2002 Éditions scientifiques et médicales Elsevier SAS. All rights reserved S0246-0203(02)01108-1/FLA
SPECTRAL GAP AND LOGARITHMIC SOBOLEV INEQUALITY FOR UNBOUNDED CONSERVATIVE
SPIN SYSTEMS
TROU SPECTRAL ET INÉGALITÉS DE SOBOLEV LOGARITHMIQUES POUR DES SYSTÉMES DE SPINS
CONSERVATIFS ET NON BORNÉS
C. LANDIMa,b, G. PANIZOa, H.T. YAUc
aIMPA, Estrada Dona Castorina 110, CEP 22460 Rio de Janeiro, Brazil bCNRS UMR 6085, Université de Rouen, 76128 Mont Saint Aignan, France cCourant Institute, New York University, 251 Mercer street, New York, NY 10 012, USA
Received 28 September 2000, revised 6 July 2001
ABSTRACT. – We consider reversible, conservative Ginzburg–Landau processes, whose potential are bounded perturbations of the Gaussian potential, evolving on ad-dimensional cube of lengthL. Following the martingale approach introduced in (S.L. Lu, H.T. Yau, Spectral gap and logarithmic Sobolev inequality for Kawasaki and Glauber dynamics, Comm. Math. Phys.
156 (1993) 433–499), we prove in all dimensions that the spectral gap of the generator and the logarithmic Sobolev constant are of orderL−2.2002 Éditions scientifiques et médicales Elsevier SAS
Keywords: Interacting particle systems; Spectral gap; Logarithmic Sobolev inequality
RÉSUMÉ. – Nous considérons des processus de Ginzburg–Landau réversibles, dont le potentiel est une perturbation bornée du potential Gaussien, évoluent sur un cube d-dimensionel de largeurL. Suivant la méthode martingale introduite dans (S.L. Lu, H.T. Yau, Spectral gap and logarithmic Sobolev inequality for Kawasaki and Glauber dynamics, Comm. Math. Phys. 156 (1993) 433–499), nous démontrons qu’en toute dimension le trou spectral et la constante de Sobolev logarithmique sont d’ordreL−2. 2002 Éditions scientifiques et médicales Elsevier SAS
E-mail addresses: landim@impa.br (C. Landim), gpanizo@impa.br (G. Panizo), yau@cims.nyu.edu (H.T. Yau).
1. Introduction
In recent years some progress has been made in the investigation of convergence to equilibrium of reversible conservative interacting particle systems [1,2,9,8,11,4,5].
In finite volume the techniques used to obtain the rate of convergence to equilibrium rely mostly on the estimation of the spectral gap of the generator. In general, one shows that the generator of the particle system restricted to a cube of lengthNhas a gap of order N−2in any dimension. This estimate together with standard spectral arguments permits to prove that the particle system restricted to a cube of sizeN decays to equilibrium in the variance sense at the exponential rate exp{−ct/N2}: for any functionf inL2,
Ptf −Eπ[f]22exp{−ct/N2}f −Eπ[f]22,
where{Pt, t0}stands for the semi-group of the process,π for the invariant measure, Eπ[f]for the expectation off with respect toπand · 2for theL2norm with respect toπ.
In infinite volume, since the spectrum of the generator of a conservative system has no gap at the origin, instead of exponential convergence to equilibrium, one expects a polynomial convergence. In this context, the main difficulty is to use the local information on the gap of the spectrum of the generator restricted to a finite cube to deduce the global behavior of the system in infinite volume.
On the other hand, the relation between the logarithmic Sobolev inequality and the hypercontractivity has long been established. The hypercontractivity in turn permits to prove upper and lower Gaussian estimates of the transition probability of a reversible Markov process (cf. [6,11]).
In this article we present a sharp estimate of the spectral gap and of the logarithmic Sobolev constant for the Ginzburg–Landau process whose potential is a bounded perturbation of the Gaussian potential. The precise assumptions are given in Section 2.
We follow here the martingale approach introduced in [14]. The main ideas are essentially the same but there are several technical difficulties coming from the unboundedness of the spins. The main ingredients are a local central limit theorem, uniform over the parameter and from which follows the equivalence of ensembles, and some sharp large deviations estimates.
The article is divided as follows. In Section 2 we state the main results and introduce the notation. In Section 3 we prove the spectral gap and in Section 4 the logarithmic Sobolev inequality. In Section 5 we prove a uniform local central limit theorem and deduce some results regarding the equivalence of ensembles. In Section 6 we obtain some large deviations estimates which play a central role in the proof of the logarithmic Sobolev inequality.
