∈ Ar0,ξ, we have
Lξ − z = LRe ξ − 2i Im ξ(t− Re ξ)− (Im ξ)2 − z ,
or, equivalently,
Lξ−z = (Id−2i Im ξ(t−Re ξ)(LRe ξ−z)−1−(Im ξ)2(LRe ξ−z)−1)(LRe ξ−z) .
230
Holomorphic extension
Thus, we have to show that
Id− 2i Im ξ(t− Re ξ)(LRe ξ − z)−1 − (Im ξ)2(LRe ξ − z)−1
is bijective. By the spectral theorem, we get
‖(LRe ξ − z)−1‖ ≤
1
dist(z, sp(LRe ξ))
≤ 1
r0
. (2.6)
Let us also estimate (t− Re ξ)(LRe ξ − z)−1. Take u ∈ L2(R+) and let
v = (LRe ξ − z)−1u.
We have
(D2t + (t− Re ξ)2 − z)v = (LRe ξ − z)v = u,
and we must estimate (t − Re ξ)v. Taking the scalar product with v, we
get
‖(t− Re ξ)v‖2 − Re z‖v‖2 ≤ 〈v, u〉 ≤ ‖u‖‖v‖ ≤ r−10 ‖u‖2 .
Since the de Gennes function satisfies µ(Re ξ) ≤ 1 when Re ξ ≥ 0 (see [3,
Proposition 3.2.2]), we get, for all z ∈ Ar0,ξ,
Re z ≤ µ(Re ξ) + 2r0 ≤ 1 + 2r0 .
We find
‖(t− Re ξ)(LRe ξ − z)−1‖ ≤
√
3r−10 + r
−2
0 =: r1 . (2.7)
From (2.6) and (2.7), it follows that
‖ − 2i Im ξ(t− Re ξ)(LRe ξ − z)−1 − (Im ξ)2(LRe ξ − z)−1‖
≤ 2|Im ξ|r1 + |Im ξ|2r−10 ≤ 2εr1 + (εr
− 12
0 )2 .
Thus, there exists ε0 > 0 such that for all ε ∈ (0, ε0), we have, for all
ξ ∈ Sε and z ∈ Ar0,ξ,
‖ − 2i Im ξ(t− Re ξ)(LRe ξ − z)−1 − (Im ξ)2(LRe ξ − z)−1‖ < 1 .
This shows that Id − 2i Im ξ(t − Re ξ)(LRe ξ − z)−1 − (Im ξ)2(LRe ξ − z)−1
is bijective and thus Lξ − z is bijective. The estimate of the difference of
the resolvents immediately follows. 
231
V. Bonnaillie-Noël, F. Hérau & N. Raymond
2.3.2. Projections
Let us now introduce
Pξ =
1
2iπ
∫
Γξ
(z − Lξ)−1 dz ,
where Γξ is the circle of center µ(Re ξ) and radius 32r0. It is classical to
show, by using the resolvent and Cauchy formulas, that Pξ is the projection
on the characteristic space associated with the eigenvalues of Lξ enlaced
by Γξ.
Lemma 2.3. There exists ε0 > 0 such that for all ε ∈ (0, ε0) and all
ξ ∈ Sε, the rank of Pξ is 1 and Pξ is holomorphic on Sε.
Proof. Let us fix ξ0 ∈ Sε. There exists δ > 0 such that, for all ξ ∈ D(ξ0, δ),
we have ξ ∈ Sε, Γξ0 ⊂ Ar0,ξ and
Pξ =
1
2iπ
∫
Γξ0
(z − Lξ)−1 dz .
