# A short review on Improvements and stability
for some interpolation inequalities

Jean Dolbeault  ,  Maria J. Esteban  CEREMADE (CNRS UMR No. 7534), PSL
University, Université Paris-Dauphine, Place de Lattre de Tassigny, 75775
Paris 16, France.
E-mails<EMAIL_ADDRESS><EMAIL_ADDRESS>,
Alessio Figalli  Mathematics Department, ETH Zürich, Ramistrasse 101, 8092
Zürich, Switzerland.
E-mail<EMAIL_ADDRESS>,  Rupert L. Frank  Department of
Mathematics, LMU Munich, Theresienstr. 39, 80333 München, Germany, and Munich
Center for Quantum Science and Technology, Schellingstr. 4, 80799 München,
Germany, and Mathematics 253-37, Caltech, Pasadena, CA 91125, USA. E-mail:
<EMAIL_ADDRESS>and  Michael Loss  School of Mathematics, Georgia Institute of
Technology Atlanta, GA 30332, United States of America. E-mail:
<EMAIL_ADDRESS>

###### Abstract.

In this paper, we present recent stability results with explicit and
dimensionally sharp constants and optimal norms for the Sobolev inequality and
for the Gaussian logarithmic Sobolev inequality obtained by the authors in
[22]. The stability for the Gaussian logarithmic Sobolev inequality is
obtained as a byproduct of the stability for the Sobolev inequality. Here we
also give a new, direct, alternative proof of this result. We also discuss
improved versions of interpolation inequalities based on the carré du champ
method.

###### Key words and phrases:

Sobolev inequality; logarithmic Sobolev inequality; interpolation; Gagliardo-
Nirenberg inequalities; stability; rearrangement; heat equation; fast
diffusion equations

###### 2020 Mathematics Subject Classification:

Primary: 49J40; Secondary: 26D10, 35A23, 58E35.

© 2024 by the authors. Reproduction of this article by any means permitted for
noncommercial purposes. CC-BY 4.0

##  1\. Introduction and main results

Let us assume that $d\geq 3$ without further notice. The classical Sobolev
inequality on ${\mathord{\mathbb{R}}}^{d}$, can be written as follows:

$\|\nabla f\|^{2}_{\mathrm{L}^{2}({\mathord{\mathbb{R}}}^{d})}\geq
S_{d}\,\|f\|^{2}_{\mathrm{L}^{2^{*}}({\mathord{\mathbb{R}}}^{d})}\quad\forall\,f\in\dot{\mathrm{H}}^{1}({\mathord{\mathbb{R}}}^{d})\,,$
(S)

where $2^{*}=\frac{2\,d}{d-2}$ is the Sobolev exponent,
$S_{d}=\tfrac{1}{4}\,d\,(d-2)\,|\mathbb{S}^{d}|^{2/d}$ is the sharp Sobolev
constant, and $|\mathbb{S}^{d}|$ stands for the volume of the unit sphere
$\mathbb{S}^{d}\subset{\mathord{\mathbb{R}}}^{d+1}$. Here
$\dot{\mathrm{H}}^{1}({\mathord{\mathbb{R}}}^{d})$ denotes the closure of
$C^{\infty}_{c}({\mathord{\mathbb{R}}}^{d})$ with respect to the seminorm
$|f|_{\dot{\mathrm{H}}^{1}({\mathord{\mathbb{R}}}^{d})}:=\|\nabla
f\|^{2}_{\mathrm{L}^{2}({\mathord{\mathbb{R}}}^{d})}$. As proved in [1, 37]
(see also [35, 36]), equality in (S) holds if $f$ is one of the Aubin-Talenti
functions, that is, one of the functions belonging to the $(d+2)$-dimensional
Aubin-Talenti manifold

$\mathcal{M}:=\left\\{g_{a,b,c}\,:\,(a,b,c)\in(0,+\infty)\times{\mathord{\mathbb{R}}}^{d}\times{\mathord{\mathbb{R}}}\right\\}\quad\mbox{with}\quad
g_{a,b,c}(x):=c\left(a+|x-b|^{2}\right)^{-\frac{d-2}{2}}\,.$

In fact, there is equality in (S) if and only if $f$ is in $\mathcal{M}$
according to [32, 29, 14].

Applying the inverse of the stereographic projection and integrating on the
sphere $\mathbb{S}^{d}$ with respect to the uniform probability measure
$d\mu_{d}$, one can rewrite the above inequality as

$\left\|{\nabla
u}\right\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}\geq\tfrac{1}{4}\,d\,(d-2)\left(\left\|{u}\right\|_{\mathrm{L}^{2^{*}}(\mathbb{S}^{d})}^{2}-\left\|{u}\right\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}\right)\quad\forall\,u\in\mathrm{H}^{1}(\mathbb{S}^{d})\,.$
(1)

and state, equivalently, that the only functions in
$\mathrm{H}^{1}(\mathbb{S}^{d})$ for which there is equality in (1) are the
functions $\omega\mapsto G_{b,c}(\omega):=c\,(1+b\cdot\omega)^{-(d-2)/2}$
where $b\in B_{1}:=\\{b\in{\mathord{\mathbb{R}}}^{d+1}\,:\,|b|<1\\}$ and
$c\in{\mathord{\mathbb{R}}}$ are constants. One can embed (1) in the following
family of Gagliardo-Nirenberg-Sobolev inequalities:

$\left\|{\nabla
u}\right\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}\geq\frac{d}{p-2}\left(\left\|{u}\right\|_{\mathrm{L}^{p}(\mathbb{S}^{d})}^{2}-\left\|{u}\right\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}\right)\quad\forall\,u\in\mathrm{H}^{1}(\mathbb{S}^{d})\,,\;\forall\,p\in(1,2)\cup(2,2^{*}]\,.$
(GNS)

As proved by Bidaut-Véron–Véron [7] and Beckner [5] for $p>2$ (see also [2, 3,
4] if $p\leq 2^{\\#}$), the constant $d/(p-2)$ is the best possible constant
in (GNS) (here $2^{\\#}:=(2\,d^{2}+1)/(d-1)^{2}<2^{*}$ denotes the Bakry-Emery
exponent). If $p<2^{*}$, the only optimizers in
$\mathrm{H}^{1}(\mathbb{S}^{d})$ for (GNS) are the constant functions. The
carré du champ method used in [2, 3, 4] relies on the linear heat equation,
which induces the limitation $p\leq 2^{\\#}$. This limitation is not technical
as shown in [25]. The whole range $p\in(1,2)\cup(2,2^{*}]$ was covered using a
carré du champ method based on nonlinear diffusion equations by Demange and
Dolbeault-Esteban-Loss in [20, 24]. The case $p=2$ (that can be obtained from
(GNS) by taking the limit $p\to 2$) is the logarithmic Sobolev inequality

$\left\|{\nabla
u}\right\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}\geq\frac{d}{2}\,\int_{\mathbb{S}^{d}}{|u|^{2}\,\ln\left(\frac{|u|^{2}}{\left\|{u}\right\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}}\right)}\;d\mu\quad\forall\,u\in\mathrm{H}^{1}(\mathbb{S}^{d})\setminus\\{0\\}\,,$
(2)

where the only optimizers are the constant functions. Inequality (2) can also
be proved by the carré du champ method based on the heat equation. A
remarkable feature of the carré du champ method is that additional terms
appear in the computations. This has been used in [23] to establish, for all
$p\in(2,2^{*})$, the improved interpolation inequalities

$\left\|{\nabla
u}\right\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}-\frac{d}{p-2}\left(\left\|{u}\right\|_{\mathrm{L}^{p}(\mathbb{S}^{d})}^{2}-\left\|{u}\right\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}\right)\geq\left\|{u}\right\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}\Psi_{d,p}\\!\left(\frac{\left\|{\nabla
u}\right\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}}{\left\|{u}\right\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}}\right)\quad\forall\,u\in\mathrm{H}^{1}(\mathbb{S}^{d})\setminus\\{0\\},$
(3)

for some convex function $\Psi_{d,p}$ such that
$\Psi_{d,p}(0)=\Psi_{d,p}^{\prime}(0)=0$ and $\Psi_{d,p}(s)>0$ if $s>0$.
Inequality (3) is obviously stronger than (GNS) and can be used to prove that
the equality case in (GNS) is realized only by constant functions. If we test
(3) with $u_{\varepsilon}(\omega)=1+\varepsilon\,\mathsf{b}\cdot\omega$ for a
given $\mathsf{b}\in B_{1}$, an elementary computation shows that there is a
cancellation of the $O(\varepsilon^{2})$ terms as $\varepsilon\to 0$ in the
left-hand side of the inequality and the first non-zero term is of the order
of $O(\varepsilon^{4})$. If we denote by $\Pi_{1}$ the
$\mathrm{L}^{2}(\mathbb{S}^{d})$ projection on the space generated by the
coordinate functions $\omega_{i}$ with $i=1$, $2$,…$d$, we learn from [10,
Theorem 6] that there is an explicit constant $\kappa=\kappa(p,d)\in(0,1)$,
depending on $d$ and $p\in[1,2)\cup(2,2^{*})$ such that, for all
$u\in\mathrm{H}^{1}(\mathbb{S}^{d})$,

$\left\|{\nabla
u}\right\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}-\frac{d}{p-2}\left(\left\|{u}\right\|_{\mathrm{L}^{p}(\mathbb{S}^{d})}^{2}-\left\|{u}\right\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}\right)\geq\kappa\left(\tfrac{\left\|{\nabla\Pi_{1}u}\right\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{4}}{\left\|{\nabla
u}\right\|_{\mathrm{L}^{+}(\mathbb{S}^{d})}\left\|{u}\right\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}}+\left\|{\nabla(\mathrm{Id}-\Pi_{1})\kern
1.0ptu}\right\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}\right)\,.$

The proof relies on (3) and on a decomposition in spherical harmonics directly
inspired by [28]. Let us also notice that Inequality (3) can take various
forms. For instance, if $p\in(2,2^{\\#})$ and
$1/\theta=1+(\frac{d-1}{d+2})^{2}\,(p-1)\,(2^{\\#}-p)/(p-2)$, we read from
[21, Ineq. (2.4)] that (3) takes the form

$\left\|{\nabla
u}\right\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}\geq\frac{d\,\theta}{p-2}\left(\left\|{u}\right\|_{\mathrm{L}^{p}(\mathbb{S}^{d})}^{2/\theta}\,\left\|{u}\right\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2-2/\theta}-\left\|{u}\right\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}\right)\quad\forall\,u\in\mathrm{H}^{1}(\mathbb{S}^{d})\,,$

which is a strict improvement compared to (GNS) as can be recovered using
Hölder inequalities. The case $p=2$ is also covered, except that in the l.h.s.
of (3), the deficit of (GNS) has to be replaced by the deficit of (2). Without
entering into details, let us quote some related results. By a direct
variational approach, an improved inequality like (3) is proved in [28,
Theorem 2], in the subcritical range, for some $\Psi_{d,p}(s)\sim s^{2}$.
Using nonlinear flows and appropriate orthogonality constraints, improved
inequalities with $\Psi_{d,p}(s)\sim s$ are known from [25]. Both results are
unified in [10]. Let us finally mention that improved inequalities are proved
in [8] with explicit constants, not on $\mathbb{S}^{d}$ but on
${\mathord{\mathbb{R}}}^{d}$, for the Gagliardo-Nirenberg-Sobolev inequalities
using entropy methods and regularization effects for fast diffusion flows, but
with some restrictions on the decay of the functions at infinity.

