[ifplatform] Sur des inégalités d’interpolation Gaussiennes G.
<EMAIL_ADDRESS>J.
<EMAIL_ADDRESS>N. Simonov]nikita.simonov@sorbonne-
universite.fr

# On Gaussian interpolation inequalities

Giovanni Brigati0000-0002-3698-9995  CEREMADE (CNRS UMR n∘ 7534), PSL
University, Université Paris-Dauphine,
Place de Lattre de Tassigny, 75775 Paris 16, France  [ ,  Jean
Dolbeault0000-0003-4234-2298  CEREMADE (CNRS UMR n∘ 7534), PSL University,
Université Paris-Dauphine,
Place de Lattre de Tassigny, 75775 Paris 16, France  [ and  Nikita
Simonov0000-0002-3241-190X  LJLL (CNRS UMR n∘ 7598) Sorbonne Université,
4 place Jussieu, 75005 Paris, France  [

###### Abstract.

This paper is devoted to Gaussian interpolation inequalities with endpoint
cases corresponding to the Gaussian Poincaré and the logarithmic Sobolev
inequalities, seen as limits in large dimensions of Gagliardo-Nirenberg-
Sobolev inequalities on spheres. Entropy methods are investigated using not
only heat flow techniques but also nonlinear diffusion equations as on
spheres. A new stability result is established for the Gaussian measure, which
is directly inspired by recent results for spheres.

###### Key words and phrases:

logarithmic Sobolev inequality, Gagliardo-Nirenberg-Sobolev inequalities,
Gaussian Poincaré inequality, sphere, spectral decomposition, entropy methods,
Ornstein-Uhlenbeck operator, nonlinear diffusions, improved inequalities,
stability

###### 2020 Mathematics Subject Classification:

26D10, 46T12, 46E35, 39B62, 43A90, 49J40, 58E35

Cet article est consacré à des inégalités d’interpolation Gaussiennes, avec
comme cas extrÃªmes l’inégalité de Poincaré Gaussienne et l’inégalité de
Sobolev logarithmique, vues comme limites en grandes dimensions des inégalités
de Gagliardo-Nirenberg-Sobolev sur les sphères. Les méthodes d’entropie sont
abordées en utilisant non seulement des techniques basée sur l’équation de la
chaleur mais aussi sur des équations de diffusion non-linéaires, comme pour
les sphères. Un nouveau résultat de stabilité est établi pour les mesures
Gaussiennes, qui s’inspire directement de résultats récents sur les sphères.

##  1\. Introduction and main results

Let us consider the _Gagliardo-Nirenberg-Sobolev inequalities_ on the unit
$d$-dimensional sphere

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

for any $p\in[1,2)\cup(2,+\infty)$ if $d=1$, $2$, and for any
$p\in[1,2)\cup(2,2^{*}]$ if $d\geq 3$. Here $d\mu_{d}$ denotes the uniform
probability measure on ${\mathbb{S}}^{d}\subset{\mathbb{R}}^{d+1}$ and, if
$d\geq 3$, $2^{*}=2\,d/(d-2)$ is the critical Sobolev exponent. By convention,
we take $2^{*}=+\infty$ if $d=1$ or $2$. The purpose of this paper is to
clarify the links of these interpolation inequalities with the family of
_Gaussian interpolation inequalities_

$\|\nabla
v\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}\geq\frac{1}{2-p}\left(\|v\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}-\|v\|_{\mathrm{L}^{p}({\mathbb{R}}^{n},d\gamma)}^{2}\right)\quad\forall\,v\in\mathrm{H}^{1}({\mathbb{R}}^{n},d\gamma){}$
(2)

where the exponent $p$ is taken in the range $1\leq p<2$. Inequality (2) is
intermediate between the Poincaré inequality corresponding to $p=1$ and the
_Gaussian logarithmic Sobolev inequality_

$\|\nabla
v\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}\geq\frac{1}{2}\int_{{\mathbb{R}}^{n}}{|v|^{2}\,\log\left(\frac{|v|^{2}}{\|v\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}}\right)}\,d\gamma\quad\forall\,v\in\mathrm{H}^{1}({\mathbb{R}}^{n},d\gamma)$

obtained as a limit case of (2) as $p\to 2_{-}$. Here
$d\gamma(y):=(2\,\pi)^{-n/2}\,e^{-\frac{1}{2}\,|y|^{2}}\,dy$ denotes the
centred normalized Gaussian probability measure and the dimension $n$ is any
positive integer.

It is somewhat classical that, if we consider the Sobolev inequality on the
sphere, _i.e._ , Inequality (1) with $p=2^{*}$ and $d\geq 3$, rescale it to
the sphere of radius $\sqrt{d}$ and fix a function depending on $n$ variables
only on this sphere, since the curvature tends to $0$ at the right order as
one takes the limit as $d\to+\infty$, then the sphere tends to the flat space
with Gaussian measure. For instance, we read (with adapted notation) from [9,
p. 4818] that _if we rescale this inequality so as to be on a sphere of radius
$\sqrt{d}$ and take the limit $d\to\infty$ or $p\to 2$ we obtain in the
Poincaré limit the Gross logarithmic inequality for the Gaussian measure since
$-(1/d)\,\Delta$ on ${\mathbb{S}}^{d}$ goes to $-\Delta+x\cdot\nabla$ on the
infinite-dimensional limit._ The last part of the sentence refers to a result
known as the MaxwellâPoincaré lemma : see [39] and [43, Remark 4, p. 254] for
some historical comments. The statement of [9] has been made more precise
later in [12, 11] using a slightly different limit. However, to our knowledge,
the infinite-dimensional limit has not been considered in the subcritical
range $p<2^{*}$.

On the sphere, Inequality (1) follows from [4, 6] for any $p\geq 1$ if $d=1$
and any $p\in[1,2^{\\#}]$ with $2^{\\#}:=(2\,d^{2}+1)/(d-1)^{2}$ if $d\geq 2$.
The proof in the range $p\in(2^{\\#},+\infty)$ if $d=2$ and
$p\in(2^{\\#},2^{*}]$ if $d\geq 3$ can be found in [15, Corollary 6.1], [16]
and [10]. Also see [37] in the case $p=2^{*}$. In the case of the Gaussian
measure, we refer to [8] (also see [34]) for a first proof of Inequality (2).
The formal analogy of (1) and (2) is striking. Although computations are
somewhat standard, our first purpose is to make this point rigorous and
recover (2) as a special limit of (1) as $d\to+\infty$.

######  Theorem 1.1.

Let $n$ be a positive integer, $p\in[1,2)$ and consider a function
$v\in\mathrm{H}^{1}(\mathrm{R}^{n},dx)$ with compact support. For any $d\geq
n$, large enough, if
$u_{d}(\omega)=v\big{(}\omega_{1}/\sqrt{d},\omega_{2}/\sqrt{d},\ldots,\omega_{n}/\sqrt{d}\big{)}$
where
$\omega=(\omega_{1},\omega_{2},\ldots,\omega_{d+1})\in{\mathbb{S}}^{d}\subset{\mathbb{R}}^{d+1}$,
then

$\lim_{d\to+\infty}d\left(\|{\nabla
u_{d}}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d},d\mu_{d})}^{2}-\frac{d}{2-p}\left(\|{u_{d}}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d},d\mu_{d})}^{2}-\|{u_{d}}\|_{\mathrm{L}^{p}({\mathbb{S}}^{d},d\mu_{d})}^{2}\right)\right)\\\
=\|\nabla
v\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}-\frac{1}{2-p}\left(\|v\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}-\|v\|_{\mathrm{L}^{p}({\mathbb{R}}^{n},d\gamma)}^{2}\right)\,.$

The _carré du champ_ method has frequently been applied to prove Gaussian
interpolation inequalities ranging between the logarithmic Sobolev inequality
and the Poincaré inequality like (2) using the linear flow associated with the
Ornstein-Uhlenbeck operator; see [4], and [7] for an overview. Still in the
case of the Gaussian measure, we adopt here a new point of view by using
_nonlinear diffusion equations_ in order to prove the same inequalities, but
with different remainder terms. This is a very natural point of view when
dealing with inequalities like (1) on the sphere, as shown for instance in
[26] (see earlier references therein). In that case, linear flows are indeed
limited to exponents $p\leq 2^{\\#}$ if $d\geq 2$. To overcome this difficulty
if either $d=2$ and $p>2^{\\#}$ or $d\geq 3$ and $p\in(2^{\\#},2^{*}]$, one
has to consider fast diffusion flows. Before explaining the results for the
Gaussian measure, let us summarize the main known results on the sphere.

On ${\mathbb{S}}^{d}$, let us consider a positive solution $u$ of

$\frac{\partial u}{\partial t}=u^{-p\,(1-m)}\left(\Delta
u+(m\,p-1)\,\frac{|\nabla u|^{2}}{u}\right)$ (3)

where $\Delta$ denotes the Laplace-Beltrami operator on ${\mathbb{S}}^{d}$. In
the case $m=1$, $u^{p}$ solves the heat equation and for this reason, we shall
call it the _linear case_. Otherwise $u^{p}$ solves a nonlinear diffusion
equation corresponding either to a fast diffusion flow with $m<1$ or to a
porous media equation with $m>1$. In any case, we claim that

$\frac{d}{dt}\left\|u\right\|_{\mathrm{L}^{p}({\mathbb{S}}^{d})}^{2}=0\quad\mbox{and}\quad\frac{d}{dt}\left\|u\right\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}=2\,(p-2)\int_{{\mathbb{S}}^{d}}{u^{-\,p\,(1-m)}\,|\nabla
u|^{2}}\,d\mu_{d}\,.$

Let us define

$m_{\pm}(d,p):=\frac{1}{(d+2)\,p}\left(d\,p+2\pm\sqrt{d\,(p-1)\,\big{(}2\,d-(d-2)\,p\big{)}}\right)\,.$
(4)

The following result can be found in [22, 26] with additional details in [27,
24, 18].

######  Proposition 1.2 ([22, 26]).

Assume that $d\geq 1$, with either $p\in[1,2)\cup(2,+\infty)$ if $d=2$ or
$p\in[1,2)\cup(2,2^{*}]$ if $d\geq 3$, and let $m\in[m_{-}(d,p),m_{+}(d,p)]$.
If $u>0$ solves (3) with an initial datum in
$\mathrm{L}^{2}\cap\mathrm{L}^{p}({\mathbb{S}}^{d},d\mu_{d})$, then

$\frac{d}{dt}\left(\|{\nabla
u}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d},d\mu_{d})}^{2}-\frac{d}{p-2}\left(\|{u}\|_{\mathrm{L}^{p}({\mathbb{S}}^{d},d\mu_{d})}^{2}-\|{u}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d},d\mu_{d})}^{2}\right)\right)\leq
0\quad\forall\,t>0\,.$

The limit, as $t\to+\infty$, of any solution of (3) is a constant. This means
that the _deficit_ , that is, the difference of the two sides in Inequality
(1), converges to $0$. Then, by Proposition 1.2, it follows that the deficit
is non-negative, which directly proves (1). The same monotonicity property
applies to the deficit of the _logarithmic Sobolev inequality on the sphere_

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

in the limit case corresponding to $p=2$. The admissible values of the
parameters are limited to $m_{-}(d,p)\leq m\leq m_{+}(d,p)$ and $1\leq p\leq
2^{*}$ if $d\geq 3$. Moreover, at the endpoints, we have $m_{\pm}(d,1)=1$ and
$m_{\pm}(d,2^{*})=(d-1)/d$ if $d\geq 3$, while $m_{+}(d,2^{\\#})=1$ so that
$m=1$ is admissible if and only if $1\leq p\leq 2^{\\#}$ when $d\geq 2$. See
Fig. 1. For appropriate initial data, it is shown in [27] that the
monotonicity property of the deficit along the flow of (3) is violated for any
$p\in[2,2^{*})$ or $p=2^{*}$ if $d\geq 3$ as soon as either $m<m_{-}(d,p)$ or
$m>m_{+}(d,p)$. The case $p=2$ corresponding to the logarithmic Sobolev
inequality is included.

Figure 1.  Case $d=5$. The admissible parameters $p$ and $m$ correspond to the
grey area. The boundary of the admissible set is tangent to the vertical lines
$p=1$ at $(m,p)=(1,1)$ and $p=2^{*}=10/3$ at $(m,p)=(4/5,10/3)$.
Qualitatively, this figure does not change as $d$ increases but gets squeezed
in the interval $1\leq p\leq 2$ as $d\to+\infty$.

In view of the results of Theorem 1.1, it is natural to ask whether there is
also a monotonicity property of the deficit associated to the Gaussian
interpolation inequalities (2) when we rely on a nonlinear diffusion flow on
${\mathbb{R}}^{n}$. Let $m_{\pm}(p):=\lim_{d\to+\infty}m_{\pm}(d,p)$ where
$m_{\pm}$ is given by (4) and notice that

$m_{\pm}(p)=1\pm\frac{1}{p}\,\sqrt{(p-1)\,(2-p)}\,.$ (5)

The diffusion operator associated to the Gaussian measure is the _Ornstein-
Uhlenbeck operator_ $\mathcal{L}=\Delta-x\cdot\nabla$ and we consider now the
nonlinear parabolic equation

$\frac{\partial v}{\partial
t}=v^{-p\,(1-m)}\left(\mathcal{L}v+(m\,p-1)\,\frac{|\nabla
v|^{2}}{v}\right)\,.$ (6)

In the definition of $\mathcal{L}$, $\Delta$ denotes the standard Laplacian on
${\mathbb{R}}^{n}$. The following result is new for $m\neq 1$ while the case
$m=1$ follows from the method of the _carré du champ_ developed by Bakry and
Emery in [4].

######  Theorem 1.3.

Assume that $n\geq 1$, $p\in[1,2)$. If $v>0$ solves (6) with
$m\in[m_{-}(p),m_{+}(p)]$ for an initial datum in
$\mathrm{L}^{2}\cap\mathrm{L}^{p}({\mathbb{R}}^{n},d\gamma)$, then

$\frac{d}{dt}\left(\|\nabla
v\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}-\frac{1}{p-2}\left(\|v\|_{\mathrm{L}^{p}({\mathbb{R}}^{n},d\gamma)}^{2}-\|v\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}\right)\right)\leq
0\quad\forall\,t>0\,.$

The limiting case $p=2$ corresponding to the Gaussian logarithmic Sobolev
inequality is also covered but it is obtained as a standard application of the
linear _carré du champ_ method known from [4] because $m_{\pm}(2)=1$. See Fig.
2.

Figure 2.  The admissible parameters $p$ and $m$ correspond to the grey area
and are independent of the dimension $n$. The boundary of the admissible set
is tangent to the vertical lines $p=1$ at $(m,p)=(1,1)$ and $p=2$ at
$(m,p)=(1,2)$. It is the limit set of the admissible parameters for
Proposition 1.2 as $d\to+\infty$.

