The Dual Hamilton–Jacobi Equation and the Poincaré Inequality

Abstract

Following the equivalence between logarithmic Sobolev inequalities and hypercontractivity shown by L. Gross, and applying the ideas and methods of the work by Bobkov, Gentil and Ledoux, we would like to establish a new connection between the logarithmic Sobolev inequalities and the hypercontractivity of solutions of dual Hamilton–Jacobi equations. In addition, Poincaré inequality is also recovered by the dual Hamilton–Jacobi equations.

1. Introduction

The logarithmic Sobolev inequality was first introduced by Gross in the fundamental work [1] as a reformulation of hypercontractivity, and provides upper bounds for the entropy. Gross’s method and technique has been further applied by the important works of [2,3,4]. Otto and Villani [5] first presented evidence of a fascinating connection between logarithmic Sobolev inequalities and transportation cost inequalities, by the infimum-convolution description of the Hamilton–Jacobi solutions. Then, Bobkov, Gentil and Ledoux [2] established a similar relationship for the solutions of Hamilton–Jacobi partial differential equations and examined some converse results from transportation cost inequalities to logarithmic Sobolev inequalities. Immediately after that, Gentil [3] provided the equivalence between the control of the Hamilton–Jacobi equations and the entropy energy inequality. Moreover, Gentil [4] developed a generalization of the $L^p$-Euclidean logarithmic Sobolev inequality, which is equivalent to an optimal control of Hamilton–Jacobi equations. Very recently, Bez, Nakamura and Tsuji [6] provided deficit estimates for the hypercontractivity inequality associated with the Hamilton–Jacobi equation, the Poincaré inequality, and for Beckner’s inequality. Conversely, refs. [7,8,9] contributed the evidence of a surprising and intimate connection between the concave function and the reverse logarithmic Sobolev inequality. In recent years, this direction has experienced rapid development. There is a vast amount of literature on logarithmic Sobolev inequalities and related topics (see the highly influential works, e.g., [1,2,3,4,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]). The sharp logarithmic Sobolev inequality shows that for the Lebesgue measure on ${\mathbb{R}}^n$ (see, e.g., [22]),

$$\begin{array}{c}\mathrm{Ent}\left(\right|f{|}^2)\le {\displaystyle \frac{n}{2}}ln\left({\displaystyle \frac{2}{\pi en}}{\int }_{{\mathbb{R}}^n}{\parallel \nabla f\parallel }^{2}dx\right),\end{array}$$

(1)

with equality if and only if $f\left(x\right)={\left(2\pi \right)}^{-{\displaystyle \frac{n}{4}}}exp(-{\displaystyle \frac{{\parallel x-b\parallel }^2}{4}})$ for $b\in {\mathbb{R}}^n$. (1) is equivalent to the logarithmic Sobolev inequality proved independently by Federbush [23], Gross [1] and Stam [24],

$$\begin{array}{c}{\int }_{supp\left(h\right)}{\left|h\right|}^{2}ln\left({\displaystyle \frac{\left|h\right|}{{\parallel h\parallel }_{L^2\left(\gamma \right)}}}\right)d\gamma \le {\int }_{{\mathbb{R}}^n}{\parallel \nabla h\parallel }^{2}d\gamma ,\end{array}$$

(2)

with equality if and only if $h\left(x\right)=c{e}^{\langle a,x\rangle }$ with $c>0$, $a\in {\mathbb{R}}^n$ and $\gamma$ is the canonical Gaussian measure on ${\mathbb{R}}^n$. Moreover, according to the isoperimetric inequality and an asymptotic argument, Beckner [10] obtained the following Gross logarithmic Sobolev inequality [1]:

$$\begin{array}{cc}\hfill {\mathrm{Ent}}_{\gamma }\left(f^2\right)& \le 2{\int }_{{\mathbb{R}}^n}{|\nabla f|}^{2}d\gamma .\hfill \end{array}$$

(3)