The Gaussian measure can be seen as an infinite dimensional limit
$d\to+\infty$ of the uniform probability measure on the sphere of radius
$\sqrt{d}$ tested against functions depending only on a finite number $N$ of
coordinates: see for instance [34, 11]. Since $\lim_{d\to+\infty}2^{*}=2$, it
also turns out that (S) has to be replaced by a Gaussian interpolation
inequality as follows. On ${\mathord{\mathbb{R}}}^{N}$, with $N\geq 1$, let us
consider the Gaussian measure $d\gamma(x)=e^{-\,\pi\,|x|^{2}}\,dx$. With
$\mathrm{L}^{2}(\gamma):=\mathrm{L}^{2}({\mathord{\mathbb{R}}}^{N},d\gamma)$,
if $\mathrm{H}^{1}(\gamma)$ denotes the space of all
$u\in\mathrm{L}^{2}(\gamma)$ with distributional gradient in
$\mathrm{L}^{2}(\gamma)$, the logarithmic Sobolev inequality is:

$\int_{{\mathord{\mathbb{R}}}^{N}}|\nabla
u|^{2}\,d\gamma\geq\pi\int_{{\mathord{\mathbb{R}}}^{N}}u^{2}\ln\left(\frac{|u|^{2}}{\|u\|_{\mathrm{L}^{2}(\gamma)}^{2}}\right)\,d\gamma\quad\forall\,u\in\mathrm{H}^{1}(\gamma)\setminus\\{0\\}\,.$
(LSI)

According to the result of Carlen [15, Theorem 5], equality holds in (LSI) if
and only if for some $a\in{\mathord{\mathbb{R}}}^{N}$ and
$c\in{\mathord{\mathbb{R}}}$,

$u(x)=c\,e^{a\cdot x}\,.$ (4)

Improved forms of the inequality are also known for instance from [27, 26],
under some restrictions.

Now let us turn our attention to stability issues for (S), (GNS) and (LSI).

$\bullet$ Stability for the Sobolev inequality. In [9] Brezis and Lieb asked
the following question:

(Q) Do there exist constants $\kappa$, $\alpha>0$ such that the _Sobolev
deficit_ $\boldsymbol{\delta}$ controls some distance $\mathbf{d}$ from the
Aubin-Talenti manifold $\mathcal{M}$ according to

$\boldsymbol{\delta}(f):=\|\nabla
f\|^{2}_{\mathrm{L}^{2}({\mathord{\mathbb{R}}}^{d})}-S_{d}\,\|f\|^{2}_{\mathrm{L}^{2^{*}}({\mathord{\mathbb{R}}}^{d})}\geq\kappa\,\mathbf{d}(f,\mathcal{M})^{\alpha}\quad?$

The ‘best possible answer’ to this question would involve finding the
strongest possible topology to define the distance $\mathbf{d}$ and the best
possible constant $\kappa$ and exponent $\alpha$. The first answer to Brezis
and Lieb’s question was given by Bianchi and Egnell in [6]: there is a
constant $C_{\rm BE}^{d}>0$ such that

$\boldsymbol{\delta}(f)\geq C_{\rm BE}^{d}\,\inf_{g\in\mathcal{M}}\|\nabla
f-\nabla
g\|^{2}_{\mathrm{L}^{2}({\mathord{\mathbb{R}}}^{d})}\quad\forall\,f\in\dot{\mathrm{H}}^{1}({\mathord{\mathbb{R}}}^{d})\,.$
(5)

Similar results for other inequalities have been proved using the strategy of
Bianchi and Egnell: see, for example, [18]. The main drawback of this strategy
is that no explicit estimate of $C_{\rm BE}^{d}$ is known nor its dependence
on $d$. Recently, in [22], we proved the following result.

######  Theorem 1 ([22, Theorem 1.1]).

Let $d\geq 3$. There is an explicit constant $\beta>0$ such that

$\|\nabla
f\|^{2}_{\mathrm{L}^{2}({\mathord{\mathbb{R}}}^{d})}-S_{d}\,\|f\|^{2}_{\mathrm{L}^{2^{*}}({\mathord{\mathbb{R}}}^{d})}\geq\;\frac{\beta}{d}\;\inf_{g\in\mathcal{M}}\|\nabla
f-\nabla
g\|^{2}_{\mathrm{L}^{2}({\mathord{\mathbb{R}}}^{d})}\quad\forall\,f\in\dot{\mathrm{H}}^{1}({\mathord{\mathbb{R}}}^{d})\,.$
(6)

This result is dimensionally sharp. Indeed, Theorem 1 can be rewritten as
$C_{\rm BE}^{d}\geq\beta/d$. On the other hand, it was proved implicitly in
[6] (see also [18]) that $C_{\rm BE}^{d}\leq 4/(4+d)$. This inequality is in
fact strict: $C_{\rm BE}^{d}<4/(4+d)$, according to [30], and we learn from
[31] that equality in (5), written with the optimal value of $C_{\rm BE}^{d}$,
is achieved. Hence, Theorem 1 captures the dimensional behavior of $C_{\rm
BE}^{d}$. For completeness, let us quote the extension of Theorem 1 in [17] to
fractional Sobolev inequalities. Using the inverse stereographic projection,
the stability result of Theorem 1 can be rewritten on $\mathbb{S}^{d}$ as

$\left\|{\nabla
u}\right\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}-\tfrac{1}{4}\,d\,(d-2)\left(\left\|{u}\right\|_{\mathrm{L}^{2^{*}}(\mathbb{S}^{d})}^{2}-\left\|{u}\right\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}\right)\geq\;\frac{\beta}{d}\;\inf_{(b,c)\in
B_{1}\times{\mathord{\mathbb{R}}}}\left\|{\nabla u-\nabla
G_{b,c}}\right\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}\quad\forall\,u\in\mathrm{H}^{1}(\mathbb{S}^{d})\,.$

$\bullet$ Stability for the Gaussian logarithmic Sobolev inequality. The
interpretation of the Gaussian measure as the limit of uniform probabililty
measures on $d$-dimensional spheres as $d\to+\infty$ and the explicit
dimensional dependence of the stability constant of Theorem 1 provides us with
a stability result for (LSI).

######  Theorem 2 ([22, Corollary 1.2]).

With $\beta>0$ as in Theorem 1, for all $N\in{\mathord{\mathbb{N}}}$, we have

$\int_{{\mathord{\mathbb{R}}}^{N}}|\nabla
u|^{2}\,d\gamma-\pi\int_{{\mathord{\mathbb{R}}}^{N}}u^{2}\ln\left(\frac{|u|^{2}}{\|u\|_{\mathrm{L}^{2}(\gamma)}^{2}}\right)\,d\gamma\geq\frac{\beta\,\pi}{2}\inf_{b\in{\mathord{\mathbb{R}}}^{N}\\!,\,c\in{\mathord{\mathbb{R}}}}\int_{{\mathord{\mathbb{R}}}^{N}}\big{(}u-c\,e^{b\cdot
x}\big{)}^{2}\,d\gamma\quad\forall\,u\in\mathrm{H}^{1}(\gamma)\,.$ (7)

In [22] this result was obtained as a corollary of Theorem 1. In this paper we
give a new, direct proof which highlights the strategy of [22] in a slightly
simpler setting.

In the next sections, we briefly describe the strategies of [23] and [22] to
prove the improvements in the case of subcritical inequalities (Section 2) and
the stability results of Theorem 1 (Section 3) and Theorem 2 (Section 4). The
new proof of Theorem 2 is given in Section 5.

##  2\. Improved interpolation inequalities

In this section we describe the method used in [23] to prove (3) and similar
improvements of the (GNS) inequalities in the subcritical case
$p\in(2,2^{*})$. Inequalities (GNS) and their limiting case corresponding to
$p=2$ can be written as:

$\mathsf{i}\geq d\,\mathsf{e}\quad\mbox{where}\quad\mathsf{i}:=\left\|{\nabla
u}\right\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}\quad\mbox{and}\quad\mathsf{e}:=\left\\{\begin{array}[]{l}\frac{1}{p-2}\left(\left\|{u}\right\|_{\mathrm{L}^{p}(\mathbb{S}^{d})}^{2}-\left\|{u}\right\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}\right)\quad\mbox{if}\quad
p\neq 2\,,\\\\[4.0pt]
\frac{1}{2}\,\int_{\mathbb{S}^{d}}{|u|^{2}\,\ln\Big{(}\frac{|u|^{2}}{\left\|{u}\right\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}}\Big{)}}\;d\mu\quad\mbox{if}\quad
p=2\,,\\\ \end{array}\right.$

for any $u\in\mathrm{H}^{1}(\mathbb{S}^{d})$ and by homogeneity we can assume
that $\left\|{u}\right\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}=1$. Let
$I_{p}:=[0,+\infty)$ if $p\geq 2$ and $I_{p}:=[0,1/(2-p))$ if $p\in(1,2)$.

