Mass transport and variants of the logarithmic Sobolev inequality

Introduction

This work deals with Sobolev inequalities and isoperimetric properties of absolutely continuous probability measures on Euclidean space or Riemannian manifolds. This subject is connected, among other fields, to analysis, probability theory, differential geometry, partial differential equations. In particular such properties are crucial in the study of the concentration of measure phenomenon and of the regularizing effect and trend to equilibrium of evolution equations. Several surveys were devoted to this quickly developing topic, see e.g. , , , , , , .

Besides the Poincaré or spectral gap inequality, the logarithmic Sobolev inequality is the best studied Sobolev property for probability measures. The basic example of a measure satisfying the logarithmic Sobolev inequality

where $F$ is some increasing function, were established for the measures $\mu_{\alpha}$ and their $d$-dimensional analogs in , , with $F(t)=\log(t)^{2-2/\alpha}$ for large $t$. The recent developments can be found in papers , , , . Inequalities of this type imply hyperboundedness of the related semigroups for certain Orlicz norms and under mild conditions isoperimetric inequalities, see . In fact, they are closely related with the Sobolev inequalities for Orlicz norm $\|f-\int f\,d\mu\|^{2}_{\Phi}\leq C\int|\nabla f|^{2}\,d\mu,$ where $\Phi$ is some Orlicz function. Details about the connections and additional semigroups properties appear in , . Let us also mention another important work of Wang devoted to the so-called super-Poincaré inequalities. It establishes a correspondence between $F$-Sobolev and super-Poincaré inequalities, and gives consequences in terms of isoperimetric and Nash inequalities as well as spectral properties of semigroups.

– Modified log-Sobolev inequality with cost function $c$

An isoperimetric inequality is a lower bound of the $\mu$-boundary measure of sets $\mu^{+}(\partial A)$ in terms of their measure $\mu(A)$. Recall that for a Borel measure on a metric space $(X,\rho)$, the boundary measure of a Borel set $A\subset X$ can be defined as the Minkowski content

The isoperimetric function of a probability measure is defined for $t\in(0,1)$ by

One easily checks that the measure $\mu_{\alpha}$, $\alpha\geq 1$ satisfy an isoperimetric inequality of the form $\mathcal{I}_{\mu_{\alpha}}(t)\geq\kappa(\alpha)L_{\alpha}(t)$, where

Indeed the sets of minimal boundary measure for given measure are half-lines for log-concave probability measures on the real line, see e.g. . It is well known that isoperimetric inequalities often imply Sobolev type inequalities. Indeed a natural way to try and unify the above functional inequalities satisfied by $\mu_{\alpha}$ is to derive them from the above isoperimetric inequality. Several papers deal with such results (see , for the log-Sobolev inequality, for $F$-Sobolev inequalities). The most general result in this direction is given in where inequalities of the following form (encompassing $F$-Sobolev and modified log-Sobolev) are deduced from isoperimetric inequalities:

However deriving isoperimetric inequalities is hard. In practice one often proves Sobolev inequalities first and then deduce the isoperimetric inequalities from a method of Ledoux (see , , , ) which applies when the curvature is bounded below to certain Sobolev inequalities with energy term $\int|\nabla f|^{2}d\mu$.

Let us mention a few successful methods to establish Sobolev type inequalities. On the real line, thanks to Hardy type inequalities, it is possible to express simple necessary and sufficient conditions for certain Sobolev inequalities to hold. This technique was first applied to the logarithmic Sobolev inequality by Bobkov and Götze . See e.g. , , for further applications.

for some $x_{0}\in M$, then $\mu$ satisfies a log-Sobolev inequality.

where $\Pi(\mu,\nu)$ is the set of probability measures on $X\times X$ with first marginal $\mu$ and second marginal $\nu$. When an optimal $\pi$ exists it is called “optimal transportation plan”. When an optimal plan is supported by the graph of a function $T:X\to X$, then $T$ pushes forward $\mu$ to $\nu$ and is called optimal transport map. The existence and the structure of such optimal plans and maps is by now quite developed, see and the reference therein.

A second purpose of this work is to develop the mass transport approach in order to get new and old results in a unified manner. To do this we had to introduce several new ways of handling the terms involved in optimal transport. Let us mention that transportation is intimately linked with the entropy functional, and therefore naturally yields modified log-Sobolev inequalities. Among other things this paper shows how to recover $F$-Sobolev inequalities, and therefore isoperimetric inequalities.

Next we describe the structure of the paper and highlight some of the main results and techniques. Section 2 is devoted to tightening techniques. A functional inequality is tight when it becomes an equality for constant functions. It is called defective otherwise. Tightness it crucial in applications of Sobolev inequalities to concentration or to hypercontractivity properties. A classical method of Rothaus allows to transform a defective log-Sobolev inequality into a tight one, by means of a Poincaré inequality. It does not apply to the modified inequalities. Theorem 2.4 develops a new simple method for tightening general “modified $F$-Sobolev inequalities” (3). This result encompasses and simplifies several existing tightness lemma for $F$-Sobolev inequalities. We also collect known facts about how to derive global Poincaré inequalities from local ones.

Section 3 gives a short account of the consequences of isoperimetric inequalities in terms of Sobolev type inequalities, with emphasis on the measures satisfying the same isoperimetric inequalities as the model measures $\mu_{\alpha}$. To do this we combine the main result of with our new tightening results.

In Section 4 we introduce a new variant of the log-Sobolev inequality, which plays a crucial role in the paper. For $\tau\in(0,1]$ we say that $\mu$ satisfies $I(\tau)$ if there exists numbers $B,C$ such that every smooth function $f$ verifies

Further results of the paper show that for $\alpha\in(1,2)$, $\mu_{\alpha}$ satisfies $I(\tau)$ for $\tau=2-2/\alpha$. The main result of the section is that $I(\tau)$ implies appropriate $F$-Sobolev inequalities and modified log-Sobolev inequalities.