In this paper, we further develop this direction and provide a new connection between the logarithmic Sobolev inequalities and the hypercontractivity of solutions of dual Hamilton–Jacobi equations. Let us first recall that the solutions of Hamilton–Jacobi partial differential equations. Suppose that f is a bounded Lipschitz function on ${\mathbb{R}}^n$, the operator ${\left({\tilde{Q}}_t\right)}_{t\ge 0}$ is defined by the following equation:

$${\tilde{Q}}_{t}f\left(x\right)=\left\{\begin{array}{cc}\underset{y\in {\mathbb{R}}^n}{inf}\left\{f\left(y\right)+{\displaystyle \frac{{\left|x-y\right|}^2}{2t}}\right\},\hfill & t>0,x\in {\mathbb{R}}^n,\hfill \\ f\left(x\right),\hfill & t=0,x\in {\mathbb{R}}^n.\hfill \end{array}\right.$$

(4)

It is well known that $v=v(x,t)={\tilde{Q}}_{t}f\left(x\right)$ is the solution of the following Hamilton–Jacobi initial value problem:

$$\left\{\begin{array}{cc}{\displaystyle \frac{\partial v}{\partial t}}(x,t)+{\displaystyle \frac{{\left|\nabla v(x,t)\right|}^2}{2}}=0,\hfill & t>0,x\in {\mathbb{R}}^n,\hfill \\ v(x,0)=f\left(x\right),\hfill & t=0,x\in {\mathbb{R}}^n.\hfill \end{array}\right.$$

(5)

More details about the Hamilton–Jacobi equations can be found in [25,26]. Bobkov, Gentil and Ledoux [2] established the following theorem by the solutions of Hamilton–Jacobi partial differential equations:

Theorem 1

(Bobkov-Gentil-Ledoux [2]). Assume that μ is absolutely continuous with respect to Lebesgue measure and that for some $\rho >0$ and all smooth enough functions f on ${\mathbb{R}}^n$,

$$\rho En{t}_{\mu }\left(f^2\right)\le 2\int {|\nabla f|}^{2}d\mu ,$$

(6)

where $En{t}_{\mu }\left(f^2\right)=\int f^{2}log{f}^{2}d\mu -\int f^{2}d\mu log\int f^{2}d\mu$. Then, for every bounded measurable function f on ${\mathbb{R}}^n$, every $t\ge 0$ and every $a\in \mathbb{R}$,

$${∥e^{{\tilde{Q}}_{t}f}∥}_{a+\rho t}\le {∥e^f∥}_a.$$

(7)

Conversely, if (7) holds for all $t\ge 0$ and some $a\ne 0$, then the logarithmic Sobolev inequality (6) holds.

In this paper, using the creative ideas and methods in the work [2], we would like to show that the logarithmic Sobolev inequalities are similarly equivalent to hypercontractivity of solutions of dual Hamilton–Jacobi equations. That is, we will extend $a+\rho t$ in (7) to any smooth function $\lambda \left(t\right):(0,+\infty )\to \mathbb{R}$ satisfies ${\lambda }^{\prime }\left(t\right)<0$. In particular, (10) will turn into the well-known Sobolev inequality (6) by making $\lambda \left(t\right)=\lambda \left(0\right)-\rho t$ for some $\rho >0$ in Theorem 2 (for details, please see Section 2). Moreover, the Poincaré inequality is also recovered.

2. The Logarithmic Sobolev Inequalities and the Dual Hamilton–Jacobi Equations

Let us now consider the dual Hamilton–Jacobi initial value problem:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\displaystyle \frac{\partial v}{\partial t}}-{\displaystyle \frac{1}{2}}{|\nabla v|}^2=0,\hfill & \mathrm{in}{\mathbb{R}}^n\times (0,\infty ),\hfill \\ v=f,\hfill & \mathrm{on}{\mathbb{R}}^n\times \{t=0\}.\hfill \end{array}\right.\end{array}$$

(8)