Said in simple words, the result of Theorem 1.3 is that the admissible range
of exponents of the nonlinear flow, for which the deficit associated to (2) is
monotone non-increasing, is obtained as the limit of the range of the
corresponding exponents on the sphere, in the large dimensions limit.
Moreover, $p=2$ appears as a _critical exponent_ for the Gaussian measure.

Let us now focus on stability results. The main result of [18] for the sphere
is a constructive stability estimate for Inequality (1), limited to the
subcritical range $p\in(1,2^{*})$, which measures the distance to optimal
functions, and distinguishes the subspaces generated by constant functions,
spherical harmonic functions associated to the first positive eigenvalue of
the Laplace-Beltrami operator, and the orthogonal directions. Optimal
exponents in the stability estimate measuring the distance to the set of
optimal functions differ, depending on the directions. Here we have the exact
counterpart in the Gaussian case. Let $\Pi_{1}$ denote the orthogonal
projection of $\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)$ onto the
$(n+1)$-dimensional function space generated by $1$ and $x_{i}$ with $i=1$,
$2$, …,$n$.

######  Theorem 1.4.

For all $n\geq 1$, and all $p\in(1,2),$ there is an explicit constant
$c_{n,p}>0$ such that, for all $v\in\mathrm{H}^{1}(d\gamma)$, it holds

$\|\nabla
v\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}-\frac{1}{p-2}\left(\|v\|_{\mathrm{L}^{p}({\mathbb{R}}^{n},d\gamma)}^{2}-\|v\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}\right)\\\
\geq\,c_{n,p}\left(\|\nabla(\mathrm{Id}-\Pi_{1})v\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}+\frac{\|\nabla\Pi_{1}v\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{4}}{\|\nabla
v\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}+\|v\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}}\right).$

Exponents $2$ and $4$ which appear in the right-hand side of the inequality
are sharp and the constant $c_{n,p}$ has an explicit although complicated
expression given in the proof. If $p=1$, the last term drops and the distance
to optimal functions is measured only by

$\|\nabla(\mathrm{Id}-\Pi_{1})v\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}\,,$

and a decomposition on Hermite polynomials shows that the estimate cannot be
improved. If $p=2^{*}$ and $n\geq 3$, the recent stability result on Sobolev’s
inequality on ${\mathbb{R}}^{n}$ of [25], which quantifies the estimate of
Bianchi and Egnell in [14] can be translated into a stability result for (1)
on ${\mathbb{S}}^{d}$, that can be recast in the form of [18, Theorem 6]. By a
large dimension argument, a stability on the Gaussian logarithmic Sobolev
inequality is also shown in [25], although the distance is measured only by an
$\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)$ 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.

This paper is organized as follows. In Section 2, we give a new proof of
Inequality (2) as a consequence of Inequality (1) by taking a large dimensions
limit, applied to the inequality written for a function depending only on a
fixed number $n$ of real variables. To our knowledge, this is new except for
the limit case $p=2$ of the logarithmic Sobolev inequality. Section 3 is
devoted to the proof of Theorem 1.3: we characterize the nonlinear diffusion
flows of porous medium or fast diffusion type such that the deficit is
monotone non-increasing and recover the picture known on the sphere in the
large dimensions limit. Moreover, by the _carré du champ_ method, we establish
improved inequalities that provide us with first stability results. The
stability result of Theorem 1.4 for the Gaussian measure is proved in Section
4 using a detailed Taylor expansion and the improved inequalities of Section
3.

##  2\. From subcritical interpolation inequalities on the sphere to Gaussian
interpolation

In this section we explain how Inequality (2) can be seen as the limit of
Inequality (1) in the large dimensions limit, that is, as $d\to+\infty$, and
prove Theorem 1.1. Comments on the limit case $p=2$ can be found at the end of
this section.

The unit sphere ${\mathbb{S}}^{d}$ is parametrized in terms of the
_stereographic coordinates_ by

$\omega_{j}=\frac{2\,x_{j}}{1+|x|^{2}}\quad\mbox{if}\quad 1\leq j\leq
d\quad\mbox{and}\quad\omega_{d+1}=\frac{1-|x|^{2}}{1+|x|^{2}}$

where $\omega=(\omega_{1},\omega_{2},\ldots,\omega_{d+1})$ denote the
coordinates in ${\mathbb{R}}^{d+1}\supset{\mathbb{S}}^{d}$ and
$x=(x_{1},x_{2},\ldots,x_{d})$ are Cartesian coordinates in
${\mathbb{R}}^{d}$. To a function $u$ on ${\mathbb{S}}^{d}$, we associate a
function $w$ on ${\mathbb{R}}^{d}$ using the stereographic projection such
that

$\left(\frac{2}{1+|x|^{2}}\right)^{\frac{d-2}{2}}u(\omega)=w(x)\quad\forall\,x\in{\mathbb{R}}^{d}\,.$

It is a standard result that

$\int_{{\mathbb{S}}^{d}}{|u|^{p}}\,d\mu_{d}=2^{\frac{\delta(p)}{2}}\,\big{|}{\mathbb{S}}^{d}\big{|}^{-1}\int_{{\mathbb{R}}^{n}}{\langle{x}\rangle^{-\delta(p)}\,|w|^{p}}\,dx$
(7)

and that

$\int_{{\mathbb{S}}^{d}}{|\nabla
u|^{2}}\,d\mu_{d}+\tfrac{1}{4}\,d\,(d-2)\int_{{\mathbb{S}}^{d}}{|u|^{2}}\,d\mu_{d}=\big{|}{\mathbb{S}}^{d}\big{|}^{-1}\int_{{\mathbb{R}}^{n}}{|\nabla
w|^{2}}\,dx\,.$

where $\langle{x}\rangle:=\sqrt{1+|x|^{2}}$ and $\delta(p):=2\,d-p\,(d-2)$.
Using the stereographic projection, Inequality (1) can be written on the
Euclidean space ${\mathbb{R}}^{d}$ as the weighted interpolation inequality

$\int_{{\mathbb{R}}^{n}}{|\nabla
w|^{2}}\,dx+\frac{d\,\delta(p)}{p-2}\int_{{\mathbb{R}}^{n}}{\frac{|w|^{2}}{\langle{x}\rangle^{4}}}\,dx\geq\frac{\mathcal{C}_{d,p}}{p-2}\left(\int_{{\mathbb{R}}^{n}}{\frac{|w|^{p}}{\langle{x}\rangle^{\delta(p)}}}\,dx\right)^{\frac{2}{p}}\quad\mbox{with}\quad\mathcal{C}_{d,p}=2^{\frac{\delta(p)}{p}}\,d\,\big{|}{\mathbb{S}}^{d}\big{|}^{1-\frac{2}{p}}.$
(8)

See [24] for details. Equality is achieved by the Aubin-Talenti function
$w_{\star}(x):=\langle{x}\rangle^{2-d}$. Assume that $d\geq 4$. Let us
consider $f=w/w_{\star}$ and notice that the inequality rewritten in terms of
$f$ is

$\int_{{\mathbb{R}}^{n}}{|\nabla
f|^{2}\,w_{\star}^{2}}\,dx+\frac{4\,d}{p-2}\int_{{\mathbb{R}}^{n}}{|f|^{2}\,w_{\star}^{2^{*}}}\,dx\geq\frac{\mathcal{C}_{d,p}}{p-2}\left(\int_{{\mathbb{R}}^{n}}{|f|^{p}\,w_{\star}^{2^{*}}}\,dx\right)^{\frac{2}{p}}\,.$

Since we are interested in the limit as $d\to+\infty$, we shall consider the
case $1\leq p<2$, which is admissible for any $d\geq 2$.

With $g(x)=f\big{(}x/\sqrt{d}\big{)}$, we obtain after changing variables that

$\frac{1}{4}\int_{{\mathbb{R}}^{d}}|\nabla
g|^{2}\,\frac{dx}{\left(1+\tfrac{1}{d}\,|x|^{2}\right)^{d-2}}+\frac{1}{p-2}\int_{{\mathbb{R}}^{d}}|g|^{2}\,\frac{dx}{\left(1+\tfrac{1}{d}\,|x|^{2}\right)^{d}}\geq\frac{\mathcal{C}_{d,p}\,d^{d\frac{p-2}{2\,p}}}{4\,d\,(p-2)}\left(\int_{{\mathbb{R}}^{d}}|g|^{p}\,\frac{dx}{\left(1+\tfrac{1}{d}\,|x|^{2}\right)^{d}}\right)^{\frac{2}{p}}\,.$

Let us assume that $n\geq 1$ is a given integer and take $d>\max\\{n,3\\}$.
With
$x=(y,z)\in{\mathbb{R}}^{n}\times{\mathbb{R}}^{d-n}\approx{\mathbb{R}}^{d}$,
we also assume that the function $g$ depends only on $y$. In other words, we
write $g=g_{d}$ where $g_{d}(y,z)=v(y)$ for some function $v$ defined on
${\mathbb{R}}^{n}$, that is,

$g_{d}(y,z)=v(y)\quad\forall\,(y,z)\in{\mathbb{R}}^{n}\times{\mathbb{R}}^{d-n}$
(9)

Here we use a subscript $d$ in order to emphasize that $g_{d}$ has to be
considered as a function on ${\mathbb{R}}^{d}$. Let us define

$c_{d}:=(d\,\pi)^{\frac{d}{2}}\,\frac{\Gamma\left(d/2\right)}{\Gamma(d)}\,.$

######  Lemma 2.1.

Let $n$ be a positive integer, $p\in[1,2)$, consider a function
$v\in\mathrm{H}^{1}({\mathbb{R}}^{n},dx)$ with compact support and define
$g_{d}$ according to (9). Then we have

$\lim_{d\to+\infty}\frac{1}{4\,c_{d}}\int_{{\mathbb{R}}^{d}}|\nabla
g_{d}|^{2}\,\frac{dx}{\left(1+\tfrac{1}{d}\,|x|^{2}\right)^{d-2}}\\\
+\frac{1}{2-p}\,\lim_{d\to+\infty}\frac{1}{c_{d}}\left(\frac{\mathcal{C}_{d,p}\,d^{d\frac{p-2}{2\,p}}}{4\,d}\left(\int_{{\mathbb{R}}^{d}}|g_{d}|^{p}\,\frac{dx}{\left(1+\tfrac{1}{d}\,|x|^{2}\right)^{d}}\right)^{\frac{2}{p}}-\int_{{\mathbb{R}}^{d}}|g_{d}|^{2}\,\frac{dx}{\left(1+\tfrac{1}{d}\,|x|^{2}\right)^{d}}\right)\\\
=\|\nabla
v\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}-\frac{1}{2-p}\left(\|v\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}-\|v\|_{\mathrm{L}^{p}({\mathbb{R}}^{n},d\gamma)}^{2}\right)\,.$

In other words, we prove that the infinite dimensional limit of (8), for
functions depending only on a finite number $n$ of real variables, is (2). The
assumption of compact support can be removed if $v$ and $\nabla v$ have
sufficient decay properties at infinity.

######  Proof 2.2.

Using

$\left(1+\tfrac{1}{d}\,|x|^{2}\right)^{2-d}=\Big{(}1+\tfrac{1}{d}\left(|y|^{2}+|z|^{2}\right)\Big{)}^{2-d}=\left(1+\tfrac{1}{d}\,|y|^{2}\right)^{2-d}\left(1+\tfrac{1}{d}\,|\zeta|^{2}\right)^{2-d}\quad\mbox{with}\quad\zeta=\frac{z}{\sqrt{1+\tfrac{1}{d}\,|y|^{2}}}\,,$

we can integrate with respect to $z$ and obtain

$\int_{{\mathbb{R}}^{d-n}}\left(1+\tfrac{1}{d}\,|x|^{2}\right)^{2-d}\,dz=\left(1+\tfrac{1}{d}\,|y|^{2}\right)^{2-\frac{d+n}{2}}\int_{{\mathbb{R}}^{d-n}}\left(1+\tfrac{1}{d}\,|\zeta|^{2}\right)^{2-d}\,d\zeta\,.$

Let $d\gamma(y):=(2\,\pi)^{-n/2}\,e^{-\frac{1}{2}\,|y|^{2}}\,dy$ be the
centred normalized Gaussian probability measure. We recall that
$|{\mathbb{S}}^{k-1}|=2\,\pi^{k/2}/\Gamma(k/2)$ for any
$k\in{\mathbb{N}}\setminus\\{0\\}$ and

$\int_{0}^{+\infty}r^{a-1}\left(1+\tfrac{1}{d}\,r^{2}\right)^{-b}\,dr=d^{\frac{a}{2}}\,\frac{\Gamma\left(\frac{a}{2}\right)\,\Gamma\left(b-\frac{a}{2}\right)}{2\,\Gamma(b)}$

if $0<a<2\,b$. Applying these formulas with $a=k=d-n$ and $b=d-2>2-n$, we find
that

$\int_{{\mathbb{R}}^{d-n}}\left(1+\tfrac{1}{d}\,|\zeta|^{2}\right)^{2-d}\,d\zeta=(d\,\pi)^{\frac{d-n}{2}}\,\frac{\Gamma\left(\frac{d+n}{2}-2\right)}{\Gamma(d-2)}\,.$

Applying these formulas with $a=b=k=d\geq 2$, we find that

$\int_{{\mathbb{R}}^{n}}{\left(1+\tfrac{1}{d}\,|x|^{2}\right)^{-d}}\,dx=c_{d}$

and, as a consequence

$\lim_{d\to+\infty}\frac{1}{c_{d}}\int_{{\mathbb{R}}^{d-n}}\left(1+\tfrac{1}{d}\,|\zeta|^{2}\right)^{2-d}\,d\zeta=\frac{4}{(2\,\pi)^{n/2}}$

using Stirling’s formula. Since

$\lim_{d\to+\infty}\left(1+\tfrac{1}{d}\,|y|^{2}\right)^{2-\frac{d+n}{2}}=e^{-\frac{1}{2}\,|y|^{2}}\,,$

we obtain

$\lim_{d\to+\infty}\frac{1}{c_{d}}\int_{{\mathbb{R}}^{n}}{|\nabla
g_{d}(y)|^{2}\left(1+\tfrac{1}{d}\,|x|^{2}\right)^{2-d}}\,dx=4\int_{{\mathbb{R}}^{n}}{|\nabla
v|^{2}}\,d\gamma\,.$

Similar computations show that

$\int_{{\mathbb{R}}^{n}}{|g_{d}(y)|^{2}\left(1+\tfrac{1}{d}\,|x|^{2}\right)^{-d}}\,dx=\int_{{\mathbb{R}}^{n}}{|g_{d}(y)|^{2}\left(1+\tfrac{1}{d}\,|y|^{2}\right)^{-\frac{d+n}{2}}}\,dy\int_{{\mathbb{R}}^{d-n}}\left(1+\tfrac{1}{d}\,|\zeta|^{2}\right)^{-d}\,d\zeta\,,\\\
\lim_{d\to+\infty}\frac{1}{c_{d}}\int_{{\mathbb{R}}^{n}}{|g_{d}(y)|^{2}\left(1+\tfrac{1}{d}\,|x|^{2}\right)^{-d}}\,dx=\int_{{\mathbb{R}}^{n}}{|v|^{2}}\,d\gamma\,,$

and

$\int_{{\mathbb{R}}^{n}}{|g_{d}(y)|^{p}\left(1+\tfrac{1}{d}\,|x|^{2}\right)^{-d}}\,dx=\int_{{\mathbb{R}}^{n}}{|g_{d}(y)|^{p}\left(1+\tfrac{1}{d}\,|y|^{2}\right)^{-\frac{d+n}{2}}}\,dy\int_{{\mathbb{R}}^{d-n}}\left(1+\tfrac{1}{d}\,|\zeta|^{2}\right)^{-d}\,d\zeta\,,\\\
\lim_{d\to+\infty}\frac{\mathcal{C}_{d,p}\,d^{d\frac{p-2}{2\,p}}}{4\,d\,c_{d}}\left(\int_{{\mathbb{R}}^{n}}{|g_{d}(y)|^{p}\left(1+\tfrac{1}{d}\,|x|^{2}\right)^{-d}}\,dx\right)^{\frac{2}{p}}=\left(\int_{{\mathbb{R}}^{n}}{|v|^{p}}\,d\gamma\right)^{\frac{2}{p}}\,.$

This completes the proof of Lemma 2.1.