This comes from the continuity of the family of circles Γξ and the holo-
morphy of the resolvent z 7→ (z−Lξ)−1. From this, we now see that Pξ is
holomorphic on the disk D(ξ0, δ). Indeed, since (Lξ)ξ∈Sε is a holomorphic
family of type (A), D(ξ0, δ) 3 ξ 7→ (z−Lξ)−1 is holomorphic, uniformly for
z ∈ Γξ0 . Finally, with (2.5), there exists ε0 > 0 such that, for all ε ∈ (0, ε0)
and all ξ ∈ Sε,
‖Pξ − PRe ξ‖ ≤
3r0
2 Cε < 1 .
By a classical lemma about pairs of projections (see [6, Section I.4.6]) and
since the rank of PRe ξ is 1, we deduce that the rank of Pξ is 1. 
Remark 2.4. We can also choose ε0 > 0 so that, for all ε ∈ (0, ε0), all
ξ ∈ Sε and all ψ ∈ L2(R+),∣∣∣∣∣
∫
R+
(Pξψ)2 dt−
∫
R+
(PRe ξψ)2 dt
∣∣∣∣∣ ≤ 12‖ψ‖2 .
2.3.3. Conclusion
Since Pξ commutes with Lξ, we can consider the restriction of Lξ to the
range of Pξ. This restriction is a linear map in dimension one. Therefore,
232
Holomorphic extension
for all ξ ∈ Sε, there exists a unique F (ξ) ∈ C such that
∀ ψ ∈ L2(R+) , Lξ(Pξψ) = F (ξ)Pξψ .
Taking the scalar product with Pξψ, we get, for all ψ ∈ L2(R+) and all
ξ ∈ Sε,
F (ξ)
∫
R+
(Pξψ)2 dt =
∫
R+
Lξ(Pξψ)Pξψ dt .
Let ξ0 ∈ Sε. By Remark 2.4, there exists ψ0 ∈ L2(R+) (a real normalized
eigenfunction of LRe ξ0 associated with µ(Re ξ0)) such that∫
R+
(Pξ0ψ0)2 dt 6= 0 .
Thus there exists a neighborhood V of ξ0 such that, for all ξ ∈ V,
F (ξ) =
∫
R+ Lξ(Pξψ0)Pξψ0 dt∫
R+(Pξψ0)
2 dt .
Thus F is holomorphic near ξ0. When ξ ∈ R, we have clearly F (ξ) = µ(ξ).
This, with (2.4), terminates the proof of Theorem 1.2.
References
[1] V. Bonnaillie-Noël, F. Hérau & N. Raymond – “Magnetic WKB
constructions”, Arch. Ration. Mech. Anal. 221 (2016), no. 2, p. 817–
891.
[2] M. Dauge & B. Helffer – “Eigenvalues variation. I. Neumann prob-
lem for Sturm-Liouville operators”, J. Differ. Equations 104 (1993),
no. 2, p. 243–262.
[3] S. Fournais & B. Helffer – Spectral methods in surface super-
conductivity, Progress in Nonlinear Differential Equations and their
Applications, vol. 77, Birkhäuser, Boston, MA, 2010.
[4] B. Helffer – Spectral theory and its applications, Cambridge Stud-
ies in Advanced Mathematics, vol. 139, Cambridge University Press,
Cambridge, 2013.
[5] P. D. Hislop & I. M. Sigal – Introduction to spectral theory, Applied
Mathematical Sciences, vol. 113, Springer, 1996, With applications to
Schrödinger operators.
233
V. Bonnaillie-Noël, F. Hérau & N. Raymond
[6] T. Kato – Perturbation theory for linear operators, Die Grundlehren
der mathematischen Wissenschaften, vol. 132, Springer, 1966.
[7] N. Popoff – “Sur l’opérateur de schrödinger magnétique dans un
domaine diédral”, Thèse, Université de Rennes 1 (France), 2012.
[8] N. Raymond – Bound states of the magnetic schrödinger operator,
EMS Tracts in Mathematics, vol. 27, European Mathematical Society,
2017.
