# Hypercontractivity for Markov semi-groups

C. Roberto and B. Zegarlinski  Université Paris Nanterre, Modal’X, FP2M, CNRS
FR 2036, 200 avenue de la République 92000 Nanterre, France  Imperial College
of London, Faculty of Natural Sciences, Department of Mathematics, Huxley
Building, South Kensington Campus, London SW7 2AZ, UK<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We investigate in a systematic way hypercontractivity property in Orlicz
spaces for Markov semi-groups related to homogeneous and non homogeneous
diffusions in $\mathbb{R}^{n}$. We provide an explicit construction of a
family of Orlicz functions for which we prove that the associated
hypercontractivity property is equivalent to a suitable functional inequality.
Connections between our results and the Talagrand Conjecture on the
regularization effect of the Ornstein-Uhlenbeck semi-group in $\mathbb{L}_{1}$
are also discussed.

###### Key words and phrases:

Hypercontractivity, Ornstein-Uhlenbeck semigroup, Markov semi-groups,
$\Gamma_{2}$ calculus, Talagrand’s conjecture

Work supported by the Labex MME-DII funded by ANR, reference
ANR-11-LBX-0023-01 and ANR-15-CE40-0020-03 - LSD - Large Stochastic Dynamics,
and the grant of the Simone and Cino Del Duca Foundation. and the grant
Research Impulse Award DRI165BZ in the UK

##  1\. Introduction

The first aim for this paper is to give a unified setting for strong
contractivity properties of Markov semi-group to be satisfied with respect to
suitable family of Luxembourg norms in Orlicz spaces.

Initiated by Nelson in the late sixties [Nel66, Nel73a, Nel73b] in quantum
field theory, the notion of hypercontractivity of the Ornstein-Ulhenbeck
process was put in light by Gross’ seminal work [Gro75]. One of the main
observation of Gross is that hypercontractivity is equivalent to the so called
log-Sobolev inequality. See also [Fed69, Sta59] for earlier paper on related
topic.

More precisely, let $\gamma_{n}$ be the standard Gaussian measure on
$\mathbb{R}^{n}$. Then the Ornstein-Uhlenbeck semi-group $(P_{t})_{t\geq 0}$,
whose infinitesimal generator is $L:=\Delta-x\cdot\nabla$ (with the dot sign
standing for the Euclidean scalar product), is reversible with respect to
$\gamma_{n}$ and satisfies the following remarkable _hypercontractivity
property_ : for any $f\colon\mathbb{R}^{n}\to\mathbb{R}$ smooth enough it
holds

$\|P_{t}f\|_{q(t)}\leq\|P_{s}f\|_{q(s)},\qquad s\leq t$

where $q(t)=1+(q(0)-1)e^{2t}$, $q(0)\geq 1$, and
$\|g\|_{p}^{p}:=\int|g|^{p}d\gamma_{n}$, $p\geq 1$. Such a contraction
property is equivalent [Gro75] to the following log-Sobolev inequality: for
any $f\colon\mathbb{R}^{n}\to\mathbb{R}$ smooth enough, it holds

$\mathrm{Ent}_{\gamma_{n}}(f^{2}):=\int f^{2}\log f^{2}d\gamma_{n}-\int
f^{2}d\gamma_{n}\log\int f^{2}d\gamma_{n}\leq 2\int|\nabla f|^{2}d\gamma_{n}.$

Using Gross’s paper and $\Gamma^{2}$-calculus of Bakry-Emery [BE85, Bak94], it
can be immediately proved that also any semi-group associated to a diffusion
of the form $L:=\Delta-\nabla V\cdot\nabla$, with $V$ satisfying
$\mathrm{Hess}(V)\geq\rho>0$, as a matrix, enjoys the hypercontractivity
property as above with reference measure having density $e^{-V}$ with respect
to the Lebesgue measure and $q(t)=1+(q(0)-1)e^{(4/\rho)t}$, $q(0)\geq 1$.

From the seventies, both the hypercontractivity property and the log-Sobolev
inequality found a huge amount of applications in various field, including
Analysis (isoperimetry, concentration of measure phenomenon, convex geometry),
Statistical mechanics, Information Theory… Giving an exhaustive presentation
of the literature is out of reach. We refer to the textbooks [ABC+00, Led01,
GZ03, BGL14, Mas07, O’D14] for an introduction and references.

Now, let $\Phi\colon\mathbb{R}_{+}\to\mathbb{R}_{+}$ be a continuous convex
function satisfying $\Phi(x)=0$ iff $x=0$. Later on we may call such a
function a _Young function_ 111Note however that, usually, one does not
require in the definition of a Young function neither the regularity
assumption, nor the condition $\phi(x)=0$ iff $x=0$. Then, given
$f\colon\mathbb{R}^{n}\to\mathbb{R}$ such that $\int\Phi(\alpha
f)d\gamma_{n}<+\infty$ for some $\alpha>0$, one can define the so-called
Luxembourg norm associated to $\Phi$ and $\gamma_{n}$ as

$\|f\|_{\Phi}=\inf\left\\{\lambda>0:\int\Phi\left(\frac{|f|}{\lambda}\right)d\gamma_{n}\right\\}.$

The power function $\Phi(x)=|x|^{p}$, $p\geq 1$, trivially corresponds to the
usual $\mathbb{L}_{p}$-norm introduced above $\|f\|_{\Phi}=\|f\|_{p}$. The
space of all functions with finite Luxembourg norm will be denoted by
$\mathbb{L}_{\Phi}(\gamma_{n})$ (or simply $\mathbb{L}_{\Phi}$ when there is
no confusion, note however that norms are always computed with an underlying
measure).

With this definition at hand, for the family of Young functions
$\Phi_{t}(x)=|x|^{q(t)}$, $x\in\mathbb{R}$, $t\geq 0$ with
$q(t)=1+(q(0)-1)e^{2t}$, $q(0)\geq 1$, the hypercontractivity above can be
restated as follows

$\|P_{t}f\|_{\Phi_{t}}\leq\|P_{s}f\|_{\Phi_{s}},\qquad s\leq t.$

In other words, the Ornstein-Uhlenbeck semi-group is a contraction along the
family of Orlicz spaces $(\mathbb{L}_{\Phi_{t}})_{t\geq 0}$.

Following Gross’ ideas, in [BCR06] the authors proved that some contraction
property along a different type of family of Orlicz spaces could hold.
Consider the following infinitesimal generator in dimension $n$,
$L:=\Delta-\nabla V\cdot\nabla$, with $V(x)=\sum_{i=1}^{n}|x_{i}|^{\alpha}$,
$\alpha\in[1,2]$, $x\in\mathbb{R}$. (More precisely one should consider a
regularized version of $|x|^{\alpha}$ in a neighborood of the origin. For the
sake of simplicity we may avoid such technical considerations in this
introduction, that are irrelevant for our purpose, and we refer to [BCR06] for
details). Denote by $(P_{t})_{t\geq 0}$ the associated semi-group and by
$\mu(dx)=Z^{-1}e^{-V(x)}dx$, $x\in\mathbb{R}^{n}$, the associated reversible
probability measure, $Z:=\int e^{-V(x)}dx$ being the normalization constant.
Finally define $\Phi_{t}(x)=|x|^{p}e^{q(t)F(x)}$ with
$F(x):=\log(1+x)^{2(\alpha-1)/\alpha}-\log(2)^{2(\alpha-1)/\alpha}$, $q(t)=Ct$
for some constant $C>0$, and $p>1$. By construction
$\mathbb{L}_{\Phi_{t}}\subset\mathbb{L}_{\Phi_{s}}\subset\mathbb{L}_{p}$ for
any $s\leq t$ and $\mathbb{L}_{\Phi_{t}}\not\subset\mathbb{L}_{p+\varepsilon}$
for any $\varepsilon>0$, $t\geq 0$ and $\alpha\in[1,2)$ (since
$e^{q(t)F(x)}\ll|x|^{\varepsilon}$ near infinity, for any $\alpha\in[1,2)$).

In [BCR06] it is proved that $(P_{t})_{t\geq 0}$ is a contraction along the
family of Orlicz spaces $(\mathbb{L}_{\Phi_{t}})_{t\geq 0}$: namely that, for
any $s\leq t$, it holds $\|P_{t}f\|_{\Phi_{t}}\leq\|P_{s}f\|_{\Phi_{s}}$.
Moreover, such a contraction property is equivalent to the following, known as
$F$-Sobolev inequality: for any $f\colon\mathbb{R}^{n}\to\mathbb{R}$ it holds

$\int f^{2}F\left(\frac{f^{2}}{\int f^{2}d\gamma_{n}}\right)d\mu\leq
C^{\prime}\int|\nabla f|^{2}d\mu$

where $C^{\prime}$ is a constant that depends on $C$ and $\alpha$. Note that
$\alpha=2$ corresponds to the Gaussian case depicted above. Such inequalities
and contraction properties were used to establish dimension free isoperimetric
inequalities and concentration properties for $\mu$ [BCR06, BCR07].

Motivated by the previous two fundamental examples, the aim of this paper is
to investigate on contraction properties
$\|P_{t}f\|_{\Phi_{t}}\leq\|P_{s}f\|_{\Phi_{s}}$, $s\leq t$, along abstract
general family of Orlicz spaces $(\mathbb{L}_{\Phi_{t}})_{t\geq 0}$, together
with possible connection with functional inequalities of $F$-Sobolev type.

The second objective of the paper is to explore a more general setting which
would include inhomogeneous diffusion operators associated to one parameter
families of probability measures that we now introduce. Consider
$L_{t}:=\Delta-\nabla V_{t}\cdot\nabla$, $t\geq 0$, on $\mathbb{R}^{n}$, with
$V_{t}$ smooth enough and such that $\int e^{-V_{t}}=1$ so that
$\mu_{t}(dx)=e^{-V_{t}(x)}dx$ is a probability measure on $\mathbb{R}^{n}$ for
all $t\geq 0$. The associated semi-group will be denoted by
$(P^{(t)}_{s})_{s\geq 0}$ (we refer to e.g. [Bak94, GZ03] for its construction
and related technicalities) which is reversible in $\mathbb{L}_{2}(\mu_{t})$.
One wishes to obtain contraction bounds of the type
$\|P_{t}^{(t)}f\|_{\Phi_{t}}\leq m(t,s)\|P_{s}^{(s)}f\|_{\Phi_{s}}$, $s\leq t$
for some function $m$ possibly equal to 1.

Thus in the more general setting we not only change with time the Orlicz
functions, but also the underlying probability measures.

Besides interesting generalisation, we hope that our results can be used in
the future to study linear and nonlinear parabolic time dependent problems.

After Section 2, that collects some technical facts about Orlicz
functions/norms, we deal in Section 3 with the homogeneous setting.

Our first main theorem is Theorem 3.9 asserts that, for any properly chosen
family of Orlicz spaces (see Section 3.2), we have

$\|P_{t}f\|_{\Phi_{t}}\leq\|P_{s}f\|_{\Phi_{s}}$

if and only if some inequality of log-Sobolev type holds. Theorem 3.9
encompasses the above two known fundamental examples and can therefore be seen
as a generalization of Gross’ theorem. We also develop some analog of Høegh-
Krohn and Simon’s result (which asserts that if $P_{t}$ is continuous from
$\mathbb{L}_{2}$ into $\mathbb{L}_{4}$, then, under a Poincaré inequality,
$P_{t}$ is hypercontractive in the $\mathbb{L}_{p}$ scale (similarly to
Gross’s result)), dealing with continuity from $\mathbb{L}_{\Phi}$ into
$\mathbb{L}_{\Psi}$, for general Orlicz spaces. This will allow us to give
some connection with the regularization effect of the Ornstein-Uhlenbeck semi-
group in $\mathbb{L}_{1}$, known as ”Talagrand’s Conjecture”.

Section 4 is devoted to the time-inhomogeneous setting. Our second main result
is Theorem 4.1 which constitutes some analog of Gross’ theorem for
inhomogeneous Markov semi-groups.

Acknowledgment. We warmly thank T. Tao for pointing out to us Voigt’s paper
[Voi92].

##  2\. Technical preparations

In this section we collect some useful technical facts on various aspects of
Young functions and Luxembourg norms.

###  2.1. Youngs functions

An even continuous convex function $\Phi\colon\mathbb{R}\to\mathbb{R}_{+}$
satisfying $\Phi(x)=0$ iff $x=0$ is called a _Young function_. If in addition
$\lim_{x\to 0}\Phi(x)/x=0$, $\lim_{x\to\infty}\Phi(x)/x=+\infty$, and
$\Phi(\mathbb{R})\subset\mathbb{R}^{+}$, $\Phi$ is called a _nice Young
function_ , or _$N$ -function_ [RR91].

Classical examples include $\Phi(x)=|x|^{p}$, $p\geq 1$ which is a nice Young
function only for $p>1$; $\Phi(x)=e^{|x|}-|x|-1$,
$\Phi(x)=e^{|x|^{\delta}}-1$, $\delta>1$ are nice Young functions.

We say that $\Phi$ satisfies the $\Delta_{2}$-condition if for some $K>0$ and
all $x\geq 0$, it holds $\Phi(2x)\leq K\Phi(x)$. A useful consequence of the
$\Delta_{2}$-condition is the fact that $x\Phi^{\prime}(x)$ compares to
$\Phi$. More precisely,

(2.1) $\Phi(x)\leq x\Phi^{\prime}(x)\leq(K-1)\Phi(x).$

The first inequality follows from the convexity property of $\Phi$ and
$\Phi(0)=0$, while the second is a consequence of the $\Delta_{2}$-condition
and $\Phi(2x)-\Phi(x)=\int_{x}^{2x}\Phi^{\prime}(t)dt\geq\Phi^{\prime}(x)x$.

Given a Young function $\Phi$ and a probability measure $\mu$, for any
$f\colon\mathbb{R}^{n}\to\mathbb{R}$ we set

$\|f\|_{\Phi}:=\inf\left\\{\lambda>0:\int\Phi\left(\frac{|f|}{\lambda}\right)d\mu\right\\}\in[0,\infty]$

with the convention that $\inf\emptyset=+\infty$. When useful we may write
$\|f\|_{\Phi,\mu}$ to emphasize the underlying measure.

###  2.2. Derivative of Luxembourg norm

Here we give an explicit expression of the derivative with respect to time of
the following function $t\mapsto\|P_{t}f\|_{\Phi_{t}}$ which will constitute
the starting point of our investigations.

In the sequel we will use the following notations. Given a family of Young
functions $(\Phi_{t})_{t\geq 0}=(\Phi(t,x))_{t\geq 0}$, we denote by
$\dot{\Phi}_{t}$ the derivative with respect to $t$, and by
$\Phi_{t}^{\prime}$ and $\Phi_{t}^{\prime\prime}$ the first and second order
derivative with respect to the second variable $x$.

Consider the inhomogeneous diffusion generator $L_{t}:=\Delta-\nabla
V_{t}\cdot\nabla$, $t\geq 0$, on $\mathbb{R}^{n}$, with $V_{t}$ sufficiently
smooth and such that $\int e^{-V_{t}}=1$ so that $\mu_{t}(dx)=e^{-V_{t}(x)}dx$
is a probability measure on $\mathbb{R}^{n}$ for all $t\geq 0$. Denote by
$(P^{(t)}_{s})_{s\geq 0}$ the associated semi-group. By construction $L_{t}$
is symmetric in $\mathbb{L}_{2}(\mu_{t})$ and the following integration by
parts formula holds for any $\Psi\colon\mathbb{R}\to\mathbb{R}$, any
$f,g\colon\mathbb{R}^{n}\to\mathbb{R}$, smooth enough

(2.2) $\int\Psi(f)L_{t}gd\mu_{t}=-\int\Psi^{\prime}(f)\nabla f\cdot\nabla
gd\mu_{t}.$

We may prove the following differential property.

######  Lemma 2.1.

Let $f\colon\mathbb{R}^{n}\to\mathbb{R}$ a smooth function and
$(\Phi_{t})_{t\geq 0}$ be a family of $\mathcal{C}^{2}$ Young functions. Let
$N(t):=\|P_{t}^{(t)}f\|_{\Phi_{t}}$ and $g:=\frac{P_{t}^{(t)}f}{N(t)}$, $t\geq
0$. Then, it holds

$\displaystyle N^{\prime}(t)\\!\int\\!g\Phi_{t}^{\prime}(g)d\mu_{t}$
$\displaystyle=N(t)\left(\int\dot{\Phi}_{t}(g)d\mu_{t}-\int\\!\phi_{t}^{\prime\prime}(g)|\nabla
g|^{2}d\mu_{t}-\int\\!\Phi_{t}(g)\dot{V_{t}}d\mu_{t}\right)$
$\displaystyle\quad+\int\int_{0}^{t}P_{t-s}^{(t)}f\nabla
P_{s}^{(t)}(\Phi_{t}^{\prime}(g))\cdot\nabla\dot{V}_{t}dsd\mu_{t}$
$\displaystyle\quad-\int\left[\nabla
V_{t}\cdot\nabla\dot{V}_{t}-\Delta\dot{V}_{t}\right]\int_{0}^{t}P_{t-s}^{(t)}fP_{s}^{(t)}(\Phi_{t}^{\prime}(g))dsd\mu_{t}$

In particular, when $V_{t}$ does not depend on $t$ (homogeneous case), the
latter reduces to

$N^{\prime}(t)\int
g\Phi_{t}^{\prime}(g)d\mu=N(t)\left(\int\dot{\Phi}_{t}(g)d\mu-\int\phi_{t}^{\prime\prime}(g)|\nabla
g|^{2}d\mu\right).$

######  Remark 2.2.

We assumed $\mathcal{C}^{2}$ for Young functions for simplicity. Most of the
results in this paper can easily be understood for any Young function using
the notion of second order derivative in the sense of Aleksandrov.

