# Discrete logarithmic Sobolev inequalities in Banach spaces

Dario Cordero-Erausquin and Alexandros Eskenazis  Institut de Mathématiques de
Jussieu, Sorbonne Université, Paris, 75252, France

###### Abstract.

Let $\mathscr{C}_{n}=\\{-1,1\\}^{n}$ be the discrete hypercube equipped with
the uniform probability measure $\sigma_{n}$. We prove that if
$(E,\|\cdot\|_{E})$ is a Banach space of finite cotype and $p\in[1,\infty)$,
then every function $f:\mathscr{C}_{n}\to E$ satisfies the dimension-free
vector-valued $L_{p}$ logarithmic Sobolev inequality

$\big{\|}f-\mathbb{E}f\big{\|}_{L_{p}(\log
L)^{p/2}(E)}\leqslant\mathsf{K}_{p}(E)\left(\rmintop\nolimits_{\mathscr{C}_{n}}\Big{\|}\sum_{i=1}^{n}\updelta_{i}\partial_{i}f\Big{\|}_{L_{p}(E)}^{p}\,\mathop{}\\!\mathrm{d}\sigma_{n}(\updelta)\right)^{1/p}.$

The finite cotype assumption is necessary for the conclusion to hold. This
estimate is the hypercube counterpart of a result of Ledoux (1988) in Gauss
space and the optimal vector-valued version of a deep inequality of Talagrand
(1994). As an application, we use such vector-valued $L_{p}$ logarithmic
Sobolev inequalities to derive new lower bounds for the bi-Lipschitz
distortion of nonlinear quotients of the Hamming cube into Banach spaces with
prescribed Rademacher type.

###### Key words and phrases:

Hamming cube, logarithmic Sobolev inequality, Rademacher type, bi-Lipschitz
embedding.

###### 2020 Mathematics Subject Classification:

Primary: 42C10; Secondary: 30L15, 46B07, 60G46.

E-mail addresses<EMAIL_ADDRESS>$\\{$alexandros.eskenazis@imj-
prg.fr<EMAIL_ADDRESS>
##  1\. Introduction

Let $\mathscr{C}_{n}$ be the Hamming cube $\\{-1,1\\}^{n}$ equipped with the
uniform probability measure $\sigma_{n}$ and let
$\uppsi:[0,\infty)\to[0,\infty)$ be a strictly increasing, convex function
with $\uppsi(0)=0$. If $(E,\|\cdot\|_{E})$ is a normed space, we shall denote
the $\uppsi$-Orlicz norm norm of an $E$-valued function $f:\mathscr{C}_{n}\to
E$ by

$\|f\|_{L_{\uppsi}(E)}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\inf\left\\{\upgamma>0:\
\rmintop\nolimits_{\mathscr{C}_{n}}\uppsi\big{(}\|f(x)\|_{E}/\upgamma\big{)}\,\mathop{}\\!\mathrm{d}\sigma_{n}(x)\leqslant
1\right\\}.$ (1)

Clearly, if $\uppsi(t)=t^{p}$, then the $L_{\uppsi}(E)$ norm reduces to the
vector-valued $L_{p}(E)$ norm. Per standard notation, if
$\uppsi(t)=t^{p}\log^{\upalpha}(e+t^{p})$ for some $\upalpha>0$, we denote
$L_{\uppsi}(E)$ by $L_{p}(\log L)^{\upalpha}(E)$. Moreover, we will denote by
$\mathbb{E}f$ the expectation of $f$ with respect to $\sigma_{n}$ and when
$E=\mathbb{C}$ we shall abbreviate $\|f\|_{L_{p}(\log
L)^{\upalpha}(\mathbb{C})}$ as $\|f\|_{L_{p}(\log L)^{\upalpha}}$. If
$i\in\\{1,\ldots,n\\}$, the $i$-th partial derivative of $f$ is defined as

$\forall\
x\in\mathscr{C}_{n},\qquad\partial_{i}f(x)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\frac{f(x)-f(x_{1},\ldots,x_{i-1},-x_{i},x_{i+1},\ldots,x_{n})}{2}.$
(2)

When $E=\mathbb{C}$, we will denote the $L_{p}$ norm of the gradient of a
function $f:\mathscr{C}_{n}\to\mathbb{C}$ by

$\|\nabla
f\|_{L_{p}}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\left(\rmintop\nolimits_{\mathscr{C}_{n}}\Big{(}\sum_{i=1}^{n}|\partial_{i}f(x)|^{2}\Big{)}^{p/2}\,\mathop{}\\!\mathrm{d}\sigma_{n}(x)\right)^{1/p}.$
(3)

