ANNALES
DE LA FACULTÉ DES SCIENCES
Mathématiques
BO’AZKLARTAG
Poincaré Inequalities and Moment Maps
Tome XXII, no1 (2013), p. 1-41.
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Annales de la Facult´e des Sciences de Toulouse Vol. XXII, n 1, 2013 pp. 1–41
Poincar´ e Inequalities and Moment Maps
Bo’az Klartag(1)
ABSTRACT.— We discuss a method for obtaining Poincar´e-type inequal- ities on arbitrary convex bodies in Rn. Our technique involves a dual version of Bochner’s formula and a certain moment map, and it also ap- plies to some non-convex sets. In particular, we generalize the central limit theoremfor convex bodies to a class of non-convex domains, including the unit balls ofp-spaces inRnfor 0< p <1.
R´ESUM´E.— Nous explorons un proc´ed´e de preuve d’in´egalit´es de type Poincar´e sur les corps convexes deRn. Notre technique utilise une version duale de la formule de Bochner et une application moment. Elle s’applique
´egalement `a certains corps non-convexes. En particulier, nous g´en´eralisons le th´eor`eme central limite pour les ensembles convexes `a une classe de domaines non-convexes, qui comprend les boules unit´es deRnmunies de
la normeppour 0< p <1.
1. Introduction
An important observation that goes back to Sudakov [28] and to Diaconis and Freedman [13] is that approximatelygaussian marginals are intimately related to thin shell inequalities. That is, let X be a random vector in Rn with mean zero and identitycovariance, where the dimensionnis assumed veryhigh. Suppose thatX satisfies a thin shell inequality, of the form
E |X|2
n −1 2
1, (1.1)
(1) School of Mathematical Sciences, Tel-Aviv University, Tel Aviv 69978, Israel. Sup- ported in part by the Israel Science Foundation and by a Marie Curie Reintegration Grant from the Commission of the European Communities.
Article propos´e par Franck Barthe.
where|·|is the standard Euclidean norm inRn. It then follows that there are plentyof vectorsθ∈Rnfor which the scalar productX, θis approximately a gaussian random variable. See von Weizs¨acker [31], Bobkov [7], Anttila, Ball and Perissinaki [3] or [22, 24] for further explanations, and Eldan and Klartag [15] for connections to the hyperplane conjecture.
In this paper, Poincar´e-type inequalities refer to inequalities in which the variance of a function is bounded in terms of an integral of a quadratic form involving the gradient of the function. One of the methods used to prove a thin shell bound such as (1.1) goes through Poincar´e-type inequal- ities in high-dimensional spaces. This approach was pursued in [23], where the Bochner formula was applied to studyoptimal thin shell bounds and Poincar´e-type inequalities for the uniform measure on high-dimensional con- vex bodies. The technique in [23] and in the related work byBarthe and Cordero-Erausquin [6] relied verymuch on symmetries of the probability distribution under consideration. The method seemed quite irrelevant for arbitraryconvex bodies, possessing no symmetries. The following twist is proposed here: Introduce additional symmetries by considering a certain transportation of measure from a space of twice or thrice the dimension.
The plan is to applyBochner’s formula in this higher dimensional space, and deduce a Poincar´e-type inequality for the original measure.
We proceed bydemonstrating the Poincar´e-type inequalities that are obtained in the simplest case, perhaps, in which the convex set we investigate is Rn+, the orthant of all x ∈ Rn with positive coordinates. A function ϕ : Rn+ → R∪ {∞} is called coordinate p-convex, for 0 < p 1, if the function
(x1, . . . , xn)→ϕ
x1/p1 , . . . , x1/pn is convex onRn+. For instanceϕ(x) =n
i=1√xi is coordinatep-convex for anyp1/2. A convex function is obviouslycoordinate 1-convex.
Theorem 1.1. — Let n 1, k > 1 be integers. Suppose that µ is a Borel measure on Rn+ with density exp(−ϕ), where ϕ : Rn+ → R∪ {∞}
is coordinate p-convex for p = 1/k. Assume that f : Rn+ → R is a locally Lipschitz, µ-integrable function with
f dµ= 0. Then,
Rn+
f2dµ k2 k−1
n
i=1
Rn+
x2i∂if(x)2dµ(x). (1.2) Here, ∂if =∂f /∂xi stands for the derivative of f with respect to the ith variable.
We emphasize that the function f in Theorem 1.1 is not assumed to satisfyanyboundaryconditions. Compare, for example, to the Hardy-type
inequalities in Brezis and Marcus [10] and Avkhadiev and Wirths [4]. We saythat a subset K⊂Rn+ is coordinatep-convex for 0< p1 when
{(xp1, . . . , xpn) ; (x1, . . . , xn)∈K}
is a convex set. In other words, K is coordinate p-convex when the func- tion that equals 0 onK and equals +∞outsideK is coordinate p-convex.
Observe that the intersection of coordinate p-convex sets is again a coor- dinate p-convex set. Dilations centered at the origin preserve coordinate p-convexity. Forp= 1, translations do not necessarilypreserve coordinate p-convexity, but coordinatep-convexityis preserved bytranslations conju- gated with the map x→(xp1, . . . , xpn). From Theorem 1.1 we immediately deduce:
Corollary 1.2. — Let n1, >1 be integers, and assume thatK ⊂ Rn+ is a coordinate (1/)-convex set with a non-empty interior. Then, for any locally Lipschitz, integrable function f :K→Rwith
Kf = 0,
K
f2dx 2 −1
n
i=1
K
x2i ∂if(x)2dx.