###### Proof.

From the very definition of the Luxembourg norm, we observe that for any
$t\geq 0$, $\int\Phi_{t}\left(\frac{P_{t}^{(t)}f}{N(t)}\right)d\mu_{t}=1$.
Therefore, taking the derivative, we get

$\displaystyle\int\dot{\Phi}_{t}(g)d\mu_{t}+\int\Phi_{t}^{\prime}(g)\frac{d}{dt}\left(\frac{P_{t}^{(t)}f}{N(t)}\right)d\mu_{t}-\int\Phi_{t}(g)\dot{V_{t}}d\mu_{t}=0$

where as already mentioned the dot stands for the derivative with respect to
the variable $t$. Observe that,

$\frac{d}{dt}\left(\frac{P_{t}^{(t)}f}{N(t)}\right)=\frac{\dot{P}_{t}^{(t)}f}{N(t)}+\frac{L_{t}P_{t}^{(t)}f}{N(t)}-\frac{P_{t}^{(t)}fN^{\prime}(t)}{N(t)^{2}}=\frac{\dot{P}_{t}^{(t)}f}{N(t)}+L_{t}g-\frac{N^{\prime}(t)}{N(t)}g$

where we set

$\displaystyle\dot{P}_{t}^{(t)}f$ $\displaystyle:=\lim_{\varepsilon\to
0}\frac{P_{t}^{(t+\varepsilon)}f-P_{t}^{(t)}f}{\varepsilon}=\lim_{\varepsilon\to
0}\frac{1}{\varepsilon}\int_{0}^{t}\frac{d}{ds}\left(P_{s}^{(t+\varepsilon)}(P_{t-s}^{(t)}f)\right)ds$
$\displaystyle=\lim_{\varepsilon\to
0}\frac{1}{\varepsilon}\int_{0}^{t}P_{s}^{(t+\varepsilon)}\left([L_{t+\varepsilon}-L_{t}]P_{t-s}^{(t)}f\right)ds$
$\displaystyle=\int_{0}^{t}P_{s}^{(t)}\left(-\nabla\dot{V}_{t}\cdot\nabla
P_{t-s}^{(t)}f\right)ds.$

Therefore, using (2.2), we get

$\displaystyle\int\Phi_{t}^{\prime}(g)\frac{d}{dt}\left(\frac{P_{t}^{(t)}f}{N(t)}\right)d\mu_{t}$
$\displaystyle=-\frac{1}{N(t)}\int\int_{0}^{t}\Phi_{t}^{\prime}(g)P_{s}^{(t)}\left(\nabla\dot{V}_{t}\cdot\nabla
P_{t-s}^{(t)}f\right)dsd\mu_{t}$
$\displaystyle\quad-\int\phi_{t}^{\prime\prime}(g)|\nabla
g|^{2}d\mu_{t}-\frac{N^{\prime}(t)}{N(t)}\int g\Phi_{t}^{\prime}(g)d\mu_{t}.$

The previous computations lead to

$\displaystyle\frac{N^{\prime}(t)}{N(t)}\int g\Phi_{t}^{\prime}(g)d\mu_{t}$
$\displaystyle=\int\dot{\Phi}_{t}(g)d\mu_{t}-\int\phi_{t}^{\prime\prime}(g)|\nabla
g|^{2}d\mu_{t}-\int\Phi_{t}(g)\dot{V_{t}}d\mu_{t}$
$\displaystyle\quad-\frac{1}{N(t)}\int\int_{0}^{t}\Phi_{t}^{\prime}(g)P_{s}^{(t)}\left(\nabla\dot{V}_{t}\cdot\nabla
P_{t-s}^{(t)}f\right)dsd\mu_{t}$

and we are left with the study of the last term on the right hand side of the
latter. By reversibility of the semi-group, we have

$\displaystyle\int\Phi_{t}^{\prime}(g)P_{s}^{(t)}\left(\nabla\dot{V}_{t}\cdot\nabla
P_{t-s}^{(t)}f\right)d\mu_{t}$ $\displaystyle=\int
P_{s}^{(t)}(\Phi_{t}^{\prime}(g))\nabla\dot{V}_{t}\cdot\nabla
P_{t-s}^{(t)}fd\mu_{t}$ $\displaystyle=\int
P_{t-s}^{(t)}f\nabla^{*}\left(P_{s}^{(t)}(\Phi_{t}^{\prime}(g))\nabla\dot{V}_{t}\right)d\mu_{t}$

where $\nabla^{*}$ is the adjoint of $\nabla$ in $\mathbb{L}_{2}(\mu_{t})$,
namely such that $\int u\nabla vd\mu_{t}=\int v\nabla^{*}ud\mu_{t}$. One can
see that $\nabla^{*}=-\nabla+\nabla V_{t}$ where $\nabla V_{t}$ acts
multiplicatively. Therefore,

$\displaystyle\int\Phi_{t}^{\prime}(g)P_{s}^{(t)}\left(\nabla\dot{V}_{t}\cdot\nabla
P_{t-s}^{(t)}f\right)d\mu_{t}$ $\displaystyle=-\int P_{t-s}^{(t)}f\nabla
P_{s}^{(t)}(\Phi_{t}^{\prime}(g))\cdot\nabla\dot{V}_{t}d\mu_{t}$
$\displaystyle\quad+\int
P_{t-s}^{(t)}fP_{s}^{(t)}(\Phi_{t}^{\prime}(g))\left[\nabla
V_{t}\cdot\nabla\dot{V}_{t}-\Delta\dot{V}_{t}\right]d\mu_{t}$

The expected result follows. ∎

###  2.3. Expansion of the square

Here we may recall the _expansion of the square_ method or, as it is called in
[HZ10], $U$-bounds. That is the bounds obtained by using Leibnitz rule
together with integration by parts as follows. Let
$U\colon\mathbb{R}^{n}\to\mathbb{R}$ be such that $\int e^{-U}dx<\infty$.
Then, for any differentiable function $f\colon\mathbb{R}^{n}\to\mathbb{R}$,
one has

(2.3) $\int f^{2}(|\nabla U|^{2}-2\Delta U)e^{-U}dx\leq 4\int|\nabla
f|^{2}e^{-U}dx.$

In fact, expanding the square, one has

$\displaystyle 0\leq\int|\nabla(fe^{-U/2})|^{2}dx$ $\displaystyle=\int|\nabla
f|^{2}e^{-U}dx-\int f\nabla f\cdot\nabla Ue^{-U}dx+\frac{1}{4}\int
f^{2}|\nabla U|^{2}e^{-U}dx.$

The expected inequality (2.3) then follows by applying an integration by parts
on the cross term.

The _expansion of the square_ method revealed to be very powerful. It can be
used for instance to prove Hardy’s inequality with optimal constant on
$\mathbb{R}^{d}$, $d\geq 3$, or Poincaré inequality for the Gaussian measure.
We refer the interested reader to [HZ10, DV12] for more results and
references.

##  3\. Hypercontractivity for homogeneous Markov semi-groups

In this section our aim is to introduce the notion of _standard Orlicz family_
that will play a key role for proving the equivalence between some functional
inequality and an hypercontractivity property along the corresponding family
of Orlicz spaces. We need first to analyze how to get an hypercontractivity
property along a general family of Orlicz spaces.

All along the section we set $L=\Delta-\nabla V\cdot\nabla$, with $V$ smooth
enough and such that $\mu(dx)=e^{-V}dx$ is a probability measure on
$\mathbb{R}^{n}$. We denote by $(P_{t})_{t\geq 0}$ the associated semi-group
which is reversible with respect to $\mu$. Orlicz spaces and their
corresponding Luxembourg norms are understood with respect to $\mu$.

###  3.1. Hypercontractivity along Orlicz spaces

Using Lemma 2.1 we first prove that hypercontractivity is a direct and
immediate consequence of some family of functional inequalities. Our second
result shows how that family can, under some assumptions, be reduced to one
single functional inequality of log-Sobolev-type.

######  Proposition 3.1.

Let $(\Phi_{t})_{t\geq 0}$ be a family of ${\mathcal{C}}^{2}$ Young functions.
Assume that for any $t\geq 0$, any sufficiently smooth function $f$, we have

(3.1)
$\|f\|_{\Phi_{t}}^{2}\int\dot{\Phi}_{t}\left(\frac{f}{\|f\|_{\Phi_{t}}}\right)d\mu\leq\int\Phi_{t}^{\prime\prime}\left(\frac{f}{\|f\|_{\Phi_{t}}}\right)|\nabla
f|^{2}d\mu.$

Then, for any $t\geq s$,

$\|P_{t}f\|_{\Phi_{t}}\leq\|P_{s}f\|_{\Phi_{s}}.$

###### Proof.

We need to prove is that $N:t\mapsto\|P_{t}f\|_{\Phi_{t}}$ is non-increasing.
Lemma 2.1 asserts that

$\displaystyle\frac{N^{\prime}(t)}{N(t)}\int
g\Phi_{t}^{\prime}\left(g\right)d\mu=\int\dot{\Phi}_{t}\left(g\right)d\mu-\int\Phi_{t}^{\prime\prime}\left(g\right)|\nabla
g|^{2}d\mu$

where $g:=\frac{P_{t}f}{N(t)}$. Since for any $t\geq 0$, $\Phi_{t}$ is a Young
function, it satisfies $x\Phi_{t}^{\prime}(x)\geq 0$ for any $x\in\mathbb{R}$.
It follows by (3.1) that $N^{\prime}(t)\leq 0$ which is the expected result. ∎

Using an isometry between $\mathbb{L}_{\Phi_{t}}$ and $\mathbb{L}_{\Phi_{s}}$,
we may translate (3.1) for $\Phi_{s}$ into a similar inequality for
$\Phi_{t}$, therefore reducing the family of inequalities (3.1) possibly to a
single one.

######  Proposition 3.2.

Let $(\Phi_{t})_{t\geq 0}$ be a family of ${\mathcal{C}}^{2}$ Young functions.
Assume that for some $t,s\geq 0$ there exist two positive constants $C(t,s)$
and $\widetilde{C}(t,s)$ such that
$(i)$

$\dot{\Phi}_{t}(\Phi_{t}^{-1})\leq C(t,s)\dot{\Phi}_{s}(\Phi_{s}^{-1}),$

$(ii)$

$\frac{\Phi_{t}^{\prime\prime}}{{\Phi_{t}^{\prime}}^{2}}\circ\Phi_{t}^{-1}\geq\widetilde{C}(t,s)\frac{\Phi_{s}^{\prime\prime}}{{\Phi_{s}^{\prime}}^{2}}\circ\Phi_{s}^{-1}.$

Assume furthermore that for some constant $c>0$ and for any $f$ (smooth
enough), it holds

(3.2)
$\|f\|_{\Phi_{s}}^{2}\int\dot{\Phi}_{s}\left(\frac{f}{\|f\|_{\Phi_{s}}}\right)d\mu\leq
c\int\Phi_{s}^{\prime\prime}\left(\frac{f}{\|f\|_{\Phi_{s}}}\right)|\nabla
f|^{2}d\mu.$

Then, for any $f$ smooth enough it holds

$\|f\|_{\Phi_{t}}^{2}\int\dot{\Phi}_{t}\left(\frac{f}{\|f\|_{\Phi_{t}}}\right)d\mu\leq
c\frac{C(t,s)}{\widetilde{C}(t,s)}\int\Phi_{t}^{\prime\prime}\left(\frac{f}{\|f\|_{\Phi_{t}}}\right)|\nabla
f|^{2}d\mu.$

###### Proof.

Let

$\begin{array}[]{rcl}I_{s,t}:\mathbb{L}_{\Phi_{t}}&\rightarrow&\mathbb{L}_{\Phi_{s}}\\\
f&\mapsto&\|f\|_{\Phi_{t}}\Phi_{s}^{-1}\circ\Phi_{t}\left(\frac{f}{\|f\|_{\Phi_{t}}}\right).\end{array}$

For any $f\in\mathbb{L}_{\Phi_{t}}$, by the very definition of the Luxembourg
norm, it holds $\|I_{s,t}(f)\|_{\Phi_{s}}=\|f\|_{\Phi_{t}}$. Therefore,
$I_{s,t}(f)$ is an isometry between the two Orlicz spaces
$\mathbb{L}_{\Phi_{t}}$ and $\mathbb{L}_{\Phi_{s}}$. Applying (3.2) to
$I_{s,t}(f)$ leads to

$\displaystyle\|f\|_{\Phi_{t}}^{2}\int\dot{\Phi}_{s}\left(\Phi_{s}^{-1}\circ\Phi_{t}\Big{(}\frac{f}{\|f\|_{\Phi_{t}}}\Big{)}\right)d\mu\leq$
$\displaystyle\qquad\qquad\qquad\qquad
c\int\Phi_{s}^{\prime\prime}\circ\Phi_{s}^{-1}\circ\Phi_{t}\left(\frac{f}{\|f\|_{\Phi_{t}}}\right)\frac{\Phi_{t}^{\prime}\left(\frac{f}{\|f\|_{\Phi_{t}}}\right)^{2}|\nabla
f|^{2}}{\Phi_{s}^{\prime}\circ\Phi_{s}^{-1}\circ\Phi_{t}\left(\frac{f}{\|f\|_{\Phi_{t}}}\right)^{2}}d\mu.$

The result follows by $(i)$ and $(ii)$. ∎

The simplest example is given by the $\mathbb{L}_{p}$ scale
$\Phi_{t}(x)=|x|^{q(t)}$ for some function $q$ that we assume to be
increasing. Then, it holds

$\dot{\Phi}_{t}(\Phi_{t}^{-1})=\frac{q^{\prime}(t)}{q(t)}x\log x\qquad\mbox{
and
}\qquad\frac{\Phi_{t}^{\prime\prime}}{{\Phi_{t}^{\prime}}^{2}}\circ\Phi_{t}^{-1}=\frac{q(t)-1}{q(t)}\frac{1}{x}.$

Therefore assumptions $(i)$ and $(ii)$ hold with
$C(t,s)=\frac{q^{\prime}(t)q(s)}{q(t)q^{\prime}(s)}$ and
$\widetilde{C}(t,s)=\frac{(q(t)-1)q(s)}{q(t)(q(s)-1)}$. In particular, the
choice $q(t)=1+e^{(4/\rho)t}$, $\rho>0$, guarantees that
$C(t,s)=\widetilde{C}(t,s)$ for all $s,t$. Hence, the family of inequalities
(3.2) are all equivalent to (3.2) with $s=0$, which reads

$\mathrm{Ent}_{\mu}(f^{2})\leq\rho\int|\nabla f|^{2}d\mu$

since $\Phi_{0}=|x|^{2}$ (ant therefore $\dot{\Phi}_{0}(x)=(2/\rho)x^{2}\log
x^{2}$ and $\Phi_{0}^{\prime\prime}(x)=2$). This is the log-Sobolev inequality
and therefore Proposition 3.3 is just one direction in Gross’ theorem [Gro75].

In the above example, both $\dot{\Phi}_{t}(\Phi_{t}^{-1})$ and
$\frac{\Phi_{t}^{\prime\prime}}{{\Phi_{t}^{\prime}}^{2}}\circ\Phi_{t}^{-1}$
are of the form $a(t)b(x)$. Based on this simple observation, we may construct
a generic family of Orlicz functions that, by construction, will automatically
satisfies assumptions $(i)$ and $(ii)$ of the latter. This is the object of
the next section.

###  3.2. The standard Orlicz family

We define a large class of family of $N$-functions that we will call the
_standard Orlicz family_.

######  Definition 3.3 (standard Orlicz family).

Let $F:(0,\infty)\to\mathbb{R}$ be a $\mathcal{C}^{2}$ increasing function
with $F(1)=0$. Assume that $(0,\infty)\ni x\mapsto xF(x)$ is convex and that
$1/xF(x)$ is not integrable at $x=0$, $x=1$ and $x=+\infty$. Let
$\mathcal{F}_{1}:(0,1)\to\mathbb{R}$ and
$\mathcal{F}_{2}:(1,+\infty)\to\mathbb{R}$ be two primitives of $x\mapsto
1/(xF(x))$.

Let $\Phi_{0}$ be a nice Young function and $x_{o}$ the unique positive point
such that $\Phi_{0}(x_{o})=1$. We assume that
$-\left(\frac{\Phi_{0}}{\Phi_{0}^{\prime}}\right)^{\prime}F(\Phi_{0})-\Phi_{0}F^{\prime}(\Phi_{0})$
is non-increasing on $\mathbb{R}^{+}$ and that $\Phi_{0}$ is of class
$\mathcal{C}^{2}$ on $(0,\infty)$.

Given an increasing function $\lambda\colon[0,\infty)\to[0,\infty)$, with
$\lambda(0)=0$, we define the family of functions $(\Phi_{t})_{t\geq 0}$ by

$\Phi_{t}(x)=\begin{cases}0&\mbox{ for }x=0\\\
\mathcal{F}_{1}^{-1}\left(\mathcal{F}_{1}(\Phi_{0}(x))+\lambda(t)\right)&\mbox{
for }x\in(0,x_{o})\\\ 1&\mbox{ for }x=x_{o}\\\
\mathcal{F}_{2}^{-1}\left(\mathcal{F}_{2}(\Phi_{0}(x))+\lambda(t)\right)&\mbox{
for }x\in(x_{o},+\infty).\end{cases}\qquad\forall t>0.$

We shall call the family $(\Phi_{t})_{t\geq 0}$ the _standard Orlicz family
built from $F$, $\Phi_{0}$ and $\lambda$_.

######  Remark 3.4.

The Lemma below will prove that all $\Phi_{t}$ are indeed Young functions and
in fact nice Young functions. This justifies the terminology ”Orlicz family”.
Also, it is not difficult to check that the definition above does not depend
on the choice of the primitives: any two different primitives lead to the same
final function $\Phi_{t}$.

######  Example 3.5.

As an example consider $F(x)=\log(x)$ and any nice Young function $\Phi_{0}$.
Then, $\mathcal{F}_{1}(x)=\log(\log(1/x))$, $x\in(0,1)$ and
$\mathcal{F}_{2}(x)=\log(\log(x))$, $x>1$ so that
$\mathcal{F}_{1}^{-1}(x)=e^{-e^{x}}$ and $\mathcal{F}_{2}^{-1}(x)=e^{e^{x}}$,
$x\in\mathbb{R}$. Hence, $\Phi_{t}(x)=\Phi_{0}^{e^{\lambda(t)}}$. This
corresponds to an $\mathbb{L}_{p}$ scale when $\Phi_{0}(x)=|x|^{q}$ for some
$q>1$. More specifically, if $q(t)=1+e^{(4/\rho)t}$ and
$\lambda(t)=\log(q(t)/2)$, with $\Phi_{0}(x)=x^{2}$, we have
$\Phi_{t}(x)=|x|^{q(t)}$ and we are back to Gross’ setting.

The more general choices $F(x)=\log(1+x)^{\beta}-\log(2)^{\beta}$,
$\beta\in(0,1)$, can also be considered, but lead to non explicit
$\mathcal{F}_{1}$ and $\mathcal{F}_{2}$. Although one can easily give an
asymptotic of the corresponding $\phi_{t}(x)$, when $x$ tends to $0$ or
$+\infty$. For instance, $\Phi_{t}$ is equivalent to
$\Phi_{0}e^{a_{\beta}\lambda(\log\phi_{0})^{\beta}}$ when $x$ tends to
infinity, where $a_{\beta}$ is a numerical constant that depends only on
$\beta$. This amounts to the family of Young functions $x^{2}e^{ctF(x)}$
considered in [BCR06, Section 7].

In the next lemma we collect some property of the standard Orlicz families.

