Functional Inequalities for Gaussian and Log-Concave

Probability Measures

Ewain Gwynne

Adviser: Professor Elton Hsu

Northwestern University

A thesis submitted for a Bachelor’s degree inMathematics (with honors)

andMathematical Methods in the Social Sciences

Abstract

We give three proofs of a functional inequality for the standard Gaussian measure originally due toWilliam Beckner. The first uses the central limit theorem and a tensorial property of the inequality.The second uses the Ornstein-Uhlenbeck semigroup, and the third uses the heat semigroup. These lattertwo proofs yield a more general inequality than the one Beckner originally proved. We then generalizeour new inequality to log-concave probability measures, study the relationship between this inequalityand a generalized logarithmic Sobolev inequality, and prove several other inequalities for log-concaveprobability measures, including Brascamp and Lieb’s sharpened Poincare inequality and Bobkov andLedoux’s sharpened logarithmic Sobolev inequality of the same form. We discuss some of the potentialapplications of our work in economics.

1

Contents

0.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1 Introduction 4

2 Proof of Beckner’s Inequality via the Central Limit Theorem 52.1 Tensorial property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Two-point inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 First proof of Beckner’s inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Extended Beckner Inequality via Semigroup Methods 123.1 The Ornstein-Uhlenbeck operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 The Ornstein-Uhlenbeck semigroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3 Proof of extended Beckner inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.4 Classical Heat semigroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4 Beckner Inequality for Log-Concave Probability Measures 224.1 Generalization of the Ornstein-Uhlenbeck operator and semigroup . . . . . . . . . . . . . . . 224.2 Commutation with the gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.3 Proof of Beckner’s inequality for log-concave probability measures . . . . . . . . . . . . . . . 26

5 Other Inequalities for Log-Concave Probability Measures 275.1 Generalized logarithmic Sobolev inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.2 Inequality for the semigroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.3 Brascamp-Lieb inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.4 Sharpened logarithmic Sobolev inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

6 Appendices 406.1 Sobolev Spaces for Log-Concave Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406.2 Existence of Semigroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426.3 Applications to Economics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2

0.1 Preface

I completed this thesis in 2013, shortly before my graduation from Northwestern University. It is intendedto fulfill the requirements for graduation with honors in Mathematics as well as the requirements for a majorin Mathematical Methods in the Social Sciences (MMSS).

This thesis consists of a mixture of results which I proved in collaboration with my adviser, and my ownexposition of material from research papers. The various sections herein were written over the course ofnearly a year, beginning in late Spring of 2012 (when I was a Junior in college) and continuing throughoutSummer 2012 and the following school year. My work on this thesis has exposed me to a broad range oftechniques and concepts in analysis and probability theory, which will be of use to me in my intended futurecareer as a mathematician. This work has also improved my intuition for problem solving and my ability toread research papers in mathematics.

I am indebted to several organizations and individuals for the successful completion of this thesis.I would like to thank my adviser, Professor Elton Hsu, for suggesting this project and for his guidance

throughout my work. He struck a perfect balance between providing enough guidance to keep me fromfollowing dead ends and to make sure I had sufficient mathematical background to tackle problems thatarose, and allowing me to explore my own ideas and learn from my mistakes. Moreover, his explanationshave been a great help in building my mathematical intuition, and his suggestions for improvements in mywriting have not only strengthened the exposition in this paper, but have also made me a stronger writer ingeneral.

I would like to thank Professor Valentino Tosatti for serving as my second reader. His review of thisdocument and careful, insightful comments on it have been a major help in the writing process.

I would like to thank Professors Joseph Ferrie and William Rogerson of the MMSS program and ProfessorMike Stein of the Math department for their flexibility in allowing me to do a thesis which would work forboth of my majors.

I would like to thank Northwestern University for funding part of my work on this project via an under-graduate research grant in the summer of 2012. The financial independence provided by this grant enabledme to devote my full attention to research during that summer, and thereby to discover more mathematicsthan would otherwise have been possible.

I would like to thank my classmates Ian Colley, Dan Douglas, and Dan Kaplan (also writing math honorstheses) and all of my classmates in MMSS for always being their to share my struggles and triumphs in thethesis process.

Finally, I would like to thank my mother, Laurie Gwynne, and my sisters, Ævalina and AramaintaGwynne, for their unceasing support throughout my work on this thesis and in my life in general.

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1 Introduction

The standard Gaussian measure on Rn is the measure

γn = (2π)−n/2e−|x|2/2dx. (1.1)

In the case n = 1, we write γ1 = γ. Two of the most fascinating and important properties of this measureare the Poincare inequality

‖f‖22 −(∫

Rnf dγn

)≤ ‖∇f‖22 (1.2)

and Gross’s [16] logarithmic Sobolev inequality∫Rnf2 log |f | dγ − ‖f‖22 log ‖f‖2 ≤ ‖∇f‖22, (1.3)

both valid for functions f in the Sobolev space W 2,1(γn) (see Appendix 6.1 for the definition of this spaceand its basic properties). The Poincare and log-Sobolev inequalities are used throughout pure and appliedmathematics, in fields as diverse as quantum mechanics, mathematical finance, infinite dimensional analysis,mathematical statistics, stochastic analysis, random matrix theory, and partial differential equations. Forexample, the logarithmic Sobolev inequality can be viewed as a sharpened form of Heisenberg’s uncertaintyprinciple. It is also used to obtain bounds for the solutions of partial differential equations, improve modelsof fluctuations in stock prices, and characterize the behavior of Brownian motion on manifolds.

Recall that for 1 ≤ p < ∞, the Lp norm of a measurable function f on Rn with respect to a measure µon Rn is defined by

‖f‖p =

(∫Rn|f |p dµ

)1/p

.

Here the measure µ will always be clear from the context. Beckner [5] has proven a functional inequality forthe standard Gaussian measure which involves the Lp norms for 1 ≤ p ≤ 2:

‖f‖22 − ‖f‖2p ≤ (2− p)‖∇f‖22. (1.4)

For p = 1, inequality (1.4) is equivalent to the Poincare inequality, as can be seen for bounded f by addinga sufficiently large constant C so that f + C is non-negative, and for the general f by approximation bybounded functions. Furthermore, if we divide both sides of (1.4) by 2− p and let p→ 2, the left side tendsto the left side of (1.3). Thus Beckner’s inequality interpolates between the Poincare inequality and thelogarithmic Sobolev inequality.

Beckner’s original proof of inequality (1.4) is based on the explicit spectral decomposition of the Ornstein-Uhlenbeck operator in terms of Hermite polynomials and Nelson’s [26] hypercontractivity inequality for theOrnstein-Uhenbeck semigroup. This latter inequality is a significant result in its own right, and is mosteasily proven using the logarithmic Sobolev inequality (see, for example, [9]).

Apparently unaware of Beckner’s work, Lata la and Oleszkiewicz [21] proved an extension of Beckner’sinequality for measures ce−|x1|r−...−|xn|rdx with 1 ≤ r ≤ 2. However, in the Gaussian case r = 2 inequality(1.4) was derived from the logarithmic Sobolev inequality and hypercontractivity of the Ornstein-Uhlenbecksemigroup, via an argument similar to that in [5]. In Section 3.1 of [22], Ledoux used non-linear PDEto prove a version of (1.4) for the invariant probability measures of Markov semigroups whose generatorssatisfy a curvature-dimension inequality; in the Gaussian case, this inequality reduces to a sharpened formof (1.4), with the right side multiplied by (n− 1)/n and the parameter p allowed to increase to 2n/(n− 1).Several other authors have also studied generalizations of Beckner’s inequality in various directions; seefor example [1], [2], [4], [11], [22], and [30]. However, the arguments in these papers also require resultscomparable in difficulty to hypercontractivity or the logarithmic Sobolev inequality. So, it is instructive tofind a direct proof of Beckner’s inequality.

We shall give three such proofs. For our first proof of Beckner’s inequality, in Chapter 2, we shalldeduce (1.4) via the central limit theorem (Theorem 2.8) and an approximation argument, beginning withan analogous inequality for a probability measure on a two-point set. This method of proof was suggested asan alternative approach in [5], and is similar to Gross’ original proof of the logarithmic Sobolev inequality.

4

In fact, approximation arguments of this sort are pervasive in probability theory. So, this proof illustratesan important technique. In the course of the proof, we prove an important tensorial property of inequality(1.4), which allows one to deduce the n-dimensional inequality from the 1-dimensional one, and implies thatthe inequality can be extended to infinite dimensional Gaussian measures.

Our second proof, in Chapter 3, uses the elementary properties of the Ornstein-Uhlenbeck operator andits associated semigroup, introduced in Sections 3.1 and 3.2. The Ornstein-Uhlenbeck operator satisfies aspecial integration by parts formula (Proposition 3.3), and its semigroup preserves Gaussian integrals. Thesetwo properties make the Ornstein-Uhlenbeck operator a natural tool for proving inequalities of the sort westudy here. Our method actually yields a new, more general version of inequality (1.4), valid with theexponent 2 replaced by any q > 2:

‖f‖2q − ‖f‖2p ≤ (q − p)‖∇f‖2q, f ∈W q,1(µ), q ≥ 2, 1 ≤ p ≤ 2. (1.5)

Our third proof, given in Section 3.4, replaces the Ornstein-Uhlenbeck semigroup with the better knownclassical heat semigroup, and also yields the extended inequality (1.5).

Beginning in Chapter 4 we shall concern ourselves with log-concave probability measures, a naturalgeneralization of Gaussian measures. We will study analogues of the Ornstein-Uhlenbeck operator andsemigroup in this more general setting. We will then use them to extend our proof of (1.5) to prove ananalogue of this inequality for general log-concave probability measures on Rn, with a multiplicative constantdepending on the measure appearing on the right side.

In Chapter 5, we will study several other inequalities for log-concave probability measures, often using thesemigroup of Subection 4.1. In Section 5.1, we will explore the implication relationships between inequality(1.5) and a generalized logarithmic Sobolev inequality:

2

q‖f‖2−qq

∫Rn|f |q log |f | dµ− 2

qlog(‖f‖q)‖f‖2q ≤ C‖∇f‖2q, f ∈W q,1(γn), q ≥ 2

in the context of a general log-concave probability measure µ on Rn.In Section 5.2, we will prove an inequality for the semigroup associated to a log-concave probability

measure, which extends an inequality which Beckner derived along with (1.4) in [5].In Section 5.3 we will use semigroup methods to prove a sharpened Poincare inequality for log-concave

probability measures due to Brascamp and Lieb [10]:

‖f‖22 −(∫

Rnf dµ

)2

≤∫Rn〈(D2v)−1∇f,∇f〉 dµ.

In the course of the proof, we also obtain invertibility of our generalization of the Ornstein-Uhlenbeckoperator on the space of functions in L2(µ) with vanishing mean (Proposition 5.14).

In Section 5.4, we shall prove an analogous sharpened logarithmic Sobolev inequality due to Bobkov andLedoux [7], under a stronger hypothesis on the measure µ; and use the Herbst argument (Proposition 5.21)to give counterexamples which show that this inequality cannot hold in general.

The inequalities we study here have potential uses in many different fields. To illustrate their applicability,we shall discuss some of their potential applications in economics in Appendix 6.3.

2 Proof of Beckner’s Inequality via the Central Limit Theorem

The main goal of this chapter is to prove the following theorem.

Theorem 2.1 (Beckner’s Inequality). If f ∈W 2,1(γn), then for each p ∈ (1, 2),

‖f‖22 − ‖f‖2p ≤ (2− p)‖∇f‖22. (2.1)

We shall first establish a tensorial property of the inequality 2.1, then proceed by way of an approximationargument using the central limit theorem and an analogous inequality for a probability measure on a two-point set. This method of proof is inspired by Gross’ [16] original proof of the logarithmic Sobolev inequality,and was suggested in [5].

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2.1 Tensorial property

In this section, we shall prove a tensorial property of Beckner’s inequality, in the setting of an arbitraryprobability measure. This property will allow us to pass from an inequality for a measure on a two-point setto the n-fold convolution of such a measure in order to apply the central limit theorem in the next section.It will also allow us to deduce the n-dimensional case of Beckner’s inequality from the 1-dimensional case:

Theorem 2.2 (Tensorial property). Let µ be a probability measure on a set Ω. Let 1 ≤ p ≤ 2. Suppose thatF is a subspace of L2(µ), B is a bilinear form on F , and µ satisfies an inequality

‖f‖22 − ‖f‖2p ≤ C(2− p)B(f, f), f ∈ F (2.2)

for some constant C > 0. Then the n-fold Cartesian product measure µn satisfies the inequality

‖f‖22 − ‖f‖2p ≤ C(2− p)Bn(f, f), f ∈ Fn,

where Fn is the space of functions f on Ωn such that xi 7→ f(x1, ..., xn) ∈ F for each fixed x1, ..., xi−1, xi+1, ..., xn ∈Ω and

Bn(f, g) :=

∫Ωn

n∑i=1

B [f(x1, ..., xi−1, ·, xi+1, ..., xn), g(x1, ..., xi−1, ·, xi+1, ..., xn)] dµn(x). (2.3)

In words, the bilinear form Bn is defined as follows: for each index i, we apply the operator B to thefunction on Ω given by xi 7→ f(x1, ..., xi−1, xi, xi+1, ..., xn) with the variables other than the ith fixed. Then,we sum over all n. Allowing the coordinates we had previously held fixed to vary, this gives a function onΩn. We then integrate this function over Ωn.

To see why we should care about this particular bilinear form, consider the most important case ofTheorem 2.2, namely where Ω = R, F = C∞c (R), and B(f, g) =

∫R f′g′dµ. Here we have

B [f(x1, ..., xi−1, ·, xi+1, ..., xn), g(x1, ..., xi−1, ·, xi+1, ..., xn)] =

∫R∂if(x1, ..., xn)∂ig(x1, ..., xn) dµ(xi)

and so

Bn(f, g) =

∫Rn〈∇f,∇g〉dµn. (2.4)

Thus, in the case of a probability measure on R, Theorem 2.2 tells us that Beckner’s inequality (2.1) indimension one implies Beckner’s inequality in dimension n for f ∈ C∞c (R), and hence, by density (seeAppendix 6.1), also for f ∈ W 2,1(µ). In particular, it will suffice to prove Theorem 2.1 only in the one-dimensional case.

Our proof of Theorem 2.2 is based on an argument by Lata la and Oleszkiewicz [21]. We first need aresult which characterizes the expression on the left side of (2.2).

Lemma 2.3. Let µ be a probability measure on a set Ω. Let q ∈ [1, 2]. For f ∈ L2(µ), f ≥ 0, set

Φ(f) =

∫Ω

fq dµ−(∫

Ω

f dµ

)q.

Then Φ is a convex functional on the non-negative functions on L2(µ), i.e. for any t ∈ [0, 1], if f, g ≥ 0,then

Φ(tf + (1− t)g) ≤ tΦ(f) + (1− t)Φ(g). (2.5)

Proof. First suppose that f and g satisfy A ≥ f, g ≥ a for constants a,A > 0. We need to show that

α(t) := Φ(tf + (1− t)g)− tΦ(f)− (1− t)Φ(g) ≤ 0

for each t ∈ [0, 1]. By our hypotheses on f , α is a twice-differentiable function of t. We have α(0) = α(1) = 0.Therefore, if α(t) is positive for some t ∈ [0, 1], then α attains a positive maximum in [0, 1]. So, it suffices to

6

show that there can be no such maximum. For this, it is enough to show that α′′(t) ≥ 0. Using dominatedconvergence to differentiate under the integral sign, we compute

α′′(t) =d2

dt2Φ(tf + (1− t)g)

= q(q − 1)

∫Ω

(tf + (1− t)g)q−2(f − g)2 dµ− q(q − 1)

(∫Ω

(tf + (1− t)g) dµ

)q−2(∫Ω

f − g dµ)2

.

Thus, we need to show that(∫Ω

(tf + (1− t)g)q−2(f − g)2 dµ

)(∫Ω

tf + (1− t)g dµ)2−q

≥(∫

Ω

f − g dµ)2

. (2.6)

Fix t and set h = (tf + (1− t)g)2−q. By Holder’s inequality,(∫Ω

f − g dµ)2

=

(∫Ω

f − g√h

√h dµ

)2

≤(∫

Ω

(f − g)2

hdµ

)(∫Ω

h dµ

)=

(∫Ω

(tf + (1− t)g)q−2(f − g)2 dµ

)(∫Ω

(tf + (1− t)g)2−q dµ

).

But, since x 7→ x2−q is concave on [0,∞), we have∫Ω

(tf + (1− t)g)2−q dµ ≤(∫

Ω

tf + (1− t)g dµ)2−q

.