######  Theorem 3 ([23, Theorem 1.1]).

Let $d\geq 3$ and $p\in(2,2^{*})$. With the above notation and conventions,
there is an explicit function $\varphi$, such that $\varphi(0)=0$,
$\varphi^{\prime}(0)=1$, and $\varphi^{\prime\prime}>0$ on $I_{p}$, for which

$\mathsf{i}\geq d\,\varphi(\mathsf{e})\,.$ (8)

Similar results can be proved in dimension $d=1$ and $d=2$: see [20, 23, 21,
10]. Since

$\mathsf{i}-d\,\mathsf{e}\geq
d\,\big{(}\varphi(\mathsf{e})-\mathsf{e}\big{)}\geq 0\,,$

it is clear that the equality case in (3) holds if and only if $\mathsf{e}=0$,
that is, if $u$ is a constant. Moreover $\varphi(\mathsf{e})-\mathsf{e}$
measures a distance to the constants, for instance in
$\mathrm{L}^{1}(\mathbb{S}^{d})$ norm using a generalized Ciszár-Kullback-
Pinsker inequality, and
$\varphi(\mathsf{e})-\mathsf{e}\sim\varphi^{\prime\prime}(0)\,\mathsf{e}^{2}/2$
as $\mathsf{e}\to 0$. To prove (3) in the stronger homogeneous Sobolev norm
$\dot{\mathrm{H}}^{1}({\mathord{\mathbb{R}}}^{d})$, it is enough to define the
convex function

$\Psi_{d,p}(s):=s-d\,\varphi^{-1}\left(\frac{s}{d}\right)\,.$

It is elementary to verify that $\Psi_{d,p}(0)=\Psi_{d,p}^{\prime}(0)=0$ and
that $\Psi_{d,p}(s)>0$ if $s\neq 0$.

Let us give a sketch of the proof of Theorem 3 based on the method of [23]. As
a first step, using Schwarz foliated symmetrization (see for instance [23,
Section 2] for references) and cylindrical coordinates on
$\mathbb{S}^{d}\subset{\mathord{\mathbb{R}}}^{d+1}$, we can reduce the problem
to functions depending only on one coordinate $z\in[-1,1]$ corresponding to
the South Pole – North Pole axis. In other words, we consider a function
$u(\omega)=f(z)$ where $\omega=(\omega_{1},\omega_{2},\ldots,\omega_{d+1})$
and $z=\omega_{d+1}$, so that

$\left\|{\nabla
u}\right\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}=\int_{-1}^{1}{|f^{\prime}|^{2}\,\big{(}1-z^{2}\big{)}}\;d\sigma_{d}\quad\mbox{and}\quad\left\|{u}\right\|_{\mathrm{L}^{q}(\mathbb{S}^{d})}^{q}=\int_{-1}^{1}{|f|^{q}}\;d\sigma_{d}\,,$

where ′ denotes the $z$\- derivative, the Laplace-Beltrami operator is reduced
to the _ultraspherical_ operator: $\Delta
u=\mathcal{L}\,f:=\big{(}1-z^{2}\big{)}\,f^{\prime\prime}-d\,z\,f^{\prime}$,
and the uniform probability measures becomes

$d\sigma_{d}(z):=\frac{\Gamma(\tfrac{d+1}{2})}{\sqrt{\pi}\,\Gamma(\tfrac{d}{2})}\,\big{(}1-z^{2}\big{)}^{\frac{d}{2}-1}\,dz\,.$

The key idea is to prove that $\mathsf{i}-d\,\varphi(\mathsf{e})$ is monotone
non-increasing under the action of

$\frac{\partial\rho}{\partial t}=\mathcal{L}\,\rho^{m},$ (9)

where $m=1$ corresponds to the heat flow, $m>1$ to the porous medium flow and
$m<1$ to the fast diffusion flow. Here we choose $\rho=|f|^{p}$ so that
$\left\|{u}\right\|_{\mathrm{L}^{p}({\mathord{\mathbb{R}}}^{d})}$ is conserved
under the action of (9). It is convenient to introduce the exponent $\beta$
such that

$m=1+\tfrac{2}{p}\,\big{(}\tfrac{1}{\beta}-1\big{)}$

and consider the function $w=f^{\beta}$, such that $w^{\beta\,p}=\rho$, which
solves

$\frac{\partial w}{\partial
t}=w^{2-2\beta}\left(\mathcal{L}w+\kappa\,\frac{|w^{\prime}|^{2}}{w}\right)\quad\mbox{with}\quad\kappa:=\beta\,(p-2)+1\,.$
(10)

The _carré du champ_ method shows that
$\frac{d}{dt}\left(\mathsf{i}-d\,\mathsf{e}\right)\leq 0$ if the function

$\gamma(\beta):=-\,\big{(}\tfrac{d-1}{d+2}\,(\kappa+\beta-1)\big{)}^{2}+\kappa\,(\beta-1)+\,\tfrac{d}{d+2}\,(\kappa+\beta-1)$

takes nonnegative values, which amounts to $m_{-}(d,p)\leq m\leq m_{+}(d,p)$
where

$m_{\pm}(d,p):=\frac{1}{(d+2)\,p}\left(d\,p+2\pm\sqrt{d\,(d-2)\,(p-1)\,(2^{*}-p)}\,\right)\,.$

Using $\lim_{t\to+\infty}\left(\mathsf{i}-d\,\mathsf{e}\right)=0$, we conclude
that $\mathsf{i}-d\,\mathsf{e}\geq 0$ for any $t\geq 0$ and, as a special
case, for $t=0$: this proves (GNS) for an arbitrary initial datum. If
$m_{-}(d,p)<m<m_{+}(d,p)$, we find that $\gamma(\beta)>0$, which leaves some
space for improvement. A more detailed and quite lengthy computation along the
flow (10) yields

$\frac{d}{dt}\big{(}\mathsf{i}-\,d\,\varphi(\mathsf{e})\big{)}\leq\frac{\gamma}{\beta^{2}}\,\frac{\mathsf{i}-\,d\,\varphi(\mathsf{e})}{\big{(}1-(p-2)\,\mathsf{e}\big{)}^{\delta}}\,\frac{d\mathsf{e}}{dt}\,,$

where $\delta=1$ if $1\leq p\leq 2$ and
$\delta:=\frac{2-(4-p)\,\beta}{2\,\beta\,(p-2)}$ if $p>2$, if $\varphi$ solves

$\frac{d\varphi}{ds}=1+\frac{\gamma}{\beta^{2}}\,\frac{\varphi(s)}{\left(1\,-\,(p-2)\,s\right)^{\delta}}\,,\quad\varphi(0)=0\,.$

This proves (8) for any $t\geq 0$ as a consequence of the monotonicity of
$\mathsf{i}-d\,\varphi(\mathsf{e})$ and the fact that
$\lim_{t\to+\infty}\big{(}\mathsf{i}-d\,\varphi(\mathsf{e})\big{)}=0$. See
[10, Appendix B.4] for detailed justifications.∎

##  3\. Stability for the Sobolev inequality. Proof of Theorem 1

In this section we explain the general ideas of the proof of Theorem 1 in
[22].

###  3.1. On the stability proof by Bianchi and Egnell

The strategy of Bianchi-Egnell to prove (5) is based on two main steps:

  1. (1)

A local stability estimate in a neighborhood of $\mathcal{M}$, obtained by a
local spectral analysis.

  2. (2)

A reduction of the global estimate to the local estimate by the concentration-
compactness method based on Lions’ analysis (see [33]).

Theorem 1 is a significant improvement of Bianchi-Egnell’s result, as it
contains a dimensionally sharp lower estimate for the best stability constant.
Our strategy is to make the second step constructive and the first one
explicit, with much more detailed estimates.

###  3.2. Strategy of the proof of Theorem 1

The proof is divided into several steps:

  1. (1)

Local analysis: prove the inequality for _nonnegative_ functions close to
$\mathcal{M}$ with an explicit remainder term. The analysis is quite involved:
it relies on “cuttings” at various heights, the use of uniform bounds on
spherical harmonics and some delicate concavity properties.

  2. (2)

Local to global extension: prove the inequality for _nonnegative_ functions
far from $\mathcal{M}$ using the _competing symmetries_ method of [16] and a
continuous Steiner symmetrization.

  3. (3)

Deduce the inequality for _sign-changing_ functions from the inequality for
_nonnegative_ functions by a concavity argument.

$\bullet$ In the first step, in order to obtain uniform estimates as
$d\to+\infty$, we need to expand

$(1+r)^{2^{*}}-1-2^{*}r$

with an accurate remainder term, for all $r\geq-\,1$. To do that, we “cut $r$
into pieces” by defining

$r_{1}:=\min\\{r,\gamma\\}\,,\quad
r_{2}:=\min\big{\\{}(r-\gamma)_{+},M-\gamma\big{\\}}\quad\mbox{and}\quad
r_{3}:=(r-M)_{+}\,,$

where $\gamma$ and $M$ are suitable parameters satisfying $0<\gamma<M.$
Furthermore, define

$\textstyle\theta:=2^{*}-2=\frac{4}{d-2}\,.$

Notice that $\theta\in(0,1]$ if $d\geq 6$ and $\lim_{d\to+\infty}\theta(d)=0$.

######  Lemma 4 ([22, Proposition 2.9]).

Given $d\geq 6$, $r\in[-1,\infty)$, and $\overline{M}\in[\sqrt{e},+\infty)$,
we have

$(1+r)^{2^{*}}-1-2^{*}r\leq\tfrac{1}{2}\,2^{*}\,(2^{*}-1)\,(r_{1}+r_{2})^{2}+2\,(r_{1}+r_{2})\,r_{3}+\left(1+C_{M}\,\theta\,\overline{M}^{-1}\ln\overline{M}\right)r_{3}^{2^{*}}\\\
+\left(\tfrac{3}{2}\,\gamma\,\theta\,r_{1}^{2}+C_{M,\overline{M}}\,\theta\,r_{2}^{2}\right)\mathbbm{1}_{\\{r\leq
M\\}}+C_{M,\overline{M}}\,\theta\,M^{2}\,\mathbbm{1}_{\\{r>M\\}}\,,$

where all the constants in the above inequality are explicit.