######  Lemma 3.6.

Let $F$, $\Phi_{0}$ and $\lambda$ satisfying the assumptions of Definition 3.3
and let $(\Phi_{t})_{t\geq 0}$ be the standard Orlicz family built from $F$,
$\Phi_{0}$ and $\lambda$. Then,
$(i)$ $\mathcal{F}_{1}$ and $\mathcal{F}_{2}$ are $\mathcal{C}^{2}$ functions
respectively on $(0,1)$ and $(1,+\infty)$. $\mathcal{F}_{1}$ is decreasing
with $\lim_{x\to 0}\mathcal{F}_{1}(x)=-\lim_{x\to
1}\mathcal{F}_{1}(x)=+\infty$. While $\mathcal{F}_{2}$ is increasing with
$\lim_{x\to
1}\mathcal{F}_{2}(x)=-\lim_{x\to+\infty}\mathcal{F}_{2}(x)=-\infty$. In
particular $\Phi_{t}$ is well defined and continuous. Moreover, for $t\geq s$,
$\Phi_{t}\leq\Phi_{s}$ on $(0,x_{0})$ and $\Phi_{t}\geq\Phi_{s}$ on
$(x_{0},+\infty)$.
$(ii)$ For any $t\geq 0$, $\Phi_{t}$ is a nice Young function of class
$\mathcal{C}^{2}$ on $(0,\infty)$ (with
$\phi_{t}^{\prime}(x_{o})=\Phi_{0}^{\prime}(x_{o})$ and
$\Phi_{t}^{\prime\prime}(x_{o})=\Phi_{0}^{\prime\prime}(x_{o})$).
$(iii)$ For any $t\geq 0$,
$\dot{\Phi}_{t}\circ\Phi_{t}^{-1}=\lambda^{\prime}(t)xF(x)$.
$(iv)$ For any $t\geq s\geq 0$.
$\frac{\Phi_{t}^{\prime\prime}}{{\Phi_{t}^{\prime}}^{2}}\circ\Phi_{t}^{-1}\geq\frac{\Phi_{s}^{\prime\prime}}{{\Phi_{s}^{\prime}}^{2}}\circ\Phi_{s}^{-1}$.
$(v)$ Assume that $\lambda$ tends to infinity at infinity. Then for any
$f\in\mathbb{L}_{\infty}$,
$\lim_{t\to+\infty}\|f\|_{\Phi_{t}}=\frac{1}{x_{0}}\|f\|_{\infty}$.

###### Proof.

Points $(i)$ and $(iii)$ are simple consequences of the definitions of the
object involved.
It is not difficult but tedious to prove that for all $t\geq 0$, $\Phi_{t}$ is
$\mathcal{C}^{2}$ (we omit the proof). Using that $\Phi_{t}\leq\Phi_{0}$ on
$(0,x_{o})$ and since $\Phi_{0}$ is a nice Young function we deduce that
$\lim_{x\to 0}\frac{\Phi_{t}(x)}{x}=0$. Similarly,
$\lim_{x\to\infty}\frac{\Phi_{t}(x)}{x}=+\infty$. In order to prove that
$\Phi_{t}$ is a nice Young function it therefore remains to show that
$\Phi_{t}$ is convex. For $x\neq x_{0}$, a simple differentiation gives

(3.3) $\displaystyle\frac{\Phi_{t}^{\prime\prime}}{{\Phi_{t}^{\prime}}^{2}}$
$\displaystyle=\mathcal{F}^{\prime}(\Phi_{t})\left(-\frac{\mathcal{F}^{\prime\prime}(\Phi_{t})}{\mathcal{F}^{\prime}(\Phi_{t})^{2}}+\frac{\mathcal{F}^{\prime\prime}(\Phi_{0})}{\mathcal{F}^{\prime}(\Phi_{0})^{2}}+\frac{\Phi_{0}^{\prime\prime}}{\mathcal{F}^{\prime}(\Phi_{0}){\Phi_{0}^{\prime}}^{2}}\right)$
$\displaystyle=\mathcal{F}^{\prime}(\Phi_{t})\left(\left(\frac{1}{\mathcal{F}^{\prime}}\right)^{\prime}(\Phi_{t})-\left(\frac{1}{\mathcal{F}^{\prime}}\right)^{\prime}(\Phi_{0})+\frac{\Phi_{0}^{\prime\prime}}{\mathcal{F}^{\prime}(\Phi_{0}){\Phi_{0}^{\prime}}^{2}}\right)$

where $\mathcal{F}=\mathcal{F}_{1}$ when $x\in(0,x_{0})$ and
$\mathcal{F}=\mathcal{F}_{2}$ when $x>x_{0}$. A Taylor expansion of
$\left(\frac{1}{\mathcal{F}^{\prime}}\right)^{\prime}$ at the first order
insures that

$\frac{\Phi_{t}^{\prime\prime}}{{\Phi_{t}^{\prime}}^{2}}=\mathcal{F}^{\prime}(\Phi_{t})\left((\Phi_{t}-\Phi_{0})\left(\frac{1}{\mathcal{F}^{\prime}}\right)^{\prime\prime}(\theta)+\frac{\Phi_{0}^{\prime\prime}}{\mathcal{F}^{\prime}(\Phi_{0}){\Phi_{0}^{\prime}}^{2}}\right)$

for some $\theta\in(\Phi_{t},\Phi_{0})$ when $x\in(0,x_{0})$ and
$\theta\in(\Phi_{0},\Phi_{t})$ when $x>x_{0}$. Since $x\mapsto xF(x)$ is
convex, $\frac{1}{\mathcal{F}^{\prime}}$ is convex. It follows that
$\left(\frac{1}{\mathcal{F}^{\prime}}\right)^{\prime\prime}(\theta)\geq 0$ and
thus that
$(\Phi_{t}-\Phi_{0})\left(\frac{1}{F}^{\prime}\right)^{\prime\prime}(\theta)+\frac{\Phi_{0}^{\prime\prime}}{\mathcal{F}^{\prime}(\Phi_{0}){\Phi_{0}^{\prime}}^{2}}$
has the same sign as $\mathcal{F}^{\prime}(\Phi_{t})$. This proves that
$\Phi_{t}$ is convex.

Next, we deal with Point $(iv)$. From (3.3) we have with the same notation as
before

$\frac{\Phi_{t}^{\prime\prime}}{{\Phi_{t}^{\prime}}^{2}}\circ\Phi_{t}^{-1}(x)=-\frac{\mathcal{F}^{\prime\prime}(x)}{\mathcal{F}^{\prime}(x)}+\mathcal{F}^{\prime}(x)\left(\frac{\mathcal{F}^{\prime\prime}(\Phi_{0})}{\mathcal{F}^{\prime}(\Phi_{0})^{2}}+\frac{\Phi_{0}^{\prime\prime}}{\mathcal{F}^{\prime}(\Phi_{0}){\Phi_{0}^{\prime}}^{2}}\right)\circ\Phi_{t}^{-1}(x).$

Note that by hypothesis,

$\frac{\mathcal{F}^{\prime\prime}(\Phi_{0})}{\mathcal{F}^{\prime}(\Phi_{0})^{2}}+\frac{\Phi_{0}^{\prime\prime}}{\mathcal{F}^{\prime}(\Phi_{0}){\Phi_{0}^{\prime}}^{2}}=-\left(\frac{\Phi_{0}}{\Phi_{0}^{\prime}}\right)^{\prime}F(\Phi_{0})-\Phi_{0}F^{\prime}(\Phi_{0})$

is non-increasing. Thus, by Point $(ii)$ and using the sign of
$\mathcal{F}^{\prime}(x)$ on each domain $(0,x_{0})$ and $(x_{0},+\infty)$, we
have

$\displaystyle\frac{\Phi_{t}^{\prime\prime}}{{\Phi_{t}^{\prime}}^{2}}\circ\Phi_{t}^{-1}(x)$
$\displaystyle\geq$
$\displaystyle-\frac{\mathcal{F}^{\prime\prime}(x)}{\mathcal{F}^{\prime}(x)}+\mathcal{F}^{\prime}(x)\left(\frac{\mathcal{F}^{\prime\prime}(\Phi_{0})}{\mathcal{F}^{\prime}(\Phi_{0})^{2}}+\frac{\Phi_{0}^{\prime\prime}}{\mathcal{F}^{\prime}(\Phi_{0}){\Phi_{0}^{\prime}}^{2}}\right)\circ\Phi_{s}^{-1}(x)$
$\displaystyle=$
$\displaystyle\frac{\Phi_{s}^{\prime\prime}}{{\Phi_{s}^{\prime}}^{2}}\circ\Phi_{s}^{-1}(x)$

which is the expected result.

Finally we will prove Point $(v)$. Let $f\in\mathbb{L}_{\infty}$. Then,
$\int\Phi_{t}\Big{(}\frac{x_{0}|f|}{\|f\|_{\infty}}\Big{)}d\mu\leq\Phi_{t}(x_{0})=1$.
Hence, by definition of the norm,
$\|f\|_{\Phi_{t}}\leq\frac{1}{x_{0}}\|f\|_{\infty}$. In order to prove the
bound from below, fix $\varepsilon>0$ small enough. Then note that for any
$x>x_{0}$, $\lim_{t\to+\infty}\Phi_{t}(x)=+\infty$. Thus

$\displaystyle\int\Phi_{t}\left(\frac{|f|x_{0}}{\|f\|_{\infty}(1-\varepsilon)}\right)d\mu$
$\displaystyle\geq\int_{\\{|f|\geq\|f\|_{\infty}(1-\frac{\varepsilon}{2})\\}}\Phi_{t}\left(\frac{|f|x_{0}}{\|f\|_{\infty}(1-\varepsilon)}\right)d\mu$
$\displaystyle\geq\Phi_{t}\left(x_{0}\Big{(}1+\frac{\varepsilon}{2(1-\varepsilon)}\Big{)}\right)\mu\left(\\{|f|\geq\|f\|_{\infty}(1-\frac{\varepsilon}{2})\\}\right)\geq
1$

provided $t$ is large enough. It follows that
$\frac{1}{x_{0}}\|f\|_{\infty}(1-\varepsilon)\leq\|f\|_{\Phi_{t}}$ for $t$
large enough. This leads to the expected result and achieves the proof of the
lemma. ∎

######  Remark 3.7.

When $\lambda(t)=\alpha t$ for some $\alpha>0$, the standard Orlicz family
enjoy a shift type property. Indeed, in that case
$\Phi_{t}=\mathcal{F}_{i}^{-1}(\mathcal{F}_{i}(\Phi_{s})+\lambda(t-s))$,
$i=1,2$. Therefore, the standard Orlicz families $(\Phi_{t})_{t\geq 0}$ built
from $\Phi_{0}$, $F$ and $\lambda$ and $(\Psi_{t})_{t\geq 0}$ built from
$\Phi_{s}$, $F$ and $\lambda$, satisfy $\Psi_{t}=\Phi_{t+s}$ for any $t\geq
0$.

######  Remark 3.8.

When $\Phi_{0}(x)=x^{2}$,

$-\left(\frac{\Phi_{0}}{\Phi_{0}^{\prime}}\right)^{\prime}F(\Phi_{0})-\Phi_{0}F^{\prime}(\Phi_{0})=-\frac{1}{2}F(x^{2})-x^{2}F^{\prime}(x^{2}).$

Thus, this function is non-increasing if and only if
$\frac{3}{2}F^{\prime}(x)+xF^{\prime\prime}(x)\geq 0$ if and only if $x\mapsto
xF(x^{2})$ is convex. Thus, in that case, one can only assume that $x\mapsto
xF(x^{2})$ is convex (which implies that $x\mapsto xF(x)$ is convex).

###  3.3. Gross-Orlicz’ theorem

Thanks to the above definition of the standard Orlicz family, we can state one
of our main results which generalizes Gross’s theorem.

######  Theorem 3.9 (Gross-Orlicz).

Let $(\Phi_{t})_{t\geq 0}$ be a standard Orlicz family built from $F$,
$\Phi_{0}$ and $\lambda$ satisfying the hypotheses of Lemma 3.6. Let $c>0$.
Then the following are equivalent
$(i)$

(3.4)
$\|f\|_{\Phi_{0}}^{2}\int\Phi_{0}\Big{(}\frac{f}{\|f\|_{\Phi_{0}}}\Big{)}F\left(\Phi_{0}\Big{(}\frac{f}{\|f\|_{\Phi_{0}}}\Big{)}\right)d\mu\leq
c\int\Phi_{0}^{\prime\prime}\left(\frac{f}{\|f\|_{\Phi_{0}}}\right)|\nabla
f|^{2}d\mu;$

for any function $f$ for which the right hand side is well defined;
$(ii)$ $t\geq s\geq 0$, it holds

$\|P_{t}f\|_{\Phi_{t}}\leq\|P_{s}f\|_{\Phi_{s}}.$

for any function $f$ for which the right hand side is well defined.
Moreover $(i)\Rightarrow(ii)$ with any (increasing) $\lambda$ such that
$\frac{\Phi_{t}^{\prime\prime}}{{\Phi_{t}^{\prime}}^{2}}\circ\Phi_{t}^{-1}\geq
c\lambda^{\prime}(t)\frac{\Phi_{0}^{\prime\prime}}{{\Phi_{0}^{\prime}}^{2}}\circ\Phi_{0}^{-1}$
for any $t\geq 0$ (in particular, any $\lambda$ satisfying
$\lambda^{\prime}(t)\leq 1/c$ would do); and $(ii)\Rightarrow(i)$ with
$c=1/\lambda^{\prime}(0)$.

###### Proof.

We first prove that $(i)$ implies $(ii)$. Item $(iv)$ of Lemma 3.6 guarantees
that
$\frac{\Phi_{t}^{\prime\prime}}{{\Phi_{t}^{\prime}}^{2}}\circ\Phi_{t}^{-1}\geq
c\lambda^{\prime}(t)\frac{\Phi_{0}^{\prime\prime}}{{\Phi_{0}^{\prime}}^{2}}\circ\Phi_{0}^{-1}$
with $\lambda(t)=t/c$. Hence, the set of functions $\lambda$, increasing,
satisfying
$\frac{\Phi_{t}^{\prime\prime}}{{\Phi_{t}^{\prime}}^{2}}\circ\Phi_{t}^{-1}\geq
c\lambda^{\prime}(t)\frac{\Phi_{0}^{\prime\prime}}{{\Phi_{0}^{\prime}}^{2}}\circ\Phi_{0}^{-1}$
for any $t\geq 0$ is non empty and we may fix one of them.

Consider the standard Orlicz family $(\Phi_{t})_{t\geq 0}$ built from $F$,
$\Phi_{0}$ and $\lambda$.

Note that by definition of the standard Orlicz family,
$\dot{\Phi}_{0}=\lambda^{\prime}(0)\Phi_{0}F(\Phi_{0})$. Thus Inequality (3.9)
reads as

$\|f\|_{\Phi_{0}}^{2}\int\dot{\Phi}_{0}\left(\frac{f}{\|f\|_{\Phi_{0}}}\right)d\mu\leq\lambda^{\prime}(0)c\int\Phi_{0}^{\prime\prime}\left(\frac{f}{\|f\|_{\Phi_{0}}}\right)|\nabla
f|^{2}d\mu.$

From the properties proved in Lemma 3.6 we can apply Proposition 3.2 with
$C(t,0)=\frac{\lambda^{\prime}(t)}{\lambda^{\prime}(0)}$ and
$\widetilde{C}(t,0)=c\lambda^{\prime}(t)$. We get that, for any $t\geq 0$, any
smooth function $f$ satisfies

$\|f\|_{\Phi_{t}}^{2}\int\\!\dot{\Phi}_{t}\left(\frac{f}{\|f\|_{\Phi_{t}}}\right)d\mu\leq\lambda^{\prime}(0)c\frac{C(t,0)}{\widetilde{C}(t,0)}\int\\!\Phi_{t}^{\prime\prime}\left(\frac{f}{\|f\|_{\Phi_{t}}}\right)|\nabla
f|^{2}d\mu=\int\\!\Phi_{t}^{\prime\prime}\left(\frac{f}{\|f\|_{\Phi_{t}}}\right)|\nabla
f|^{2}d\mu.$

The result of Point $(ii)$ follows by Proposition 3.1.

Now we prove that $(ii)$ implies $(i)$. Let $N(t)=\|P_{t}f\|_{\Phi_{t}}$. By
Lemma 2.1 at $t=0$, we infer that

$\displaystyle\frac{N^{\prime}(0)}{N(0)}\int\frac{f}{\|f\|_{\Phi_{0}}}\Phi_{0}^{\prime}\left(\frac{f}{\|f\|_{\Phi_{0}}}\right)d\mu=\int\dot{\Phi}_{0}\left(\frac{f}{\|f\|_{\Phi_{0}}}\right)d\mu-\int\Phi_{0}^{\prime\prime}\left(\frac{f}{\|f\|_{\Phi_{0}}}\right)|\nabla\frac{f}{\|f\|_{\Phi_{0}}}|^{2}d\mu.$

The hypercontractivity property of Point $(ii)$ insures that
$N^{\prime}(0)\leq 0$. Thus, since $x\Phi_{0}^{\prime}(x)\geq 0$, we get that

$\|f\|_{\Phi_{0}}^{2}\int\dot{\Phi}_{0}\left(\frac{f}{\|f\|_{\Phi_{0}}}\right)d\mu\leq\int\Phi_{0}^{\prime\prime}\left(\frac{f}{\|f\|_{\Phi_{0}}}\right)|\nabla
f|^{2}d\mu.$

The result follows by the formula
$\dot{\Phi}_{0}=\lambda^{\prime}(0)\Phi_{0}F(\Phi_{0})$ proved in Lemma 3.6. ∎

######  Remark 3.10.

When $\Phi_{0}(x)=x^{2}$, Inequality (3.4) reads as

$\int f^{2}F\left(\frac{f^{2}}{\mu(f^{2})}\right)d\mu\leq 2c\int|\nabla
f|^{2}d\mu.$

This is the usual $F$-Sobolev inequality introduced by Wang [Wan97], and
corresponds to the log-Sobolev inequality when $F(x)=\log x$.

When $F=\log$, one can consider $\lambda(t)=\log(1+e^{(4/\rho)t})-\log 2$,
with $\rho=2c$. Then, as already mentioned in Example 3.5, the standard Orlicz
family built from $\Phi_{0}(x)=x^{2}$, $F$ and $\lambda$, is
$\Phi_{t}(x)=|x|^{q(t)}$ with $q(t)=1+e^{(4/\rho)t}$. In that case Theorem 3.9
is nothing but Gross’ equivalence between the log-Sobolev inequality and the
hypercontractivity in $\mathbb{L}_{p}$ scale recalled in the introduction.

Theorem 3.9 has to be compared to [BCR06, Theorem 6]. When $F=\log$, [BCR06,
Theorem 6] asserts that $\|P_{t}f\|_{\widetilde{q}(t)}\leq\|f\|_{2}$ with
$\widetilde{q}(t)=2e^{\rho t}$ which is off by a factor of $2$ in the
exponential (though capturing the exponential character of the
$\mathbb{L}_{p}$ scale).