Solutions of (8) can be represented by the maximum convolutions. That is, given a bounded Lipschitz function f on ${\mathbb{R}}^n$, the maximum convolution of f is defined as

$$\begin{array}{c}\hfill Q_{t}f\left(x\right)=\left\{\begin{array}{cc}\underset{y\in {\mathbb{R}}^n}{sup}\left\{f\left(y\right)-{\displaystyle \frac{{\left|x-y\right|}^2}{2t}}\right\},\hfill & t>0,x\in {\mathbb{R}}^n,\hfill \\ f\left(x\right),\hfill & t=0,x\in {\mathbb{R}}^n.\hfill \end{array}\right.\end{array}$$

(9)

Then, the family ${\left(Q_t\right)}_{t\ge 0}$ defines a semigroup. In other words, $v=v(x,t)=Q_{t}f\left(x\right)$ is a solution of the dual Hamilton–Jacobi initial value problem (8).

The operators ${\tilde{Q}}_t$ and $Q_t$ are related by the property that $Q_t(-f)=-{\tilde{Q}}_{t}f$. It is not difficult to find that for any f and $t>0$, $Q_{t}f$ is lower semicontinuous. If f is bounded (resp. Lipschitz), $Q_{t}f$ is bounded and Lipschitz (resp. Lipschitz). Given a bounded function f, $Q_{t}f\left(x\right)\to f\left(x\right)$ as $t\to 0$ if and only if f is upper semicontinuous at x. The definitions (4) and (9) also imply that $Q_t{\tilde{Q}}_{t}f\le f\le {\tilde{Q}}_t{Q}_{t}f$ for any function f (see, e.g., [2,25,26]).

Now, we have prepared enough to prove the main theorem and establish a new connection between the logarithmic Sobolev inequalities and the hypercontractivity of solutions of dual Hamilton–Jacobi equations, as follows:

Theorem 2.

If μ is absolutely continuous with respect to the Lebesgue measure and the smooth function $\lambda \left(t\right):(0,+\infty )\to \mathbb{R}$ satisfies ${\lambda }^{\prime }\left(t\right)<0$. Then, the following two statements are equivalent:

(1) For all smooth enough functions f on ${\mathbb{R}}^n$,

$$\begin{array}{cc}\hfill -{\lambda }^{\prime }\left(t\right)En{t}_{\mu }\left(f^2\right)& \le {2\int |\nabla f|}^{2}d\mu .\hfill \end{array}$$

(10)

(2) If f is the bounded Lipschitz and sufficiently smooth function on ${\mathbb{R}}^n$, then

$$\begin{array}{cc}\hfill {∥e^f∥}_{\lambda \left(0\right)}& \le {∥e^{Q_{t}f}∥}_{\lambda \left(t\right)},\lambda \left(0\right)\ne 0,\hfill \end{array}$$

(11)

where $t\in (0,\delta )$, δ is a small enough positive constant.

Proof. 

(1) Firstly, it is sufficient to check that (11) holds, provided that the logarithmic Sobolev inequality (10) is satisfied for the smooth function $\lambda \left(t\right):(0,+\infty )\to \mathbb{R}$ with ${\lambda }^{\prime }\left(t\right)<0$.

By a simple density argument, the logarithmic Sobolev inequality (10) holds for all (locally) Lipschitz functions. By regularization, it may be assumed that f is compactly supported with bounded derivatives of any orders. Let thus f be a bounded Lipschitz and sufficiently smooth function on ${\mathbb{R}}^n$ and introduce an important new function

$$\begin{array}{c}\hfill F\left(s\right)=\parallel e^{{\tilde{Q}}_{s}f}{\parallel }_{\lambda (t-s)},\end{array}$$

