Sharp Stability for LSI

Abstract

A fundamental tool in mathematical physics is the logarithmic Sobolev inequality. A quantitative version proven by Carlen with a remainder involving the Fourier–Wiener transform is equivalent to an entropic uncertainty principle more general than the Heisenberg uncertainty principle. In the stability inequality, the remainder is in terms of the entropy, not a metric. Recently, a stability result for $H^1$ was obtained by Dolbeault, Esteban, Figalli, Frank, and Loss in terms of an $L^p$ norm. Afterward, Brigati, Dolbeault, and Simonov discussed the stability problem involving a stronger norm. A full characterization with a necessary and sufficient condition to have $H^1$ convergence is identified in this paper. Moreover, an explicit $H^1$ bound via a moment assumption is shown. Additionally, the $L^p$ stability of Dolbeault, Esteban, Figalli, Frank, and Loss is proven to be sharp.

1. Introduction

The LSI appears in statistical mechanics, mathematical statistics, and quantum field theory. There exist many formulations of the classical Gaussian logarithmic Sobolev inequality, which states

$${\int }_{{\mathbb{R}}^n}f\left(x\right)logf\left(x\right)d\gamma \left(x\right)\le \frac{1}{2}{\int }_{{\mathbb{R}}^n}\frac{{|\nabla f\left(x\right)|}^2}{f\left(x\right)}d\gamma \left(x\right)$$

(1)

for positive, smooth, normalized functions, where $d\gamma :={\left(2\pi \right)}^{-n/2}{e}^{-{|x|}^2/2}dx$, and is equivalent to Nelson’s hypercontractive inequality. The term on the left-hand side is the entropy and represented by $Ent\left(f\right)$, and the integral term on the right side of the inequality is known as the Fisher information (denoted by $I\left(f\right)$). Several proofs have appeared that utilize the central limit theorem, Prekopka–Leindler inequality, Ornstein–Uhlenbeck semigroup, harmonic analysis, and optimal transport theory. The historical evolution started with Stam [1] and Federbush [2]. Gross obtained the LSI in 1975 [3] by utilizing probabilistic methods, and Cordero-Erausquin discovered a very elegant and simple proof via optimal transport theory in 2002 [4].

An important application is via the Ising model [3,5] (with the Glauber–Langevin dynamic). In addition, the LSI appeared in Perelman’s proof of the Poincaré conjecture and also plays a fundamental role in the calculus of variations and partial differential equations [6].

Although simple to state, the LSI is delicate: a simple computation shows that equality holds if $f\left(x\right)=e^{a·x+b_a}$; however, the fact that these are the only functions achieving equality was solved by Eric Carlen [7] in two ways: first, if $g\in L^p\left({\mathbb{R}}^{2n}\right)$ is a product function in $\left(\frac{x+y}{\sqrt{2}},\frac{x-y}{\sqrt{2}}\right)$ and $(x,y)$, then g and its factors are Gaussian; subsequently, a Minkowski-type inequality and the strict superadditivity of the Fisher information were combined with factorization.

The other method is based on the Beckner–Hirschman entropic uncertainty principle: let

$$U:L^2({\mathbb{R}}^n,dx)\to L^2({\mathbb{R}}^n,dm)$$

be defined by $Uh\left(x\right)=2^{-\frac{n}{4}}{e}^{{\pi |x|}^2}h\left(x\right)$ and observe that the Fourier transform is defined by

$$\widehat{f}\left(\xi \right)=\mathcal{F}f\left(\xi \right)=\int e^{-2\pi i\xi ·x}f\left(x\right)dx.$$

With this, the Fourier–Wiener transform is

$$\mathcal{W}=U\mathcal{F}{U}^{*},$$

($U^{*}$ is the adjoint of U). Assuming that $dm=2^{\frac{n}{2}}{e}^{-{2\pi |x|}^2}dx$ and $f\in L^2\left(dm\right)$ are normalized, Carlen showed that the entropic uncertainty principle is equivalent to

$${Ent}_{dm}{(|\mathcal{W}f|}^2{):=\int |\mathcal{W}f|}^2{log|\mathcal{W}f|}^{2}dm\le \frac{1}{2\pi }{\int |\nabla f|}^{2}dm-{\int |f|}^{2}log{|f|}^{2}dm=:{\delta }_c\left(f\right),$$

(2)

and if $f\ge 0$, ${Ent}_{dm}{(|\mathcal{W}f|}^2)=0$ only if $f_a\left(x\right)=e^{2\pi (a·x-\frac{{|a|}^2}{2})}$, where $a\in {\mathbb{R}}^n$.

The uncertainty principle conjectured by Hirschman [8] was eventually substantiated by Beckner [9]: the sum of the entropies of a function and its Fourier transform has a lower bound via $n(1-log2)$. A rescaling of (2) improves (1). Moreover, Carlen’s quantitative LSI in the previously stated form improves upon Heisenberg’s uncertainty principle via the Beckner–Hirschman inequality.

In the quantitative inequality, the remainder is via the entropy, not a metric. The first Wasserstein-Kantorovich metric result was shown by myself and Marcon [10]: with $M>0$ and $ϵ>0$, consider

$$\mathcal{F}(ϵ,M):=\{e^{-h}:(-1+ϵ)\le D^{2}h\le M\}$$

(where the inequalities are in the sense of the constants times the identity matrix); there exists $C=C(ϵ,M)>0$ so that for $f\in \mathcal{F}(ϵ,M)$ with unit mass and zero barycenter,

$$W_2(fd\gamma ,d\gamma )\le C\delta {\left(f\right)}^{\frac{1}{2}}.$$

Additionally, we constructed examples that show that the $\frac{1}{2}$-exponent is sharp and obtained new bounds on the entropy that show that $W_2$-stability is not true without extra assumptions.