Plugging this into the last line proves (2.6).For the general non-negative f and g in L2, one can find sequences (fj) and (gj) which are bounded

above and bounded away from 0 and which converge to f and g, respectively, in the L2 norm. Take the limitin (2.5) for fj and gj to obtain the result for f and g.

Lemma 2.4. Let µ be a measure on a set Ω. Then if q ∈ (1, 2) and f ∈ L2(µn) is non-negative,∫Ωnfq dµn −

(∫Ωnf dµn

)q≤

n∑i=1

∫Ωn

[∫Ωi

fq dµi −(∫

Ωi

f dµi

)q]dµn

where ∫Ω

f dµi :=

∫Ω

f(x1, ..., xi−1, xi, xi+1, ..., xn) dµ(xi).

Proof. First observe that the formula concerns only the absolute value of f , so we may assume that f ≥ 0.Suppose first that n = 2, in which case the desired formula is∫

Ω

∫Ω

fq dµ1dµ2 −(∫

Ω×Ω

f dµ1dµ2

)q≤∫

Ω

∫Ω

[∫Ω

fq dµ1 −(∫

Ω

f dµ1

)q+

∫Ω

fq dµ2 −(∫

Ω

f dµ2

)q]dµ1dµ2. (2.7)

Note that since µ1 and µ2 are probability measures, integrating a second time with respect to the samemeasure has no effect. So, we can rearrange terms to obtain the equivalent inequality∫

Ω

(∫Ω

f dµ2

)qdµ1 −

(∫Ω

∫Ω

f dµ1dµ2

)q≤∫

Ω

∫Ω

fq dµ1dµ2 −∫

Ω

(∫Ω

f dµ1

)qdµ2. (2.8)

Define Φ as in Lemma 2.3. We apply the general form of Jensen’s inequality, with the L1(Ω)-valued randomvariable y 7→ f(·, y) and the convex function Φ on L1(Ω) to obtain

Φ

(∫Ω

f(·, y) dµ2(y)

)≤∫

Ω

Φ(f(·, y)) dµ2(y).

7

Expanding both sides of this inequality, we obtain (2.8). This proves the result in the n = 2 case. Thegeneral case follows by a simple induction argument.

Proof of Theorem 2.2. Let f ∈ Fn. Apply Lemma 2.4 to find

‖f‖22 − ‖f‖2p ≤∫

Ωn

n∑i=1

[∫Ω

f2 dµi −(∫

Ω

|f |p dµi)2/p

]dµn. (2.9)

By hypothesis, for fixed x1, ..., xi−1, xi+1, ..., xn ∈ Ω, each term in the sum on the right side of (2.9) isbounded above by

C(2− p)B [f(x1, ..., xi−1, ·, xi+1, ..., xn), f(x1, ..., xi−1, ·, xi+1, ..., xn)] .

Summing over all i and integrating, we get that the right side of (2.9) is bounded above by

C(2− p)∫

Ωn

n∑i=1

B [f(x1, ..., xi−1, ·, xi+1, ..., xn), f(x1, ..., xi−1, ·, xi+1, ..., xn)] dµn(x) = C(2− p)Bn(f, f).

Remark 2.5. The tensorial property proven in Lemma 2.4 also allows one to extend Beckner’s inequalityto infinite dimensional Gaussian measures. See [9] for a discussion of these measures.

Remark 2.6. Later, we shall prove an inequality for log-concave probability measures of the form

‖f‖2q − ‖f‖2p ≤ C(q − p)‖∇f‖2q, q ≥ 2, 1 ≤ p ≤ q, f ∈W 2,1(µ).

In the case q > 2, this inequality does not, to our knowledge, possess the tensorial property of this section.We must therefore prove this inequality in all dimensions, rather than just deducing the n-dimensional casefrom the one-dimensional case as we do for Beckner’s inequality below.

2.2 Two-point inequality

Let Xn = −1, 1n, with the uniform probability measure mn which assigns measure 1/2n to each point.In this subsection, we shall prove an inequality for X = X1 which is analogous to that in Theorem 2.1.This inequality will eventually enable us to deduce Theorem 2.1 in the one-dimensional case via a limitingargument based on the central limit theorem (Theorem 2.8).

Proposition 2.7. Define a bilinear form on X by

B(f, g) =1

4(f(1)− g(−1))2. (2.10)

Then for any function f on X,‖f‖22 − ‖f‖2p ≤ (2− p)B(f, f), (2.11)

where here the norms are with respect to m.

Proof. We begin with some reductions. Every function on X is of the form f(x) = ax + b for constantsa and b. If b = 0, then |f | is constant, so the left side of (2.11) vanishes and the result is obvious. So,we may assume that b 6= 0. Then since the desired inequality is invariant under rescaling, we may takef(x) = fa(x) = ax + 1 for some a. The measure m is symmetric about the origin, so it suffices to assumethat a ≥ 0. Move all of the terms in (2.11) to one side and treat the difference as a function of a:

φ(a) = (2− p)B(fa)− (‖fa‖22 − ‖fa‖2p).

We shall show that φ ≥ 0, whence the desired inequality. The proof is essentially a direct calculation. Wetreat two cases.

8

Case I: 0 ≤ a ≤ 1. Then

φ(a) = −(p− 1)a2 − 1 +

((1 + a)p + (1− a)p

2

)2/p

.

Observe that φ(0) = 0. We shall show that φ′(0) = 0 and that φ is convex on [0, 1], so has a unique minimumvalue of 0 at the origin. We have

φ′(a) = −2(p− 1)a+ ‖fa‖2p((1 + a)p−1 − (1− a)p−1

)(2.12)

and φ′(0) = 0. Moreover,

φ′′(a) = −2(p− 1) + 2−p2 ‖fa‖

2−2pp

((1 + a)p−1 − (1− a)p−1

)2+(p− 1)‖fa‖2−pp

((1 + a)p−2 + (1− a)p−2

)(2.13)

The second term here is always non-negative. By Jensen’s inequality,

‖fa‖2−pp ≥(

1 + a+ 1− a2

)2−p

= 1.

By convexity of the function x 7→ xp−2 for x > 0,

(1 + a)p−2 + (1− a)p−2 ≥ 2

(1 + a+ 1− a

2

)p−2

= 2.

Thus the third term in (2.13) satisfies

(p− 1)‖fa‖2−pp

((1 + a)p−2 + (1− a)p−2

)≥ 2(p− 1).

Therefore φ′′ ≥ 0 on [0, 1], so φ is convex on [0, 1], as desired.

Case II: a ≥ 1. Then

φ(a) = −(p− 1)a2 − 1 +

((1 + a)p + (1− a)p

2

)2/p

.

We know from the previous case that φ(1) ≥ 0, so to show that φ is non-negative it suffices to show thatφ′(a) ≥ 0 for a ≥ 1. We compute

φ′(a) = −2(p− 1)a+ ‖fa‖2−p2

((1 + a)p−1 + (a− 1)p−1

). (2.14)

For the first factor of the second term in (2.14), we have by Jensen’s inequality that

‖fa‖2−p2 ≥(

1 + a+ a− 1

2

)2−p

= a2−p. (2.15)

We claim that the second factor in (2.14) satisfies

(1 + a)p−1 + (a− 1)p−1 ≥ 2(p− 1)ap−1. (2.16)

If we can demonstrate this, then (2.14), (2.15), and (2.16) will imply

φ′(a) ≥ −2(p− 1)a+ a2−p · 2(p− 1)ap−1 = 0 (2.17)

which is what we need to show.To prove the claim, set

ψ(a) = (1 + a)p−1 + (a− 1)p−1 − 2(p− 1)ap−1.

9

We must show that ψ is non-negative on [1,∞). It suffices to show that ψ(1) ≥ 0 and that ψ is increasingon [1,∞). We have ψ(1) = 2p−1 − 2(p− 1) ≥ 0. Furthermore

ψ′(a) = (p− 1)((1 + a)p−2 + (a− 1)p−2 − 2(p− 1)ap−2

)≥ (p− 1)

((a+ 1)p−2 + (a− 1)p−2 − 2ap−2

).

The function x 7→ xp−2 is convex on (0,∞), so

(a+ 1)p−2 + (a− 1)p−2 ≥ 2

(a+ 1

2+a− 1

2

)p−2

= 2ap−2

and hence ψ′(a) ≥ 0 and ψ is increasing in a. It follows that ψ(a) ≥ ψ(1) ≥ 0 for a ≥ 1. As we remarkedabove, this proves (2.16), and completes the proof of (2.11).

2.3 First proof of Beckner’s inequality

Theorem 2.8 (Central Limit Theorem). Let µ be a probability measure on R, normalized so that∫R x dµ(x) =

0 and∫R x

2 dµ(x) = 1. Define a function φn on Rn by

φn(x1, ..., xn) =1√n

n∑j=1

xj .

Let µn be the product of n copies of µ on Rn, and let µn∗ be the law of φn, i.e. the probability measure onR such that

µn∗(a, b] = µnx ∈ Rn : a ≤ φn(x) ≤ b

for each interval (a, b]. Then as n→∞,

µn∗(a, b]→ γ(a, b]

for each interval (a, b], and ∫Rg dµn∗ →

∫Rg dγ

for any bounded continuous function g. In other words, µn∗ → γ in the weak-* topology on the set of finitemeasures on R.

A proof of the central limit theorem can be found in Ch. 3 of [13]. In probabilistic terms, the centrallimit theorem is the statement that the normalized means φn of a sequence of independent identicallydistributed random variables converge in distribution to a variable with a Gaussian distribution. In practicalapplications, the central limit theorem is used to estimate the probability that the mean of large sample liesin a given range, using known probabilities for the Gaussian measure. It is the fundamental tool behindmost elementary hypothesis tests and confidence intervals.

We shall apply the central limit with the measure m from Section 2.2. Note that we can and do viewm as a measure on R, which assigns zero measure to sets which do not contain 1 or −1. First we applyTheorem 2.2 in this setting. For B as in (2.3), we have

Bn(f, f) =

∫Xn

(n∑i=1

B [f(x1, ..., xi−1, ·, xi+1, ..., xn)]

)dµn

=1

4

∫Xn

n∑i=1

(f(x1, ..., xi−1, 1, xi+1, ..., xn)− f(x1, ..., xi−1,−1, xi+1, ..., xn))2dmn. (2.18)

By Proposition 2.7 and Theorem 2.2,

‖f‖2L2(mn) − ‖f‖2Lp(mn) ≤ C(2− p)Bn(f, f), 1 ≤ p ≤ 2 (2.19)

10

for each function f on Xn.Now suppose that f is a function on R. For each integer n, define a function fn on Xn by

fn(x1, ..., xn) = f

1√n

n∑j=1

xj

. (2.20)

In the notation of Theorem 2.8, fn = f φn, so the theorem suggests that fn will converge in some sense tof . We seek a simple expression for Bn(fn, fn). Fix an integer i = 1, ..., n, and define Xn

−i to be the productof the n−1 copies of X in Xn other than the ith. Write

∑y for the sum of the n−1 coordinates of y ∈ Xn

−i.Then the integral of the ith summand in (2.18) is∫

Xn(fn(x1, ..., xi−1, 1, xi+1, ..., xn)− fn(x1, ..., xi−1,−1, xi+1, ..., xn))2 dmn

=∑y∈Xn−i

1

2n−1

[f

(∑y + 1√n

)− f

(∑y − 1√n

)]2

. (2.21)

If the tuple y has k ones and n − 1 − k negatives ones, then∑y = 2k − n + 1. For each k = 0, ..., n − 1,

there are n− 1 choose k tuples y ∈ Xn−i satisfying this condition. Thus the integral in (2.21) equals

1

2n−1

n−1∑k=0

(n− 1

k

)[f

(2k − n√

n+

2√n

)− f

(2k − n√

n

)].

Observe that this is independent of i. We therefore have

Bn(fn, fn) =

n−1∑k=0

n

2n+1

(n− 1

k

)[f

(2k − n√

n+

2√n

)− f

(2k − n√

n

)]2

=

n−1∑k=0

1

2n−1

(n− 1

k

)f(

2k−n√n

+ 2√n

)− f

(2k−n√

n

)2/√n

2

. (2.22)

We are now ready to prove the one-dimensional version of Theorem 2.1.

Proof of Theorem 2.1. By Theorem 2.2 and the remarks following it, it will suffice to prove Theorem 2.1 inthe case n = 1. Furthermore, since C∞c (Rn) is dense in W 2,1(µ) (c.f. Appendix 6.1), it suffices to prove theresult for f ∈ C∞(R). Define fn as in (2.20). We shall deduce (2.1) as the limit as n→∞ of (2.11) for fn.

The functions f2 and |f |p are continuous with compact support, so by Theorem 2.8, as n→∞,

‖fn‖L2(mn) = ‖f‖L2(mn∗) → ‖f‖L2(γ), ‖fn‖Lp(mn) = ‖f‖Lp(mn∗) → ‖f‖Lp(γ). (2.23)

This proves convergence of the left side of (2.11).It remains to treat the right side. Let gn be the function on R which equalsf

(2k−n√

n+ 2√

n

)− f

(2k−n√

n

)2/√n

2

on [ 2k−n√n, 2(k+1)−n√

n] for each k = 0, ..., n− 1 and which equals zero elsewhere.

Let mn∗ be the law of φn, as in Theorem 2.8. By elementary probability theory, the measure mn∗ is abinomial distribution with parameters n− 1 and 1/2. Therefore

∫Rgn dm(n−1)∗ =

n−1∑k=0

1

2n−1

(n− 1

k

)f(

2k−n√n

+ 2√n

)− f

(2k−n√

n

)2/√n

2

= Bn(fn, fn).

11

On the other hand, gn is the square of a difference quotient of f on each interval [ 2k−n√n, 2(k+1)−n√

n]. Since

f ∈ C∞c (R), it is readily verified that gn → (f ′)2 uniformly as n → ∞. Thus, given ε > 0, we may chooseN so large that |gn − (f ′)2| < ε uniformly over R whenever n ≥ N . By the central limit theorem, we maychoose N ′ ≥ N such that ∣∣∣∣∫

Rn(f ′)2 dm(n−1)∗ −

∫R

(f ′)2 dγ

∣∣∣∣ < ε

whenever n ≥ N ′. Hence for each such n,∣∣∣∣Bn(fn, fn)−∫R

(f ′)2 dγ

∣∣∣∣ ≤ ∣∣∣∣∫Rgn − (f ′)2 dm(n−1)∗

∣∣∣∣+

∣∣∣∣∫Rn

(f ′)2dm(n−1)∗ −∫R

(f ′)2 dγ

∣∣∣∣ < 2ε

whence Bn(fn, fn)→∫R(f ′)2dγ. Thus, if we take the limit in (2.11) for fn, we get (2.1).

3 Extended Beckner Inequality via Semigroup Methods

In this chapter, we introduce the Ornstein-Uhlenbeck operator and its associated semigroup, and later theclassical heat semigroup, and use them to prove an extended version of Beckner’s inequality. The followingbit of notation will be useful here and in the remainder of the paper. Let q ≥ 1. If 1 ≤ p ≤ q, we say that aprobability measure µ on Rn satisfies inequality Bec(q, p) with constant C if for each f ∈W q,1(µ),

‖f‖2q − ‖f‖2p ≤ C(q − p)‖∇f‖2q. (3.1)

Notice that Bec(2, p) for 1 ≤ p ≤ 2 is Beckner’s original inequality. Our goal is to prove the following.

Theorem 3.1. The measure γn satisfies Bec(q, p) with constant 1 whenever q ≥ 2 and 1 ≤ p ≤ q. Further-more, equality holds if and only if f is constant, a.e.

3.1 The Ornstein-Uhlenbeck operator

In this section, we will introduce the Ornstein-Uhlenbeck operator and discuss the basic theory of semigroups.A more detailed discussion of the Ornstein Uhlenbeck operator can be found in [9]. A more detailed discussionof semigroups can be found in [8].

Definition 3.2. The Ornstein-Uhlenbeck operator is the densely defined unbounded operator A : L2(γn)→L2(γn) defined by

Af(x) = ∆f(x)− 〈x,∇f(x)〉 (3.2)

with domain consisting of those f ∈ L2(γn) such that this formula defines an L2 function. Here ∆ =∑nj=1 ∂

2j

denotes the Laplacian.

The Ornstein-Uhlenbeck operator has a number of interesting properties, as well as applications in partialdifferential equations and probability theory. For our purposes, the importance of this operator lies in thefollowing integration by parts formula, which will play an essential role in what follows.