(12)

where $\lambda \left(t\right):(0,+\infty )\to \mathbb{R}$ is a smooth function that satisfies the condition ${\lambda }^{\prime }\left(t\right)<0$. For all $s>0$ and almost every x, the partial derivatives ${\displaystyle \frac{\partial }{\partial s}}{\tilde{Q}}_{s}f\left(x\right)$ exist; therefore, F is differentiable at every point $s>0$, where $\lambda \left(t\right)\ne 0$. Thus, differentiating $F{\left(s\right)}^{\lambda (t-s)}=\int e^{\lambda (t-s){\tilde{Q}}_{s}f}$ at time s and using the chain rule to yield

$$\begin{array}{cc}\hfill {\left(F{\left(s\right)}^{\lambda (t-s)}\right)}_s^{\prime }& =F{\left(s\right)}^{\lambda (t-s)}\left(-{\lambda }^{\prime }(t-s)lnF\left(s\right)+{\displaystyle \frac{\lambda (t-s)}{F\left(s\right)}}{F}^{\prime }\left(s\right)\right)\hfill \\ \hfill \mathrm{and}{\left(\int e^{\lambda (t-s){\tilde{Q}}_{s}f}\right)}_s^{\prime }& =\int e^{\lambda (t-s){\tilde{Q}}_{s}f}\left(-{\lambda }^{\prime }(t-s){\tilde{Q}}_{s}f+\lambda (t-s){\displaystyle \frac{\partial {\tilde{Q}}_{s}f}{\partial s}}\right),\hfill \end{array}$$

it follows that

$$\begin{array}{cc}\hfill & -{\lambda }^{\prime }(t-s)FlnF+\lambda (t-s)F^{\prime }\left(s\right)\hfill \\ \hfill & =F{\left(s\right)}^{1-\lambda (t-s)}\int e^{\lambda (t-s){\tilde{Q}}_{s}f}\left(-{\lambda }^{\prime }(t-s){\tilde{Q}}_{s}f+\lambda (t-s){\displaystyle \frac{\partial {\tilde{Q}}_{s}f}{\partial s}}\right),\hfill \end{array}$$

which is equivalent to

$$\begin{array}{cc}\hfill {\lambda }^2(t-s)F{\left(s\right)}^{\lambda (t-s)-1}{F}^{\prime }\left(s\right)& =-{\lambda }^{\prime }(t-s)En{t}_{\mu }\left(e^{\lambda (t-s){\tilde{Q}}_{s}f}\right)\hfill \\ \hfill & +\int {\lambda }^2(t-s)e^{\lambda (t-s){\tilde{Q}}_{s}f}{\displaystyle \frac{\partial {\tilde{Q}}_{s}f}{\partial s}}.\hfill \end{array}$$

(13)

From the above definition of $F\left(t\right)$, with the logarithmic Sobolev inequality (10), followed by the fact that ${\displaystyle \frac{\partial }{\partial s}}{\tilde{Q}}_{s}f\left(x\right)=-{\displaystyle \frac{{\left|\nabla {\tilde{Q}}_{s}f\left(x\right)\right|}^2}{2}}$ almost everywhere in x, then we pass to the limit $s\to 0^{+}$ on both sides and obtain

$$\begin{array}{cc}\hfill & {\lambda }^2(t-s)F{\left(s\right)}^{\lambda (t-s)-1}{F}^{\prime }\left(s\right)\hfill \\ \hfill & =\left(-{\lambda }^{\prime }(t-s)En{t}_{\mu }\left(e^{\lambda (t-s){\tilde{Q}}_{s}f}\right)+\int {\lambda }^2(t-s)e^{\lambda (t-s){\tilde{Q}}_{s}f}{\displaystyle \frac{\partial {\tilde{Q}}_{s}f}{\partial s}}\right)\hfill \\ \hfill & \underrightarrow{s\to 0^{+}}-{\lambda }^{\prime }\left(t\right)En{t}_{\mu }\left(e^{\lambda \left(t\right)f}\right)+\int {\lambda }^2\left(t\right)e^{\lambda \left(t\right)f}·-{\displaystyle \frac{{\left|\nabla f\left(x\right)\right|}^2}{2}}\hfill \\ \hfill & \le 2\int {\left|\nabla e^{{\displaystyle \frac{\lambda \left(t\right)f}{2}}}\right|}^{2}d\mu -\int {\lambda }^2\left(t\right)e^{\lambda \left(t\right)f}{\displaystyle \frac{{\left|\nabla f\left(x\right)\right|}^2}{2}}=0\hfill \end{array}$$