Under additional integrability assumptions we prove corresponding modified log-Sobolev inequalities and Inequalities $I(\tau)$. The main part of the work consists in estimating the last term involving the convexity defect. This can be done when

$\mathcal{D}_{V}(x,y)$ has an upper bound of the type $c_{0}(x-y)$ and $\mu$ satisfied certain integrability assumption,

$V$ is controlled by a function of the type $G(\nabla V)$ and $\Delta V$ grows slower than $|\nabla V|^{2}$,

$V$ satisfies V(x)\leq C_{1}\bigl{<}\nabla V(x),x\bigr{>}+C_{2} and certain integrability assumption on $\nabla V$,

$V$ is obtained by a perturbation of some convex potential.

We recover and extend the Euclidean version of Wang’s result. A simple new result asserts that when $\mathcal{D}_{V}(x,y)$ is upper bounded by $\lambda c(x-y)$ where $c$ is a strictly convex cost and $\lambda\geq 0$ then $\mu$ satisfies a defective modified log-Sobolev inequality with cost $c$ provided there exists $\varepsilon>0$ such that

In Section 7 we generalize some results obtained in this paper to Riemannian manifolds. We apply the manifold version of (4) obtained by Cordero-Erausquin, McCann and Schmuckenschläger in for quadratic transportation cost. We consider a smooth complete connected Riemannian manifold without boundary $M$ with a probability measure $\mu=e^{-V}\,d\mbox{\rm vol}$. In particular, we establish Sobolev type inequalities and isoperimetric inequalities for $\mu$ such that $D^{2}V+\mbox{\rm Ric}\geq 0$ and $e^{\varepsilon\rho^{\alpha}(x_{0},x)}\in L^{1}(\mu)$, where $\rho$ is the Riemannian distance, $\varepsilon>0$ and $x_{0}$ is an arbitrary point on $M$ and $\alpha\in(1,2]$.

Most results of the paper apply to measures with the tail behavior of the order $e^{-|x|^{\alpha}}$ with $1<\alpha\leq 2$ (apart from Section 3, Subsection 5.4 (Corollary 5.14), Theorem 5.16 and Theorem 7.2). Nevertheless, some of our results for $\alpha\leq 2$ can be adapted to $\alpha\geq 2$.

List of the main objects considered in this paper

Duality: If $\alpha>1$ we denote by $\alpha^{*}$ the number such that $\frac{1}{\alpha}+\frac{1}{\alpha^{*}}=1$. This is consistent with the definition of the convex conjugate (or Fenchel-Legendre transform) of a function $c^{*}(x)=\sup_{y}\langle x,y\rangle-c(y)$, since the conjugate of $x\mapsto|x|^{\alpha}/\alpha$ is $x\mapsto|x|^{\alpha^{*}}/\alpha^{*}$.

Note that $c_{\alpha}^{*}=c_{\alpha^{*}}$. For $\alpha\in(1,2]$, up to multiplicative constants $c_{\alpha}(t)\approx\min(t^{2},|t|^{\alpha})$ whereas for $\alpha>2$, $c_{\alpha}(t)\approx\max(t^{2},|t|^{\alpha})$.

Special generalized entropies: $F_{\tau}(t)=\log^{\tau}(1+t)-\log^{\tau}(2)$, $t\geq 0$.

Modified log-Sobolev inequality (MLSI) for a cost function $c$:

More general functions of $\frac{\nabla f}{f}$ are sometimes considered.

The $q$-$F$-Sobolev inequality (qFSI) is defined with the same formula, replacing $f^{2}$ by $|f|^{q}$ and $|\nabla f|^{2}$ by $|\nabla f|^{q}$. When $F=\log$ this is the classical log-Sobolev inequality (LSI).

For the $q$-Poincaré inequality (qP), write $q$ instead of $2$.

Acknowledgements: We would like to thanks Dominique Bakry, Jérome Bertrand, Michel Ledoux, Assaf Naor and Zhongmin Qian for useful discussions and for communicating several references to us. The second author would like to express his gratitude to the research team of the Laboratoire de Statistique et Probabilités from the Université Paul Sabatier in Toulouse, where this work was partially done.

How to tighten the inequalities

The results of this section apply in rather general settings. For simplicity we assume that $\mu$ is a probability measure on a Riemannian manifold, and is absolutely continuous with respect to the volume measure.

The following result of Bobkov and Zegarlinski is an extension to $q\neq 2$ of an argument going back to Rothaus .

Let $q\in(1,2]$. Assume that a probability measure $\mu$ satisfies a defective $q$-log-Sobolev inequality as well as a $q$-Poincaré inequality:

Then it automatically satisfies a tight $q$-logarithmic Sobolev inequality

This is a simple consequence from the following inequality (see for its proof)

The change of functions $f^{q}=g^{2}$ turns the $q$-log-Sobolev inequality into a modified-log Sobolev inequality with function $c^{*}(t)$ proportional to $|t|^{q}$.

2. Modified energies

$(i)$ the function $\Phi(x)=xF(x)$ extends to a $\mathcal{C}^{2}$-function on $[0,A^{2}]$,

$(ii)$ there exists a constant $d\geq 0$ such that for all $x\in(0,A^{2}]$, $F(x)\leq d(x-1)$,

Assume that a probability measure $\mu$ satisfies a defective modified $F$-Sobolev inequality: for all $f$,

If $\mu$ also satisfies a Poincaré inequality, then there exists a constant $C$ such that for every $f$,

Let $f$ be a function such that $\int f^{2}d\mu=1$. Let $A>1$. It holds

If $F$ is as in Theorem 2.4 above, then there exists a constant $\gamma$ depending on $A$ and $F$ only such that

Since $\mathbf{Var}_{\mu}(|f|)\leq\mathbf{Var}_{\mu}(f)$ we may assume that $f\geq 0$. In this case, when $f\geq A$

Next we establish Inequality (6) when $F$ satisfies Hypothesis $(i)$ of Theorem 2.4. Let $f\geq 0$ with $\mu(f^{2})=1$. Noting that $\Phi(1)=0$ and $\Phi^{\prime}(1)\geq 0$, we have by Taylor’s formula

where the latter inequality follows from the bound \int\big{(}f-\mu(f^{2})^{\frac{1}{2}}\big{)}^{2}\,d\mu\leq 2\mathbf{Var}_{\mu}(f), which is readily checked by expanding the square.