2. Notation and results
For L1, denote by L the cube {1, . . . , L}d. Configurations of the state space RL are denoted by the Greek letters η, ξ, so that ηx indicates the value of the spin at x∈L for the configuration η. The configuration η undergoes a diffusion on RL
whose infinitesimal generatorLL is given by LL=1
2
x,y∈L
|x−y|=1
(∂ηx −∂ηy)2−1 2
x,y∈L
|x−y|=1
V(ηy)−V(ηx)(∂ηy −∂ηx).
V :R→R represents the potential V (a)=(1/2)a2+F (a), where F:R→R is a smooth bounded function such thatF∞<∞,
e−V (x)dx=1.
We assumed the convex part of the potential to be Gaussian for simplicity. All proofs of the results presented in Section 5 on the uniform local central limit theorems rely strongly on this hypothesis. We believe, however, that the approach presented here extend to the case where we have a bounded perturbation of a convex potential. In this respect, it was recently observed by Caputo [3] that when the potential is a purely convex function, theL2behavior of the inverse of the spectral gap and of the logarithmic Sobolev constant can be easily obtained by techniques introduced for models with convex interactions (see [13] and references therein).
Denote byZ:R→Rthe partition function Z(λ)=
∞
−∞
eλa−V (a)da, (2.1)
byR:R→Rthe density function∂λlogZ(λ), which is smooth and strictly increasing, and bythe inverse ofR so that
α= 1 Z((α))
∞
−∞
ae(α)a−V (a)da for eachαinR.
ForλinR, denote byν¯λL the product measure onRL defined by
¯
νλL(dη)=
x∈L
1
Z(λ)eληx−V (ηx)dηx
and letναL= ¯ν(α)L . Notice thatEνα[ηx] =αfor allαinR,xinL. Most of the times omit the superscript L. For each M inR, denote byνL,M the canonical measure on Lwith total spin equal toM:
νL,M(·)=ναL
·
x∈L
ηx=M
. Expectation with respect toνL,M is denoted byEL,M.
An elementary computation shows that the product measures {¯νλ, λ ∈ R} are reversible for the Markov process with generator LL. The Dirichlet form DL
associated toLL is given by
DL(µ, f )=1 2
x,y∈L
|x−y|=1
(Tx,yf )2µ.
In this formula and below, for a probability measureµ,·µstands for expectation with respect toµ. Furthermore, forx,yinZd,Tx,yrepresents the operator that acts on smooth functionsf as
Tx,yf = ∂f
∂ηx
− ∂f
∂ηy
andµstands for the invariant measuresνα,νL,M.
For a positive integerLandM inR, denote byW (L, M)the inverse of the spectral gap of the generatorLL with respect to the measureνL,M:
W (L, M)=sup
f
f;fνL,M
DL(νL,M, f ).
In this formula the supremum is carried over all smooth functions f inL2(νL,M)and f;fµ stands for the variance off with respect toµ. We also denote this variance by the symbol Var(µ, f ). Let
W (L)=sup
M∈RW (L, M).
THEOREM 2.1. – There exists a finite constantC0depending only onF such that W (L)C0L2
for allL2.
A lower bound of the same order is easy to derive. Fix a smooth function H:[0,1]d → R such that H (u) du =0 and let fH(η) =x∈LH (x/L)ηx. An elementary computation shows that
fH;fHνL,M=
x
H (x/L) 2
η2e1;ηe1νL,M
+
x
H (x/L)2ηe1;ηe1νL,M− η2e1;ηe1νL,M
,
DL(νL,M, fH)=(1/2)
|x−y|=1
H (y/L)−H (x/L)2.
In this formula {ej, 1 j d} stands for the canonical basis of Rd. By Corol- lary 5.3, as L ↑ ∞, M/Ld → α, fH;fHνL,M/L2DL(νL,M, fH) converges to
ηe1;ηe1να
H (u)2du/ (∇H )(u)2du. This proves that lim inf
L→∞ L−2W (L) >0.
ForL2, a probability measureν on RL and a function f such that f2ν =1, denote bySL(ν, f )the entropy off2dνwith respect toν:
SL(ν, f )= f2logf2dν
and byθ (L, M)the inverse of the logarithmic Sobolev constant of the Ginzburg–Landau process on the cubeLwith respect to the measureνL,M:
θ (L, M)=sup
f
SL(νL,M, f ) DL(νL,M, f ).