One can then prove that there exist computable constants $\epsilon_{1}$,
$\epsilon_{2}$, $k_{0}$, and $\epsilon_{0}\in(0,1/\theta)$, such that

$\left\|{\nabla
r}\right\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}+\mathrm{A}\,\left\|{r}\right\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}-\mathrm{A}\,\left\|{1+r}\right\|_{\mathrm{L}^{2^{*}}(\mathbb{S}^{d})}^{2}\geq\frac{4\,\epsilon_{0}}{d-2}\left(\left\|{\nabla
r}\right\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}+\mathrm{A}\,\left\|{r}\right\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}\right)+\sum_{k=1}^{3}I_{k}\,,$

with $\mathrm{A}:=\tfrac{1}{4}\,d\,(d-2)$ and

$\displaystyle I_{1}$
$\displaystyle\textstyle:=(1-\theta\,\epsilon_{0})\int_{\mathbb{S}^{d}}{\left(|\nabla
r_{1}|^{2}+\mathrm{A}\,r_{1}^{2}\right)}\,d\mu_{d}-\mathrm{A}\,(2^{*}-1+\epsilon_{1}\,\theta)\int_{\mathbb{S}^{d}}{r_{1}^{2}}\,d\mu_{d}+\mathrm{A}\,k_{0}\,\theta\int_{\mathbb{S}^{d}}{(r_{2}^{2}+r_{3}^{2})}\,d\mu_{d}\,,$
$\displaystyle I_{2}$
$\displaystyle\textstyle:=(1-\theta\,\epsilon_{0})\int_{\mathbb{S}^{d}}{\left(|\nabla
r_{2}|^{2}+\mathrm{A}\,r_{2}^{2}\right)}\,d\mu_{d}-\mathrm{A}\,\big{(}2^{*}-1+(k_{0}+C_{\epsilon_{1},\epsilon_{2}})\,\theta\big{)}\int_{\mathbb{S}^{d}}{r_{2}^{2}}\,d\mu_{d}\,,$
$\displaystyle I_{3}$
$\displaystyle\textstyle:=(1-\theta\,\epsilon_{0})\int_{\mathbb{S}^{d}}{\left(|\nabla
r_{3}|^{2}+\mathrm{A}\,r_{3}^{2}\right)}\,d\mu_{d}-\tfrac{2}{2^{*}}\,\mathrm{A}\,(1+\epsilon_{2}\,\theta)\int_{\mathbb{S}^{d}}{r_{3}^{2^{*}}}\,d\mu_{d}-\mathrm{A}\,k_{0}\,\theta\int_{\mathbb{S}^{d}}{r_{3}^{2}}\,d\mu_{d}\,.$

Next, one can use spectral gap estimates to prove $I_{1}\geq 0$ and the
Sobolev inequality to prove $I_{3}\geq 0$, noting that the extra coefficient
$2/2^{*}<1$ gives enough room to accommodate all error terms. Finally, using
that $\mu\big{(}\\{r_{2}>0\\}\big{)}$ is small, an improved spectral gap
inequality allows us to show $I_{2}\geq 0$.

If $d=3$, $4$, or $5$, we can rely on a simpler Taylor expansion that can be
found in [22, Proposition 2.7]. As a consequence, the following result has
been proved.

######  Theorem 5 ([22, Theorem 2.1]).

There are explicit constants $\epsilon_{0}\in(0,1/3)$ and $\delta\in(0,1/2)$
such that for all $d\geq 3$ and for all nonnegative
$u=1+r\in\mathrm{H}^{1}(\mathbb{S}^{d})$ with

$\left\|{r}\right\|_{\mathrm{L}^{2^{*}}(\mathbb{S}^{d})}^{2}\leq\frac{\delta}{1-\delta}\,,\quad\int_{\mathbb{S}^{d}}{r}\,d\mu_{d}=0\quad\mbox{and}\quad\int_{\mathbb{S}^{d}}{\omega\,r}\,d\mu_{d}=0\,,$

one has

$\left\|{\nabla
u}\right\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}+\mathrm{A}\,\left\|{u}\right\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}-\mathrm{A}\,\left\|{u}\right\|_{\mathrm{L}^{2^{*}}(\mathbb{S}^{d})}^{2}\geq\frac{4\,\epsilon_{0}}{d-2}\left(\left\|{\nabla
r}\right\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}+\mathrm{A}\,\left\|{r}\right\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}\right)\,.$

Let us define the stability quotient

$\mathcal{E}(f):=\frac{\|\nabla
f\|^{2}_{\mathrm{L}^{2}({\mathord{\mathbb{R}}}^{d})}-S_{d}\,\|f\|^{2}_{\mathrm{L}^{2^{*}}({\mathord{\mathbb{R}}}^{d})}}{\inf_{g\in\mathcal{M}}\|\nabla
f-\nabla g\|^{2}_{\mathrm{L}^{2}({\mathord{\mathbb{R}}}^{d})}}$

and consider the infimum

$\mathscr{I}(\delta):=\inf\Big{\\{}\mathcal{E}(f)\,:\,0\leq
f\in\dot{\mathrm{H}}^{1}({\mathord{\mathbb{R}}}^{d})\setminus\mathcal{M}\,,\quad\inf_{g\in\mathcal{M}}\|\nabla
f-\nabla
g\|^{2}_{\mathrm{L}^{2}({\mathord{\mathbb{R}}}^{d})}\leq\delta\,\|\nabla
f\|^{2}_{\mathrm{L}^{2}({\mathord{\mathbb{R}}}^{d})}\Big{\\}}\,.$

For a given $f$ satisfying $\inf_{g\in\mathcal{M}}\|\nabla f-\nabla
g\|^{2}_{\mathrm{L}^{2}({\mathord{\mathbb{R}}}^{d})}\leq\delta\,\|\nabla
f\|^{2}_{\mathrm{L}^{2}({\mathord{\mathbb{R}}}^{d})}$, up to a conformal
transformation, we can assume that $\inf_{g\in\mathcal{M}}\|\nabla f-\nabla
g\|^{2}_{\mathrm{L}^{2}({\mathord{\mathbb{R}}}^{d})}$ is realized by the
Aubin-Talenti function $g=g_{*}$ with

$g_{*}(x):=|{\mathord{\mathbb{S}}}^{d}|^{-\frac{d-2}{2\,d}}\left(\frac{2}{1+|x|^{2}}\right)^{\frac{d-2}{2}}\quad\forall\,x\in{\mathord{\mathbb{R}}}^{d}\,,$
(11)

use the inverse stereographic projection to transform $f$ and $g$ respectively
into $u=1+r$ and $1$, and notice that

$\displaystyle\mathrm{A}\,\left\|{r}\right\|_{\mathrm{L}^{2^{*}}(\mathbb{S}^{d})}^{2}\leq\left\|{\nabla
r}\right\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}+\mathrm{A}\,\left\|{r}\right\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}=\inf_{g\in\mathcal{M}}\|\nabla
f-\nabla g\|^{2}_{\mathrm{L}^{2}({\mathord{\mathbb{R}}}^{d})}\,,$
$\displaystyle\|\nabla
f\|^{2}_{\mathrm{L}^{2}({\mathord{\mathbb{R}}}^{d})}=\left\|{\nabla
u}\right\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}+\mathrm{A}\,\left\|{u}\right\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}=\left\|{\nabla
r}\right\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}+\mathrm{A}\,\left\|{r}\right\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}+\mathrm{A}\,,$

where the first line follows from (1). We deduce from the condition
$\inf_{g\in\mathcal{M}}\|\nabla f-\nabla
g\|^{2}_{\mathrm{L}^{2}({\mathord{\mathbb{R}}}^{d})}\leq\delta\,\|\nabla
f\|^{2}_{\mathrm{L}^{2}({\mathord{\mathbb{R}}}^{d})}$ that

$\mathrm{A}\,\left\|{r}\right\|_{\mathrm{L}^{2^{*}}(\mathbb{S}^{d})}^{2}\leq\left\|{\nabla
r}\right\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}+\mathrm{A}\,\left\|{r}\right\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}\leq\frac{\delta\,\mathrm{A}}{1-\delta}$

and apply Theorem 5 to obtain

$\mathcal{E}(f)=\frac{\left\|{\nabla
u}\right\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}+\mathrm{A}\,\left\|{u}\right\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}-\mathrm{A}\,\left\|{u}\right\|_{\mathrm{L}^{2^{*}}(\mathbb{S}^{d})}^{2}}{\left\|{\nabla
r}\right\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}+\mathrm{A}\,\left\|{r}\right\|_{\mathrm{L}^{2}(\mathbb{S}^{d})}^{2}}\geq\frac{4\,\epsilon_{0}}{d-2}\,.$

This completes the local analysis where, by homogeneity, the scale is fixed in
terms of $\|\nabla f\|_{\mathrm{L}^{2}({\mathord{\mathbb{R}}}^{d})}^{2}$. With
the notation of Theorem 5,

$\mathscr{I}(\delta)\geq\frac{4\,\epsilon_{0}}{d-2}\,.$

$\bullet$ In Step 2, we deal with nonnegative functions $f$ that are not close
to the manifold $\mathcal{M}$, i.e., such that

$\inf_{g\in\mathcal{M}}\|\nabla f-\nabla
g\|_{\mathrm{L}^{2}({\mathord{\mathbb{R}}}^{d})}^{2}>\delta\,\|\nabla
f\|_{\mathrm{L}^{2}({\mathord{\mathbb{R}}}^{d})}^{2}\,.$ (12)

The first ingredient is the method of competing symmetries [16] of Carlen and
Loss. Consider any nonnegative function
$f\in\dot{\mathrm{H}}^{1}({\mathord{\mathbb{R}}}^{d})$ and let

$(Uf)(x):=\left(\frac{2}{|x-e_{d}|^{2}}\right)^{\frac{d-2}{2}}f\left(\frac{x_{1}}{|x-e_{d}|^{2}},\ldots,\frac{x_{d-1}}{|x-e_{d}|^{2}},\frac{|x|^{2}-1}{|x-e_{d}|^{2}}\right)\quad\mbox{where}\quad
e_{d}=(0,\ldots,0,1)\in{\mathord{\mathbb{R}}}^{d}\,,$

and notice that $\mathcal{E}(Uf)=\mathcal{E}(f)$. We also consider the
symmetric decreasing rearrangement $\mathcal{R}f=f^{*}$, with the properties
that $f$ and $f^{*}$ are equimeasurable, and that $\|\nabla
f^{*}\|_{\mathrm{L}^{2}({\mathord{\mathbb{R}}}^{d})}\leq\|\nabla
f\|_{\mathrm{L}^{2}({\mathord{\mathbb{R}}}^{d})}$.