Furthermore, for $F(x):=\log(1+x)^{\beta}-\log(2)^{\beta}$, $\beta\in(0,1)$,
[BCR06, Theorem 6 and Corollary 34] does not give an hypercontractivity
property, but only hyper-boundedness (see section 3.5 below for more on hyper-
boundedness). One of the main difference comes from the fact that in [BCR06]
the authors deals with an explicit family of Young functions which imposes in
some situation stronger assumptions. This happens for the second assumption of
[BCR06, Theorem 6] which reads in our setting as
$\Phi_{t}(x)F(x^{2})\leq\ell(t)\Phi_{t}(F(\Phi_{t}(x)))+m$. We do not need
such an assumption here.

To conclude with the comparison between the two theorems, we observe that the
first assumption of [BCR06, Theorem 6] is implied by $x\mapsto xF(x^{2})$
convex, see remark 3.8 above and [BCR06, Proposition 7].

Notice that, for the following smooth version of $|x|^{\alpha}$,
$\alpha\in(1,2)$,

$u_{\alpha}(x)=\left\\{\begin{array}[]{ll}|x|^{\alpha}&\mbox{for }|x|>1\\\
\frac{\alpha(\alpha-2)}{8}x^{4}+\frac{\alpha(4-\alpha)}{4}x^{2}+(1-\frac{3}{4}\alpha+\frac{1}{8}\alpha^{2})&\mbox{for
}|x|\leq 1\end{array}\right.$

it has been proved in [BCR06, Proposition 33] that the probability measure
$d\mu_{\alpha}^{n}(x)=\prod_{i=1}^{n}Z_{\alpha}^{-1}e^{-u_{\alpha}(x_{i})}dx_{i}$
on $\mathbb{R}^{n}$ satisfies (3.4) with
$F(x)=\log(1+x)^{\beta}-\log(2)^{\beta}$ with $\beta=2(1-\frac{1}{\alpha})$,
$\Phi_{0}(x)=x^{2}$ and some constant $c=c(\alpha)>0$ (that does not depend on
$n$). This in turn leads to an hypercontractivity property for the standard
Orlicz family built from $F$, $\Phi_{0}$ and any $\lambda$ satisfying
$\lambda^{\prime}(t)\leq 1/c$.

###  3.4. Perturbation of Orlicz families and hypercontractivity

In this next section we show how to translate the hypercontractivity property
for one family of Young functions to another.

######  Proposition 3.11.

Let $(\Psi_{t})_{t\geq 0}$ and $(\Phi_{t})_{t\geq 0}$ be two families of Young
functions and $(P_{t})_{t\geq 0}$ be a linear semi-group acting on a set of
functions $\mathcal{A}$ onto itself. Assume that for some $t\geq 0$,
$(i)$ any $f\in{\mathcal{A}}$ satisfies
$\|P_{t}f\|_{\Psi_{t}}\leq\|f\|_{\Psi_{0}}$,
$(ii)$ the function $\Psi_{t}^{-1}\circ\Phi_{t}$ is convex, satisfies
$\Psi_{t}^{-1}\circ\Phi_{t}\leq\Psi_{0}^{-1}\circ\Phi_{0}$ and
$\Psi_{t}^{-1}\circ\Phi_{t}(\mathcal{A})\subset\mathcal{A}$,
$(iii)$ for any function $f\in{\mathcal{A}}$, any convex function $F$,
$F(P_{t}f)\leq P_{t}(F(f))$.
Then, any $f\in{\mathcal{A}}$ satisfies

$\|P_{t}f\|_{\Phi_{t}}\leq\|f\|_{\Phi_{0}}.$

###### Proof.

By definition of the norm and Jensen’s inequality given in $(iii)$, together
with $(ii)$, we have

$\displaystyle 1$
$\displaystyle=\int\Phi_{t}\left(\frac{P_{t}f}{\|P_{t}f\|_{\Phi_{t}}}\right)d\mu=\int\Psi_{t}\circ\Psi_{t}^{-1}\circ\Phi_{t}\left(P_{t}\frac{f}{\|P_{t}f\|_{\Phi_{t}}}\right)d\mu$
$\displaystyle\leq\int\Psi_{t}\left(P_{t}\Psi_{t}^{-1}\circ\Phi_{t}\left(\frac{f}{\|P_{t}f\|_{\Phi_{t}}}\right)\right)d\mu.$

This implies by definition of the norm and the hypercontractivity for the
family $\Psi_{t}$ (given in $(i)$) that

$1\leq\|P_{t}\Psi_{t}^{-1}\circ\Phi_{t}\Big{(}\frac{f}{\|P_{t}f\|_{\Phi_{t}}}\Big{)}\|_{\Psi_{t}}\leq\|\Psi_{t}^{-1}\circ\Phi_{t}\Big{(}\frac{f}{\|P_{t}f\|_{\Phi_{t}}}\Big{)}\|_{\Psi_{0}}.$

It follows that
$1\leq\int\Psi_{0}\circ\Psi_{t}^{-1}\circ\Phi_{t}\Big{(}\frac{f}{\|P_{t}f\|_{\Phi_{t}}}\Big{)}d\mu$.
Hence by point $(ii)$,
$1\leq\int\Phi_{0}\Big{(}\frac{f}{\|P_{t}f\|_{\Phi_{t}}}\Big{)}d\mu$. In turn,
$\|f\|_{\Phi_{0}}\geq\|P_{t}f\|_{\Phi_{t}}$. This ends the proof. ∎

######  Example 3.12.

Assume that $\Phi_{t}=\Psi_{t}\circ F$ for a fixed Young function $F$. Then
hypotheses $(ii)$ and $(iii)$ are automatically satisfied. For instance, it is
known that the linear semi-group with diffusion generator $L=\Delta-\nabla
U\nabla$ with $\mathrm{Hess}(U)\geq\rho>0$ satisfies log-Sobolev inequality
with constant $2/\rho$ and in turn is hypercontractive in the sense that

$\|P_{t}f\|_{q(t)}\leq\|f\|_{2},\qquad\qquad\mbox{with
}q(t):=1+e^{(4/\rho)t}.$

Now let $\Psi_{t}(x)=|x|^{q(t)}$. Hence, for any Young function $F$, the
previous proposition asserts that

$\|P_{t}f\|_{F^{q(t)}}\leq\|f\|_{F^{2}}.$

Similarily from [BCR06] we learn that the semi-group associated to
$L=\Delta-\nabla U\nabla$ with $U(x)=|x|^{\alpha}$ (more precisely a smoothed
version of $|x|^{\alpha}$), $1\leq\alpha\leq 2$, is hypercontractive in the
Orlicz’ spaces family $\mathbb{L}_{\Phi_{t}}$ with
$\Phi_{t}=x^{2}e^{ct\log(1+|x|^{2})^{2(1-\frac{1}{\alpha})}}$ for some
constant $c$. It follows that for any Young function $F$, the semi-group
$(P_{t})_{t\geq 0}$ is also hypercontractive in the Orlicz’ spaces family
$\mathbb{L}_{\Psi_{t}}$ with
$\Psi_{t}=F(x)^{2}e^{ct\log(1+|F(x)|^{2})^{2(1-\frac{1}{\alpha})}}$.

###  3.5. Hypercontractivity versus hyper-boundedness

In this section, we deal with perturbation arguments that allows one to get
some hyper-boundedness property starting from hypercontractivity.

######  Theorem 3.13.

Let $(\Phi_{t})_{t\geq 0}$ and $(\Psi_{t})_{t\geq 0}$ be two standard Orlicz
families built respectively from $F$ and $\widetilde{F}$, $\Phi_{0}$ and
$\lambda$, both satisfying the hypotheses of Definition 3.3.

Assume that for any $\varepsilon>0$ there exists $D(\varepsilon)\geq 0$ such
that all $x\geq 0$ satisfy

$\widetilde{F}(x)\leq\varepsilon F(x)+D(\varepsilon).$

Suppose also that for any $f$ and any $t\geq 0$, it holds

$\|P_{t}f\|_{\Phi_{t}}\leq\|f\|_{\Phi_{0}}.$

Then, for any $s_{2}\geq s_{1}\geq 0$, any $t\geq 0$, any $\mathcal{C}^{1}$
increasing function $q:\mathbb{R}^{+}\to\mathbb{R}^{+}$ with $q(0)=s_{1}$ and
$q(t)=s_{2}$, it holds

$\|P_{t}f\|_{\Psi_{s_{2}}}\leq\|f\|_{\Psi_{s_{1}}}\exp\left\\{\int_{0}^{t}q^{\prime}(u)\lambda^{\prime}(q(u))D\left(\frac{\lambda^{\prime}(0)}{q^{\prime}(u)\lambda^{\prime}(q(u))}\right)du\right\\}\qquad\forall
f\in\mathbb{L}_{\Psi_{s_{1}}}.$

###### Proof.

Our aim is to use the hypercontractivity property in Orlicz spaces
$\mathbb{L}_{\Phi_{t}}$ together with Theorem 3.9 to get a functional
inequality involving the Young functions $\Phi_{t}$, and then use the
assumption on $F$ and $\widetilde{F}$ to get a similar inequality for
$\widetilde{F}$.

Fix $s_{2}\geq s_{1}\geq 0$, $t\geq 0$ and a $\mathcal{C}^{1}$ increasing
function $q:\mathbb{R}^{+}\to\mathbb{R}^{+}$ with $q(0)=s_{1}$ and
$q(t)=s_{2}$. Fix $f\in\mathbb{L}_{\Psi_{s_{1}}}$ and let
$N(u):=\|P_{u}f\|_{\Psi_{q(u)}}$ and $g:=P_{u}f/N(u)$. Applying Lemma 2.1 to
$\widetilde{\Psi}(t,x):=\Phi_{q(t)}(x)$, and observing that
$\frac{\partial}{\partial
t}\widetilde{\Psi}(t,x):=q^{\prime}(t)\dot{\Psi}_{q(t)}(x)$, we get

$\displaystyle\frac{N^{\prime}(u)}{N(u)}\int
g\Psi_{q(u)}^{\prime}\left(g\right)d\mu$
$\displaystyle=q^{\prime}(u)\int\dot{\Psi}_{q(u)}\left(g\right)d\mu-\int\Psi_{q(u)}^{\prime\prime}\left(g\right)\left|\nabla
g\right|^{2}d\mu.$

Since $\Psi_{q(u)}$ is a nice Young function,
$x\Psi_{q(u)}^{\prime}(x)\geq\Psi_{q(u)}(x)$ for any $x\geq 0$, any $u$.
Therefore $\int g\Psi_{q(u)}^{\prime}\left(g\right)d\mu\geq 1$ and in turn,
when $N^{\prime}(u)\geq 0$, we have

$\frac{N^{\prime}(u)}{N(u)}\leq
q^{\prime}(u)\int\dot{\Psi}_{q(u)}\left(g\right)d\mu-\int\Psi_{q(u)}^{\prime\prime}\left(g\right)\left|\nabla
g\right|^{2}d\mu.$

Now, thanks to Theorem 3.9, the hypercontractivity assumption guarantees that

$\|f\|_{\Phi_{0}}^{2}\int\Phi_{0}\left(\frac{f}{\|f\|_{\Phi_{0}}}\right)F\left(\Phi_{0}\left(\frac{f}{\|f\|_{\Phi_{0}}}\right)\right)d\mu\leq\frac{1}{\lambda^{\prime}(0)}\int\Phi_{0}^{\prime\prime}\left(\frac{f}{\|f\|_{\Phi_{0}}}\right)|\nabla
f|^{2}d\mu.$

By our assumption, item $(iii)$ of Lemma 3.6 (recall that
$\Psi_{0}=\Phi_{0}$), we have

$\dot{\Psi}_{0}=\lambda^{\prime}(0)\Phi_{0}\widetilde{F}(\Phi_{0})\leq\lambda^{\prime}(0)\varepsilon\Phi_{0}F(\Phi_{0})+\lambda^{\prime}(0)D(\varepsilon)\Phi_{0}.$

Therefore, for any $f$ with $\|f\|_{\Phi_{0}}=1$,

$\displaystyle\int\dot{\Psi}_{0}\left(f\right)d\mu$
$\displaystyle\leq\lambda^{\prime}(0)\varepsilon\int\Phi_{0}\left(f\right)F\left(\Phi_{0}\left(f\right)\right)d\mu+\lambda^{\prime}(0)D(\varepsilon)$
$\displaystyle\leq\varepsilon\int\Phi_{0}^{\prime\prime}\left(f\right)\left|\nabla
f\right|^{2}d\mu+\lambda^{\prime}(0)D(\varepsilon)=\varepsilon\int\Psi_{0}^{\prime\prime}\left(f\right)\left|\nabla
f\right|^{2}d\mu+\lambda^{\prime}(0)D(\varepsilon).$

Recall the isometry

$\begin{array}[]{rcl}I_{s,t}:\mathbb{L}_{\Psi_{t}}&\rightarrow&\mathbb{L}_{\Psi_{s}}\\\
f&\mapsto&\|f\|_{\Psi_{t}}\Psi_{s}^{-1}\circ\Psi_{t}\left(\frac{f}{\|f\|_{\Psi_{t}}}\right)\end{array}$

from Proposition 3.2 that we may use with $s=0$ and $t=q(u)$. The previous
inequality applied to $I_{0,q(u)}(f)$ ensures that for any $f$ with
$\|f\|_{\Psi_{q(u)}}=\|I_{0,q(u)}(f)\|_{\Psi_{0}}=\|I_{0,q(u)}(f)\|_{\Phi_{0}}=1$,
it holds

$\int\dot{\Psi}_{0}\circ\Psi_{0}^{-1}\circ\Psi_{q(u)}(f)d\mu\leq\varepsilon\int\frac{\Psi_{0}^{\prime\prime}}{{\Psi_{0}^{\prime}}^{2}}\circ\Psi_{0}^{-1}\circ\Psi_{q(u)}(f)\left|\nabla
f\right|^{2}{\Psi_{q(u)}^{\prime}}^{2}(f)d\mu+\lambda^{\prime}(0)D(\varepsilon).$

It follows from items $(iii)$ and $(iv)$ of Lemma 3.6 that

$\frac{\lambda^{\prime}(0)}{\lambda^{\prime}(q(u))}\int\dot{\Psi}_{q(u)}(f)d\mu\leq\varepsilon\int\Psi_{q(u)}^{\prime\prime}(f)\left|\nabla
f\right|^{2}d\mu+\lambda^{\prime}(0)D(\varepsilon).$

This leads to

$\frac{N^{\prime}(u)}{N(u)}\leq\frac{q^{\prime}(u)\lambda^{\prime}(q(u))}{\lambda^{\prime}(0)}\left(\varepsilon\int\Psi_{q(u)}^{\prime\prime}(f)\left|\nabla
f\right|^{2}d\mu+\lambda^{\prime}(0)D(\varepsilon)\right)-\int\Psi_{q(u)}^{\prime\prime}(f)\left|\nabla
f\right|^{2}d\mu$

for any $u$ such that $N^{\prime}(u)\geq 0$. Therefore, provided that
$q^{\prime}(u)\lambda^{\prime}(q(u))\varepsilon=\lambda^{\prime}(0)$ and
$N^{\prime}(u)\geq 0$ it holds

$\frac{N^{\prime}(u)}{N(u)}\leq
q^{\prime}(u)\lambda^{\prime}(q(u))D\left(\frac{\lambda^{\prime}(0)}{q^{\prime}(u)\lambda^{\prime}(q(u))}\right).$

This bound trivially holds when $N^{\prime}(u)<0$. Hence

$\displaystyle\log\|P_{t}f\|_{\Psi_{s_{2}}}-\log\|P_{0}f\|_{\Psi_{s_{1}}}$
$\displaystyle=\int_{0}^{t}\frac{d}{du}\log\|P_{u}f\|_{\Psi_{q(u)}}du=\int_{0}^{t}\frac{N^{\prime}(u)}{N(u)}du$
$\displaystyle\leq\int_{0}^{t}q^{\prime}(u)\lambda^{\prime}(q(u))D\left(\frac{\lambda^{\prime}(0)}{q^{\prime}(u)\lambda^{\prime}(q(u))}\right)du.$

The result follows. ∎

As an example of application, consider, for $\beta\in(0,1]$,
$F_{\beta}(x)=(\log(1+x))^{\beta}-(\log 2)^{\beta}$. It is not difficult to
check that for any $\varepsilon>0$ and any $\beta^{\prime}<\beta$, it holds

$F_{\beta^{\prime}}(x)\leq\varepsilon F_{\beta}(x)+D(\varepsilon)$

with

$D(\varepsilon):=-(\log 2)^{\beta^{\prime}}+\varepsilon(\log
2)^{\beta}+\left(\frac{\beta^{\prime}}{\beta}\right)^{\frac{\beta^{\prime}}{\beta-\beta^{\prime}}}\frac{\beta-\beta^{\prime}}{\beta}\left(\frac{1}{\varepsilon}\right)^{\frac{\beta^{\prime}}{\beta-\beta^{\prime}}}.$

Now let

$u_{\alpha}(x)=\left\\{\begin{array}[]{ll}|x|^{\alpha}&\mbox{for }|x|>1\\\
\frac{\alpha(\alpha-2)}{8}x^{4}+\frac{\alpha(4-\alpha)}{4}x^{2}+(1-\frac{3}{4}\alpha+\frac{1}{8}\alpha^{2})&\mbox{for
}|x|\leq 1\end{array}\right.$

be a smooth version of $|x|^{\alpha}$, $\alpha\in(1,2)$. Define the
probability measure
$d\mu_{\alpha}^{n}(x)=\prod_{i=1}^{n}Z_{\alpha}^{-1}e^{-u_{\alpha}(x_{i})}dx_{i}$
on $\mathbb{R}^{n}$. As already mentioned in Remark 3.10, it follows from
[BCR06, Proposition 33] that Inequality (3.4) holds for $F_{\beta}$,
$\Phi_{0}(x)=x^{2}$ and some $c=c(\alpha)>0$ and therefore that the semi-group
$(P_{t})_{t\geq 0}$ associated to $\mu_{\alpha}^{n}$ is hypercontractive along
the standard Orlicz family $(\Phi_{t})_{t\geq 0}$ built from $F_{\beta}$,
$\Phi_{0}$ and any $\lambda$ satisfying $\lambda^{\prime}(t)\leq 1/c$. Fix for
simplicity $\lambda(t)=t/c$.

Consider the standard Orlicz families $(\Psi_{t})_{t\geq 0}$ built from
$F_{\beta^{\prime}}$, $\Phi_{0}(x)=x^{2}$ and $\lambda$.