In this formula, the supremum is carried over all smooth functionsf inL2(νL,M)such thatf2νL,M =1. Let
θ (L)=sup
M∈Rθ (L, M).
THEOREM 2.2. – Assume that F∞ < ∞. There exists a finite constant C depending only onF such thatθ (L)CL2for allL2.
We follow here the martingale method developed by Lu and Yau [14] to prove the spectral gap and a bound on the logarithmic Sobolev constant for a conservative interacting particle system. This approach relies on two a-priori estimates. First, a local central limit theorem for i.i.d. random variables with marginals equal to the marginals of the product measureν¯λ, uniform over the parameter λinR. Second, a spectral gap or a logarithmic Sobolev inequality, uniform over the density, for a Glauber dynamics on one site which is reversible with respect to the one-site marginal of the canonical invariant measure.
3. Spectral gap
To fix ideas, we prove Theorem 2.1 in dimension 1. The reader can find in Section A.3.3 of [10] the arguments needed to extend the proof to higher dimensions. To detach the main ideas, we divide the proof in four steps. The proof goes by induction on L. We start withL=2.
In this section all constants are allowed to depend onF∞,F∞. In the case they depend on some other parameter, the dependence is stated explicitly.
Step 1. One-site spectral gap. Consider a smooth functionf:R2 →R. We want to estimatef;fν2,M in terms of the Dirichlet form off. Since for the measureν2,M
the total spin is fixed to be equal toM, letg(a)=f (M−a, a)and notice thatf;fν2,M
is equal tog;gν2,M.
The following result will be of much help. FixL2 andMinR. Denote byν1L,Mthe marginal distribution ofηLwith respect toνL,M. The Glauber dynamics has a positive spectral gap which is uniform with respect toM:
LEMMA 3.1. – There is a finite constantC0depending only onF∞such that Var(ν1L,M, f )C0Eν1
L,M
∂f
∂ηL
2
for everyL2, everyMinRand every smooth functionf:R→RinL2(ν1L,M).
Remark 3.2. – In the case of grand canonical measures, this result is true under the more general hypothesis of strict convexity at infinity of the potential (cf. [13]
and references therein). In case of canonical measures the main problem is to obtain a good approximation of the one-site marginal in terms of the one-site marginal of grand canonical measures.
Before proving this result, we conclude the first step. Applying this result to the func- tion g defined above, we obtain that its variance is bounded by C0Eν1
2,M[(∂g/∂η2)2]. Since∂g/∂η2=(∂f/∂η2−∂f/∂η1), we have that
f;fν2,M= g;gν2,M = g;gν1
2,M
C0Eν1
2,M
∂g
∂η2
2
=C0Eν2,M
∂g
∂η2
2
=C0Eν
2,M
∂f
∂η2
− ∂f
∂η1
2 .
This shows that W (2)C0, proving Theorem 2.1 in the case L=2. We conclude this step with the
Proof of Lemma 3.1. – We first prove the lemma for the grand canonical measure. Fix λinRand denote byν¯λ1the one-site marginal of the product measureν¯λL. Fixxλ inR, that will be specified later, andf inL2(ν¯λ1). The variance off is bounded above by
R
f (x)−f (xλ)2e−Vλ(x)dx,
whereVλ(x)= −λx+logZ(λ)+V (x). By Schwarz inequality, the previous expression is less than or equal to
∞ xλ
dx[f(x)]2e−Vλ(x)
eVλ(x) ∞ x
dy (y−xλ)e−Vλ(y)
+
xλ
−∞
dx[f(x)]2e−Vλ(x)
eVλ(x) x
−∞
dy (xλ−y)e−Vλ(y)
.
It remains to show that the expressions inside braces are uniformly bounded inx andλ for an appropriate choice ofxλ. Both expressions are handled in the same way and we consider, to fix ideas, the first one where we need to estimate
sup
xxλ
e(1/2)(x−λ)2+F (x) ∞ x
dy (y−xλ)e−(1/2)(y−λ)2−F (y)
.
Choosexλ=λand change variables to reduce the previous expression to
sup
x0
e(x2/2)+Fλ(x) ∞ x
dy ye−(y2/2)−Fλ(y)
,
whereFλ(a)=F (a+λ). In the case whereF =0, this expression is bounded above by some universal constant C0. SinceF is bounded, this expression is less than or equal to C0exp{2F∞}uniformly overλ. This concludes the proof of the lemma in the case of grand canonical measures.