Forx, y∈Rn+ we writexy when xi yi fori= 1, . . . , n. A function ϕ:Rn+→R∪ {∞} isincreasingwhen
xy =⇒ ϕ(x)ϕ(y) (forx, y∈Rn+).
It is simple to see that whenf is increasing and coordinatep-convex, it is also coordinateq-convex for any0< q < p. A functionϕ:Rn →R∪ {∞}
is said to be unconditionalwhen
ϕ(x1, . . . , xn) =ϕ(|x1|, . . . ,|xn|) (x∈Rn).
Observe that when ϕ is an unconditional, convex function on Rn, the re- striction ϕ|Rn+ is necessarilyincreasing and hence coordinate p-convex for any0 < p 1. Thus Corollary1.2 recovers the Poincar´e-type inequalities from [23]: Quite unexpectedly, the unconditionality is used only to infer that when ϕ|Rn+ is coordinate 1-convex, it is also coordinate (1/2)-convex.
Theorem 1.1 maybe generalized to measures on Rn whose densityis un- conditional, as follows:
Theorem 1.3. — Let µ be a probability measure on Rn with density exp(−ϕ), whereϕ:Rn→R∪ {∞}is unconditional, andϕ|Rn+ is increasing and coordinate 1/k-convex for an integerk >1. Denote
Vi=
Rnx2idµ(x) (i= 1, . . . , n).
Then, for any µ-integrable, locally Lipschitz function f : Rn → R with f dµ= 0,
Rnf2dµ
Rn n
i=1
k2
k−1x2i +Vi ∂if(x)2dµ(x). (1.3) Furthermore, when the function f is unconditional, we may eliminate the Vi’s on the right-hand side of (1.3).
For 0< p < 1, denote by µp the uniform probabilitymeasure on the non-convex set
Bpn=
x∈Rn ;
n
i=1
|xi|p1
.
Theorem 1.3 applies for the measure µp, with anyk1/p. Substituting f(x) =|x|2−
|y|2dµ(y) into Theorem 1.3 yields thin shell bounds, which maybe used to infer the existence of approximatelygaussian marginals.
Once Theorem 1.1 and Corollary1.2 are formulated, one is tempted to try and find a more direct proof of these inequalities. In Section 6 we discuss such a direct argument, based on the Brascamp-Lieb inequality[8], and obtain generalizations of Theorem 1.1 and Theorem 1.3 in whichk > 1 is not necessarilyan integer. Similarly, > 1 does not have to be an integer in Corollary1.2.
Next, supposeK⊂Rnis a convex body, i.e., a bounded, open convex set.
We turn to the details of the Poincar´e-type inequalities that are obtained for K. Recall that a function on Rn is log-concave if it takes the form exp(−H) for a convex functionH :Rn→R∪ {∞}. A Borel measure onRn is log-concave if its densityis log-concave, and in particular, the uniform probabilitymeasure on an open, convex set is log-concave. We saythat aC3- smooth, convex functionψ:Rn→Rinduces a “log-concave transportation toK” if the following two conditions hold:
(a) The functionρψ(x) = det∇2ψ(x) is positive and log-concave onRn, where∇2ψis the Hessian of ψ.
(b) We have∇ψ(Rn) =K, where∇ψ(Rn) ={∇ψ(x);x∈Rn}.
Observe that the mapx→ ∇ψ(x) pushes forward the measure whose densityis ρψ, to the uniform measure on the convex bodyK. For a given convex bodyK⊂Rn, there are plentyof convex functionsψthat induce a log-concave transportation toK. In fact, for anylog-concave functionρon Rn whose integral equals the volume of K, there exists a convex function ψ which satisfies (a) and (b) withρψ =ρ. This follows from the standard
theoryof optimal transportation of measure (e.g., Villani [30]). For indices i, j, k= 1, . . . , n we abbreviate
ψi= ∂ψ
∂xi
, ψij = ∂2ψ
∂xi∂xj
, ψijk= ∂3ψ
∂xi∂xj∂xk
. We also write
ψij
i,j=1,...,n for the inverse matrix to the Hessian matrix
∇2ψ = (ψij)i,j=1,...,n. The Legendre transform of ψ is the function ψ∗ : K→Rdefined via
ψ∗(x) = sup
y∈Rn[x, y −ψ(y)].
Then ∇ψ∗ is the inverse map to ∇ψ. With any x ∈ K we associate the quadratic formQ∗ψ,x onRn defined by
Q∗ψ,x(V) =
n
i,j,k,,m,p=1
ViVjψmψjkmψkpψip
where V = (V1, . . . , Vn)∈Rn and where the functionsψij, ψm, ψjkm etc.
are evaluated at the point∇ψ∗(x). Forx∈K andU ∈Rn, set
Qψ,x(U) = sup
4
n
i,j=1
ψijUiVj
2
; V ∈Rn, Q∗ψ,x(V)1
, whereψij is evaluated at the point ∇ψ∗(x). It could occur thatQψ,x(U) is finite onlyforU in a certain subspaceE⊂Rn. Note thatQψ,xis a quadratic form on that subspaceE.
There is one technical assumption that we must make. In Section 3 we define the notion ofregularity at infinityof the functionψ, and throughout the analysis below we conveniently assume theψis indeed regular at infinity.
This assumption seems to hold in the examples that we consider. In the case whereK⊂Rn is a simple rational polytope, regularity at infinity was investigated in the works of Abreu [2], Donaldson [14] and Guillemin [21], who explained that it holds under fairlymild assumptions.