######  Proof 2.3 (Proof of Theorem 1.1).

Applied to the function $u_{d}$, Inequality (1) is transformed into Inequality
(8) applied to

$g_{d}(x)=u_{d}\left(\frac{2\,y}{1+\tfrac{1}{d}\,|x|^{2}}\right)\quad\forall\,x=(y,z)\in{\mathbb{R}}^{n}\times{\mathbb{R}}^{d-n}\,.$

Notice that the factor $2^{\delta(p)/2}$, which arises from the stereographic
projection and appears in (7), plays a role in the computation of the constant
$\mathcal{C}_{d,p}$ and is taken into account in the limit as $d\to+\infty$.
Since the right-hand side uniformly converges to $v(y)$ for any smooth and
compactly supported function $v$, the same conclusion holds for Theorem 1.1 as
for Lemma 2.1.

It is a natural question to ask what happens in (1) to the marginals depending
only on a finite number $n$ of variables if $p=2$ or in the case $2<p\leq
2^{*}=2\,d/(d-2)$. We may notice that $\lim_{d\to+\infty}2\,d/(d-2)=2$ and it
is known, for instance from [9], that one recovers the _Gaussian logarithmic
Sobolev inequality_ as a limit case of Sobolev’s inequality on
${\mathbb{S}}^{d}$ corresponding to $p=2^{*}$ when $d\to+\infty$. This is also
true if we consider a sequence $(p_{d})_{d\in{\mathbb{N}}}$ with
$1<p_{d}<2^{*}$, depending on $d$, if its limit is also $2$, as shown next. By
convention, when $p_{d}=2$, we consider the Gaussian logarithmic Sobolev
inequality instead of (2).

######  Proposition 2.4.

Let $n$ be a positive integer and consider a function
$v\in\mathrm{H}^{1}({\mathbb{R}}^{n},dx)$ with compact support. For any $d\geq
n$, large enough, let
$u_{d}(\omega)=v\big{(}\omega_{1}/\sqrt{d},\omega_{2}/\sqrt{d},\ldots,\omega_{n}/\sqrt{d}\big{)}$
where
$\omega=(\omega_{1},\omega_{2},\ldots,\omega_{d})\in{\mathbb{R}}^{d+1}\supset{\mathbb{S}}^{d}$
is such that $|\omega|=1$, as in Theorem 1.1. Then we have

$\lim_{d\to+\infty}d\left(\|{\nabla
u_{d}}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d},d\mu_{d})}^{2}-\frac{d}{2}\int_{{\mathbb{S}}^{d}}{|u_{d}|^{2}\,\log\left(\frac{|u_{d}|^{2}}{\left\|u_{d}\right\|_{\mathrm{L}^{2}({\mathbb{S}}^{d})}^{2}}\right)}\,d\mu_{d}\right)\\\
=\|\nabla
v\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}-\frac{1}{2}\int_{{\mathbb{R}}^{n}}{|v|^{2}\,\log\left(\frac{|v|^{2}}{\|v\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}}\right)}\,d\gamma\,.$

If $(p_{d})_{d\geq 3}$ is a sequence of real numbers such that
$p_{d}\in(1,2)\cup(2,2^{*})$ and $\lim_{d\to+\infty}p_{d}=2$, then

$\lim_{d\to+\infty}d\left(\|{\nabla
u_{d}}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d},d\mu_{d})}^{2}-\frac{d}{2-p_{d}}\left(\|{u_{d}}\|_{\mathrm{L}^{2}({\mathbb{S}}^{d},d\mu_{d})}^{2}-\|{u_{d}}\|_{\mathrm{L}^{p_{d}}({\mathbb{S}}^{d},d\mu_{d})}^{2}\right)\right)\\\
=\|\nabla
v\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}-\frac{1}{2}\int_{{\mathbb{R}}^{n}}{|v|^{2}\,\log\left(\frac{|v|^{2}}{\|v\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}}\right)}\,d\gamma\,.$

######  Proof 2.5.

The proof is an adaptation of the proof of Theorem 1.1 and, in the case
$p_{d}\neq 2$, relies on the standard observation that

$\lim_{p\to
2}\frac{\|v\|_{\mathrm{L}^{p}({\mathbb{R}}^{n},d\gamma)}^{2}-\|v\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}}{p-2}=\frac{1}{2}\int_{{\mathbb{R}}^{n}}{|v|^{2}\,\log\left(\frac{|v|^{2}}{\|v\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}}\right)}\,d\gamma\,.$

As this computation raises no special difficulty, details are omitted.

##  3\. Entropy methods and nonlinear flows for Gaussian measures

In this section, we prove the result of Theorem 1.3 for the Gaussian measure
$d\gamma$ and extend it to the slightly more general framework of a uniformly
strictly log-concave measure $d\mu$, before drawing some consequences. Most of
the results are similar to computations usually done on the sphere, but we are
not aware of the use of nonlinear flows ($m\neq 1$) in the context of Gaussian
measures. This approach is very natural in the perspective of spheres in the
large-dimensional limit.

###  3.1. Gaussian interpolation inequalities: a proof by the _carré du champ_
method

On ${\mathbb{R}}^{n}$, let us consider the probability measure

$d\mu=Z^{-1}\,e^{-\phi}\,dy\quad\mbox{with}\quad
Z=\int_{{\mathbb{R}}^{n}}{e^{-\phi}}\,dy$ (10)

and redefine the _Ornstein-Uhlenbeck operator_ by

$\mathcal{L}:=\Delta-\nabla\phi\cdot\nabla$ (11)

on $\mathrm{L}^{2}({\mathbb{R}}^{n},d\mu)$. This generalizes the case of the
harmonic potential $\phi(y)=\frac{1}{2}\,|y|^{2}$ considered in the
introduction. We assume that $\phi$ satisfies the _Bakry-Emery condition_

$\mathrm{Hess}\,\phi\geq\lambda_{\star}\,\mathrm{Id}\quad\mbox{a.e.}$ (12)

for some $\lambda_{\star}>0$. The harmonic potential corresponds to the
equality case with $\lambda_{\star}=1$. Under Assumption (12), it is well
known (see for instance [7, Section 7.6.2]) that, with
$\lambda=\lambda_{\star}$ and for any $p\in[1,2)$,

$\|\nabla
f\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\mu)}^{2}\geq\frac{\lambda}{2-p}\left(\|f\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\mu)}^{2}-\|f\|_{\mathrm{L}^{p}({\mathbb{R}}^{n},d\mu)}^{2}\right)\quad\forall\,f\in\mathrm{H}^{1}({\mathbb{R}}^{n},d\mu)$
(13)

and also, by taking the limit as $p\to 2_{-}$, that

$\|\nabla
f\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\mu)}^{2}\geq\frac{\lambda}{2}\int_{{\mathbb{R}}^{n}}{|f|^{2}\,\log\left(\frac{|f|^{2}}{\|f\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\mu)}^{2}}\right)}\,d\mu\quad\forall\,f\in\mathrm{H}^{1}({\mathbb{R}}^{n},d\mu)\,.$

The classical proof by the carré du champ, as in [5, 4, 3], relies on the
_Ornstein-Uhlenbeck flow_ $\partial_{t}\rho=\mathcal{L}\rho$ applied to the
solution with initial datum $\rho(t=0,\cdot)=|f|^{p}$. Here we consider a more
general strategy and compute as in the carré du champ method using the
_nonlinear diffusion flow_

$\frac{\partial\rho}{\partial
t}=\frac{1}{m}\,\mathcal{L}\rho^{m}\quad\forall\,t\geq
0\,,\quad\rho(t=0,\cdot)=|f|^{p}\,.$ (14)

There is no difficulty in proving global existence of a positive bounded
solution. The case of a non-negative solution is also covered by a standard
approximation of the initial datum. In the case of the harmonic potential, the
key of the existence proof is to combine an $\mathrm{L}^{1}$-contraction
property and the Maximum Principle as in the classical theory of nonlinear
diffusions: see for instance [42, Chapter 9] for a general reference on this
topic. In the case of a general potential $\phi$, for existence and uniqueness
we refer to [19, Section 3] and to [33] for a general overview of the topic.
Concerning the $t\to+\infty$ limit, it is enough to consider initial data
bounded from above and from below by two positive constants: this property is
preserved along the flow and the nonlinear diffusion term can then be
estimated using linear operators, so that the solution converges to a unique
constant determined by mass conservation. The proofs can be adapted from [17,
19] and from the results on the sphere on the other hand: see [18] and
references therein. As they present no essential difficulty, they will be
omitted. Our goal here is to understand the range of $m$ for which we have a
monotonicity property of the deficit as in the case $m=1$. Let $m_{\pm}(p)$ be
defined as in (5).

######  Theorem 3.1.

Assume that $n\geq 1$, $p\in[1,2)$ and $m\in[m_{-}(p),m_{+}(p)]$. We consider
the measure $d\mu$ as in (10) such that (12) holds for some
$\lambda_{\star}>0$. If $\rho>0$ solves (14) with $\mathcal{L}$ defined by
(11) for an initial datum $\rho(t=0,\cdot)=|f|^{p}$ in
$\mathrm{L}^{2/p}\cap\mathrm{L}^{1}({\mathbb{R}}^{n},d\mu)$, then

$\frac{d}{dt}\left(\|\nabla\rho^{1/p}\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\mu)}^{2}-\frac{\lambda_{\star}}{2-p}\left(\|\rho^{1/p}\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\mu)}^{2}-\|\rho\|_{\mathrm{L}^{1}({\mathbb{R}}^{n},d\mu)}^{2/p}\right)\right)\leq
0\quad\forall\,t>0\,.$

The result of Theorem 3.1 is new for $m\neq 1$. Theorem 1.3 corresponds to the
special case of the harmonic potential $\phi(y)=\frac{1}{2}\,|y|^{2}$ with
$v=\rho^{1/p}$ in Theorem 3.1. The range of the admissible parameters $(m,p)$
is the same in Theorems 1.3 and 3.1, and shown in Fig. 2. With the additional
observation that $\rho(t,\cdot)$ converges to a constant as $t\to+\infty$ so
that the limit of the deficit is $0$, the monotonicity of the deficit of
Theorem 3.1 provides us with a proof of (13). Indeed, as a monotone non-
increasing function with limit $0$ as $t\to+\infty$, the deficit is non-
negative for any $t\geq 0$ and, as a special case, for the deficit written for
the initial datum. This, written for $\rho(t=0,\cdot)=|f|^{p}$, is precisely
Inequality (13).

######  Proof 3.2 (Proof of Theorem 3.1).

In order to do computations, a very convenient reformulation is obtained with
the substitution

$w^{\beta\,p}=\rho\,.$

Then, $w$ is a solution of the PDE

$\frac{\partial w}{\partial
t}=w^{2-2\beta}\left(\mathcal{L}w+\kappa\,\frac{|\nabla w|^{2}}{w}\right)$
(15)

for any $t\geq 0$, with initial datum $w(t=0,\cdot)=|f|^{1/\beta}$, where

$\kappa:=\beta\,(p-2)+1\quad\mbox{and}\quad\beta:=\frac{2}{2-p\,(1-m)}\,.$
(16)

A first computation shows that $\int_{{\mathbb{R}}^{n}}{w^{\beta
p}}\,d\mu=\int_{{\mathbb{R}}^{n}}{|f|^{p}}\,d\mu$ is independent of $t$
because

$\frac{d}{dt}\int_{{\mathbb{R}}^{n}}{w^{\beta
p}}\,d\mu=\beta\,p\int_{{\mathbb{R}}^{n}}{w^{\kappa}\left(\mathcal{L}w+\kappa\,\frac{|\nabla
w|^{2}}{w}\right)}\,d\mu=0\,.$

A second useful computation goes as follows:

$\displaystyle-\frac{1}{2\,\beta^{2}}\,$
$\displaystyle\,\frac{d}{dt}\int_{{\mathbb{R}}^{n}}{\left(\big{|}\nabla
w^{\beta}\big{|}^{2}+\frac{\lambda_{\star}}{p-2}\,w^{2\beta}\right)}\,d\mu$
$\displaystyle=$
$\displaystyle\int_{{\mathbb{R}}^{n}}{\left(\mathcal{L}w+(\beta-1)\,\frac{|\nabla
w|^{2}}{w}-\frac{\lambda_{\star}\,w}{\beta\,(p-2)}\right)\left(\mathcal{L}w+\kappa\,\frac{|\nabla
w|^{2}}{w}\right)}\,d\mu$ $\displaystyle=$
$\displaystyle\int_{{\mathbb{R}}^{n}}{(\mathcal{L}w)^{2}}\,d\mu+(\kappa+\beta-1)\int_{{\mathbb{R}}^{n}}{(\mathcal{L}w)\,\frac{|\nabla
w|^{2}}{w}}\,d\mu+\kappa\,(\beta-1)\int_{{\mathbb{R}}^{n}}{\frac{|\nabla
w|^{4}}{w^{2}}}\,d\mu-\lambda_{\star}\int_{{\mathbb{R}}^{n}}{|\nabla
w|^{2}}\,d\mu\,.$