We turn now to the case of canonical measures. We need to introduce some notation.
For λ in R, let {Xjλ, j 1} be a sequence of i.i.d. random variables with density Z(λ)−1exp{λx − V (x)}. For a positive integer L, denote by fλ,L the density of (σ (λ)2L)−1/21jL{Xλj−γ1(λ)}, whereγk(λ)is thekth truncated moment ofX1λand σ (λ)2is its variance:γ1(λ)=E[X1λ],γk(λ)=E[(Xλ1−γ1(λ))k]. We prove in Section 5 an Edgeworth expansion forfλ,L uniform over the parameterλ.
We may write the measure νL,M1 (dx) in terms of the density fλ,L. Choose λ so that γ1(λ)=M/L: λ=(M/L). Then, νL,M1 (dx)=√
L/(L−1)gλ(x)fλ,L−1([γ1− x]/σ√
L−1) fλ,L(0)−1dx, where gλ stands for the density Z(λ)−1exp{λx −V (x)}.
Hereafter, we will omit the dependence ofγj andσ onλ.
Denote the Radon–Nikodym derivative of νL,M1 (dx) with respect to the Lebesgue measure byR(x)=RL,M(x). Fix a functionf inL2(ν1L,M)andxλinRto be specified later. Following the proof for the grand canonical measure, we bound the variance off by
R
f (x)−f (xλ)2R(x) dx.
We now repeat the arguments presented in the case of the grand canonical measures.
After few steps, we reduce the proof of the lemma to the proof that
sup
xxλ
R(x)−1
∞ x
dy(y−xλ)R(y)
is bounded, uniformly in M. Choose xλ=λ, change variables and recall the notation introduced above to rewrite the previous expression as
sup
x0
∞
x
dy y gλ(y+λ) gλ(x+λ)
fλ,L−1([γ1−y−λ]/σ√ L−1) fλ,L−1([γ1−x−λ]/σ√
L−1)
.
By the explicit formula for the densitygλand sinceF is bounded, this expression is less than or equal to
e2F∞sup
x0
ex2/2
∞ x
dy ye−y2/2fλ,L−1([γ1−y−λ]/σ√ L−1) fλ,L−1([γ1−x−λ]/σ√
L−1)
.
We need now to estimate the ratio of the densities inside the integral. For a positive integer L, denote by gλ,L(x) the density of 1jLXλj. An elementary induction argument shows that gλ,L(x) = Z(λ)−Lexp{λx}g0,L(x) so that gλ,L(x)/gµ,L(x) = (Z(µ)/Z(λ))Lexp{(λ−µ)x} for any parameterµ. Chooseµ so thatγ1(λ)−γ1(µ)= x/(L−1) and notice thatµλbecause x0 andγ1is an increasing function. The previous identity gives that
fλ,L−1([γ1(λ)−y−λ]/σ (λ)√ L−1) fλ,L−1([γ1(λ)−x−λ]/σ (λ)√
L−1)
=fµ,L−1([γ1(λ)−λ+x−y]/σ (µ)√ L−1) fµ,L−1([γ1(λ)−λ]/σ (µ)√
L−1) e(λ−µ)(x−y).
The exponential is bounded by 1 becauseµλandxy. To conclude the proof of the lemma it is therefore enough to show that the previous ration is bounded.
In the proof of Lemma 5.1 we show that|γ1(λ)−λ|is bounded, uniformly inλ, by a constantC1which depends only onF∞and thatσ (µ)is bounded above and below by a finite positive constant for allλinRandxinR+. In particular, by Theorem 5.2, there existsL0such that forLL0, the ratio on the right hand side of the previous formula is bounded by a constant that depends only onF∞. On the other hand, for 2LL0, by Lemma 5.6 and explicit computations to express fµ,L in terms off˜µ,L, this ratio is bounded by exp{CL}for some constantCdepending only onF∞. This concludes the proof of the lemma. ✷
Step 2. Decomposition of the variance. We will obtain now a recursive equation forW (L). Assume that we already estimatedW (K)for 2KL−1. Let us write the identity
f −EL,M[f] =f −EL,M[f |ηL]+EL,M[f |ηL] −EL,M[f]. Through this decomposition we may express the variance off as
EL,M
(f −EL,M[f])2
=EL,M
(f −EL,M[f |ηL])2+EL,M
(EL,M[f |ηL] −EL,M[f])2. (3.1)
The first term on the right-hand side is easily analyzed through the induction assumption and a simple computation on the Dirichlet form. We write
EL,M
(f −EL,M[f|ηL])2=EL,M
EL,M
(f −EL,M[f |ηL])2|ηL
=EL,M
EL−1,M−ηL
(fηL−EL−1,M−ηL[fηL])2. Here we used the fact that EL,M[· |ηL] =EL−1,M−ηL[·]. In this formula and below fηL stands for the real function defined on RL−1 whose value at (ξ1, . . . , ξL−1) is given byfηL(ξ1, . . . , ξL−1)=f (ξ1, . . . , ξL−1, ηL). By the induction assumption this last expectation is bounded above by
W (L−1)EL,M
DL−1(νL−1,M−ηL, fηL)W (L−1)DL(νL,M, f ).