Virginie Bonnaillie-Noël
Département de mathématiques et
applications,
École normale supérieure, CNRS
PSL Research University
75005 Paris, France
bonnaillie@math.cnrs.fr
Frédéric Hérau
LMJL - UMR6629
Université de Nantes, CNRS
2 rue de la Houssinière, BP 92208
44322 Nantes cedex 3, France
herau@univ-nantes.fr
Nicolas Raymond
IRMAR, Univ. Rennes 1, CNRS
Campus de Beaulieu
35042 Rennes cedex, France
nicolas.raymond@univ-rennes1.fr
234


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Reduced $L_{q,p}$-Cohomology of Some Twisted Products

Olivier Ramaré

Modified truncated Perron formulae

David Zimmermann

ANNALES MATHÉMATIQUES
BLAISE PASCAL
David Zimmermann
Elementary proof of logarithmic Sobolev inequalities for
Gaussian convolutions on R
Volume 23, no 1 (2016), p. 129-140.
<http://ambp.cedram.org/item?id=AMBP_2016__23_1_129_0>
© Annales mathématiques Blaise Pascal, 2016, tous droits réservés.
L’accès aux articles de la revue « Annales mathématiques Blaise Pas-
cal » (http://ambp.cedram.org/), implique l’accord avec les condi-
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utilisation commerciale ou impression systématique est constitutive
d’une infraction pénale. Toute copie ou impression de ce fichier doit
contenir la présente mention de copyright.
Publication éditée par le laboratoire de mathématiques
de l’université Blaise-Pascal, UMR 6620 du CNRS
Clermont-Ferrand — France
cedram
Article mis en ligne dans le cadre du
Centre de diffusion des revues académiques de mathématiques
http://www.cedram.org/
Annales mathématiques Blaise Pascal 23, 129-140 (2016)
Elementary proof of logarithmic Sobolev
inequalities for Gaussian convolutions on R
David Zimmermann
Abstract
In a 2013 paper, the author showed that the convolution of a compactly
supported measure on the real line with a Gaussian measure satisfies a logarithmic
Sobolev inequality (LSI). In a 2014 paper, the author gave bounds for the optimal
constants in these LSIs. In this paper, we give a simpler, elementary proof of this
result.
Une preuve élémentaire des inégalités de Sobolev logarithmiques
pour des convolutions gaussiennes sur R
Résumé
Dans un article de 2013, l’auteur a montré que la convolution d’une mesure
à support compact sur la droite réelle avec une mesure gaussienne satisfait une
inégalité de Sobolev logarithmique. Dans un article de 2014, l’auteur a donné des
bornes pour les constantes optimales dans ces inégalités de Sobolev logarithmiques.
Dans cet article, nous donnons une preuve élémentaire simple de ce résultat.
1. Introduction
A probability measure µ on Rn is said to satisfy a logarithmic Sobolev
inequality (LSI) with constant c ∈ R if
Entµ(f2) ≤ c E (f, f)
for all locally Lipschitz functions f : Rn → R+, where Entµ, called the
entropy functional, is defined as
Entµ(f) :=
∫
f log f∫
f dµ
dµ
Keywords: Logarithmic Sobolev inequality, convolutions.
Math. classification: 26D10.
129
David Zimmermann
and E (f, f), the energy of f , is defined as
E (f, f) :=
∫
|∇f |2dµ,
with |∇f | defined as
|∇f |(x) := lim sup
y→x
|f(x)− f(y)|
|x− y|
so that |∇f | is defined everywhere and coincides with the usual notion
of gradient where f is differentiable. The smallest c for which a LSI with
constant c holds is called the optimal log-Sobolev constant for µ.
LSIs are a useful tool that have been applied in various areas of mathe-
matics, such as geometry [1, 2, 5, 8, 9, 10, 14], probability [6, 11, 12, 13, 16],
optimal transport [15, 17], and statistical physics [19, 20, 21]. In [23], the
present author showed that the convolution of a compactly supported
measure on R with a Gaussian measure satisfies a LSI, and an application
of this fact to random matrix theory was given; that result, however, did
not provide any quantitative information about the optimal log-Sobolev
constants. In [22, Thms. 2 and 3], bounds for the optimal constants in
these LSIs were given (stated as Theorem 1.1 below), and the results were
extended to Rn. (See [18] for statements about LSIs for convolutions with
more general measures).