######  Theorem 6 ([16, Theorem 3.3]).

Let $f\in\mathrm{L}^{2^{*}}({\mathord{\mathbb{R}}}^{d})$ be a non-negative
function with $\|f\|_{\mathrm{L}^{2^{*}}({\mathord{\mathbb{R}}}^{d})}=1$. The
sequence $f_{n}=(\mathcal{R}U)^{n}f$ is such that
$\lim_{n\to+\infty}\|f_{n}-g_{*}\|_{\mathrm{L}^{2^{*}}({\mathord{\mathbb{R}}}^{d})}=0$.
If $f\in\dot{\mathrm{H}}^{1}({\mathord{\mathbb{R}}}^{d})$, then $(\|\nabla
f_{n}\|_{\mathrm{L}^{2}({\mathord{\mathbb{R}}}^{d})})_{n\in{\mathord{\mathbb{N}}}}$
is a non-increasing sequence.

Whether $f_{n}$ satisfies (12) for all $n\in{\mathord{\mathbb{N}}}$ or not, we
face an alternative.

######  Lemma 7 ([22, Lemma 3.5]).

Let $f$ be as in Theorem 6 and such that (12) holds and let
$f_{n}=(\mathcal{R}U)^{n}f$. Then either $\inf_{g\in\mathcal{M}}\|\nabla
f_{n}-\nabla
g\|_{\mathrm{L}^{2}({\mathord{\mathbb{R}}}^{d})}^{2}\geq\delta\,\|\nabla
f_{n}\|_{\mathrm{L}^{2}({\mathord{\mathbb{R}}}^{d})}^{2}$ for all $n$, or
there exists $n_{0}\in{\mathord{\mathbb{N}}}$ such that

$\inf_{g\in\mathcal{M}}\|\nabla f_{n_{0}}-\nabla
g\|_{\mathrm{L}^{2}({\mathord{\mathbb{R}}}^{d})}^{2}>\delta\,\|\nabla
f_{n_{0}}\|_{\mathrm{L}^{2}({\mathord{\mathbb{R}}}^{d})}^{2}\quad\mbox{and}\quad\inf_{g\in\mathcal{M}}\|\nabla
f_{n_{0}+1}-\nabla
g\|_{\mathrm{L}^{2}({\mathord{\mathbb{R}}}^{d})}^{2}<\delta\,\|\nabla
f_{n_{0}+1}\|_{\mathrm{L}^{2}({\mathord{\mathbb{R}}}^{d})}^{2}\,.$

In the first case, we have

$\lim_{n\to+\infty}\|\nabla
f_{n}\|^{2}_{2}\leq\frac{1}{\delta}\,\lim_{n\to+\infty}\,\inf_{g\in\mathcal{M}}\|\nabla
f_{n}-\nabla g\|_{2}^{2}=\frac{1}{\delta}\left(\lim_{n\to+\infty}\|\nabla
f_{n}\|^{2}_{2}-S_{d}\,\|f\|_{2^{*}}^{2}\right)$

where the last equality arises as a consequence of the properties of
$(f_{n})_{n\in{\mathord{\mathbb{N}}}}$ (see [22, Lemma 3.4]). Combined with
the simple estimate

$\mathcal{E}(f)=\frac{\|\nabla
f\|^{2}_{2}-S_{d}\,\|f\|_{2^{*}}^{2}}{\inf_{g\in\mathcal{M}}\|\nabla f-\nabla
g\|^{2}_{2}}\geq\frac{\|\nabla f\|^{2}_{2}-S_{d}\,\|f\|_{2^{*}}^{2}}{\|\nabla
f\|^{2}_{2}}\geq\frac{\|\nabla
f_{n}\|^{2}_{2}-S_{d}\,\|f\|_{2^{*}}^{2}}{\|\nabla f_{n}\|^{2}_{2}}\,,$

we can take the limit as $n\to+\infty$ and obtain $\mathcal{E}(f)\geq\delta$.
In the second case, we adapt a strategy due to Christ in [19], by building a
continuous rearrangement _flow_ $(\mathsf{f}_{\tau})_{n_{0}\leq\tau<n_{0}+1}$
with $\mathsf{f}_{n_{0}}=Uf_{n}$ such that

$\|f_{\tau}\|_{\mathrm{L}^{2^{*}}({\mathord{\mathbb{R}}}^{d})}=\|f\|_{\mathrm{L}^{2^{*}}({\mathord{\mathbb{R}}}^{d})}\,,\;\tau\mapsto\|\nabla
f_{\tau}\|_{\mathrm{L}^{2}({\mathord{\mathbb{R}}}^{d})}\;\mbox{\it is
nonincreasing, and}\;\lim_{\tau\to n_{0}+1}\mathsf{f}_{\tau}=f_{n_{0}+1}\,.$

Choosing the smallest $\tau_{0}\in(n_{0},n_{0}+1)$ such that
$\inf_{g\in\mathcal{M}}\|\nabla(\mathsf{f}_{\tau_{0}}-g)\|_{\mathrm{L}^{2}({\mathord{\mathbb{R}}}^{d})}^{2}=\delta\,\|\nabla
f_{\tau_{0}}\|_{\mathrm{L}^{2}({\mathord{\mathbb{R}}}^{d})}^{2}$ and using
$\|f\|_{\mathrm{L}^{2^{*}}({\mathord{\mathbb{R}}}^{d})}=\|f_{n_{0}}\|_{\mathrm{L}^{2^{*}}({\mathord{\mathbb{R}}}^{d})}=\|\mathsf{f}_{\tau_{0}}\|_{\mathrm{L}^{2^{*}}({\mathord{\mathbb{R}}}^{d})}$,
this gives:

$\mathcal{E}(f)\geq
1-S_{d}\,\frac{\|f\|^{2}_{\mathrm{L}^{2^{*}}({\mathord{\mathbb{R}}}^{d})}}{\|\nabla
f\|^{2}_{\mathrm{L}^{2}({\mathord{\mathbb{R}}}^{d})}}\geq
1-S_{d}\,\frac{\|f_{n_{0}}\|^{2}_{\mathrm{L}^{2^{*}}({\mathord{\mathbb{R}}}^{d})}}{\|\nabla
f_{n_{0}}\|^{2}_{\mathrm{L}^{2}({\mathord{\mathbb{R}}}^{d})}}\geq
1-S_{d}\,\frac{\|\mathsf{f}_{\tau_{0}}\|^{2}_{\mathrm{L}^{2^{*}}({\mathord{\mathbb{R}}}^{d})}}{\|\nabla\mathsf{f}_{\tau_{0}}\|^{2}_{\mathrm{L}^{2}({\mathord{\mathbb{R}}}^{d})}}=\delta\,\mathcal{E}(f_{\tau_{0}})\geq\delta\,\mathscr{I}(\delta)\,.$

To build the flow, we refer to [12, 13] and to [22] for further references.
The existence of $\tau_{0}$ requires a discussion that can be found in [22,
Section 3.1.2].

Finally, it is simple to prove that for all $\delta\in(0,1)$,
$\mathscr{I}(\delta)\leq 1$. Therefore, in all cases,
$\mathcal{E}(f)\geq\delta\,\mathscr{I}(\delta)\,.$

$\bullet$ The third step is to remove the positivity assumption of Theorem 1
as in [22, Section 3.2]. Take $f=f_{+}-f_{-}$ with
$\left\|{f}\right\|_{\mathrm{L}^{2^{*}}({\mathord{\mathbb{R}}}^{d})}=1$ and
define
$m:=\left\|{f_{-}}\right\|_{\mathrm{L}^{2^{*}}({\mathord{\mathbb{R}}}^{d})}^{2^{*}}$.
Without loss of generality one may assume that
$1-m=\left\|{f_{+}}\right\|_{\mathrm{L}^{2^{*}}({\mathord{\mathbb{R}}}^{d})}^{2^{*}}>1/2$.
The positive concave function

$h_{d}(m):=m^{\frac{d-2}{d}}+(1-m)^{\frac{d-2}{d}}-1$

satisfies

$2\,h_{d}(1/2)\,m\leq h_{d}(m)\,,\quad h_{d}(1/2)=2^{2/d}-1\,.$

With $\boldsymbol{\delta}(f)=\left\|{\nabla
f}\right\|_{\mathrm{L}^{2}({\mathord{\mathbb{R}}}^{d})}^{2}-S_{d}\left\|{f}\right\|_{\mathrm{L}^{2^{*}}({\mathord{\mathbb{R}}}^{d})}^{2}$,
one finds $g_{+}\in\mathcal{M}$ such that

$\boldsymbol{\delta}(f)\geq C_{\rm BE}^{d,\rm pos}\left\|{\nabla f_{+}-\nabla
g_{+}}\right\|_{\mathrm{L}^{2}({\mathord{\mathbb{R}}}^{d})}^{2}+\frac{2\,h_{d}(1/2)}{h_{d}(1/2)+1}\left\|{\nabla
f_{-}}\right\|_{\mathrm{L}^{2}({\mathord{\mathbb{R}}}^{d})}^{2}\,,$

and therefore

$C^{d}_{\rm
BE}\geq\tfrac{1}{2}\,\min\left\\{\max_{0<\delta<1/2}\,\delta\,\mathscr{I}(\delta),\frac{2\,h_{d}(1/2)}{h_{d}(1/2)+1}\right\\}\,.$

$\bullet$ Combining all estimates of the three previous steps completes the
proof of Theorem 1.∎