The previous theorem shows that (for $s_{1}=0$ and $s_{2}=s$)

$\|P_{t}f\|_{\Psi_{s}}\leq m(t)\|f\|_{2}$

where

$m(t)=\inf_{q}\exp\left\\{\frac{1}{c}\int_{0}^{t}q^{\prime}(u)D\left(\frac{1}{q^{\prime}(u)}\right)du\right\\}$

where the infimum is running over all increasing
$q\colon[0,t]\to\mathbb{R}_{+}$ with $q(0)=0$ and $q(t)=s$. We stress that the
Luxembourg norm is computed here with reference measure $\mu_{\alpha}^{n}$.

One has

$q^{\prime}(u)D\left(\frac{1}{q^{\prime}(u)}\right)=-q^{\prime}(u)(\log
2)^{\beta^{\prime}}+(\log
2)^{\beta}+C_{\beta,\beta^{\prime}}(q^{\prime}(u))^{\frac{\beta}{\beta-\beta^{\prime}}}$

where we set, for simplicity,
$C_{\beta,\beta^{\prime}}:=\left(\frac{\beta^{\prime}}{\beta}\right)^{\frac{\beta^{\prime}}{\beta-\beta^{\prime}}}\frac{\beta-\beta^{\prime}}{\beta}$.
Therefore,

$\inf_{q}\left\\{\int_{0}^{t}q^{\prime}(u)D\left(\frac{1}{q^{\prime}(u)}\right)du\right\\}=-s(\log
2)^{\beta^{\prime}}+t(\log
2)^{\beta}+C_{\beta,\beta^{\prime}}\inf_{q}\left\\{\int_{0}^{t}(q^{\prime}(u))^{\frac{\beta}{\beta-\beta^{\prime}}}du\right\\}.$

Since $\beta/(\beta-\beta^{\prime})\geq 1$, by Holder’s inequality (and
equality cases in Holder’s inequality), it is easy to see that

$\inf_{q}\left\\{\int_{0}^{t}(q^{\prime}(u))^{\frac{\beta}{\beta-\beta^{\prime}}}du\right\\}=s^{\frac{\beta}{\beta-\beta^{\prime}}}t^{-\frac{\beta^{\prime}}{\beta-\beta^{\prime}}}.$

As a conclusion

$m(t)=\exp\left\\{\frac{1}{c}\left(-s(\log 2)^{\beta^{\prime}}+t(\log
2)^{\beta}+C_{\beta,\beta^{\prime}}s^{\frac{\beta}{\beta-\beta^{\prime}}}t^{-\frac{\beta^{\prime}}{\beta-\beta^{\prime}}}\right)\right\\}.$

Note in particular that the factor in the exponential explodes for a fixed
$t$, when $s$ goes to infinity. This must be since the semi-group associated
to $\mu_{\alpha}^{n}$ can not be ultracontractive.

###  3.6. Høegh-Krohn and Simon’s type result

In the sequel, $\Phi$ and $\Psi$ denote two Young functions such that
$\mathbb{L}_{\Psi}\subset\mathbb{L}_{\Phi}$, $(P_{t})_{t\geq 0}$ is a Markov
semi-group on $\mathbb{R}^{n}$ associated with the infinitesimal generator
$L=\Delta-\nabla V\cdot\nabla$ which is symmetric in $\mathbb{L}_{2}(\mu)$
with $\mu$ of density $e^{-V}$ with respect to the Lebesgue measure on
$\mathbb{R}^{n}$ and $V$ smooth enough and satisfying $\int e^{-V}=1$.

After the pionner work by Nelson [Nel66], Høegh-Krohn and Simon [SHK72] proved
that, if $P_{t_{o}}$ is continuous from $\mathbb{L}_{2}$ into $\mathbb{L}_{4}$
for some fixed time $t_{o}>0$, then $\mu$ satisfies a defective log-Sobolev
inequality. Moreover, if in addition a Poincaré inequality holds for $\mu$,
$P_{t}$ is continuous from $\mathbb{L}_{p}$ into $\mathbb{L}_{q}$ for any
$p>1$ and some $q=q(p,t)>p$, which in turn implies a tight log-Sobolev
inequality.

In this section, we explore similar questions with general Orlicz spaces.
Namely, we investigate the consequences of the fact that $P_{t_{o}}$ is
continuous from $\mathbb{L}_{\Phi}$ into $\mathbb{L}_{\Psi}$ for some fixed
time $t_{o}>0$. Our first statement deals with a very general situation. Then
we will focus to some specific choices of $\Phi$ and $\Psi$.

######  Proposition 3.14.

Assume that $\Phi$ is $\mathcal{C}^{2}$ and satisfies the
$\Delta_{2}$-condition with constant $K$ (recall Section 2.1). Assume that
there exist $t_{0}>0$ and $C\in[1,\infty)$ such for any
$f\in\mathbb{L}_{\Phi}(\mu)$, $\|P_{t_{o}}f\|_{\Psi}\leq C\|f\|_{\Phi}$ (i.e.
$P_{t_{o}}$ is continuous from $\mathbb{L}_{\Phi}$ into $\mathbb{L}_{\Psi}$).
Then, for any $f$ non-negative satisfying $\|f\|_{\Phi}=1$, it holds

(3.5) $\int
f\Phi^{\prime}(f)\log\left(\frac{f}{\Psi^{-1}\circ\Phi(f)}\right)d\mu\leq
t_{o}\int\Phi^{\prime\prime}(f)|\nabla f|^{2}d\mu+(K-1)\log C.$

###### Proof.

For $s\in[0,1]$, let $\Phi_{s}$ be the Young function defined by
$\Phi_{s}^{-1}:=(\Phi^{-1})^{1-s}(\Psi^{-1})^{s}$. As for the classical result
of Høegh-Krohn and Simon [SHK72] we need to use a complex interpolation
theorem. We follow [Bak94, Proof of Theorem 3.6]. Self-adjointness of $L$
guarantees that its spectrum is contained in $(-\infty,0]$ with the
corresponding spectral decomposition $L=-\int_{0}^{\infty}\lambda
dE_{\lambda}$. From this we can define an extension of $P_{t}$ to the half
complex plane $z=t+iy$, $t\geq 0$, by setting
$P_{z}:=\int_{0}^{\infty}\exp(-z\lambda)dE_{\lambda}$. The operator $P_{z}$
coincides with $P_{t}$ for $y=0$ and satisfies the semi-group property
$P_{z_{1}}\circ P_{z_{2}}=P_{z_{1}+z_{2}}$. This is an analytic family of
operators in the sense that for all $f,g\in\mathbb{L}_{2}(\mu)$, $z\mapsto\int
fP_{z}gd\mu$ is analytic. Also it is a contraction in $\mathbb{L}_{\phi}$
since $|\exp(-\lambda z)|\leq 1$. Therefore, for any $f\in\mathbb{L}_{\Phi}$,
$\|P_{iy}f\|_{\Phi}\leq\|f\|_{\Phi}$, and, by assumption, for any
$f\in\mathbb{L}_{\Phi}(\mu)$, $\|P_{t_{o}+iy}f\|_{\Psi}=\|P_{t_{o}}\circ
P_{iy}f\|_{\Psi}\leq C\|f\|_{\Phi}$. Applying the generalized version of
Stein’s interpolation theorem [Voi92, Theorem 2.1], for any $f\in\|f\|_{\Phi}$
and any $s\in[0,1]$ it holds $\|P_{st_{o}}f\|_{\Phi_{s}}\leq
C^{s}\|f\|_{\Phi}$. The latter being true for all $s$, with equality at $s=0$,
we are guaranteed that

$\frac{d}{ds}\left(\frac{\|P_{st_{o}}f\|_{\Phi_{s}}}{C^{s}}\right)_{|_{s=0}}=\frac{\|f\|_{\Phi}\left(\int\dot{\Phi}_{0}(g)d\mu-
t_{o}\int\Phi^{\prime\prime}(g)|\nabla g|^{2}d\mu\right)}{\int
g\Phi^{\prime}(g)d\mu}-\|f\|_{\Phi}\log C\leq 0$

where we have used Lemma 2.1 and set $g:=f/\|f\|_{\Phi}$. Therefore, for any
$f$ with $\|f\|_{\Phi}=1$, it holds

$\int\dot{\Phi}_{0}(f)d\mu\leq t_{o}\int\Phi^{\prime\prime}(f)|\nabla
f|^{2}d\mu+\log C\int f\Phi^{\prime}(f)d\mu.$

Now observe that, since $\Phi_{s}^{-1}:=(\Phi^{-1})^{1-s}(\Psi^{-1})^{s}$,

$\dot{\Phi}_{0}(x)=x\Phi^{\prime}(x)\log\left(\frac{x}{\Psi^{-1}\circ\Phi(x)}\right),\qquad
x>0.$

Moreover, since $\Phi$ satisfies the $\Delta_{2}$ condition with constant $K$,
by (2.1), $\int f\phi^{\prime}(f)d\mu\leq(K-1)\int\Phi(f)d\mu=K-1$. This leads
to the desired inequality (3.5). ∎

######  Remark 3.15.

We notice that, in [BCR06], following [Cat04], a semi-group technique is used
to obtain an inequality similar (but different) to (3.5). We were not able to
generalize such a technique to our general setting and therefore exploited
instead the original idea of Høegh-Krohn and Simon (Stein’s interpolation
theorem).

In the next subsections we explore the meaning of Inequality (3.5) on specific
examples.

####  3.6.1. $\mathbb{L}_{p}$ scale: Høegh-Krohn and Simon classical result.

Proposition 3.14 recovers the classical result by Høegh-Krohn and Simon
[SHK72]. Indeed, for $\Phi(x)=x^{2}$ and $\Psi(x)=x^{4}$, say, Inequality
(3.5) reads (observe that $K=4$) for all $f$ with $\int f^{2}d\mu=1$

$\mathrm{Ent}_{\mu}(f^{2})\leq 4t_{o}\int|\nabla f|^{2}d\mu+6\log C$

This is a defective log-Sobolev inequality. Thanks to Rothaus’ lemma [Rot85,
Lemma 10], for any smooth enough $f$, it holds
$\mathrm{Ent}_{\mu}(f^{2})\leq\mathrm{Ent}_{\mu}\left((f-\int
fd\mu)^{2}\right)+2\int\left(f-\int fd\mu\right)^{2}d\mu$. Therefore thanks to
Miclo’s proof [Mic15] of Høegh-Krohn and Simon’s conjecture asserting that, as
soon as $P_{t_{o}}$ is continuous from $\mathbb{L}_{2}$ into $\mathbb{L}_{p}$,
with $p>2$, a poincaré inequality holds (namely that $\left(f-\int
fd\mu\right)^{2}d\mu\leq D\int|\nabla f|^{2}d\mu$), one can tightened the
latter and get $\mathrm{Ent}_{\mu}(f^{2})\leq(4t_{o}+(6\log C+2)D)\int|\nabla
f|^{2}d\mu$ from which we deduce that $P_{t}$ is hypercontractive in the
classical $\mathbb{L}_{p}$ sense.

####  3.6.2. $F$-Sobolev inequalities.

A second example, already considered in [BCR06, Section 4.1], is given by
$\Phi(x)=x^{2}$ and $\Psi(x)=x^{2}e^{(\log x)^{\beta}}$ for $x$ large, and
$\beta\in(0,1)$ (observe that the values of $\Psi$ around the origin makes no
qualitative difference). In that case after some algebra, inequality (3.5)
reads

$\int f^{2}F(f^{2})\leq c\int|\nabla f|^{2}d\mu+c^{\prime}$

for some $c,c^{\prime}>0$ possibly depending on $\beta$ and $C$ (but not on
$f$), and any $f$ with $\int f^{2}d\mu=1$, and
$F(x)=\log(1+x)^{\beta}-\log(2)^{\beta}$. This is the so-called $F$-Sobolev
inequality considered in the introduction. In [BCR06, Lemma 9], see also
[LZ07], an analogous result to Rothaus’ lemma is proved allowing one to
tighten the latter, since $\mu$ satisfies a Poincaré inequality as established
by Miclo in [Mic15] in the setting of this section, and therefore obtain that
$P_{t}$ is hypercontractive in the family of Orlicz spaces
$\Phi_{t}(x)=x^{2}e^{q(t)F(x)}$.

####  3.6.3. $\mathbb{L}_{1}\log\mathbb{L}_{1}$ and the Talagrand Conjecture.

In this section we will consider the following $\mathcal{C}^{2}$ Orlicz
function

$\Phi(x)=\begin{cases}2x\log x+1&\mbox{ for }|x|\geq 1\\\ x^{2}&\mbox{ for
}|x|\leq 1.\end{cases}$

Our aim is to derive some functional inequality from a continuity assumption
of $P_{t_{o}}$ from $\mathbb{L}_{\Phi}$ into $\mathbb{L}_{\Psi}$. Then we may
specify on the one hand on the particular choice $\Psi(x)=|x|^{p}$ with
general measure $\mu$, obtaining that $(P_{t})_{t\geq 0}$ is then
ultracontactive; and on the other hand on general $\Psi$ satisfying
$\lim_{x\to\infty}\frac{\Psi(x)}{\Phi(x)}=\infty$, but with the Gaussian
measure and the Ornstein-Uhlenbeck semi-group that is proved to be not
continuous from $\mathbb{L}_{\Phi}$ into $\mathbb{L}_{\Psi}$, for any $t\geq
0$.

Many results of this section could be generalized to other Young functions
$\Phi$, but at the price of somehow unpleasant technicalities and many
assumptions. This is why we prefer digging on the latter specific choice of
Young function $\Phi$ that already contains many interesting features. As
possible extensions, we could consider asymptotically $\Phi(x)\simeq
x\log(x)^{p_{1}}\log_{2}(x)^{p_{2}}...\log_{i}(x)^{p_{i}}$ with $p_{1}\geq 1$
and $p_{k}>0$, where $\log_{k}=\log(\log(\dots(\log)))$ stands for the
iterated $k$ times of the logarithm. On the other hand, $\Phi(x)\simeq
x\log(\log(x))$ would not be reached by our approach. However we believe that
all the results of this section should hold for any $1$-regular Young function
satisfying $\lim_{x\to\infty}\frac{\Phi(x)}{x}=\infty$.

For $x\geq 0$, set

$H(x):=\int_{0}^{x}\sqrt{\Phi^{\prime\prime}(t)}dt=\begin{cases}\sqrt{2}x&\mbox{for
}x\in[0,1]\\\ 2\sqrt{2x}-\sqrt{2}&\mbox{for }x\geq 1\end{cases},\qquad
H(-x)=H(x)\mbox{ for }x\geq 0$

and observe that its reciprocal function satisfies $H^{-1}(x)=x/\sqrt{2}$ for
$0\leq x\leq\sqrt{2}$ and $H^{-1}(x)=(x+\sqrt{2})^{2}/8$ for $x\geq\sqrt{2}$.
We introduce the following Young function

$U(x):=\Phi(H^{-1}(|x|))$

and $\theta(x):=\log\Phi^{-1}(x)/\Psi^{-1}(x))$, $x\geq 0$. We recall that
$A_{-}:=\max(-A,0)$ and $A_{+}:=\max(A,0)$ denote the negative and positive
part of $A$ (which are positive). We first prove that (3.5) implies some
$F$-Sobolev type inequality that will be the starting point of our forthcoming
examples.

######  Proposition 3.16.

Assume that there exists $t_{o}>0$ and $C\geq 1$ such that for all
$f\in\mathbb{L}_{\Phi}$, $\|P_{t}f\|_{\Psi}\leq C\|f\|_{\Phi}$. Assume also
that $\Theta:=\sup_{x}\theta(x)_{-}<\infty$. Then, there exists a constant $c$
that depends only $t_{o}$, $C$ and $\Theta$ such that for all $f$ smooth
enough, it holds

(3.6) $\int
f^{2}\log_{+}\left(\frac{f^{2}}{\|f\|_{U}^{2}}\right)\theta_{+}\left(U\left(\frac{f}{\|f\|_{U}}\right)\right)d\mu\leq
c\int|\nabla f|^{2}d\mu+c\|f\|_{U}^{2}.$

###### Proof.

Since $\Phi(x)\leq x\Phi^{\prime}(x)\leq 2\Phi(x)$ (and $\Phi$ satisfies the
$\Delta_{2}$-condition with constant $K=4$),
$\theta(x)=\theta(x)_{+}-\theta(x)_{-}$, Inequality (3.5) implies

$\int\Phi(f)\theta\left(\Phi(f)\right)_{+}d\mu\leq
t_{o}\int\Phi^{\prime\prime}(f)|\nabla f|^{2}d\mu+3\log C+4\Theta$

for any $f$ with $\int\Phi(f)d\mu=\|f\|_{\Phi}=1$. (There is no issue with the
sign of $f$ here. Indeed applying (3.5) to $f_{+}$ and $f_{-}$ and then
summing up the obtained inequalities lead to the latter.) Observe that
$U(x)\geq\frac{1}{8}x^{2}\log_{+}(x^{2})$. Whence, the latter implies that,
for any $g$ smooth enough satisfying $\|g\|_{U}=1$ (set $g=H(f)$), it holds

$\int g^{2}\log_{+}(g^{2})\theta_{+}(U(g))d\mu\leq 8t_{o}\int|\nabla
g|^{2}d\mu+8(3\log C+4\Theta).$

Applying the latter to $g=h/\|h\|_{U}$ leads to the desired conclusion. ∎

Now we may consider two different situations according to two different
choices of the function $\Psi$. As a first possible natural choice, consider
$\Psi(x)=|x|^{p}$, with $p>1$. We will prove that, in that case and under the
additional assumption that a log-Sobolev inequality holds, the semi-group is
ultra-contractive.

######  Theorem 3.17.

Assume that there exists $t_{o}>0$ and $C>0$ such that for all
$f\in\mathbb{L}_{\Phi}$, $\|P_{t}f\|_{\Psi}\leq C\|f\|_{\Phi}$, with $\Phi$
given at the beginning of this section, and $\Psi(x)=|x|^{p}$, $p>1$. Then,
for any $t>0$, there exists a constant $m(t)$ such that
$\|P_{t}f\|_{\infty}\leq m(t)\|f\|_{1}$ for any $f\in\mathbb{L}_{1}$.

######  Remark 3.18.

The same conclusion would occur for more general choices of $\Psi$. For
example one could consider asymptotically
$\Psi(x)=|x|^{p}(\log(x))^{p_{1}}\dots(\log_{i}(x))^{p_{i}}$,
$p_{1},\dots,p_{i}\in\mathbb{R}$.
Tight coercive inequalities for such functions were recently studied in [WZ].