We prove this using the fact that $\lambda \left(t\right)$ is continuous and ${\tilde{Q}}_{s}f\left(x\right)\to f\left(x\right)$ as $s\to 0^{+}$. According to the contradiction method, it follows that $F^{\prime }\left(s\right)\le 0(s\to 0^{+})$. In fact, if $F^{\prime }\left(s\right)>0$ for some fixed $s\in (0,\delta )$, there is a contradiction in the above inequality since

$$\begin{array}{cc}\hfill & {\lambda }^2(t-s)F{\left(s\right)}^{\lambda (t-s)-1}{F}^{\prime }\left(s\right)\hfill \\ \hfill & \underrightarrow{s\to 0^{+}}-{\lambda }^{\prime }\left(t\right)En{t}_{\mu }\left(e^{\lambda \left(t\right)f}\right)+\int {\lambda }^2\left(t\right)e^{\lambda \left(t\right)f}·-{\displaystyle \frac{{\left|\nabla f\left(x\right)\right|}^2}{2}}\le 0,\hfill \end{array}$$

which holds true for every $t>0$ and arbitrary $\lambda \left(t\right)$, satisfying

$${\lambda }^{\prime }\left(t\right)En{t}_{\mu }\left(e^{\lambda \left(t\right)f}\right)+\int {\lambda }^2\left(t\right)e^{\lambda \left(t\right)f}·{\displaystyle \frac{{\left|\nabla f\left(x\right)\right|}^2}{2}}\ge 0.$$

This means that there exists a small enough constant $\delta >0$ such that $F\left(t\right)\le F\left(0\right)$ for every $t\in (0,\delta )$, which is equivalent to

$$\begin{array}{c}\hfill {\parallel e^{{\tilde{Q}}_{t}f}\parallel }_{\lambda \left(0\right)}\le {∥e^f∥}_{\lambda \left(t\right)}.\end{array}$$

Please notice the fact that $f\left(g\right)={\left(\int g{\left(x\right)}^{a}d\mu \left(x\right)\right)}^{{\displaystyle \frac{1}{a}}}$ is an increasing function on $(0,+\infty )$; more precisely, if $g_1\le g_2$ with $g_1,g_2\in (0,+\infty )$, then $f\left(g_1\right)\le f\left(g_2\right)$. These, together with $f\le {\tilde{Q}}_t{Q}_{t}f$, show that

$$\begin{array}{cc}\hfill \parallel e^f{\parallel }_{\lambda \left(0\right)}& \le {∥e^{Q_{t}f}∥}_{\lambda \left(t\right)},\hfill \end{array}$$

where $t\in (0,\delta )$, $\delta$ is a small enough positive constant.

(2) Now, we move on to the opposite direction. Let us notice that f is the bounded Lipschitz and sufficiently smooth function satisfying (11) for every small enough $t>0$ and some $\lambda \left(0\right)\ne 0$. Applying the fact that $f\left(g\right)={\left(\int g{\left(x\right)}^{a}d\mu \left(x\right)\right)}^{{\displaystyle \frac{1}{a}}}$ is an increasing function on $(0,+\infty )$ and $Q_t{\tilde{Q}}_{t}f\le f$, we see that