Theorem 1.4. — LetK⊂Rn be a convex body. Suppose thatψ:Rn→ Rinduces a log-concave transportation toK. Assume further thatψis reg- ular at infinity. Then, for any Lipschitz function f :K→R,
K
f = 0 ⇒
K
f2
K
Qψ,x(∇f(x))dx.
In order to applyTheorem 1.4 one needs to select a functionψ which induces a log-concave transportation toK. Unfortunately, we are currently unaware of a general method for constructing a “reasonable” function ψ that satisfies (a) and (b), with good control over derivatives up to order three. In simple cases, such as when K ⊂Rn is the cube or the simplex, Theorem 1.4 does yield meaningful inequalities as is explained below.
The rest of the paper is organized as follows: We present the proof of Theorem 1.1 in Section 2. The argument relies on the analysis of a particular moment map, fromRkn to Rn+. We then proceed in Section 3 and discuss general moment maps from toric K¨ahler manifolds, and prove Theorem 1.4.
Next, in Section 4 we applyTheorem 1.4 for the case of the simplex. In particular, Theorem 4.5 below provides certain Poincar´e-type inequalities for a class of distributions on the regular simplex. In Section 5 we deduce Theorem 1.3 from Theorem 1.1 via a rather standard argument.
Acknowledgements. — Thanks to Semyon Alesker, Franck Barthe, Ha¨ım Brezis, DmitryFaifman, Uri Grupel, Greg Kuperberg, Emanuel Milman, Yaron Ostrover, Leonid Polterovich, Yanir Rubinstein and Mikhail Sodin for interesting related discussions. Thanks also to the anonymous referee for encouraging me to learn about K¨ahler-Einstein metrics.
2. Non-Linear Measure Projection
In this section we prove Theorem 1.1. The analysis in this section is also intended to serve as a preparation for Section 3. Let n, k 1 be positive integers, fixed throughout this section. Denotem=nk. We use
z= (z1, . . . , zn)∈(Rk)n =Rkn
as coordinates inRkn, wherez1, . . . , zn arek-dimensional vectors. Consider the mapπ:Rm→Rn+ defined by
π(z) = (|z1|k, . . . ,|zn|k) (z1, . . . , zn)∈(Rk)n.
Here, Rn+ is the closure of Rn+ in Rn, and |zi| stands for the standard Euclidean norm of zi ∈ Rk. The continuous map π is proper, meaning that π−1(K) is compact wheneverK ⊂ Rn+ is compact. Let Sk−1 ={y ∈ Rk;|y|= 1}denote the unit sphere inRk, and more generally, letSk−1(R) = {y∈Rk;|y|=R}. We writeσRfor the uniform probabilitymeasure on the sphere Sk−1(R). With anyx∈ Rn+ we associate the Cartesian product of spheres,
π−1(x) :=Sk−1(x1/k1 )×Sk−1(x1/k2 )×. . .×Sk−1(x1/kn )⊆(Rk)n =Rm.
We denote byσxthe uniform probabilitymeasure onπ−1(x), that is, the di- rect product of the uniform probabilitymeasures on the spheresSk−1(x1/kj ) forj= 1, . . . , n.
We view the map π as a kind of moment map. The case k = 2 fits verywell with the standard terminology, as in this caseπis related to the moment map associated with the symplectic action of the group (SO(2))n on (R2)n (see, e.g., Cannas da Silva [11]). In the following lemma we verify that indeed the uniform measure onRm is pushed forward to the uniform measure on Rn+ via the map π, up to a normalizing coefficient. We write V olk for the standardk-dimensional volume measure.
Lemma 2.1. — For any integrable function f :Rn+→R,
Rmf(π(z))dV olm(z) =ωn,k
Rn+
f(x)dV oln(x) (2.1) where ωn,k =
πk/2/Γ(k/2 + 1)n
is the nth power of the volume of the k-dimensional unit ball. Furthermore, for any Borel set A⊆Rm,
V olm(A) =ωn,k
Rn+
σx(A)dV oln(x). (2.2)
Proof. — Integrating in polar coordinates for eachzj∈Rk(j= 1, . . . , n), we find that
Rmf(|z1|k, . . . ,|zn|k)dz1. . . dzn=ωkn
Rn+
f(xk1, . . . , xkn)
n j=1
xkj−1
dx1. . . dxn,
where ωk =kπk/2/Γ(k/2 + 1) is the surface area of the unit sphere inRk. Applying the change of variables (t1, . . . , tn) = (xk1, . . . , xkn) we obtain
Rn+
f(xk1, . . . , xkn)
n j=1
xkj−1
dx1. . . dxn =k−n
Rn+
f(t1, . . . , tn)dt1. . . dtn
and (2.1) follows. The relation (2.2) is proven in a similar fashion.
Supposeν is a Borel measure onRm. For a functionf ∈L2(ν) we define fH−1(ν)= sup
Rmf gdν;
Rm|∇g|2dν 1
, (2.3)
where the supremum runs over allC1-smooth functions g :Rm →R that belong to L2(ν). Note thatfH−1(ν)= +∞ when
f dν= 0. The square of theH−1(ν)-norm is sub-additive inν, as will be proven next:
Lemma 2.2. — Supposeν is a Borel measure onRmthat takes the form
ν=
Ω
ναdλ(α) (2.4)
for Borel measures {να}α∈Ω on Rm and a measure λon Ω. Then, for any f ∈L2(ν),
f2H−1(ν)
Ωf2H−1(να)dλ(α).