###### Proof.

Our first aim is to briefly prove a Poincaré and a log-Sobolev inequality (we
may give the main lines, leaving the details to the reader). Fix $q\in(1,p)$.
Then it is not difficult to see that there exists a constant $c_{q}$ such that
for any $f\in\mathbb{L}_{q}$, it holds $\|f\|_{\phi}\leq c_{q}\|f\|_{q}$.
Therefore for all $f\in\mathbb{L}_{q}$, $\|P_{t}f\|_{p}\leq Cc_{q}\|f\|_{p}$.
By [Glu20], which is a generalization of [Mic15], $\mu$ satisfies a Poincaré
inequality: namely for some constant $C_{P}$ and all $f$ smooth enough,

$\int f^{2}d\mu-(\int fd\mu)^{2}\leq C_{P}\int|\nabla f|^{2}d\mu.$

On the other hand, by the classical result by Høegh-Krohn and Simon [SHK72] a
defective log-Sobolev inequality holds, that, with the Poincaré inequality
together with Rothaus’ lemma [Rot85], implies a tight log-Sobolev inequality,
namely that for all smooth $f$ it holds

(3.7) $\int f^{2}\log\left(\frac{f^{2}}{\int f^{2}d\mu}\right)d\mu\leq
C_{LS}\int|\nabla f|^{2}d\mu$

for some constant $C_{LS}$ that depends on $q,C$ and $C_{P}$.

After some algebra and using some approximations arguments, inequality (3.6)
implies that, for any $f$ smooth enough, it holds

$\int f^{2}(\log(f^{2}/\|f\|_{U}^{2}))^{2}d\mu\leq c\int|\nabla
f|^{2}d\mu+c\|f\|_{U}^{2}.$

for some constant $c$ depending on $t_{o}$, $C$ and $p$ (recall that
$U(x):=\Phi(H^{-1}(|x|))$ is defined at the beginning of this section). Set

$F(x):=\begin{cases}(\log(x))^{2}&\mbox{if }x\geq 1\\\ (x-1)^{2}&\mbox{if
}0\leq x\leq 1\end{cases}$

which is concave on $[e,\infty)$, $\mathcal{C}^{2}$ and satisfies $F(1)=0$.
Set also $G(x)=F(x)\mathds{1}_{[1,\infty)}$. Our aim is to prove that, for any
$f$ smooth enough, it holds

(3.8) $\int f^{2}G\left(\frac{f^{2}}{\mu(f^{2})}\right)d\mu\leq c\int|\nabla
f|^{2}d\mu.$

To do so we may proceed in different steps. We first change the normalization,
using $\mathbb{L}_{2}$-norm instead of $\mathbb{L}_{U}$-norm. Then, using the
log-Sobolev inequality above (Inequality (3.7)), we will remove the defective
part of the inequality by means of a Lemma à la Rothaus. Such a step will need
more regularity ($\mathcal{C}^{2}$), this is why we may first move to (3.8)
with $G$ replaced by $F$.

We introduce the $\mathbb{L}_{2}(\mu)$-norm of the function as follows: for
any $\varepsilon>0$ it holds

$\displaystyle\int
f^{2}\left(\log(f^{2}/\|f\|_{2}^{2})+\log(\|f\|^{2}/\|f\|_{U}^{2})\right)^{2}d\mu$
$\displaystyle\geq(1-\varepsilon)\int
f^{2}\left(\log(f^{2}/\|f\|_{2}^{2})\right)^{2}d\mu$
$\displaystyle\quad-\left(\frac{1-\varepsilon}{\varepsilon}\right)\int
f^{2}(\log(\|f\|_{2}^{2}/\|f\|_{U}^{2}))^{2}d\mu.$

By Lemma 3.19, $\frac{\|f\|_{2}^{2}}{\|f\|_{U}^{2}}\leq 2$. Therefore, since
$x(\log x)^{2}\leq 1$ for all $x\in(0,2)$, we are guaranteed that $\int
f^{2}(\log(\|f\|_{2}^{2}/\|f\|_{U}^{2}))^{2}d\mu\leq\|f\|_{U}^{2}$. Hence, for
any $f$ smooth enough, it holds

$\int f^{2}(\log(f^{2}/\|f\|_{2}^{2}))^{2}d\mu\leq c\int|\nabla
f|^{2}d\mu+c\|f\|_{U}^{2}$

for some other constant $c$ that, in the sequel, may change from line to line.
Next, we introduce the function $F$. Since $xF(x)\leq ax(\log x)^{2}+a$,
$x>0$, for some constant $a>0$, increasing $c$ if necessary, the later implies

(3.9) $\int f^{2}F(f^{2}/\|f\|_{2}^{2}))d\mu\leq c\int|\nabla
f|^{2}d\mu+c\|f\|_{U}^{2}+c\|f\|_{2}^{2}$

which is a defective $F$-Sobolev inequality. Our next step is to use the log-
Sobolev inequality to tightened Inequality (3.9).

A tedious but straightforward adaptation of the proof of [BCR06, Lemma 9]
shows that there exists a constant $b$ such that for any $f$ smooth enough it
holds (Rothaus’s type Lemma [Rot85])

(3.10) $\int f^{2}F\left(\frac{f^{2}}{\mu(f^{2})}\right)d\mu\leq
b\int\tilde{f}^{2}F\left(\frac{\tilde{f}^{2}}{\mu(\tilde{f}^{2})}\right)d\mu+b\int\tilde{f}^{2}d\mu$

where $\tilde{f}:=f-\int fd\mu$. Following [BG99] we introduce the following
Young function $N(x):=x^{2}\log(1+x^{2})$. Now Inequality (3.9) together with
(3.10) imply that, for any $f$ smooth enough, and since
$\|h\|_{U}\leq\sqrt{2}\|h\|_{N}$ by Lemma 3.19, it holds

$\int f^{2}F\left(\frac{f^{2}}{\mu(f^{2})}\right)d\mu\leq c\int|\nabla
f|^{2}d\mu+c\|\tilde{f}\|_{N}^{2}+c\|\tilde{f}\|_{2}^{2}$

for some constant $c$. On the one hand, the Poincaré inequality with constant
$C_{P}$ asserts that $\|\tilde{f}\|_{2}^{2}\leq\frac{C_{LS}}{2}\int|\nabla
f|^{2}d\mu$. On the other hand, Bobkov and Gotze proved in [BG99] (see their
remark after Proposition 4.1) that the log-Sobolev inequality is equivalent to
$\|\tilde{f}\|_{N}^{2}\leq a\int|\nabla f|^{2}d\mu$ for some constant $a$.
Therefore for any $f$ smooth enough, the following tight $F$-Sobolev
inequality holds

$\int f^{2}F\left(\frac{f^{2}}{\mu(f^{2})}\right)d\mu\leq c\int|\nabla
f|^{2}d\mu.$

Since $G\leq F$, inequality (3.8) follows.

Our next step is to use Capacity formalism to reduce (3.8) to some Nash-type
inequality. We may use known theorems from the literature. Introduce first the
notion of Capacity of a set $A\subset\mathbb{R}^{n}$ with $\mu(A)\leq 1/2$
with respect to a probability measure

$\mathrm{Cap}_{\mu}(A):=\inf\left\\{\int|\nabla f|^{2}d\mu:0\leq f\leq
1,f\mathds{1}_{A}=1,\mu(f=0)\geq 1/2\right\\}.$

The notion of electrostatic Capacity goes back to Maz’ya [Maz85] while the
above proper adaptation to probability settings is due to [BR03]. We refer the
interested reader to [BCR06, Section 5.2] for a historical presentation. Now,
using [BCR06, Theorem 22], Inequality (3.8) implies that for all
$A\subset\mathbb{R}^{n}$ with $\mu(A)\leq 1/2$, it holds

$\mu(A)G\left(\frac{1}{\mu(A)}\right)\leq c\;\mathrm{Cap}_{\mu}(A)$

for some constant $c$. Since $\|\tilde{f}\|_{2}^{2}\leq C_{P}\int|\nabla
f|^{2}d\mu$ by assumption, by [BR03, Section 4], one has $\mu(A)\leq
c\mathrm{Cap}_{\mu}(A)$ for all $A\subset\mathbb{R}^{n}$ with $\mu(A)\leq
1/2$. Therefore, setting
$\alpha(x):=G(x)\mathds{1}_{[e^{2},\infty)}+4\mathds{1}_{[1,e^{2}]}$ (devised
so that $x\mapsto x/\alpha(x)$ is non-decreasing), we are guaranteed that

$\mu(A)\alpha\left(\frac{1}{\mu(A)}\right)\leq c\;\mathrm{Cap}_{\mu}(A)$

for all $A\subset\mathbb{R}^{n}$ with $\mu(A)\leq 1/2$ (and some constant
$c$). Applying [BCR07, Corollary 6] (with $\beta:=c/\alpha$), we can conclude
that, for any $f$ smooth enough, it holds

$\int f^{2}d\mu\leq\frac{8c}{\alpha(s)}\int|\nabla
f|^{2}d\mu+s\left(\int|f|d\mu\right)^{2}$

for all $s\geq 1$. This is a Nash-type inequality that, optimizing over $s\geq
1$, can be recast as

$\theta(\|f\|_{2}^{2})\leq\int|\nabla f|^{2}d\mu$

for all $f$ satisfying $\|f\|_{1}=1$, with $\theta(x):=\frac{1}{8c}\sup_{s\geq
1}\alpha(s)(x-s)$, $x\geq 1$. It is easy to see that $\theta(x)\geq
cx\mathds{1}_{[1,x_{o}]}+cx(\log x)^{2}\mathds{1}_{[x_{o},\infty)}$, $x\geq
1$, for some constant $c>0$ and $x_{o}\geq 1$. Hence
$n(t):=\int_{t}^{\infty}\frac{1}{\theta(x)}dx<\infty$. The expected ultra-
contractivity property follows from [Cou96, Theorem II.5] with
$m(t)=n^{-1}(t)$. This ends the proof. ∎

In the proof of Theorem 3.17, we used the following lemma (recall that
$N(x):=x^{2}\log(1+x^{2})$).

######  Lemma 3.19.

For any $h\in\mathbb{L}_{N}$ it holds
$\frac{1}{\sqrt{2}}\|h\|_{2}\leq\|h\|_{U}\leq\sqrt{2}\|h\|_{N}$.

###### Proof.

Observe first that (after some algebra)

$U(x)\leq\begin{cases}\frac{x^{2}}{2}&\mbox{ if }|x|\leq\sqrt{2}\\\
N(x)&\mbox{ if }|x|\geq\sqrt{2}.\end{cases}$

Also, it is easy to see that, for any $x>0$, $N(x/\sqrt{2})\leq N(x)/2$ (a
property known as $\nabla_{2}$-condition, see [RR91]). Therefore, for
$h\in\mathbb{L}_{N}$, since $U(x)\leq\frac{x^{2}}{2}+N(x)$, one has

$\displaystyle\int U\left(\frac{h}{\sqrt{2}\|h\|_{N}}\right)d\mu$
$\displaystyle\leq\int\frac{h^{2}}{4\|h\|_{N}^{2}}d\mu+\int
N\left(\frac{h}{\sqrt{2}\|h\|_{N}}\right)d\mu$
$\displaystyle\leq\frac{\|h\|_{2}^{2}}{4\|h\|_{N}^{2}}+\frac{1}{2}\int
N\left(\frac{h}{\|h\|_{N}}\right)d\mu=\frac{\|h\|_{2}^{2}}{4\|h\|_{N}^{2}}+\frac{1}{2}\leq\frac{5}{16}+\frac{1}{2}<1$

where we used that $\|h\|_{2}\leq\frac{\sqrt{5}}{2}\|h\|_{N}$, a property
proved in [BG99, Lemma 4.3]. The second inequality follows from the very
definition of the Luxembourg norm.

The first inequality follows at once from the fact that
$U(x)\geq\frac{x^{2}}{2}$, $x\in\mathbb{R}$, and the definition of the
Luxembourg norm. ∎

Now we turn to a second example of application. Consider the Gaussian measure
$\mu=\gamma_{n}$ associated to the Ornstein-Ulhenbeck semi-group
$(P_{t})_{t\geq 0}$ with any $\Psi$. We will prove the following statement
that is related to a conjecture by Talagrand.

######  Proposition 3.20.

Let $\Psi$ be a Young function satisfying
$\lim_{x\to\infty}\frac{\Psi(x)}{\Phi(x)}=+\infty$. Then, the Ornstein-
Ulhenbeck semi-group $P_{t}$ is not continuous from $\mathbb{L}_{\Phi}$ into
$\mathbb{L}_{\Psi}$ for any $t\geq 0$.

######  Remark 3.21.

This complement a similar statement in [BBB+13] when $\Phi(x)=|x|$, namely,
the authors proved that the Ornstein-Ulhenbeck semi-group is not continuous
from $\mathbb{L}_{1}$ into $\mathbb{L}_{\Psi}$ with $\Psi$ any Young function
satisfying $\lim_{x\to\infty}\frac{\Psi(x)}{x}=+\infty$.

Such properties are in fact related to a conjecture by Talagrand [Tal89] that
we now briefly describe in the context of the Ornstein-Ulhenbeck semi-group
(the original conjecture was stated for a discrete semi-group on the hypercube
$\\{0,1\\}^{n}$). Since the Ornstein-Ulhenbeck semi-group satisfies the
hypercontractivity property $\|P_{t}f\|_{q}\leq\|f\|_{p}$, for any $p\geq 1$
and $q\leq 1+(p-1)e^{2t}$, the question posed by Talagrand is: what happens
for $p=1$? In other words is the semi-group regularizing in some sense when
starting with functions only in $\mathbb{L}_{1}$? The conjecture is actually
more precise, Talagrand ([Tal89, Conjecture 1]) asks for proving
$\lim_{s\to\infty}\sup_{n}\sup_{f\geq 0:\int
fd\gamma_{n}=1}s\gamma_{n}(\\{x:Ptf(x)\geq s\\})=0$. In [Tal89, Conjecture 1],
a precise decay is given: prove that $\gamma_{n}(\\{x:Ptf(x)\geq
s\\})\leq\frac{c}{s\sqrt{\log s}}$ for any $s\geq 2$, say, and with a constant
$c$ that is independent on $n$ and $f\geq 0$ such that $\int fd\gamma_{n}=1$
(but depending possibly on $t$). Both conjectures were solved by Eldan-Lee
[EL15] and Lehec [Leh16] after partial answers from [BBB+13]. We refer the
reader to [GMRS17, GLM+] for an historical presentation and related results.
It should be noticed that, given $f$ and $n$, by Lebesgue’s theorem,
$\lim_{s\to\infty}s\gamma_{n}(\\{x:Ptf(x)\geq s\\})=0$. Therefore the
conjecture is all contained in the uniformity on the function and the
dimension. Also, Markov’s inequality insures by invariance that
$s\gamma_{n}(\\{x:Ptf(x)\geq s\\})\leq 1$ for all $f\geq 0$ such that $\int
fd\gamma_{n}=1$.

What Proposition 3.20 suggests is that a similar regularization effect should
hold for functions in $\mathbb{L}_{1}\log\mathbb{L}_{1}$. Using Eldan-Lee and
Lehec’s result together with Jensen’s inequality, observe that, for any non-
negative $f$ with $\|f\|_{\Phi}=1$ (which is equivalent to saying that
$\|\Phi(f)\|_{1}=1$) and $s\geq 2$, it holds

$\displaystyle\gamma_{n}(\\{P_{t}f\geq s\\})$
$\displaystyle=\gamma_{n}(\\{\Phi(P_{t}f)\geq\Phi(s)\\})\leq\gamma_{n}(\\{P_{t}\Phi(f)\geq\Phi(s)\\})\leq\frac{c}{\Phi(s)\sqrt{\log(\phi(s))}}$
$\displaystyle\leq\frac{c^{\prime}}{s\log(s)^{\frac{3}{2}}}.$

A natural question is therefore: is the latter can be improved, or the upper
bound (asymptotically) optimal?

As already mentioned Proposition 3.20 may hold for any $1$-regular Young
function with $\lim_{x\to\infty}\frac{\Phi(x)}{x}=\infty$. Therefore one can
ask for similar question for any such $\Phi$’s.

###### Proof of Proposition 3.20.

Fix $\Psi$, a Young function satisfying
$\lim_{x\to\infty}\frac{\Psi(x)}{\Phi(x)}=+\infty$. Assume by contradiction
that $P_{t}$ is continuous from $\mathbb{L}_{\Phi}$ into $\mathbb{L}_{\Psi}$,
that is, there exists a constant $C>0$ such that $\|P_{t}f\|_{\Psi}\leq
C\|f\|_{\Phi}$ for all $f\in\mathbb{L}_{\Phi}$. Observe that we can modify
$\Psi$ on any finite interval $[0,A]$ without modifying the qualitative
character of the Luxembourg norm $\|\cdot\|_{\Psi}$. Therefore, we can assume
without loss of generality that
$\Theta:=\sup_{x}\log\Phi^{-1}(x)/\Psi^{-1}(x)<\infty$. By Proposition (3.16)
it holds

$\int
f^{2}\log_{+}\left(\frac{f^{2}}{\|f\|_{U}^{2}}\right)\theta_{+}\left(U\left(\frac{f}{\|f\|_{U}}\right)\right)d\gamma_{n}\leq
c\int|\nabla f|^{2}d\gamma_{n}+c\|f\|_{U}^{2}$

for some constant $c$ that does not depend on $f$. Let $a\geq 1$ and
$f_{a}(x)=e^{ax_{1}/2}$, $x=(x_{1},\dots,x_{n})\in\mathbb{R}^{2}$. A simple
computation shows that $\int|\nabla f_{a}|^{2}d\gamma_{n}=\frac{a^{2}}{4}\int
f_{a}^{2}d\gamma_{1}(x_{1})=\frac{a^{2}}{4}e^{a^{2}/2}$. Therefore, by Lemma
3.22, monotonicity and positivity, the latter applied to $f_{a}$ implies that

$e^{a^{2}/2}\inf_{y\geq\frac{e^{a^{2}/2}}{\|f_{a}\|_{U}}}\left\\{\theta_{+}\left(U\left(y\right)\right)\right\\}\int_{a}^{\infty}e^{-\frac{1}{2}(x_{1}-a)^{2}}\log_{+}\left(\frac{e^{\frac{a}{2}(2x_{1}-a)}}{4a^{2}}\right)dx_{1}\leq
c^{\prime}a^{2}e^{a^{2}/2}$

for some constant $c^{\prime}$ independent on $a$. For $a$ sufficiently large,
the integral can be bounded below by $c^{\prime\prime}a^{2}$ where
$c^{\prime\prime}$ is independent on $a$. Therefore, using again Lemma 3.22,
it holds

$\inf_{y\geq\frac{e^{a^{2}/4}}{2a}}\left\\{\theta_{+}\left(U\left(y\right)\right)\right\\}\leq
c^{\prime\prime\prime}.$

Since $\lim_{\infty}U=\lim_{\infty}\theta=+\infty$, this is a contradiction
when $a\to\infty$. The proof is complete. ∎

The proof of Proposition 3.20 relied on the following lemma.