Proposition 3.3. If Af exists and g ∈W 2,1(γn), then∫Rn〈∇f,∇g〉 dγn = −

∫RngAf dγn. (3.3)

Proof. By converting to an ordinary Lebesgue integral and integrating by parts, we obtain the elementaryformula ∫

R∂jf(x) dγn(x) =

∫Rxjf(x) dγn(x), f ∈ C∞c (Rn). (3.4)

12

By approximation, the same holds for f ∈ W 2,1(γn). We now apply this and the product rule to the term〈x, g(x)∇f(x)〉 in each coordinate to get∫

RngAf dγn =

∫Rn−〈x, g(x)∇f(x)〉+ g(x)∆f(x) dγn(x)

=

∫Rn−〈∇f(x),∇g(x)〉 − g(x)∆f(x) + g(x)∆f(x) dγn(x)

= −∫Rn〈∇f,∇g〉 dγn.

We are interested in solving the analogue of the heat equation for the operator A, namely

∂tu(t, x) = Au(t, x), u(0, x) = f(x), (3.5)

for sufficiently regular functions f ∈ L2(µ). In order to do so, we will need the following notion fromfunctional analysis.

Definition 3.4. A contraction semigroup is a family of operators Tt : t ≥ 0 on a Banach space X satisfying

1. T0 is the identity on X;

2. Tt Ts = Tt+s.

3. t 7→ Tt is strongly continuous in t. In other words, if tj → t then for each x ∈ X, Ttjx → Ttx in thenorm of X.

4. ‖Ttx‖ ≤ ‖x‖ for all x ∈ X.

Perhaps the simplest nontrivial example of a contraction semigroup is the family of maps on R definedby Tt(x) = e−tx.

Definition 3.5. The infinitesimal generator of Tt is the operator S defined by

S(x) = limt→0

Tt(x)− xt

,

with domain D(S) consisting of those x ∈ X such that this limit exists.

In our above example, the infinitesimal generator of x 7→ e−tx is x 7→ −x. In general, abstract theory(see, for example, Section 7.4 of [14]) implies that the infinitesimal generator of a contraction semigroupon a Banach space X always exist on a dense subspace of X. However, it is difficult in general to find itsprecise domain. For many purposes (including our own), one can settle for a semigroup whose generator isan extension of a given operator, i.e. one which agrees with the given operator on its domain, but may havea larger domain.

The relation between the infinitesimal generator and the semigroup is made clear by the following.

Lemma 3.6. If x ∈ D(S), then for each t ≥ 0,

d

dtTtx = TtSx = STtx. (3.6)

Proof. Since t 7→ Ttx is continuous, we have

d

dtTtx = lim

s→0

Ts+tx− Ttxs

= Tt

(lims→0

Tsx− xs

)= TtSx

andd

dtTtx = lim

s→0

TsTtx− Ttxs

= STtx.

13

3.2 The Ornstein-Uhlenbeck semigroup

By Lemma 3.6, if we can find a semigroup Tt whose infinitesimal generator is an extension of the OrnsteinUhlenbeck operator A, then u(t, x) = Ttf(x) will be a solution to (3.5). We could simply state the formula(called the Mehler Formula) for Tt and prove that it has the necessary properties, but for pedagogical reasonswe first make the following informal derivation.

We assume all functions involved are sufficiently regular that our calculations make sense. Taking theFourier transform, defined by

g(ξ) =

∫Rne−i〈x,ξ〉g(x) dx, g ∈ L1(Rn)

in (3.5) gives

∂tu(t, ξ) = −|ξ|2u(t, ξ) + i

n∑j=1

∂ξj (−iξj u(t, ξ)) = −|ξ|2u(t, ξ) +

n∑j=1

ξj∂ξj u(t, ξ) + nu(t, ξ),

with initial condition u(0, ξ) = f(ξ). Equivalently,

∂tu(t, ξ)−n∑j=1

ξj∂ξj u(t, ξ) + (|ξ|2 − n)u(t, ξ) = 0, u(0, ξ) = f(ξ).

For fixed ξ0 ∈ Rn, the characteristic ODE for this PDE are

t(s) = 1

ξ(s) = −ξz(s) = (n− |ξ|2)z(s)

t(0) = 0, ξ(0) = ξ0, z(0) = f(ξ0),

where z is the variable we substitute for u. Solving the first two equations and substituting into the third,we obtain

t = s

ξ(s) = e−sξ0

z(s) = (n− |ξ0|2e−2s)z(s)

z(0) = f(ξ0).

The last two equations yield the solution

z(s) = ese−12 |ξ0|

2(1−e−2s)f(ξ0).

Given (t, ξ) ∈ [0,∞) × Rn, we select s ∈ [0,∞) and ξ0 ∈ Rn such that (t, ξ) = (t(s), ξ(s)) = (s, e−sξ0), i.e.s = t, ξ0 = etξ. Then

u(t, ξ) = z(s) = ete−12 |ξ|

2e2t(1−e−2t)f(etξ).

The inverse Fourier transform u(t, x) of this function will be et times the convolution of the inverse Fourier

transform of f(etξ) and the inverse Fourier transform of exp(− 12 |ξ|

2e2t(1 − e−2t)). From the elementary

properties of the Fourier transform, the inverse Fourier transform of f(etξ) is (2π)n/2e−tf(e−tξ) and theinverse Fourier transform of exp(− 1

2 |ξ|2e2t(1− e−2t)) is

(2π)n/2

(et√

1− e−2t)nexp(−1

2|ξ|2e−2t(1− e−2t)−1).

Thus, upon taking the inverse Fourier transform and cancelling a factor of (2π)net, we get

u(t, x) =1

(et√

1− e−2t)n

∫Rn

exp(−1

2|z|2e−2t(1− e−2t)−1)f(e−tx− e−tz) dz.

14

Substitute z = et√

1− e−2ty and use symmetry of the Gaussian to find

u(t, x) =

∫Rnf(e−tx+

√1− e−2ty) dγn(y).

We are therefore lead to define

Definition 3.7. The Ornstein-Uhlenbeck Semigroup is the family of operators Tt defined for t ≥ 0 by

Ttf(x) =

∫Rnf(e−tx+

√1− e−2ty) dγn(y),

for all functions f on Rn for which this integral exists for a.e. x ∈ Rn.

We claim that Tt is a contraction semigroup on Lp, 1 ≤ p ≤ ∞, with infinitesimal generator A. Toprove our claim, we shall require the following elementary lemma.

Lemma 3.8 (Change of Variables Formula). If a2 + b2 = 1, then for any f ∈ L1(γn),∫Rnf(u) dγn(u) =

∫Rn

∫Rnf(ax+ by) dγn(x) dγn(y).

Proof. Since γn is a probability measure, we may insert a second integral to get∫Rf(u) dγn(u) =

∫Rn

∫Rf(u) dγn(u) dγn(v) =

1

(2π)n

∫R

∫Rf(u)e−|u|

2/2−|v|2/2dudv.

Make the change of variables u = ax + by, v = bx − ay. Since a2 + b2 = 1, the Jacobian matrix for thischange of variables has determinant one. So, by the change of variables formula from elementary calculus,our integral becomes

1

(2π)n

∫Rn

∫Rnf(ax+ by)e−|ax+by|2/2−|bx−ay|2/2dxdy =

1

(2π)n

∫Rn

∫Rnf(ax+ by)e−|x|

2/2−|y|2/2dxdy

=

∫Rn

∫Rnf(ax+ by) dγn(x) dγn(y).

From this lemma, we can deduce some basic properties of the Ornstein-Uhenbeck semigroup.

Proposition 3.9. Let s, t ≥ 0. Then:

1. Tt preserves integrals: if f ∈ L1(γn), then∫RnTtf dγ

n =

∫Rnf dγn.

2. ‖Ttf‖p ≤ ‖f‖p for all p ≥ 1; in particular, Tt is a bounded, equivalently continuous, map from Lp(γn)to itself.

3. Tt Ts = Tt+s.

4. If f ∈W 2,1(γn), then so is Ttf and for each k = 1, ..., n, ∂k(Ttf) = e−tTt(∂kf).

5. t 7→ Tt is a continuous function of t. That is, if tj → t, then for each f ∈ Lp(γn), Ttjf → Ttf in Lp.

6. If f ∈ L2(γn), then limt→∞ Ttf =∫Rn f dγ

n in Lp for any 1 ≤ p <∞.

Clearly, T0 is the identity operator, so items 2, 3, and 5 mean that Tt is a contraction semigroup onLp(γn). Item 1 implies that γn is the invariant measure for Tt.

15

Proof. 1. If we take a = e−t and b =√

1− e−2t, then Lemma 3.8 gives∫RnTtf dγ

n =

∫Rn

∫Rnf(e−tx+

√1− e−2ty) dγn(y) dγn(x) =

∫Rnf dγn. (3.7)

2. From item 1, together with Jensen’s inequality, we get for f ∈ L1(γn) that

‖Ttf‖pp =

∫Rn

∣∣∣∣∫Rnf(e−tx+

√1− e−2ty) dγn(y)

∣∣∣∣p dγn(x)

≤∫Rn

∫Rn|f(e−tx+

√1− e−2ty)|p dγn(y) dγn(x) = ‖f‖pp.

Since L1(γn) is dense in Lp(γn) for p ≥ 1, this proves item 2.

3. Observe that

Tt+sf(x) =

∫Rnf(e−t−sx+

√1− e−2t−2sy) dγn(y).

Apply Proposition 3.8 in the y variable, with

a = e−s√

1− e−2t

√1− e−2t−2s

, b =

√1− e−2s

√1− e−2t−2s

.

This gives us

Tt+sf(x) =

∫Rn

∫Rnf(e−te−sx+ e−s

√1− e−2tw +

√1− e−2sz) dγn(w) dγn(z) = TtTsf(x).

4. This can be obtained for f ∈ C∞(Rn) by differentiating under the integral sign, and follows for thegeneral f ∈W 2,1(γn) by the usual density argument.

5. Suppose tj → t. First consider f ∈ Cc(Rn). Then Ttjf → Ttf a.e., and it follows from dominatedconvergence that Ttjf → Ttf in Lp(γn). For the general f ∈ Lp(γn), one can find a sequence (fi) ∈Cc(Rn) converging to f in Lp(γn). Then

‖Ttjf − Ttf‖p ≤ ‖Ttjf − Ttjfi‖p + ‖Ttjfi − Ttfi‖p + ‖Ttfi − Ttf‖p.

Taking the limit, first in i then in j, proves item 5.

6. If f is bounded, the result is immediate from the dominated convergence theorem. In the general case,let ε > 0 and choose a bounded measurable g such that ‖f − g‖p < ε. Then

lim supt→∞

∥∥∥∥Ttf − ∫Rnf dγn

∥∥∥∥p

≤ lim supt→∞

(‖Ttf − Ttg‖p +

∥∥∥∥Ttg − ∫Rng dγn

∥∥∥∥p

+

∥∥∥∥Ttg − ∫Rng dγn

∥∥∥∥p

)< 2ε

which proves the result in general.

Finally, we can relate the semigroup Tt to the Ornstein-Uhlenbeck operator A.

Proposition 3.10. Let f ∈ D(A). Then t 7→ Ttf is a differentiable function of t and

d

dtTtf = ATtf = TtAf.

In particular, taking t = 0, the infinitesimal generator of Tt is an extension of A.

16

Proof. We have

d

dtTt(f)(x) = −e−t

∫Rn〈x,∇f(e−tx+

√1− e−2ty)〉 dγn(y)

+e−2t

√1− e−2t

∫Rn〈y,∇f(e−tx+

√1− e−2ty)〉 dγn(y). (3.8)

By item 4 of Proposition 3.9, the first term on the right side of (3.8) equals

−〈x, e−tTt∇f(x)〉 = −〈x,∇Ttf(x)〉,

where the action of Tt on vector valued functions is componentwise. By (3.4) applied in each coordinate ofy, the second term in (3.8) equals

e−2t

∫Rn

∆f(e−tx+√

1− e−2ty) dγ(y) = e−2tTt∆f(x) = ∆Ttf(x).

Substituting these two relations into (3.8) proves the first desired equality, and taking t = 0 shows that theinfinitesimal generator of Tt is an extension of A (its domain may be larger than the domain of A). Thesecond equality follows from Lemma 3.6.

3.3 Proof of extended Beckner inequality

The integration by parts formula 3.3 makes the operator A and its associated semigroup Tt a natural toolfor the proof of Theorem 3.1. Here we give the proof, and also prove some sharpness results.

Proof of Theorem 3.1. By density, we may take f ∈ C2(Rn) with f bounded. Then |f | ∈ W q,1(γn) with|∇|f || = |∇f | a.e. (see Appendix 6.1). So, it suffices to suppose that f is non-negative. Furthermore, byreplacing f with f + ε and letting ε→ 0, we may take f ≥ c > 0 for some constant c. Let

φ(t) =

(∫Rn

[Ttfp]q/p dγn

)2/q

.

Since T0 is the identity and limt→∞ Ttf =∫Rn f dγ

n, we have that the left side of Bec(q, p) is given by

‖f‖2q − ‖f‖2p =

∫ ∞0

φ′(t) dt.

We shall estimate φ′. To simplify notation in what follows, put

α(t) =

(∫Rn

[Ttfp]q/p dγn

)2/q−1

. (3.9)

Using the relation ∂tTtf = ATtf , we compute

φ′(t) =2

pα(t)

∫Rn

[Ttfp]q/p−1ATtf

p dγn.

By the integration by parts formula for A, this equals

2

pα(t)

∫Rn〈∇[Ttf

p]q/p−1,∇ (Ttfp)〉 dγn =

2

p

(q

p− 1

)α(t)

∫Rn

[Ttfp]q/p−2|∇Ttfp|2 dγn. (3.10)

We have∇Ttfp = e−tTt(∇(fp)) = e−tpTt(f

p−1∇f). (3.11)

Therefore, (3.10) equals

2e−2t (q − p)α(t)

∫Rn

[Ttfp]q/p−2|Tt(fp−1∇f)|2 dγn. (3.12)

17

By Holder’s inequality applied inside the definition of Tt,

|Tt(fp−1∇f)|2 ≤(Tt(f

p−1|∇f |))2 ≤ (Ttf

p)2−2/p

(Tt|∇f |p)2/p. (3.13)

Plugging this into (3.12) yields

φ′(t) ≤ 2e−2t (q − p)α(t)

∫Rn

(Ttfp)q/p−2/p

(Tt|∇f |p)2/pdγn. (3.14)

The q = 2 case can be handled by trivial modifications of what follows, so we henceforth assume q > 2.Apply Holder’s inequality a second time, this time with the exponents q/(q − 2) and q/2, to get∫

Rn(Ttf

p)q/p−2/p

(Tt|∇f |p)2/pdγn

≤(∫

Rn(Ttf

p)q/p

dγn)1−2/q (∫

Rn(Tt|∇f |p)q/p dγn

)2/q

.

The last term here is precisely α(t)−1, so upon plugging this into (3.14), we obtain

φ′(t) ≤ 2e−2t (q − p)(∫

Rn(Tt|∇f |p)q/p dγn

)2/q

.

Since 1 ≤ p ≤ q, we have

(Tt|∇f |p)q/p ≤ Tt|∇f |q (3.15)

So, since Tt preserves integrals,

φ′(t) ≤ 2e−2t (q − p)(∫

Rn|∇f |q dγn

)2/q

. (3.16)

Integrating this from 0 to ∞ proves Bec(q, p).If equality holds in Bec(q, p), then it must hold for almost every t in (3.16). Therefore it must hold for

almost every t in (3.13). By the conditions for equality in Holder’s inequality, this means that

fp = c|∇f |p

for some constant c. On the other hand, the condition for equality in Jensen’s inequality, together with(3.15), imply that |∇f |p is constant. Therefore if f satisfies the hypotheses we imposed at the beginning ofthe proof, then equality in Bec(q, p) implies that f is constant. The L2 limit of constant functions is constanta.e., so upon passing to the limit we get the result for the general non-negative f . The result for the generalf ∈ W 2,1(µ) is obtained by replacing f with |f | and applying the positive case, together with Proposition6.3 of Appendix 6.1. So, by the non-negative case, equality for f implies that |f | is constant a.e. But, sincef ∈W 2,1(µ), this means that f itself is constant a.e.

Remark 3.11. If we apply Holder’s inequality in each coordinate of ∇f separately after obtaining (3.12),essentially the same proof shows that γn satisfies Bec(q, p) with the right side replaced by

(q − p)n∑k=1

‖∂kf‖2q.

This alternative inequality does not appear to be either weaker or stronger in general than the originalinequality Bec(q, p).

The condition q ≥ 2 in Theorem 3.1 is essential. To see this, we need a lemma.

Lemma 3.12. If µ satisfies inequality Bec(q, p) for any 1 ≤ p < q, and any constant C, then µ satisfies

‖f‖22 −(∫

Rnf dµ

)2

≤ C‖∇f‖2q. (3.17)

for all f ∈W q,1(µ).