In conclusion, we proved that EL,M
(f −EL,M[f |ηL])2W (L−1)DL(νL,M, f ). (3.2) The second term in (3.1) is nothing more than the variance of EL,M[f |ηL], a function of one variable. Lemma 3.1 provides an estimate for this expression:
EL,M
(EL,M[f |ηL] −EL,M[f])2C0EL,M
∂
∂ηL
EL,M[f |ηL] 2
(3.3) for some constantC0depending only onF∞.
Step 3. Bounds on Glauber dynamics, small values of L. We now estimate the right hand side of (3.3), which is the Glauber Dirichlet form ofEL,M[f |ηL], in terms of the Kawasaki Dirichlet form off. A straightforward computation gives that
∂
∂ηL
EL,M[f |ηL] = 1 L−1
L−1
x=1
EL,M
∂f
∂ηL
− ∂f
∂ηx
ηL
+EL,M
f; 1
L−1
L−1
x=1
V(ηx)ηL
. (3.4)
In this formula E[g;h|F] stands for the conditional covariance of gand h:E[g;h| F] =E[gh|F] −E[g|F]E[h|F]. We examine these two terms separately.
The first expression on the right hand side of (3.4) is easily estimated. Recall the definition of the operator Tx,yf. Since TL,xf =xyL−1Ty+1,yf, by Schwarz inequality, we have that
EL,M
EL,M
1 L−1
L−1
x=1
TL,xf ηL
2
1
L−1
L−1
x=1
(L−x)
L−1
y=x
EL,M[(Ty,y+1f )2]
L
L−1
x=1
EL,M[(Tx,x+1f )2] =LDL(νL,M, f ). (3.5)
The second term in (3.4) is also easy to handle for small values ofL. SinceV (ϕ)= (1/2)ϕ2+F (ϕ) and since 1xL−1ηx is fixed for the measure EL,M[· | ηL], the square of the second term on the right hand side of (3.4) is equal to
EL,M
f; 1
L−1
L−1
x=1
F(ηx)ηL
2
=EL−1,M−ηL
fηL; 1 L−1
L−1
x=1
F(ηx) 2
EL−1,M−ηL[fηL;fηL]EL−1,M−ηL
1 L−1
L−1
x=1
F (η x) 2
.
In this formula, F stands for F − FνL−1,M−ηL. The second term is bounded by 4F2∞. On the other hand, by the induction assumption, the first term is bounded by W (L−1)DL−1(νL−1,M−ηL, fηL). Hence, taking expectation with respect toνL,M, we obtain that
EL,M
EL,M
f; 1
L−1
L−1
x=1
V(ηx)ηL
2
C0W (L−1)DL(νL,M, f )
for some constant C0 depending on F only. Without much effort and using the local central limit theorem, one can obtain an estimate of typeC0W (L−1)L−1DL(νL,M, f ) for the left hand side of the last expression. However, for small values of L this improvement is irrelevant.
From this estimate and (3.5) we get that the left hand side of (3.3), which is the second term of (3.1), is bounded above by
C0{L+W (L−1)}DL(νL,M, f ).
Putting together this estimate with (3.2), we obtain that
f;fνL,M [1+C0]W (L−1)+C0LDL(νL,M, f ) or, taking a supremum over smooth functionsf, that
W (L)C1W (L−1)+C0L. (3.6) This inequality permits to iterate the estimate W (2)C obtained in Step 1 to derive estimates ofW (L)for small values ofL. We now consider large values ofL.