######  Lemma 3.22.

Let $a\geq 1$, $f_{a}(t)=e^{at/2}$, $t\in\mathbb{R}$ and $\gamma$ be the
standard Gaussian measure on the line. Then, it holds

$\|f_{a}\|_{U}\leq 2ae^{a^{2}/4}.$

###### Proof.

From Lemma 3.19 we have $\|f_{a}\|_{U}\leq\sqrt{2}\|f_{a}\|_{N}$. Therefore we
need to find an upper bound on $\|f_{a}\|_{N}$. Fix $\lambda>0$. Since
$N(x)=x^{2}\log(1+x^{2})\leq 1+x^{2}\log(x^{2})$, it holds

$\displaystyle\int N\left(\frac{f_{a}}{\lambda}\right)d\gamma\leq
1+\frac{1}{\lambda^{2}}\int
ate^{at-\frac{t^{2}}{2}}\frac{dt}{\sqrt{2\pi}}-\frac{2\log\lambda}{\lambda^{2}}\int
f_{a}^{2}d\gamma=1+\frac{e^{a^{2}/2}}{\lambda^{2}}\left(a^{2}-2\log\lambda\right).$

Choose $\lambda=ae^{a^{2}/4}$ (which is greater than $1$) so that $\int
N\left(\frac{f_{a}}{\lambda}\right)d\gamma\leq 2$. Finally note that
$N(x/\sqrt{2})\leq\frac{1}{2}N(x)$, $x\in\mathbb{R}$ so that
$\|f_{a}\|_{N}\leq\sqrt{2}\lambda$ from which the expected result follows. ∎

##  4\. Contraction property for inhomogeneous Markov semi-groups

In this section we deal with the time-dependent diffusion operators
$L_{t}:=\Delta-\nabla V_{t}\cdot\nabla$, $t\geq 0$, on $\mathbb{R}^{n}$, with
$V_{t}$ smooth enough and such that $\int e^{-V_{t}}=1$. Recall that the
associated semi-group $(P^{(t)}_{s})_{s\geq 0}$ is reversible with respect to
the probability measure $\mu_{t}(dx):=e^{-V_{t}(x)}dx$. All along this section
Luxembourg norms are understood with respect to $\mu_{t}$.

In order to obtain contraction bounds for $P^{(t)}_{t}f$, one could try to use
the following natural simple strategy. For $t$ fixed, one may assume that
$\mathrm{Hess}(V_{t})\geq\rho_{t}$ for some $\rho_{t}>0$ so that Gross’
theorem applies and leads to

$\|P_{s}^{(t)}f\|_{q_{t}(s),\mu_{t}}\leq\|f\|_{2,\mu_{t}}\qquad s,t\geq 0,$

with, say, $q_{t}(s)\leq 1+e^{(4/\rho_{t})s}$ where we set
$\|g\|_{q,\mu_{t}}:=\left(\int|g|^{q}d\mu_{t}\right)^{\frac{1}{q}}$ for the
$\mathbb{L}_{q}$ norm of $g$ with respect to $\mu_{t}$ (we choose $q_{t}(0)=2$
for simplicity). Applying the latter at time $s=t$ leads to

$\|P_{t}^{(t)}f\|_{q_{t}(t),\mu_{t}}\leq\|f\|_{2,\mu_{t}}.$

Now observe that the latter might be very weak if $\rho_{t}$ is small.
Moreover and more essentially one would like to deal with a norm on the right
hand side independent of $t$ (say related to $\mu_{0}$). Before achieving this
program, by means of Lemma 2.1, let us end this introduction with an easy (and
very specific) example of inhomogeneous Markov semi-group whose
hypercontractivity property can be derived from the results of the previous
section.

Consider for instance $V_{t}=U_{1}$ for $t\in[0,T]$ and $V_{t}=U_{2}$ for
$t>T$ where $U_{1}$ and $U_{2}$ are associated to hypercontractivity
properties in Orlicz spaces $\mathbb{L}_{\Phi_{t}^{(1)}}$ and
$\mathbb{L}_{\Phi_{t}^{(2)}}$ respectively. Then, we can argue that
$\|P_{t}^{(t)}f\|_{\Phi_{t}^{(1)}}\leq\|P_{s}^{(s)}f\|_{\Phi_{s}^{(1)}}$, for
any $s\leq t\leq T$, and then
$\|P_{t}^{(t)}f\|_{\Phi_{t}^{(2)}}\leq\|P_{s}^{(s)}f\|_{\Phi_{s}^{(2)}}$ for
any $T<s\leq t$. Therefore, if the two families of Orlicz spaces coincide at
time T, i.e. $\Phi_{T}^{(1)}=\Phi_{T}^{(2)}$, and potentially modulo some
extra assumptions on the Young functions $\Phi_{t}^{(1)}$, $\Phi_{t}^{(2)}$,
if we set $\Phi_{t}=\Phi_{t}^{(1)}$ for $t\in[0,T]$ and
$\Phi_{t}:=\Phi_{t}^{(2)}$ for $t\geq T$, one has
$\|P_{t}^{(t)}f\|_{\Phi_{t}}\leq\|P_{s}^{(t)}f\|_{\Phi_{s}}$ for all $s\leq
t$. As already mentioned this is however very specific and somehow
artificially inhomogeneous. We would like to deal with examples of potentials
$V_{t}$ that evolve all along the time $t$.

In the next section we will deal under the restricted hypothesis of the log-
Sobolev inequality (4.1) related to the Orlicz family $\Phi_{t}(x)=|x|^{q(t)}$
($\mathbb{L}_{p}$ scale). This makes the presentation more precise and easier
by reducing some technicalities. However, it already encompasses many of the
difficulties. The last section (that comes after) will finally deal with a
more general setting.

We stress that the results below are a first step in the understanding of
contraction properties for inhomogeneous Markov semi-groups. Many remains to
be discovered and we believe that our investigations open new lines of
research with possible application, as mentioned in the introduction, to non-
linear parabolic time dependent problems (in infinite dimension).

###  4.1. $\mathbb{L}_{p}$-scales

In order to give the flavor of what is happening in the inhomogeneous setting
(and avoid some technicalities), in this section we may only deal with
contractivity properties in $\mathbb{L}_{p}$-scales. Recall that
$X_{-}=\max(-X,0)$ stands for the negative part.

######  Theorem 4.1.

Consider the inhomogeneous diffusion operator $L_{t}$ as above. Set
$a_{t}:=\|(\dot{V}_{t})_{-}\|_{\infty}$,
$b_{t}:=\||\nabla\dot{V}_{t}|\|_{\infty}$, $c_{t}:=\|(\nabla
V_{t}\cdot\nabla\dot{V}_{t}-\Delta\dot{V}_{t})_{-}\|_{\infty}$ and assume that
$a_{t},b_{t},c_{t}<\infty$ for all $t\geq 0$. Assume also that, for all $t\geq
0$ there exists $\rho_{t}\in\mathbb{R}$ such that
$\mathrm{Hess}(V_{t})\geq\rho_{t}$. Finally, assume that there exists
$\bar{\rho}_{t}>0$ such that the following log-Sobolev inequality holds

(4.1) $\int f^{2}\log(f^{2})d\mu_{t}\leq\bar{\rho}_{t}\int|\nabla
f|^{2}d\mu_{t}.$

for all $f$ with $\int f^{2}d\mu_{t}=1$ for which the right hand side is well
defined. Then, for any $f\colon\mathbb{R}^{n}\to\mathbb{R}_{+}$ smooth enough,
any $p>1$ and any $s\leq t$, it holds

$\|P_{t}^{(t)}f\|_{\Phi_{t}}\leq m(s,t)\|P_{s}^{(s)}f\|_{\Phi_{s}}$

where $\Phi_{t}(x)=|x|^{q(t)}$,
$q(t)=1+(p-1)\exp\\{\int_{0}^{t}(2/\bar{\rho}_{s})ds\\}$, $t\geq 0$ and

$m(s,t):=\exp\left\\{\int_{s}^{t}\frac{a_{u}}{q(u)}+uc_{u}+b_{u}^{2}\frac{1-e^{-\rho_{u}u}}{2\rho_{u}}(q(u)-1)du\right\\}.$

######  Remark 4.2.

As already mentioned, one possible criterion for the log-Sobolev inequality
(4.1) to hold, is $\mathrm{Hess}(V_{t})\geq\rho_{t}>0$ (as a matrix), which
implies $\bar{\rho}_{t}\leq\frac{2}{\rho_{t}}$. Alternatively, as will be used
below, one can apply a perturbation argument _à la_ Holley & Stroock [HS87].

If $V_{t}$ does not depend on $t$ then $m(t,s)=1$ and
$q(t)=1+(p-1)e^{(2/\bar{\rho})t}$, which is Gross’ theorem off by a factor of
$2$ in the exponential (see the introduction). This is coming from a technical
computation that uses Cauchy-Schwarz’ inequality. One can actually improve
this and get
$q(t)=1+(p-1)\exp\\{(2-\varepsilon)\int_{0}^{t}(2/\bar{\rho}_{s})ds\\}$, for
any $\varepsilon>0$, but at the price of a factor $m(s,t)$ that depends on
$\varepsilon$, and that increases when $\varepsilon$ decreases.

Modulo such a factor 2, the above theorem can therefore be seen as an
inhomogeneous counter part of Gross’ theorem.

######  Example 4.3.

The above theorem contains some non trivial examples. For instance one can
consider potentials of the form $V_{t}=U(x)+\alpha(t)V+\beta(t)$ with $V$
unbounded and $\beta(t):=\log\int e^{-U-\alpha V}dx$ so that $\mu_{t}$ indeed
defines a probability measure.

Take for instance $U(x)=\frac{|x|^{2}}{2}$,
$\alpha\colon[0,\infty)\to[0,\infty)$ non-decreasing and
$V(x)=(1+|x|^{2})^{\frac{\beta}{2}}$, with $\beta\in(0,1]$ (this is an
unbounded (time-dependent) perturbation of the standard Gaussian potential
$U$).

Then, $\dot{V}_{t}=\alpha^{\prime}(t)(1+|x|^{2})^{\frac{\beta}{2}}$ so that
$a_{t}=0$;
$\nabla\dot{V}_{t}=\alpha^{\prime}(t)\beta(1+|x|^{2})^{\frac{\beta}{2}-1}x$,
whence
$b_{t}=\alpha^{\prime}(t)\beta\sqrt{\frac{(1-\beta)^{1-\beta}}{(2-\beta)^{2-\beta}}}$
(which is understood as its limit when $\beta=1$, namely
$b_{t}=\alpha^{\prime}(t)$ if $\beta=1$). Using crude estimates, it is not
difficult to prove that $c_{t}\leq n^{2}\alpha^{\prime}(t)(\alpha(t)+2)$. On
the other hand (again we omit details) $\mathrm{Hess}(V_{t})\geq 1$ so that
$\rho_{t}=1$ and (4.1) holds with $\bar{\rho}_{t}=2$. Theorem 4.1 then implies
that the corresponding inhomenegous semi-group $(P_{t}^{(t)})_{t\geq 0}$ is
hyper-bounded in the $\mathbb{L}_{q(t)}$ scale, with $q(t)=1+(p-1)e^{t}$.

For $V(x)=\log(1+|x|^{2})$ one can easily see that $a_{t}=0$,
$b_{t}=\alpha^{\prime}(t)$ and $c_{t}<\infty$. The issue is coming from
estimating $\bar{\rho}_{t}$. In fact $\mathrm{Hess}(V_{t})(x)$ is bounded
below by a positive matrix only outside a ball (of radius proportional to
$\alpha$). Therefore, one can write $V_{t}=H+R$, with $H$ strictly convex, in
the sense that $\mathrm{Hess}(H)\geq 1/2$, say, and $R$ is bounded. Then
Bakry-Emery criterion applies to the measure with density proportional to
$e^{-H}$, leading to a log-Sobolev constant at most $4$, and then we use
Holley-Stroock perturbation lemma, see e.g. [ABC+00, THeorem 3.4.3] to get
Inequality (4.1) with constant $\bar{\rho}_{t}$ at most $4e^{\mathrm{Osc}(R)}$
(therefore potentially exponentially big in $\alpha$) where
$\mathrm{Osc}(R)=\sup R-\inf R$ is the oscillation of $R$. Theorem 4.1 applies
and finally leads to some contraction property with $q(t)\to\infty$ for
$\alpha$ bounded or slowly growing to infinity (for instance
$\alpha(t)=\log\log t$ would do).

###### Proof of Theorem 4.1.

Let $\Phi_{t}(x):=|x|^{q(t)}$ with $q\colon[0,\infty)\to[0,\infty)$ increasing
and satisfying $q(0)>1$. Set $N(t):=\|P_{t}^{(t)}f\|_{\Phi_{t}}$ for some non-
negative smooth $f$, and $g:=\frac{P_{t}^{(t)}f}{N}$ so that
$\int\Phi_{t}(g)d\mu_{t}=1$. From Lemma 2.1, we have

$\displaystyle N^{\prime}$ $\displaystyle(t)\int
g\Phi_{t}^{\prime}(g)d\mu_{t}\leq
N(t)\left(\int\dot{\Phi}_{t}(g)d\mu_{t}-\int\phi_{t}^{\prime\prime}(g)|\nabla
g|^{2}d\mu_{t}+a_{t}\right)$
$\displaystyle+b_{t}\int\int_{0}^{t}P_{t-s}^{(t)}f|\nabla
P_{s}^{(t)}(\Phi_{t}^{\prime}(g))|dsd\mu_{t}+c_{t}\int\int_{0}^{t}P_{t-s}^{(t)}fP_{s}^{(t)}(\Phi_{t}^{\prime}(g))dsd\mu_{t}.$

We observe that $x\Phi_{t}^{\prime}(x)=q\Phi_{t}(x)$ so that $\int
g\Phi_{t}^{\prime}(g)d\mu_{t}=q$. Also, by reversibility, the last term of the
latter satisfies

$\displaystyle\int\int_{0}^{t}P_{t-s}^{(t)}fP_{s}^{(t)}(\Phi_{t}^{\prime}(g))dsd\mu_{t}$
$\displaystyle=\int_{0}^{t}\int
P_{s}^{(t)}(P_{t-s}^{(t)}f)\Phi_{t}^{\prime}(g)d\mu_{t}ds$
$\displaystyle=tN(t)\int g\Phi_{t}^{\prime}(g)d\mu_{t}=tN(t)q(t).$

Since $\dot{\Phi}_{t}(x)=q^{\prime}(t)|x|^{q}\log(|x|)$ and
$\Phi_{t}^{\prime\prime}(x)=q(q-1)|x|^{q-2}$, we get

$\displaystyle q(t)\frac{N^{\prime}(t)}{N(t)}$
$\displaystyle\leq\frac{q^{\prime}(t)}{q(t)}\mathrm{Ent}_{\mu_{t}}(g^{q})-q(q-1)\int
g^{q-2}|\nabla g|^{2}d\mu_{t}+a_{t}+tc_{t}q(t)$
$\displaystyle\quad+\frac{b_{t}}{N(t)}\int\int_{0}^{t}P_{t-s}^{(t)}f|\nabla
P_{s}^{(t)}(\Phi_{t}^{\prime}(g))|dsd\mu_{t}.$

The condition $\mathrm{Hess}(V_{t})\geq\rho_{t}$ ensures that $|\nabla
P_{s}^{(t)}h|\leq e^{-\rho_{t}s}P_{s}^{(t)}(|\nabla h|)$ for all $s\geq 0$ and
all $h$ (see e.g. [ABC+00][Proposition 5.4.5]). Hence, by reversibility

$\displaystyle\int\int_{0}^{t}P_{t-s}^{(t)}f|\nabla
P_{s}^{(t)}(\Phi_{t}^{\prime}(g))|dsd\mu_{t}$
$\displaystyle\leq\int\int_{0}^{t}e^{-\rho_{t}s}P_{t-s}^{(t)}fP_{s}^{(t)}(|\nabla\Phi_{t}^{\prime}(g)|)dsd\mu_{t}$
$\displaystyle=\int\int_{0}^{t}e^{-\rho_{t}s}P_{t}^{(t)}f\Phi_{t}^{\prime\prime}(g)|\nabla
g|dsd\mu_{t}$ $\displaystyle=N(t)\frac{1-e^{-\rho_{t}t}}{\rho_{t}}q(q-1)\int
g^{q-1}|\nabla g|d\mu_{t}.$

Using the inequality
$uv\leq\frac{1}{2\varepsilon}u^{2}+\frac{\varepsilon}{2}v^{2}$ with
$\varepsilon=b_{t}\frac{1-e^{-\rho_{t}t}}{\rho_{t}}$,
$u=g^{\frac{q}{2}-1}|\nabla g|$ and $v=g^{\frac{q}{2}}$ we get

$\frac{1-e^{-\rho_{t}t}}{\rho_{t}}\int g^{q-1}|\nabla
g|d\mu_{t}\leq\frac{1}{2b_{t}}\int g^{q-2}|\nabla
g|^{2}d\mu_{t}+b_{t}\frac{1-e^{-\rho_{t}t}}{2\rho_{t}}\int g^{q}d\mu_{t}$

so that, since $\int g^{q}d\mu_{t}=\int\Phi_{t}(g)d\mu_{t}=1$,

$\displaystyle q(t)\frac{N^{\prime}(t)}{N(t)}$
$\displaystyle\leq\frac{q^{\prime}(t)}{q(t)}\mathrm{Ent}_{\mu_{t}}(g^{q})-\frac{q(q-1)}{2}\int
g^{q-2}|\nabla g|^{2}d\mu_{t}+a_{t}+tc_{t}q(t)$
$\displaystyle\quad+b_{t}^{2}\frac{1-e^{-\rho_{t}t}}{2\rho_{t}}q(t)(q(t)-1).$

Next, we observe that $\int g^{q-2}|\nabla
g|^{2}d\mu_{t}=\frac{4}{q^{2}}\int|\nabla g^{\frac{q}{2}}|^{2}d\mu_{t}$.
Hence, for $q(t):=1+(p-1)\exp\\{\int_{0}^{t}(2/\bar{\rho}_{s})ds\\}$ which
satisfies $\frac{2(q-1)}{q^{\prime}}=\bar{\rho}_{t}$, we are guaranteed by
(4.1) that

$\displaystyle\frac{q^{\prime}(t)}{q(t)}\mathrm{Ent}_{\mu_{t}}(g^{q})$
$\displaystyle-\frac{q(q-1)}{2}\int g^{q-2}|\nabla g|^{2}d\mu_{t}$
$\displaystyle=\frac{q^{\prime}(t)}{q(t)}\left(\mathrm{Ent}_{\mu_{t}}(g^{q})-\frac{2(q-1)}{q^{\prime}(t)}\int|\nabla
g^{\frac{q}{2}}|^{2}d\mu_{t}\right)$
$\displaystyle=\frac{q^{\prime}(t)}{q(t)}\left(\mathrm{Ent}_{\mu_{t}}(g^{q})-\bar{\rho}_{t}\int|\nabla
g^{\frac{q}{2}}|^{2}d\mu_{t}\right)\leq 0.$

It follows that

$\displaystyle\frac{N^{\prime}(t)}{N(t)}$
$\displaystyle\leq\frac{a_{t}}{q(t)}+tc_{t}+b_{t}^{2}\frac{1-e^{-\rho_{t}t}}{2\rho_{t}}(q(t)-1)$

which leads to the desired conclusion. ∎

###  4.2. General result

In this section we establish a more general result than Theorem 4.1, at the
price of somehow heavy hypothesis.