Theorem 1.1. Let µ be a probability measure on R whose support is
contained in an interval of length 2R, and let γδ be the centered Gaussian
of variance δ > 0, i.e., dγδ(t) = (2πδ)−1/2 exp(− t
2
2δ )dt. Then for some
absolute constants Ki, the optimal log-Sobolev constant c(δ) for µ ∗ γδ
satisfies
c(δ) ≤ K1
δ3/2R
4R2 + δ exp
(
2R2
δ
)
+K2 (
√
δ + 2R)2.
In particular, if δ ≤ R2, then
c(δ) ≤ K3
δ3/2
R
exp
(
2R2
δ
)
.
The Ki can be taken in the above inequalities to be K1 = 6905,K2 =
4989,K3 = 7803.
Theorem 1.1 was proved in [22] using the following theorem due to
Bobkov and Götze [4, p. 25, Thm 5.3]:
130
log-Sobolev inequalities for convolutions
Theorem 1.2 (Bobkov, Götze). Let µ be a Borel probability measure on
R with distribution function F (x) = µ((−∞, x]). Let p be the density of
the absolutely continuous part of µ with respect to Lebesgue measure, and
let m be a median of µ. Let
D0 = sup
x<m
(
F (x) · log 1
F (x) ·
∫ m
x
1
p(t)dt
)
,
D1 = sup
x>m
(
(1− F (x)) · log 11− F (x) ·
∫ x
m
1
p(t)dt
)
,
defining D0 and D1 to be zero if µ((−∞,m)) = 0 or µ((m,∞)) = 0,
respectively, and using the convention 0 · ∞ = 0. Then the optimal log
Sobolev constant c for µ satisfies 1150(D0 +D1) ≤ c ≤ 468(D0 +D1).
Remark 1.3. In Rn, the analogue of Theorem 1.1 holds for measures
supported in a ball of radius R, with optimal log-Sobolev constant c(δ)
bounded by
c(δ) ≤ KR2 exp
(
20n+ 5R
2
δ
)
for some absolute constant K and for δ ≤ R2. This was proved in [22]
using a theorem due to Cattiaux, Guillin, and Wu [7, Thm. 1.2] that gives
satisfaction of a LSI under a Lyapunov condition.
The goal of the present paper is to provide an elementary proof of
Theorem 1.1. The result proved is the following:
Theorem 1.4. Let µ be a probability measure on R whose support is
contained in an interval of length 2R, and let γδ be the centered Gaussian
of variance δ > 0, i.e., dγδ(t) = (2πδ)−1/2 exp(− t
2
2δ )dt. Then the optimal
log-Sobolev constant c(δ) for µ ∗ γδ satisfies
c(δ) ≤ max
(
2δ exp
(
4R2
δ
+ 4R√
δ
+ 14
)
, 2δ exp
(
12R2
δ
))
.
In particular, if δ ≤ 3R2, we have
c(δ) ≤ 2δ exp
(
12R2
δ
)
.
131
David Zimmermann
Remark 1.5. The bound in Theorem 1.4 is worse than the bound in Theo-
rem 1.1 for small δ, but still has an order of magnitude that is exponential
in R2/δ. (It is shown in [22, Example 21] that one cannot do better than
exponential in R2/δ for small δ.)
Remark 1.6. In fact, the proof of Theorem 1.4 yields a slightly stronger
result. The proof is based upon showing that the convolution of the com-
pactly supported measure with the Gaussian is the push-forward of the
Gaussian under a Lipschitz map. This fact, together with the Gaussian
isoperimetric inequality, yields the isoperimetric inequality for the convo-
lution measure, which implies the logarithmic Sobolev inequality; see [3],
in which Bakry and Ledoux show that a probability measure on R satis-
fies this isoperimetric inequality if and only if the measure is a Lipshitz
push-forward of the Gaussian.)