Motivated by a quantitative version of Margulis’ graph connectivity theorem
[30], Talagrand [41] proved an important functional inequality asserting that
for every $p\in[1,\infty)$ there exists $\mathsf{C}_{p}\in(0,\infty)$ such
that for any $n\in\mathbb{N}$, every function $f:\mathscr{C}_{n}\to\mathbb{C}$
satisfies

$\|f-\mathbb{E}f\|_{L_{p}(\log L)^{p/2}}\leqslant\mathsf{C}_{p}\|\nabla
f\|_{L_{p}}.$ (4)

When $p=2$, this estimate is equivalent to the usual logarithmic Sobolev
inequality on the discrete hypercube up to the value of the implicit constant
which follows from the classical works [8, 19] of Bonami and Gross (see also
Section 4.3 for comparison with the more classical formulation of this
inequality in terms of entropy). It is worth pointing out that $L_{p}(\log
L)^{p/2}$ is the optimal Orlicz space guaranteeing the validity of an
inequality in the spirit of (4). As a direct consequence of the $L_{p}$
logarithmic Sobolev inequality (4), Talagrand also discovered the first proof
of the dimension-free $L_{p}$ Poincaré inequality on the hypercube, according
to which every $f:\mathscr{C}_{n}\to\mathbb{C}$ satisfies

$\|f-\mathbb{E}f\|_{L_{p}}\leqslant\mathsf{C}_{p}\|\nabla f\|_{L_{p}}.$ (5)

Refinements and extensions of (5) have been a central topic of research in
discrete analysis after Talagrand’s original proof (see, e.g., the works [5,
6, 36, 20, 21, 16, 22, 13, 10]).

In a striking recent article, Ivanisvili, van Handel and Volberg [22],
obtained a vector-valued extension of (5) with optimal assumptions on the
target space. In [22, Theorem 1.5], they showed that if a Banach space $E$ has
finite cotype (see [28, Chapter 9]) then for any $p\in[1,\infty)$ there exists
a constant $\mathsf{C}_{p}(E)$ such that for any $n\in\mathbb{N}$, every
function $f:\mathscr{C}_{n}\to E$ satisfies

$\|f-\mathbb{E}f\|_{L_{p}}\leqslant\mathsf{C}_{p}(E)\left(\rmintop\nolimits_{\mathscr{C}_{n}}\Big{\|}\sum_{i=1}^{n}\updelta_{i}\partial_{i}f\Big{\|}_{L_{p}(E)}^{p}\,\mathop{}\\!\mathrm{d}\sigma_{n}(\updelta)\right)^{1/p}.$
(6)

Observe that this inequality indeed extends (5) since, when $E=\mathbb{C}$,
the right-hand side of (6) is equivalent to $\|\nabla f\|_{L_{p}}$ up to
constants depending only on $p$ by Khintchine’s inequality [24]. Vector-valued
inequalities of the form (6) where first studied in classical work of Pisier
[37], who showed that (6) is valid with $\mathsf{C}_{p}(E)$ replaced by a
constant multiple of $\log n$ for an arbitrary target Banach space $E$. In
[41, Section 6], Talagrand constructed a Banach space for which the
logarithmic factor cannot be removed and thus the finite cotype assumption is
necessary for the validity of the dimension-free inequality (6) in view of the
Maurey–Pisier theorem [32].

In contrast to the Poincaré inequality (5) which has been extensively studied
in the context of analysis in Banach spaces, vector-valued versions of
Talagrand’s logarithmic Sobolev inequality (4) have been elusive (even for the
most well-known special case $p=2$). The main purpose of the present article
is to unify Talagrand’s work [41] with the work of Ivanisvili, van Handel and
Volberg [22] by proving an _optimal_ vector-valued version of (4). Our main
result reads as follows.

######  Theorem 1.

Let $(E,\|\cdot\|_{E})$ be a Banach space of finite cotype and
$p\in[1,\infty)$. There exists a constant $\mathsf{K}_{p}(E)\in(0,\infty)$
such that for any $n\in\mathbb{N}$, every function $f:\mathscr{C}_{n}\to E$
satisfies

$\big{\|}f-\mathbb{E}f\big{\|}_{L_{p}(\log
L)^{p/2}(E)}\leqslant\mathsf{K}_{p}(E)\left(\rmintop\nolimits_{\mathscr{C}_{n}}\Big{\|}\sum_{i=1}^{n}\updelta_{i}\partial_{i}f\Big{\|}_{L_{p}(E)}^{p}\,\mathop{}\\!\mathrm{d}\sigma_{n}(\updelta)\right)^{1/p}.$
(7)

Our proof in fact yields a stonger dimension-free inequality in an arbitrary
Banach space (see Remark 7) in the spirit of [22, Theorem 1.4]. This allows us
to derive the following refinement of the classical Pisier inequality [37,
Lemma 7.3] which may be of independent interest.