Proof. — We mayassume thatfH−1(να)<∞forλ-almost anyα∈Ω, as otherwise there is nothing to prove. Letg be a smooth function onRm which belongs toL2(ν). Sincef, g∈L2(να) forλ-almost any α∈Ω, then
Rmf gdνα
fH−1(να)
Rm|∇g|2dνα
forλ-almost anyα∈Ω. From (2.4) and the Cauchy-Schwartz inequality,
Rmf gdν
ΩfH−1(να)
Rm|∇g|2dνα
1/2 dλ(α)
Rmf2H−1(να)dλ(α)·
Rm|∇g|2dν.
Recall that we use (z1, . . . , zn)∈(Rk)n as coordinates inRm=Rkn. Let us furthermore denotez= (z,1, . . . , z,k)∈Rk, for any= 1, . . . , n.
Lemma 2.3. — Assume k2. Let x∈ Rn+. Let 1 n,1j k, and denote f(z) =z,j forz∈Rm. Then,
fH−1(σx) x2/k k(k−1).
Proof. — We claim that for anysmooth functionh:Rk →R andθ ∈ Sk−1,
Sk−1y, θh(y)dσ1(y)
1 k(k−1) ·
Sk−1|∇h|2dσ1. (2.5) Indeed, (2.5) simplyexpresses the standard fact that y → √
k(y·θ) is a normalized eigenfunction of the Laplace-Beltrami operator onSk−1, corre- sponding to the eigenvalue k−1 (see, e.g., M¨uller [26]). Byscaling, we see that for anyR >0 andθ∈Sk−1,
Sk−1(R)y, θh(y)dσR(y) R2 k(k−1) ·
Sk−1(R)|∇h|2dσR. (2.6) According to (2.6), for anyfixedz1, . . . , z−1, z+1, . . . , zn∈Rkand a smooth functiong:Rm→R,
Sk−1(R)
z,jg(z1, . . . , zn)dσR(z) x2/k k(k−1)
Sk−1(R)|∇g(z)|2dσR(z), whereR=x1/k . Recall that the probabilitymeasureσx is a product mea- sure, and thatσR is thethfactor in this product. Integrating with respect to the remaining variablesz1, . . . , z−1, z+1, . . . , zn, and using the Cauchy- Schwartz inequality, we obtain
π−1(x)
z,jg(z)dσx(z) x2/k k(k−1)
π−1(x)|∇g(z)|2dσx(z).
The lemma follows from the definition of theH−1(σx)-norm.
The following lemma is one of the reasons for considering the higher- dimensional space Rm, rather than working in the original spaceRn+. The extra dimensions translate to “extra symmetries”, which substitute for the explicit symmetries assumed in [23, Corollary 5] and in Barthe and Cordero- Erausquin [6, Section 3].
Lemma 2.4. — Assumek2, let1n,1jk and letx∈Rn+. Suppose that f :Rn+ →R is differentiable atx. Denote g(z) =f(π(z)) for z∈Rm. Then,
∂g
∂z,j
H−1(σx)
k
k−1 ·x
∂f(x).
Proof. — Note that for z∈π−1(x),
∂g
∂z,j
(z1, . . . , zn) = k|z|k−2z,j·∂f(|z1|k, . . . ,|zn|k)
=
kx(k −2)/k∂f(x1, . . . , xn) z,j.
That is, the function ∂g/∂z,j is proportional to the linear function z → z,j on the support of σx, and the proportion coefficient is exactly kx(k −2)/k∂f(x1, . . . , xn). According to Lemma 2.3,
∂g
∂z,j
H−1(σ
x)
= kx(k −2)/k∂f(x1, . . . , xn)· z,jH−1(σx)
kx(k −2)/k∂f(x1, . . . , xn)· x2/k k(k−1).
Suppose Ω ⊂ Rm is a bounded, open set. We saythat a Ck-smooth functionu: Ω→RisCk-smooth up to the boundaryif all of its derivatives of order up to k maybe continuouslyextended to the boundaryof Ω. In other words, the boundaryvalues ofu and its derivatives are well-defined on∂Ω, bycontinuity. ForR >1 denote
ΩR=!
(z1, . . . , zn)∈(Rk)n ; R−1<|zi|< R for i= 1, . . . , n"
. We denote by∂regΩR the regular part of the boundary∂ΩR. That is,
∂regΩR=
# n
$
i=1
A−i
%
∪
# n
$
i=1
A+i
%
where A±i =!
z∈(Rk)n ; log|zi|=±logR, R−1<|zj|< Rfor allj=i"
. (2.7) We write DR for the collection of all functionsu: ΩR → R, that are C2- smooth up to the boundary, and that satisfy Neumann’s condition:
(∇u)i, zi= 0 for any i= 1, . . . , n, z∈A±i . (2.8) Here, ∇u= ((∇u)1, . . . ,(∇u)n)∈(Rk)n. LetG= (O(k))n, where O(k) is the group of all orthogonal transformations in Rk. The group G acts on Rm= (Rk)n, via
g.(z1, . . . , zn) = (g1(z1), . . . , gn(zn))
for g = (g1, . . . , gn) ∈G=O(k)n and z = (z1, . . . , zn)∈(Rk)n. A subset U ⊆RmisG-invariantifg.z∈U for anyz∈U, g∈G. SupposeU ⊆Rmis G-invariant andf :U →R. We saythatf isG-invariant if
f(g.z) =f(z) for g∈G, z∈U.