The author would like to thank his Ph.D. advisor, Todd Kemp, for his
valuable insights and discussions regarding this topic. The author would
also like to thank the anonymous referee for his or her corrections and
suggestions (especially Remark 1.6) that have improved the overall pre-
sentation and exposition of this paper.
2. Proof of Theorem 1.4
The proof of Theorem 1.4 is based on two facts: first, the Gaussian mea-
sure γ1 of unit variance satisfies a LSI with constant 2. Second, Lipshitz
functions preserve LSIs. We give a precise statement of this second fact
below.
Proposition 2.1. Let µ be a measure on R that satisfies a LSI with
constant c, and let T : Rn → Rn be Lipschitz. Then the push-forward
measure T∗µ also satisfies a LSI with constant c||T ||2Lip.
Proof. Let g : Rn → R be locally Lipschitz. Then g◦T is locally Lipschitz,
so by the LSI for µ,∫
(g ◦ T )2 log (g ◦ T )
2∫
(g ◦ T )2 dµ dµ ≤ c
∫
|∇(g ◦ T )|2dµ. (2.1)
But since T is Lipschitz,
|∇(g ◦ T )| ≤ (|∇g| ◦ T )||T ||Lip.
132
log-Sobolev inequalities for convolutions
So by a change of variables, (2.1) simply becomes∫
g2 log g
2∫
g2 dT∗µ
dT∗µ ≤ c||T ||2Lip
∫
|∇g|2dT∗µ.
as desired. 
We now prove Theorem 1.4.
Proof of Theorem 1.4. In light of Proposition 2.1, we will establish the
theorem by showing that µ∗γδ is the push-forward of γ1 under a Lipschitz
map. By translation invariance of LSI, we can assume that supp(µ) ⊆
[−R,R]. We will also first assume that δ = 1 (the general case will be
handled at the end of the proof by a scaling argument).
Let F and G be the cumulative distribution functions of γ1 and µ ∗ γ1,
i.e.,
F (x) =
∫ x
−∞
p(t) dt, G(x) =
∫ x
−∞
q(t) dt,
where
p(t) = 1√
2π
exp
(
− t
2
2
)
and q(t) =
∫ R
−R
p(t− s) dµ(s).
Notice that q is smooth and strictly positive, so that G−1 ◦ F is well-
defined and smooth. It is readily seen that (G−1 ◦ F )∗(γ1) = µ ∗ γ1, so to
establish the theorem we simply need to bound the derivative of G−1 ◦F .
Now
(G−1 ◦ F )′(x) = 1
G′((G−1 ◦ F )(x)) · F
′(x) = p(x)
q((G−1 ◦ F )(x)) .
We will bound the above derivative in cases – when x ≥ 2R, when
−2R ≤ x ≤ 2R, and when x ≤ −2R.
We first consider the case x ≥ 2R. Define
Λ(x) =
∫ R
−R
exsdµ(s), K(x) = log Λ(x) +R
x
.
Note Λ and K are smooth for x 6= 0.
Lemma 2.2. For x ≥ 2R,
exp
(
−2R2 − 2R− 18
)
p(x) ≤ q(x+K(x)) ≤ e−R p(x).
133
David Zimmermann
Proof. By definition of q, p,Λ, and K,
q(x+K(x)) =
∫ R
−R
p(x+K(x)− s) dµ(s)
= p(x) · e−xK(x)
∫ R
−R
exp
(
−(K(x)− s)
2
2
)
· exs dµ(s)
= e
−R p(x)
Λ(x)
∫ R
−R
exp
(
−(K(x)− s)
2
2
)
· exs dµ(s)
≤ e
−R p(x)
Λ(x)
∫ R
−R
exs dµ(s)
= e−R p(x).
To get the other inequality, first note that e−Rx ≤ Λ(x) ≤ eRx. (These
are just the maximum and minimum values in the integrand defining Λ.)