18

Proof. It suffices to prove the implication for bounded functions f ∈ W q,1(µ). Replace f with 1 + εf inBec(q, p), and divide by ε2:

‖1 + εf‖2q − ‖1 + εf‖2pε2

≤ (q − p)C‖∇f‖2q. (3.18)

Apply L’hopital’s rule twice to get that as ε→ 0 the left side tends to

1

2

d2

dε2(‖1 + εf‖2q − ‖1 + εf‖2p

)|ε=0.

If ε is so small that 1 + εf > 0, we have

d2

dε2‖1 + εf‖2q = 2(q − 1)‖1 + εf‖2−qq

∫Rnf2(1 + εf)q−2 dµ+ 2(2− q)‖1 + εf‖2−2q

q

(∫Rnf(1 + εf)q−1 dµ

)2

.

Replacing q with p, then evaluating both expressions at ε = 0, we see that the limit of the left side of (3.18)is

(q − 1)

∫Rnf2 dµ+ (2− q)

(∫Rnf dµ

)2

− (p− 1)

∫Rnf2dµ− (2− p)

(∫Rnf dµ

)2

= (q − p)‖f‖22 − (q − p)(∫

Rnf dµ

)2

.

Dividing by q − p then proves (3.17).

Proposition 3.13. The standard Gaussian measure γn does not satisfy inequality Bec(q, p) for any 1 ≤ p <q < 2 and any constant C.

Proof. By the Lemma, it will suffice to show that γn does not satisfies inequality (3.17). Take ft(x1, ..., xn) =etx1 , for t > 0. One has the formula∫

Rneax1 dγn(x) = ea

2/2, a ∈ R.

From this, one sees that (3.17) for ft with constant C is equivalent to

e2t2 − et2

≤ t2Ceqt2

⇔ e(2−q)t2 − e(1−q)t2

t2≤ C.

But, the left side of this last inequality tends to∞ as t→∞. Thus γn cannot satisfy (3.17) for any constantC, so by Proposition 3.12 it cannot satisfy Bec(q, p) for any q < 2, any p < q, and any constant C.

By a similar argument, we can also get that our result in Theorem 3.1 is sharp.

Proposition 3.14. Let q ≥ 2 and 1 ≤ p ≤ 2. The standard Gaussian measure γn does not satisfy inequalityBec(q, p) for constant C < 1.

Proof. As in the proof of Proposition 3.13, set ft(x1, ..., xn) = etx1 for t > 0. Then Bec(q, p) with constantC for ft is equivalent to

eqt2

− ep2

≤ t2C(q − p)eqt2

⇔ 1− e(p−q)t2

t2≤ C(q − p).

As t→ 0, the left side of this last inequality tends to q − p, so it cannot hold with C < 1.

19

3.4 Classical Heat semigroup

Inequality Bec(q, p) for the Gaussian measure can also be proven by means of the classical heat semigroup,rather than the Ornstein-Uhlenbeck semigroup. This is the method used by E. Hsu and the author in [17].The proof itself is slightly longer, but the classical heat semigroup is better known than the Ornstein-Uhlenbeck semigroup, and less work is required to establish its basic properties. We give this alternativeproof here.

Definition 3.15. The classical heat semigroup Ps : s > 0 defined by

Psf(x) =1

(2πs)n/2

∫Rnf(y)e−|x−y|

2/2sdy (3.19)

for bounded continuous functions f on Rn.

We shall require the following elementary properties of Ps.

Proposition 3.16. Suppose f : Rn → R is bounded and continuous. Then

1. Psf → f as s→ 0.

2. P1f(0) =∫Rn f dγ

n.

3. For each s, t ≥ 0, Ps Pt = Ps+t.

4. Psf solves the heat equation: ∂sPsf = 12∆Psf .

5. If ∇f is bounded and continuous, then ∇Psf = Ps∇f , where the action of Ps on ∇f is componentwise.

Note that items 1 and 3 imply that Ps is a semigroup. Item 4 implies that its infinitesimal generator isan extension of the half-Laplacian 1

2∆. The heat semigroup is not, however, a contraction semigroup. Item2 provides the motivation for using the heat semigroup in the context of inequalities for γn.

Proof. 1. Substitute y = x−√sz in (3.19) to obtain the alternative formula

Psf(x) =1

(2π)n/2

∫Rnf(x−

√sz)e−|z|

2/2dz =

∫Rnf(x−

√sz) dγn(z). (3.20)

Since f is bounded, we can use dominated convergence to find that this tends to∫Rn f(x) dγn(z) = f(x)

as s→ 0.

2. This is immediate from (3.19) and (1.1).

3. Using the alternative formula (3.20) for Ps, we have

PsPtf(x) =

∫Rn

∫Rnf(x−

√sy −

√tz) dγn(z) dγn(y).

By Lemma 3.8 with a =√s√s+t

, b =√t√s+t

, this equals∫Rnf(x−

√s+ tu) dγn(u) = Ps+tf(x).

4. Let

ρ(x, y, s) =1

(2πs)n/2e−|x−y|

2/2s

be the heat kernel. For j = 1, ..., n, we have

∂2j ρ(x, y, s) =

((xj − yj)2

s2− 1

s

)ρ(x, y, s)

20

and so

∆ρ(x, y, s) =

(|x− y|2

s2− n

s

)ρ(x, y, s) = 2∂sρ(x, y, s).

Differentiating under the integral sign is justified since ∆ρ ∈ L1(Rn) and f is bounded, so we obtain

∆Psf(x) =

∫Rnf(y)∆ρ(x, y, s) dy = 2

∫Rnf(y)∂sρ(x, y, s) dy = 2∂sf(y).

5. This follows from differentiation under the integral sign in (3.20).

We now employ these elementary properties of the classical heat semigroup Ps to give an alternativeproof of inequality Bec(q, p).

Proof of Theorem 3.1. By a standard approximation argument, it is enough show the inequality (3.1) for asmooth function f such that 0 < c ≤ f ≤ C and ∇f is bounded. For 0 ≤ s ≤ 1, consider the function

φs(x) =[Ps (P1−sf

p)q/p

(x)]2/q

. (3.21)

The idea of considering such a function in the context of functional inequalities can be traced back toNeveu [27]. We can write the left side of (3.1) as

‖f‖2q − ‖f‖2p = φ1(0)− φ0(0) =

∫ 1

0

∂sφs(0) ds.

The technical part of our proof is the computation of the derivative of 3.21 with respect to s.From the definition (3.21) of φs we have

∂sφs = ∂s

[Psg

q/ps

]2/q=

2

qas∂s(Psgs)

q/p,

where, to simplify the notation here and later, we have introduced the functions

gs = P1−sfp and as =

(Psg

q/ps

)2/q−1

.

We compute

∂sφs =2

qas(∂sPs)g

q/ps +

2

pasPs

(gq/p−1s ∂sgs

). (3.22)

Using the relation ∂sPs = (1/2)Ps∆, we may rewrite the first term on the right side as (1/q)asPs∆(gq/ps

),

which equals1

p

(q

p− 1

)asPs

(gq/p−2s |∇gs|2

)+

1

pasPs

(gq/p−1s ∆gs

)(3.23)

by the identity

∆(hq/p

)=q

p

(q

p− 1

)hq/p−2|∇h|2 +

q

phq/p−1∆h

applied with h = gs. From ∂sP1−s = −(1/2)∆P1−s we have ∂sgs = −(1/2)∆gs, so the second term in thesum (3.23) exactly cancels the second term in (3.22). In the remaining term, we use the fact that P1−scommutes with ∇ and write ∇gs = pP1−s(f

p−1∇f), which gives

∂sφs = (q − p)asPs(gq/p−2s |P1−s(f

p−1∇f)|2). (3.24)

Note that P1−s is an integral with respect to a (probability) measure, so we can use Holder’s inequality withthe exponents p/(p− 1) and p to get

|P1−s(fp−1∇f)| ≤ P1−s(f

p−1|∇f |) ≤ (P1−sfp)

(p−1)/p(P1−s|∇f |p)1/p

.

21

Thus, by (3.24),

∂sφs ≤ (q − p)asPs(gq/p−2/ps (P1−s|∇f |p)2/p

). (3.25)

The case q = 2 is covered by trivial modifications to what follows, so in the remainder of the proof we assumeq > 2. Holder’s inequality with the exponents q/(q − 2) and q/2 yields

Ps

(gq/p−2/ps (P1−s|∇f |p)2/p

)≤(Psg

q/ps

)1−2/q (Ps (P1−s|∇f |p)q/p

)2/q

.

The first factor on the right side is exactly a−1s , which cancels the factor as in (3.25). We now have

∂sφs ≤ (q − p)(Ps (P1−s|∇f |p)q/p

)2/q

.

From 1 ≤ p ≤ q we have P1−s|∇f |p ≤ (P1−s|∇f |q)p/q. This together with the semigroup property PsP1−s =P1 gives (

Ps (P1−s|∇f |p)q/p)2/q

≤ (PsP1−s|∇f |q)2/q=

(∫Rn|∇f |q dγ

)2/q

.

The last equality holds at x = 0. It follows that

∂sφs ≤ (q − p)(∫

Rn|∇f |q dγ

)2/q

.

Integrating from 0 to 1 yields (3.1).

4 Beckner Inequality for Log-Concave Probability Measures

We now turn our attention to a general log-concave probability measure µ on Rn.

Definition 4.1. A probability measure µ on Rn is called log-concave with concavity b > 0 if

µ = e−v(x)dx,

where v ∈ C2(Rn) and the matrix D2v of second order partial derivatives of v satisfies 〈D2v(x)ξ, ξ〉 ≥ b|ξ|2for each x, ξ ∈ Rn.

The most important log-concave probability measure is the standard Gaussian measure γn, which corre-sponds to the case where v(x) = |x|2/2 + log(2π)n/2. Log-concave probability measures satisfy many of thesame properties as Gaussian measures do, so it is natural to ask to what extent the inequalities we study forthe Gaussian measure can be generalized to this setting. The goal of this chapter is to prove the following.

Theorem 4.2. Let µ be a log-concave probability measure on Rn with concavity b > 0. Then µ satisfiesinequality Bec(q, p) with constant 1/b for all q ≥ 2 and all 1 ≤ p ≤ q. Furthermore, equality holds if andonly if f is constant a.e.

To prove Theorem 4.2, we proceed as in Chapter 3. We first define analogues of the Ornstein-Uhlenbeckoperator and the Ornstein-Uhlenbeck semigroup, then use these operators to prove the result via an argumentsimilar to that in Section 3.3.

4.1 Generalization of the Ornstein-Uhlenbeck operator and semigroup

Let µ = e−v(x)dx be a log-concave probability measure on Rn with concavity b > 0. Define an operator Aon L2(µ) by

A(f) := ∆f − 〈∇f,∇v〉, (4.1)

with domain D(A) consisting of all functions f such that ∆f − 〈∇f,∇v〉 defines a function in L2(µ).This operator A is a generalization of the Ornstein-Uhlenbeck operator, as suggested by the following

generalization of Proposition 3.3.

22

Lemma 4.3. If g ∈W 2,1(µ) and f ∈ D(A), then∫RngA(f) dµ = −

∫Rn〈∇f,∇g〉 dµ. (4.2)

Proof. By converting to an integral with respect to Lebesgue measure and integrating by parts, we obtain∫Rnf∂kvdµ =

∫Rn∂kfdµ, f ∈ C∞c (Rn), k = 1, ..., n. (4.3)

By approximation, this also holds for f ∈W 2,1(µ). Using (4.3), we then calculate∫RngA(f) dµ =

n∑k=1

∫Rng∂2kf − g∂kv∂kf dµ =

n∑k=1

∫Rng∂2kf − ∂kg∂kf − g∂2

kf dµ = −∫Rn〈∇f,∇g〉 dµ.

In analogy with Chapter 3, we now seek a contraction semigroup whose infinitesimal generator is anextension of the operator A. To formalize some of the basic properties we need this semigroup to possess,we make the following definition.

Definition 4.4. Suppose Tt is a contraction semigroup on a Banach space X of functions on a probabilityspace Ω. Then Tt is said to be a Markov semigroup if Tt1 = 1 (where 1 denotes the constant functionω 7→ 1); and Tt preserves positivity: if f ≥ 0 a.e., then Ttf ≥ 0 a.e.

From the Mehler formula, it is clear that the Ornstein-Uhlenbeck semigroup of Chapter 3 is a Markovsemigroup.

In Appendix 6.2, it is shown, using abstract functional analytic methods, that there exists a Markovsemigrup Tt, consisting of symmetric operators, whose infinitesimal generator is an extension of A. Unlikein Chapter 3, we do not have an explicit formula for Tt. Nevertheless, it turns out that this semigroupsatisfies many of the same properties as the Ornstein-Uhlenbeck semigroup.

Proposition 4.5. Let Tt be the symmetric Markov semigroup with generator a self-adjoint extension ofA. Then

1. Each Tt preserves integrals: if f ∈ L1(µ), then∫RnTtf dµ =

∫Rnf dµ.

2. Suppose c ≤ f ≤ C for some constants c and C. Then for each x ∈ Rn, c ≤ Ttf(x) ≤ C.

3. Ttf(x) is given by integration against a Borel probability measure νt,x for a.e. x ∈ Rn.

4. Each Tt defines a norm-decreasing operator Lp(µ)→ Lp(µ), for each 1 ≤ p ≤ ∞.

5. If f ∈ C2(Rn) ∩ L2(µ), then so is Ttf .

Proof. 1. Apply (4.2) to get

d

dt

∫RnTtf dµ =

∫RnA(Ttf) dµ =

∫Rn〈∇Ttf,∇1〉 dµ = 0.

Thus the integral of Ttf is independent of t, and in particular it equals the integral of T0f = f .

2. Since Tt fixes the constant functions and preserves positivity, the fact that C − f ≥ 0 implies thatC − Ttf ≥ 0, and similarly for c.

23

3. By item 2, f 7→ Ttf(x) defines a positive linear functional on C0(Rn). So, the Riesz representationtheorem implies that Ttf(x) is given by integrating against a Borel measure νt,x for each f ∈ C0(Rn).For a non-negative f ∈ L2(µ), one can find a sequence (fj) ∈ C0(Rn) which increases to f a.e., so bydominated convergence Ttf(x) is also given by integration against νt,x. For the general f ∈ L2(µ), theresult follows by considering positive and negative parts separately. Finally, from Tt1 = 1, we get thatνt,x is a probability measure.

4. If f is bounded, then |f | and |f |p are each in L2(µ), and by item 3, we can apply Jensen’s inequalityto get

|Ttf |p ≤ Tt|f |p.Therefore item 1 gives ∫

Rn|Ttf |p dµ ≤

∫RnTt|f |p dµ =

∫Rn|f |p dµ.

The set of bounded f is dense in Lp(µ) for each 1 ≤ p ≤ ∞, so we get a unique extension of Tt to eachLp.

5. We have that u(t, x) := Ttf(x) solves the differential equation

∂tu = ∆u− 〈∇v,∇u〉 (4.4)

with initial condition u(0, x) = f(x). The coefficient vector ∇v in this PDE is C1, so by standardregularity results for parabolic equations (see, for example, Ch. 7 of [14]) applied on each boundedsubset of Rn, it follows that Ttf lies in C2(Rn) as well.

4.2 Commutation with the gradient

The only missing ingredient in our proof of Bec(q, p) is an inequality relating |∇Ttf |2 and Tt|∇f |2. To obtainsuch an estimate, we first make the following definition, which we take from [22].

Definition 4.6. We define the carre du champ of A to be the unbounded bilinear form Γ : L2(µ)×L2(µ)→L2(µ) given by

2Γ(f, g) := A(fg)− fA(g)− gA(f).

We define the curvature operator of A to be the unbounded bilinear form Γ2 : L2(µ)×L2(µ)→ L2(µ) givenby

2Γ2(f, g) := AΓ(f, g)− Γ(f,Ag)− Γ(g,Af). (4.5)

The domains of Γ and Γ2 are the subsets of L2(µ)× L2(µ) on which the above formulas produce functionsin L2(µ). We put Γ(f) = Γ(f, f) and similarly for Γ2(f).

Proposition 4.7. The carre du champ of A is given by

Γ(f, g) = 〈∇f,∇g〉.

Proof. Due to bilinearity, it suffices to prove the result in the case f = g, i.e. we must show

Γ(f) = |∇f |2.

We have

A(f2) = ∆(f2)−∇(f2) · ∇v= 2|∇f |2 + 2f∆f − 2f∇f · ∇v

and2fA(f) = 2f∆f − 2f∇f · ∇v.

Subtracting, we getΓ(f) = |∇f |2

which is the desired formula.

24

Proposition 4.8. The curvature operator of A satisfies

Γ2(f) = |D2f |2 + 〈(D2v)∇f,∇f〉.