We write π−1(Rn+) for the collection of all z ∈ (Rk)n with zi = 0 for all i. Assume that ψ : π−1(Rn+) → R is a C2-smooth function, and denote byν the measure onπ−1(Rn+) whose densityis exp(−ψ). For a C2-smooth functionu:π−1(Rn+)→Rwrite
νu=eψdiv(e−ψ∇u) =u− ∇ψ,∇u,
where div stands for the usual divergence operator in Rm. Integrating by parts, we see that for any u, f : ΩR → R that are C2-smooth up to the boundary,
ΩR
∇u,∇fdν=−
ΩR
f(νu)dν+
∂regΩR
f∇u, Ne−ψ, where N is the outer unit normal. It follows that when f : ΩR → R is C1-smooth up to the boundaryandu∈ DR,
ΩR
∇u,∇fdν=−
ΩR
f(νu)dν. (2.9) The well-known Bochner identitystates that for anyC3-smooth function u: ΩR→R,
1
2ν|∇u|2=∇u,∇(νu)+
m
i=1
|∇∂iu|2+&
(∇2ψ)∇u,∇u'
, (2.10) as maybe verified directly. In the following lemma we integrate Bochner’s formula (2.10), and use theG-invariance in order to eliminate a term.
Lemma 2.5. — Let R > 1 and let u∈ DR be a G-invariant function.
Then,
ΩR
|νu|2dν=
ΩR
m
i=1
|∇∂iu|2dν+
ΩR
&
(∇2ψ)∇u,∇u' dν.
Proof. — It suffices to prove the lemma under the additional assumption thatuisC3-smooth up to the boundaryin ΩR, as such functions are dense
in DR in theC2-topology. We integrate the identity (2.10) over ΩR. From (2.9),
1 2
ΩR
ν|∇u|2dν+
ΩR
|νu|2dν=
ΩR
m
i=1
|∇∂iu|2dν+
ΩR
&
(∇2ψ)∇u,∇u' dν, sinceu∈ DR. To conclude the lemma, it suffices to show that
ΩR
ν|∇u|2dν= 0.
This would follow from (2.9) once we show that|∇u|2∈ DR. Hence, in order to conclude the lemma, we need to prove that
(∇ |∇u|2
i, zi
)= 0 for any i= 1, . . . , n, z∈A±i . (2.11)
So far we did not applythe G-invariance of u. It will playa role in the proof of (2.11). Fix i= 1, . . . , n. Sinceu∈ DR, then according to (2.8), for z∈A±i ,
(∇u)i, zi= 0.
However, sinceuisG-invariant, then (∇u)i is always a vector proportional tozi. We conclude that
(∇u)i= 0 on A±i . (2.12) We maydifferentiate (2.12) in the direction of ∇u, since∇uis tangential to∂regΩR, and obtain
(∇2u)∇u
i= 0 on A±i . (2.13)
Observe that
∇ |∇u|2= 2(∇2u)∇u. (2.14)
From (2.13) and (2.14) we deduce (2.11).
Lemma 2.6. — Suppose that ϕ : Rn+ → R is C2-smooth, and that the function
(x1, . . . , xn)→ϕ(xk1, . . . , xkn)
is convex in Rn+. For z ∈π−1(Rn+)denote ψ(z) = ϕ(π(z)). Then, for any G-invariant functionu:Rm→R,
&
(∇2ψ)∇u,∇u'
0 (2.15)
at any pointz∈π−1(Rn+)in whichuis differentiable.
Proof. — Fix a pointz= (z1, . . . , zn)∈(Rk)nwithzi= 0 for alli. Then the function
Rn+(a1, . . . , an)→ψ(a1z1, . . . , anzn)∈R
is convex on Rn+, byour assumption. In particular, ∇2ψ(z)|E is positive semi-definite, where
E={(a1z1, . . . , anzn) ; a1, . . . , an ∈R} ⊂Rm
is ann-dimensional subspace. SinceuisG-invariant and differentiable atz,
then∇u(z)∈E, and (2.15) follows.
Write νR for the restriction of the measure ν to ΩR. We will use the following standard fact from the theoryof stronglyelliptic partial differential equations:
Lemma 2.7. — Suppose R >1. Let f : ΩR →Rbe a G-invariant Lip- schitz function with
f dνR = 0. Then, there exists a G-invariant function u∈ DR with
udνR= 0 such that
νu=f in ΩR. (2.16)
Proof sketch. — Denote QR = [−1/R, R]n ⊂Rn and g(|z1|, . . . ,|zn|) = f(z1, . . . , zn) forz∈ΩR. Theng is Lipschitz inQR. Denote byη the finite Borel measure on QR which is the push-forward of the measure νR under the map (z1, . . . , zn) → (|z1|, . . . ,|zn|). Then η has a densityof the form exp(−θ) onQR, whereθis Lipschitz . Furthermore,
gdη= 0. The task of solving (2.16) is reduced to the task of findingu:QR →R,C2-smooth up to the boundarywith
udη= 0, such that
u=g+∇u,∇θ, (2.17)
and such that u satisfies Neumann’s boundarycondition on ∂QR. Trans- lating and rescaling, we mayreplace the cube QR and work in the cube Q = [0,1/2]n ⊂ Rn. Let ˜g : Rn → R be the Zn-periodic function which satisfies
˜
g(x1, . . . , xn) =g(|x1|, . . . ,|xn|) for allx∈[−1/2,1/2]n. Then ˜gis a Lipshitz function onRn, that is even in all of the coordinates. We similarlyconstruct the function ˜θwhich is the uniqueZn-periodic extension ofθthat is even in all of the coordinates. We seek for aC2-smooth function
˜
u:Rn →R, that is Zn-periodic and even in all of the coordinates, which satisfies
u˜= ˜g+(
∇u,˜ ∇θ˜)
. (2.18)
Note that the restriction of ˜uto [0,1/2]nwill solve (2.17) and will automat- icallysatisfyNeumann’s boundarycondition. Our task is thus reduced to solving a stronglyelliptic partial differential equation on the torus, with Lip- schitz coefficients. The usual interior regularityestimates (see, e.g., Gilbarg and Trudinger [19, Chapter 4]) show that there exists a uniqueZn-periodic solution of integral zero which isC2-smooth. Moreover, this unique solution inherits of all the symmetries of the problem, and is therefore even in all of
the coordinates, as required.