This implies that −R+R/x ≤ K(x) ≤ R+R/x, so for −R ≤ s ≤ R and
x ≥ 2R, we have
−2R− R
x
≤ −2R+ R
x
≤ K(x)− s ≤ 2R+ R
x
so that
exp
(
−(K(x)− s)
2
2
)
≥ exp
(
−(2R+R/x)
2
2
)
≥ exp
(
−(2R+R/(2R))
2
2
)
= exp
(
−2R2 −R− 18
)
.
Therefore
q(x+K(x)) = e
−R p(x)
Λ(x)
∫ R
−R
exp
(
−(K(x)− s)
2
2
)
· exs dµ(s)
≥ exp
(
−2R2 − 2R− 18
)
p(x). 
Lemma 2.3. K ′(x) ≤ R for x > 0.
134
log-Sobolev inequalities for convolutions
Proof. Recall that e−Rx ≤ Λ(x). (Again, e−Rx is the minimum value in
the integrand defining Λ). We therefore have
K ′(x) = Λ
′(x)
xΛ(x) −
log Λ(x)
x2
− R
x2
=
∫ R
−R s e
sx dµ(s)
xΛ(x) −
log Λ(x)
x2
− R
x2
≤
R
∫ R
−R e
sx dµ(s)
xΛ(x) +
Rx
x2
− R
x2
= 2R
x
− R
x2
.
By elementary calculus, the above has a maximum value of R. 
Lemma 2.4. For x ≥ 2R,
x−R ≤ (G−1 ◦ F )(x) ≤ x+K(x).
Proof. Since G and G−1 are increasing, the lemma is equivalent to
G(x−R) ≤ F (x) ≤ G(x+K(x)).
The first inequality follows from the definition of G and the Fubini-Tonelli
Theorem:
G(x−R) =
∫ x−R
−∞
q(t) dt =
∫ x−R
−∞
∫ R
−R
p(t− s) dµ(s) dt
=
∫ R
−R
∫ x−R
−∞
p(t− s) dt dµ(s)
=
∫ R
−R
∫ x−R−s
−∞
p(u) du dµ(s)
where u = t− s
≤
∫ R
−R
∫ x
−∞
p(u) dt dµ(s)
= F (x).
135
David Zimmermann
To establish the other inequality, we use Lemmas 2.2 and 2.3:
1−G(x+K(x)) =
∫ ∞
x+K(x)
q(t) dt =
∫ ∞
x
q(u+K(u))(1 +K ′(u)) du
where t = u+K(u)
≤
∫ ∞
x
p(u)e−R(1 +R) du
by Lemmas 2.2 and 2.3
≤
∫ ∞
x
p(u) du
since eR ≥ 1 +R
= 1− F (x),
so that F (x) ≤ G(x+K(x)), as desired. 
We are almost ready to bound (G−1 ◦ F )′(x) for x ≥ 2R. The last
observation to make is that q is decreasing on [R,∞) since
q′(t) =
∫ R
−R
p′(t− s) dµ(s) =
∫ R
−R
−(t− s)p(t− s) dµ(s) ≤ 0 for t ≥ R.
So for x ≥ 2R we have, by Lemma 2.4,
q((G−1 ◦ F )(x)) ≥ q(x+K(x)).
Combining this with Lemma 2.2, we get
(G−1 ◦ F )′(x) = p(x)
q((G−1 ◦ F )(x)) ≤
p(x)
q(x+K(x)) ≤ exp
(
2R2 + 2R+ 18
)
for x ≥ 2R.
In the case where −2R ≤ x ≤ 2R, first note that for all x,
x−R ≤ (G−1 ◦ F )(x) ≤ x+R;
136
log-Sobolev inequalities for convolutions
the first inequality above was done in Lemma 2.4, and the second inequal-
ity is proven in the same way. So
sup
−2R≤x≤2R
(G−1 ◦ F )′(x) = sup
−2R≤x≤2R
p(x)
q((G−1 ◦ F )(x))
≤ sup
−2R≤x≤2R
−R≤y≤R
p(x)
q(x+ y)
=
 inf
−2R≤x≤2R
−R≤y≤R
q(x+ y)
p(x)

−1
.