Finally, if the function $F$ satisfies Hypothesis $(ii)$ of Theorem 2.4 we simply observe that $\int f^{2}F(f^{2})\,d\mu=\int f^{2}F(f^{2})-d(f^{2}-1)\,d\mu$ and note that on $\{f^{2}<A^{2}\}$,

Consider functions $F$ and $H$ as in Theorem 2.4 and set $m=-\inf_{t\in(0,1]}tF(t)\geq 0$. Let $\mu$ be a probability measure $\mu$ such that for all $f$,

Then for $\eta>0$ and all functions $f$ with $\mu(f^{2})=1$ it holds

It is enough to work with non-negative functions. The change of function $f^{2}=g^{q}$ yields

Since for $t>0$, $tF(t)\geq tF_{+}(t)-m$, we get

Given a non-negative function $\varphi$ with $\mu(\varphi^{q})=1$, we apply the latter inequality to $g=\theta(\varphi)$ where for $x\geq 0$

Obviously for $x\geq 0$, $\theta(x)\leq x\mathbf{1}_{x\geq 1+\eta}\leq x$. Hence

This estimate, together with the fact that $\varphi=g$ when $\gamma\geq 1+2\eta$, yields

Finally, since $\nabla g=0$ when $\varphi<1+\eta$, and $|\nabla g|\leq 2|\nabla\varphi|$ when $\varphi\geq 1+\eta$

where the last inequality follows from $g\leq\varphi$ and $x\mapsto x^{q}H(1/x)$ non-decreasing on $(0,+\infty)$. From the above three estimates, Inequality (7) gives for $\varphi$ with $\mu(\varphi^{q})=1$

The claim follows from the change of functions $f^{2}=\varphi^{q}$. ∎

By homogeneity we may assume that $\int f^{2}\,d\mu=1$. Let $\eta>0$ such that $A^{2}=(1+2\eta)^{q}$. Combining the previous two lemmas

where we have used again that $H(x)/x^{q}$ is non-increasing. Finally we apply Poincaré’s inequality and the bound $H(x)\geq cx^{2}$,

3. Tightening for free: local inequalities

Local inequalities are easy to derive for locally bounded potentials by standard perturbation techniques. In many cases they allow to tighten defective inequalities. They are defined below.

Let $q\geq 1$ and $\mu$ be a probability measure. One says that $\mu$ satisfies a local $q$-Poincaré inequality if for every $\eta\in(0,1)$, there exists a set $A$ with $\mu(A)\geq\eta$ such that the measure $\mu_{A}=\frac{\mathbf{1}_{A}}{\mu(A)}.\mu$ satisfies a $q$-Poincaré inequality, meaning that there exists $C_{A}<+\infty$ such that for every smooth $f$,

When $q=2$ we just say that $\mu$ verifies a local Poincaré inequality.

The goal of this paragraph is to show how to extend isoperimetric inequalities when they are known only for sets of small or large measure. The argument is based on local Cheeger’s inequalities (which are equivalent to local $1$-Poincaré inequalities). One gets the following convenient result.

Then there exists a constant $c$ such that arbitrary sets satisfy $\mu^{+}(\partial A)\geq c\,I(\min(\mu(A),\mu(A^{c}))).$

The proof is based on the following easy fact:

First recall that a probability measure $\nu$ satisfies Cheeger’s isoperimetric inequality with constant $c$ means that for every set $c\,\nu^{+}(\partial A)\geq\min(\nu(A),\nu(A^{c}))$. This is equivalent to the functional inequality

Using the variational expression of the median

one easily checks that the above inequality for $\nu$ can be transfered to any perturbed probability $\eta=e^{g}\cdot\nu$ as

Since the uniform probability measure on $B_{r}$ satisfies Cheeger’s isoperimetric inequality, so does the measure $\mu_{B_{r}}$ (indeed $V$ is bounded from above and below on $B_{r}$). ∎

Consider an arbitrary set $A$ with $\mu(A)\in[\varepsilon,1-\varepsilon]$. It is enough to find a universal constant $C>0$ for which $\mu^{+}(\partial A)\geq C$. To do this, choose $R$ such that $\mu(B_{R})=1-\varepsilon/2$. Plainly $\mu_{B_{R}}(A)\leq(1-\varepsilon)/(1-\varepsilon/2)<1$ and

By the previous lemma, $\mu_{B_{R}}$ satisfies Cheeger’s isoperimetric inequality. Hence there is a constant $K>0$ (depending only on $\varepsilon$ and $\mu$) such that $\mu_{B_{R}}^{+}(\partial A)\geq K$. Finally $\mu^{+}(\partial A)\geq\mu(B_{R})\mu_{B_{R}}^{+}(\partial A)\geq(1-\varepsilon/2)K.$ ∎

3.2. Sobolev inequalities

Next we deal with defective $F$-Sobolev inequalities. In the case $q=2$ the following result is a consequence of several existing results in the literature (Röckner-Wang show that a local Poincaré inequality implies a weak Poincaré inequality, Wang shows that a weak Poincaré inequality and a specific super Poincaré inequality implies a Poincaré inequality, and that defective $F$-Sobolev inequalities imply super-Poincaré inequalities. See also Aida .) However these results do not provide explicit constants. The next proposition gives a concrete bound with a straightforward proof.