The purely technical purpose of introducing the exponent $\beta$ is to make
the last line of the above computation $2$-homogeneous in $w$, which makes the
discussion easier to read. Inserting the two following estimates,

$\displaystyle\int_{{\mathbb{R}}^{n}}{(\mathcal{L}w)^{2}}\,d\mu=-\int_{{\mathbb{R}}^{n}}{\nabla
w\cdot\nabla(\mathcal{L}w)}\,d\mu$
$\displaystyle=-\int_{{\mathbb{R}}^{n}}{\nabla w\cdot(\mathcal{L}\nabla
w)}\,d\mu+\int_{{\mathbb{R}}^{n}}{\nabla w\cdot[\mathcal{L},\nabla]\,w}\,d\mu$
$\displaystyle=\int_{{\mathbb{R}}^{n}}{\|\mathrm{Hess}w\|^{2}}\,d\mu+\int_{{\mathbb{R}}^{n}}{\nabla
w\cdot[\mathcal{L},\nabla]\,w}\,d\mu$
$\displaystyle=\int_{{\mathbb{R}}^{n}}{\|\mathrm{Hess}w\|^{2}}\,d\mu+\int_{{\mathbb{R}}^{n}}{\mathrm{Hess}\,\phi:\nabla
w\otimes\nabla w}\,d\mu$
$\displaystyle\geq\int_{{\mathbb{R}}^{n}}{\|\mathrm{Hess}w\|^{2}}\,d\mu+\lambda_{\star}\int_{{\mathbb{R}}^{n}}{|\nabla
w|^{2}}\,d\mu$

and

$\int_{{\mathbb{R}}^{n}}{(\mathcal{L}w)\,\frac{|\nabla
w|^{2}}{w}}\,d\mu=-\,2\int_{{\mathbb{R}}^{n}}{\mathrm{Hess}w:\frac{\nabla
w\otimes\nabla w}{w}}\,d\mu+\int_{{\mathbb{R}}^{n}}{\frac{|\nabla
w|^{4}}{w^{2}}}\,d\mu\,,$

we obtain that

$\frac{d}{dt}\int_{{\mathbb{R}}^{n}}{\left(\big{|}\nabla
w^{\beta}\big{|}^{2}+\frac{\lambda_{\star}}{p-2}\,w^{2\beta}\right)}\,d\mu\leq
0$

if, for any function $w$, we have

$\mathcal{Q}_{\beta}[w]:=\int_{{\mathbb{R}}^{n}}{\|\mathrm{Hess}w\|^{2}}\,d\mu-2\,(\kappa+\beta-1)\int_{{\mathbb{R}}^{n}}{\mathrm{Hess}w:\frac{\nabla
w\otimes\nabla w}{w}}\,d\mu\\\
+\big{(}\kappa\,(\beta-1)+\kappa+\beta-1\big{)}\int_{{\mathbb{R}}^{n}}{\frac{|\nabla
w|^{4}}{w^{2}}}\,d\mu\geq 0\,.$

A sufficient condition is obtained if the reduced discriminant is negative,
that is, if

$(\kappa+\beta-1)^{2}-\big{(}\kappa\,(\beta-1)+\kappa+\beta-1\big{)}\leq 0\,.$

Altogether, this gives the condition

$\beta_{-}(p)\leq\beta\leq\beta_{+}(p)\quad\mbox{with}\quad\beta_{\pm}(p):=\frac{1\pm\sqrt{(p-1)\,(2-p)}}{1-(p-1)\,(2-p)}\,.$
(17)

See Fig. 3. Equivalently, written in terms of $m$, the condition is
$m_{-}(p)\leq m\leq m_{+}(p)$ with $m_{\pm}(p)$ defined by (5). Summarizing
our computations, we learn that

$\frac{d}{dt}\left(\|\nabla
w\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\mu)}^{2}-\frac{\lambda_{\star}}{p-2}\left(\|w\|_{\mathrm{L}^{p}({\mathbb{R}}^{n},d\mu)}^{2}-\|w\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\mu)}^{2}\right)\right)=-\,2\,\beta^{2}\,\mathcal{Q}_{\beta}[w]\leq
0$ (18)

because $\mathcal{Q}_{\beta}[w]$ is non-negative under Condition (17).

To make these computations rigorous, one has to justify all integrations by
parts. This is by now rather standard and can be done using the following
scheme.

  1. (1)

Let $h>0$ be large enough and consider
$\Omega_{h}:=\left\\{x\in{\mathbb{R}}^{n}\,:\,\phi(x)<h\right\\}$. We can
consider the evolution equation restricted to $\Omega_{h}$, with no flux
boundary conditions. Then apply the _carré du champ_ method and keep track of
the boundary terms. Since $\Omega_{h}$ is a bounded convex domain, these terms
have a sign due to Grisvard’s lemma: see for instance [29, Lemma 5.2], [38,
Proposition 4.2], [31, 23] and [33, Lemma A.3].

  2. (2)

Extend Inequality (13) written on $\Omega_{h}$ to ${\mathbb{R}}^{d}$ by taking
the limit as $h\to+\infty$ and then argue by density.

This completes the proof of Theorem 3.1.

Figure 3.  The admissible parameters $p$ and $\beta$ correspond to the grey
area and are independent of the dimension $n$. The boundary of the admissible
set is tangent to the vertical lines $p=1$ at $(\beta,p)=(1,1)$ and $p=2$ at
$(\beta,p)=(1,2)$. This figure corresponds to Fig. 2 up to the transformation
of $m\mapsto\beta=2/\big{(}2-p\,(1-m)\big{)}$ according to (16).

######  Remark 3.3.

For any $p\in[1,2)$, if the condition $m_{-}(p)\leq m\leq m_{+}(p)$ is not
satisfied, then one can find a positive initial datum such that the function
$v=w^{\beta}$ where $w$ solves (15) with $m$ and $\beta$ related by (16) is
such that

$\frac{d}{dt}\left(\|\nabla
v\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\mu)}^{2}-\frac{d}{p-2}\left(\|v\|_{\mathrm{L}^{p}({\mathbb{R}}^{n},d\mu)}^{2}-\|v\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\mu)}^{2}\right)\right)_{|t=0}>0\,.$

See [27] for a similar statement on the sphere and its proof.

###  3.2. Improved inequalities based on the _carré du champ_ method

In the proof of Theorem 3.1, using only $\mathcal{Q}_{\beta}[w]\geq 0$ is a
crude estimate. Let us explain how one can obtain improved estimates by making
a better use of $\mathcal{Q}_{\beta}[w]\geq 0$. Under Condition (17), we can
indeed rewrite $\mathcal{Q}_{\beta}[w]$ as an integral of a sum of squares,

$\mathcal{Q}_{\beta}[w]=\int_{{\mathbb{R}}^{n}}{\left\|\mathrm{Hess}w-(\kappa+\beta-1)\,\frac{\nabla
w\otimes\nabla
w}{w}\right\|^{2}}\,d\mu+\delta\int_{{\mathbb{R}}^{n}}{\frac{|\nabla
w|^{4}}{w^{2}}}\,d\mu$ (19)

with

$\delta:=\kappa\,(\beta-1)+\kappa+\beta-1-(\kappa+\beta-1)^{2}=\big{(}\beta-\beta_{-}(p)\big{)}\,\big{(}\beta_{+}(p)-\beta\big{)}>0\,.$
(20)

As in [2, Theorem 2], let us consider the special case $m=\beta=1$ of the
linear flow. Let us define the _entropy_ and the _Fisher information_ by

$\mathcal{E}[w]:=\frac{1}{2-p}\left(\|w\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\mu)}^{2}-\|w\|_{\mathrm{L}^{p}({\mathbb{R}}^{n},d\mu)}^{2}\right)\quad\mbox{and}\quad\mathcal{I}[w]:=\|\nabla
w\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\mu)}^{2}\,.$

Inequality (13) amounts simply to

$\mathcal{I}[w]-\lambda_{\star}\,\mathcal{E}[w]\geq 0\,.$

We can now state a first _improved entropy–entropy production inequality_.

######  Proposition 3.4.

Let $n$ be any positive integer. We consider the measure $d\mu$ as in (10)
such that (12) holds for some $\lambda_{\star}>0$. For any $p\in(1,2)$, let
$\varphi(s):=1+s-(1+s)^{p-1}$. With the above notation, we have

$\mathcal{I}[f]\geq\frac{\lambda_{\star}}{(2-p)^{2}}\,\|f\|_{\mathrm{L}^{p}({\mathbb{R}}^{n},d\mu)}^{2}\,\varphi\left(\frac{(2-p)\,\mathcal{E}[f]}{\|f\|_{\mathrm{L}^{p}({\mathbb{R}}^{n},d\mu)}^{2}}\right)\quad\forall\,f\in\mathrm{H}^{1}({\mathbb{R}}^{n},d\mu)\,.$
(21)

Since $p\in(1,2)$, the function $\varphi$ is convex with $\varphi(0)=0$ and
$\varphi^{\prime}(0)=2-p$, which implies in particular that
$\varphi(s)\geq(2-p)\,s$ for any $s\geq 0$, so that Inequality (21) is
stronger than (13). Inequality (21) amounts to

$\|\nabla
f\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\mu)}^{2}\geq\frac{\lambda_{\star}}{(2-p)^{2}}\left(\|f\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\mu)}^{2}-\|f\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\mu)}^{2\,(p-1)}\,\|f\|_{\mathrm{L}^{p}({\mathbb{R}}^{n},d\mu)}^{2\,(2-p)}\right)$

and can also be rewritten as

$\|\nabla
f\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\mu)}^{2}-\frac{\lambda_{\star}}{2-p}\left(\|f\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\mu)}^{2}-\|f\|_{\mathrm{L}^{p}({\mathbb{R}}^{n},d\mu)}^{2}\right)\\\
\geq\frac{\lambda_{\star}}{(2-p)^{2}}\left((p-1)\,\|f\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\mu)}^{2}+(2-p)\,\|f\|_{\mathrm{L}^{p}({\mathbb{R}}^{n},d\mu)}^{2}-\|f\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\mu)}^{2\,(p-1)}\,\|f\|_{\mathrm{L}^{p}({\mathbb{R}}^{n},d\mu)}^{2\,(2-p)}\right)\,.$
(22)

We claim no originality in either Proposition 3.4 or Inequality (22) and refer
to [2, Theorem 2] and [1, Ineq. (3.3)] for earlier results. Also see [24,
Theorem 2.1] for an improved inequality like (22) in the case of the sphere.
Let us give the main ideas of the proof for completeness. We refer to [18,
Appendix B.4] for a fully detailed estimates in the similar case of the
sphere.

######  Proof 3.5 (Sketch of a proof).

With $\beta=1$, notice that (19) holds with $\delta=(2-p)\,(p-1)$. Using the
Cauchy-Schwarz inequality, we obtain

$\big{(}\mathcal{I}[w]\big{)}^{2}=\left(\int_{{\mathbb{R}}^{n}}{|\nabla
w|^{2}}\,d\mu\right)^{2}\leq\int_{{\mathbb{R}}^{n}}{|w|^{2}}\,d\mu\int_{{\mathbb{R}}^{n}}{\frac{|\nabla
w|^{4}}{w^{2}}}\,d\mu$

and, with $M:=\left(\int_{{\mathbb{R}}^{n}}{|w|^{p}}\,d\mu\right)^{2/p}$, we
can also write that

$\int_{{\mathbb{R}}^{n}}{|w|^{2}}\,d\mu=(2-p)\,\mathcal{E}[w]+M\,.$

Altogether, with $\mathsf{e}(t):=(2-p)\,M^{-1}\,\mathcal{E}[w(t,\cdot)]$, if
$w$ solves (15) with $\beta=1$, then

$\mathsf{e}^{\prime}=-\,2\,\beta^{2}\,(2-p)\int_{{\mathbb{R}}^{n}}{|\nabla
w|^{2}}\,d\mu$ (23)

and we have the differential inequality

$\mathsf{e}^{\prime\prime}+2\,\lambda_{\star}\,\mathsf{e}^{\prime}-(p-1)\,\frac{(\mathsf{e}^{\prime})^{2}}{1+\mathsf{e}}\geq
0\,.$

We claim that
$(2-p)\,\mathsf{e}^{\prime}+2\,\lambda_{\star}\left(1+\mathsf{e}-(1+\mathsf{e})^{p-1}\right)\leq
0$, which follows from the observation that the equation

$y^{\prime\prime}+\mathsf{a}\,y^{\prime}-\mathsf{b}\,\frac{(y^{\prime})^{2}}{1+y}=0$

can be solved using the ansatz
$y^{\prime}=\frac{\mathsf{a}}{\mathsf{b}-1}\,\varphi(y)$ with $\varphi(0)=0$
if

$\varphi^{\prime}-\mathsf{b}\,\frac{\varphi}{1+s}=1-\mathsf{b}\,.$ (24)

It is straightforward to check that the unique solution is
$\varphi(s)=1+s-(1+s)^{\mathsf{b}}$. With $\mathsf{a}=2\,\lambda_{\star}$ and
$\mathsf{b}=p-1$, we obtain

$\left(\mathsf{e}^{\prime}+\frac{2\,\lambda_{\star}}{2-p}\,\varphi(\mathsf{e})\right)^{\prime}\geq(p-1)\,\frac{\mathsf{e}^{\prime}}{1+\mathsf{e}}\left(\mathsf{e}^{\prime}+\frac{2\,\lambda_{\star}}{2-p}\,\varphi(\mathsf{e})\right)\,.$

By integrating from $t=0$ to $+\infty$ using the facts that
$\mathsf{e}^{\prime}\leq 0$,
$\mathsf{e}^{\prime}+\frac{2\,\lambda_{\star}}{2-p}\,\mathsf{e}\leq 0$ and
$\lim_{t\to+\infty}\mathsf{e}^{\prime}(t)=\lim_{t\to+\infty}\mathsf{e}(t)=0$,
we conclude that

$\frac{2-p}{M}\left(\mathcal{I}[w]-\frac{\lambda_{\star}\,M}{(2-p)^{2}}\,\varphi\left(\frac{(2-p)\,\mathcal{E}[w]}{M}\right)\right)=-\,\frac{1}{2}\left(\mathsf{e}^{\prime}+\frac{2\,\lambda_{\star}}{2-p}\,\varphi(\mathsf{e})\right)\geq
0\quad\forall\,t\geq 0$

Inspired once more by the results on the sphere (see [24] and earlier
references therein), it is a very natural question to wonder if _improved
entropy–entropy production inequalities_ can be achieved with $\beta\neq 1$.
The answer goes as follows. Let us consider the function $\varphi_{\beta}$
given by

$\chi_{\beta}(s)=\big{(}1-\mathsf{b}(\beta)\big{)}\left(1+s-(1+s)^{\mathsf{b}(\beta)}\right)\quad\mbox{where}\quad\mathsf{b}(\beta)=\frac{\delta(\beta)}{\beta^{2}}\,\frac{2-p}{\lambda_{\star}}$

and $\delta(\beta)$ is defined by (20).