######  Corollary 2.

For every $p\in[1,\infty)$ there exists $\mathsf{K}_{p}\in(0,\infty)$ such
that the following claim holds for an arbitrary Banach space
$(E,\|\cdot\|_{E})$. For any $n\in\mathbb{N}$, every function
$f:\mathscr{C}_{n}\to E$ satisfies

$\begin{split}\big{\|}f-\mathbb{E}f\big{\|}_{L_{p}(\log
L)^{p/2}(E)}\leqslant\mathsf{K}_{p}&\left(\rmintop\nolimits_{\mathscr{C}_{n}}\Big{\|}\sum_{i=1}^{n}\updelta_{i}\partial_{i}f\Big{\|}_{L_{p}(E)}^{p}\,\mathop{}\\!\mathrm{d}\sigma_{n}(\updelta)\right)^{1/p}\\\
&+2(\log
n+1)\rmintop\nolimits_{\mathscr{C}_{n}}\Big{\|}\sum_{i=1}^{n}\updelta_{i}\partial_{i}f\Big{\|}_{L_{1}(E)}\,\mathop{}\\!\mathrm{d}\sigma_{n}(\updelta).\end{split}$
(8)

It is worth emphasizing that (8) is a two-fold refinement of Pisier’s
inequality. Besides the fact that it allows us to derive an upper bound for
the $L_{p}(\log L)^{p/2}$ Orlicz norm of the function $f-\mathbb{E}f$ instead
of the usual $L_{p}$ norm, the dimensional factor $\log n$ is multiplied by
the vector-valued $L_{1}$ norm of the gradient of $f$ rather than the vector-
valued $L_{p}$ norm as in [37].

###  1.1. Self improvement of Poincaré inequalities for hypercontractive
semigroups

As we shall explain in detail in Section 2, the main intricacies arising in
the proof of Theorem 1 are due to the appearance of the vector-valued gradient
on the right-hand side of inequality (7). In contrast to that, much simpler
reasoning allows us to prove that _metric_ Poincaré inequalities (that is,
inequalities in which the right-hand side depends only on distances between
values of the function) for vector-valued functions can be boosted to metric
logarithmic Sobolev type inequalities provided that the additional scalar-
valued inequality holds. This implication is summarized as

scalar log-Sobolev + vector Poincaré $\Longrightarrow$ vector log-Sobolev

and will be formalized in Proposition 9 below. For now, we register one
consequence of it which is the first vector-valued logarithmic Sobolev
inequality on the symmetric group $\mathscr{S}_{n}$ generated by
transpositions for targets of martingale type 2 (see Section 4 for all the
relevant definitions).

######  Theorem 3.

Let $\mathscr{S}_{n}$ be the symmetric group on $n$ elements equipped with the
uniform probability measure $\upnu_{n}$ and
$\mathscr{T}_{n}\subseteq\mathscr{S}_{n}$ the generating set consisting of all
transpositions. If $(E,\|\cdot\|_{E})$ is a Banach space of martingale type 2,
then there exists a constant $\mathsf{M}(E)\in(0,\infty)$ such that for any
$n\in\mathbb{N}$, every function $f:\mathscr{S}_{n}\to E$ satisfies

$\big{\|}f-\mathbb{E}f\big{\|}_{L_{2}(\log
L)(E)}^{2}\leqslant\frac{\mathsf{M}(E)\log
n}{n-1}\sum_{\uptau\in\mathscr{T}_{n}}\rmintop\nolimits_{\mathscr{S}_{n}}\big{\|}f(\uppi)-f(\uptau\circ\uppi)\big{\|}_{E}^{2}\,\mathop{}\\!\mathrm{d}\upnu_{n}(\uppi).$
(9)

This corollary improves upon the corresponding Poincaré inequality in the same
setting which follows from the work of Khot and Naor [25, Section 6] up to the
(necessary) logarithmic factor.