Then it satisfies the following $q$-Poincaré inequality: for all smooth $f$

with K=\kappa\left(\max\Big{(}\big{(}1+(2\cdot 3^{1/(q-1)})^{-1}\big{)}^{-1},1-\big{(}4F_{+}^{-1}(4(D+M))\big{)}^{-1}\Big{)}\right), where $\kappa(r)=\inf\{c_{A};\;\mu(A)\geq r\}$ for $r\in(0,1)$, and $F_{+}^{-1}$ is the generalized left inverse of $F_{+}=\max(F,0)$.

Since $F\geq F_{+}-M$ the hypothesis implies, for all $f$

Without loss of generality we consider a function $f$ with $\mu(f)=0$ and $\mu(|f|^{q})=1$. Given a set $A$ to be specified later, we write

We bound the first term by means of the local $q$-Poincaré inequality, noting that $\int f\,d\mu=0$ implies that $\int f\mathbf{1}_{A}\,d\mu=-\int f\mathbf{1}_{A^{c}}\,d\mu$. By the convexity relation $|x+y|^{q}\leq 2^{q-1}(|x|^{q}+|y|^{q})$, we get for any probability measure

The local $q$-Poincaré inequality hence guarantees

Using both estimates and recalling that $\int|f|^{q}d\mu=1$ gives

To conclude we choose $\varepsilon=1/(4(D+M))$, and $A$ large enough to ensure \big{(}1-\mu(A)\big{)}F_{+}^{-1}\Big{(}\frac{1}{\varepsilon}\Big{)}\leq\frac{1}{4} and $\left(2\frac{1-\mu(A)}{\mu(A)}\right)^{q-1}\leq\frac{1}{3}$. Using again $\int|f|^{q}d\mu=1$ we obtain for $f$ with $\int f\,d\mu=0$

provided \mu(A)\geq 1-1/\big{(}4F_{+}^{-1}(4(D+M))\big{)} and \mu(A)\geq 1/\big{(}1+(2\cdot 3^{1/(q-1)})^{-1}\big{)}. Optimizing on such sets yields the claimed result. ∎

When $q=2$ the estimates can be improved since (11) can be replaced by the variance identity. Also when $F=\log$, the duality of entropy may be used to get a more precise bound

The translation invariance of the energy term was implicitly but crucially used. If $\mu$ satisfies a local Poincaré inequality and a defective modified log-Sobolev inequality with function $H(x)\geq cx^{2}$ then the above method yields $\int f^{2}d\mu\leq D\int f^{2}H(|\nabla f|/f)\,d\mu$ for functions $f$ with $\mu(f)=0$.

Here is a direct consequence of Propositions 2.10 and 2.1:

Let $q\in(1,2]$. If a probability measure $\mu$ satisfies a defective $q$-log-Sobolev inequality as well as a local $q$-Poincaré inequality, then it satisfies a tight $q$-log-Sobolev inequality.

The next classical result yields local Poincaré inequalities under mild conditions.

Let $(M,g)$ be a connected smooth and complete Riemannian manifold. Let $d\mu(x)=e^{-V(x)}dv(x)$ be a Borel probability measure on $M$ (here $v$ is the Riemannian volume). If $V$ is locally bounded, then $\mu$ enjoys a local Poincaré inequality.

Functional inequalities via isoperimetry

Then there exist $C,B>0$ such that for every locally Lipschitz $f$

If in addition, there exists $q>0$ such that $x\to c^{*}(x)/x^{q}$ is non-increasing, then the inequality can be made tight in the following way: the last term $\int f^{2}d\mu$ can be replaced by $\mbox{\rm Var}_{\mu}(f)$.

Assume that the probability measure $\mu$ verifies

If $\alpha\in(1,2]$ then there exists $C$ such that for all $f$

If $\alpha\geq 2$ then there exists $C$ such that all $f$ verify

Applying Theorem 3.1 for $F=\log$ and $c(x)=|x|^{\alpha}$ shows that for all $f$

When $\alpha\leq 2$, there exists a constant $\kappa$ such that $|x|^{\alpha^{*}}\leq\kappa c_{\alpha^{*}}(x)$. Applying this bound yields an inequality which can be tightened thanks to Theorem 2.4 and the Poincaré inequality again. When $\alpha\geq 2$, $\alpha^{*}\in(1,2]$, making the change of function $f^{2}=g^{\alpha^{*}}$ in the above inequality yields a defective $\alpha^{*}$-Sobolev inequality. Cheeger’s isoperimetric inequality also implies that $\mu$ satisfies $\alpha^{*}$-Poincaré inequality (see, for example, ). By Proposition 2.1 this is enough to tighten the $\alpha^{*}$-log-Sobolev inequality. ∎

One can also establish functional inequalities interpolating between the above $F$-Sobolev inequalities and modified log-Sobolev inequalities. In particular the above theorem implies the next result, which was proved in , with the restriction $1<\alpha\leq 2$: if $\mu$ satisfies (13) for some $\alpha>1$ and if $\tau\alpha^{*}\geq 2$ then there exists $C^{\prime}$ such that for all $f$

The techniques of allow to show that every measure $\mu$ satisfying the isoperimetric inequality (13) for some $\alpha\in(1,2]$ also verifies the inequality $I(\tau)$ introduced in the next section, when $\tau=2/\alpha^{*}$.

Inequality I​(τ)𝐼𝜏I(\tau)

In this section we introduce a new variant of the logarithmic Sobolev inequality. For $\tau\in(0,1]$ we say that a measure $\mu$ satisfies Inequality $I(\tau)$ if for some constants $B,C$ and all $f$

We show next that any probability measure satisfying $I(\tau)$ and a local Poincaré inequality, automatically satisfies an $F_{\tau}$-Sobolev inequality as well as the corresponding modified log-Sobolev inequality. Recall that for $F_{\tau}(t)=\log^{\tau}(1+t)-\log^{\tau}(2)$ and that for $\beta\geq 2$, $c_{\beta}(t)$ is comparable to $\max(t^{2},t^{\beta})$.