######  Proposition 3.6.

Let $n$ be any positive integer. We consider the measure $d\mu$ as in (10)
such that (12) holds for some $\lambda_{\star}>0$. For any $p\in(1,2)$, take
$\beta\in\big{[}1,\beta_{+}(p)\big{]}$. With the same notation as in
Proposition 3.4 and $\chi_{\beta}$ as above, we have

$\mathcal{I}[f]\geq\|f\|_{\mathrm{L}^{p}({\mathbb{R}}^{n},d\mu)}^{2}\,\chi_{\beta}\left(\frac{(2-p)\,\mathcal{E}[f]}{\|f\|_{\mathrm{L}^{p}({\mathbb{R}}^{n},d\mu)}^{2}}\right)\quad\forall\,f\in\mathrm{H}^{1}({\mathbb{R}}^{n},d\mu)\,.$

######  Proof 3.7.

With the notation

$\mathsf{e}:=\frac{\big{\|}w^{\beta}\big{\|}_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\mu)}^{2}}{\big{\|}w^{\beta}\big{\|}_{\mathrm{L}^{p}({\mathbb{R}}^{n},d\mu)}^{2}}\quad\mbox{and}\quad\mathsf{i}:=\frac{\big{\|}\nabla\big{(}w^{\beta}\big{)}\big{\|}_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\mu)}^{2}}{\big{\|}w^{\beta}\big{\|}_{\mathrm{L}^{p}({\mathbb{R}}^{n},d\mu)}^{2}}\,,$

we have now to consider the differential inequality

$\left(\mathsf{i}^{\prime}-\frac{\lambda_{\star}}{2-p}\,\mathsf{e}\right)^{\prime}\leq-\,\delta\,\beta^{2}\int_{{\mathbb{R}}^{n}}{\frac{|\nabla
w|^{4}}{w^{2}}}\,d\mu\,,$

which directly comes from the carré du champ computation (18) with
$\mathcal{Q}_{\beta}$ and $\delta$ defined respectively in (19) and (20). The
key ingredient is to replace an estimate due to Demange in the case of the
sphere for $p>2$ (see [21, 22] and also [Lemma 15, (iii)]) by its counterpart
for log-concave measures and $p\in(1,2)$. Compared to the linear case, the
Cauchy-Schwarz inequality in the proof of Proposition 3.4 has to be replaced
by the Hölder inequalities

$\displaystyle\int_{{\mathbb{R}}^{n}}{|\nabla
w|^{2}}\,d\mu=\int_{{\mathbb{R}}^{n}}{\left(\frac{|\nabla
w|^{2}}{w}\,w\,1\right)}\,d\mu\leq\left(\int_{{\mathbb{R}}^{n}}{\frac{|\nabla
w|^{4}}{w^{2}}}\,d\mu\right)^{\frac{1}{2}}\left(\int_{{\mathbb{R}}^{n}}{|w|^{2\,\beta}}\,d\mu\right)^{\frac{1}{2\,\beta}}1^{\frac{\beta-1}{2\,\beta}}\,,$
$\displaystyle\frac{1}{\beta^{2}}\int_{{\mathbb{R}}^{n}}{\big{|}\nabla
w^{\beta}\big{|}^{2}}\,d\mu=\int_{{\mathbb{R}}^{n}}{\left(\frac{|\nabla
w|^{2}}{w}\,w^{2\,\beta-1}\,1\right)}\,d\mu\leq\left(\int_{{\mathbb{R}}^{n}}{\frac{|\nabla
w|^{4}}{w^{2}}}\,d\mu\right)^{\frac{1}{2}}\left(\int_{{\mathbb{R}}^{n}}{|w|^{2\,\beta}}\,d\mu\right)^{\frac{2\,\beta-1}{2\,\beta}}1^{\frac{1}{2\,\beta}}\,,$

after observing that $\beta\geq\beta_{-}(p)\geq\beta_{-}(3/2)=2/3>1/2$, so
that

$\int_{{\mathbb{R}}^{n}}{\frac{|\nabla
w|^{4}}{w^{2}}}\,d\mu\geq\frac{\int_{{\mathbb{R}}^{n}}{|\nabla
w|^{2}}\,d\mu\,\int_{{\mathbb{R}}^{n}}{\big{|}\nabla
w^{\beta}\big{|}^{2}}\,d\mu}{\beta^{2}\int_{{\mathbb{R}}^{n}}{w^{2\beta}}\,d\mu}=-\,\frac{\mathsf{i}\,\mathsf{e}^{\prime}}{\beta^{4}\,(1+\mathsf{e})}\,.$

where the last equality is a consequence of the definition of $\mathsf{i}$ and
(23). Hence,

$\left(\mathsf{i}-\frac{\lambda_{\star}}{2-p}\,\mathsf{e}\right)^{\prime}\leq\frac{\delta}{\beta^{2}}\,\frac{\mathsf{i}\,\mathsf{e}^{\prime}}{1+\mathsf{e}}\,.$

Let us compute

$\left(\mathsf{i}-\frac{\lambda_{\star}}{2-p}\,\chi(\mathsf{e})\right)^{\prime}=\left(\mathsf{i}-\frac{\lambda_{\star}}{2-p}\,\mathsf{e}\right)^{\prime}+\frac{\lambda_{\star}}{2-p}\left(\mathsf{e}-\chi(\mathsf{e})\right)^{\prime}\leq\frac{\delta}{\beta^{2}}\,\frac{\mathsf{e}^{\prime}}{1+\mathsf{e}}\left(\mathsf{i}-\frac{\lambda_{\star}}{2-p}\,\chi(\mathsf{e})\right)$

on the condition that $\chi$ solves

$\chi^{\prime}(s)=1+\frac{\delta}{\beta^{2}}\,\frac{2-p}{\lambda_{\star}}\,\frac{\chi(s)}{1+s}\,,\quad\chi(0)=0\,.$

The solution $\chi(s)=(1-\mathsf{b})\,\varphi(s)$ is such that $\varphi$
solves (24) with $\mathsf{b}=\delta\,\beta^{-2}\,(2-p)/\lambda_{\star}$, which
shows that $\chi=\chi_{\beta}$. The proof then follows from the same
considerations as for Proposition 3.4 (also see [18, Appendix B.4] for
details).

If $\phi(y)=|y|^{2}/2$ is the harmonic potential so that $d\mu=d\gamma$, by
testing (2) with $f_{\varepsilon}(y)=1+\varepsilon\,y_{1}$ where $y_{1}$
denotes the first coordinate of
$y=(y_{1},y_{2},\ldots,y_{n})\in{\mathbb{R}}^{n}$, we find that

$\|\nabla
f_{\varepsilon}\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}-\frac{1}{2-p}\left(\|f_{\varepsilon}\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}-\|f_{\varepsilon}\|_{\mathrm{L}^{p}({\mathbb{R}}^{n},d\gamma)}^{2}\right)=\frac{1}{2}\,(p-1)\,\varepsilon^{4}+O(\varepsilon^{5})\quad\mbox{as}\quad\varepsilon\to
0\,.$

This is the standard computation for checking that $\lambda=\lambda_{\star}=1$
is the optimal constant in (2). Since

$\mathcal{I}[f_{\varepsilon}]-\frac{1}{(2-p)^{2}}\,\|f_{\varepsilon}\|_{\mathrm{L}^{p}({\mathbb{R}}^{n},d\gamma)}^{2}\,\varphi\left(\frac{(2-p)\,\mathcal{E}[f_{\varepsilon}]}{\|f_{\varepsilon}\|_{\mathrm{L}^{p}({\mathbb{R}}^{n},d\gamma)}^{2}}\right)=\frac{1}{2}\,(p-1)^{2}\,\varepsilon^{4}+O(\varepsilon^{5})\quad\mbox{as}\quad\varepsilon\to
0\,,$

we also learn that (21) involves the optimal exponent at least in the limit as
$\varepsilon\to 0$. After observing that $\|\nabla
f_{\varepsilon}\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}=\varepsilon^{2}$,
we may wonder whether the _deficit_

$\|\nabla
f\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}-\frac{1}{2-p}\left(\|f\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}-\|f\|_{\mathrm{L}^{p}({\mathbb{R}}^{n},d\gamma)}^{2}\right)$

measures the distance in terms of $\|\nabla
f\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{4}$. The detailed answer is
not limited to the case $\mu=\gamma$ and goes as follows. For simplicity, we
take $\beta=1$ and consider $\varphi(s):=1+s-(1+s)^{p-1}$ as in Proposition
3.4. We recall that $\varphi$ is monotone increasing and convex on
${\mathbb{R}}^{+}$, such that $\varphi^{\prime}(0)=2-p$, hence invertible of
inverse $\varphi^{-1}$ such that $\psi(t):=t-(2-p)\,\varphi^{-1}(t)$ is also a
convex, non-negative, monotone increasing function.

######  Corollary 3.8.

Let $p\in(1,2)$ and $n$ be a positive integer. We consider the measure $d\mu$
as in (10) such that (12) holds for some $\lambda_{\star}>0$. With $\psi$ as
above, for any $f\in\mathrm{H}^{1}(d\gamma)$ we have

$\|\nabla
f\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\mu)}^{2}-\frac{\lambda_{\star}}{2-p}\left(\|f\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\mu)}^{2}-\|f\|_{\mathrm{L}^{p}({\mathbb{R}}^{n},d\mu)}^{2}\right)\geq\frac{\lambda_{\star}}{2-p}\,\|f\|_{\mathrm{L}^{p}({\mathbb{R}}^{n},d\mu)}^{2}\,\psi\left(\frac{2-p}{\lambda_{\star}}\,\frac{\|\nabla
f\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\mu)}^{2}}{\|f\|_{\mathrm{L}^{p}({\mathbb{R}}^{n},d\mu)}^{2}}\right)\,.$

Moreover, there is some $\kappa>0$ such that

$\|\nabla
f\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\mu)}^{2}-\frac{\lambda_{\star}}{2-p}\left(\|f\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\mu)}^{2}-\|f\|_{\mathrm{L}^{p}({\mathbb{R}}^{n},d\mu)}^{2}\right)\geq\frac{\kappa\,\|\nabla
f\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\mu)}^{4}}{\|\nabla
f\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\mu)}^{2}+\frac{\lambda_{\star}}{2-p}\,\|f\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\mu)}^{2}}\,.$
(25)

The constant $\kappa$ depends only on $p$ and its value is estimated in the
proof below.

######  Proof 3.9.

Let $M=\|f\|_{\mathrm{L}^{p}({\mathbb{R}}^{n},d\mu)}^{2}$. We deduce from
Proposition 3.4 that

$\mathsf{i}:=\frac{(2-p)^{2}}{\lambda_{\star}\,M}\,\mathcal{I}[f]\geq\varphi(\mathsf{e})\quad\mbox{where}\quad\mathsf{e}:=\frac{(2-p)\,\mathcal{E}[f]}{\|f\|_{\mathrm{L}^{p}({\mathbb{R}}^{n},d\mu)}^{2}}\,,$

which is equivalent to $-\,\mathsf{e}\geq-\,\varphi^{-1}(\mathsf{i})$, so that

$\mathsf{i}-(2-p)\,\mathsf{e}\geq\mathsf{i}-(2-p)\,\varphi^{-1}(\mathsf{i})=\psi(\mathsf{i})$

and, as a consequence,

$\mathcal{I}[f]-\lambda_{\star}\,\mathcal{E}[f]=\frac{\lambda_{\star}\,M}{(2-p)^{2}}\,\big{(}\mathsf{i}-(2-p)\,\mathsf{e}\big{)}\geq\frac{\lambda_{\star}\,M}{(2-p)^{2}}\,\psi(\mathsf{i})=\frac{\lambda_{\star}\,M}{(2-p)^{2}}\,\psi\left(\frac{2-p}{\lambda_{\star}\,M}\,\|\nabla
f\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\mu)}^{2}\right)\,.$

Since $\varphi(s)\sim s$ as $s\to+\infty$, we deduce that
$\psi(t)\sim(p-1)\,t$ as $t\to+\infty$. On the other hand, since

$\psi^{\prime\prime}(t)=(2-p)\,\frac{\varphi^{\prime\prime}(s)}{\big{(}\varphi^{\prime}(s)\big{)}^{3}}\quad\mbox{with}\quad
s=\varphi^{-1}(t)\,,$

we learn that $\psi^{\prime\prime}(0)=(p-1)/(2-p)>0$ and
$t\mapsto\psi^{\prime\prime}(t)$ is non-increasing because
$\varphi^{\prime}(s)^{5}\,\psi^{\prime\prime\prime}(t)=\varphi^{\prime}(s)\,\varphi^{\prime\prime\prime}(s)-\varphi^{\prime\prime}(s)^{2}<0$.
This allows us to define

$\kappa:=\inf_{t>0}t^{-2}\,(1+t)\,\psi(t)\,.$

Using $\psi(0)=\psi^{\prime}(0)=0$, we know that $\kappa>0$ and
$\psi(t)\geq\kappa\,t^{2}/(1+t)$ concludes the proof using
$\|f\|_{\mathrm{L}^{p}({\mathbb{R}}^{n},d\mu)}\leq\|f\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\mu)}$.

######  Remark 3.10.

If $\phi(y)=|y|^{2}/2$ is the harmonic potential so that $d\mu=d\gamma$, then
$\lambda=\lambda_{\star}=1$ is the optimal constant in (2) and the results of
Proposition 3.4 and Corollary 3.8 are both stability estimates. Even in the
general case of a measure $d\mu$ as in (10) such that (12) holds for some
$\lambda_{\star}>0$, Proposition 3.4 and Corollary 3.8 provide improvements to
the basic inequality with a (generically non-optimal) constant
$\lambda_{\star}$.