##  4\. Stabitity for the Gaussian logarithmic Sobolev inequality. Proof of
Theorem 2

In this section we describe the main steps in the proof of the stability
estimate for the Gaussian logarithmic Sobolev inequality (2) by considering
the large dimensional limit of (6), as it appears in [22, Section 4]. With
$u=f/{a,\,c}$, where $g_{\star}$ is given by (11), Inequality (6) can be
rewritten as

$\int_{{\mathord{\mathbb{R}}}^{d}}{|\nabla
u|^{2}\,g_{\star}^{2}\normalcolor}\,dx+d\,(d-2)\,\int_{{\mathord{\mathbb{R}}}^{d}}{|u|^{2}\,g_{\star}^{2^{*}}\normalcolor}\,dx-d\,(d-2)\,\left\|{g_{\star}}\right\|_{\mathrm{L}^{2^{*}}({\mathord{\mathbb{R}}}^{d})}^{2^{*}-2}\normalcolor\left(\int_{{\mathord{\mathbb{R}}}^{d}}{|u|^{2^{*}}\,g_{\star}^{2^{*}}\normalcolor}\,dx\right)^{2/2^{*}}\hskip
56.9055pt\\\
\geq\frac{\beta}{d}\left(\int_{{\mathord{\mathbb{R}}}^{d}}{|\nabla
u|^{2}\,g_{\star}^{2}\normalcolor}\,dx+d\,(d-2)\,\int_{{\mathord{\mathbb{R}}}^{d}}{|u-g_{d}/g_{\star}|^{2}\,g_{\star}^{2^{*}}}\,dx\right),$

where $g_{d}\in\mathcal{M}$ realizes $\inf_{g\in\mathcal{M}}\|\nabla f-\nabla
g\|^{2}_{\mathrm{L}^{2}({\mathord{\mathbb{R}}}^{d})}$. For some parameters
$a_{d}$, $b_{d}$ and $c_{d}$, we can write that
$g_{d}(x)=c_{d}\,(a_{d}+|x-b_{d}|^{2})^{1-d/2}$. We rescale the function $u$
according to

$u(x)=v\left(r_{d}\,x\right)\quad\forall\,x\in{\mathord{\mathbb{R}}}^{d}\,,\quad
r_{d}=\sqrt{\tfrac{d}{2\,\pi}}$

and consider the function $w_{v}^{d}$ such that
$w_{v}^{d}\left(r_{d}\,x\right)=g_{d}(x)/g_{\star}(x)$. Hence

$\int_{{\mathord{\mathbb{R}}}^{d}}{|\nabla
v|^{2}\left(1+\tfrac{1}{r_{d}^{2}}\,|x|^{2}\right)^{2}}\,d\mu_{d}\geq
2\,\pi\,(d-2)\left(\left(\int_{{\mathord{\mathbb{R}}}^{d}}{|v|^{2^{*}}}\,d\mu_{d}\right)^{2/2^{*}}-\int_{{\mathord{\mathbb{R}}}^{d}}{|v|^{2}}\,d\mu_{d}\right)\hskip
56.9055pt\\\ +\frac{\beta}{d}\left(\int_{{\mathord{\mathbb{R}}}^{d}}{|\nabla
v|^{2}}\,d\mu_{d}+2\,\pi(d-2)\normalcolor\,\int_{{\mathord{\mathbb{R}}}^{d}}{|v-w_{v}^{d}|^{2}}\,d\mu_{d}\right)$

where $d\mu_{d}=Z_{d}^{-1}\,g_{\star}^{2^{*}}\,dx$ is the probability measure
given by

$d\mu_{d}(x):=\frac{1}{Z_{d}}\left(1+\tfrac{1}{r_{d}^{2}}\,|x|^{2}\right)^{-d}\,dx\quad\mbox{with}\quad
Z_{d}=\frac{2^{1-d}\,\sqrt{\pi}}{\Gamma\left(\tfrac{d+1}{2}\right)}\left(\frac{d}{2}\right)^{\frac{d}{2}}\,.$

Our goal is to take the limit $d\to+\infty$ when one considers functions
$v(x)$ depending only on $y\in{\mathord{\mathbb{R}}}^{N}$, with
$x=(y,z)\in{\mathord{\mathbb{R}}}^{N}\times{\mathord{\mathbb{R}}}^{d-N}\approx{\mathord{\mathbb{R}}}^{d}$,
for some fixed integer $N$. With $|x|^{2}=|y|^{2}+|z|^{2}$, notice that

$1+\tfrac{1}{r_{d}^{2}}\,|x|^{2}=1+\tfrac{1}{r_{d}^{2}}\,\big{(}|y|^{2}+|z|^{2}\big{)}=\left(1+\tfrac{1}{r_{d}^{2}}\,|y|^{2}\right)\left(1+\tfrac{|z|^{2}}{r_{d}^{2}+|y|^{2}}\right)$

and, as a consequence,

$\displaystyle\textstyle\lim_{d\to+\infty}\left(1+\tfrac{1}{r_{d}^{2}}\,|y|^{2}\right)^{-\frac{N+d}{2}}=e^{-\pi\,|y|^{2}}\,,$
$\displaystyle\textstyle\lim_{d\to+\infty}\int_{{\mathord{\mathbb{R}}}^{d}}{|v(y)|^{2}}\,d\mu_{d}=\int_{{\mathord{\mathbb{R}}}^{N}}{|v|^{2}}\,d\gamma\,,$
$\displaystyle\textstyle\lim_{d\to+\infty}\int_{{\mathord{\mathbb{R}}}^{d}}{|\nabla
v|^{2}\left(1+\tfrac{1}{r_{d}^{2}}\,|x|^{2}\right)^{2}}\,d\mu_{d}=4\,\int_{{\mathord{\mathbb{R}}}^{N}}{|\nabla
v|^{2}}\,d\gamma\,,$

where $d\gamma(y):=e^{-\pi\,|y|^{2}}\,dy$ is the standard _Gaussian
probability measure_. However, the function $w_{v}^{d}$ depends on $d$ and the
main difficulty is to obtain enough estimates on the parameters $a_{d}$,
$b_{d}$ and $c_{d}$ to pass to the limit after integrating in all integrals
with respect to $z$, which completes the proof of (7). See [22, Section 4] for
further details. ∎

##  5\. Stability of the logarithmic Sobolev inequality: a new proof of
Theorem 2

Instead of proving Theorem 2 as a consequence of Theorem 1, one can give a
direct proof of (7). Just like in Section 3, we prove the quantitative version
of the sharp logarithmic Sobolev inequality (Theorem 2) in two steps, one
close to and one far from the set of optimizers.

Let us start with a consequence of Theorem 5.

######  Theorem 8.

There are explicit constants $\eta>0$ and $\delta\in(0,1/2)$ such that for all
$N\in{\mathord{\mathbb{N}}}$ and for all for all nonnegative
$u=1+r\in\mathrm{H}^{1}(\mathbb{S}^{d})$ satisfying

$\int_{{\mathord{\mathbb{R}}}^{N}}r^{2}\,d\gamma\leq\frac{\delta}{1-\delta}$
(13)

and

$\int_{{\mathord{\mathbb{R}}}^{N}}r\,d\gamma=0=\int_{{\mathord{\mathbb{R}}}^{N}}x_{j}\,r\,d\gamma\,,\quad
j=1,\,2,\ldots,N\,,$ (14)

one has

$\int_{{\mathord{\mathbb{R}}}^{N}}|\nabla
u|^{2}\,d\gamma-\pi\int_{{\mathord{\mathbb{R}}}^{N}}|u|^{2}\ln\left(\frac{|u|^{2}}{\|u\|_{\mathrm{L}^{2}(\gamma)}^{2}}\right)\,d\gamma\geq\eta\int_{{\mathord{\mathbb{R}}}^{N}}r^{2}\,d\gamma\,.$

The constant $\delta$ coincides with the corresponding constant in Theorem 5
and $\eta=2\,\pi\,\epsilon_{0}$.

###### Proof.

Notice that $x\in\mathrm{L}^{2}(\gamma)$, so the orthogonality constraints
raise no integration issues. We denote
$\Sigma_{d}:=\\{x\in{\mathord{\mathbb{R}}}^{d+1}:\,|x|=\rho_{d}\\}$ with
$\rho_{d}:=\sqrt{d/(2\,\pi)}$. The factor of $1/(2\,\pi)$ in the definition of
$\rho_{d}$ is necessary to get the $\pi$ in the exponent of the Gaussian
density. We integrate on $\Sigma_{d}$ with respect to the uniform probability
measure $d\mu_{d}$. By rescaling our result in Theorem 5 we find that

$\int_{\Sigma_{d}}|\nabla
R|^{2}\,d\mu_{d}-\pi\,\frac{d-2}{2}\left(\left(\int_{\Sigma_{d}}(1+R)^{\frac{2\,d}{d-2}}\,d\mu_{d}\right)^{\frac{d-2}{d}}-\int_{\Sigma_{d}}(1+R)^{2}\,d\mu_{d}\right)\\\
\geq
2\,\pi\,\epsilon_{0}\int_{\Sigma_{d}}\left(\frac{1}{\pi}\,\frac{2}{d-2}\,|\nabla
R|^{2}+R^{2}\right)\,d\mu_{d}\,.$ (15)

This inequality is valid for all $R\in\mathrm{H}^{1}(\Sigma_{d})$ such that

$\left(\int_{\Sigma_{d}}R^{\frac{2\,d}{d-2}}\,d\mu_{d}\right)^{\frac{d-2}{d}}\leq\frac{\delta}{1-\delta}$
(16)

and

$\int_{\Sigma_{d}}R\,d\mu_{d}=0=\int_{\Sigma_{d}}x_{j}\,R\,d\mu_{d}\,,\quad
j=1,\ldots,d+1\,.$ (17)

Given a function $r\in\mathrm{H}^{1}(\gamma)$ and $d>N$, we apply this
inequality to the function

$R_{d}(x):=r(x_{1},\ldots,x_{N})-\int_{\Sigma_{d}}r\,d\mu_{d}-2\,\pi\,\frac{d+1}{d}\sum_{n=1}^{N}x_{n}\int_{\Sigma_{d}}y_{n}\,r(y_{1},\ldots,y_{N})\,d\mu_{d}(y)$

for $x\in\Sigma_{d}$. This function satisfies the orthogonality conditions
(17). Note here that the functions $\sqrt{2\,\pi}\,\sqrt{(d+1)/d}\,x_{j}$ are
$\mathrm{L}^{2}$-normalized on $\Sigma_{d}$.