Let $\tau\in(0,1]$ and $\alpha\in(1,2]$ be related by $\tau=\frac{2(\alpha-1)}{\alpha}$. Let $\mu$ be a probability measure satisfying Inequality $I(\tau)$. Then there exist constants $B_{i},C_{i}$ such that for all $f$

If $\mu$ also verifies a local Poincaré inequality, then (14) and (15) can be tightened (i.e. one can take $B_{i}=0$).

Let $\tau\in(0,1)$. First we deduce (15) from $\mathcal{I}(\tau)$. Assume as we may that $f$ is non-negative with $\int f^{2}d\mu=1$. Our task is to bound from above the quantity $\int|\nabla f|^{2}\log^{1-\tau}(e+f^{2})\,d\mu$. Since $\tau\in(0,1)$, we may apply Young’s inequality in the form $xy\leq\tau x^{1/\tau}+(1-\tau)y^{1/(1-\tau)}$ and the easy inequality $x\log(e+x)\leq x\log x+e$:

Taking integrals and using the fact that up to constants $c_{\alpha^{*}}(t)$ is comparable to $\max(t^{2},|t|^{2/\tau})$, we obtain that for some constant $B_{0}$ depending on $\alpha$

If we choose $\varepsilon>0$ small enough to have $C\varepsilon^{2}(1-\tau)\leq 1/2$, the above inequality can be combined with Inequality $\mathcal{I}(\tau)$ to obtain the defective modified log-Sobolev inequality (15).

In order to show that $\mathcal{I}(\tau)$ implies a defective $F_{\tau}$-Sobolev inequality, we consider

Let us fix a positive Lipschitz function $g$. We denote by $L$ the Luxembourg norm of $g$ related to $\Phi$:

Thus by definition \displaystyle\int\Phi\Bigl{(}\frac{g}{L}\Bigr{)}\,d\mu=1. Since $\Phi(x)\leq x^{2}$, one has

Set $f=\varphi(g/L):=\sqrt{\Phi(g/L)}$. Note that $\int f^{2}\,d\mu=1$. Thus by hypothesis,

The left hand side of this inequality equals to

It is not hard to check that there exists a constant $\kappa\geq 0$ depending on $\tau\in(0,1)$ such that for all $x\geq 0$,

for instance the existence of a finite $\kappa$ for $x\in$ is obvious by continuity, whereas for $x\geq 4$ one may use $\sqrt{x}\geq\log(e+x)$ and the bound $x\log x\geq x\log(e+x)-e$. Hence there are constants $\kappa_{1},\kappa_{2}>0$ depending on $\tau$ such that

Now let us estimate the gradient term in (16). Recall that $f=\varphi(g/L)$, where

Elementary estimates show that there exists $M>0$ such that for every $x\geq 0$

Applying this bound together with the estimate $f^{2}\leq\frac{g^{2}}{L^{2}}$ we obtain

Combining the latter inequality with (16) and (17) we get that

The claim follows from the estimate $L^{2}\leq\int g^{2}\,d\mu$ and monotonicity of $F_{\tau}$.

If $\mu$ satisfies a local Poincaré inequality, then the defective $F_{\tau}$-Sobolev inequality is enough to apply Proposition 2.10. Hence $\mu$ satisfies a Poincaré inequality. By Theorem 2.4, this is enough to tighten both (14) and (15). ∎

Optimal transportation and functional inequalities.

The optimal transport theory is widely represented in surveys and monographs and the reader can consult for definitions and main results.

The application of the optimal transportation techniques to functional inequalities is based on the following remarkable estimate called ”above-tangent lemma”. It has numerous applications to functional inequalities (especially Sobolev-type inequalities). From a more general point of view this inequality comes from convexity of some special functional (the so-called ”displacement convexity”). This notion has been introduced by McCann in . For more details about displacement convexity, above-tangent inequalities and applications, see .

Without loss of generality one can assume that $g$ and $h$ are smooth and bounded. By the change of variables formula

Integrating with respect to $g\cdot\mu$ and changing variables, one gets

The claim follows from the fact that the last integrand is non-positive. This is due to the structure of the optimal transport $T$ which ensures that pointwise, $DT$ can be diagonalized, with a non-negative spectrum. ∎

This lemma tells that the convexity type information about the potential $V$, i.e. an estimate of $\mathcal{D}_{V}$ in

passes to the entropy functional on the space of probability measures

The second term of the right-hand side is linear in the displacement $\theta(x)=T(x)-x$. It can be thought of as the linear part in the tangent approximation of the entropy functional. We will call it the ”linear term”.

Our aim is to derive modified LSI inequalities using the ”above-tangent” lemma for $g=f^{2}$ and $h=1$:

We will show that the ”linear” term in the above inequality can be estimated by assuming the integrability of $\exp(\varepsilon|x|^{p})$ for some $\varepsilon>0,p>1$. Estimating the term involving $\mathcal{D}_{V}$ is more difficult and can be done by different methods, under various assumptions. When $\mathcal{D}_{V}(x,y)\leq c(y-x)$, the integral involving $\mathcal{D}_{V}$ can be upper-bounded by the transportation cost from $f^{2}\cdot\mu$ to $\mu$ when the unit cost is $c$. This argument was already used many times. We will see in the next subsections that estimates of the form $\mathcal{D}_{V}(x,y)\leq\varphi(x)+\psi(y)$ are even more convenient.