Proof. In the case f = g formula (4.5) is

2Γ2(f) = AΓ(f)− 2Γ(f,Af) (4.6)

By Proposition 4.7 the first term in (4.6) is given by

AΓ(f) = ∆|∇f |2 −∇v · ∇|∇f |2.

Expanding out the second term above, we get

∇v · ∇|∇f |2 = 2

n∑j,k=1

∂kf∂jkf∂jv.

For the second term in (4.6), we have

Γ(f,Af) = ∇f · ∇(∆f −∇v · ∇f).

We compute

∇f · ∇(∇v · ∇f) =

n∑j,k=1

∂kf∂jkv∂jf +

n∑j,k=1

∂kf∂jkf∂jv

= 〈(D2v)∇f,∇f〉+

n∑j,k=1

∂kf∂jkf∂jv.

Plugging our calculations into (4.6 and cancelling the terms 2∑nj,k=1 ∂kf∂jkf∂jv, we find

2Γ2(f) = ∆|∇f |2 − 2∇f · ∇∆f + 2〈(D2v)∇f,∇f〉. (4.7)

We have

∆|∇f |2 = 2

n∑j,k=1

∂kf∂kjjf + 2

n∑j,k=1

(∂jkf)2 = 2

n∑j,k=1

∂kf∂kjjf + 2|D2f |2

and

∇f · ∇∆f =

n∑j,k=1

∂kf∂kjjf.

Plugging these two expressions into (4.7), cancelling a pair of terms, and dividing by 2, we get

Γ2(f) = |D2f |2 + 〈(D2v)∇f,∇f〉.

From the preceding proposition and our hypothesis on D2v, one has the inequality

Γ2(f) ≥ 〈(D2v)∇f,∇f〉 ≥ b|∇f |2. (4.8)

In the terminology of Markov semigroups, the operator A has positive curvature (see [22] for further discus-sion). From this bound, we can deduce our desired inequality for the gradient of Ttf . Our proof follows thatin [22].

Proposition 4.9. For function f ∈ C∞(Rn) ∩ L2(µ) and t ≥ 0,

|∇Ttf |2 ≤ e−2btTt|∇f |2.

25

Proof. Fix t > 0 and for t ≥ s ≥ 0 define

φ(s) = e−2bsTs|∇Tt−sf |2.

Using the relation ∂sTs = TsA, we compute

φ′(s) = −2be−2bsTs|∇Tt−sf |2 + e−2bsTs(A|∇Tt−sf |2 + ∂s|∇Tt−sf |2

). (4.9)

By Proposition 4.7 one has

∂s|∇Tt−sf |2 = −2〈∇Tt−sf,∇ATt−sf〉 = −2Γ(Tt−sf,ATt−sf〉,

and so from (4.9) and the definition of Γ2,

φ′(s) = −2e−2bsTs [−bΓ(Tt−sf) + Γ2(Tt−sf)) .

By (4.8), the argument of Ts is non-negative, so since Ts preserves positivity, φ′ is non-negative. Thus φ isincreasing. In particular,

|∇Ttf |2 = φ(0) ≤ φ(t) = e−2btTt|∇f |2

which proves the result.

Corollary 4.10. If f ∈ L1(µ), then

limt→∞

Ttf =

∫Rnf dµ

in L1.

Proof. If f ∈ C1c (Rn), we infer from Proposition 4.9 that |∇Ttf |2 → 0 uniformly as t → ∞, so Ttf must

converge to a constant function. Since Tt preserves integrals, this constant must be∫Rn f dµ. For the general

f ∈ L1(µ), given ε > 0 we can find g ∈ C1c (Rn) with ‖f − g‖1 < ε and

∫Rn g dµ =

∫Rn f dµ. Then

lim supt→∞

∥∥∥∥Ttf − ∫Rnf dµ

∥∥∥∥1

≤ lim supt→∞

(‖Ttf − Ttg‖1 +

∥∥∥∥Ttg − ∫Rng dµ

∥∥∥∥1

)< ε,

which proves the result for f .

4.3 Proof of Beckner’s inequality for log-concave probability measures

We are now ready to prove Theorem 4.2. The proof is essentially the same as that in Section 3.3.

Proof of Theorem 4.2. Define A as in (4.1) and let Tt be the semigroup of Proposition 4.5. By a standardapproximation argument, we may take f ∈ C2(Rn) with f ≥ a > 0 for some constant a. Then by Proposition4.5, Ttf has the same properties for each t ≥ 0. Set

φ(t) =

(∫Rn

[Ttfp]q/p dµ

)2/q

.

Since T0 is the identity and limt→∞ Ttf =∫Rn f dµ, we have that the left side of Bec(q, p) is given by

‖f‖2q − ‖f‖2p =

∫ ∞0

φ′(t) dt.

We shall estimate φ′. To simplify notation in what follows, put

α(t) =

(∫Rn

[Ttfp]q/p dµ

)2/q−1

. (4.10)

26

Using the relation ∂tTt = ATt, we compute

φ′(t) =2

pα(t)

∫Rn

[Ttfp]q/p−1ATtf

p dµ.

By the integration by parts formula for A, this equals

2

pα(t)

∫Rn〈∇[Ttf

p]q/p−1,∇ (Ttfp)〉dµ =

2

p

(q

p− 1

)α(t)

∫Rn

[Ttfp]q/p−2|∇Ttfp|2 dµ. (4.11)

By Proposition 4.9 applied with f replaced by fp,

|∇Ttfp|2 ≤ e−2bt[Tt|∇(fp)|]2 = e−2btp2[Tt(fp−1|∇f |)]2.

Therefore,

φ′(t) ≤ 2e−2bt (q − p)α(t)

∫Rn

[Ttfp]q/p−2[Tt(f

p−1|∇f |)]2 dµ. (4.12)

Now apply Holder’s inequality to get[Tt(f

p−1|∇f |)]2 ≤ [Ttf

p]2−2/p (Tt|∇f |p)2/p.

Thus,

φ′(t) ≤ 2e−2bt (q − p)α(t)

∫Rn

[Ttfp]q/p−2/p (Tt|∇f |p)2/p

dµ. (4.13)

The case q = 2 is covered by trivial modifications to what follows, so in the remainder of the proof we assumeq > 2. Apply Holder’s inequality a second time, this time with the exponents q/(q − 2) and q/2, to get∫

Rn[Ttf

p]q/p−2/p (Tt|∇f |p)2/pdµ

≤(∫

Rn[Ttf

p]q/p dµ

)1−2/q (∫Rn

(Tt|∇f |p)q/p dµ)2/q

.

The first factor here is precisely α(t)−1, so upon plugging this into (4.13), we obtain

φ′(t) ≤ 2e−2bt (q − p)(∫

Rn(Tt|∇f |p)q/p dµ

)2/q

.

Since 1 ≤ p ≤ q,(Tt|∇f |p)q/p ≤ Tt|∇f |q.

So, since Tt preserves integrals, we get

φ′(t) ≤ 2e−2bt (q − p)(∫

Rn|∇f |q dµ

)2/q

.

Integrating this from 0 to ∞ proves Bec(q, p) with constant 1/b.The condition for equality follows just as in the proof of Theorem 3.1.

5 Other Inequalities for Log-Concave Probability Measures

In the remainder of the paper, we shall study several inequalities for log-concave probability measures whichare related to inequality Bec(q, p).

27

5.1 Generalized logarithmic Sobolev inequality

Let q ≥ 1. We say that µ satisfies inequality LSI(q) with constant C if whenever f ∈W q,1(µ),

2

q‖f‖2−qq

∫Rn|f |q log |f | dµ− 2

qlog(‖f‖q)‖f‖2q ≤ C‖∇f‖2q. (5.1)

Note that LSI(2) is the ordinary logarithmic Sobolev inequality (1.3), with a constant C on the right. Weshall explore the relationship between inequality LSI(q) and the inequality Bec(q, p) from Chapters 3 and 4.

First we have an implication relation within inequality LSI(q):

Proposition 5.1. If µ satisfies LSI(q) with constant C, then for each r > q, µ satisfies LSI(r) with constantC.

Proof. If f ∈ W r,1(µ), then |f | ∈ W r,1(µ) with |∇|f || = |∇f | a.e., so it suffices to consider f ≥ 0. ApplyLSI(q) to the function fr/q:

2r

q2‖f‖2r/q−rr

∫Rnfr log f dµ− 2r

q2log(‖f‖r)‖f‖2r/qr ≤ C r

2

q2

(∫Rnfr−q|∇f |q dµ

)2/q

. (5.2)

Apply Holder’s inequality on the right side, with exponents r/q and r/(r − q), to get(∫Rnfr−q|∇f |q dµ

)2/q

≤(‖f‖r−qr ‖∇f‖q/rr

)2/q

= ‖f‖2r/q−2r ‖∇f‖2r.

Plugging this into (5.2) and dividing by (r/q)2‖f‖2r/q−2r on both sides yields LSI(r) with constant C.

Proposition 5.2. If µ satisfies Bec(q, p) with some constant Cp for each p ∈ [1, q), then µ also satisfiesLSI(q) with constant C := lim supp→q Cp.

Proof. By the usual approximation arguments, it suffices to prove the inequality for f ∈ C∞(Rn)∩W 2,1(µ)with f ≥ c > 0 for some constant c. Divide both sides of Bec(q, p) by q − p to get

‖f‖2q − ‖f‖2pq − p

≤ Cp‖∇f‖2q. (5.3)

Our hypotheses on f imply that p 7→ ‖f‖2p is a differentiable function of p, so as p → q−, the left side of(5.3) tends to

d

dp‖f‖2p|p=q =

2

q‖f‖2−qq

∫Rnfq log(f) dµ− 2

qlog(‖f‖q)‖f‖2q,

which is precisely the left side of LSI(q). The right side of (5.3) is bounded above by C‖∇f‖2q as p → q−,which proves the desired inequality.

By Theorem 4.2, we immediately get

Corollary 5.3. For q ≥ 2, the measure µ satisfies inequality inequality LSI(q) with constant 1/b.

There is a partial converse to Proposition 5.2. In order to prove it, we will need to consider how thequotient

‖f‖2q − ‖f‖2pq − p

changes as q and p vary. It turns out that a slight variant of this quantity is better behaved. Namely,

θ(q, p) :=‖f‖2q − ‖f‖2p

1/p− 1/q= qp

‖f‖2q − ‖f‖2pq − p

, 1 ≤ p < q. (5.4)

Of course, θ depends on f , but the function will always be clear from the context, so is not indicated in thenotation. The key feature of θ is the following.

28

Lemma 5.4. The function θ is increasing in q and p.

This lemma and its proof are based on a result in [21].

Proof. The function φ is the negative of a difference quotient:

θ(q, p) = −β(1/p)− β(1/q)

1/p− 1/q,

whereβ(t) := ‖f‖21/t (5.5)

for t ∈ (0, 1]. We claim that β is convex. Given the claim, we have that the difference quotients of β areincreasing in both arguments. Hence

β(1/p)− β(1/q)

1/p− 1/q

is decreasing in both arguments, so its negative θ(q, p) is increasing in both arguments, which proves ourclaim.

It remains to prove that β is convex. We first claim that

α(t) :=1

2log(β(t)) = log(‖f‖1/t) = t log

(∫Rn|f |1/t dµ

).

is a convex function of t. Indeed, if t, s ∈ [0, 1], we can apply Holder’s inequality with exponents t/(t + s)and s/(t+ s) to get

α

(t+ s

2

)=

t+ s

2log

(∫Rnf2/(t+s) dµ

)≤ t+ s

2log(‖f1/(t+s)‖(t+s)/t‖f1/(t+s)‖(t+s)/s

)=

t+ s

2log(‖f‖1/(t+s)1/t

)+t+ s

2log(‖f‖1/(t+s)1/s

)=

1

2α(t) +

1

2α(s).

Now, the convexity of α and the fact that the exponential function is increasing and convex gives us

β

(t+ s

2

)≤ e2(α(t)/2+α(s)/2) ≤ 1

2e2α(t) +

1

2e2α(s) =

1

2β(t) +

1

2β(s),

which proves that β is convex.

We can now prove a partial converse of Proposition 5.2.

Proposition 5.5. Suppose that µ satisfies LSI(q) with constant C. Then for all p ∈ [1, q), µ satisfiesBec(q, p) with constant (q/p)C.

Proof. As before, it suffices to prove the implication for f ∈ C∞(Rn) with f ≥ c > 0 for some constant c.Since θ(q, p) is increasing in p ≤ q, we have

‖f‖2q − ‖f‖2pq − p

=θ(q, p)

qp≤ 1

p

limp→q θ(q, p)

q=q

plimp→q

θ(q, p)

qp=q

plimp→q

‖f‖2q − ‖f‖2pq − p

.

This last limit is precisely

d

dp‖f‖2p|p=q =

2

q‖f‖2−qq

∫Rnfq log(f) dµ− 2

qlog(‖f‖q)‖f‖2q.

By LSI(q), this is less than or equal to C‖∇f‖2q, so

‖f‖2q − ‖f‖2pq − p

≤ q

pC‖∇f‖2q,

which is Bec(q, p) with the claimed constant.

29

Remark 5.6. Propositions 5.2 and 5.5, together with the results of Chapter 3, tell us that the standardGaussian measure satisfies LSI(q) with constant 1 for q ≥ 2, and does not satisfy LSI(q) with any constantfor q < 2. We also see that for q ≥ 2 and 1 ≤ p ≤ q, Bec(q, p) with constant q/p can be deduced from thelogarithmic Sobolev inequality via the “soft” argument

LSI(2) (constant 1) ⇒ LSI(q) (constant 1) ⇒ Bec(q, p) (constant q/p)

where the first implication is by Proposition 5.1 and the second by Proposition 5.5. However, the sharpconstant 1 we obtained in Chapter 3 cannot be deduced in this indirect manner.

Lemma 5.4 also tells us something about the relationship between inequality Bec(q, p) for different valuesof the parameter q.

Corollary 5.7. Suppose 1 ≤ p ≤ q ≤ r. If µ satisfies Bec(r, p) with constant C, one has

‖f‖2q − ‖f‖2p ≤r

qC(q − p)‖∇f‖2r.

Proof. Since θ is increasing in q we have that for any p ≤ q,

‖f‖2q − ‖f‖2pq − p

=1

qpθ(q, p) ≤ 1

qpθ(r, p) =

r

q

‖f‖2r − ‖f‖2pr − p

.

By hypothesis, this is less than or equal to C‖∇f‖2r, which proves our result.

Taking µ = γ, r = 2, q ≤ 2, we see that Beckner’s original inequality Bec(2, p) for γn yield the estimate

‖f‖2q − ‖f‖2p ≤2

q(q − p)‖∇f‖22,

for 1 ≤ p ≤ q ≤ 2.From what we proved above, whenever 1 ≤ q ≤ r we have the implications

Bec(q, p) (all p < q, constant C)⇒ LSI(q) (constant C)

⇒ LSI(r) (constant C)⇒ Bec(r, p) (all p < r, constant (r/p)C)

There is another implication within inequality Bec(q, p).

Proposition 5.8. Suppose 1 ≤ q ≤ r and µ satisfies Bec(q, p) with constant C for all p ≤ q. Then for eachr > q and each p ∈ [r/q, r), µ satisfies Bec(r, p) with constant (r/q)C.

Proof. If f ∈ W r,1(µ), then |f | ∈ W r,1(µ) with |∇|f || = |∇f | a.e., so it suffices to consider f ≥ 0. ApplyBec(q, p) to the function fr/q:

‖f‖2r/qr − ‖f‖2r/qrp/q ≤ C(q − p)r2

q2

(∫Rnfr−q|∇f |q dµ

)2/q

. (5.6)

Apply Holder’s inequality on the right side, with exponents r/q and r/(r − q), to get(∫Rnfr−q|∇f |q dµ

)2/q

≤(‖f‖r−qr ‖∇f‖q/rr

)2/q

= ‖f‖2r/q−2r ‖∇f‖2r.

Plugging this into (5.6), dividing by ‖f‖2r/q−2r on both sides, and rewriting the constant on the right yields

‖f‖2r − ‖f‖2−2r/qr ‖f‖2r/qrp/q ≤ C

(r − rp

q

)r

q

(∫Rnfr−q|∇f |q dµ

)2/q

. (5.7)

Since rp/q ≤ r, we have

‖f‖2r/q−2r ≥ ‖f‖2r/q−2

rp/q ⇒ ‖f‖2−2r/qr ≤ ‖f‖2−2r/q

rp/q .

Therefore, the left side of (5.7) is greater than or equal to ‖f‖2r − ‖f‖2rp/q. As p ranges from 1 to q, rp/q

ranges from r/q to r. Thus, we see that (5.7) implies Bec(r, p) with the asserted constant and range for theauxiliary parameter.