For convenience, let S = {(x, y) : −2R ≤ x ≤ 2R,−R ≤ y ≤ R}. Now
inf
(x,y)∈S
q(x+ y)
p(x) = inf(x,y)∈S
1
p(x)
∫ R
−R
p(x+ y − s) dµ(s).
Since p has no local minima, the minimum value of the above integrand
occurs at either s = R or s = −R. Without loss of generality, we assume
the minimum is achieved at s = R (otherwise, we can replace (x, y) with
(−x,−y) by symmetry of S and p). So
inf
(x,y)∈S
q(x+ y)
p(x) ≥ inf(x,y)∈S
1
p(x) · p(x+ y +R).
Elementary calculus shows that the above infimum is equal to e−6R2
(achieved at x = 2R, y = R). Therefore
sup
−2R≤x≤2R
(G−1 ◦ F )′(x) ≤
(
inf
(x,y)∈S
q(x+ y)
p(x)
)−1
≤ e6R2 .
The case x ≤ −2R is dealt with in the same way as the case x ≥ 2R,
the analagous statements being:
exp
(
−2R2 − 2R− 18
)
p(x) ≤ q(x+K(x)) ≤ e−R p(x),
K ′(x) ≤ R,
x+K(x) ≤ (G−1 ◦ F )(x) ≤ x+R,
and q is increasing for x ≤ −2R. The upper bound for (G−1 ◦ F )′(x)
obtained in this case is the same as the one in the case x ≥ 2R.
137
David Zimmermann
We therefore have
||G−1 ◦ F ||Lip ≤ max
(
exp
(
2R2 + 2R+ 18
)
, e6R
2
)
So by Proposition 2.1, µ ∗ γ1 satisfies a LSI with constant c(1) satisfying
c(1) ≤ 2||G−1 ◦ F ||2Lip ≤ max
(
2 exp
(
4R2 + 4R+ 14
)
, 2 e12R2
)
.
This proves the theorem for the case δ = 1.
To establish the theorem for a general δ > 0, first observe that
µ ∗ γδ = (h√δ)∗
(
((h1/√δ)∗µ) ∗ γ1
)
,
where hλ denotes the scaling map with factor λ, i.e., hλ(x) = λx. Now
(h1/√δ)∗µ is supported in [−R/
√
δ,R/
√
δ], so by the case δ = 1 just
proven, ((h1/√δ)∗µ) ∗ γ1 satisfies a LSI with constant
max
(
2 exp
(
4(R/
√
δ)2 + 4(R/
√
δ) + 14
)
, 2 e12(R/
√
δ)2
)
.
Finally, since ||h√δ||
2
Lip = δ, we have by Proposition 2.1,
c(δ) ≤ max
(
2δ exp
(
4R2
δ
+ 4R√
δ
+ 14
)
, 2δ exp
(
12R2
δ
))
.
In particular, when δ ≤ 3R2 (in fact when δ ≤ (160 − 64
√
6)R2 ≈
3.23R2), we have
2δ exp
(
4R2
δ
+ 4R√
δ
+ 14
)
≤ 2δ exp
(
12R2
δ
)
so the above bound on c(δ) simplifies to
c(δ) ≤ 2δ exp
(
12R2
δ
)
. 
References
[1] D. Bakry – “L’hypercontractivité et son utilisation en thorie des
semigroupes”, in Lectures on probability theory (Saint-Flour, 1992),
Lecture Notes in Math., vol. 1581, Springer, Berlin, 1994, p. 1–114.
138
log-Sobolev inequalities for convolutions
[2] , “On Sobolev and logarithmic Sobolev inequalities for Markov
semigroups”, in New trends in stochastic analysis, World Sci. Publ.,
River Edge, NJ, 1997, p. 43–75.