I am grateful to Ha¨ım Brezis for explaining to me that even in the case where the functionf in Lemma 2.7 isC∞-smooth up to the boundary, the solutionuis not necessarilyC4-smooth up to the boundary.
Lemma 2.8. — Let ϕ be as in Lemma 2.6. Suppose that µ is a Borel measure onRn+ with densityexp(−ϕ). Then, for any locally Lipschitz func- tionf ∈L2(µ)∩L1(µ),
V arµ(f) k2 k−1
n
i=1
Rn+
x2i∂if(x)2dµ(x). (2.19) Here,V arµ(f) =
(f−E)2dµ, whereE∈Ris such that
(f−E)dµ= 0.
Proof. — Denoteψ(z) =ϕ(π(z)) forz∈π−1(Rn+). Letνbe the measure onRmwhose densityis
z→ωn,k−1exp(−ψ(z)) (z∈π−1(Rn+))
where ωn,k is as in Lemma 2.1. Then π pushes the measure ν forward to the measureµ, as we learn from Lemma 2.1, and in fact,
ν=
Rn+
σxdµ(x). (2.20)
FixR > 1 and denoteg(z) =f(π(z)). The function g is Lipschitz on ΩR. Let ER ∈ R be such that
(g−ER)dνR = 0. According to Lemma 2.7, there exists a G-invariant function u ∈ DR with
udνR = 0 such that νu=−(g−ER). Lemma 2.5 and Lemma 2.6 implythat
ΩR
|νu|2dν
ΩR
m
i=1
|∇∂iu|2dν. (2.21)
We repeat the dualityargument from [23, Section 2]:
(g−ER)2dνR (2.22)
=−
gνudνR=
m
i=1
∂ig∂iudνR
m
i=1
∂igH−1(νR)
|∇∂iu|2dνR
*+ +,
m
i=1
∂ig2H−1(νR)
*+ +, m
i=1
|∇∂iu|2dνR
*+ +,
m
i=1
∂ig2H−1(νR)
|νu|2dνR,
where we used (2.21) in the last inequality. Therefore,
ΩR
(g−ER)2dνR
m
i=1
∂ig2H−1(νR)=
n
=1 k
j=1
∂g
∂z,j
2
H−1(νR)
. (2.23) According to Lemma 2.2 and to (2.20), for any= 1, . . . , nandj= 1, . . . , k,
∂g
∂z,j
2
H−1(νR)
Rn+
∂g
∂z,j
2
H−1(σx)
dµ(x) k k−1
Rn+
x2∂f(x)2dµ(x), (2.24) where the last inequalityis the content of Lemma 2.4. Bycombining (2.23) and (2.24), and lettingRtend to infinity, we obtain
V arµ(f) =V arν(g) k2 k−1
n
i=1
Rn+
x2i ∂if(x)2dµ(x).
Proof of Theorem 1.1. — Assume first that ϕ is finite and C2-smooth.
All we need in order to deduce (1.2) from (2.19) is to remove the assumption thatf ∈L2(µ). To that end, given a locallyLipschitzf ∈L1(µ) andM >0, we consider the truncation
fM = max{min{f, M},−M}.
Then fM ∈ L2(µ) is locallyLipschitz. We apply(2.19) for fM and letM tend to infinity, and obtain (1.2) from the monotone convergence theorem.
This completes the proof in the case whereϕis finite and smooth. For the
general case, a standard approximation argument is needed. One possibility is to observe that it is enough to prove the theorem where the integrals over Rn+ are replaced byintegrals over the cube
-R−1, R.n
⊂Rn+,
for any R >1. On the bounded cube, it is straightforward to approximate exp(−ϕ) bya finite, smooth density, such that both the left-hand side and the right-hand side of (1.2) are well-approximated, for a given locallyLips-
chitz functionf.
Remark 2.9. — Supposek1, . . . , kn 2 are integers, and that the func- tionϕ:Rn+→R∪ {∞}is such that
(x1, . . . , xn)→ϕ(xk11, . . . , xknn)
is convex onRn+. It is straightforward to adapt the proof of Theorem 1.1 to this case. We obtain a variant of Theorem 1.1, in which the inequality(1.2) is modified as follows: The factor k2/(k−1) is inserted into the sum, and replaced byk2i/(ki−1). See Theorem 6.1 below.
3. Toric K¨ahler Manifolds
This section provides a proof of Theorem 1.4. Throughout this section, we assume that we are given a convex body K ⊂ Rn, and a sufficiently smooth convex function ψ : Rn → R with ∇ψ(Rn) = K. (Say, ψ is C∞- smooth. In the formulation of Theorem 1.4, it is assumed that ψ is C3, yet the approximation argument is straightforward). Most of the argument generalizes to anyopen, convex set K ⊂ Rn. In particular, the analysis in Section 2 for k = 2 is parallel to the case where K equals Rn+ and ψ(x) =n
i=1exp(xi).