30

5.2 Inequality for the semigroup

In this section, we prove an inequality for the semigroup Tt whose generator is an extension of A, with Adefined as in (4.1). This inequality is closely related to Beckner’s p-norm inequality Bec(2, p), and generalizesBeckner’s inequality for the Ornstein-Uhlenbeck in [5], this time for q ≤ 2.

Theorem 5.9 (Inequality for the Semigroup). Let µ = e−v(x)dx be a log-concave probability measure onRn with concavity b > 0. Let A be as in (4.1). Let f ∈ W 2,1(µ). For p ∈ (1, 2], let t(p) be such thate−2bt(p) = p− 1. Then whenever 1 ≤ p ≤ q ≤ 2,

‖Tt(q)‖22 − ‖Tt(p)f‖22 ≤ (q − p)1

b‖∇f‖22. (5.8)

Remark 5.10. If we take q = 2, then t(q) = 0 and T0f = f , so inequality (5.8) yields to the relation

‖f‖22 − ‖Tt(p)f‖22 ≤ (2− p)1

b‖∇f‖22. (5.9)

In the Gaussian case (for which b = 1), this is the inequality for the Ornstein-Uhlenbeck semigroup whichBeckner proved in [5]. Our result here generalizes this inequality for other exponents and other measures.

Proof. By a standard approximation argument, we can assume that f ∈ C∞0 (Rn) with C > f > c for someC, c > 0. Define

φ(p) = ‖Tt(p)f‖22.Then (5.8) is equivalent to the relation

φ(q)− φ(p)

q − p≤ 1

b‖∇f‖22.

The left side is a difference quotient of φ. Furthermore, the hypotheses on f imply that φ is differentiable.So, by the mean value theorem, to prove our inequality it suffices to show that

φ′(p) ≤ 1

b‖∇f‖22, (5.10)

for each p ∈ [1, 2]. This is the inequality we shall prove.If we differentiate φ(p) in p and integrate by parts, we get

φ′(p) =

(d

dpt(p)

)(d

dt

∫Rn

(Tt(p)f)2 dµ

)= − 1

b(p− 1)

(∫RnA(Tt(p)f)Tt(p)f dµ

)=

1

b(p− 1)

(∫Rn|∇Tt(p)f |2 dµ

). (5.11)

By Proposition 4.9, this is less than or equal to

e−2bt(p)

b(p− 1)

(∫RnTt(p)|∇f |2 dµ

)=

1

b‖∇f‖22,

as required.

Remark 5.11. Equation (5.11) shows that φ(p) = ‖Tt(p)f‖22 is an increasing function of p. In fact, if weapply the same argument n times, with ∂jf in place of f , we get

φ′′(p) =1

b2(p− 1)2

n∑j,k=1

∫Rn|∂jkTt(p)f |2 dµ ≥ 0,

whence φ(p) is a convex function of p. It follows that

φ(q)− φ(p)

q − pis increasing in q and p, so the inequality get sharper as q and p increase. In particular, Beckner’s originalinequality (5.9) with q = 2 is stronger than the inequality for smaller q.

31

Remark 5.12. In the Gaussian case, Nelson’s hypercontractivity inequality is essentially the statement that

‖Tt(p)f‖2 ≤ ‖f‖p.

Thus, together with hypercontractivity, (5.8) immediately implies the inequality

‖Tt(q)f‖22 − ‖f‖2p ≤ (2− p)‖∇f‖22.

In particular, if we take q = 2, we get Beckner’s p-norm inequality Bec(2, p). This is how Beckner originallyobtained this inequality. Since Bec(2, p) implies the logarithmic Sobolev inequality LSI(2) and LSI(2)can be used to prove hypercontractivity [9], we see that the inequalities of this section, Beckner’s p-norminequalities, LSI(2), and hypercontractivity are all logically equivalent in the Gaussian case.

5.3 Brascamp-Lieb inequality

In this section, we prove a sharpened form of the Poincare inequality for log-concave probability measureson Rn, originally due to Brascamp and Lieb [10]:

Theorem 5.13. Let µ = e−v(x)dx be a log-concave probability measure on Rn with concavity b > 0. Letf ∈W 2,1(µ). Then

‖f‖22 −(∫

Rnf dµ

)2

≤∫Rn〈(D2v)−1∇f,∇f〉 dµ. (5.12)

Observe that if x ∈ Rn, and y =√

(D2v)x, then

〈(D2v)−1x, x〉 = 〈y, y〉 ≤ 1

b〈(D2v)y, y〉 =

1

b|x|2.

Therefore inequality (5.12) is sharper than the inequality Bec(2, 1) with constant 1/b, which we proved inChapter 4.

The original proof in [10] is by a direct, albeit lengthy, calculation and an induction argument. We take analternative, functional-analytic approach which yields several intermediate results which are of independentinterest. The first of these is:

Proposition 5.14. Define A := ∆ − 〈∇v,∇〉 as in (4.1). Let A be the self-adjoint extension of A whichis the infinitesimal generator of the semigroup Tt of Chapter 4 (c.f. Appendix 6.2). Then A is invertiblefrom the set of f ∈ D(A) with

∫Rn f dµ = 0 to the set of g ∈ L2(µ) with

∫Rn g dµ = 0. Furthermore the

inverse of A is continuous with respect to the L2 norm, and we have the inequality

‖f‖2 ≤1

b‖Af‖2. (5.13)

To prove this, we need two lemmas.

Lemma 5.15. Let Tt be a contraction semigroup on a Hilbert space X with generator S. Let x ∈ X. LetY be a dense linear subspace of X, and suppose that for each y ∈ Y ,

lims→0

1

s〈Tsx− x, y〉 = 〈z, y〉 (5.14)

for some z ∈ X. Then Sx exists and equals z.

Proof. Since Y is dense, (5.14) holds for each y ∈ X, not just for y ∈ Y . For each t ≥ 0, Tt is symmetricand so

lims→0

1

s〈Tt+sx− Ttx, y〉 = lim

s→0

1

s〈Tsx− x, Tty〉 = 〈z, Tty〉 = 〈Ttz, y〉.

Thusd

dt〈Ttx, y〉 = 〈Ttz, y〉.

32

Integrating, we get

〈Ttx− x, y〉 =

∫ t

0

〈Tsz, y〉ds

for each y ∈ X. Therefore Ttx− x =∫ t

0Tszds. As t→ 0,

1

t

∫ t

0

Tszds→ z

strongly, so t−1(Ttx− x)→ z strongly, i.e. Sx = z.

Now let Tt be the semigroup of Proposition 5.14.

Lemma 5.16. Let f ∈ L2(µ) with∫Rn f dµ = 0. Then

‖Ttf‖2 ≤ e−bt‖f‖2.

Proof. By density it suffices to prove the formula for f ∈ D(A). Fix t > 0 and for t ≥ s ≥ 0 define

φ(s) = e−2bs

∫Rn

(Tt−sf)2 dµ.

We have φ(0) = ‖Ttf‖22, φ(t) = e−2bt‖f‖22, and

φ′(s) = −2be−2bs

∫Rn

(Tt−sf)2 dµ+ e−2bs

∫Rn∂s(Tt−sf)2 dµ. (5.15)

We compute ∫Rn∂s(Tt−sf)2 dµ = −

∫Rn

2(ATt−sf)(Tt−sf) dµ = 2

∫Rn|∇Tt−sf |2 dµ.

Plugging this into (5.15), we find

φ′(s) = 2e−2bs(‖∇Tt−s‖22 − b‖Tt−sf‖22).

By the Poincare inequality, this is non-negative. Thus φ′ is increasing, so in particular

‖Ttf‖22 = φ(0) ≤ φ(t) = e−2bt‖f‖22.

Proof of Proposition 5.14. For g ∈ L2(µ) with∫Rn g dµ = 0, define

Bg = −∫ ∞

0

Ttg dt. (5.16)

We claim that B = A−1. First we need to check that B is well defined and continuous. Let s > 0. ByMinkowski’s inequality for integrals,(∫

Rn

(∫ s

0

Ttg dt

)2

dµ

)1/2

≤∫ s

0

‖Ttg‖2 dt (5.17)

By Lemma 5.16, this is less than or equal to∫ s

0

e−bt‖g‖2 dt =

(1

b− e−2s

)‖g‖2.

Letting s→∞, we find

‖Bg‖2 ≤1

b‖g‖2. (5.18)

33

Thus B : L2(µ)→ L2(µ) continuously.Now, if g = Af , then

∫Rn g dµ =

∫Rn fA(1) dµ = 0, and

Bg = −∫ ∞

0

Tt(Af) dt = −∫ ∞

0

d

dtTtf dt = f.

Thus BA = Id.On the other hand, if g ∈ L2(µ) with

∫Rn g dµ = 0, then

1

s(TsBg −Bg) =

∫ ∞0

1

s(Ttg − Tt+sg) dt,

so if φ ∈ C∞c (Rn), we get from Fubini’s theorem and symmetry of Tt that∫Rn

1

s(TsBg −Bg)φdµ =

∫ ∞0

∫Rn

1

s(Ttg − Tt+sg)φdµdt

=

∫ ∞0

∫Rn

1

s(Ttφ− Tt+sφ) g dµdt.

Note that Lemma 5.16 and Holder’s inequality show that Ttgφ is jointly integrable in t and over Rn, so thatthe application of Fubini’s theorem is justified. As s→ 0, this tends to

−∫ ∞

0

∫RnAφg dµdt =

∫RnA

(−∫ ∞

0

Ttφdt

)g dµ =

∫Rnφg dµ.

Thus by Lemma 5.15, ABg exists and equals g, as required. Thus B = A−1 and estimate (5.13) is immediatefrom (5.18).

Now we proceed to prove Theorem 5.13. Our proof is based on that of B. Heffler [18]. Like the proof ofTheorem 4.2, this proof relies on a commutation relation between the operator A defined in (4.1) and thegradient operator.

Let L2(µ)n be the space of n-component vector valued functions with components in L2(µ), with innerproduct 〈F,G〉L2(µ)n =

∫Rn〈F,G〉 dµ. Let L : (L2(µ))n → (L2(µ))n be the unbounded operator defined by

LF = (D2v)F −AF, (5.19)

where A acts componentwise on vector-valued functions. Then if f ∈ C∞c (Rn),

∇Af = ∇∆f − (D2f)∇v − (D2v)∇f = −L∇f. (5.20)

Proof of Theorem 5.13. By approximation, it suffices to prove the inequality for f ∈ C∞(Rn), with f con-stant outside a compact set. Since the inequality is invariant under adding a constant to f , we may alsoassume that

∫Rn f dµ = 0. Then by Lemma 5.14, g := A−1f is well defined. By the integration by parts

formula for A, one has

‖f‖22 =

∫Rn

(Ag)2 dµ = −∫Rn〈∇Ag,∇g〉 dµ =

∫Rn〈L∇g,∇g〉 dµ. (5.21)

We have∇f = ∇Ag = −L∇g. (5.22)

For any F in the domain of L, we have

〈LF,F 〉L2(µ)n = 〈(D2v)F, F 〉L2(µ)n − 〈AF, F 〉L2(µ)n = 〈(D2v)F, F 〉L2(µ)n + ‖∇F‖22. (5.23)

The second term on the right in (5.23) non-negative, so

〈LF,F 〉L2(µ)n ≥ 〈(D2v)F, F 〉L2(µ)n ≥ b‖F‖22. (5.24)

34

Therefore the operator L := bI − L is non-positive on L2(µ)n. Clearly this operator is symmetric, andits domain contains C∞c (Rn)n, so is dense in L2(µ)n. So, exactly as we do for A in Appendix 6.2, wecan apply the Friedrichs extension theorem to extend L to a self-adjoint operator on L2(µ), still denotedby L, then apply the spectral theorem to get a contraction semigroup with generator L. It then followsfrom the Hille-Yosida theorem that the extension L := bI − L of L is invertible with inverse satisfying‖L−1F‖ ≤ (1/b)‖F‖L2(µ)n for each F ∈ L2(µ)n.

Since F 7→ 〈LF, F 〉L2(µ)n defines an inner product on the domain of L, we have by the Cauchy-Schwarz

inequality that whenever F ∈ L2(µ)n and G is in the domain of L,

|〈F,G〉L2(µ)n |2 = |〈LL−1F,G〉L2(µ)n |2 ≤ 〈L−1F, F 〉L2(µ)n〈LG,G〉L2(µ)n

with equality iff G = L−1F . Therefore

〈L−1F, F 〉L2(µ)n = sup

|〈F,G〉L2(µ)n |2

〈LG,G〉L2(µ)n: G ∈ D(L)

.

By (5.24), this is less than or equal to

sup

|〈F,G〉L2(µ)n |2

〈(D2v)G,G〉L2(µ)n: G ∈ D(L)

.

By the same argument above with L replaced by D2v (and the fact that D(L) is dense in L2(µ)n), thisequals 〈(D2v)−1F, F 〉L2(µ)n , and hence

〈L−1F, F 〉L2(µ)n ≤ 〈(D2v)−1F, F 〉L2(µ)n . (5.25)

From (5.22) one has∇g = −L−1∇f. (5.26)

Then applying this and (5.25) with F = ∇f , we get

〈L∇g,∇g〉L2(µ)n = 〈L−1∇f,∇f〉L2(µ)n ≤ 〈(D2v)−1∇f,∇f〉L2(µ)n

and plugging this into (5.21) gives our desired inequality.

Remark 5.17. Inequality (5.12) suggests that one might look for an analogous version of inequality Bec(2, p)or LSI(2) for log-concave probability measures, i.e. an inequality of the form

2

q‖f‖2−qq

∫Rn|f |q log |f | dµ− 2

qlog(‖f‖2)‖f‖2q ≤ C

(∫Rn〈(D2v)−1∇f,∇f〉q/2 dµ

)2/q

. (5.27)

or

‖f‖2q − ‖f‖2p ≤q

pC(q − p)

(∫Rn〈(D2v)−1∇f,∇f〉q/2 dµ

)2/q

(5.28)

Bobkov and Ledoux [7] have demonstrated that this is not possible in general, although (5.27) holds forq = 2 under additional regularity hypotheses on the measure. In the next section, we give their proof, anddeduce (5.27) and (5.28) for the general q, with a suitable constant, as a corollary.

5.4 Sharpened logarithmic Sobolev inequality

In this section, we shall prove a sharpened logarithmic Sobolev inequality for a restricted class of log-concaveprobability measures due to Bobkov and Ledoux [7], which is analogous to the sharpened Poincare inequalityof Section 5.3. The key tool is the following deep convexity inequality:

35

Theorem 5.18 (Prekopa-Leindler). Let f, g, h be non-negative, measurable functions on Rn. Suppose that0 < t < 1 and for each x, y ∈ Rn,

f((1− t)x+ ty) ≥ g(x)1−th(y)t.

Then ∫Rnf(x) dx ≥

(∫Rng(x) dx

)1−t(∫Rnh(x) dx

)t.

For the proof, see [24]. We remark that Bobkov and Ledoux [7] have also used the Prekopa-Leindlertheorem to prove the Brascamp-Lieb inequality of Section 5.3, via an argument similar to the one below.

Theorem 5.19 (Bobkov-Ledoux). Let µ = e−v(x)dx be a log-concave probability measure on an open convexset Ω ⊂ Rn.1 Let f ∈ W 2,1(µ). Suppose that for any h ∈ Rn, the function x 7→ 〈D2v(x)h, h〉 is concave onΩ. Then one has the sharpened logarithmic Sobolev inequality∫

Ω

f2 log |f | dµ− ‖f‖22 log (‖f‖2) ≤ C∫

Ω

〈(D2v)−1∇f,∇f〉 dµ (5.29)

with constant C = 3/2.

Note that, unlike our previous results, this inequality does not hold for general log-concave probabilitymeasures, even with a different constant. A counterexample is given below. The minimal hypotheses on µrequired to obtain inequality (5.29) are, to our knowledge, not known.

Furthermore, the constant 3/2 in inequality (5.29) is not sharp in all cases. For example, if µ = γn is thestandard Gaussian measure, then (5.29) is just the ordinary logarithmic Sobolev inequality with constant C,and holds with C = 1. Below, we give a counterexample to the effect that the constant cannot be improvedto C = 1 in all cases. The sharpest possible constants in (5.29) are not known in general.

Proof. By the usual approximation arguments, we can take f to be smooth with 0 < c ≤ f ≤ C < ∞. Infact, by approximating and rescaling, we can arrange that, in addition, f ≡ 1 outside of a compact set. Thenwe can write f2 = eg, where g ∈ C∞c (Ω).

Let t, s > 0 with t+ s = 1. Set

gt(z) = supg(x)− [tv(x) + sv(y)− v(z)] : x, y ∈ Ω, z = tx+ sy.