$$\begin{array}{c}\hfill F^{\prime }\left(0\right)\le 0.\end{array}$$

This, together with (13) and (5), provides that

$$\begin{array}{cc}\hfill & {\lambda }^2(t-s)F{\left(s\right)}^{\lambda (t-s)-1}{F}^{\prime }\left(s\right){|}_{s=0}\hfill \\ \hfill & =\left(-{\lambda }^{\prime }(t-s)En{t}_{\mu }\left(e^{\lambda (t-s){\tilde{Q}}_{s}f}\right)+\int {\lambda }^2(t-s)e^{\lambda (t-s){\tilde{Q}}_{s}f}{\displaystyle \frac{\partial {\tilde{Q}}_{s}f}{\partial s}}\right){|}_{s=0}\hfill \\ \hfill & =-{\lambda }^{\prime }\left(t\right)En{t}_{\mu }\left(e^{\lambda \left(t\right)f}\right)-\int \lambda {\left(t\right)}^2{e}^{\lambda \left(t\right)f}{\displaystyle \frac{{\left|\nabla f\left(x\right)\right|}^2}{2}}\hfill \\ \hfill & =-{\lambda }^{\prime }\left(t\right)En{t}_{\mu }\left(e^{\lambda \left(t\right)f}\right)-2\int {\left|\nabla e^{{\displaystyle \frac{\lambda \left(t\right)f}{2}}}\right|}^2\le 0.\hfill \end{array}$$

If f is used to replace $e^{{\displaystyle \frac{\lambda \left(t\right)f}{2}}}$, the desired Formula (10) follows immediately; the proof of Theorem 2 is complete. □

In Theorem 2, we present the equivalence between logarithmic Sobolev inequalities and hypercontractivity of solutions of dual Hamilton–Jacobi equations for any smooth function $\lambda \left(t\right)$ with ${\lambda }^{\prime }\left(t\right)<0$ in the right neighborhood of the origin. Inequalities (10) and (11) are interdependent. The logarithmic Sobolev inequality (10) depends on t, which satisfies (11). The optimal value of $-{\lambda }^{\prime }\left(t\right)$ is its maximum value for (10).

In particular, taking $\lambda \left(t\right)=\lambda \left(0\right)-\rho t$ for some $\rho >0$ in (10), then we can obtain the well-known Sobolev inequality (6). More generally, if $-inf{\lambda }^{\prime }\left(t\right)=\rho$, it follows that (10) turns into inequality (6). According to Theorem 2.1, (11) holds true for any function $\lambda \left(t\right)$ satisfying $-inf{\lambda }^{\prime }\left(t\right)=\rho$.

The Brunn–Minkowski inequality is a central notion in contemporary convex geometry (see the important works, e.g., [27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83]). The famous Prékopa–Leindler inequality is the functional version of the Brunn–Minkowski inequality, discovered by Prékopa and Leindler. Giving three non-negative measurable functions $u,v,w:{\mathbb{R}}^n\to {\mathbb{R}}^{+}$ and $a,b>0,a+b=1$, Prékopa and Leindler proved that when

$$\begin{array}{cc}\hfill w(ax+by)& \ge u{\left(x\right)}^{a}v{\left(y\right)}^b,\hfill \end{array}$$

(14)

holds for every $x,y\in {\mathbb{R}}^n$, then

$$\begin{array}{cc}\hfill \int wdx& \ge {\left(\int udx\right)}^a{\left(\int vdx\right)}^b.\hfill \end{array}$$

(15)

Applying this inequality, one can prove the famous Brunn–Minkowski inequality.

How to derive the Poincaré inequality from the logarithmic Sobolev inequality has been studied since the 1980s. Recently, Dolbeault, Nazaret and Savaré [84] explored Poincaré and logarithmic Sobolev inequalities by a gradient flow approach. Next, Menz and Schlichting [85] proved the Eyring–Kramers formula for the optimal constant in the Poincaré and logarithmic Sobolev inequality for the associated generator of the diffusion. Also, many other related important works can be found in, e.g., [86,87,88,89].

Theorem 2 shows that (10) is equivalent to (11). In the following, we would like to consider the case $\lambda \left(t\right)=\lambda \left(0\right)-ct:\left(0,{\displaystyle \frac{\lambda \left(0\right)}{c}}\right)\to (0,+\infty )$ for some $c>0$ and $\lambda \left(0\right)>0$, prove that (11) holds true by the Taylor formula and Prékopa–Leindler inequality, and recover the Poincaré inequality from (16) with details, as follows.

Theorem 3.