###  3.3. From the _carré du champ_ method to Obata’s theorem

As a side result, we consider an improvement of the _carré du champ_ method as
in [36, Theorem 2.1] or [26], which goes as follows. Let us consider the
optimal constant $\lambda_{1}>0$ in the Poincaré inequality

$\int_{{\mathbb{R}}^{n}}{|\nabla
w|^{2}}\,d\mu\geq\lambda_{1}\int_{{\mathbb{R}}^{n}}{|w-\bar{w}|^{2}}\,d\mu\quad\forall\,w\in\mathrm{H}^{1}({\mathbb{R}}^{n},d\mu)\,,$
(26)

where $\bar{w}:=\int_{{\mathbb{R}}^{n}}{w}\,d\mu$. By expanding
$\int_{{\mathbb{R}}^{n}}{\big{(}\mathcal{L}w+\lambda_{1}\,(w-\bar{w})\big{)}^{2}}\,d\mu\geq
0$, we obtain

$\int_{{\mathbb{R}}^{n}}{(\mathcal{L}w)^{2}}\,d\mu\geq\lambda_{1}\int_{{\mathbb{R}}^{n}}{|\nabla
w|^{2}}\,d\mu\,.$ (27)

On the other hand, by the computation of Section 3.1 with $\beta=p=1$, we know
that

$-\frac{1}{2}\,\frac{d}{dt}\int_{{\mathbb{R}}^{n}}{\left(|\nabla
w|^{2}+\frac{\lambda_{\star}}{p-2}\,w^{2}\right)}\,d\mu=\int_{{\mathbb{R}}^{n}}{(\mathcal{L}w)^{2}}\,d\mu-\lambda_{\star}\int_{{\mathbb{R}}^{n}}{|\nabla
w|^{2}}\,d\mu=\mathcal{Q}_{1}[w]\geq 0\,.$

which proves, for instance using (18), that

$\lambda_{1}\geq\lambda_{\star}\,.$

This result goes back to the work of Obata and the _carré du champ_ method is
in fact very close to the spirit of the historical proof: see [41, p. 327].

######  Lemma 3.11.

Assume that $n\geq 1$, $p\in[1,2)$ and consider the measure $d\mu$ as in (10)
such that (12) holds for some $\lambda_{\star}>0$. Then (13) holds with

$\lambda=(2-p)\,\lambda_{1}+(p-1)\,\lambda_{\star}\,.$ (28)

As a consequence of (28), we have $\lambda\geq\lambda_{\star}$ with equality
for the optimal value of $\lambda$ in (13) if and only if
$\lambda_{\star}=\lambda_{1}$ and $\phi(y)=\lambda_{\star}\,|y-y_{0}|^{2}/2$
for some $y_{0}\in{\mathbb{R}}^{n}$.

######  Proof 3.12.

The _carré du champ_ method applied as in Section 3.1 with $p=1$ shows that
$\lambda_{1}\geq\lambda_{\star}$. Coming back to the computations of Section
3.1, we can rearrange the integrals in the expression $\mathcal{Q}_{\beta}[w]$
differently and get

$\displaystyle-\frac{1}{2\,\beta^{2}}\,\frac{d}{dt}\int_{{\mathbb{R}}^{n}}{\left(\big{|}\nabla
w^{\beta}\big{|}^{2}+\frac{\lambda}{p-2}\,w^{2\beta}\right)}\,d\mu$
$\displaystyle=(1-\theta)\int_{{\mathbb{R}}^{n}}{(\mathcal{L}w)^{2}}\,d\mu-\lambda\int_{{\mathbb{R}}^{n}}{|\nabla
w|^{2}}\,d\mu$ $\displaystyle\hskip
28.45274pt+\theta\int_{{\mathbb{R}}^{n}}{(\mathcal{L}w)^{2}}\,d\mu+(\kappa+\beta-1)\int_{{\mathbb{R}}^{n}}{(\mathcal{L}w)\,\frac{|\nabla
w|^{2}}{w}}\,d\mu+\kappa\,(\beta-1)\int_{{\mathbb{R}}^{n}}{\frac{|\nabla
w|^{4}}{w^{2}}}\,d\mu$
$\displaystyle=\big{(}(1-\theta)\,\lambda_{1}+\theta\,\lambda_{\star}-\lambda\big{)}\int_{{\mathbb{R}}^{n}}{|\nabla
w|^{2}}\,d\mu$ $\displaystyle\hskip
28.45274pt+\theta\left(\int_{{\mathbb{R}}^{n}}{\|\mathrm{Hess}w\|^{2}}\,d\mu+\lambda_{\star}\int_{{\mathbb{R}}^{n}}{|\nabla
w|^{2}}\,d\mu\right)$ $\displaystyle\hskip
28.45274pt-2\,(\kappa+\beta-1)\int_{{\mathbb{R}}^{n}}{\mathrm{Hess}w:\frac{\nabla
w\otimes\nabla\,w}{w}}\,d\mu+\big{(}\kappa\,(\beta-1)+\kappa+\beta-1\big{)}\int_{{\mathbb{R}}^{n}}{\frac{|\nabla
w|^{4}}{w^{2}}}\,d\mu$

where the $(1-\theta)\,\lambda_{1}$ term factor from (27). With the choice of
$\theta$ such that

$(\kappa+\beta-1)^{2}-\theta\,\big{(}\kappa\,(\beta-1)+\kappa+\beta-1\big{)}=0\,,$

which means $\theta=\theta(\beta)$ with

$\theta(\beta):=\frac{(p-1)^{2}\,\beta^{2}}{(p-2)\,\beta^{2}+2\,\beta-1}\,,$

we can write that

$\theta\int_{{\mathbb{R}}^{n}}{\|\mathrm{Hess}w\|^{2}}\,d\mu-2\,(\kappa+\beta-1)\int_{{\mathbb{R}}^{n}}{\mathrm{Hess}w:\frac{\nabla
w\otimes\nabla\,w}{w}}\,d\mu+\big{(}\kappa\,(\beta-1)+\kappa+\beta-1\big{)}\int_{{\mathbb{R}}^{n}}{\frac{|\nabla
w|^{4}}{w^{2}}}\,d\mu\\\
=\theta\int_{{\mathbb{R}}^{n}}{\left\|\mathrm{Hess}w-\frac{\beta\,(p-1)}{\theta}\,\frac{\nabla
w\otimes\nabla\,w}{w}\right\|^{2}}\,d\mu\geq 0\,.$

Altogether, we have shown that

$\frac{d}{dt}\int_{{\mathbb{R}}^{n}}{\left(\big{|}\nabla
w^{\beta}\big{|}^{2}+\frac{\lambda}{p-2}\,w^{2\beta}\right)}\,d\mu\leq 0$

if $\lambda=(1-\theta)\,\lambda_{1}+\theta\,\lambda_{\star}$. Recall that
$\|w^{\beta}\|_{\mathrm{L}^{p}({\mathbb{R}}^{n},d\mu)}$ is independent of $t$
so that the deficit functional associated to (13) is monotone non-increasing.
Also notice that $\theta\big{(}\beta_{\pm}(p)\big{)}=1$ for any $p\in[1,2)$.
The observation that
$\min_{\beta\in[\beta_{-}(p),\beta_{+}(p)]}\theta(\beta)=\theta(1)=p-1$
completes the proof of (28).

With $\beta=1$ and $p=1$, the computation in the proof of Theorem 3.1 shows
that, if the initial datum $w(t=0,\cdot)$ is an optimal function for the
Poincaré inequality (26), then

$0=\frac{d}{dt}\int_{{\mathbb{R}}^{n}}{\left(|\nabla
w|^{2}-\lambda_{\star}\,w^{2}\right)}\,d\mu=-\,2\left(\mathcal{Q}_{1}[w]+\int_{{\mathbb{R}}^{n}}{\left(\mathrm{Hess}\,\phi-\lambda_{\star}\mathrm{Id}\right):\nabla
w\otimes\nabla w}\,d\mu\right)$

at $t=0$. Here we keep all terms and in particular do not use the fact that
$\mathrm{Hess}\,\phi-\lambda_{\star}\mathrm{Id}\geq 0$ a.e. in the sense of
positive matrices. Since $\nabla w\neq 0$ a.e. and $\mathcal{Q}_{1}[w]\geq 0$,
we find that $\mathrm{Hess}\,\phi-\lambda_{\star}\mathrm{Id}=0$ a.e. This
completes the proof in the equality case $\lambda_{\star}=\lambda_{1}$.

######  Remark 3.13.

The proof of Lemma 3.11 is reminiscent of [36, 26]. The result when
$\lambda_{\star}=\lambda_{1}$ points in the direction of _Obata’s theorem_
(also known as the _ObataâLichnerowicz theorem_) and in some sense, it is the
analogue for Gaussian measures of the result of [35, p. 135] (also see for
instance [13, p. 179]) on the sphere. The case $\lambda_{\star}=\lambda_{1}$
is easy to understand in dimension $d=1$: with $\beta=1$ and $p=1$, we apply
the computation of the proof of Lemma 3.11 to a function $u$ in the eigenspace
associated with $\lambda_{1}$ and obtain that $u^{\prime\prime}=0$ almost
everywhere. This means that $u(y)=a\,y+b$ for some real constants $a\neq 0$
and $b$, and there is no loss of generality if we take $a=1$. Using now the
eigenvalue equation $\mathcal{L}u+\lambda_{1}\,u=0$, we read that
$\phi^{\prime}(y)=\lambda_{1}\,(y-b)$, which means that $\phi$ is a harmonic
potential. In higher dimensions, one has to remember that Inequality (13) can
be tensorized on product spaces: see for instance [34, 20, 28]. This is
however responsible for some technicalities, which are dealt with in greatest
generality, _e.g._ , in [30].

###  3.4. Improved inequalities under orthogonality conditions

Let $\Pi_{1}$ be the $\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)$orthogonal
projection onto the space generated by the constants and the coordinate
functions, corresponding to the Hermite polynomials of order less or equal
than $1$. The following result was recently proved in [18, Appendix A] on the
basis of Nelson’s hypercontractivity estimate in [40, Theorem 3] and its
relation with Gross’ logarithmic Sobolev inequality in [32] (also see [34, 1]
for earlier results).

######  Proposition 3.14.

Let $n\geq 1$ and $p\in[1,2)$. For any
$f\in\mathrm{H}^{1}({\mathbb{R}}^{n},d\gamma)$, we have

$\|\nabla
f\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}-\frac{1}{p-2}\left(\|f\|_{\mathrm{L}^{p}({\mathbb{R}}^{n},d\gamma)}^{2}-\|f\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}\right)\geq\frac{1}{2}\,(2-p)\,\|\nabla(\mathrm{Id}-\Pi_{1})f\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}\,.$

Compared to (2), this result provides us with an _improved entropy–entropy
production inequalities_ under orthogonality conditions. As noted in [18],
such an improvement is not optimal. There are other possible approaches. For
instance, a finer analysis of entropy methods has been used in [27] on the
sphere, that could probably be adapted to the case of the Gaussian measure.
Alternatively, one could use the convex interpolation of [34], with the
possible advantage that the result would not degenerate in the limit as $p\to
2$ using the recent stability result of [25, Theorem 2].

##  4\. Stability results for the Gaussian measure in the subcritical range

The whole Section is devoted to the proof of Theorem 1.4. We split it into
four lemmas. The key estimate is obtained in Lemma 4.7.

① Let us start with the easy case, far away from the optimizers of (2) in the
sense that for some $\theta>0$, we assume

$\|\nabla
f\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}\geq\theta\,\|f\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}\,.$
(29)

By homogenity of the inequalities, we can fix
$\|f\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}=1$ without loss of
generality.

######  Lemma 4.1.

Let $n\geq 1$ and $\theta\in(0,1)$. For any function
$f\in\mathrm{H}^{1}({\mathbb{R}}^{n},d\gamma)$ such that
$\|f\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}=1$ and $\|\nabla
f\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}\geq\theta$, we have the
estimate

$\|\nabla
f\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}-\frac{1}{2-p}\left(\|f\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}-\|f\|_{\mathrm{L}^{p}({\mathbb{R}}^{n},d\gamma)}^{2}\right)\geq\kappa_{\star}(\theta)\,\|\nabla
f\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}$

In case (29), this already proves the result of Theorem 1.4 with $c_{n,p}\leq
c_{n,p}^{(1)}:=\kappa_{\star}(\theta)$. The result of Lemma 4.1 directly
follows from (25), but we give a proof based on the convexity properties of
the entropy, which directly explains where the constant comes from.

######  Proof 4.2.

From Corollary 3.8 with $\lambda_{\star}=1$ and Hölder’s inequality, we obtain

$\|\nabla
f\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}-\frac{1}{2-p}\left(\|f\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}-\|f\|_{\mathrm{L}^{p}({\mathbb{R}}^{n},d\gamma)}^{2}\right)\\\
\geq\frac{\|f\|_{\mathrm{L}^{p}({\mathbb{R}}^{n},d\gamma)}^{2}}{2-p}\,\frac{1}{2}\,\psi\left((2-p)\,\theta\right)\left((2-p)\,\frac{\|\nabla
f\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}}{\|f\|_{\mathrm{L}^{p}({\mathbb{R}}^{n},d\gamma)}^{2}}\right)^{2}\\\
\geq\frac{1}{2}\,(2-p)\,\psi\left((2-p)\,\theta\right)\,\frac{\|\nabla
f\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{4}}{\|f\|_{\mathrm{L}^{p}({\mathbb{R}}^{n},d\gamma)}^{2}}\geq\kappa_{\star}(\theta)\,\|\nabla
f\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}$

with
$\kappa_{\star}(\theta):=\frac{1}{2}\,(2-p)\,\psi\left((2-p)\,\theta\right)\,\theta$
because $t\mapsto\psi^{\prime\prime}(t)$ is non-increasing, $\|\nabla
f\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}\geq\theta$ and
$\|f\|_{\mathrm{L}^{p}({\mathbb{R}}^{n},d\gamma)}^{2}\leq\|f\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}$.

② From now on we work in a neighbourhood of the constants which, by
homogeneity of the inequalities, is defined as

$\|\nabla
f\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}\leq\theta\,\|f\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}\,.$

With $\theta>0$ small, we claim that $\int_{{\mathbb{R}}^{n}}{f}\,d\gamma$ is
close to $1$ if $\|f\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}=1$.

######  Lemma 4.3.

Let $n\geq 1$ and $\theta\in(0,1)$. For any non-negative function

$f\in\mathrm{H}^{1}({\mathbb{R}}^{n},d\gamma)\quad\mbox{such
that}\quad\|f\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}=1\quad\mbox{and}\quad\|\nabla
f\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}\leq\theta\,,$ (30)

we have the estimate

$\sqrt{1-\theta}\leq\int_{{\mathbb{R}}^{n}}{f}\,d\gamma\leq 1\,.$

######  Proof 4.4.

With $\bar{f}:=\int_{{\mathbb{R}}^{n}}{f}\,d\gamma$, the result follows from
the Gaussian Poincaré inequality according to

$1=\|f\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}=\int_{{\mathbb{R}}^{n}}{|f-\bar{f}|^{2}}\,d\gamma+\bar{f}^{2}\leq\|\nabla
f\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}+\bar{f}^{2}\leq\theta+\bar{f}^{2}\,.$

③ Assume that $f$ is as in (30) and let us decompose
$u_{f}(x):=f(x)/\int_{{\mathbb{R}}^{n}}{f}\,d\gamma$ as

$u_{f}(x)=1+\varepsilon\,x\cdot\nu+\eta\,r(x)$

where $\nu\in{\mathbb{S}}^{n-1}$ is such that
$\varepsilon\,\nu=\int_{{\mathbb{R}}^{n}}{x\,u_{f}(x)}\,d\gamma$ with
$\varepsilon>0$, $\eta$ is a positive number and $r$ is a function in
$\mathrm{H}^{1}({\mathbb{R}}^{n},d\gamma)\cap(\mathrm{Id}-\Pi_{1})\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)$
such that $\|\nabla r\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}=1$ and
$\|r\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}\leq 1/2$ by the Gaussian
Poincaré inequality after taking into account the additional orthogonality
condition $\int_{{\mathbb{R}}^{n}}{r\,x_{i}}\,d\gamma=0$ for any $i=1$,
$2$,…,$n$.