We shall apply the Prekopa-Leindler theorem to the functions

egt−vχΩ, eg/t−vχΩ, e−vχΩ,

where χ denotes an indicator function. By definition of gt, we have

egt(tx+sy)−v(tx+sy) ≥ eg(x)−[tv(x)+sv(y)] = (eg(x)/t−v(x))t(e−v(y))s

so the hypotheses of the theorem are satisfied and we get(∫Ω

egt dµ

)≥(∫

Ω

eg/t dµ

)t. (5.30)

We shall take the limit as t→ 1, s→ 0 to obtain our desired inequality. To simplify notation in what follows,we denote the left side of (5.29) by Ent(f) (for “entropy”).

First we shall see how Ent(f) arises from the right side of (5.30). By logarithmic differentiation, one has

d

dt‖f‖1/t = −1

t‖f‖1−1/t

1/t

∫Ω

f1/t log(f) dµ+ log(‖f‖1/t)‖f‖1/t

1A log-concave probability measure on Ω is defined in an analogous manner to a log-concave probability measure on Rn.Sobolev spaces for these measures are defined as in Appendix 6.1, but with C∞

c (Ω) replaced by the set of smooth functions fon Ω whose derivatives up to order m are in L2(µ).

36

If we replace f by f2 and evaluate at t = 1, we get

d

dt|t=1‖f2‖1/t = −2

∫Ω

f2 log(f) dµ+ 2 log (‖f‖2) ‖f‖22 = −2 Ent(f).

Thus, recalling that eg = f2, we get by Taylor expansion at t = 1 that(∫Ω

eg/t dµ

)t=

∫Ω

eg dµ+ 2sEnt(f) +O(s2). (5.31)

We now need a suitable estimate on gt. Let

L(s) := tv(x) + sv(y)− v(z)

be the quantity subtracted from g(x) in the formula defining gt, where z = tx+ sy. Put k = x− y. One has

d

dr〈∇v(rz + (1− r)x), k〉 = −s〈D2v(rz + (1− r)x)k, k〉

d

dr

1

sv(rz + (1− r)x) = 〈∇v(rz + (1− r)x), k〉.

So, integrating by parts, we find that∫ 1

0

rs〈D2v(rz + (1− r)x)k, k〉dr = −〈∇v(z), k〉+

∫ 1

0

r〈∇v(rz + (1− r)x), k〉dr

= −〈∇v(z), k〉 − 1

sv(z) +

1

sv(x).

Similarly, ∫ 1

0

rt〈D2v(rz + (1− r)y)k, k〉dr = 〈∇v(z), k〉 − 1

tv(z) +

1

tv(y).

Thus

L(s) = ts

∫ 1

0

rs〈D2v(rz + (1− r)x)k, k〉+ rt〈D2v(rz + (1− r)y)k, k〉dr. (5.32)

By our hypothesis on v,

〈D2v(rz + (1− r)x)k, k〉 ≥ r〈D2v(z)k, k〉+ (1− r)〈D2v(x)k, k〉 ≥ r〈D2v(z)k, k〉

and similarly with x and y interchanged. Thus we find

L(s) ≥ ts∫ 1

0

r2〈D2v(z)k, k〉dr =ts

3〈D2v(z)k, k〉.

Hence,

gt(z) ≤ supg(x)− ts

3〈D2v(z)k, k〉 : x, y ∈ Ω, z = tx+ sy.

We have x = z + sk, so by Taylor expansion about s = 0 we obtain

gt(z) ≤ g(z) + s supk∈Ω〈∇g(z), k〉 − 1

3〈D2v(z)k, k〉+O(s2).

Now fix z and make the evaluation at this z implicit. Differentiating in k, we find that the quantity insidethe supremum is maximized by choosing k such that

∇g − 2

3(D2v)k = 0.

37

This is equivalent to k = 32 (D2v)−1∇g, and hence

gt ≤ g +3s

2〈(D2v)−1∇g,∇g〉 − 3s

4〈(D2v)−1∇g,∇g〉+O(s2)

= g +3s

4〈(D2v)−1∇g,∇g〉+O(s2).

From the Taylor expansion of the exponential function, we then get

egt ≤ ege 3s4 〈(D

2v)−1∇g,∇g〉+O(s2) = eg +3s

4eg〈(D2v)−1∇g,∇g〉+O(s2).

Substitute this inequality and (5.31) into (5.30) to obtain∫Ω

eg dµ+ 2sEnt(f) +O(s2) ≤∫

Ω

eg dµ+3s

4

∫Ω

eg〈(D2v)−1∇g,∇g〉 dµ+O(s2).

Cancelling∫

Ωeg dµ on both sides, dividing by s, and then letting s→ 0, we find

2 Ent(f) ≤ 3

4

∫Ω

eg〈(D2v)−1∇g,∇g〉 dµ = 3

∫Ω

〈(D2v)−1∇f,∇f〉 dµ.

This completes the proof.

Recall that in Section 5.1, we established the chain of implications:

LSI(2) (constant C) ⇒ LSI(q) (constant C) ⇒ Bec(q, p) (constant (q/p)C)

for q ≥ 2, 1 ≤ p < q. (The first implication is Proposition 5.1 and the second is Proposition 5.5.) Exactly the

same arguments used there, with Rn replaced by Ω and ‖∇f‖2q replaced by(∫

Ω〈(D2v)−1∇f,∇f〉q/2 dµ

)2/q,

yield

Corollary 5.20. Let µ satisfy the hypotheses of Theorem 5.19, q ≥ 2, 1 ≤ p < q, f ∈W q,1(µ). Then

2

q‖f‖2−qq

∫Ω

|f |q log |f | dµ− 2

qlog(‖f‖2)‖f‖2q ≤ C

(∫Ω

〈(D2v)−1∇f,∇f〉q/2 dµ)2/q

. (5.33)

and

‖f‖2q − ‖f‖2p ≤q

pC(q − p)

(∫Ω

〈(D2v)−1∇f,∇f〉q/2 dµ)2/q

(5.34)

with constant C = 3/2.

To see that Theorem 5.19 cannot hold without the additional concavity hypothesis on v, and that theconstant in this theorem cannot be sharpened to C = 1, we proceed by way of the following two results,which appear in [23].

Proposition 5.21 (Herbst Argument). Let µ be a log-concave probability measure which satisfies (5.29).Then for function f ∈W 2,1(µ) with 〈(D2v)−1∇f,∇f〉 ≤ 1 a.e. and

∫Ωf dµ := α <∞, and for any t > 0,∫

e2tf dµ ≤ e2Ct2+αt. (5.35)

Proof. By approximating the general f satisfying the conditions of the proposition with continuously differ-entiable functions in the C0,1 norm, we may assume that f is continuously differentiable. Let φ(t) = ‖e2tf‖22,g = etf/φ(t)1/2. By (5.29), ∫

Ω

g2 log g dµ ≤ C∫

Ω

〈(D2v)−1∇g,∇g〉 dµ

38

Note that the second term on the left vanishes since ‖g‖2 = 1. In terms of f and φ, this reads

tφ′(t)

2φ(t)− 1

2log φ(t) ≤ C

φ(t)

∫Ω

t2〈(D2v)−1∇f,∇f〉e2tf dµ.

After dividing by t2, the left side of this inequality equals 12ddt

log φ(t)t . On the right we have 〈(D2v)−1∇f,∇f〉 ≤

1, so1

2

d

dt

log φ(t)

t≤ C.

We have limt→0log φ(t)

t =∫Rn f dµ = α, so if we integrate both sides and use the fundamental theorem of

calculus we getlog φ(t)

2t≤ Ct+ α/2.

Rearranging terms gives (5.35).

Corollary 5.22. If f satisfies the hypotheses of Proposition 5.21 and 0 < λ < 1/2C, then∫Ω

eλf2

<∞.

Proof. By Chebyshev’s inequality and (5.35), for any t, r > 0,

µ2f(x) ≥ α+ r ≤ µe2tf(x) ≥ eαt+rt ≤ e2Ct2+αt−αt−rt = e2Ct2−rt.

The right side is minimized when t = r/4C, in which case we have

µ2f(x) ≥ α+ r ≤ e−r2/8C

By Fubini’s theorem, one then has∫Ω

eλf2

dµ =

∫Ω

e(λ/4)(2f)2 dµ

= 1 +λ

2

∫ ∞0

re(λ/4)r2µf(x) ≥ rdr

≤ 1 +λ

2

∫ ∞0

re(λ/4)r2−r2/8Cdr,

which is finite provided λ < 1/2C.

For example, consider v(x) = − log(2x) on (0, 1). The function v is strictly convex on (0, 1) and theintegral e−v(x) over (0, 1) is 1, so µ = e−v(x)dx = 2xdx defines a log-concave probability measure on (0, 1).This measure is also introduced as a counterexample in [7]. Let f(x) = log x on (0, 1). We have f ′2/v′′ ≡ 1and ∫ 1

0

f dµ = 2

∫ 1

0

x log xdx = −1/2.

So, if µ satisfies (5.29) for some constant C, then Corollary 5.22 implies that we must in particular have∫ 1

0eλf

2

dµ <∞ for any 0 < λ < 2/C. But, for any such λ,∫ 1

0

eλf2

dµ = 2

∫ 1

0

xeλ(log x)2dx =∞.

Thus µ cannot satisfy (5.29) for any constant C. Note further that (5.34) with q = 2 implies (5.29) with thesame constant, so µ cannot satisfy (5.34) with q = 2 and any constant either.

39

As another example, take v(x) = xp on (1,∞) for 2 ≤ p < 3. Then v is strictly convex, so µ = Ce−v(x)dx,

where C =(∫∞

1e−v(x)dx

)−1, defines a log-concave probability measure on (1,∞). Furthermore v′′ is concave,

so µ satisfies the hypotheses of Theorem 5.19. Let

f(x) =2√p(p− 1)

pxp/2.

Then f ′2/v′′ ≡ 1,∫∞

1f dµ <∞, so the hypotheses of Proposition 5.35 are satisfied. For λ > 0,∫ ∞

1

eλf2

dµ =

∫ ∞1

exp

(4(p− 1)

pλxp − xp

)dx.

This is finite if and only if λ < p4(p−1) . Therefore Corollary 5.22 implies that µ cannot satisfy (5.29) for any

constant C < 2(p− 1)/p. In particular, µ cannot satisfy (5.29) with constant 1 unless p = 2.

Remark 5.23. To our knowledge it is not known what hypotheses on the measure are required to obtain(5.33) and (5.34) in general, nor what the sharpest possible constants in these inequalities are.

6 Appendices

6.1 Sobolev Spaces for Log-Concave Measures

In this appendix, we shall define the measures and function spaces which are the settings for our inequalities,and prove some of their basic properties. Recall that a multi-index is an element α = (α1, ..., αn) of Nn. Wewrite |α| =

∑nk=1 αk, and ∂α =

∏nk=1 ∂

αjj .

Given a log-concave probability measure µ on Rn, p ∈ [1,∞), and m ∈ N, define a norm on the spaceC∞c (Rn) of smooth functions with compact support by

‖f‖Wp,m(µ) :=∑|α|≤m

‖∂αf‖p.

The completion of C∞c (Rn) under this norm is called the mth Sobolev space with exponent p, and is denotedby W p,m(µ).

A sequence (fj) in C∞c (Rn) converges to f ∈ C∞(Rn) in W p,m(µ) if and only if ∂αfj → ∂αf in Lp foreach α with |α| ≤ m. Likewise, if (fj) is a Cauchy sequence in C∞c (Rn), then whenever |α| ≤ m, (∂αf) isa Cauchy sequences in Lp. Since Lp is complete, (fj) converges to a function f ∈ Lp(µ) in the Lp norm,and so does each (∂αfj). We want to define the Sobolev partial derivatives ∂αf of f as the Lp limits of thesequences (∂αfj). However, we first need to check these these limits do not depend on the choice of sequence(fj) ∈ C∞c (Rn). By induction on m, it will suffice to establish the following:

Proposition 6.1. Suppose that two sequences (fj) and (gj) in C∞c (Rn) converge to the same function f inLp(µ), and for each k = 1, ..., n, (∂kfj) and (∂kgj) converge in Lp(µ). Then for each k,

limj→∞

∂kfj = limj→∞

∂kgj .

Proof. By replacing fj by fj − gj , it suffices to assume that fj → 0 in Lp and to show that hk :=limj→∞ ∂kfj = 0 for each k = 1, ..., n. Let φ ∈ C∞c (Rn). Then by the integration by parts formula(4.3) and the product rule,∫

Rnhkφdµ = lim

j→∞

∫Rn

(∂kfj)φdµ = limj→∞

∫Rnfj(φ∂kv − ∂kφ) dµ.

But, since fj → 0 in Lp(µ), this limit is zero. Therefore∫Rn hkφdµ = 0 for each φ ∈ C∞c (Rn), which implies

that hk = 0 a.e.

40

In particular, Proposition 6.1 implies that the Sobolev partial derivatives of a function in C∞c (Rn) agreewith its ordinary partial derivatives, so there is no ambiguity in using the same notation for both. We remarkthat, by definition, C∞c (Rn) is dense in W p,m(µ), with respect to both its own norm and the Lp norm. Assuch, by approximating more general functions by smooth, compactly supported ones, it often suffices toprove results only on C∞c (Rn).

Next we establish that the spaces W p,m(µ) are sufficiently large to be of interest. Recall that Cm(Rn)denotes the set of m-times continuously differentiable functions on Rn.

Proposition 6.2. If f ∈ Cm(Rn) and f and each of its partial derivatives up to order m are in Lp(µ), thenf ∈W p,m(µ).

Proof. First we show that the set Cmc (Rn) of m-times continuously differentiable functions with compactsupport is contained in W p,m(µ). Given f ∈ Cmc (Rn), we need to approximate f in the W p,m normby functions in C∞c (Rn). For this, let g ∈ C∞c (Rn) be non-negative with Lebesgue integral is 1. Setgt(x) = t−ng(x/t), and let

f ∗ gt(x) =

∫Rngt(x− y)f(y) dy

be the convolution of f and gt. Since f and gt are compactly supported, so is f ∗ gt. By the mean valuetheorem and the dominated convergence theorem, we can differentiate under the integral sign to obtain thatf ∗ gt is smooth, and that for each multi-index α with |α| ≤ m,

∂α(f ∗ gt) = f ∗ (∂αgt) = (∂αf) ∗ gt.

It is a standard theorem in the theory of Lp spaces that the functions gt are an approximate identity, in thesense that h ∗ gt → h in Lp(dx) (and hence also in Lp(µ)) for any h ∈ Lp(dx). In particular,

(∂αf) ∗ gt → ∂αf

in Lp(µ) as t→ 0.So, f ∗ gt → f in W p,m(µ), which implies f ∈W p,m(µ).Now suppose f satisfies the hypotheses of the theorem, but may not be compactly supported. Since

W p,m is complete, in light of the above it will suffice to approximate f in the W p,m norm by functions inCkc (Rn). To do so, set

Bj = x ∈ Rn : |x| ≤ j.

Take a sequence of smooth bump functions βj , equal to 1 on Bj , 0 on Rn\Bj+1, and with uniformly boundedpartial derivatives up to order α. Then if fj := βjf , we have ∂αfj → ∂αf pointwise for each |α| ≤ m. Theproduct rule allows us to bound the integrals of |∂αfj |p over Bj+1 \Bj in terms of the integrals of the partialderivatives of f over this region. Since each ∂αf ∈ Lp, said integrals tend to zero, and we get fj → f inW p,m.

The following proposition will often allow us to reduce proofs to the case of non-negative functions.

Proposition 6.3. Let f ∈ C1(Rn) ∩W p,1(µ). Then |f | ∈W p,1(µ) and |∇f | = |∇|f || a.e.

Proof. We first define a family of functions which will act as a C1 approximation of the absolute valuefunction. For y ∈ R, t > 0, set

Ft(y) = (y2 + t2)1/2 − t.

We claim that Ft f → |f | in W p,1(µ) as t→ 0. Indeed, we have Ft f → |f | pointwise, and

‖Ft f‖∞ ≤ t+ ‖f‖∞.

By dominated convergence, Ft f → |f | in Lp. Furthermore, for each index k = 1, ..., n,

∂k(Ft f) =f∂kf

(f2 + t2)1/2.

41

Now, if either ∂kf(x) = 0 or f(x) 6= 0, then as t→ 0, this converges to sgn(f(x))∂kf(x). So, the only pointswhere convergence can fail to occur are those in

D = x ∈ Rn : f(x) = 0,∇f(x) 6= 0.