2. Estimation of the linear term

The classical estimate is recalled in the next two lemmas

Let $c$ be a convex cost function. Assume that $T$ pushes forward $f^{2}\cdot\mu$ to $\mu$, and is optimal for the cost $c$. Then for every $\alpha>0$,

We simply apply Young’s inequality $\langle u,v\rangle\leq\alpha c(u)+\alpha c^{*}(v/\alpha)$ to $u=T(x)-x$ and $v=-2\nabla f(x)/f(x)$, and integrate with respect to $f^{2}.\mu$. The conclusion comes from \int c\big{(}T(x)-x\big{)}f(x)^{2}\,d\mu(x)=W_{c}(f^{2}\cdot\mu,\mu). ∎

It is well known that the transportation cost $W_{c}$ in the above lemma can be estimated in terms of the entropy of $f^{2}$ if $\mu$ has strong integrability properties. This is recalled now:

Let $\mu,\,f\cdot\mu$ and $g\cdot\mu$ be probability measures and $c$ a cost function. Then for all $\alpha>0$,

where $\mu_{A}$ is the conditional measure $\mu_{A}=\frac{\mu|_{A}}{\mu(A)}$.

We bound the transportation cost from above by using the product coupling:

The next result provides a new way to deal with the linear term in the above tangent inequality, in relation with Inequality $I(\tau)$. It is quite flexible, as it does not require the transport to be optimal.

Let $\alpha\in(1,2],\delta>0$ and let $\mu$ be a probability measure such that

Let $T$ be a map which pushes forward a probability measure $f^{2}\cdot\mu$ to $\mu$. Then for all $\varepsilon>0$ there exist $C_{1},C_{2}>0$ depending on $\delta,\alpha$ and the above integral such that

where we have set $\tau=\frac{2\alpha-2}{\alpha}=\frac{2}{\alpha^{*}}\in(0,1]$.

Note that 2\int\bigl{<}\nabla f(x),x-T(x)\bigr{>}f(x)\,d\mu(x) is not bigger than

for arbitrary $\varepsilon$. Since $|x-T(x)|^{2}\leq 2|x|^{2}+2|T(x)|^{2}$ and $1-\tau\geq 0$, we get

We apply the inequality $ab\leq a\varphi(a)+b\varphi^{-1}(b)$, $a,b\geq 0$ for the function $\varphi(t)=e^{\frac{\delta}{2}t^{\alpha/2}}-1$, and get

Using $1+f^{2}\leq e+f^{2}$ with the relation $\frac{2}{\alpha}+\tau-1=1$ and Assumption (20), we get constants $\kappa_{i}$ depending on $\alpha,\delta,\mu$ but not on $f$ such that the above quantity is at most

The last term in (21) is controlled by the change of variable formula and the integrability assumption again:

3. Basic facts about convexity defect

In the next two subsections, we will work with potentials $V$ such that there exists a function $c$ such that

In other words, the defect of convexity satisfies $\mathcal{D}_{V}(x,y)\leq c(y-x)$. When $c$ is a negative function, $V$ is uniformly convex. We will focus on the case when $c$ is positive. In this case we say that $V$ is weakly convex. This subsection provides concrete examples of such potentials.

The first example is as follows: if $V$ is $\mathcal{C}^{2}$, the condition $D^{2}V\geq-\lambda$ for some $\lambda\geq 0$, is equivalent to condition (22) with $c(u)=\frac{\lambda}{2}|u|^{2}.$ It is also equivalent to the fact that the function $V(x)+\frac{\lambda}{2}|x|^{2}$ is convex. Next we present other possible conditions, extending the latter.

Choose $c(x)=\|x\|^{p}$ where $\|\cdot\|$ is a strictly convex norm and $p>1$. Note that $p>2$ is not very interesting in our case since for a smooth $V$

dominates $-\lambda\|u\|^{p}$ only when $\lambda=0$ and $V$ is convex since $-|u|^{p}>>-|u|^{2}$ for small $u$. The case $p\in(1,2)$ contains new examples. We need some preparation.

For $p\in$, a norm on a vector space $X$ has a modulus of smoothness of power-type $p$ or for short is $p$-smooth with constant $S$ if for all $x,y\in X$ it satisfies

As shown in , this formulation is equivalent to the more standard definition given in . We need the following classical fact, see Lemma 4.1 in

Let $(X\|\cdot\|)$ be a Banach space with a $p$-smooth norm with constant $S$. Let $X$ be a random vector with values in $X$ such that $E\|Z\|^{p}<+\infty$. Then

Applying the above lemma when the law of $Z$ is $(1-t)\delta_{x}+t\delta_{y}$ for $t\in(0,1]$, $x,y\in X$ yields

Letting $t$ to zero yields for almost every $x$ and for all $u$

In this form the meaning of $p$-smoothness is very clear: the application $\|\cdot\|^{p}$ is not too much above its tangent map and the distance is measured by $\|u\|^{p}$. Therefore the application $-\|\cdot\|^{p}$ is not too much below its tangent. Hence we obtain

Actually when $u>-1$ it is even true that $|1+u|^{p}\leq 1+pu+|u|^{p}$. To see this we start with the case $u\geq 0$ and consider the function $\varphi$ defined on $[0,+\infty)$ by $\varphi(u)=1+pu+u^{p}-(1+u)^{p}$. Clearly $\varphi(0)=\varphi^{\prime}(0)=0$. Moreover $\varphi$ is convex since \varphi^{\prime\prime}(u)=p(p-1)\big{(}u^{p-2}-(1+u)^{p-2}\big{)}\geq 0, using $p-2\leq 0$. Hence $\varphi$ is nonnegative.

Next we prove the stronger inequality when $u\in$. Setting $t=-u$ we have to show that the function $\psi$ defined on $$by$\psi(t)=1-pt+t^{p}-(1-t)^{p}$is nonnegative. This is clear since$\psi(0)=0$and$\psi^{\prime}(t)=p\big{(}u^{p-1}+(1-u)^{p-1}-1\big{)}\geq 0$since$u,1-u$and$p-1$are in$$(so$u^{p-1}\geq u$,$(1-u)^{p-1}\geq 1-u$).