We now use the well-known fact that, as $d\to+\infty$, the marginal of
$d\mu_{d}$ corresponding to the first $N$ coordinates converges to $d\gamma$.
Thus,

$\displaystyle\lim_{d\to+\infty}\int_{\Sigma_{d}}|\nabla
r|^{2}\,d\mu_{d}=\int_{{\mathord{\mathbb{R}}}^{N}}|\nabla r|^{2}\,d\gamma\,,$
$\displaystyle\lim_{d\to+\infty}\int_{\Sigma_{d}}r^{2}\,d\mu_{d}=\int_{{\mathord{\mathbb{R}}}^{N}}r^{2}\,d\gamma\,,$
$\displaystyle\lim_{d\to+\infty}\int_{\Sigma_{d}}r\,d\mu_{d}=\int_{{\mathord{\mathbb{R}}}^{N}}r\,d\gamma=0\,,$
$\displaystyle\lim_{d\to+\infty}\int_{\Sigma_{d}}y_{n}\,r(y_{1},\ldots,y_{N})\,d\mu_{d}(y)=\int_{{\mathord{\mathbb{R}}}^{N}}y_{n}\,r\,d\gamma=0\,.$

From this we conclude easily that

$\lim_{d\to+\infty}\int_{\Sigma_{d}}|\nabla
R_{d}|^{2}\,d\mu_{d}=\int_{{\mathord{\mathbb{R}}}^{N}}|\nabla
r|^{2}\,d\gamma\,,\quad\lim_{d\to+\infty}\int_{\Sigma_{d}}R_{d}^{2}\,d\mu_{d}=\int_{{\mathord{\mathbb{R}}}^{N}}r^{2}\,d\gamma\,.$

With some modest amount of effort one also finds that

$\lim_{d\to+\infty}\int_{\Sigma_{d}}R_{d}^{\frac{2\,d}{d-2}}\,d\mu_{d}=\int_{{\mathord{\mathbb{R}}}^{N}}r^{2}\,d\gamma\,.$

Assuming that the inequality in (13) is strict, the same is true for the left
side when $d$ is sufficiently large, and consequently the smallness condition
(16) holds when $d$ is sufficiently large. Thus, inequality (15) is valid for
all sufficiently large $d$. The equality case in (13) can be obtained at the
very end by a simple approximation argument.

Now, we drop the gradient term in the right side and letting $d\to+\infty$ we
infer that

$\int_{{\mathord{\mathbb{R}}}^{N}}|\nabla
r|^{2}\,d\gamma-\pi\,\limsup_{d\to+\infty}\frac{d-2}{2}\left(\left(\int_{\Sigma_{d}}(1+R_{d})^{\frac{2\,d}{d-2}}\,d\mu_{d}\right)^{\frac{d-2}{d}}\kern-12.0pt-\int_{\Sigma_{d}}(1+R_{d})^{2}\,d\mu_{d}\right)\geq
2\,\pi\,\epsilon_{0}\int_{{\mathord{\mathbb{R}}}^{N}}r^{2}\,d\gamma\,.$

Finally, we verify that

$\limsup_{d\to+\infty}\frac{d-2}{2}\left(\left(\int_{\Sigma_{d}}(1+R_{d})^{\frac{2\,d}{d-2}}\,d\mu_{d}\right)^{\frac{d-2}{d}}\kern-12.0pt-\int_{\Sigma_{d}}(1+R_{d})^{2}\,d\mu_{d}\right)=\int_{{\mathord{\mathbb{R}}}^{N}}(1+r)^{2}\,\ln\left(\frac{(1+r)^{2}}{\|1+r\|_{\mathrm{L}^{2}(\gamma)}^{2}}\right)\,d\gamma\,.$

In fact, if the orthogonality conditions were not present and the marginals
would already be equal to their limit, this would follow from the fact that

$\lim_{p\to
1^{+}}\frac{1}{p-1}\left(\left(\int_{{\mathord{\mathbb{R}}}^{N}}h^{p}\,d\gamma\right)^{1/p}-\int_{{\mathord{\mathbb{R}}}^{N}}h\,d\gamma\right)=\int_{{\mathord{\mathbb{R}}}^{N}}h\ln\left(\frac{h}{\int_{{\mathord{\mathbb{R}}}^{N}}h\,d\gamma}\right)d\gamma\,,$

valid on any measure space for any nonnegative function $h$ that satisfies
$h\in\mathrm{L}^{1}\cap\mathrm{L}^{p_{0}}(\gamma)$ for some $p_{0}>1$. Proving
the latter fact is simple, as well as including the effect of the
orthogonality conditions and the convergence of the marginals, so we shall
omit it. These remarks complete the proof Theorem 8. ∎

We emphasize that in the previous proof we did not use Theorem 1, but rather
Theorem 5. In this way we avoid having to control the distance to the set of
optimizers in the high-dimensional limit, which seems harder than verifying
the orthogonality conditions.

###### Proof of Theorem 2.

As in the proof of Theorem 1, we first prove the result for nonnegative
functions and then extend it to sign changing solutions. Let us denote by
$\kappa^{\rm pos}$ the stability constant in the stability inequality
restricted to nonnegative functions.

_Step 1._ Let $\eta$ and $\delta$ be as in Theorem 8. For $0\leq
u\in\mathrm{H}^{1}(\gamma)$ we distinguish two cases.

$\bullet$ The first case is where

$\inf_{a\in{\mathord{\mathbb{R}}}^{N}\\!,\,c\in{\mathord{\mathbb{R}}}}\int_{{\mathord{\mathbb{R}}}^{N}}(u-c\,e^{a\cdot
x})^{2}\,d\gamma\leq\delta\int_{{\mathord{\mathbb{R}}}^{N}}u^{2}\,d\gamma\,.$

The infimum on the left-hand side is attained at some
$a\in{\mathord{\mathbb{R}}}^{N}$ and $c\in{\mathord{\mathbb{R}}}$, as can be
checked by optimizing
$\int_{{\mathord{\mathbb{R}}}^{N}}\big{|}v-c\,e^{|a|^{2}/(2\,\pi)-\pi\,|x-a/\pi|^{2}/2}\big{|}^{2}\,dx$
where $v(x):=u(x)\,e^{-\,\pi\,|x|^{2}/2}$. Let

$\tilde{u}(y):=e^{-\,\,y\cdot
a-\frac{|a|^{2}}{2\,\pi}}\,u\big{(}y+\tfrac{a}{\pi}\big{)}\,.$

Then, by a simple computation involving an integration by parts and a change
of variables,

$\int_{{\mathord{\mathbb{R}}}^{N}}|\nabla\tilde{u}|^{2}\,d\gamma-\pi\int_{{\mathord{\mathbb{R}}}^{N}}\tilde{u}^{2}\ln\left(\frac{\tilde{u}^{2}}{\|\tilde{u}\|_{\mathrm{L}^{2}(\gamma)}^{2}}\right)\,d\gamma=\int_{{\mathord{\mathbb{R}}}^{N}}|\nabla
u|^{2}\,d\gamma-\pi\int_{{\mathord{\mathbb{R}}}^{N}}u^{2}\ln\left(\frac{u^{2}}{\|u\|_{\mathrm{L}^{2}(\gamma)}^{2}}\right)\,d\gamma\,.$

Therefore, the deficit of $\tilde{u}$ coincides with that of $u$, while the
infimum for $\tilde{u}$ among all functions of the form (4) is attained at the
constant $c\,\exp\big{(}|a|^{2}/(2\,\pi)\big{)}$. Finally, by multiplying
$\tilde{u}$ with a constant, we may assume that this constant is equal to one.
To summarize, we may assume without loss of generality that the infimum in the
theorem is attained at $a=0$ and $c=1$.

Let us set $r:=u-1$. Then the minimality implies that $r$ satisfies the
orthogonality conditions (14). Moreover, we have

$\int_{{\mathord{\mathbb{R}}}^{N}}r^{2}\,d\gamma\leq\delta\int_{{\mathord{\mathbb{R}}}^{N}}u^{2}\,d\gamma=\delta\left(1+\int_{{\mathord{\mathbb{R}}}^{N}}r^{2}\,d\gamma\right),$

so the smallness condition (13) is satisfied and we can apply Theorem 8. This
yields the inequality in the theorem with a stability constant $\eta$.

$\bullet$ Next, we consider the case where

$\inf_{a\in{\mathord{\mathbb{R}}}^{N}\\!,\,c\in{\mathord{\mathbb{R}}}}\int_{{\mathord{\mathbb{R}}}^{N}}(u-c\,e^{x\cdot
a})^{2}\,d\gamma>\delta\int_{{\mathord{\mathbb{R}}}^{N}}u^{2}\,d\gamma\,.$

We argue similarly as we did in Section 3 concerning the Sobolev inequality,
but there are some differences in this case.

For $f\in\mathrm{L}^{2}(\gamma)$ we denote by $Uf$ its Gaussian rearrangement,
that is, the function on ${\mathord{\mathbb{R}}}^{N}$ whose superlevel sets
have the form $\\{x\in{\mathord{\mathbb{R}}}^{N}:\,x_{1}<\mu\\}$ for some
$\mu\in{\mathord{\mathbb{R}}}$ and have the same $\gamma$-measure as the
corresponding superlevel sets of $f$. Moreover, we denote

$Vf:=e^{\frac{\pi}{2}\,|x|^{2}}\,\mathcal{R}\left(e^{-\,\frac{\pi}{2}\,|x|^{2}}\,f\right)\,,$

where $\mathcal{R}$ is, as before, the Euclidean rearrangement. Then, as shown
in [16, Theorem 4.1], for any $0\leq f\in\mathrm{L}^{2}(\gamma)$ one has

$f_{n}:=(VU)^{n}f\to\|f\|_{\mathrm{L}^{2}(\gamma)}\quad\text{in}\quad\mathrm{L}^{2}(\gamma)\,.$

Moreover, $\|f_{n}\|_{\mathrm{L}^{2}(\gamma)}=\|f\|_{\mathrm{L}^{2}(\gamma)}$
and

$n\mapsto\int_{{\mathord{\mathbb{R}}}^{N}}|\nabla
f_{n}|^{2}\,d\gamma-\pi\int_{{\mathord{\mathbb{R}}}^{N}}f_{n}^{2}\ln\left(\frac{f_{n}^{2}}{\|f_{n}\|_{\mathrm{L}^{2}(\gamma)}^{2}}\right)\,d\gamma$

is nonincreasing. This is the analogue of Theorem 6 in the present case.