The proof of Theorem 1.4 is essentiallyan interpretation of the dual Bochner inequalityin a certain toric K¨ahler manifold. We begin with a quick review of the the basic definitions, see e.g. Tian [29, Chapter 1] for more information. SupposeXis a complex manifold of complex dimensionn. The induced almost complex structure is a certain smooth mapJ :T X→T X, such that for anyp∈X the restrictionJ|TpX is a linear operator ontoTpX with
J2|TpX =−I.
In fact, in an open set U ⊂ Cn containing the origin, consider the map f(z) = √
−1z defined in a neighborhood of zero. Its derivative at zero is
J|T0U. One verifies that this construction ofJ does not depend on the choice of the chart, as the transition functions are holomorphic. A closed 2-formω onX isK¨ahlerif the bilinear form
gω(u, v) =ω(u, Jv) (p∈X, u, v∈TpX)
is a Riemannian metric, which is alsoJ-invariant (i.e.,gω(u, v) =gω(Ju, Jv) for any p ∈ X and u, v ∈ TpX). Next, we specialize to the case of toric K¨ahler manifolds, see also Abreu [1] and Gromov [20]. We consider the complex torus
TnC=Cn/(√
−1Zn) =! x+√
−1y ; x∈Rn, y∈Rn/Zn"
.
(Perhaps it is more common to saythat (C∗)n is the complex torus, where C∗=C\{0}. Note that exp(2πz) is a biholomorphism betweenT1CandC∗).
The real torusTn =Rn/Zn acts on the complex manifoldTnC via t.(x+√
−1y) =x+√
−1(y+t)
t∈Tn, x+√
−1y∈TnC . Functions, vector fields and differential forms on Rn have toric-invariant extensions toTnC. For instance, we extend the convex functionψtoTnC by
ψ(x+√
−1y) =ψ(x) forx+√
−1y∈TnC.
Then ψ is a Tn-invariant function on the complex manifold TnC. With a slight abuse of notation, we use the same letter to denote a function onRn, and its toric-invariant extension to TnC. Consider the K¨ahler form on TnC defined by
ωψ= 2√
−1∂∂ψ¯ =
√−1 2
n
i,j=1
ψijdzi∧d¯zj,
where we use the notation ψij =∂2ψ/(∂xi∂xj) explained in the Introduc- tion. Abbreviating gψ=gωψ, we have
gψ
∂
∂xi
, ∂
∂xj
=gψ
∂
∂yi
, ∂
∂yj
=ψij (i, j= 1, . . . , n) whilegψ
∂
∂xi,∂y∂j
= 0 for anyi, j. Furthermore, observe that ωnψ=ρψV ol2n
whereV ol2nis the standard volume form onTnCandρψ(x) = det∇2ψ(x) for x∈Rn. It is customaryto call the mapx+√
−1y→ ∇ψ(x) the associated moment map, see Abreu [1] and Gromov [20].
Below we review in great detail some of the standard formulae of Rie- mannian geometryin the case of a toric K¨ahler manifold. As much as pos- sible, we prefer real formulae in real variables. One reason for this is that the complex notation fits well onlywith the casek= 2 in Section 2. For a smooth functionu:Rn →Rwe write
∇ψu=
n
i,j=1
ψijui
∂
∂xj =
n
j=1
uj ∂
∂xj
for the Riemannian gradient of u, where we abbreviate uj =n
i=1ψijui. Next, we describe the connection ∇ψ that corresponds to the Riemannian metricgψ. As is computed, e.g., in Tian [29],
∇ψ∂
∂yj
∂
∂xk
= 1 2
n
=1
ψjk ∂
∂y
, ∇ψ∂
∂xj
∂
∂xk
=1 2
n
=1
ψjk ∂
∂x
where ψjk = n
m=1ψmψjkm. We view the Hessian ∇ψ,2h of a smooth functionh:Rn →Ras a linear operator onTpX, specifically,
TpX U → ∇ψU∇ψh∈TpX.
In coordinates, for a smooth functionh:Rn→R,
∇ψ,2h ∂
∂xi
=
n
j,k=1
ψjkhik−1 2ψijkhk
∂
∂xj
,
∇ψ,2h ∂
∂yi
= 1 2
n
j,k=1
ψijkhk
∂
∂yj
, (3.1)
where ψjki =n
,m=1ψjψmkψim. It is unfortunate that we have to work with the real Hessian, and not with the simpler complex Hessian. We denote byψthe Riemmanian Laplacian onTnC, corresponding to the Riemmanian metric gψ. Then ψh is the trace of ∇ψ,2h, and for a smooth function h:Rn→R,
ψh=
n
i,j=1
ψijhij.
The Bochner-Weitzenb¨ock formula from Riemannian geometry(e.g. Pe- tersen [27, Section 7.3.1]) states that for anysmooth functionu:Rn→R,
1
2ψ|∇ψu|2=∇ψu,∇ψ(ψu)+|∇ψ,2u|2HS+Ricψ(∇ψu,∇ψu) (3.2)
where |∇ψ,2u|2HS is the Hilbert-Schmidt norm of the Hessian, and where Ricψ is the Ricci form, which is the bilinear form given by
Ricψ
∂
∂xj, ∂
∂xk
=−1 2
∂2logρψ
∂xj∂xk
forj, k= 1, . . . , n. Note thatRicψ(∇ψu,∇ψu)0 whenρψ is log-concave.