######  Lemma 4.5.

Let $n\geq 1$ and $\theta\in(0,1)$. Let
$f\in\mathrm{H}^{1}({\mathbb{R}}^{n},d\gamma)$ be such that (30) holds. With
the above notation, we have

$\|u_{f}\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}=1+\varepsilon^{2}+\eta^{2}\,\|r\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}\leq
1+\theta\quad\mbox{and}\quad\|\nabla
u_{f}\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}=\varepsilon^{2}+\eta^{2}\leq\frac{\theta}{1-\theta}$

and, if $\,\eta>t\,\varepsilon^{2}$ for some $t>0$, then

$\|\nabla
u_{f}\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}-\frac{1}{2-p}\left(\|u_{f}\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}-\|u_{f}\|_{\mathrm{L}^{p}({\mathbb{R}}^{n},d\gamma)}^{2}\right)\geq\frac{1}{4}\,(2-p)\left(\eta^{2}+\frac{t^{2}\,\varepsilon^{4}}{1+\varepsilon^{2}+\eta^{2}}\right)\,.$

By homogeneity, if (30) holds and $\eta>t\,\varepsilon^{2}$ for some given
$t>0$, using Lemma 4.3, we obtain

$\|\nabla
f\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}-\frac{1}{2-p}\left(\|f\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}-\|f\|_{\mathrm{L}^{p}({\mathbb{R}}^{n},d\gamma)}^{2}\right)\\\
\geq\frac{1}{4}\,(2-p)\,(1-\theta)\left(\|\nabla(\mathrm{Id}-\Pi_{1})\,f\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}+\frac{t^{2}\,\|\nabla(\Pi_{1}\,f)\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{4}}{\|\nabla
f\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}+\|f\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}}\right)\,.$

This case already covers the result of Theorem 1.4 with

$c_{n,p}\leq
c_{n,p}^{(1)}:=\frac{1}{4}\,(2-p)\,(1-\theta)\min\big{\\{}t,1\big{\\}}\,.$
(31)

######  Proof 4.6.

The result follows from Proposition 3.14 and from the chain of elementary
inequalities

$\eta^{2}\geq\frac{1}{2}\left(\eta^{2}+t^{2}\,\varepsilon^{4}\right)\geq\frac{1}{2}\left(\eta^{2}+\frac{t^{2}\,\varepsilon^{4}}{1+\varepsilon^{2}+\eta^{2}}\right)\,.$

④ The next part of the proof relies on a Taylor expansion of
$\|u_{f}\|_{\mathrm{L}^{p}({\mathbb{R}}^{n},d\gamma)}^{2}$. With no loss of
generality, by rotational invariance, we can assume that $\nu=(1,0,\ldots,0)$
so that with Cartesian coordinates
$x=(x_{1},x_{2},\ldots,x_{n})\in{\mathbb{R}}^{n}$, we write
$u_{f}(x)=1+\varepsilon\,x_{1}+\eta\,r(x)$. The following result is at the
core of our strategy. It heavily relies on the Gaussian logarithmic Sobolev
inequality and new estimates for the remainder terms based on the boundedness
of $\int_{{\mathbb{R}}^{n}}{r^{2}\,\log r^{2}}\,d\gamma$.

######  Lemma 4.7.

Let $n\geq 1$ and $f\in\mathrm{H}^{1}({\mathbb{R}}^{n},d\gamma)$ be a non-
negative function such that (30) holds. We keep the same notation as in Lemma
4.5 and further assume that $\eta\leq t\,\varepsilon^{2}$ for some $t>0$. Then
there is a constant $\mathcal{C}>0$, depending only on $n$, $p$ and $t$, such
that

$\|u_{f}\|_{\mathrm{L}^{p}({\mathbb{R}}^{n},d\gamma)}^{2}\geq
1+(p-1)\left(\varepsilon^{2}+\eta^{2}\|r\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}\right)\\\
+(p-1)\,(2-p)\left(\frac{1}{2}\,\varepsilon^{4}+\varepsilon^{2}\,\eta\int_{{\mathbb{R}}^{n}}{x_{1}^{2}\,r(x)}\,d\gamma\right)-\frac{\mathcal{C}\,\varepsilon^{4}}{\log\left(\frac{1}{\varepsilon}\right)}\quad\mbox{as}\quad\varepsilon\to
0_{+}\,.$

######  Proof 4.8.

This proof is elementary although a little bit lengthy. Several steps are
similar to those of [18, Section 4] and will be only sketched. Let us split it
into three steps.

Step 1. Let us start with a list of preliminary remarks.

$\bullet$ Let $B_{\varepsilon}$ be the centred ball of radius
$1/(2\,\varepsilon)$, that is,

$B_{\varepsilon}:=\big{\\{}x\in{\mathbb{R}}^{n}\,:\,2\,\varepsilon\,|x|<1\big{\\}}$

and let $B_{\varepsilon}^{c}={\mathbb{R}}^{n}\setminus B_{\varepsilon}$. We
observe that

$\gamma(B_{\varepsilon}^{c})=\big{|}{\mathbb{S}}^{n-1}\big{|}\int_{1/(2\,\varepsilon)}^{+\infty}r^{n-1}\,e^{-\frac{r^{2}}{2}}\,dr=c_{n}\,\varepsilon^{2-n}\,e^{-\frac{1}{8\,\varepsilon^{2}}}\left(1+O\left(\varepsilon^{2}\right)\right)\quad\mbox{as}\quad\varepsilon\to
0_{+}$

with $c_{n}=2^{3\,(2-n)/2}/\Gamma(n/2)$. Let
$\xi_{p}:=\sup_{\varepsilon\in(0,1/2)}\varepsilon^{-5}\big{(}\gamma(B_{\varepsilon}^{c})\big{)}^{(2-p)/2}$.
Hence we have

$\int_{B_{\varepsilon}^{c}}|g|^{p}\,d\gamma\leq\|g\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{p}\,\big{(}\gamma(B_{\varepsilon}^{c})\big{)}^{(2-p)/2}\leq\xi_{p}\,\|g\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{p}\,\varepsilon^{5}$

for any $g\in\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)$, by Hölder’s inequality
and, as a consequence,

$\int_{B_{\varepsilon}}|g|^{p}\,d\gamma\leq\int_{{\mathbb{R}}^{n}}{|g|^{p}}\,d\gamma\leq\int_{B_{\varepsilon}}|g|^{p}\,d\gamma+\xi_{p}\,\|g\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{p}\,\varepsilon^{5}\quad\forall\,\varepsilon\in(0,1/2)\,.$
(32)

From now on, we assume without further notice that $x\in B_{\varepsilon}$
unless it is specified.

$\bullet$ An expansion in Taylor series of $(1+s)^{p}-1-p\,s$ for $s\leq 0$
shows that all terms are non-negative:

$(1+s)^{p}\geq 1+p\,s+\frac{1}{2}\,p\,(p-1)\,s^{2}\quad\forall\,s\in(-1,0]\,.$

Applied to $u_{f}=1+\varepsilon\,x_{1}+\eta\,r$ whenever
$1+\varepsilon\,x_{1}>0$ and $r\leq 0$, we obtain

$|u_{f}|^{p}=\left(1+\varepsilon\,x_{1}+\eta\,r\right)^{p}=\left(1+\varepsilon\,x_{1}\right)^{p}\left(1+\frac{\eta\,r}{1+\varepsilon\,x_{1}}\right)^{p}\\\
\geq\left(1+\varepsilon\,x_{1}\right)^{p}\left(1+p\,\frac{\eta\,r}{1+\varepsilon\,x_{1}}+\frac{1}{2}\,p\,(p-1)\left(\frac{\eta\,r}{1+\varepsilon\,x_{1}}\right)^{2}\right)\\\
=\left(1+\varepsilon\,x_{1}\right)^{p}+p\left(1+\varepsilon\,x_{1}\right)^{p-1}\,\eta\,r+\frac{1}{2}\,p\,(p-1)\left(1+\varepsilon\,x_{1}\right)^{p-2}\,\eta^{2}\,r^{2}\,.$

$\bullet$ Let us consider the case $1+\varepsilon\,x_{1}>0$ and $r>0$. The
function

$\rho(s):=\frac{1}{s^{2}}\left((1+s)^{p}-1-p\,s-\frac{1}{2}\,p\,(p-1)\,s^{2}\right)\quad\,\forall\,s\geq
0$

is bounded. Let us extend $\rho$ by $0$ on $(-1,0)$ and define

$\psi_{\varepsilon,\eta,r}(x):=\left(1+\varepsilon\,x_{1}\right)^{p-2}\,\rho\left(\frac{\eta\,r(x)}{1+\varepsilon\,x_{1}}\right)\,.$

With this definition, using $u_{f}\geq 0$ by hypothesis, we obtain

$\displaystyle|u_{f}|^{p}\,$
$\displaystyle\geq\left(1+\varepsilon\,x_{1}+\eta\,r\right)^{p}$
$\displaystyle=\left(1+\varepsilon\,x_{1}\right)^{p}+p\left(1+\varepsilon\,x_{1}\right)^{p-1}\,\eta\,r+\frac{1}{2}\,p\,(p-1)\left(1+\varepsilon\,x_{1}\right)^{p-2}\,\eta^{2}\,r^{2}+\frac{1}{2}\,p\,(p-1)\,\eta^{2}\,r(x)^{2}\,\psi_{\varepsilon,\eta,r}(x)$

with equality whenever $r\geq 0$.

As $\eta\to 0_{+}$, $\psi_{\varepsilon,\eta,r}$ converges a.e. to $0$ on
$B_{\varepsilon}$ uniformly with respect to $\varepsilon\in(0,1/2)$. The
dominated convergence theorem is enough to conclude that

$\lim_{\eta\to
0_{+}}\int_{B_{\varepsilon}}r(x)^{2}\,\psi_{\varepsilon,\eta,r}(x)\,d\gamma=0$

for a given function $r$, but this is not enough to conclude uniformly with
respect to $r$. To do this, we need more detailed estimates. Notice however
that

$\|\psi_{\varepsilon,\eta,r}\|_{\mathrm{L}^{\infty}(B_{\varepsilon})}\leq
M:=2^{2-p}\,\|\rho\|_{\mathrm{L}^{\infty}({\mathbb{R}}^{+})}$

where $M$ is independent of $\varepsilon$, $\eta$ and $r$.

$\bullet$ Since
$\int_{B_{\varepsilon}}x_{1}^{2\,k}\,d\gamma\geq\int_{B_{\varepsilon}}x_{1}^{2\,k+1}\,d\gamma=0$
and
$\int_{{\mathbb{R}}^{n}}{x_{1}^{2\,k}}\,d\gamma\geq\int_{{\mathbb{R}}^{n}}{x_{1}^{2\,k+1}}\,d\gamma=0$
for any $k\in{\mathbb{N}}$, an expansion in Taylor series of
$\left(1+\varepsilon\,x_{1}\right)^{p}$ gives

$\int_{{\mathbb{R}}^{n}}{\left(1+\varepsilon\,x_{1}\right)^{p}}\,d\gamma\geq\int_{{\mathbb{R}}^{n}}{\left(1+\frac{1}{2}\,p\,(p-1)\,\varepsilon^{2}\,x_{1}^{2}+\frac{1}{24}\,p\,(p-1)\,(p-2)\,(p-3)\,x_{1}^{4}\,\varepsilon^{4}\right)}\,d\gamma\\\
=1+\frac{1}{2}\,p\,(p-1)\,\varepsilon^{2}+\frac{1}{8}\,p\,(p-1)\,(p-2)\,(p-3)\,\varepsilon^{4}\,.$

By applying (32) with $g=1+\varepsilon\,x_{1}$, we obtain

$\int_{{\mathbb{R}}^{n}}{|1+\varepsilon\,x_{1}|^{p}}\,d\gamma\leq\int_{B_{\varepsilon}}|1+\varepsilon\,x_{1}|^{p}\,d\gamma+\xi_{p}\left(1+\varepsilon^{2}\right)^{p/2}\,\varepsilon^{5}\,.$

Summing up with $\varepsilon^{2}\leq\theta$, we have

$\int_{B_{\varepsilon}}|1+\varepsilon\,x_{1}|^{p}\,d\gamma\geq
1+\frac{1}{2}\,p\,(p-1)\,\varepsilon^{2}+\frac{1}{8}\,p\,(p-1)\,(p-2)\,(p-3)\,\varepsilon^{4}-\xi_{p}\,(1+\theta)^{p/2}\,\varepsilon^{5}\,.$

$\bullet$ Let us estimate
$\|u_{f}\|_{\mathrm{L}^{p}({\mathbb{R}}^{n},d\gamma)}^{p}$ using

$\int_{{\mathbb{R}}^{n}}{|u_{f}|^{p}}\,d\gamma\geq\int_{B_{\varepsilon}}{|u_{f}|^{p}}\,d\gamma$

and

$\int_{B_{\varepsilon}}{|u_{f}|^{p}}\,d\gamma\geq\int_{B_{\varepsilon}}{|1+\varepsilon\,x_{1}|^{p}}\,d\gamma\\\
+p\,\eta\int_{B_{\varepsilon}}{\left(1+\varepsilon\,x_{1}\right)^{p-1}\,r}\,d\gamma+\frac{1}{2}\,p\,(p-1)\,\eta^{2}\int_{B_{\varepsilon}}{\left(1+\varepsilon\,x_{1}\right)^{p-2}\,r^{2}}\,d\gamma\\\
+\frac{1}{2}\,p\,(p-1)\,\eta^{2}\int_{B_{\varepsilon}}{r(x)^{2}\,\psi_{\varepsilon,\eta,r}(x)}\,d\gamma\,.$

We obtain

$\int_{{\mathbb{R}}^{n}}{|u_{f}|^{p}}\,d\gamma\geq
1+\frac{1}{2}\,p\,(p-1)\,\varepsilon^{2}+\frac{1}{8}\,p\,(p-1)\,(p-2)\,(p-3)\,\varepsilon^{4}-\xi_{p}\,(1+\theta)^{p/2}\,\varepsilon^{5}\\\
+p\,\eta\int_{B_{\varepsilon}}{\left(1+\varepsilon\,x_{1}\right)^{p-1}\,r}\,d\gamma+\frac{1}{2}\,p\,(p-1)\,\eta^{2}\int_{B_{\varepsilon}}{\left(1+\varepsilon\,x_{1}\right)^{p-2}\,r^{2}}\,d\gamma\\\
+\frac{1}{2}\,p\,(p-1)\,\eta^{2}\int_{B_{\varepsilon}}{r(x)^{2}\,\psi_{\varepsilon,\eta,r}(x)}\,d\gamma\,.$

Step 2. We prove that
$\eta^{2}\int_{B_{\varepsilon}}{r(x)^{2}\,\psi_{\varepsilon,\eta,r}(x)}\,d\gamma$
is of order $o\big{(}\varepsilon^{4}\big{)}$ as $\eta\leq
t\,\varepsilon^{2}\to 0$ for a given $t>0$.