We apply this procedure to our function $u$, which we assume here to be
nonnegative, and obtain a sequence of functions $u_{n}$ with constant
$\mathrm{L}^{2}(\gamma)$-norm. Moreover, since

$\inf_{a,\,c}\|u_{n}-c\,e^{a\cdot
x}\|_{\mathrm{L}^{2}(\gamma)}\leq\big{\|}u_{n}-\|u\|_{\mathrm{L}^{2}(\gamma)}\big{\|}_{\mathrm{L}^{2}(\gamma)}\to
0\quad\mbox{as}\quad n\to+\infty\,,$

there is an $n_{0}\in{\mathord{\mathbb{N}}}$ such that

$\inf_{a,\,c}\|u_{n_{0}}-c\,e^{a\cdot
x}\|_{\mathrm{L}^{2}(\gamma)}^{2}\geq\delta\,\|u\|_{\mathrm{L}^{2}(\gamma)}^{2}>\inf_{a,\,c}\|u_{n_{0}+1}-c\,e^{a\cdot
x}\|_{\mathrm{L}^{2}(\gamma)}^{2}\,.$

This replaces Lemma 7. We have

$\displaystyle\frac{\int_{{\mathord{\mathbb{R}}}^{N}}|\nabla
u|^{2}\,d\gamma-\pi\int_{{\mathord{\mathbb{R}}}^{N}}u^{2}\ln\left(\frac{u^{2}}{\|u\|_{\mathrm{L}^{2}(\gamma)}^{2}}\right)\,d\gamma}{\inf_{a,\,c}\|u-c\,e^{a\cdot
x}\|_{\mathrm{L}^{2}(\gamma)}^{2}}$
$\displaystyle\geq\frac{\int_{{\mathord{\mathbb{R}}}^{N}}|\nabla
u|^{2}\,d\gamma-\pi\int_{{\mathord{\mathbb{R}}}^{N}}u^{2}\ln\left(\frac{u^{2}}{\|u\|_{\mathrm{L}^{2}(\gamma)}^{2}}\right)\,d\gamma}{\|u\|_{\mathrm{L}^{2}(\gamma)}^{2}}$
$\displaystyle\geq\frac{\int_{{\mathord{\mathbb{R}}}^{N}}|\nabla
u_{n_{0}}|^{2}\,d\gamma-\pi\int_{{\mathord{\mathbb{R}}}^{N}}u_{n_{0}}^{2}\ln\left(\frac{u_{n_{0}}^{2}}{\|u_{n_{0}}\|_{\mathrm{L}^{2}(\gamma)}^{2}}\right)\,d\gamma}{\|u\|_{\mathrm{L}^{2}(\gamma)}^{2}}\,.$

We now use a continuous rearrangement flow to connect $u_{n_{0}}$ to
$u_{n_{0}+1}$. More precisely, we a consider a family of functions
$(\mathsf{u}_{\tau})_{n_{0}\leq\tau\leq n_{0}+1}$, where
$\mathsf{u}_{n_{0}}:=Uu_{n_{0}}$ and $\mathsf{u}_{n_{0}+1}:=u_{n_{0}+1}$. We
define $\mathsf{u}_{\tau}$ as $e^{\frac{\pi}{2}\,|x|^{2}}$ times the
continuous (Euclidean) rearrangement of
$e^{-\frac{\pi}{2}\,|x|^{2}}\,Uu_{n_{0}}$ at parameter $\tau$. In the same way
as in [22, Lemma 36], one sees that

$\tau\mapsto\inf_{a,\,c}\|\mathsf{u}_{\tau}-c\,e^{a\cdot
x}\|_{\mathrm{L}^{2}(\gamma)}^{2}$

is continuous, and therefore there is a $\tau_{0}\in[n_{0},n_{0}+1)$ such that

$\inf_{a,\,c}\|\mathsf{u}_{\tau_{0}}-c\,e^{a\cdot
x}\|_{\mathrm{L}^{2}(\gamma)}^{2}=\delta\,\|u\|_{\mathrm{L}^{2}(\gamma)}^{2}\,.$

It follows that

$\frac{\int_{{\mathord{\mathbb{R}}}^{N}}|\nabla
u_{n_{0}}|^{2}\,d\gamma-\pi\int_{{\mathord{\mathbb{R}}}^{N}}u_{n_{0}}^{2}\ln\left(\frac{u_{n_{0}}^{2}}{\|u_{n_{0}}\|_{\mathrm{L}^{2}(\gamma)}^{2}}\right)\,d\gamma}{\|u\|_{\mathrm{L}^{2}(\gamma)}^{2}}\\\
\geq\frac{\int_{{\mathord{\mathbb{R}}}^{N}}|\nabla\mathsf{u}_{\tau_{0}}|^{2}\,d\gamma-\pi\int_{{\mathord{\mathbb{R}}}^{N}}\mathsf{u}_{\tau_{0}}^{2}\ln\left(\frac{\mathsf{u}_{\tau_{0}}^{2}}{\|\mathsf{u}_{\tau_{0}}\|_{\mathrm{L}^{2}(\gamma)}^{2}}\right)\,d\gamma}{\|u\|_{\mathrm{L}^{2}(\gamma)}^{2}}\\\
=\delta\,\frac{\int_{{\mathord{\mathbb{R}}}^{N}}|\nabla\mathsf{u}_{\tau_{0}}|^{2}\,d\gamma-\pi\int_{{\mathord{\mathbb{R}}}^{N}}\mathsf{u}_{n_{0}}^{2}\ln\left(\frac{\mathsf{u}_{\tau_{0}}^{2}}{\|\mathsf{u}_{\tau_{0}}\|_{\mathrm{L}^{2}(\gamma)}^{2}}\right)\,d\gamma}{\inf_{a,\,c}\|\mathsf{u}_{\tau_{0}}-c\,e^{a\cdot
x}\|_{\mathrm{L}^{2}(\gamma)}^{2}}\,.$

We can apply the result in the first case to the function
$\mathsf{u}_{\tau_{0}}$ and infer that the right side is larger or equal than
$\kappa^{\rm pos}:=\delta\,\eta$. This concludes the proof in the case of
nonnegative functions.

_Step 2._ We now prove the theorem in the general case, that is, for sign-
changing functions.

We shall use the notation

$D(v):=\int_{{\mathord{\mathbb{R}}}^{N}}|\nabla
v|^{2}\,d\gamma-\pi\int_{{\mathord{\mathbb{R}}}^{N}}v^{2}\ln\left(\frac{v^{2}}{\|v\|_{\mathrm{L}^{2}(\gamma)}^{2}}\right)\,d\gamma\quad\text{for}\,v\in\mathrm{H}^{1}(\gamma)\,.$

Let $u=u_{+}-u_{-}\in\mathrm{H}^{1}(\gamma)$. By homogeneity we can assume
$\|u\|_{\mathrm{L}^{2}(\gamma)}=1$. Replacing $u$ by $-\,u$ if necessary, we
can also assume that

$m:=\|u_{-}\|_{\mathrm{L}^{2}(\gamma)}^{2}\in[0,\tfrac{1}{2}]\,.$

Then

$D(u)=D(u_{+})+D(u_{-})+\pi\,h(m)$

with

$h(p):=-\,\big{(}p\ln p+(1-p)\ln(1-p)\big{)}\,.$

Since the function $p\mapsto h(p)$ is monotone increasing and concave on
$[0,1/2]$, it holds that

$h(p)\geq(2\ln 2)\,p\quad\forall\,p\in[0,\tfrac{1}{2}]\,.$

Thus, with $\kappa^{\rm pos}$ denoting the constant from Step 1,

$\displaystyle D(u)$ $\displaystyle\geq D(u_{+})+(2\,\pi\,\ln
2)\,m\geq\kappa^{\rm pos}\inf_{a,\,c}\|u_{+}-c\,e^{a\cdot
x}\|_{\mathrm{L}^{2}(\gamma)}^{2}+(2\,\pi\,\ln
2)\,\|u_{-}\|_{\mathrm{L}^{2}(\gamma)}^{2}$
$\displaystyle\geq\frac{1}{2}\,\min\big{\\{}\kappa^{\rm pos},\,2\,\pi\,\ln
2\big{\\}}\,\inf_{a,\,c}\|u-c\,e^{a\cdot x}\|_{\mathrm{L}^{2}(\gamma)}^{2}\,.$

This proves the inequality in the general case, with
$\kappa=\frac{1}{2}\,\min\big{\\{}\kappa^{\rm pos},\,2\,\pi\,\ln 2\big{\\}}$.
It is straightforward to verify that $\kappa=\beta\,\pi/2$, $\beta$ being the
constant in the statement of Theorem 1. This ends our second proof of Theorem
2 ∎

Acknowledgements. Partial support through US National Science Foundation
grants DMS-1954995 (R.L.F.) and DMS-2154340 (M.L.), as well as through the
Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) Germany’s
Excellence Strategy EXC-2111-390814868 (R.L.F.) and through TRR 352 – Project-
ID 470903074 (R.L.F.), the French National Research Agency (ANR) project
Conviviality (ANR-23-CE40-0003, J.D.) and the European Research Council under
the Grant Agreement No. 721675 (RSPDE) Regularity and Stability in Partial
Differential Equations (A.F.).
© 2024 by the authors. Reproduction of this article by any means permitted for
noncommercial purposes. CC-BY 4.0

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