Finally we prove (24) when $u=-t\in(-\infty,-1]$ by studying the function $\xi$ defined on $[1,+\infty)$ by $\xi(t)=1-pt+2^{2-p}t^{p}-(t-1)^{p}$. First $\xi(1)\geq 0$ and we shall prove that $\xi$ is nondecreasing. To see this we compute

The latter quantity is nonpositive on $$and nonnegative on$[2,+\infty)$. Therefore$\xi^{\prime}$achieves its minimum at$t=2$where$\xi^{\prime}(2)=0$. So$\xi$ is nondecreasing as claimed. The proof is complete. ∎

Let $p\in(1,2]$. Assume that there exists $\lambda\geq 0$ such that the function $x\mapsto V(x)+\lambda\|x\|_{p}^{p}$ is convex. Then for almost every $x$ and every $u$,

In other words $\mathcal{D}_{V}(x,y)\leq\lambda 2^{2-p}\|y-x\|_{p}^{p}.$

4. Isoperimetric inequalities for weakly convex potentials

Let $T$ be the optimal map pushing forward \big{(}f/\mu(f)\big{)}.\mu to $\mu_{B_{r}}$ for the cost function $c(y-x)$. If we apply Lemma 5.1, we get after multiplication by $\int f\,d\mu$

Then there exist $D>0$ and $a_{0}>0$ such that the following is true:

for any Borel set $A$ such that a:=\min\big{(}\mu(A),\mu(A^{c})\big{)} verifies $a\leq a_{0}$, it holds

where $r$ is chosen so that $a=\mu(B_{r}^{c})$.

Let us assume that $\mu(A)\leq 1/2$ (the case $\mu(A^{c})<1/2$ follows from the same method since $A$ and its complement play symmetric roles in our estimates). Hence by hypothesis $a=\mu(A)=\mu(B_{r}^{c})$. We apply Lemma 5.9. Our task is to bound the transportation costs involved in its conclusion. Set $\eta=1/(3+\varepsilon)$ and $K(\eta)=\eta\log\left(\int\exp(c(y-x)/\eta)\,d\mu(x)d\mu(y)\right)$. It is finite by hypothesis. Lemma 5.3 gives

Applying the corresponding bound for $\mu(A^{c})W_{c}(\mu_{A^{c}},\mu_{B_{r}})$ would give a term of order $1-a$ which is too big. To avoid this problem, we consider another coupling. Let $S$ be the $c$-optimal map pushing forward $\mu_{A^{c}\cap B_{r}^{c}}$ to $\mu_{A\cap B_{r}}$. We define the map $T:A^{c}\to B_{r}$ by $T(x)=x$ when $x\in A^{c}\cap B_{r}$ and by $T(x)=S(x)$ for $x\in A^{c}\cap B_{r}^{c}$. One readily checks that $T$ pushes $\mu_{A^{c}}$ forward to $\mu_{B_{r}}$ (this uses the relation $\mu(A^{c})=\mu(B_{r})$ and its consequence $\mu(A^{c}\cap B^{c}_{r})=\mu(A\cap B_{r})$). Hence

Apply Lemma 5.3 to the latter transportation cost yields

Note that $-x\log x$ is increasing for $x\leq 1/e$. Thus $\mu(A)=a\leq 1/e$ ensures that

Combining Lemma 5.9 with (25), (26) and the relation $\eta=1/(3+\varepsilon)$ yields

When $a$ is small enough, $2K(\eta)a+\eta a\log\frac{1}{1-a}$ is less than half of the left hand side, hence $\frac{\varepsilon}{3+\varepsilon}a\log\frac{1}{a}\leq 4r\mu^{+}(\partial A)$. ∎

then there exist $D,a_{0}>$ such that every Borel set $A$ such that a:=\min\big{(}\mu(A),\mu(A^{c})\big{)}\leq a_{0} verifies

Proposition 5.10 gives $Dr\mu^{+}(\partial A)\geq a\log\frac{1}{a}$ when $a\leq a_{0}$ is such that $a=\mu(B_{r}^{c})$. Using Markov’s inequality in exponential form gives

hence $r\leq\psi^{-1}(\log\frac{K}{a}).$ ∎

The restriction on the value of $a$ may be weakened or removed by making more precise calculations in concrete situations, or in general situation by applying Proposition 2.8.

Assume that $c$ is not the zero function. Jensen’s inequality yields

Next we give an application to potentials with Hessian bounded from below and with a strong integrability property. A similar statement holds when $\int e^{\psi(|x|)}d\mu(x)<\infty$ for an increasing $\psi$ with $\lim_{+\infty}\frac{\psi(t)}{t^{2}}=+\infty$.

There there exists $\kappa>0$ such that the isoperimetric profile of $\mu$ satisfies

By hypothesis $\mathcal{D}_{V}(x,y)\leq\frac{K}{2}|x-y|^{2}$. We need to check that \int\exp\big{(}\beta|x-y|^{2}\big{)}d\mu(x)d\mu(y) is finite for some $\beta>3K/2$. However this is true for every $\beta$. Indeed for every $\delta>0$ there is a constant such that for all $x$, $|x|^{2}\leq\delta|x|^{\alpha}+N(\alpha,\delta)$ (e.g. using Young’s inequality $xy\leq x^{p}/p+y^{p^{*}}/p^{*}$ for $p=\alpha/2>1$). Hence

is finite by choosing $\delta<\varepsilon/(2\beta)$. Therefore we may apply the previous corollary with $\psi(t)=t^{\alpha}$. This gives the claimed isoperimetric inequalities for small values of $t$. Since $V$ is locally bounded we apply Proposition 2.8 to extend the result to all $t\in(0,1)$. ∎