$\bullet$ By the logarithmic Sobolev inequality,

$\int_{{\mathbb{R}}^{n}}{h^{2}\,\log h^{2}}\,d\gamma\leq
2\int_{{\mathbb{R}}^{n}}{|\nabla
h|^{2}}\,d\gamma+\int_{{\mathbb{R}}^{n}}{h^{2}}\,d\gamma\,\log\left(\int_{{\mathbb{R}}^{n}}{h^{2}}\,d\gamma\right)\,,$

applied to $h=1+(r-1)_{+}$, we learn that

$\int_{{\mathbb{R}}^{n}}{h^{2}\,\log h^{2}}\,d\gamma\leq
2\int_{{\mathbb{R}}^{n}}{|\nabla
r|^{2}}\,d\gamma+\left(1+\int_{{\mathbb{R}}^{n}}{r^{2}}\,d\gamma\right)\,\log\left(1+\int_{{\mathbb{R}}^{n}}{r^{2}}\,d\gamma\right)\leq
2\,\log(2\,e)\,.$

Let $\chi:=\mathbb{1}_{\\{\eta\,h>s_{0}\\}}$ for any $s_{0}>1$ and consider
$A_{\varepsilon,\eta,r,s_{0}}:=\left\\{x\in B_{\varepsilon}\,:\,\eta\,r(x)\leq
s_{0}\right\\}$. Then we have

$\int_{{\mathbb{R}}^{n}}{h^{2}\,\log
h^{2}}\,d\gamma\geq\int_{B_{\varepsilon}\setminus
A_{\varepsilon,\eta,r,s_{0}}}h^{2}\,\log
h^{2}\,d\gamma=\int_{B_{\varepsilon}}{\chi\,h^{2}\,\log
h^{2}}\,d\gamma\geq\log\left(\frac{s_{0}}{\eta}\right)^{2}\int_{B_{\varepsilon}}{\chi\,r^{2}}\,d\gamma$

because $h\geq 1$ a.e. and $h>s_{0}/\eta$ on $B_{\varepsilon}\setminus
A_{\varepsilon,\eta,r,s_{0}}$ and, as a consequence,

$\eta^{2}\int_{B_{\varepsilon}\setminus
A_{\varepsilon,\eta,r,s_{0}}}r(x)^{2}\,\psi_{\varepsilon,\eta,r}(x)\,d\gamma\geq-\,\frac{\log(2\,e)\,M}{\log
s_{0}+\log\left(\frac{1}{\eta}\right)}\,\eta^{2}\,.$

$\bullet$ Let us notice that

$\mathcal{M}=\sup_{s>0}\frac{|\rho(s)|}{\log(1+s)}$

is finite. This allows us to write that

$\eta^{2}\int_{A_{\varepsilon,\eta,r,s_{0}}}r(x)^{2}\,\psi_{\varepsilon,\eta,r}(x)\,d\gamma\geq-\,2^{2-p}\,\mathcal{M}\int_{A_{\varepsilon,\eta,r,s_{0}}}\eta^{2}\,r^{2}\,\log(1+2\,\eta\,r)\,\mathbb{1}_{\\{r>0\\}}\,d\gamma\,,$

where the restriction to the set $\\{r>0\\}$ comes from the fact that
$\psi_{\varepsilon,\eta,r}(x)=0$ whenever $r(x)\leq 0$. Now we estimate
$\log(1+2\,\eta\,r)$ by

$\displaystyle\log(1+2\,\eta\,r)\leq\log\left(1+2\,\sqrt{\eta}\right)\quad$
$\displaystyle\mbox{if}\quad 0\leq r\leq\frac{1}{\sqrt{\eta}}\,,$
$\displaystyle\log(1+2\,\eta\,r)\leq\frac{\log(1+2\,s_{0})}{\log\left(\frac{1}{\eta}\right)}\,\log
r^{2}\quad$ $\displaystyle\mbox{if}\quad\frac{1}{\sqrt{\eta}}\leq
r\leq\frac{s_{0}}{\eta}\,,$

and conclude using $\int_{{\mathbb{R}}^{n}}{r^{2}}\,d\gamma\leq 1$ and
$\int_{{\mathbb{R}}^{n}}{r^{2}\,\log r^{2}}\,d\gamma\leq
2\int_{{\mathbb{R}}^{n}}{|\nabla r|^{2}}\,d\gamma=2$ by the logarithmic
Sobolev inequality that

$\eta^{2}\int_{A_{\varepsilon,\eta,r,s_{0}}}r(x)^{2}\,\psi_{\varepsilon,\eta,r}(x)\,d\gamma\geq-\,2^{2-p}\,\mathcal{M}\,\eta^{2}\left(\theta\,\log\left(1+2\,\sqrt{\eta}\right)+\frac{2\,\log(1+2\,s_{0})}{\log\left(\frac{1}{\eta}\right)}\right)\,.$

Step 3. We compute the contribution of

$p\,\eta\int_{B_{\varepsilon}}{\left(1+\varepsilon\,x_{1}\right)^{p-1}\,r}\,d\gamma\quad\mbox{and}\quad\frac{1}{2}\,p\,(p-1)\,\eta^{2}\int_{B_{\varepsilon}}{\left(1+\varepsilon\,x_{1}\right)^{p-2}\,r^{2}}\,d\gamma$

to the expansion of $\int_{{\mathbb{R}}^{n}}{|u_{f}|^{p}}\,d\gamma$.

$\bullet$ Using (32) applied to $g=|r|^{1/p}$ and the orthogonality
constraints on $r$, we obtain

$p\,\eta\int_{B_{\varepsilon}}(1+\varepsilon\,x_{1})^{p-1}\,r\,d\gamma\geq
p\,\eta\left(\int_{{\mathbb{R}}^{n}}{(1+\varepsilon\,x_{1})^{p-1}\,r}\,d\gamma-2^{1-p}\,\xi_{p}\,\varepsilon^{5}\right)\\\
\geq
p\,\eta\,\varepsilon^{2}\left(\frac{1}{2}\,(p-1)\,(p-2)\int_{{\mathbb{R}}^{n}}{x_{1}^{2}\,r}\,d\gamma-\mathsf{c}_{1}\,\varepsilon^{3}-2^{1-p}\,\xi_{p}\,\varepsilon^{5}\right)$

where
$\mathsf{c}_{1}:=\sqrt{15}\,\sup_{s\in(-1,0)\cup(0,+\infty)}\left|(1+s)^{p-1}-1-(p-1)\,s-\frac{1}{2}\,(p-1)\,(p-2)\,s^{2}\right|/s^{3}$

$\bullet$ A similar computation shows that

$\frac{1}{2}\,p\,(p-1)\,\eta^{2}\int_{B_{\varepsilon}}{\left(1+\varepsilon\,x_{1}\right)^{p-2}\,r^{2}}\,d\gamma=\frac{1}{2}\,p\,(p-1)\,\eta^{2}\left(\|r\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}-\mathsf{c}_{2}\,\varepsilon\right)$

where
$\mathsf{c}_{2}:=\sup_{s\in(-1,0)\cup(0,+\infty)}\left|(1+s)^{p-2}-1\right|/s$

Step 4. Collecting all terms, we have

$\int_{{\mathbb{R}}^{n}}{|u_{f}|^{p}}\,d\gamma\geq
1+\frac{1}{2}\,p\,(p-1)\,\varepsilon^{2}+\frac{1}{8}\,p\,(p-1)\,(p-2)\,(p-3)\,\varepsilon^{4}+\frac{1}{2}\,p\,(p-1)\,\eta^{2}\,\|r\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}\\\
+\frac{1}{2}\,p\,(p-1)\,(p-2)\,\eta\,\varepsilon^{2}\,\int_{{\mathbb{R}}^{n}}{x_{1}^{2}\,r}\,d\gamma-\,C\,\frac{\varepsilon^{4}}{\log\left(\frac{1}{\varepsilon}\right)}$

for some constant $C>0$ that is explicitly given in terms of $t$, $\xi_{p}$,
$M$, $\mathcal{M}$, $\mathsf{c}_{1}$ and $\mathsf{c}_{2}$. In order to
conclude, we notice that for any $s>-1$,
$\frac{d^{4}}{ds^{4}}(1+s)^{\frac{2}{p}}>0$ implies that

$(1+s)^{\frac{2}{p}}\geq
1+\frac{2}{p}\,s+\frac{1}{p^{2}}\,(2-p)\,s^{2}-\frac{1}{3\,p^{3}}\,(2-p)\,(4-p)\,s^{3}\,.$

Applied to

$\|u_{f}\|_{\mathrm{L}^{p}({\mathbb{R}}^{n},d\gamma)}^{2}=\left(\int_{{\mathbb{R}}^{n}}{|u_{f}|^{p}}\,d\gamma\right)^{\frac{2}{p}}=\left(\int_{{\mathbb{R}}^{n}}{\left(1+\varepsilon\,x_{1}+\eta\,r\right)^{p}}\,d\gamma\right)^{\frac{2}{p}}\,,$

this completes the proof of Lemma 4.7.

######  Proof 4.9 (Proof of Theorem 1.4).

The strategy of proof is similar to [18, Theorem 7]. Up to the replacement of
$v$ by $f=|v|$, we can assume $f\geq 0$. With

$u_{f}(x):=\frac{f(x)}{\int_{{\mathbb{R}}^{n}}{f}\,d\gamma}=1+\varepsilon\,x_{1}+\eta\,r(x)$

as in Lemma 4.5, we first notice that the case $\eta>t\,\varepsilon^{2}$ is
already covered in Lemma 4.5. Otherwise let us assume that $\eta\leq
t\,\varepsilon^{2}$ and consider the Hermite polynomial
$h_{1}(x):=x_{1}^{2}-1$ and decompose $r$ according to

$r(x):=\alpha\,h_{1}(x)+\beta\,\tilde{r}(x)$

with $\|\tilde{r}\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}=1$ so that
$\|r\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}=\alpha^{2}+\beta^{2}\leq
1/2$ and $\int_{{\mathbb{R}}^{n}}{x_{1}^{2}\,r(x)}\,d\gamma=\alpha$. With this
notation we have

$\displaystyle\|\nabla
u_{f}\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}=\varepsilon^{2}+\eta^{2}\,,$
$\displaystyle\|u_{f}\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}=1+\varepsilon^{2}+\eta^{2}\left(\alpha^{2}+\beta^{2}\right)\,,$
$\displaystyle\|u_{f}\|_{\mathrm{L}^{p}({\mathbb{R}}^{n},d\gamma)}^{2}\geq
1+(p-1)\left(\varepsilon^{2}+\eta^{2}\left(\alpha^{2}+\beta^{2}\right)\right)+(p-1)\,(2-p)\left(\frac{1}{2}\,\varepsilon^{4}+\alpha\,\varepsilon^{2}\,\eta\right)-\frac{\mathcal{C}\,\varepsilon^{4}}{\log\left(\frac{1}{\varepsilon}\right)}\,,$

where the estimate of
$\|u_{f}\|_{\mathrm{L}^{p}({\mathbb{R}}^{n},d\gamma)}^{2}$ comes from Lemma
4.7. Hence, for some $\lambda>0$ to be fixed,

$\displaystyle\|\nabla
u_{f}\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}-\frac{1}{2-p}\left(\|u_{f}\|_{\mathrm{L}^{2}({\mathbb{R}}^{n},d\gamma)}^{2}-\|u_{f}\|_{\mathrm{L}^{p}({\mathbb{R}}^{n},d\gamma)}^{2}\right)-\lambda\left(\eta^{2}+\frac{t^{2}\,\varepsilon^{4}}{1+\varepsilon^{2}+\eta^{2}}\right)$
$\displaystyle\geq\eta^{2}\left(1-\alpha^{2}-\beta^{2}-\lambda\right)+\left(\frac{p-1}{2}-\frac{\lambda\,t^{2}}{1+\varepsilon^{2}+\eta^{2}}\right)\varepsilon^{4}+(p-1)\,\alpha\,\varepsilon^{2}\,\eta-\frac{1}{2-p}\,\frac{\mathcal{C}\,\varepsilon^{4}}{\log\left(\frac{1}{\varepsilon}\right)}$
$\displaystyle\geq\left(\frac{p-1}{2}-\frac{\lambda\,t^{2}}{1+\varepsilon^{2}+\eta^{2}}-\frac{\alpha^{2}\,(p-1)^{2}}{4\,\left(1-\alpha^{2}-\beta^{2}-2\lambda\right)}\right)\varepsilon^{4}-\frac{1}{2-p}\,\frac{\mathcal{C}\,\varepsilon^{4}}{\log\left(\frac{1}{\varepsilon}\right)}$

where we applied Young’s inequality in the last line. By an appropriate choice
of the parameters, for instance $\lambda\in(0,1/16)$ and $t>0$ such that
$t^{2}<8\,(p-1)\,(5-2\,p)/3$, we obtain a positive coefficient in front of
$\varepsilon^{4}$, and the last line is non-negative if $\varepsilon>0$ is
taken small enough. Hence, the result with
$c_{n,p}=\min\big{\\{}c_{n,p}^{(1)},c_{n,p}^{(2)}\big{\\}}$ with
$c_{n,p}^{(1)}$ given by (31) and $c_{n,p}^{(2)}:=(1-\theta)\,\lambda$.

## Acknowledgements

G.B. has been funded by the European Union’s Horizon 2020 research and
innovation program under the Marie Sklodowska-Curie grant agreement No.
754362. The authors thank M.J. Esteban for useful comments. The authors
warmfully thank a referee who replied very quickly with a detailed report and
a list of suggestions which greatly improved the revised version of this
paper.

© 2023 by the authors. This paper may be reproduced, in its entirety, for non-
commercial purposes.

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