The following quantitative LSI was proven in 2022 [11]: there exists a dimensionless $\kappa >0$ such that, assuming

$$u\in H^1\left(e^{-{\pi |x|}^2}dx\right)=W^{1,2}({\mathbb{R}}^n,e^{-{\pi |x|}^2}dx),$$

where $W^{m,p}$ is the usual Sobolev space (the functions have m (weak) derivatives in $L^p$) and $W^{m,p}({\mathbb{R}}^n,e^{-{\pi |x|}^2}dx)$ is the weighted Sobolev space with the measure $e^{-{\pi |x|}^2}dx$,

$$\pi {\delta }_{*}\left(u\right):={\int |\nabla u|}^2{e}^{-{\pi |x|}^2}dx-{\pi \int |u|}^{2}ln\left(\frac{{|u|}^2}{{||u||}_{L^2}^2}\right)e^{-{\pi |x|}^2}dx\ge \kappa \underset{a,c}{inf}\int {|u-c{e}^{a·x}|}^2{e}^{-{\pi |x|}^2}dx.$$

(3)

In [12], p. 5, the stability problem relative to a stronger norm is stated:

“a stability on the Gaussian logarithmic Sobolev inequality is shown in [13], although the distance is measured only by an $L^2({\mathbb{R}}^n,d\gamma )$ norm. Whether a stronger estimate can be obtained in the limiting case $p=2$, eventually under some restriction, is therefore so far an open question.”

The authors suspected that, unlike $L^2$-stability (3), quantitative stability in a stronger norm may not hold for all of $H^1$. This is actually the case. The optimal condition to have $H^1$ convergence is identified in Theorem 1. Moreover, there exists an explicit $H^1$ bound via a moment assumption; the moment assumption is sharp: the inequality is not true if $\alpha =\infty$. Additionally, (3) is sharp via Theorem 2.

Theorem 1.

1. Let $\left\{u_k\right\}$ be normalized and centered in $L^2\left(e^{-{\pi |x|}^2}dx\right)$ and suppose ${\delta }_{*}\left(u_k\right)\to 0$ as $k\to \infty$; then,

$$|u_k|\to 1$$

in $H^1\left(e^{-{\pi |x|}^2}dx\right)$ if and only if

$$m_2\left(u_k\right):={\int |x|}^2|u_k\left(x\right){|}^2{e}^{-{\pi |x|}^2}dx\to \int {|x|}^2{e}^{-{\pi |x|}^2}dx=m_2\left(1\right).$$

2. If u is normalized and centered in $L^2\left(e^{-{\pi |x|}^2}dx\right)$ and

$$m_4\left(u\right):={\int |x|}^4{|u\left(x\right)|}^2{e}^{-{\pi |x|}^2}dx\le \alpha <\infty ,$$

then

$$|||u|-1|{|}_{H^1\left(e^{-{\pi |x|}^2}dx\right)}\le a_{\alpha }{\left({\delta }_{*}^{\frac{1}{2}}\left(u\right)+{\delta }_{*}\left(u\right)\right)}^{\frac{1}{2}},$$

$\infty >a_{\alpha }>0$.

3. There are densities $\left\{u_k\right\}$ normalized and centered in $L^2\left(e^{-{\pi |x|}^2}dx\right)$, ${\delta }_{*}\left(u_k\right)\to 0$, and

$$||u_k{||}_{H^1\left(e^{-{\pi |x|}^2}dx\right)}\to \infty .$$

Theorem 2.

Expression (3) has the optimal rate.

A simple version of (3) appears in the next lemma. Note that thanks to this reduction, one may, without loss of generality, assume the functions to be centered and normalized in Theorem 1.

Lemma 1.

Expression (3) is equivalent to

$${\int |\nabla w|}^2{e}^{-{\pi |x|}^2}dx-{\pi \int |w|}^{2}ln{|w|}^2{e}^{-{\pi |x|}^2}dx\ge \kappa \int |w-1{|}^2{e}^{-{\pi |x|}^2}dx$$

in the space of non-negative functions ($w\ge 0$) that satisfy

$${||w||}_{L^2\left(e^{-{\pi |x|}^2}dx\right)}=1$$

$${\int x|w|}^2{e}^{-{\pi |x|}^2}dx=0.$$

In particular, a completely equivalent version of (3) with a moment assumption and modulus $\omega$ was already proven in [13] by utilizing a combination of optimal transport theory and Fourier analysis. Observe also that the non-negativity assumption appeared in Carlen’s proof of the equality cases [7]. Suppose, without loss of generality, that

$${||u||}_{L^2\left(e^{-{\pi |x|}^2}dx\right)}=1$$

$${\int x|u|}^2{e}^{-{\pi |x|}^2}dx=0.$$

Now, set $dm=2^{\frac{n}{2}}{e}^{-{2\pi |x|}^2}dx$, $d\gamma ={\left(2\pi \right)}^{-\frac{n}{2}}{e}^{-\frac{{|x|}^2}{2}}dx$,

$$w\left(x\right)=u\left(\sqrt{2}x\right)$$

$$f\left(x\right)={|u|}^2\left(\frac{x}{\sqrt{2\pi }}\right)$$

and observe that

$${\int |w|}^{2}dm=\int {|u|}^2{e}^{-{\pi |x|}^2}dx=\int fd\gamma$$

$$\int xfd\gamma =0={\int x|w|}^{2}dm.$$

Suppose

$${\int |x|}^2{|u|}^2{e}^{-{\pi |x|}^2}dx\le M_{\alpha },$$

$M_{\alpha }\ge \frac{n}{2\pi }$. Note that $|\nabla |w||=|\nabla w|$ a.e.; therefore, assume $w\ge 0$. An application of ([13] Corollary 1.21) then implies that there exists a modulus $\omega$ such that

$${\int |w-1|}^{2}dm\le a\omega \left({\delta }_c\left(w\right)\right)=a\omega \left({\delta }_{*}\left(u\right)\right)$$

where $a=a\left(M_{\alpha }\right)>0$,

$${\delta }_c\left(w\right):=\frac{1}{2\pi }{\int |\nabla w|}^{2}dm-{\int |w|}^{2}ln{|w|}^{2}dm.$$

Thanks to

$${\int |w-1|}^{2}dm=\int {|u-1|}^2{e}^{-{\pi |x|}^2}dx,$$

$$\omega \left({\int |\nabla u|}^2{e}^{-{\pi |x|}^2}dx-{\pi \int |u|}^{2}ln{|u|}^2{e}^{-{\pi |x|}^2}dx\right)\ge \frac{1}{\overline{a}}\int {|u-1|}^2{e}^{-{\pi |x|}^2}dx,$$

$\overline{a}>0$. Moreover, if $n=1$,

$$\delta \left(f\right):=\frac{1}{2}\mathrm{I}\left(f\right)-\left(f\right)=\frac{1}{2}\int \frac{{|\nabla f|}^2}{f}d\gamma -\int flnfd\gamma ,$$

(recall

$$\mathrm{I}\left(f\right)=\int \frac{{|\nabla f|}^2}{f}d\gamma$$

is the Fisher information and

$$\mathrm{H}\left(f\right)=\int flnfd\gamma$$

is the entropy), ([13] Theorem 1.1) yields

$$\int |f-1|d\gamma \le {\overline{a}}_1{\delta }^{\frac{1}{4}}\left(f\right),$$

with ${\overline{a}}_1={\overline{a}}_1\left(M_{\alpha }\right)>0$. Therefore, assuming $u\ge 0$,

$$\begin{array}{cc}\hfill {\int |u-1|}^2{e}^{-{\pi |x|}^2}dx& =\int {\left|\sqrt{f}-1\right|}^{2}d\gamma \hfill \\ \hfill & \le \int |f-1|d\gamma \hfill \\ \hfill & \le {\overline{a}}_1{\delta }^{\frac{1}{4}}\left(f\right)={\overline{a}}_1{\delta }_{*}^{\frac{1}{4}}\left(u\right).\hfill \end{array}$$

In addition, higher-dimensional quantitative inequalities that also included an explicit modulus appeared in [10,13,14].

The optimal inequality via Theorem 2 is

$$||u-{1||}_{L^2\left(e^{-{\pi |x|}^2}dx\right)}\lesssim {\delta }_{*}^{\frac{1}{2}}\left(u\right),$$

and the more general stability

$${||\nabla u||}_{L^2\left(e^{-{\pi |x|}^2}dx\right)}\le a_1{\delta }_{*}^{\frac{1}{2}}\left(u\right)$$

(4)

with the sharp exponent $\frac{1}{2}$ was proved in [10] for probability measures that are absolutely continuous with respect to the Gaussian measure $d\gamma$, and the density $f\left(x\right)={|u|}^2\left(\frac{x}{\sqrt{2\pi }}\right)$ satisfies a log-$C^{1,1}$ assumption. This was achieved via optimal transport theory ([10] Theorem 1.1, Remark 4.3): with the assumptions, there exists $\alpha \in (0,\frac{1}{2})$ such that

$$\int flnfd\gamma \le \alpha \int \frac{{|\nabla f|}^2}{f}d\gamma .$$

Hence,

$$\delta \left(f\right)=\frac{1}{2}\int \frac{{|\nabla f|}^2}{f}d\gamma -\int flnfd\gamma \ge (\frac{1}{2}-\alpha )\int \frac{{|\nabla f|}^2}{f}d\gamma ,$$

thus proving (4) with a simple change of variables. A surprising Ornstein–Uhlenbeck semigroup proof enables the explicit calculation of the sharp $a_1$ with a Poincaré assumption on $fd\gamma$, $f:{\mathbb{R}}^n\to {\mathbb{R}}_{+}$ [14]. Observe that the theorem with a Poincaré assumption implies the theorem with a log-$C^{1,1}$ assumption, where one of the inequalities for the log-$C^{1,1}$ assumption is precluded. In particular, it is one of the rare inequalities that highlight the sharp exponent and the constant of proportionality. However, the 4th moment assumption in Theorem 1 includes the Poincaré assumption and approximates the optimal moment assumption. There exists a sequence $\left\{u_k\right\}$ where

$${\delta }_{*}\left(u_k\right)\to 0$$

$${\int |x|}^2{|u_k\left(x\right)|}^2{e}^{-{\pi |x|}^2}dx\to \infty .$$

A more general stability inequality for probability measures that are absolutely continuous with respect to the Gaussian measure was obtained with a combination of a Wasserstein metric and entropy. The techniques in [10,13,14,15] involve optimal transport, semigroup theory, Fourier analysis, and probability. The recent proof of (3) in [11] is a fundamental achievement. One interesting feature is the lack of additional assumptions for (3) via the Bianchi–Egnell method. Since the logarithmic Sobolev inequality has appeared in different fields, e.g., optimal transport theory, probability, statistical mechanics, quantum field theory, Riemannian geometry, thermodynamics, and information theory, a large collection of articles recently investigated various stability formulations of similar inequalities: see, for instance, [7,10,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31].

2. Proofs

Proof of Lemma 1.

Assume

$${||u||}_{L^2\left(e^{-{\pi |x|}^2}dx\right)}=1$$

$$\int x|u{|}^2{e}^{-{\pi |x|}^2}dx=0;$$

set

$$u=c{e}^{a·x};$$

then, note that $a=0$, $c=1$ (via $u\ge 0$). In particular, $u=1$ is the only normalized and centered minimizer. Additionally, assuming the analog

$${\int |\nabla w|}^2{e}^{-{\pi |x|}^2}dx-{\pi \int |w|}^2{ln|w|}^2{e}^{-{\pi |x|}^2}dx\ge \kappa \int {|w-1|}^2{e}^{-{\pi |x|}^2}dx,$$

(5)

if $w\ge 0$,

$${||w||}_{L^2\left(e^{-{\pi |x|}^2}dx\right)}=1,$$

$${\int x|w|}^2{e}^{-{\pi |x|}^2}dx=0,$$

one also obtains (3): let

$$||u||={||u||}_{L^2\left(e^{-{\pi |x|}^2}dx\right)}$$

$$\alpha :={\int x|u|}^2{e}^{-{\pi |x|}^2}dx$$

$$w\left(x\right):=\frac{u(x+\frac{\alpha }{{||u||}^2})e^{-\frac{\pi }{{||u||}^2}(\alpha ·x+\frac{{|\alpha |}^2}{{2||u||}^2})}}{||u||}$$

and observe

$$||w||={||w||}_{L^2\left(e^{-{\pi |x|}^2}dx\right)}=1$$

$${\int x|w|}^2{e}^{-{\pi |x|}^2}dx=0$$

$$\begin{array}{cc}\hfill {\int |w-1|}^2{e}^{-{\pi |x|}^2}dx& =\frac{1}{{||u||}^2}\int |u(x+\frac{\alpha }{{||u||}^2})e^{-(\frac{\pi \alpha }{{||u||}^2}·x+\frac{{\pi |\alpha |}^2}{{2||u||}^4})}-||u||{|}^2{e}^{-{\pi |x|}^2}dx\hfill \\ \hfill & =\frac{1}{{||u||}^2}\int |u(x+\frac{\alpha }{{||u||}^2})-e^{(\frac{\pi \alpha }{{||u||}^2}·x+\frac{{\pi |\alpha |}^2}{{2||u||}^4}+ln||u||)}{|}^2{e}^{-2(\frac{\pi \alpha }{{||u||}^2}·x+\frac{{\pi |\alpha |}^2}{{2||u||}^4})}{e}^{-{\pi |x|}^2}dx\hfill \\ \hfill & =\frac{1}{{||u||}^2}\int |u(x+\frac{\alpha }{{||u||}^2})-e^{(\frac{\pi \alpha }{{||u||}^2}·x+\frac{{\pi |\alpha |}^2}{{2||u||}^4}+ln||u||)}{|}^2{e}^{-\pi |x+\frac{\alpha }{{||u||}^2}{|}^2}dx\hfill \\ \hfill & =\frac{1}{{||u||}^2}\int |u\left(x\right)-e^{(\frac{\pi \alpha }{{||u||}^2}·x-\frac{{\pi |\alpha |}^2}{{2||u||}^4}+ln||u||)}{|}^2{e}^{-{\pi |x|}^2}dx.\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill & {\int |\nabla w|}^2{e}^{-{\pi |x|}^2}dx\hfill \\ \hfill & =\frac{1}{{||u||}^2}\int |\nabla u(x+\frac{\alpha }{{||u||}^2})e^{-\frac{\pi }{{||u||}^2}(\alpha ·x+\frac{{|\alpha |}^2}{{2||u||}^2})}-\frac{\pi \alpha }{{||u||}^2}u(x+\frac{\alpha }{{||u||}^2})e^{-\frac{\pi }{{||u||}^2}(\alpha ·x+\frac{{|\alpha |}^2}{{2||u||}^2})}{|}^2{e}^{-{\pi |x|}^2}dx\hfill \\ \hfill & =\frac{1}{{||u||}^2}(\int |\nabla u(x+\frac{\alpha }{{||u||}^2}){|}^2{e}^{-\pi |x+\frac{\alpha }{{||u||}^2}{|}^2}dx-〈\frac{2\pi \alpha }{{||u||}^2},\int \nabla u(x+\frac{\alpha }{{||u||}^2})u(x+\frac{\alpha }{{||u||}^2})e^{-\pi |x+\frac{\alpha }{{||u||}^2}{|}^2}dx〉\hfill \\ \hfill & +\frac{{\pi }^2{|\alpha |}^2}{{||u||}^4}\int \left|u(x+\frac{\alpha }{{||u||}^2}){|}^2{e}^{-\pi |x+\frac{\alpha }{{||u||}^2}{|}^2}dx\right)\hfill \\ \hfill & =\frac{1}{{||u||}^2}\left({\int |\nabla u\left(x\right)|}^2{e}^{-{\pi |x|}^2}dx-〈\frac{2\pi \alpha }{{||u||}^2},\int \nabla u\left(x\right)u\left(x\right)e^{-{\pi |x|}^2}dx〉+\frac{{\pi }^2{|\alpha |}^2}{{||u||}^4}\int {|u\left(x\right)|}^2{e}^{-{\pi |x|}^2}dx\right).\hfill \end{array}$$

(7)

Moreover,

$$\int \nabla u\left(x\right)u\left(x\right)e^{-{\pi |x|}^2}dx=-\int \nabla u\left(x\right)u\left(x\right)e^{-{\pi |x|}^2}dx+2\pi \int x{|u\left(x\right)|}^2{e}^{-{\pi |x|}^2}dx$$

yields

$$\int \nabla u\left(x\right)u\left(x\right)e^{-{\pi |x|}^2}dx=\pi \int x{|u\left(x\right)|}^2{e}^{-{\pi |x|}^2}dx=\pi \alpha .$$

Observe

$$〈\frac{2\pi \alpha }{{||u||}^2},\int \nabla u\left(x\right)u\left(x\right)e^{-{\pi |x|}^2}dx〉=\frac{2{\pi }^2{|\alpha |}^2}{{||u||}^2};$$

therefore, (7) implies

$$\begin{array}{cc}\hfill & \frac{1}{{||u||}^2}\left({\int |\nabla u\left(x\right)|}^2{e}^{-{\pi |x|}^2}dx-〈\frac{2\pi \alpha }{{||u||}^2},\int \nabla u\left(x\right)u\left(x\right)e^{-{\pi |x|}^2}dx〉+\frac{{\pi }^2{|\alpha |}^2}{{||u||}^4}\int {|u\left(x\right)|}^2{e}^{-{\pi |x|}^2}dx\right)\hfill \\ \hfill & =\frac{1}{{||u||}^2}\left({\int |\nabla u\left(x\right)|}^2{e}^{-{\pi |x|}^2}dx-\frac{{\pi }^2{|\alpha |}^2}{{||u||}^2}\right)=\int {|\nabla w|}^2{e}^{-{\pi |x|}^2}dx.\hfill \end{array}$$

(8)

Analogously,

$$\begin{array}{cc}\hfill & {\int |w|}^{2}ln{|w|}^2{e}^{-{\pi |x|}^2}dx=\frac{1}{{||u||}^2}(\int |u(x+\frac{\alpha }{{||u||}^2}){|}^{2}ln\left(\frac{|u(x+\frac{\alpha }{{||u||}^2}){|}^2}{{||u||}^2}\right)e^{-\pi |x+\frac{\alpha }{{||u||}^2}{|}^2}dx\hfill \\ \hfill & -〈\frac{2\pi \alpha }{{||u||}^2},\int x|u(x+\frac{\alpha }{{||u||}^2}){|}^2{e}^{-\pi |x+\frac{\alpha }{{||u||}^2}{|}^2}dx〉-\frac{{\pi |\alpha |}^2}{{||u||}^2});\hfill \end{array}$$

$$\int x|u(x+\frac{\alpha }{{||u||}^2}){|}^2{e}^{-\pi |x+\frac{\alpha }{{||u||}^2}{|}^2}dx=\int \left(x-\frac{\alpha }{{||u||}^2}\right){|u\left(x\right)|}^2{e}^{-{\pi |x|}^2}dx=0;$$

thus,

$${\int |w|}^{2}ln{|w|}^2{e}^{-{\pi |x|}^2}dx=\frac{1}{{||u||}^2}\left({\int |u\left(x\right)|}^{2}ln\left(\frac{{|u\left(x\right)|}^2}{{||u||}^2}\right)e^{-{\pi |x|}^2}dx-\frac{{\pi |\alpha |}^2}{{||u||}^2}\right).$$

(9)

Observe that (9) and (8) imply

$${\int |\nabla w|}^2{e}^{-{\pi |x|}^2}dx-{\pi \int |w|}^{2}ln{|w|}^2{e}^{-{\pi |x|}^2}dx=\frac{1}{{||u||}^2}\left({\int |\nabla u|}^2{e}^{-{\pi |x|}^2}dx-\pi \int {|u|}^{2}ln\left(\frac{{|u|}^2}{{||u||}^2}\right)e^{-{\pi |x|}^2}dx\right)$$

(10)

and therefore, (10) and (6) combine with (5):

$${\int |\nabla u|}^2{e}^{-{\pi |x|}^2}dx-{\pi \int |u|}^{2}ln\left(\frac{{|u|}^2}{{||u||}_{L^2}^2}\right)e^{-{\pi |x|}^2}dx\ge \kappa \underset{a,c}{inf}\int {|u-c{e}^{a·x}|}^2{e}^{-{\pi |x|}^2}dx.$$

 □

Proof of Theorem 1.

1. Observe that with $dm=2^{\frac{n}{2}}{e}^{-{2\pi |x|}^2}dx$ and $f\in L^2\left(dm\right)$ normalized, setting

$$f=u\left(\sqrt{2}x\right),$$

$${\int |f|}^{2}dm=\int {|u|}^2{e}^{-{\pi |x|}^2}dx=1$$

and

$$\frac{1}{2\pi }{\int |\nabla f|}^{2}dm-{\int |f|}^2{ln|f|}^{2}dm=\frac{1}{\pi }{\int |\nabla u|}^2{e}^{-{\pi |x|}^2}dx-{\int |u|}^{2}ln{|u|}^2{e}^{-{\pi |x|}^2}dx.$$

Thus, ref. [13] (Theorem 1.17) implies

$$\begin{array}{cc}\hfill & {\pi \int |x|}^2{e}^{-{\pi |x|}^2}dx-{\pi \int |x|}^2{|u\left(x\right)|}^2{e}^{-{\pi |x|}^2}dx+\frac{1}{\pi }\int {|\nabla u\left(x\right)|}^2{e}^{-{\pi |x|}^2}dx\hfill \\ \hfill & \le \sqrt{2n}{\left(\frac{1}{\pi }{\int |\nabla u|}^2{e}^{-{\pi |x|}^2}dx-{\int |u|}^{2}ln{|u|}^2{e}^{-{\pi |x|}^2}dx\right)}^{\frac{1}{2}}\hfill \\ \hfill & +\left(\frac{1}{\pi }{\int |\nabla u|}^2{e}^{-{\pi |x|}^2}dx-{\int |u|}^{2}ln{|u|}^2{e}^{-{\pi |x|}^2}dx\right).\hfill \end{array}$$

(11)

Note that supposing

$$\left({\int |x|}^2{e}^{-{\pi |x|}^2}dx-{\int |x|}^2{|u_k\left(x\right)|}^2{e}^{-{\pi |x|}^2}dx\right)\to 0,$$

and ${\delta }_{*}\left(u_k\right)\to 0$, one then obtains, thanks to (11),

$$\int |\nabla u_k{\left(x\right)|}^2{e}^{-{\pi |x|}^2}dx\to 0$$

and this implies $H^1\left(e^{-{\pi |x|}^2}dx\right)$ convergence. Conversely, assuming $\sqrt{f}\in W^{1,2}({\mathbb{R}}^n,d\gamma )$$d\gamma ={\left(2\pi \right)}^{-\frac{n}{2}}{e}^{-\frac{{|x|}^2}{2}}dx$, the end of the proof of ([13] Proposition C.1) implies

$$\frac{1}{2}\int \frac{{|\nabla f|}^2}{f}d\gamma -\int flnfd\gamma \ge \frac{1}{4n}{\left(2\int flnfd\gamma +(m_2\left(\gamma \right)-m_2\left(fd\gamma \right))\right)}^2.$$

Thus, set

$$f_k={|u_k|}^2\left(\frac{x}{\sqrt{2\pi }}\right).$$

Then,

$$\int f_k{e}^{-\frac{{|x|}^2}{2}}{\left(2\pi \right)}^{-\frac{n}{2}}dx=\int {|u_k|}^2{e}^{-{\pi |x|}^2}dx=1,$$

$$\frac{1}{2}\int \frac{|\nabla f_k{|}^2}{f_k}d\gamma -\int f_{k}ln{f}_{k}d\gamma =\frac{1}{\pi }\int |\nabla u_k{|}^2{e}^{-{\pi |x|}^2}dx-\int |u_k{|}^{2}ln{|u_k|}^2{e}^{-{\pi |x|}^2}dx.$$

In particular,

$$\begin{array}{cc}\hfill & \frac{1}{4n}{\left(2\int |u_k{|}^{2}ln|u_k{|}^2{e}^{-{\pi |x|}^2}dx+{2\pi \int |x|}^2{e}^{-{\pi |x|}^2}dx-{2\pi \int |x|}^2{|u_k|}^2{e}^{-{\pi |x|}^2}dx\right)}^2\hfill \\ \hfill & \le \frac{1}{\pi }\int |\nabla u_k{|}^2{e}^{-{\pi |x|}^2}dx-\int |u_k{|}^{2}ln{|u_k|}^2{e}^{-{\pi |x|}^2}dx.\hfill \end{array}$$

(12)

Observe via $H^1\left(e^{-{\pi |x|}^2}dx\right)$ convergence and the LSI that

$$\int |u_k{|}^{2}ln{|u_k|}^2{e}^{-{\pi |x|}^2}dx\to 0,$$

and thus, thanks to

$${\delta }_{*}\left(u_k\right)=\left(\frac{1}{\pi }\int |\nabla u_k{|}^2{e}^{-{\pi |x|}^2}dx-\int |u_k{|}^{2}ln{|u_k|}^2{e}^{-{\pi |x|}^2}dx\right)\to 0,$$

(12) implies

$$\left({\int |x|}^2{e}^{-{\pi |x|}^2}dx-{\int |x|}^2{|u_k|}^2{e}^{-{\pi |x|}^2}dx\right)\to 0.$$

2.

$${\int |x|}^4{|u\left(x\right)|}^2{e}^{-{\pi |x|}^2}dx\le A$$

implies

$$\begin{array}{cc}\hfill & {|\pi \int |x|}^2{e}^{-{\pi |x|}^2}dx-{\pi \int |x|}^2{|u\left(x\right)|}^2{e}^{-{\pi |x|}^2}dx|={\pi |\int |x|}^2[1-|u|][1+|u|]e^{-{\pi |x|}^2}dx|\hfill \\ \hfill & \le \pi {\left[{\int |x|}^4{|1+|u||}^2{e}^{-{\pi |x|}^2}dx\right]}^{1/2}{\left[{\int |1-|u||}^2{e}^{-{\pi |x|}^2}dx\right]}^{1/2}\hfill \\ \hfill & \le \pi {\left[2(A+\int |x{|}^4{e}^{-{\pi |x|}^2}dx)\right]}^{1/2}{\left[{\int |1-|u||}^2{e}^{-{\pi |x|}^2}dx\right]}^{1/2}.\hfill \end{array}$$

In particular, ref. [13] (Theorem 1.17), Lemma 1 ($|u|\ge 0$, $|\nabla |u||=|\nabla u|$ a.e.), and ([11] Theorem 2) imply

$$\begin{array}{cc}\hfill & \frac{1}{\pi }\int {|\nabla u\left(x\right)|}^2{e}^{-{\pi |x|}^2}dx\hfill \\ \hfill & \le {|\pi \int |x|}^2{e}^{-{\pi |x|}^2}dx-{\pi \int |x|}^2{|u\left(x\right)|}^2{e}^{-{\pi |x|}^2}dx|+\sqrt{2n}{\left(\frac{1}{\pi }{\int |\nabla u|}^2{e}^{-{\pi |x|}^2}dx-{\int |u|}^{2}ln{|u|}^2{e}^{-{\pi |x|}^2}dx\right)}^{\frac{1}{2}}\hfill \\ \hfill & +\left(\frac{1}{\pi }{\int |\nabla u|}^2{e}^{-{\pi |x|}^2}dx-{\int |u|}^{2}ln{|u|}^2{e}^{-{\pi |x|}^2}dx\right)\hfill \\ \hfill & \le \pi {\left[2(A+\int |x{|}^4{e}^{-{\pi |x|}^2}dx)\right]}^{1/2}{\left[{\int |1-|u||}^2{e}^{-{\pi |x|}^2}dx\right]}^{1/2}+\hfill \\ \hfill & \sqrt{2n}{\left(\frac{1}{\pi }{\int |\nabla u|}^2{e}^{-{\pi |x|}^2}dx-{\int |u|}^{2}ln{|u|}^2{e}^{-{\pi |x|}^2}dx\right)}^{\frac{1}{2}}+\left(\frac{1}{\pi }{\int |\nabla u|}^2{e}^{-{\pi |x|}^2}dx-{\int |u|}^{2}ln{|u|}^2{e}^{-{\pi |x|}^2}dx\right)\hfill \\ \hfill & \le \pi {\left[2(A+\int |x{|}^4{e}^{-{\pi |x|}^2}dx)\right]}^{1/2}\sqrt{\frac{\pi }{\kappa }}{\left[\frac{1}{\pi }{\int |\nabla u|}^2{e}^{-{\pi |x|}^2}dx-{\int |u|}^{2}ln{|u|}^2{e}^{-{\pi |x|}^2}dx\right]}^{1/2}+\hfill \\ \hfill & \sqrt{2n}{\left(\frac{1}{\pi }{\int |\nabla u|}^2{e}^{-{\pi |x|}^2}dx-{\int |u|}^{2}ln{|u|}^2{e}^{-{\pi |x|}^2}dx\right)}^{\frac{1}{2}}+\left(\frac{1}{\pi }{\int |\nabla u|}^2{e}^{-{\pi |x|}^2}dx-{\int |u|}^{2}ln{|u|}^2{e}^{-{\pi |x|}^2}dx\right).\hfill \end{array}$$

3. Suppose that $fd\gamma$ is a probability measure and $T=\nabla \mathsf{\Phi }$ is the Brenier map, which pushes forward $fd\gamma$ to $d\gamma$. The proof of the LSI via optimal transport [4] implies

$${\int |T\left(x\right)-x+\nabla lnf|}^{2}fd\gamma \le 2\delta \left(f\right).$$

The argument is as follows: Define $\theta :=\mathsf{\Phi }-\frac{1}{2}{|x|}^2$. Note that

$$\det (I+D^2\theta \left(x\right))e^{{-|x+\nabla \theta \left(x\right)|}^2/2}=f\left(x\right)e^{-{|x|}^2/2}.$$

Next, taking the log and then integrating:

$$\begin{array}{cc}\hfill \int flnfd\gamma & \le \int f\left[\Delta \theta -x·\nabla \theta \right]d\gamma -\frac{1}{2}\int {|\nabla \theta |}^{2}fd\gamma \hfill \\ \hfill & =-\int \nabla \theta ·\nabla fd\gamma -\frac{1}{2}\int {|\nabla \theta |}^{2}fd\gamma \hfill \\ \hfill & =-\frac{1}{2}\int |\nabla \theta +\frac{\nabla f}{f}{|}^{2}fd\gamma +\frac{1}{2}\int \frac{{|\nabla f|}^2}{f}d\gamma .\hfill \end{array}$$

Therefore,

$$\frac{1}{2}\int \frac{{|\nabla f|}^2}{f}d\gamma -\int flnfd\gamma \ge \frac{1}{2}\int |T-x+\nabla lnf{|}^{2}fd\gamma .$$

Thus, Jensen’s inequality yields

$$\delta \left(f\right)\ge \frac{1}{2}\int {|T(x)-x+\nabla lnf|}^{2}fd\gamma \ge \frac{1}{2}{\left(\int |T(x)-x+\nabla lnf|fd\gamma \right)}^2;$$

in particular,

$$\int |T\left(x\right)-x+\nabla lnf|fd\gamma \le \sqrt{2\delta \left(f\right)}.$$

Observe that T is the Brenier map; nevertheless, if one considers the Monge cost, it yields an upper bound on $W_1$ (in the one-dimensional case, the inequality is an equality):

$$W_1(fd\gamma ,\gamma )\le \int |T\left(x\right)-x|fd\gamma .$$

This then implies

$$\begin{array}{cc}\hfill W_1(fd\gamma ,\gamma )& \le \sqrt{2\delta \left(f\right)}+\int |\nabla f|d\gamma \hfill \\ \hfill & \le \sqrt{2\delta \left(f\right)}+{\left(\int \frac{{|\nabla f|}^2}{f}d\gamma \right)}^{\frac{1}{2}}.\hfill \end{array}$$

Therefore, let $\left\{f_{k}d\gamma \right\}$ be a sequence of probability measures with

$$\underset{k}{lim\; inf}{W}_1(f_{k}d\gamma ,\gamma )\to \infty$$

$$\delta \left(f_k\right)\to 0,$$

Ref. [25]; thus,

$$\underset{k}{lim\; inf}\int \frac{|\nabla f_k{|}^2}{f_k}d\gamma \to \infty .$$

Set

$$f_k={|u_k|}^2\left(\frac{x}{\sqrt{2\pi }}\right).$$

One then has

$$\int f_k{e}^{-\frac{{|x|}^2}{2}}{\left(2\pi \right)}^{-\frac{n}{2}}dx=\int {|u_k|}^2{e}^{-{\pi |x|}^2}dx=1$$

$$\int \frac{|\nabla f_k{|}^2}{f_k}d\gamma =\frac{2}{\pi }\int {|\nabla u_k|}^2{e}^{-{\pi |x|}^2}dx.$$

Therefore,

$$\underset{k}{lim\; inf}\int {|\nabla u_k|}^2{e}^{-{\pi |x|}^2}dx\to \infty$$

and this directly implies

$$||u_k{||}_{H^1\left(e^{-{\pi |x|}^2}dx\right)}\to \infty ,$$

$${\delta }_{*}\left(u_k\right)\to 0.$$

 □

Remark 1.

Observe that $H^1$ convergence is equivalent to $W_2$ convergence (when ${\delta }_{*}\left(u_k\right)\to 0$). The first $W_2$ bound was obtained in [10].

Proof of  Theorem 2.

If $a>0$, define

$$u_a\left(x\right)={(2a+1)}^{\frac{n}{4}}{e}^{-{a\pi |x|}^2}.$$

Now,

$$||u_a{||}_{L^2\left(e^{-{\pi |x|}^2}dx\right)}=1,$$

$$\int x|u_a{|}^2{e}^{-{\pi |x|}^2}dx=0,$$

$$\begin{array}{cc}\hfill & \frac{\left(\int |\nabla u_a{|}^2{e}^{-{\pi |x|}^2}dx-\pi \int |u_a{|}^{2}ln{|u_a|}^2{e}^{-{\pi |x|}^2}dx\right)}{\int |u_a{-1|}^2{e}^{-{\pi |x|}^2}dx}\hfill \\ \hfill & =\pi \left(\frac{\frac{2n{a}^2}{2a+1}-\frac{n}{2}ln(2a+1)+\frac{na}{2a+1}}{2-2\frac{{(2a+1)}^{\frac{n}{4}}}{{(a+1)}^{\frac{n}{2}}}}\right)\hfill \\ \hfill & =\frac{\pi }{2}\frac{2n{a}^2-\frac{n}{2}(2a+1)ln(2a+1)+na}{2a+1-\frac{{(2a+1)}^{\frac{n+4}{4}}}{{(a+1)}^{\frac{n}{2}}}},\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \underset{a\to 0^{+}}{lim}\frac{\left(\int |\nabla u_a{|}^2{e}^{-{\pi |x|}^2}dx-\pi \int |u_a{|}^{2}ln{|u_a|}^2{e}^{-{\pi |x|}^2}dx\right)}{\int |u_a{-1|}^2{e}^{-{\pi |x|}^2}dx}\hfill \\ \hfill & \underset{a\to 0^{+}}{lim}\frac{\pi }{2}\frac{4na-nln(2a+1)}{2-\left(\frac{n+4}{2}\frac{{(2a+1)}^{\frac{n}{4}}}{{(a+1)}^{\frac{n}{2}}}-\frac{n}{2}\frac{{(2a+1)}^{\frac{n+4}{4}}}{{(a+1)}^{\frac{n}{2}+1}}\right)}\hfill \\ \hfill & \underset{a\to 0^{+}}{lim}-\pi \frac{2n-n\frac{1}{2a+1}}{\left(\left(\frac{{(a+1)}^{\frac{n}{2}}\frac{n}{2}\frac{n+4}{2}{(2a+1)}^{\frac{n-4}{4}}-{(2a+1)}^{\frac{n}{4}}\frac{n+4}{2}\frac{n}{2}{(a+1)}^{\frac{n-2}{2}}}{{(a+1)}^n}\right)-\left(\frac{{(a+1)}^{\frac{n}{2}+1}\frac{n}{2}\frac{n+4}{2}{(2a+1)}^{\frac{n}{4}}-{(2a+1)}^{\frac{n+4}{4}}\frac{n}{2}{(a+1)}^{\frac{n}{2}}(\frac{n}{2}+1)}{{(a+1)}^{n+2}}\right)\right)}\hfill \\ \hfill & =2\pi .\hfill \end{array}$$

In particular, assume by contradiction that

$$\omega \left({\int |\nabla u|}^2{e}^{-{\pi |x|}^2}dx-{\pi \int |u|}^{2}ln{|u|}^2{e}^{-{\pi |x|}^2}dx\right)\ge \kappa \int {|u-1|}^2{e}^{-{\pi |x|}^2}dx$$

when u is normalized and centered, with

$$\omega \left(a\right)=o\left(a\right).$$

Hence,

$$\begin{array}{cc}\hfill & \kappa \le \frac{\omega \left(\int |\nabla u_a{|}^2{e}^{-{\pi |x|}^2}dx-\pi \int |u_a{|}^{2}ln{|u_a|}^2{e}^{-{\pi |x|}^2}dx\right)}{\int |u_a{-1|}^2{e}^{-{\pi |x|}^2}dx}\hfill \\ \hfill & =\frac{\omega \left(\int |\nabla u_a{|}^2{e}^{-{\pi |x|}^2}dx-\pi \int |u_a{|}^{2}ln{|u_a|}^2{e}^{-{\pi |x|}^2}dx\right)}{\int |\nabla u_a{|}^2{e}^{-{\pi |x|}^2}dx-\pi \int |u_a{|}^{2}ln{|u_a|}^2{e}^{-{\pi |x|}^2}dx}\frac{\int |\nabla u_a{|}^2{e}^{-{\pi |x|}^2}dx-\pi \int |u_a{|}^{2}ln{|u_a|}^2{e}^{-{\pi |x|}^2}dx}{\int |u_a{-1|}^2{e}^{-{\pi |x|}^2}dx}\hfill \\ \hfill & \to 0\hfill \end{array}$$

as $a\to 0^{+}$; this therefore contradicts $\kappa >0$. □

3. Conclusions

• A necessary and sufficient condition to have $H^1$ convergence via the logarithmic Sobolev deficit is identified in 1, Theorem 1.
• A sharp $H^1$-estimate is obtained in 2, Theorem 1.
• An explicit example is constructed in 3, Theorem 1, proving that, in general, there is blow-up in $H^1$; in particular, one may not preclude the $\alpha$ in $a_{\alpha }$ from appearing in $2,$ Theorem 1.
• An explicit example is constructed in Theorem 2, proving that the $L^p$ stability of Dolbeault, Esteban, Figalli, Frank, and Loss [11] is sharp.