If μ is a log-concave probability measure on ${\mathbb{R}}^n$ given by ${\displaystyle \frac{d\mu }{dx}}=e^{-U}$ satisfying $Hess\left(U\right)\ge cId$ for some $c>0$, and f is the bounded Lipschitz and sufficiently smooth function on ${\mathbb{R}}^n$, then the following occur:

(1) Suppose, in addition, that the function $\lambda \left(t\right)=\lambda \left(0\right)-ct:\left(0,{\displaystyle \frac{\lambda \left(0\right)}{c}}\right)\to (0,+\infty )$ with $\lambda \left(0\right)>0$, then we have

$$\begin{array}{cc}\hfill {∥e^f∥}_{\lambda \left(0\right)}& \le {∥e^{Q_{t}f}∥}_{\lambda \left(0\right)-ct}.\hfill \end{array}$$

(16)

(2) Moreover, the following Poincaré inequality holds:

$$\begin{array}{cc}\hfill cVa{r}_{\mu }\left(f\right)& \le {\int |\nabla f|}^{2}d\mu ,\hfill \end{array}$$

where $Va{r}_{\mu }\left(f\right)=\int f^{2}d\mu -{\left(\int fd\mu \right)}^2$.

Proof. 

(1) Applying the second-order Taylor expansion for U around $x_0$, we see that

$$\begin{array}{cc}\hfill U\left(x\right)& =U\left(x_0\right)+U^{\prime }\left(x_0\right)(x-x_0)+{\displaystyle \frac{U^{″}\left({\xi }_1\right)}{2}}{(x-x_0)}^2,\hfill \\ \hfill U\left(y\right)& =U\left(x_0\right)+U^{\prime }\left(x_0\right)(y-x_0)+{\displaystyle \frac{U^{″}\left({\xi }_2\right)}{2}}{(y-x_0)}^2,\hfill \end{array}$$

where ${\xi }_1$ is between x and $x_0$, and ${\xi }_2$ is between y and $x_0$. These, together with $x_0=ax+by$ and the convexity condition $Hess\left(U\right)\ge cId$, immediately provide that

$$\begin{array}{cc}\hfill aU\left(x\right)+bU\left(y\right)-U(ax+by)& \ge {\displaystyle \frac{abc}{2}}{\left|x-y\right|}^2,\hfill \end{array}$$

(17)

where $a,b>0,a+b=1$ and $x,y\in {\mathbb{R}}^n$.

Since f is a bounded Lipschitz function on ${\mathbb{R}}^n$, let us consider the following three functions:

$$\begin{array}{cc}\hfill u\left(x\right)& =e^{{\displaystyle \frac{1}{a}}f\left(x\right)-U\left(x\right)},\hfill \\ \hfill v\left(y\right)& =e^{-U\left(y\right)},\hfill \\ \hfill w\left(z\right)& =e^{Q_{b/ac}f\left(z\right)-U\left(z\right)}.\hfill \end{array}$$

This, together with (17), (4) and (9), immediately shows that

$$\begin{array}{cc}\hfill w(ax+by)& =e^{Q_{b/ac}f(ax+by)-U(ax+by)}\hfill \\ \hfill & \ge e^{Q_{b/ac}f(ax+by)-aU\left(x\right)-bU\left(y\right)+{\displaystyle \frac{cab}{2}}{\left|x-y\right|}^2}\hfill \\ \hfill & =e^{Q_{b/ac}f(ax+by)+{\displaystyle \frac{ca}{2b}}{|x-(ax+by)|}^2-aU\left(x\right)-bU\left(y\right)}\hfill \\ \hfill & \ge e^{{\tilde{Q}}_{b/ac}{Q}_{b/ac}f\left(x\right)-aU\left(x\right)-bU\left(y\right)}\hfill \\ \hfill & \ge u{\left(x\right)}^{a}v{\left(y\right)}^b.\hfill \end{array}$$