Modified log-Sobolev inequalities are established in the next subsection under the weaker integrability assumption $\exp((1+\varepsilon)c(x-y))\in L^{1}(\mu\otimes\mu)$ (see Theorem 5.16). Thus, one may ask whether the results of this subsection remain valid when $3+\varepsilon$ is replaced by $1+\varepsilon$. This is indeed the case for the statement of Remark 5.13 when $c(x)=\frac{K}{2}|x|^{2}$. If $D^{2}V\geq K$ and $\exp((\varepsilon+K/2)|x-y|^{2})\in L_{1}(\mu\otimes\mu)$, then Wang’s result yields a logarithmic Sobolev inequality. But when the Hessian is bounded from below, Ledoux showed that an appropriate (Gaussian) isoperimetric inequality follows. Apart from the factor $3+\varepsilon$, another feature of the method of this subsection is not completely satisfactory: it does not seem to extend to the Riemannian setting.

5. Modified LSI via weak convexity and integrability

In this section we derive log-Sobolev inequalities when the potential $V$ satisfies $\mathcal{D}_{V}(x,y)\leq c_{0}(y-x)$, or equivalently

If $c_{0}$ is negative then $V$ is strictly uniformly convex and log-Sobolev inequalities have been proved (Bakry-Emery for quadratic $c_{0}$, Bobkov and Ledoux in general). When $c_{0}$ is positive, $V$ is not convex anymore and an additional integrability assumption is needed to balance the convexity defect.

If there exists $\varepsilon>0$ such that

then there exists $K_{1},K_{2},K_{3}\geq 0$ such that every nonnegative smooth function $f$ verifies,

Let $\eta_{1},\eta_{2}\in(0,1)$. Assume that $\int f^{2}\,d\mu=1$. Let $T(x)=x+\theta(x)$ be the optimal transport from $f^{2}\cdot\mu$ to $\mu$ for the unit cost $c(x-y)$. Applying Lemma 5.1 to $g=f^{2}$ and $h=1$ and Young’s inequality as in Lemma 5.2 gives

where the last inequality comes from Lemma 5.3 for $\alpha=(1-\eta_{2})/(\lambda+\eta_{1})$. Rearranging the entropy terms and tuning $\eta_{1},\eta_{2}$ to ensure that $\frac{\lambda+\eta_{1}}{1-\eta_{2}}\leq\lambda+\varepsilon$ completes the proof. ∎

Theorem 5.16 yields defective modified log-Sobolev inequalities. Under suitable conditions, the methods of Section 2 allow to tighten them. This is illustrated by two of the following corollaries.

Then there exists constants $K_{1},K_{2}$ such that for every nonnegative smooth function $g$ it holds

The convexity type hypothesis on $V$ and Corollary 5.8 ensure that $\mathcal{D}_{V}(x,y)\leq 2^{2-p}\|y-x\|_{p}^{p}$. The integrability condition allows to apply Theorem 5.16 with $c(u)=\|u\|_{p}^{p}$ for which $c^{*}(u)=\|u\|_{q}^{q}$. We conclude the change of function $f=g^{\frac{q}{2}}$. ∎

Combining Theorem 5.16 for $c(u)=u^{2}/2$ and Corollary 2.13 gives the claim inequality under the slightly stronger assumption \int\exp\big{(}\frac{\lambda+\varepsilon}{2}|x-y|^{2}\big{)}d\mu(x)d\mu(y)<+\infty. Following the proof of Theorem 5.16 in our specific context, we come across a term $(\eta_{1}+\lambda)\int\frac{|x-T(x)|^{2}}{2}f(x)^{2}d\mu(x)$ where $T$ is the optimal map from $f^{2}\cdot\mu$ to $\mu$ for the quadratic cost. In order to get the full result we estimate it a bit differently. In particular, the optimality of $T$ is not used. Since for all $\eta_{2}>0$, $|x+y|^{2}\leq(1+\eta_{2})|x|^{2}+(1+\eta_{2}^{-1})|y|^{2}$:

For small enough $\eta_{i}>0$ the first term is finite. The second one is finite by the stronger integrability condition. Hence a defective LSI has been proved. It can be tightened since the potential is locally bounded. ∎

Wang’s original proof yields a better control on the constant. It is based on semigroup interpolation and seems hard to apply for integrability conditions of the form $\int\exp c(x-y)\,d\mu(x)d\mu(y)<+\infty$ with non quadratic $c$. This is possible with the transportation approach, but a limitation remains: the function $c$ in the integrability condition is the same as the one which controls the lack of convexity of $V$. Nevertheless when the potential is convex, $\lambda=0$ and $c$ disappears from the convexity hypothesis, hence any integrability assumption can be used. This is similar to what happened with applications of Bobkov’s isoperimetric inequality.

The techniques of the above proof also have the advantage to work in more general conditions:

then $\mu$ satisfies a log-Sobolev inequality as well as (q-LSI) for $q=\frac{p+2}{p+1}$.

Applying Taylor’s formula with integral remainder

where we have applied for $\eta>0$ the bound $|a+b|^{p}\leq(1+\eta)|a|^{p}+N(p,\eta)|b|^{p}$ and have computed the integrals. Next we apply the bound $|y-x|^{2}\leq(1+\eta)|x|^{2}+(1+\eta^{-1})|y|^{2}$ and develop all the products. The terms of the form $|y|^{2},|x|^{2},|x|^{2}|y|^{p}$ or $|x|^{p}|y|^{2}$ are controlled by applying Young’s inequality in the form $a^{2}b^{p}\leq\eta a^{p+2}+M(p,\eta)b^{p+2}$ or the similar upper bound of $a^{p}b^{2}$ (but each time the small $\eta$ factor should appear in front of $|x|$). Eventually