We claim that D has measure zero with respect to Lebesgue measure, and hence also with respect to µ. Indimension 1, D consists of countably many endpoints of the open intervals where f 6= 0, so has measurezero. In higher dimensions, if x ∈ D, the implicit function theorem allows us to solve f(y) = 0 locally forone of the coordinates of y as a C1 function of the others, and thereby express D ∩Ux as the image of a C1

function from Rn−1 to Rn for some neighborhood Ux of x. The image of such a function has measure zero,and countably many such Ux cover D. It follows that D has measure zero.

Thus ∂kFt f → sgn(f)∂kf a.e., and we have |∂kFt f | ≤ ‖∂kf‖∞ a.e. so a second application ofdominated convergence yields ∂kFt f → ∂kf in Lp. It follows that |f | ∈ W p,1(µ) with |∂k|f || = |∂kf | (inthe Sobolev sense).

6.2 Existence of Semigroup

In this appendix, we prove the existence of a symmetric Markov semigroup Tt whose infinitesimal generatoris an extension of the operator A defined in (4.1). We shall require the following theorems from functionalanalysis:

Theorem 6.4 (Friedrichs Extension). Let S be a densely defined symmetric operator on a Hilbert space X.If S is non-negative (or non-positive), in the sense that 〈Sx, x〉 ≥ 0 (resp. 〈Sx, x〉 ≤ 0) for x in the domainof S, then there is a unique non-negative (resp. non-positive), self-adjoint extension of S.

Theorem 6.5 (Spectral Theorem). Let S be a bounded self-adjoint operator on a Hilbert space X. Thenthere is a measure space (Ω, ν) and a linear isometry U : X → L2(ν) such that USU−1 is multiplication Mλ

by some measurable function λ on Ω.

For f, g ∈ L2(µ), write

〈f, g〉L2(µ) =

∫Rnfg dµ. (6.1)

The operator A in (4.1) is densely defined on L2, for its domain clearly contains C∞c (Rn). Symmetry of Afollows by applying Lemma 4.3 twice. Furthermore, A is non-positive, for if f ∈ D(A), then Lemma 4.3shows that

〈f,Af〉L2(µ) = −‖∇f‖2 ≤ 0.

Thus the Friedrichs extension theorem implies that A has a non-positive self-adjoint extension A.The spectral theorem implies that there is a measure ν on a set Ω and a linear isometry U : L2(µ)→ L2(ν)

such that UAU−1 is multiplication by some measurable function λ on Ω:

UAU−1 = Mλ.

For t ≥ 0, putTt = U−1MetλU : D(A)→ L2(µ). (6.2)

Proposition 6.6. Tt extends to a symmetric contraction semigroup on L2(µ) with infinitesimal generatorA.

Proof. From the definition (6.2), it is immediate that T0 is the identity operator and Tt Ts = Tt+s onD(A). Furthermore, the mapping t 7→Meλtg is continuous for fixed g ∈ L2(ν), so since U and its inverse arecontinuous, so is t 7→ Ttf for each f ∈ L2(µ). Taking the transpose of both sides of (6.2) shows that each Ttis symmetric.

Now we show that each Tt does not increase L2-norms on D(A), so extends to a norm-decreasing operatoron all of L2(µ). Since A is non-positive, we have for each f ∈ D(A) that

0 ≥ 〈Af, f〉L2(µ) = 〈U−1MλUf, f〉L2(µ) = 〈λUf, Uf〉L2(ν).

42

Since D(A) is dense in L2(µ), U−1D(A) is dense in L2(ν). Therefore this inequality implies that λ ≤ 0 a.e.Hence etλ ≤ 1 a.e., and it follows that

〈Ttf, Ttf〉L2(µ) = 〈MetλUf,MetλUf〉L2(ν) ≤ 〈Uf,Uf〉L2(ν) = 〈f, f〉L2(µ).

Therefore Tt is norm-decreasing on D(A). Since D(A) is dense in L2(µ), Tt extends to a norm-decreasingoperator on L2(µ), and by density of D(A) we still have T0 = Id and Tt Ts = Tt+s. Thus Tt is acontraction semigroup on L2(µ).

It remains to check that the infinitesimal generator of this semigroup is A. Since U is linear and contin-uous, it commutes with differentiation in t. That is, if f ∈ D(A), then in the L2 sense,

d

dtTtf = U−1 d

dtMetλUf = U−1MλMetλUf = U−1MλUU

−1MetλUf = ATtf.

Evaluating at t = 0 shows that ddt |t=0Ttf = Af whenever f ∈ D(A). Conversely, suppose f ∈ L2(µ) and

ddt |t=0Ttf exists in L2(µ). Then for any g ∈ D(A),

〈 ddt|t=0Ttf, g〉L2(µ) = lim

t→0

1

t〈Ttf − f, g〉L2(µ).

From the formula (6.2), it is clear that Tt is symmetric for each t, so this equals

limt→0

1

t〈f, Ttg − g〉L2(µ) = 〈f, Ag〉L2(µ).

Since A is self-adjoint, it follows that f ∈ D(A) with Af = ddt |t=0Ttf . Therefore the infinitesimal generator

of Tt is A.

Of course, in the Gaussian case, our semigroup Tt is just the Ornstein-Uhlenbeck semigroup.To show that Tt is a Markov semigroup, we need the following result, which characterizes contraction

and Markov semigroups in terms of their generators:

Theorem 6.7 (Hille-Yosida Theorem for Markov Semigroups). Let S be a closed linear operator defined ona domain D(S) of a Banach space X. Then S generates a contraction semigroup if and only if

1. D(S) is dense in X;

2. For every λ > 0, λI−S is invertible and the resolvent (λI−S)−1 exists and satisfies ‖(λI−S)−1‖ ≤ 1/λ.

This semigroup is Markov if and only if S(1) = 0 and (λI − S)−1 preserves positivity for all λ > 0.

For a proof, see Ch. 8 of [8].In order to apply the Hille-Yosida theorem to A, we need to check that A is closed. Indeed, if (fj) ∈ D(A),

fj → f ∈ L2(µ), and Afj → g ∈ L2(µ), then for any φ ∈ D(A),

〈g, φ〉L2(µ) = limj→∞〈Afj , φ〉L2(µ) = lim

j→∞〈fj , Aφ〉L2(µ) = 〈f, Aφ〉L2(µ).

Therefore f belongs to the domain of the adjoint of A. But, A is self-adjoint, so f ∈ D(A), and by symmetry〈g, φ〉L2(µ) = 〈Af, φ〉L2(µ) for each φ ∈ D(A). Since D(A) is dense in L2(µ), it follows that Af = g, so that

A is closed.From Proposition 6.6, A generates a contraction semigroup, so by the Hille-Yosida theorem, for each

λ > 0, (λI − A)−1 exists as a continuous operator on L2(µ). To show that our semigroup Tt is Markov, ittherefore remains to check the last two conditions in Theorem 6.7.

Proposition 6.8. The semigroup Tt of Proposition 6.6 is Markov.

43

Proof. Clearly, A(1) = 0. First we check that (λI − A)−1 is positivity preserving on (λI − A)C2c (Rn). We

must show that if f ∈ C2c (Rn) and

g = (λI − A)f ≥ 0,

then f ≥ 0. In this case, f attains a minimum at some point x0 ∈ Rn. At this point, ∇f = 0 and ∆f ≥ 0.We have

0 ≤ (λI − A)f(x0) = (λI −A)f(x0) = λf(x0)−∆f(x0),

so λf(x0) ≥ ∆f(x0) ≥ 0, whence f ≥ 0.In the general case, let g ∈ L2(µ) be non-negative. We must show that (λI − A)−1g ≥ 0. We claim that

C2c (Rn) ⊂ (λI − A)C2

c (Rn). (6.3)

Given the claim, if g ∈ L2(µ) is non-negative a.e., then we can select a non-negative sequence (gj) ∈ C2c (Rn)

which converges to g in L2. Then by the first case and (6.3), (λI − A)−1gj is non-negative a.e., and since

(λI− A)−1 is continuous, these functions converge to (λI−A)−1g in L2. Hence (λI− A)−1g is non-negativea.e. Thus, by the Hille-Yosida theorem, the semigroup generated by A is Markov.

It remains to prove (6.3). Consider a fixed compact set K ⊂ Rn which equals the closure of its interior,and put

L2K(µ) = f ∈ L2(µ) : supp(f) ⊂ K.

The space L2K(µ) is a closed subspace of L2(µ), hence is itself a Hilbert space. Furthermore A maps L2

K(µ)

to L2K(µ) and the restriction of A to D(A) ∩ L2

K(µ) is symmetric and non-positive. Since D(A) ∩ L2K(µ)

contains C2c (Ko) (where Ko is the interior of K), D(A) ∩ L2

K(µ) is dense in L2K(µ). Therefore we can

apply the Friedrichs extension theorem, the spectral theorem, and the Hille Yosida theorem just as wedid on L2(µ) to find that (λI − A)−1 exists as an operator from L2

K(µ) to L2K(µ). This resolvent must

necessarily agree with the ordinary resolvent on C2c (Ko), so we find that (λI− A)−1 maps C2

c (Ko) to L2K(µ)

for each compact K ⊂ Rn. That is, if f ∈ C2c (Rn), then (λI − A)−1f is compactly supported. On the

other hand, elliptic regularity (see [14], Ch. 6) implies, in particular, that (λI − A)−1f ∈ C2c (Rn). Thus

(λI − A)−1C2c (Rn) ⊂ C2

c (Rn), and so C2c (Rn) ⊂ (λI − A)C2

c (Rn).

6.3 Applications to Economics

Inequalities like those we study here have applications in a vast array of different fields. One such field iseconomics. Many economic phenomena, including fluctuations stock prices, machine failure rates, and shiftsin unemployment can be modelled using log-concave probability measures. Gaussian measures are ubiquitousin mathematical modelling of all forms. Other log-concave measures allow for more flexible models whichcan better match the data, or more precise calculations with certain statistics. Among these, some of themost commonly used are the Gamma and Weibull distributions.

Such measures are especially important in finance. In a 1959 paper, M.F.M. Osborne showed that thelogarithms of many stock prices follow a Brownian motion, a stochastic process with independent, Gaussianincrements [28]. Osborne’s discovery sparked widespread interest Gaussian and other log-concave measuresin finance. The ability to model stock price fluctuations mathematically enables researchers and investorsto quantify the risk associated with investing in the market, and thereby to model investment decisions in aformal manner. One of the best known applications of this idea is the Black-Scholes equation for the priceof an option. This is a partial differential equation used to optimize pricing and portfolio allocation [6].

Another application of log-concave measures is to reliability functions in industrial engineering. Considera machine which has some positive probability of breaking down. A reliability function is a measure µ on[0,∞), with the interpretation that the measure of a set E is the probability that the machine breaks downat a a time t ∈ E. Log-concave measures arise naturally in this context. For example, if F (t) = µ[0, t] is thecumulative distribution function of µ, and f is its density, then the quantity

MRL(x) =

∫ ∞t

tf(t)

1− F (x)dt− x

44

is called the mean residual lifetime function of the machine, and represents the expected time before amachine will break down, given that it has survived to time x. Naturally, one would want MRL(x) to bedecreasing in x, and it turns out that this is the case if and only if the measure µ is log-concave. Manysimilar desirable properties of a reliability function are also equivalent to log-concavity [3].

In order to use models like these, one must understand the behavior of the measures on which they rely.Of particular interest is the variance of a random variable—its average distance from its mean—and itsentropy—a measure of the uncertainty in its value. If f is a function on Rn representing a random variable,then its variance with respect to a measure µ is given by∫

Rnf2dµ−

(∫Rnf dµ

)2

and its entropy (a measure of the uncertainty in its value) is given by∫Rn|f | log |f |dµ−

∫Rnf2 dµ log

(∫Rn|∇f |2 dµ

)1/2

.

The Poincare and logarithmic Sobolev inequalities bound these two quantities, respectively, in terms of theL2 norm of the gradient of f .

These two inequalities are of much use in economic models which use Gaussian or other log-concavemeasures. For example, they can be used to estimate the total long-term variance of a stock price aboutits mean, or the expected amount of time before the process modelled by a log-concave reliability functionbreaks down. These inequalities are also used in quantitative estimates for the variance and entropy of prices,cash flows, inflation and other processes which follow a log-concave distribution [12].

Furthermore, the logarithmic Sobolev inequality can be used to prove concentration of measure inequal-ities [22], which bound the probability that a 1-Lipschitz random variable f deviates from its mean by atleast t > 0:

µx ∈ Rn : |f(x)−∫Rnf dµ| ≥ t ≤ φ(t)

for some rapidly decaying function φ. Given a concentration of measure inequality for µ, one can often obtaina similar or even sharper inequality for the n-fold product measure µn. As such, concentration of measureinequalities are indispensable in studying measures on high- or infinite-dimensional spaces. In economics, asin many other fields, large data sets are often represented as vectors in which each observation correspondsto a coordinate. A statistic of interest, e.g. the sample mean, is a function on the many-dimensional setof possible vectors of observations. Concentration of measure inequalities estimate how close the samplestatistic is likely to be to its population counterpart (typically equal to its expected value). Moreover,concentration of measure inequalities can be used to deduce generalized central limit theorems (as in [20]),which are also useful in economics and statistics for estimating large sample probabilities and proving theconvergence of estimators.

The intermediate Beckner inequality and its generalization for q > 2, which we prove in Sections 3 and4 has indirect applications in that it can be used to prove versions of the Poincare and logarithmic Sobolevinequalities, as we do in Subsection 5.1. It also has more direct uses. The quantity∫

Rnf2dµ−

(∫Rn|f |p dµ

)2/p

appearing on the left side of Beckner’s inequality is called the p-variance of f , and represents the dispersionof f about its Lp norm in much the same sense that the usual variance represents the dispersion of f aboutits mean. Beckner’s inequality might be used to estimate the rate of inflation or value of an investmentin terms of its Lp norms, instead of its mean. Such estimates might be useful, for example, if one seeksan intermediate measure of the “average” of such a quantity, between the more commonly used mean andL2 norm. Indeed, Lp norms for general p 6= 2 arise naturally in many economic applications, such as themeasurement of economic welfare [25]. As we show in Section 2, these inequalities possess a tensorisationproperty similar to concentration of measure inequalities, which makes them useful for studying large datasets. Furthermore, as with the logarithmic Sobolev inequality, Beckner’s inequality has been applied to prove

45

concentration of measure inequalities, e.g. in [21]. Generalizations of this inequality like the ones we obtainhere are likely to be used for this purpose in the future.

Sharper and more general inequalities, as well as new proofs of existing inequalities, lead to a betterunderstanding of the measures in question. This in turn leads to more precise estimates and thereby moreaccurate models and better statistical tests. So, all of our results here have potential uses in the economicscenarios discussed above. For example, our extended Beckner inequalities might someday be used to provenew concentration of measure inequalities (and thereby lead to better stochastic models) or to new methodsof estimating of the rates of convergence of stock prices or the lifetimes of industrial processes. Likewise, thesharpened Beckner and Poincare inequalities for log-concave measures of Section 5 could be used to obtainsharper estimates for variance and entropy of such processes.

Discrete analogous of Beckner, Poincare, and log-Sobolev inequalities are also of economic interest. Theseinequalities play a similar role to their continuous counterparts in models which rely on discrete randomvariables, such as Bernoulli trials or discrete random walks. Such models include those for market entry,demand for relatively small quantities of goods, and a plethora of scenarios in game theory [12]. Consequently,the inequality for Bernoulli trials we derive in Subsection 2.2 and the tensorial property of Subsection 2.1,which allows it to be extended to higher dimensions, may have economic applications in their own right aswell.

Furthermore, the Ornstein-Uhlenbeck operator and its associated semigroup, which we study in Subsec-tion 3.1 and 3.2, is often used in stochastic models for interest rates and commodity prices. For example, S.Rampertshammer [29] has developed a model for assessing the value of pairs trading based on the Ornstein-Uhlenbeck operator. Pairs trading is an investment strategy whereby one simultaneously purchases shares inan asset which is below its normal historical price and a short-sells a second asset which is above its normalhistorical price. The idea is that the prices are likely to return to their historical mean values, and, even ifthey don’t, the investor will not lose money if the market as a whole improves or worsens. Rampertshammeruses the Ornstein-Uhlenbeck process (the stochastic process generated by the Ornstein-Uhlenbeck operator)to model the likelihood that two assets will rise or fall in price at the same time, and thereby to determinethe optimal portfolio allocation in a pair trade. Plausibly, the generalization of the Ornstein-Uhlenbeck op-erator and semigroup to general log-concave probability measures which we study in Sections 4 and 5 couldbe used in similar models with different log-concave probability measures in place of the Gaussian measure.Inequalities for the Ornstein-Uhlenbeck semigroup and its generalization to log-concave probability measureslike the ones we derive here could be used to bound the variance, p-variance, and entropy of the prices andestimators involved in these models.

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