######  Theorem 4.4.

Consider the inhomogeneous diffusion operator $L_{t}$ as above. Assume that
for all $t\geq 0$, $b_{t}:=\||\nabla\dot{V}_{t}|\|_{\infty}<\infty$ and that
there exists $\rho_{t}\in\mathbb{R}$ such that
$\mathrm{Hess}(V_{t})\geq\rho_{t}$ (as a matrix).

Let $(\Phi_{t})_{t\geq 0}$ be a family of Young functions satisfying
$\Phi_{t}(x)\leq x\Phi_{t}^{\prime}(x)\leq B_{t}\Phi_{t}(x)$,
${\Phi_{t}^{\prime}}^{2}\leq C_{t}\Phi_{t}\Phi_{t}^{\prime\prime}$ and
$x^{2}\Phi_{t}^{\prime\prime}(x)\leq D_{t}\Phi_{t}(x)+E_{t}$ for all $x\geq 0$
and some constants $B_{t},C_{t},D_{t},E_{t}$.

Assume that for all $t\geq 0$ there exist $\delta_{t}\in[0,1)$ and
$F_{t}\in\mathbb{R}$ such that
$(\dot{V}_{t})_{-}\leq\frac{\delta_{t}}{4C_{t}}\left(|\nabla
V_{t}|^{2}-2\Delta V_{t}\right)+F_{t}$.

Set $W_{t}:=(\nabla V_{t}\cdot\nabla\dot{V}_{t}-\Delta\dot{V}_{t})_{-}$ and
denote by $\bar{\rho}_{t}\in(0,\infty]$ the best constant such that for all
$f$ with $\|f\|_{\Phi_{0}}=1$ it holds

(4.2)
$\int\dot{\Phi}_{t}(f)d\mu_{t}\leq\bar{\rho}_{t}\int\Phi_{t}^{\prime\prime}(f)|\nabla
f|^{2}d\mu_{t}.$

Finally, assume either that
$(i)$ $c_{t}:=\|W_{t}\|_{\infty}<\infty$ and $\bar{\rho}_{t}<1-\delta_{t}$;
or
$(ii)$ $c^{\prime}_{t}:=\max\left(2\||\nabla
W_{t}|\|_{\infty}/b_{t},\sup_{x:W_{t}(x)\neq
0}\left(\frac{L_{t}W_{t}}{W_{t}}-\rho_{t}\right)_{-}\right)<\infty$ and that
for all $t\geq 0$ there exists $\delta^{\prime}_{t}\in[0,1)$ and
$F^{\prime}_{t}\in[0,\infty)$ such that
$\delta^{\prime}_{t}\int_{0}^{t}e^{(c^{\prime}_{s}-\rho_{s})s}ds<1$ and
$W_{t}\leq\frac{\delta^{\prime}_{t}}{4B_{t}C_{t}}\left(|\nabla
V_{t}|^{2}-2\Delta V_{t}\right)+F^{\prime}_{t}$.

Then, for any $f\colon\mathbb{R}^{n}\to\mathbb{R}_{+}$ smooth enough, it holds

$\|P_{t}^{(t)}f\|_{\Phi_{t}}\leq m(s,t)\|P_{s}^{(s)}f\|_{\Phi_{s}}$

where under assumption $(i)$,

$m(s,t)=\exp\left\\{\int_{s}^{t}F_{u}+\left(b_{u}\frac{1-e^{-\rho_{u}u}}{\rho_{u}}\right)^{2}\frac{D_{u}+E_{u}}{2(1-\delta_{u}-\bar{\rho}_{u})}+c_{u}B_{u}udu\right\\},$

and under assumption $(ii)$,

$\displaystyle m(t,s)=\exp$
$\displaystyle\left\\{\int_{s}^{t}\left(\int_{0}^{u}e^{(c^{\prime}_{v}-\rho_{v})v}dv\right)^{2}\frac{b_{u}(D_{u}+E_{u})}{2(1-\delta_{u}-\bar{\rho}_{u}-\delta^{\prime}_{u}\int_{0}^{u}e^{(c^{\prime}_{v}-\rho_{v})v}dv)}\right.$
$\displaystyle\qquad\qquad\left.+\int_{0}^{u}e^{(c^{\prime}_{v}-\rho_{v})v}dv\left(B_{u}F^{\prime}_{u}+F_{u}\right)du\right\\}.$

######  Remark 4.5.

Observe that, when $\delta_{t}=0$, the assumption
$(\dot{V}_{t})_{-}\leq\frac{\delta_{t}}{C_{t}}\left(|\nabla V_{t}|^{2}-2\Delta
V_{t}\right)+D_{t}$ amounts to $a_{t}:=\|(\dot{V}_{t})_{-}\|_{\infty}<\infty$
which is the assumption that we used in Theorem 4.1. Also, it might be that
$\dot{V}_{t}\geq 0$ so that, in that case, one chooses $\delta_{t}=D_{t}=0$.

Observe also that the first inequality in the assumption $\Phi_{t}(x)\leq
x\Phi_{t}^{\prime}(x)\leq B_{t}\Phi_{t}(x)$ is satisfied by all Young
functions, while the second inequality is a consequence of the
$\Delta_{2}$-condition.

Finally we observe that, although we weakened most of the hypothesis of
Theorem 4.1, one key assumption one would like to remove/reduce is
$b_{t}=\||\nabla\dot{V}_{t}|\|_{\infty}<\infty$. Indeed one interesting
example one would like to deal with is for instance
$V_{t}(x)=(1-t)_{+}^{2}|x|^{2}+|x|^{\alpha}$, with, say, $\alpha\in(1,2)$.
Such an example is not covered by Theorem 4.4 since $b_{t}=\infty$.

###### Proof of Theorem 4.4.

We start as in the proof of Theorem 4.1. Set
$N(t):=\|P_{t}^{(t)}f\|_{\Phi_{t}}$, $g:=\frac{P_{t}^{(t)}f}{N}$ so that from
Lemma 2.1 for some non-negative smooth function $f$, it holds

$\displaystyle N^{\prime}(t)\int g\Phi_{t}^{\prime}(g)d\mu_{t}$
$\displaystyle\leq
N(t)\left(\int\dot{\Phi}_{t}(g)d\mu_{t}-\int\phi_{t}^{\prime\prime}(g)|\nabla
g|^{2}d\mu_{t}-\int\Phi_{t}(g)\dot{V}_{t}d\mu_{t}\right)$ (4.3)
$\displaystyle\quad+\int\int_{0}^{t}P_{t-s}^{(t)}f\nabla
P_{s}^{(t)}(\Phi_{t}^{\prime}(g))\cdot\nabla\dot{V}_{t}dsd\mu_{t}$
$\displaystyle\quad-\int\int_{0}^{t}[\nabla
V_{t}\cdot\nabla\dot{V}_{t}-\Delta\dot{V}_{t}]P_{t-s}^{(t)}fP_{s}^{(t)}(\Phi_{t}^{\prime}(g))dsd\mu_{t}.$

We analyze each term separately.

Since $x\Phi_{t}^{\prime}(x)\geq\Phi_{t}(x)$, it holds $\int
g\Phi_{t}^{\prime}(g)d\mu_{t}\geq\int\Phi_{t}(g)d\mu_{t}=1$. Hence, if
$N^{\prime}(t)\geq 0$, the left hand side of the latter is bounded below by
$N^{\prime}(t)$.

Since $(\dot{V}_{t})_{-}\leq\frac{\delta_{t}}{4C_{t}}\left(|\nabla
V_{t}|^{2}-2\Delta V_{t}\right)+F_{t}$, we can use the expansion of the
square, namely Inequality (2.3) with $f=\sqrt{\Phi_{t}(g)}$, to get that

$\displaystyle-\int\Phi_{t}(g)\dot{V}_{t}d\mu_{t}$
$\displaystyle\leq\int\Phi_{t}(g)(\dot{V}_{t})_{-}d\mu_{t}\leq\frac{\delta_{t}}{C_{t}}\int\frac{{\Phi_{t}^{\prime}}^{2}(g)}{\Phi_{t}(g)}|\nabla
g|^{2}d\mu_{t}+F_{t}$
$\displaystyle\leq\delta_{t}\int\Phi_{t}^{\prime\prime}(g)|\nabla
g|^{2}d\mu_{t}+F_{t}.$

Now assume first that assumption $(i)$ holds, namely that $c_{t}=\|(\nabla
V_{t}\cdot\nabla\dot{V}_{t}-\Delta\dot{V}_{t})_{-}\|_{\infty}<\infty$. In that
case we can proceed as in the proof of Theorem 4.1 to get

$\displaystyle\int\int_{0}^{t}P_{t-s}^{(t)}f\nabla
P_{s}^{(t)}(\Phi_{t}^{\prime}(g))\cdot\nabla\dot{V}_{t}dsd\mu_{t}-\int\int_{0}^{t}[\nabla
V_{t}\cdot\nabla\dot{V}_{t}-\Delta\dot{V}_{t}]P_{t-s}^{(t)}fP_{s}^{(t)}(\Phi_{t}^{\prime}(g))dsd\mu_{t}$
$\displaystyle\leq b_{t}\int\int_{0}^{t}P_{t-s}^{(t)}f|\nabla
P_{s}^{(t)}(\Phi_{t}^{\prime}(g))|dsd\mu_{t}+c_{t}\int\int_{0}^{t}P_{t-s}^{(t)}fP_{s}^{(t)}(\Phi_{t}^{\prime}(g))dsd\mu_{t}$
$\displaystyle\leq b_{t}N(t)\frac{1-e^{-\rho_{t}t}}{\rho_{t}}\int
g\Phi_{t}^{\prime\prime}(g)|\nabla g|d\mu_{t}+c_{t}tN(t)\int
g\Phi_{t}^{\prime}(g)d\mu_{t}.$

Using our assumption on $\Phi_{t}$ and
$uv\leq\frac{1}{2\varepsilon}u^{2}+\frac{\varepsilon}{2}v^{2}$, the latter is
bounded above, for any $\varepsilon>0$, by

$\displaystyle
b_{t}N(t)\frac{1-e^{-\rho_{t}t}}{\rho_{t}}\left(\frac{1}{2\varepsilon}\int
g^{2}\Phi_{t}^{\prime\prime}(g)d\mu_{t}+\frac{\varepsilon}{2}\int\Phi_{t}^{\prime\prime}(g)|\nabla
g|^{2}d\mu_{t}\right)+c_{t}B_{t}tN(t)\int\Phi_{t}(g)d\mu_{t}$
$\displaystyle\leq
b_{t}N(t)\frac{1-e^{-\rho_{t}t}}{\rho_{t}}\frac{1}{2\varepsilon}(D_{t}+E_{t})+c_{t}B_{t}tN(t)+\varepsilon\frac{b_{t}N(t)(1-e^{-\rho_{t}t})}{2\rho_{t}}\int\Phi_{t}^{\prime\prime}(g)|\nabla
g|^{2}d\mu_{t}.$

Choose $\varepsilon$ so that
$\varepsilon\frac{b_{t}(1-e^{-\rho_{t}t})}{2\rho_{t}}=[1-\delta-\bar{\rho}_{t}]/2$
so that, collecting the above computations together with inequality (4.2), we
can conclude that for any $t$ such that $N^{\prime}(t)\geq 0$,

$N^{\prime}(t)\leq
N(t)\left(F_{t}+\left(b_{t}\frac{1-e^{-\rho_{t}t}}{\rho_{t}}\right)^{2}\frac{D_{t}+E_{t}}{2(1-\delta_{t}-\bar{\rho}_{t})}+c_{t}B_{t}t\right)$

from which the conclusion under assumption $(i)$ follows.

Now we turn to assumption $(ii)$. We need to bound the last two terms in
(4.2). Using Proposition 4.6 with $W:=W_{t}/b_{t}$ (observe that, since
$\mu_{t}$ is a probability measure, $b_{t}\neq 0$), it holds

$\displaystyle\int\int_{0}^{t}P_{t-s}^{(t)}f\nabla
P_{s}^{(t)}(\Phi_{t}^{\prime}(g))\cdot\nabla\dot{V}_{t}dsd\mu_{t}-\int\int_{0}^{t}[\nabla
V_{t}\cdot\nabla\dot{V}_{t}-\Delta\dot{V}_{t}]P_{t-s}^{(t)}fP_{s}^{(t)}(\Phi_{t}^{\prime}(g))dsd\mu_{t}$
$\displaystyle\leq b_{t}\int\int_{0}^{t}P_{t-s}^{(t)}f\left(|\nabla
P_{s}^{(t)}(\Phi_{t}^{\prime}(g))|+\frac{W_{t}}{b_{t}}P_{s}^{(t)}(\Phi_{t}^{\prime}(g))\right)dsd\mu_{t}$
$\displaystyle\leq
b_{t}\int\int_{0}^{t}e^{(c_{s}-\rho_{s})s}P_{t-s}^{(t)}fP_{s}^{(t)}\left(|\nabla\Phi^{\prime}(g)|+\frac{W_{t}}{b_{t}}\Phi^{\prime}(g)\right)$
$\displaystyle=N(t)\int_{0}^{t}e^{(c_{s}-\rho_{s})s}ds\left(b_{t}\int
g\Phi_{t}^{\prime\prime}(g)|\nabla g|d\mu_{t}+\int
W_{t}g\Phi_{t}^{\prime}(g)d\mu_{t}\right)$

where we used the reversibility in the last inequality. For the first term in
the right hand side of the latter, we proceed as for aasumption $(i)$. Namely,
it holds for all $\varepsilon>0$

$\displaystyle\int g\Phi_{t}^{\prime\prime}(g)|\nabla g|d\mu_{t}$
$\displaystyle\leq\frac{1}{2\varepsilon}\int
g^{2}\Phi_{t}^{\prime\prime}(g)d\mu_{t}+\frac{\varepsilon}{2}\int\Phi_{t}^{\prime\prime}(g)|\nabla
g|^{2}d\mu_{t}$
$\displaystyle\leq\frac{1}{2\varepsilon}(D_{t}+E_{t})+\frac{\varepsilon}{2}\int\Phi_{t}^{\prime\prime}(g)|\nabla
g|^{2}d\mu_{t}.$

For the second term, we use the expansion of the square (inequality (2.3) with
$f=\sqrt{\Phi_{t}(g)}$) to get that

$\displaystyle\int W_{t}g\Phi_{t}^{\prime}(g)d\mu_{t}$ $\displaystyle\leq
B_{t}\int
W_{t}\Phi_{t}(g)d\mu_{t}\leq\frac{\delta^{\prime}}{4C_{t}}\int\Phi_{t}(g)\left(|\nabla
V_{t}|^{2}-2\Delta V_{t}\right)d\mu_{t}+B_{t}F^{\prime}_{t}$
$\displaystyle\leq\frac{\delta^{\prime}}{4C_{t}}\int\frac{\Phi_{t}^{\prime}(g)^{2}}{\Phi_{t}(g)}|\nabla
g|^{2}d\mu_{t}+B_{t}F^{\prime}_{t}\leq\delta^{\prime}\int\Phi_{t}^{\prime\prime}(g)^{2}\|\nabla
g|^{2}d\mu_{t}+B_{t}F^{\prime}_{t}.$

Summarizing, under assumption $(ii)$, when $N^{\prime}(t)>0$, we obtain

$\displaystyle\frac{N^{\prime}(t)}{N(t)}$
$\displaystyle\leq\int\dot{\Phi}_{t}(g)d\mu_{t}-\left(1-\delta_{t}-\frac{\varepsilon}{2}\int_{0}^{t}e^{(c^{\prime}_{s}-\rho_{s})s}ds+\delta^{\prime}_{t}\int_{0}^{t}e^{(c^{\prime}_{s}-\rho_{s})s}ds\right)\int\phi_{t}^{\prime\prime}(g)|\nabla
g|^{2}d\mu_{t}$
$\displaystyle\quad+\int_{0}^{t}e^{(c^{\prime}_{s}-\rho_{s})s}ds\left(\frac{b_{t}(D_{t}+E_{t})}{2\varepsilon}+B_{t}F^{\prime}_{t}+F_{t}\right)$
$\displaystyle\leq\int_{0}^{t}e^{(c^{\prime}_{s}-\rho_{s})s}ds\left(\frac{b_{t}(D_{t}+E_{t})}{2\varepsilon}+B_{t}F^{\prime}_{t}+F_{t}\right)\qquad\mbox{
(thanks to }\eqref{eq:Fsob2})$
$\displaystyle=\left(\int_{0}^{t}e^{(c^{\prime}_{s}-\rho_{s})s}ds\right)^{2}\frac{b_{t}(D_{t}+E_{t})}{2(1-\delta_{t}-\bar{\rho}_{t}-\delta^{\prime}_{t}\int_{0}^{t}e^{(c^{\prime}_{s}-\rho_{s})s}ds)}+\int_{0}^{t}e^{(c^{\prime}_{s}-\rho_{s})s}ds\left(B_{t}F^{\prime}_{t}+F_{t}\right)$

for $\varepsilon$ so that
$\frac{\varepsilon}{2}\int_{0}^{t}e^{(c^{\prime}_{s}-\rho_{s})s}ds=\frac{1-\delta_{t}-\bar{\rho}_{t}-\delta^{\prime}_{t}\int_{0}^{t}e^{(c^{\prime}_{s}-\rho_{s})s}ds}{2}$.
The desired conclusion follows. ∎

In the proof of Theorem 4.4 we used the following results borrowed from [RZ].

######  Proposition 4.6.

Let $L=\Delta-\nabla U\cdot\nabla$, on $\mathbb{R}^{n}$, and denote by
$(P_{t})_{t\geq 0}$ its associated semi-group. Assume that
$U\colon\mathbb{R}^{n}\to\mathbb{R}$ is smooth enough and satisfies $\int
e^{-U}=1$ so that $\mu(dx)=e^{-U(x)}dx$ is a probability measure on
$\mathbb{R}^{n}$ and $\mathrm{Hess}(U)\geq\rho$ (as a matrix) for some
$\rho\in\mathbb{R}$. Let $W\colon\mathbb{R}\to\mathbb{R}_{+}$ be such that
$c:=\max\left(2\||\nabla W|\|_{\infty},\sup_{x:W(x)\neq
0}\left(\frac{LW}{W}-\rho\right)_{-}\right)<\infty$. Then, for all $f$ non
negative

$|\nabla P_{t}f|+WP_{t}f\leq e^{(c-\rho)t}P_{t}\left(|\nabla f|+Wf\right).$

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