[3] D. Bakry & M. Ledoux – “Lévy-Gromov’s isoperimetric inequality
for an infinite dimensional diffusion generator”, Inventiones mathe-
maticae 123 (1996), p. 259–281.
[4] S. Bobkov & F. Götze – “Exponential Integrability and Trans-
portation Cost Related to Logarithmic Sobolev Inequalities”, J.
Funct. Anal. 163 (1999), p. 1–28.
[5] S. Bobkov & C. Houdré – “Some connections between isoperi-
metric and Sobolev-type inequalities”, Mem. Amer. Math. Soc. 129
(1995), no. 616, p. 1–28.
[6] S. Bobkov & P. Tetali – “Modified logarithmic Sobolev inequali-
ties in discrete settings”, J. Theoret. Probab. 19 (2006), no. 2, p. 289–
336.
[7] P. Cattiaux, A. Guillin & L. Wu – “A note on Talagrand’s trans-
portation inequality and logarithmic Sobolev inequality”, Probab.
Theory Relat. Fields 148 (2010), p. 285–304.
[8] E. B. Davies – “Explicit constants for Gaussian upper bounds on
heat kernels”, Amer. J. Math. 109 (1987), no. 2, p. 319–333.
[9] , Heat kernels and spectral theory, Cambridge University
Press, 1990.
[10] E. B. Davies & B. Simon – “Ultracontractivity and the heat kernel
for Schrödinger operators and Dirichlet Laplacians”, J. Funct. Anal.
59 (1984), p. 335–395.
[11] P. Diaconis & L. Saloff-Coste – “Logarithmic Sobolev inequal-
ities for finite Markov chains”, Ann. Appl. Probab. 6 (1996), p. 695–
750.
[12] L. Gross & O. Rothaus – “Herbst inequalities for supercontractive
semigroups”, J. Math. Kyoto Univ. 38 (1998), no. 2, p. 295–318.
[13] A. Guionnet & B. Zegarlinski – “Lectures on logarithmic Sobolev
inequalities”, in Séminaire de Probabilités, XXXVI, Lecture Notes in
Math., vol. 1801, Springer, Berlin, 2003, p. 1–134.
139
David Zimmermann
[14] M. Ledoux – “Isoperimetry and Gaussian analysis”, in Lectures on
probability theory and statistics, Lecture Notes in Math., vol. 1648,
Springer, Berlin, 1996, p. 165–294.
[15] , The concentration of measure phenomenon, American Math-
ematical Society, Providence, RI, 2001.
[16] , “A remark on hypercontractivity and tail inequalities for the
largest eigenvalues of random matrices”, in Séminaire de Probabilités
XXXVII, Lecture Notes in Math., vol. 1832, Springer, Berlin, 2003,
p. 360–369.
[17] C. Villani – Topics in optimal transportation, American Mathemat-
ical Society, Providence, RI, 2003.
[18] F.-Y. Wang & J. Wang – “Functional inequalities for convolution
probability measures”, http://arxiv.org/abs/1308.1713.
[19] H. Yau – “Logarithmic Sobolev inequality for the lattice gases with
mixing conditions”, Commun. Math. Phys. 181 (1996), p. 367–408.
[20] , “Log-Sobolev inequality for generalized simple exclusion pro-
cesses”, Probab. Theory Related Fields 109 (1997), p. 507–538.
[21] B. Zegarlinski – “Dobrushin uniqueness theorem and logarithmic
Sobolev inequalities”, J. Funct. Anal. 105 (1992), p. 77–111.
[22] D. Zimmermann – “Bounds for logarithmic Sobolev constants for
Gaussian convolutions of compactly supported measures”, http://
arxiv.org/abs/1405.2581.
[23] , “Logarithmic Sobolev inequalities for mollified complactly
supported measures”, J. Funct. Anal. 265 (2013), p. 1064–1083.
David Zimmermann
Department of Mathematics
University of California, San Diego
9500 Gilman Drive
La Jolla, CA 92093, USA
dszimmer@ucsd.edu
140


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