Therefore, by applying the Prékopa–Leindler inequality, we see that

$$\begin{array}{cc}\hfill \int e^{Q_{b/ac}f}d\mu & \ge {\left(\int e^{{\displaystyle \frac{1}{a}}f}d\mu \right)}^a,\hfill \end{array}$$

where $d\mu =e^{-U}dx$. Replacing f by $\left(\lambda \left(0\right)-ct\right)f$, and taking $a={\displaystyle \frac{\lambda \left(0\right)-ct}{\lambda \left(0\right)}}$, $Q_{t}mf=m{Q}_{tm}f$, we have

$$\begin{array}{cc}\hfill {∥e^f∥}_{\lambda \left(0\right)}& \le {∥e^{Q_{t}f}∥}_{\lambda \left(0\right)-ct}.\hfill \end{array}$$

(2) By applying Formula (16), we can infer that

$$\begin{array}{cc}\hfill {\left(\int e^{(\lambda \left(0\right)-ct)Q_{t}f}\right)}^{{\displaystyle \frac{1}{\lambda \left(0\right)-ct}}}& \ge {\left(\int e^{\lambda \left(0\right)f}\right)}^{{\displaystyle \frac{1}{\lambda \left(0\right)}}},\hfill \end{array}$$

(18)

where $\lambda \left(0\right)-ct>0$. In the following, we will focus on the case that $\lambda \left(0\right)$ and t are all small enough. Please also notice that for the bounded Lipschitz function f on ${\mathbb{R}}^n$, one has

$$\begin{array}{c}\hfill \underset{s\to 0}{lim}{\left(\int e^{sf}\right)}^{{\displaystyle \frac{1}{s}}}=\underset{s\to 0}{lim}{\left(\int e^{-sf}\right)}^{{\displaystyle \frac{1}{-s}}}.\end{array}$$

(19)

According to (19), it is reasonable to make $\lambda \left(0\right)\to 0$ in (18). These, together with (18), show that for $\lambda \left(0\right)\to 0$,

$$\begin{array}{c}\hfill \int e^{-ct{Q}_{t}f}d\mu \le e^{-ct\int fd\mu },\end{array}$$

which, combined with $Q_{t}f=Q_{0}f+{\displaystyle \frac{\partial Q_{t}f}{\partial t}}t+o\left(t\right)=f+{\displaystyle \frac{t}{2}}{|\nabla f|}^2+o\left(t\right)$ and the Taylor formula, follows immediately that

$$\begin{array}{cc}\hfill & 1-ct\int fd\mu -{\displaystyle \frac{c{t}^2}{2}}\int {\left|\nabla f\right|}^{2}d\mu +{\displaystyle \frac{c^2{t}^2}{2}}\int f^{2}d\mu +o\left(t^2\right)\hfill \\ \hfill & \le 1-ct\int fd\mu +{\displaystyle \frac{c^2{t}^2}{2}}{\left(\int fd\mu \right)}^2+o\left(t^2\right).\hfill \end{array}$$

This provides that

$$\begin{array}{cc}\hfill cVa{r}_{\mu }\left(f\right)& \le {\int |\nabla f|}^{2}d\mu ,\hfill \end{array}$$

where $Va{r}_{\mu }\left(f\right)=\int f^{2}d\mu -{\left(\int fd\mu \right)}^2$. Theorem 3 is thus established. □

Recall the fact that $\lambda \left(t\right)=\lambda \left(0\right)-ct:\left(0,{\displaystyle \frac{\lambda \left(0\right)}{c}}\right)\to (0,+\infty )$ is continuous and $Q_{t}f\left(x\right)\to f\left(x\right)$ as $t\to 0^{+}$. Let us consider the special case $t=0$, then (16) obviously holds true, and Theorem 3 ensures that the Poincaré inequality holds, that is, $cVa{r}_{\mu }\left(f\right)\le \int {|\nabla f|}^{2}d\mu .$

3. Conclusions

In this paper, we provide a new connection between the logarithmic Sobolev inequalities and the hypercontractivity of solutions of dual Hamilton–Jacobi equations. In addition, the Poincaré inequality is also recovered by the dual Hamilton–Jacobi equations.