Step 4. Bounds on Glauber dynamics, large values ofL. Here again we want to estimate the second term of (3.1). Applying Lemma 3.1, we bound this expression by the right hand side of (3.3). The first term of (3.4) is handled as before, giving (3.5). The
second one requires a deeper analysis. Its square is equal to
EL,M
f; 1
L−1
L−1
x=1
F(ηx)ηL
2
=EL−1,M−ηL
f; 1
L−1
L−1
x=1
F(ηx) 2
. (3.7)
Here and below we omit the subscriptηLoff. Fix 1K√
Land divide the interval {1, . . . , L−1}into4= (L−1)/Kadjacent intervals of lengthKorK+1, wherea represents the integer part ofa. Denote byIj thejth interval, byMj the total spin on Ij:Mj=x∈Ijηxand byEIj,Mj the expectation with respect to the canonical measure νIj,Mj. The right hand side of the previous formula is bounded above by
2EL−1,M−ηL
f; 1
L−1 4
j=1
x∈Ij
{F(ηx)−EIj,Mj[F]}
2
+2EL−1,M−ηL
f; 1
L−1 4
j=1
|Ij|EIj,Mj[F] 2
. (3.8)
Taking conditional expectation with respect toMj, we rewrite the first term as
2 1
L−1 4
j=1
EL−1,M−ηL
EIj,Mj
f;
x∈Ij
F(ηx) 2
24
(L−1)2 4
j=1
EL−1,M−ηL
Var(νIj,Mj, f )Var
νIj,Mj,
x∈Ij
F(ηx)
. By the induction assumption, Var(νIj,Mj, f )is bounded above byW (|Ij|)DIj(νIj,Mj, f ).
On the other hand, by Corollary 5.4, the variance of|Ij|−1x∈IjF(ηx)with respect to νIj,Mj is bounded above by C0|Ij|−1F2∞ uniformly over Mj, where C0 is a finite constant depending only onF∞. The previous expression is thus less than or equal to
C14 L2
4
j=1
W (|Ij|)|Ij|EL−1,M−ηL[DIj(νIj,Mj, f )]
C2
L 4
j=1
W (|Ij|)EL−1,M−ηL[DIj(νIj,Mj, f )].
Since W (K + 1) CW (K), which follows from (3.6) and from the bound W (K)CK2, and since the previous sum is bounded by the global Dirichlet form DL−1(νL−1,M−ηL, f ), we proved that the first term of (3.8) is bounded above by
C3W (K)
L DL−1(νL−1,M−ηL, f ). (3.9)
We turn now to the second term of (3.8). It is equal to 2
EL−1,M−ηL
f; 1
L−1 4
j=1
|Ij|(EIj,Mj[F] −a−b[mj−m]) 2
,
where mj =Mj/|Ij|, m=(M −ηL)/(L−1) and a, b are constants to be chosen later. We were allowed to add the terms a, b[mj −m] in the covariance because a, b4j=1|Ij|[mj −m] are constants. Let G(mj)=EIj,Mj[F] −a −b[mj −m]. By Schwarz inequality, the previous expression is bounded above by
2EL−1,M−ηL[f;f]EL−1,M−ηL
1 L−1
4
j=1
|Ij|G(mj) 2
.
We claim that
EL−1,M−ηL
1 L−1
4
j=1
|Ij|G(mj) 2
C0
KL (3.10)
for some finite constantC0. Indeed, developing the square, we write this expectation as 1
(L−1)2 4
j=1
|Ij|2EL−1,M−ηL[G(mj)2]
+ 1
(L−1)2
i=j
|Ij||Ii|EL−1,M−ηL[G(mi)G(mj)]. (3.11) Recall that
m=(M−ηL)/(L−1), mj=Mj/|Ij|. By Corollary 5.3,EL−1,M−ηL[G(mj)2]is bounded above by
Eνm[G(mj)2] + C0|Ij| L
Eνm[G(mj)4]. (3.12) Let A(α)=Eνα[F(η1)] and set a =A(m), b=A(m). With this choice, G(mj)= EIj,Mj[F(ηx)] − A(mj) + A(mj) − A(m) − A(m)[mj − m]. By Corollary 5.3,
|EIj,Mj[F(ηx)] −A(mj)| is less than or equal to CF∞/|Ij|. On the other hand, A(mj)−A(m)−A(m)[mj−m]is bounded in absolute value by(1/2)A∞[mj−m]2. In particular,
1 (L−1)2
4
j=1
|Ij|2Eνm[G(mj)2]C04
L2 + A2∞ 2L2
4
j=1
|Ij|2Eνm[(mj−m)4] (3.13)
for some constant C0 depending only on F. By Lemma 5.1, since νm is a product measure, the expectation on the right hand side of the previous inequality is bounded