Definition 3.1. — Suppose(M, g)is a Riemannian manifold,∇is the standard Levi-Civita connection, and ν a Borel measure onM. LetV be a vector field on M, which is locallyν-integrable. We set
VH−1(ν)= sup
MV,∇hdν ;
M|∇2h|2HSdν1
(3.3) where the supremum runs over all smooth functions h:M →Rsuch that V,∇hisν-integrable.
The proof of Lemma 2.2 immediatelygeneralizes to ν =
Ω
ναdλ(α) ⇒ V2H−1(ν)
ΩV2H−1(να)dλ(α). (3.4) Next, we use theTn-invariance and obtain a lower bound for|∇ψ,2u|2HS
in terms of the first derivatives ofu. Suppose thatu:Rn →Ris a smooth function. Denote byEp⊂TpX the subspace spanned by ∂y∂j (j= 1, . . . , n).
As in anyRiemannian manifold, the operator ∇ψ,2u is symmetric with respect to the Riemmannian metric gψ. Furthermore, from (3.1) we learn that Ep is an invariant subspace of the operator ∇ψ,2u, and the matrix representing the operator ∇ψ,2u|Ep in the basis ∂y∂k (k= 1, . . . , n) is
1 2
n
j=1
ujψjk
k,=1,...,n
.
Consequently, ∇ψ,2u2
HS
∇ψ,2u
Ep
2
HS=T race/
∇ψ,2u
Ep
20
=1 4
n
i,j,m,p=1
uiujψpjmψipm. (3.5) For x∈Rn we denote byσx the uniform probabilitymeasure on the real torus{x+√
−1y;y∈Tn}. These measures are “the same up to translation”.
For a vector fieldU =n
i=1Ui ∂∂xi set Q˜ψ,x(U) = sup
n
j=1
ψijUjVj
2
;1 4
n
i,j,k,,m,p=1
ViVjψmψjkmψkpψip1
, where the supremum runs over all V1, . . . , Vn ∈ Rn. Here, ψm, ψjkm etc.
are evaluated atx. Observe that ˜Qψ,xis essentiallythe same quadratic form asQψ,∇ψ(x)mentioned in the Introduction. That is, ifh=f(∇ψ(x)), then
Q˜ψ,x
∇ψh
=Qψ,∇ψ(x)(∇f).
Lemma 3.2. — Letu:Rn→R. Then, for any pointx∈Rn in which u is differentiable,
∇ψu2H−1(σx)Q˜ψ,x(∇ψu).
Proof. — The vector field∇ψuonTnCisTn-invariant. It therefore suffices to restrict our attention to Tn-invariant functionshin the definition (3.3) of∇ψuH−1(σx)(i.e., ifhis notTn-invariant, then average it with respect to the Tn-action). Suppose that h : Rn → R is a smooth function. From (3.5),
TnC|∇ψ,2h|2HSdσx 1 4
n
i,j,k,,m,p=1
hihjψmψjkmψkpψip
where the functions on the right-hand side are evaluated at the point x.
Since
TCn
∇ψu,∇ψhdσx=
n
i,j=1
ψijuihj,
the lemma follows from the definition of theH−1 norm.
Supposeϕ : Rn → R is a smooth function on Rn, with infϕ > −∞. Consider the finite Borel measure µ onTnC that is induced bythe volume form exp(−ϕ)ωψn. That is,µis the measure onTnCwhose densitywith respect to the standard Lebesgue measure onTnCis
exp(−ϕ(x))ρψ(x).
Observe that for anyµ-integrable functionu:Rn→R,
TnC udµ=
Rnu(x)e−ϕ(x)det∇2ψ(x)dx.
More generally, we can say that µ=
Rnσxe−ϕ(x)ρψ(x)dx. (3.6) For a smooth functionu:Rn→Rdenote
µu=ψu−
n
i,j=1
ψijuiϕj. (3.7) Integrating byparts, we see that whenu, h:Rn→Rare smooth functions, with at least one of them compactly-supported,
TnC h(µu)dµ=−
TnC∇ψu,∇ψhdµ. (3.8) We assume that the following Bakry-´Emery-Ricci condition holds true:
(?) For any x∈Rn, the matrix
# ϕi−1
2
n
k=1
ψkiϕk−1 2
∂2logρψ
∂xi∂x
%
i,=1,...,n
is positive semi-definite.
Condition (?) is equivalent to the pointwise inequality,
&
(∇ψ,2ϕ)U, U'
+Ricψ(U, U)0 (3.9)
for anyvector field of the formU =n
i=1Ui ∂∂xi. In the terminologyof Bakry and ´Emery[5], condition (?) means that the Bakry-´Emery-Ricci tensor (also known as Γ2or the “second carr´e du champ”) is positive semi-definite, when restricted to the subspace spanned by ∂x∂1, . . . ,∂x∂n. The onlycase that is relevant for Theorem 1.4, is whenρψ is log-concave andϕ≡1. Condition (?) clearlyholds true in this case. Theorem 1.1 is related to the case where ψ(x) =n
i=1exi, and condition (?) amounts to the convexityof the function ϕ(2 logx1, . . . ,2 logxn) in the interior ofRn+.
As explained in the Introduction, we have to impose certain restrictions on the behavior of ψ and ϕ at infinity. We say that the pair of functions (ψ, ϕ) isregular at infinity if there exists a linear space X of C∞-smooth functions u:Rn →Rwhich has the following properties:
