# Stability of the Poincaré–Korn inequality

Thomas A. Courtade111University of California, Berkeley, United States,
Department of Electrical Engineering and Computer Sciences.
<EMAIL_ADDRESS>Max Fathi222Université Paris Cité and Sorbonne
Université, CNRS, Laboratoire Jacques-Louis Lions and Laboratoire de
Probabilités, Statistique et Modélisation, F-75013 Paris, France; and DMA,
École normale supérieure, Université PSL, CNRS, 75005 Paris, France; and
Institut Universitaire de France.
<EMAIL_ADDRESS>
(May 2nd 2024)

###### Abstract

We resolve a question of Carrapatoso et al. [7] on Gaussian optimality for the
sharp constant in Poincaré-Korn inequalities, under a moment constraint. We
also prove stability, showing that measures with near-optimal constant are
quantitatively close to standard Gaussian.

##  1 Introduction and Main Result

Let $\mu$ be a centered Borel probability measure on $\mathbb{R}^{n}$, $n\geq
2$. Let $\mathcal{A}$ denote the set of antisymmetric linear maps from
$\mathbb{R}^{n}$ to itself. That is,

$\mathcal{A}:=\\{x\mapsto Ax~{};A\in M_{n\times n}(\mathbb{R}),A=-A^{T}\\}.$

Further define the linear space of vector-valued functions

$\mathcal{C}:=\\{u:\mathbb{R}^{n}\to\mathbb{R}^{n}~{};~{}\mbox{$u$
differentiable, $\int ud\mu=0$, and
$\|\nabla_{s}u\|_{L^{2}(\mu)}^{2}<\infty$}\\},$

where $\nabla_{s}u:=\frac{1}{2}((\nabla u)-(\nabla u)^{T})$ is the symmetrized
gradient of the vector-valued function $u:\mathbb{R}^{n}\to\mathbb{R}^{n}$.
Since $\mu$ is centered333The persistent centering assumption comes without
any loss of generality, and is only made for convenience., $\mathcal{A}$ is a
closed linear subspace of $\mathcal{C}$.

######  Definition 1.

A centered Borel probability measure $\mu$ on $\mathbb{R}^{n}$ satisfies a
Poincaré–Korn inequality with constant $C$ if

$\displaystyle\inf_{a\in\mathcal{A}}\|u-a\|^{2}_{L^{2}(\mu)}\leq
2C\|\nabla_{s}u\|_{L^{2}(\mu)}^{2},~{}~{}\mbox{for all $u\in\mathcal{C}$.}$
(1)

The Poincaré–Korn constant associated to $\mu$, denoted $C_{PK}(\mu)$, is the
smallest constant $C$ such that (1) holds.

This type of inequality was introduced in [7]. It is inspired by Poincaré
inequalities, which control variances of scalar valued functions by the
$L^{2}$ norm of their gradient, and the Korn inequality from continuous
mechanics, which controls the $L^{2}$ norm of the gradient of a vector field
satisfying some boundary condition by its symmetric part. Both inequalities
have found many applications in analysis. They were both originally introduced
for uniform measures on domains, but can be extended to general probability
densities. In this form, Poincaré inequalities are

$\int{f^{2}d\mu}-\left(\int{fd\mu}\right)^{2}\leq C_{P}(\mu)\int{|\nabla
f|^{2}d\mu},\hskip 8.53581pt\forall
f:\mathbb{R}^{d}\longrightarrow\mathbb{R}^{d},$ (2)

where $C_{P}(\mu)$ is the Poincaré constant of $\mu$, and the right-hand side
is to be understood as $+\infty$ if it is not well-defined for the function
$f$. Korn inequalities are of the form

$\inf_{a\in\mathcal{A}}\|\nabla(u-a)\|^{2}_{L^{2}(\mu)}\leq
C_{K}\|\nabla_{s}u\|_{L^{2}(\mu)}^{2},~{}~{}\mbox{for all $u\in\mathcal{C}$.}$
(3)

Poincaré inequalities are now a very classical tool in probability and
functional analysis, and applications include concentration of measure
inequalities and rates of convergence to equilibrium for Markov processes. We
refer to the monograph [2] for background and many developments. On the other
hand, classical Korn inequalities are a tool in kinetic theory and fluid
mechanics, going back to [19]. We shall make no attempt to survey the vast
literature, and refer to [18] for some background. Best constants were
investigated for example in [20]. Weighted Korn inequalities were recently
introduced in [7], motivated by hypocoercivity problems in kinetic theory.

Our definition of the Poincaré–Korn constant differs from that in [7] by a
factor of 2. This is done to give unit normalization with respect to the
standard Gaussian measure $\gamma$, defined by

$d\gamma(x):=\frac{1}{(2\pi)^{n/2}}e^{-|x|^{2}/2}dx,~{}~{}~{}~{}x\in\mathbb{R}^{n}.$

Probability measures with sufficiently regular potentials admit a finite
Poincaré–Korn constant if they satisfy a Poincaré inequality with finite
constant.

######  Proposition 1.

[7, Theorem 1] We have $C_{PK}(\gamma)=1$ in any dimension $n\geq 2$.
Moreover, if a centered probability measure with density $d\mu=e^{-\phi}dx$ of
class $C^{2}$ satisfies

$\forall\epsilon>0,\exists
C_{\epsilon}>0:~{}~{}\|\nabla^{2}\phi(x)\|^{2}\leq\epsilon|\nabla\phi(x)|^{2}+C_{\epsilon},\hskip
8.53581pt\forall x\in\mathbb{R}^{n},$

then $C_{P}(\mu)<\infty\Rightarrow C_{PK}(\mu)<\infty$.

In [7], the following conjecture is proposed regarding the rigidity of the
Poincaré–Korn inequality:

######  Conjecture 1.

If $d\mu=e^{-\phi}dx$ is centered and isotropic (i.e., $\int xd\mu=0$ and
$\int xx^{T}d\mu=\operatorname{Id}$), and satisfies
$\nabla^{2}\phi\geq\operatorname{Id}$, then $C_{PK}(\mu)\geq C_{PK}(\gamma)$,
with equality only if $\mu=\gamma$.

It turns out that the conjectured statement is indeed true, but the hypotheses
are too strong to capture salient rigidity properties of the Poincaré–Korn
constant. Namely, the following can be derived as a consequence of known
stability results for the Bakry–Émery theorem on $\mathbb{R}^{n}$ [10], or
using Caffarelli’s contraction theorem (see Appendix A). We make no claim of
originality for this statement, which was known in some communities.

######  Proposition 2.

If $d\mu=e^{-\phi}dx$ is centered, isotropic, and satisfies
$\nabla^{2}\phi\geq\operatorname{Id}$, then $\mu=\gamma$.

Therefore, to study rigidity of the Poincaré–Korn inequality, the assumption
of a uniformly convex potential in Conjecture 1 should be replaced by
something else. A natural choice is a moment assumption, which we now define.

######  Definition 2 (Moment Assumption).

We say that $\mu$ satisfies the moment assumption if, for all $1\leq i,j,k\leq
n$,

$\displaystyle\int x_{i}d\mu$ $\displaystyle=\int x_{i}d\gamma=0;$
$\displaystyle\int x_{i}x_{j}d\mu$ $\displaystyle=\int
x_{i}x_{j}d\gamma=\delta_{ij};$ $\displaystyle\int x_{i}x_{j}x_{k}d\mu$
$\displaystyle=\int x_{i}x_{j}x_{k}d\gamma=0;$

and, when $i\neq j$,

$\displaystyle\int(x_{i}^{2}+x_{j}^{2})x_{j}^{2}d\mu$
$\displaystyle=\int(x_{i}^{2}+x_{j}^{2})x_{j}^{2}d\gamma=4.$

######  Remark 1.

The first two lines of the moment assumption correspond to $\mu$ being
centered and isotropic. The third and fourth lines in the moment assumption
ensure that $\mu$ and $\gamma$ share mixed third moments and select mixed
fourth moments, respectively.

There are many interesting probability measures that satisfy the moment
assumption. For example, any product measure whose individual factors share
moments up to order 4 with the standard normal will satisfy the moment
assumption (and so will mixtures of these measures, and so forth…). Thus, a
nontrivial reformulation of Conjecture 1 is as follows:

######  Question 1.

If $\mu$ satisfies the moment assumption, is the lower bound $C_{PK}(\mu)\geq
C_{PK}(\gamma)$ true, with equality only if $\mu=\gamma$?

The moment assumption is motivated by the form of extremal functions in the
Gaussian Poincaré-Korn inequality. In particular, by considering these as test
functions in the Poincaré–Korn inequality for $\mu$, the inequality
$C_{PK}(\mu)\geq C_{PK}(\gamma)$ is a consequence of the moment assumption
(see Proposition 3 in the sequel). So, it is the rigidity phenomenon that is
interesting. We remark that the work of Serres [22] already highlights that
given a reference measure satisfying a Poincaré inequality with known sharp
constant and extremal function, it is possible to study stability of
functional inequalities within classes of measures for which the moments of
the extremal function match with those under the reference measure.

In this note, we resolve Question 1 in the affirmative, and further establish
quantitative stability of the Poincaré–Korn constant. Such stability results
on functional inequalities have been the subject of some recent attention in
analysis. For example, there have been many results on stability for sharp
functions in classical functional inequalities, including Sobolev inequalities
and isoperimetric inequalities, see [16, 17] for surveys. Stability results
under moment constraints have been studied for Poincaré inequalities [24, 5,
12, 22], eigenvalues of diffusion operators [23], as well as fractional [1]
and free [9] Poincaré inequalities. These have been obtained using the
combination of Stein’s method and variational arguments that we shall use
here. In another direction, there have been stability results for sharp
constants under convexity or curvature assumptions [8, 11, 15, 21].

Our stability result is with respect to the Zolotarev distance of order 2,
which controls the same topology as the more familiar $W_{2}$
Kantorovich–Wasserstein distance [3].

######  Definition 3.

For two Borel probability measures $\mu,\nu$ on $\mathbb{R}^{n}$, the
Zolotarev distance of order 2 is defined as

$d_{Zol,2}(\mu,\nu):=\sup_{\sup_{x}\|\nabla^{2}f(x)\|_{2}\leq 1}\int
fd\mu-\int fd\nu,$

where the supremum is over all $f:\mathbb{R}^{n}\to\mathbb{R}$ in $C^{2}$ with
$\sup_{x\in\mathbb{R}^{n}}\|\nabla^{2}f(x)\|_{2}\leq 1$.

Our main result is as follows.

######  Theorem 1 (Stability).

If $\mu$ satisfies the moment assumption, then $C_{PK}(\mu)\geq
C_{PK}(\gamma)=1$, and

$d_{Zol,2}(\mu,\gamma)\leq
c\,n^{2}\sqrt{C_{PK}(\mu)(C_{PK}(\mu)-C_{PK}(\gamma))},$

where $c$ is a universal constant.

The following is now immediate, and affirmatively answers Question 1.

######  Corollary 1 (Rigidity).

If $\mu$ satisfies the moment assumption, then $C_{PK}(\mu)\geq
C_{PK}(\gamma)$, with equality only if $\mu=\gamma$.

Acknowledgments. We thank Jean Dolbeault for telling us about this problem. TC
acknowledges NSF-CCF 1750430, the hospitality of the Laboratoire de
Probabilités, Statistique et Modélisation (LPSM) at the Université Paris Cité,
and the Invited Professor program of the Fondation Sciences Mathématiques de
Paris (FSMP). MF was supported by the Agence Nationale de la Recherche (ANR)
Grant ANR-23-CE40-0003 (Project CONVIVIALITY). This work has also received
support under the program “Investissement d’Avenir” launched by the French
Government and implemented by ANR, with the reference ANR‐18‐IdEx‐0001 as part
of its program Emergence.

##  2 Proof of Main Result

###  2.1 Notation

For a vector-valued function $u:\mathbb{R}^{n}\to\mathbb{R}^{n}$, we define

$\|u\|^{2}_{L^{2}(\mu)}:=\int|u|^{2}d\mu,$

where $|x|$ denotes the Euclidean length of $x\in\mathbb{R}^{n}$. Likewise,
for a matrix-valued function $U:\mathbb{R}^{n}\to M_{n\times n}(\mathbb{R})$,
we define

$\|U\|^{2}_{L^{2}(\mu)}:=\int U\cdot Ud\mu,$

where ‘$\cdot$’ will denote the scalar product between matrices (i.e., $A\cdot
B:=\sum_{ij}[A]_{ij}[B]_{ij}$, for identically dimensioned matrices $A,B\in
M_{m\times n}(\mathbb{R})$). The identity matrix is denoted by
$\operatorname{Id}$. All vectors are represented in matrix form as column
vectors.

In what follows, for $1\leq i,j\leq n$, we let $\delta_{ij}$ denote the usual
Kronecker delta function:

$\delta_{ij}:=\begin{cases}1&\mbox{if $i=j$}\\\
0&\mbox{otherwise.}\end{cases}$

###  2.2 Remarks on the general approach

In [11], the authors introduced a general approach for establishing stability
of functional inequalities based on approximate integration-by-parts
identities and Stein’s method; the ideas can also be found in the one-
dimensional results of [24, 5]. Interested readers are referred to [11] for an
overview of the method in abstract settings. We follow this general approach
here. Namely, the first step of the proof (Subsection 2.3) is dedicated to
proving an approximate Stein identity, which mimics the integration by parts
formula for the Gaussian measure on Hermite polynomials of degree 2. The
second step (Subsection 2.4) describes our implementation of Stein’s method
that yields the main stability result.

###  2.3 An approximate Stein identity

For a differentiable matrix-valued function $V:\mathbb{R}^{n}\to M_{n\times
n}(\mathbb{R})$ and $x\in\mathbb{R}^{n}$, we write $(x\nabla^{T})\cdot V$ to
denote the scalar product of the operator $(x\nabla^{T})$ and the function
$V$; that is,

$(x\nabla^{T})\cdot
V=\sum_{i,j=1}^{n}[x\nabla^{T}]_{ij}[V]_{ij}.:=\sum_{i,j=1}^{n}x_{i}\partial_{j}[V]_{ij}.$

The following lemma is the main result of this section.

######  Lemma 1 (Approximate Stein Identity).

Let $\mu$ satisfy the moment assumption. If $V:\mathbb{R}^{n}\to M_{n\times
n}(\mathbb{R})$ is a matrix-valued function such that each coordinate
$[V]_{ij}:\mathbb{R}^{n}\to\mathbb{R}$ is integrable and differentiable, with
$\|\nabla[V]_{ij}\|_{L^{2}(\mu)}<\infty$, then

$\displaystyle\left|\int(xx^{T}-\operatorname{Id})\cdot
Vd\mu-\int(x\nabla^{T})\cdot Vd\mu\right|\leq
K\sqrt{C_{PK}(C_{PK}-1)}\sum_{i,j=1}^{n}\|\nabla[V]_{ij}\|_{L^{2}(\mu)},$

for $K:=(1+10/\sqrt{3})<7$.

To provide some perspective, we remark that the classical Stein identity

$\int x\cdot\phi d\gamma=\int\operatorname{Id}\cdot\nabla\phi
d\gamma,~{}~{}~{}\phi:\mathbb{R}^{n}\to\mathbb{R}^{n}$

applied to the test function $\phi(x)=V^{T}(x)x$ gives

$\displaystyle\int(xx^{T}-\operatorname{Id})\cdot
Vd\gamma=\int(x\nabla^{T})\cdot Vd\gamma.$

Under certain moment assumptions, this consequence of the classical Stein
identity also characterizes the Gaussian measure, so may therefore be regarded
as one of many “Stein identities”. It is for this reason that we refer to
Lemma 1 as an “approximate Stein identity”, which becomes more faithful as
$C_{PK}(\mu)$ approaches 1.

The proof of the approximate Stein identity rests on an approximate
integration by parts formula enjoyed by near-extremizers of the Poincaré–Korn
inequality. To develop it, we assume henceforth that $\mu$ has finite second
moments. For a function $u\in\mathcal{C}$, define

$a_{u}:=\arg\min_{a\in\mathcal{A}}\|u-a\|_{L^{2}(\mu)}.$

Since $\mathcal{A}$ is a closed linear subspace of $L^{2}(\mu)$ and
$\mathcal{C}\subset L^{2}(\mu)$, the function $a_{u}$ exists and is unique.
Moreover, as a projection onto $\mathcal{A}$, the map $u\mapsto a_{u}$ is
linear and equal to identity on $\mathcal{A}$.

The following is an approximate integration by parts formula satisfied by
near-extremizers of the Poincaré–Korn inequality; it does not require the
moment assumption, and may therefore be of independent interest.

######  Lemma 2.

Let $\mu$ be a centered Borel probability measure on $\mathbb{R}^{n}$ with
finite second moments, and $C_{PK}(\mu)<\infty$. If $\epsilon\geq 0$ and
$u\in\mathcal{C}$ satisfy

$\displaystyle(2-(\epsilon/2)^{2})C_{PK}(\mu)\|\nabla_{s}u\|_{L^{2}(\mu)}^{2}\leq\inf_{a\in\mathcal{A}}\|u-a\|^{2}_{L^{2}(\mu)},$
(4)

then for every $v\in\mathcal{C}$, we have

$\left|\int(u-a_{u})\cdot(v-a_{v})d\mu-2C_{PK}(\mu)\int(\nabla_{s}u)\cdot(\nabla_{s}v)d\mu\right|\leq\epsilon
C_{PK}(\mu)\|\nabla_{s}u\|_{L^{2}(\mu)}\|\nabla_{s}v\|_{L^{2}(\mu)}.$

We briefly remark that if $u\in\mathcal{C}$ is an extremizer in the
Poincaré–Korn inequality, then we have the following (exact) integration by
parts formula:

$\int(u-a_{u})\cdot(v-a_{v})d\mu=2C_{PK}(\mu)\int(\nabla_{s}u)\cdot(\nabla_{s}v)d\mu,~{}~{}\forall
v\in\mathcal{C}.$

However, we see no reason to expect that nontrivial extremizers exist in the
Poincaré–Korn inequality for general $\mu$, which motivates the approximation
in Lemma 2. Note that there are examples of measures for which the classical
Poincaré inequality has no non-trivial extremal function, including for
example the exponential measure, see [2, Section 4.4.1].

###### Proof.

We’ll abbreviate $C_{PK}:=C_{PK}(\mu)$ for convenience. Begin by defining the
quotient space $\mathcal{Q}:=\mathcal{C}/\mathcal{A}$. Note that
$\mathcal{A}\subset\ker(\nabla_{s})$, so we may define a linear operator $D$
on $\mathcal{Q}$ via

$D[u]:=\nabla_{s}u,~{}~{}~{}u\in\mathcal{C},$

where $[u]\in\mathcal{Q}$ denotes the coset of $u$. Observe that

$\langle[u],[v]\rangle:=\int D[u]\cdot D[v]d\mu,~{}~{}[u],[v]\in\mathcal{Q},$

defines an inner product on $\mathcal{Q}\times\mathcal{Q}$. Bilinearity and
symmetry are self-evident, and positive-definiteness follows from the
Poincaré–Korn inequality, which states

$\|[u]\|^{2}:=\langle[u],[u]\rangle\geq\frac{1}{2C_{PK}}\inf_{a\in\mathcal{A}}\|u-a\|^{2}_{L^{2}(\mu)}.$

Since the quantity on the right is the quotient norm, positive-definiteness
follows. Hence, we are justified in defining a Hilbert space $\mathcal{H}$ as
the completion of $\mathcal{Q}$ in the norm $\|\cdot\|$, and extending the
inner product $\langle\cdot,\cdot\rangle$ to $\mathcal{H}$.

Now, define the operator $T:q\in\mathcal{Q}\mapsto Tq$ by

$T[u]:=u-a_{u},~{}~{}~{}u\in\mathcal{C}.$

The operator $T$ is linear, and well-defined on $\mathcal{Q}$ since $u=a_{u}$
for $u\in\mathcal{A}$. Now, fix $u\in\mathcal{C}$. The operator

$[v]\in\mathcal{Q}\mapsto\int T[u]\cdot T[v]d\mu$

is a bounded linear operator on $\mathcal{Q}$. Indeed, boundedness follows by
the Cauchy–Schwarz and Poincaré–Korn inequalities as

$\int T[u]\cdot T[v]d\mu\leq 2C_{PK}\|[u]\|\|[v]\|,~{}~{}\mbox{for
all~{}}v\in\mathcal{C}.$

Linearity now follows by linearity of $T$ and the integral. By density of
$\mathcal{Q}$ in $\mathcal{H}$ and the Riesz representation theorem, there is
$h\in\mathcal{H}$ with $\|h\|\leq\|[u]\|$ such that

$\displaystyle\int T[u]\cdot T[v]d\mu=2C_{PK}\langle
h,[v]\rangle,~{}~{}~{}\mbox{for all~{}}v\in\mathcal{C}.$

Hence, for any $v\in\mathcal{C}$, we have

$\displaystyle\int T[u]\cdot T[v]d\mu-2C_{PK}(\mu)\langle[u],[v]\rangle$
$\displaystyle=2C_{PK}(\mu)\langle h-[u],[v]\rangle$ $\displaystyle\leq
2C_{PK}(\mu)\|h-[u]\|\|[v]\|.$

Opening the square, we have

$\displaystyle C_{PK}\|h-[u]\|^{2}$
$\displaystyle=C_{PK}\|h\|^{2}-2C_{PK}\langle h,[u]\rangle+C_{PK}\|[u]\|^{2}$
$\displaystyle\leq 2C_{PK}\|[u]\|^{2}-\int T[u]\cdot T[u]d\mu$
$\displaystyle=2C_{PK}\|\nabla_{s}u\|_{L^{2}(\mu)}^{2}-\inf_{a\in\mathcal{A}}\|u-a\|^{2}_{L^{2}(\mu)}.$

The claim follows. ∎

######  Proposition 3.

Let $\mu$ satisfy the moment assumption.

  1. i)

For each $u\in\mathcal{C}$, we have $a_{u}(x)=A_{u}x$, with

$A_{u}:=\frac{1}{2}\int\left(ux^{T}-xu^{T}\right)d\mu.$

  2. ii)

Fix $i,j\in\\{1,\dots,n\\}$ with $i\neq j$. The function
$u=(u_{1},\dots,u_{n})\in\mathcal{C}$ defined by

$\displaystyle
u_{k}(x)=\delta_{ik}(1-x_{j}^{2})+\delta_{jk}x_{i}x_{j},~{}~{}~{}1\leq k\leq
n$ (5)

satisfies $a_{u}=0$ and

$[\nabla_{s}u]_{k\ell}=(\delta_{j\ell}\delta_{jk})x_{i}-(\delta_{ik}\delta_{j\ell}+\delta_{i\ell}\delta_{jk})\frac{x_{j}}{2},~{}~{}1\leq
k,\ell\leq n.$

  3. iii)

We have $C_{PK}(\mu)\geq 1$.

###### Proof.

$i)$ Let $A=-A^{T}$. By the cyclic property of trace and the isotropic
condition in the moment assumption, we may compute

$\int(A_{u}x)\cdot(Ax)d\mu=\operatorname{Tr}(A^{T}_{u}A)=\frac{1}{2}\int\operatorname{Tr}\left(xu^{T}A-ux^{T}A\right)d\mu=\int
u\cdot(Ax)d\mu,$

where we used antisymmetry of $A$ in the last step. It follows that

$\int(u-A_{u}x)\cdot ad\mu=0,~{}~{}\forall a\in\mathcal{A}.$

An application of the Hilbert projection theorem proves $i)$.

$ii)$ Fix $i,j\in\\{1,\dots,n\\}$ with $i\neq j$. For $u$ given by (5), we use
$i)$ to evaluate

$\displaystyle 2[A_{u}]_{k\ell}$
$\displaystyle=\int\left(u_{k}x_{\ell}-x_{k}u_{\ell}\right)d\mu$
$\displaystyle=\int\left(\delta_{ik}(1-x_{j}^{2})x_{\ell}+\delta_{jk}x_{i}x_{j}x_{\ell}-\delta_{i\ell}(1-x_{j}^{2})x_{k}-\delta_{j\ell}x_{i}x_{j}x_{k}\right)d\mu,$

which vanishes by the moment assumption444Some simple casework shows that,
under the assumption that $\mu$ is centered and isotropic, $a_{u}=0$ for every
choice of $i,j$ if and only if all (mixed) third moments vanish.. Next, let
$\partial_{k}$ denote partial derivative with respect to $x_{k}$, and observe

$\displaystyle 2[\nabla_{s}u]_{k\ell}$
$\displaystyle=\partial_{k}u_{\ell}+\partial_{\ell}u_{k}$
$\displaystyle=\delta_{i\ell}\partial_{k}(1-x_{j}^{2})+\delta_{j\ell}\partial_{k}x_{i}x_{j}+\delta_{ik}\partial_{\ell}(1-x_{j}^{2})+\delta_{jk}\partial_{\ell}x_{i}x_{j}$
$\displaystyle=\delta_{i\ell}\delta_{jk}(-2x_{j})+\delta_{j\ell}(\delta_{ik}x_{j}+\delta_{jk}x_{i})+\delta_{ik}\delta_{j\ell}(-2x_{j})+\delta_{jk}(\delta_{i\ell}x_{j}+\delta_{j\ell}x_{i})$
$\displaystyle=2\delta_{j\ell}\delta_{jk}x_{i}-(\delta_{ik}\delta_{j\ell}+\delta_{i\ell}\delta_{jk})x_{j}.$

$iii)$ For the choice of $u$ given by (5), we use $ii)$ and the moment
assumption to evaluate

$\inf_{a\in\mathcal{A}}\|u-a\|^{2}_{L^{2}(\mu)}=\|u\|^{2}_{L^{2}(\mu)}=\int
x_{i}^{2}x_{j}^{2}d\mu+\int(1-x_{j}^{2})^{2}d\mu=3$

and

$2\|\nabla_{s}u\|^{2}_{L^{2}(\mu)}=2\int
x_{i}^{2}d\mu+4\int(x_{j}/2)^{2}d\mu=3.$

It now follows by definitions that $C_{PK}(\mu)\geq 1$. ∎

With the necessary ingredients established, we turn our attention to the proof
of Lemma 1.

###### Proof of Lemma 1.

Abbreviate $C_{PK}:=C_{PK}(\mu)$. We can assume $C_{PK}<\infty$, else the
claim is trivial. Also, the statement is invariant to adding constants to $V$,
so we assume without loss of generality that $\int Vd\mu=0$.

To start, fix $i\neq j$, and let $u$ be given by (5). By the moment
assumption, this choice of $u$ satisfies (4) with

$\displaystyle\epsilon=2\sqrt{2}\sqrt{1-\frac{1}{C_{PK}}}.$ (6)

Next, let $\psi:\mathbb{R}^{n}\to\mathbb{R}$ be integrable and differentiable,
satisfying $\int\psi d\mu=0$ and $\|\nabla\psi\|_{L^{2}(\mu)}<\infty$. Fix
$m\in\\{1,\dots,n\\}$ and define $v:=(\delta_{1m},\dots,\delta_{nm})\psi$.
Since $\int\psi d\mu=0$ and
$\|\nabla_{s}v\|_{L^{2}(\mu)}\leq\|\nabla\psi\|_{L^{2}(\mu)}<\infty$, it
follows that $v\in\mathcal{C}$. For $A_{v}$ defined as in Proposition 3, we
may compute

$\displaystyle\int(u-a_{u})$ $\displaystyle\cdot(v-a_{v})d\mu=\int
u\cdot(v-a_{v})d\mu$ $\displaystyle=\int u_{m}\psi d\mu-\sum_{k,\ell}\int
u_{k}[A_{v}]_{k\ell}x_{\ell}d\mu$
$\displaystyle=\int(\delta_{im}(1-x_{j}^{2})\psi+\delta_{jm}x_{i}x_{m}\psi)d\mu-\sum_{\ell}\int\left([A_{v}]_{i\ell}(1-x_{j}^{2})+[A_{v}]_{j\ell}x_{i}x_{j}x_{\ell}\right)d\mu$
$\displaystyle=\int(\delta_{im}(1-x_{j}^{2})\psi+\delta_{jm}x_{i}x_{m}\psi)d\mu,$

where the last line follows from the moment assumption. Next, note that

$\displaystyle\int(\nabla_{s}u)\cdot(\nabla_{s}v)d\mu$
$\displaystyle=\int\left(x_{i}[\nabla_{s}v]_{jj}-x_{j}[\nabla_{s}v]_{ij}\right)d\mu$
$\displaystyle=\int\left(x_{i}\delta_{jm}\partial_{j}\psi-\frac{1}{2}x_{j}[\delta_{jm}\partial_{i}\psi+\delta_{im}\partial_{j}\psi]\right)d\mu.$

Define
$\mathcal{E}(\psi):=2\sqrt{3}\sqrt{C_{PK}(C_{PK}-1)}\|\nabla\psi\|_{L^{2}(\mu)}$
for convenience. An application of Lemma 2 with $\epsilon$ given in (6) yields

$\displaystyle\left|\int(\delta_{im}(1-x_{j}^{2})\psi+\delta_{jm}x_{i}x_{m}\psi)d\mu-2C_{PK}\int\left(x_{i}\delta_{jm}\partial_{j}\psi-\frac{1}{2}x_{j}[\delta_{jm}\partial_{i}\psi+\delta_{im}\partial_{j}\psi]\right)d\mu\right|\leq\mathcal{E}(\psi)$

for all $i,j,m$ with $i\neq j$. Taking $m=i$ and $\psi=[V]_{jj}$ gives

$\displaystyle\left|\int(x_{j}^{2}-1)[V]_{jj}d\mu-C_{PK}\int
x_{j}\partial_{j}[V]_{jj}d\mu\right|\leq\mathcal{E}([V]_{jj}).$

On the other hand, taking $m=j$ and $\psi=[V]_{ij}$ gives

$\displaystyle\left|\int
x_{i}x_{j}[V]_{ij}d\mu-2C_{PK}\int\left(x_{i}\partial_{j}[V]_{ij}-\frac{1}{2}x_{j}\partial_{i}[V]_{ij}\right)d\mu\right|$
$\displaystyle\leq\mathcal{E}([V]_{ij}).$

These can evidently be combined into the single matrix inequality

$\displaystyle\left|\int(x_{i}x_{j}-\delta_{ij})[V]_{ij}d\mu-2C_{PK}\int\left(x_{i}\partial_{j}[V]_{ij}-\frac{1}{2}x_{j}\partial_{i}[V]_{ij}\right)d\mu\right|$
$\displaystyle\leq\mathcal{E}([V]_{ij}),$

holding for all $1\leq i,j\leq n$. Summing over all $1\leq i,j\leq n$ and
applying the triangle inequality gives

$\displaystyle\left|\int(xx^{T}-\operatorname{Id})\cdot
Vd\mu-2C_{PK}\int\left((x\nabla^{T})\cdot V-\frac{1}{2}(x\nabla^{T})\cdot
V^{T}\right)d\mu\right|$ $\displaystyle\leq\mathcal{E}(V),$

where $\mathcal{E}(V):=\sum_{i,j=1}^{n}\mathcal{E}([V]_{ij})$. The same is
true when $V$ is replaced by $V^{T}$. However, the matrix
$(xx^{T}-\operatorname{Id})$ is symmetric, so we have

$\displaystyle\left|\int(xx^{T}-\operatorname{Id})\cdot
Vd\mu-2C_{PK}\int\left((x\nabla^{T})\cdot V^{T}-\frac{1}{2}(x\nabla^{T})\cdot
V\right)d\mu\right|$ $\displaystyle\leq\mathcal{E}(V),$

An application of the triangle inequality gives

$\displaystyle\left|C_{PK}\int(x\nabla^{T})\cdot Vd\mu-
C_{PK}\int(x\nabla^{T})\cdot V^{T}d\mu\right|\leq\frac{2}{3}\mathcal{E}(V),$

and another gives

$\displaystyle\left|\int(xx^{T}-\operatorname{Id})\cdot Vd\mu-
C_{PK}\int(x\nabla^{T})\cdot Vd\mu\right|$
$\displaystyle\leq\frac{5}{3}\mathcal{E}(V).$

Two final applications of the triangle inequality followed by Cauchy–Schwarz
gives the desired conclusion

$\displaystyle\left|\int(xx^{T}-\operatorname{Id})\cdot
Vd\mu-\int(x\nabla^{T})\cdot Vd\mu\right|$
$\displaystyle\leq\frac{5}{3}\mathcal{E}(V)+(C_{PK}-1)\left|\int(x\nabla^{T})\cdot
Vd\mu\right|$
$\displaystyle\leq\frac{5}{3}\mathcal{E}(V)+(C_{PK}-1)\sum_{i,j=1}^{n}\int\left|x_{i}\partial_{j}[V]_{ij}\right|d\mu$
$\displaystyle\leq\frac{5}{3}\mathcal{E}(V)+(C_{PK}-1)\sum_{i,j=1}^{n}\|\partial_{j}[V]_{ij}\|_{L^{2}(\mu)}$
$\displaystyle\leq
K\sqrt{C_{PK}(C_{PK}-1)}\sum_{i,j=1}^{n}\|\nabla[V]_{ij}\|_{L^{2}(\mu)}.$

∎

###  2.4 Implementation of Stein’s method

For a sufficiently smooth function $f:\mathbb{R}^{n}\to\mathbb{R}^{m}$, let
$D^{k}f$ denote the tensor of $k$-th order derivatives. The tensor $D^{k}f(x)$
can be regarded as a vector in a space of dimension $m\times n^{k}$, which we
equip with its natural Euclidean norm $\|\cdot\|_{2}$.

We’ll need the following Lemma. It combines Barbour’s solution to the
classical Stein equation in terms of the Ornstein–Uhlenbeck semigroup
$(P_{t})_{t\geq 0}$, defined by

$P_{t}f(x):=\int_{\mathbb{R}^{n}}f(e^{-t}x+(1-e^{-2t})^{1/2}z)d\gamma(z),~{}~{}~{}~{}f\in
L^{1}(\gamma),$

and the higher-order regularity estimate, that was for example derived in
[14].

######  Lemma 3.

For $f:\mathbb{R}^{n}\to\mathbb{R}$ with $\int fd\gamma<\infty$, the function
$\varphi_{f}:\mathbb{R}^{n}\to\mathbb{R}^{n}$ defined by

$\displaystyle\varphi_{f}(x):=\nabla\int_{0}^{\infty}P_{t}f(x)dt$ (7)

solves the Poisson equation

$\displaystyle f-\int
fd\gamma=x\cdot\varphi_{f}-\operatorname{Tr}(\nabla\varphi_{f}),$ (8)

and satisfies

$\displaystyle\sup_{x}\|D^{k}\varphi_{f}(x)\|_{2}\leq\sup_{x}\|D^{k}f(x)\|_{2},~{}~{}k\geq
1.$ (9)

We are now ready to implement Stein’s method to prove our main result. In
particular, for a given test function $f:\mathbb{R}^{n}\to\mathbb{R}$ with
uniformly bounded second derivatives, we’ll bootstrap the solution
$\varphi_{f}$ to the Stein equation (8) to construct a solution
$V:\mathbb{R}^{n}\to M_{n\times n}(\mathbb{R})$ to the integrated second-order
Stein equation

$\displaystyle\int fd\mu-\int fd\gamma$
$\displaystyle=\int\left((xx^{T}-\operatorname{Id})\cdot V-(x\nabla^{T})\cdot
V\right)d\mu.$

The main stability result will then follow from the approximate Stein identity
of Lemma 1, regularity estimates on $V$, and definition of the Zolotarev
distance.

###### Proof of Theorem 1.

Fix any $f:\mathbb{R}^{n}\to\mathbb{R}$ satisfying
$\sup_{x}\|\nabla^{2}f(x)\|_{2}\leq 1$. Using the classical identity
$\nabla(P_{t}f)=e^{-t}P_{t}(\nabla f)$ for $P_{t}$, we have

$\nabla\varphi_{f}(x)=\int_{0}^{\infty}\nabla^{2}P_{t}f(x)dt=\int_{0}^{\infty}e^{-2t}P_{t}(\nabla^{2}f)(x)dt.$

By the triangle and Jensen inequalities, (9), and boundedness of
$\nabla^{2}f$, (7) implies

$\displaystyle\sup_{x}\|\nabla\varphi_{f}(x)\|_{2}\leq\frac{1}{2}\sup_{x}\|\nabla^{2}\varphi_{f}(x)\|_{2}\leq\frac{1}{2}\sup_{x}\|\nabla^{2}f(x)\|_{2}\leq\frac{1}{2}.$
(10)

In particular, $\varphi_{f}$ is $1/2$-Lipschitz and $\nabla\varphi_{f}$ is
1-Lipschitz. Now, define $a\in\mathbb{R}^{n}$ and $Q\in M_{n\times
n}(\mathbb{R})$ by

$a:=\varphi_{f}(0),~{}~{}~{}[Q]_{ij}:=(1+\delta_{ij})[\nabla\varphi_{f}(0)]_{ij},$

and put

$g(x):=f(x)-a^{T}x-\frac{1}{2}x^{T}Qx.$

Using the fact that Hermite polynomials are eigenfunctions of the
Ornstein–Uhlenbeck semigroup, we can check that the solution
$\varphi_{g}:=\nabla\int_{0}^{\infty}P_{t}gdt$ to the Poisson equation

$\displaystyle g-\int
gd\gamma=x\cdot\varphi_{g}-\operatorname{Tr}(\nabla\varphi_{g}),$ (11)

is equal to

$\varphi_{g}(x)=\varphi_{f}(x)-\varphi_{f}(0)-\nabla\varphi_{f}(0)x,~{}~{}~{}x\in\mathbb{R}^{n}.$

In particular, $\varphi_{g}$ satisfies

$\displaystyle\varphi_{g}(0)=0,~{}~{}\nabla\varphi_{g}(0)=0,~{}~{}\mbox{and}~{}~{}\nabla^{2}\varphi_{g}(x)=\nabla^{2}\varphi_{f}(x).$
(12)

We now establish some basic regularity properties of $\varphi_{g}$. Combining
(12) with the Lipschitz estimates established for $\varphi_{f}$, we have

$|\varphi_{g}(x)|\leq\frac{1}{2}|x|,~{}~{}\mbox{and}~{}~{}\|\nabla\varphi_{g}(x)\|_{2}\leq|x|.$

Additionally, by a Taylor expansion around $x=0$, the properties (12) together
with boundedness of second-derivatives of $\varphi_{f}$ imply the quadratic
growth estimate

$|\varphi_{g}(x)|\leq\frac{1}{2}\|\nabla^{2}\varphi_{f}(0)\|_{2}|x|^{2}\leq\frac{1}{2}|x|^{2}.$

Next, we define a matrix-valued function $V:\mathbb{R}^{n}\to M_{n\times
n}(\mathbb{R})$ by

$V(x):=\begin{cases}\frac{1}{|x|^{2}}x\varphi_{g}^{T}(x)&\mbox{if $x\neq
0$}\\\ 0&\mbox{if $x=0$}.\end{cases}$

By definition of $V$ and the fact that $\varphi_{g}(0)=0$, we have

$V^{T}(x)x=\varphi_{g}(x),~{}~{}x\in\mathbb{R}^{n}.$

Now, we check the regularity of $V$. Since $\varphi_{g}$ inherits continuity
properties from $\varphi_{f}$, it follows that $V$ is continuous on
$\mathbb{R}^{n}\setminus\\{0\\}$. It is also continuous at $x=0$, which
follows since $|\varphi_{g}(x)|\leq\frac{1}{2}|x|^{2}$, and therefore
$\lim_{x\to 0}V(x)=0=V(0)$.

Evidently, $V$ is differentiable on $\mathbb{R}^{n}\setminus\\{0\\}$. For
$x\neq 0$, we compute

$\displaystyle\nabla([V(x)]_{ij})=\nabla\frac{x_{i}[\varphi_{g}(x)]_{j}}{|x|^{2}}=-2x\frac{x_{i}[\varphi_{g}(x)]_{j}}{|x|^{4}}+\frac{1}{|x|^{2}}(e_{i}[\varphi_{g}(x)]_{j}+x_{i}\nabla[\varphi_{g}(x)]_{j}).$

Using the regularity estimates on $\varphi_{g}$, we obtain

$|\nabla([V(x)]_{ij})|\leq|x_{i}|+\frac{1}{2}+\frac{|x_{i}|}{|x|}\leq|x_{i}|+\frac{3}{2}.$

Thus, using the moment assumption, we have

$\displaystyle\|\nabla([V]_{ij})\|_{L^{2}(\mu)}$
$\displaystyle=\left(\int|\nabla([V(x)]_{ij})|^{2}d\mu(x)\right)^{1/2}$
$\displaystyle\leq\left(\int(|x_{i}|+3/2)^{2}d\mu(x)\right)^{1/2}\leq 5/2.$
(13)

Finally, we put everything together to obtain

$\displaystyle\int fd\mu-\int fd\gamma$ $\displaystyle=\int gd\mu-\int
gd\gamma$ (14)
$\displaystyle=\int\left(x\cdot\varphi_{g}-\operatorname{Tr}(\nabla\varphi_{g})\right)d\mu$
(15) $\displaystyle=\int\left((xx^{T}-\operatorname{Id})\cdot
V-(x\nabla^{T})\cdot V\right)d\mu$ (16) $\displaystyle\leq
K\sqrt{C_{PK}(C_{PK}-1)}\sum_{i,j=1}^{n}\|\nabla[V]_{ij}\|_{L^{2}(\mu)}$ (17)
$\displaystyle<20n^{2}\sqrt{C_{PK}(C_{PK}-1)}.$ (18)

In the above, (14) follows by definition of $g$ and the moment assumption;
(15) is (11), integrated with respect to $\mu$; (16) follows since
$V^{T}(x)x=\varphi_{g}(x)$; (17) follows from the approximate Stein identity
of Lemma 1 applied to (smooth approximations of) $V$; (18) is the bound (13).
Taking supremum over $f$ proves the theorem. ∎

##  Appendix A Proof of Proposition 2

We first recall Caffarelli’s contraction theorem [6]: if a probability measure
$d\mu=e^{-\phi}dx$ satisfies $\nabla^{2}\phi\geq\operatorname{Id}$, then there
exists a transport map $T$ from $\gamma$ onto $\mu$ that is $1$-Lipschitz.
This map $T$ is the Brenier map from optimal transport theory.

For $T$ as above, we have $|T(x)-T(y)|\leq|x-y|$ for all
$x,y\in\mathbb{R}^{n}$. Moreover,

$\displaystyle 2n=\int{|x-y|^{2}d\mu(x)d\mu(y)}$
$\displaystyle=\int{|T(x)-T(y)|^{2}d\gamma(x)d\gamma(y)}$
$\displaystyle\leq\int{|x-y|^{2}d\gamma(x)d\gamma(y)}=2n.$

Hence there is equality throughout, and $|T(x)-T(y)|=|x-y|$, $\gamma^{\otimes
2}$-a.s. Since $T$ is continuous, the equality holds everywhere. In the case
where $\mu$ has full support, $T$ is surjective, and therefore $T$ is a
surjective isometry. By the Mazur–Ulam theorem, we conclude that $T$ is
affine. Since $\mu$ is centered and isotropic by assumption, it must be
standard Gaussian.

If $\mu$ does not have a full support, we can take a convolution with a
standard Gaussian, rescaled so that the new measure $\nu$ is still isotropic.
Since $\nu$ has full support and $1$-uniform log-concavity is preserved by
this operation, we can apply the previous case to deduce that $\nu$ is
Gaussian. Since a convolution of two measures is Gaussian iff both are
Gaussian, it follows that $\mu$ is also Gaussian.

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