Recall that if $(\Omega,\upmu)$ is a measure space, the entropy of a
nonnegative function $h:\Omega\to\mathbb{R}_{+}$ is

$\mathrm{Ent}(h)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\rmintop\nolimits_{\Omega}h\log
h\,\mathop{}\\!\mathrm{d}\upmu-\left(\rmintop\nolimits_{\Omega}h\,\mathop{}\\!\mathrm{d}\upmu\right)\log\left(\rmintop\nolimits_{\Omega}h\,\mathop{}\\!\mathrm{d}\upmu\right).$
(10)

Logarithmic Sobolev inequalities like (9) are usually stated in terms of
entropy rather than Orlicz norms but these formulations can be seen to be
equivalent due to the elementary estimates

$\frac{1}{2}\max\big{\\{}\|f\|_{L_{2}(E)}^{2},\mathrm{Ent}\big{(}\|f\|_{E}^{2}\big{)}\big{\\}}\leqslant\|f\|_{L_{2}(\log
L)(E)}^{p}\leqslant
14\max\big{\\{}\|f\|_{L_{2}(E)}^{2},\mathrm{Ent}\big{(}\|f\|_{E}^{2}\big{)}\big{\\}},$
(11)

which hold for every function $f:\Omega\to E$. We refer to Section 4.3 for
more on this topic.

###  1.2. Distortion lower bounds for quotients

If $(\mathcal{M},d)$ and $(\mathcal{N},\updelta)$ are two metric spaces, the
bi-Lipschitz distortion of $\mathcal{M}$ into $\mathcal{N}$, denoted by
$\mathsf{c}_{\mathcal{N}}(\mathcal{M})$, is the least $D\geqslant 1$ for which
there exists $\sigma\in(0,\infty)$ and a mapping $f:\mathcal{M}\to\mathcal{N}$
satisfying

$\forall\ x,y\in\mathcal{M},\qquad\ \sigma
d(x,y)\leqslant\updelta(f(x),f(y)\big{)}\leqslant\sigma Dd(x,y).$ (12)

Let $\mathscr{C}_{n}$ be the hypercube equipped with the Hamming metric
$\varrho(x,y)=\|x-y\|_{1}$. A geometrically important application of the
vector-valued Poincaré inequality (6) is that it yields optimal distortion
lower bounds for embeddings of $(\mathscr{C}_{n},\varrho)$ into normed spaces
with prescribed Rademacher type. Recall that $(E,\|\cdot\|_{E})$ has
Rademacher type $p\in[1,2]$ with constant $T\in(0,\infty)$ if for every
$n\in\mathbb{N}$ and every vectors $x_{1},\ldots,x_{n}\in E$, we have

$\rmintop\nolimits_{\mathscr{C}_{n}}\Big{\|}\sum_{i=1}^{n}\varepsilon_{i}x_{i}\Big{\|}_{E}^{p}\,\mathop{}\\!\mathrm{d}\sigma_{n}(\varepsilon)\leqslant
T^{p}\sum_{i=1}^{n}\|x_{i}\|_{E}^{p}.$ (13)

A classical argument due to Enflo [14, 15] combined with [22] shows that when
a normed space $E$ has Rademacher type $p$ with constant $T$, then
$\mathsf{c}_{E}(\mathscr{C}_{n})\geqslant cT^{-1}n^{1-1/p}$, where $c>0$ is
universal.

Let $\mathscr{R}\subseteq\mathscr{C}_{n}\times\mathscr{C}_{n}$ be an arbitrary
equivalence relation and denote by $\mathscr{C}_{n}/\mathscr{R}$ the set of
all equivalence classes of $\mathscr{R}$ equipped with the quotient metric,
which is given by

$\forall\
[\varepsilon],[\updelta]\in\mathscr{C}_{n}/\mathscr{R},\quad\varrho_{\mathscr{C}_{n}/\mathscr{R}}\big{(}[\varepsilon],[\updelta]\big{)}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\min\big{\\{}\varrho(\upeta_{1},\upzeta_{1})+\cdots+\varrho(\upeta_{k},\upzeta_{k})\big{\\}};$
(14)

here the minimum is taken over all $k\geqslant 1$ and
$\upeta_{1},\ldots,\upeta_{k},\upzeta_{1},\ldots,\upzeta_{k}\in\mathscr{C}_{n}$
with $\upeta_{1}\in[\varepsilon]$, $\upzeta_{k}\in[\updelta]$ and
$[\upzeta_{j}]=[\upeta_{j+1}]$ for every $j\in\\{1,\ldots,k-1\\}$. We will
also denote by $\partial_{i}\mathscr{R}$ the boundary of $\mathscr{R}$ in the
direction $i$, that is

$\forall\
i\in\\{1,\ldots,n\\},\quad\partial_{i}\mathscr{R}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\big{\\{}\varepsilon\in\mathscr{C}_{n}:\
\big{(}\varepsilon,(\varepsilon_{1},\ldots,\varepsilon_{i-1},-\varepsilon_{i},\varepsilon_{i+1},\ldots,\varepsilon_{n})\big{)}\notin\mathscr{R}\big{\\}}$
(15)

Enflo’s argument combined with the results of [22] yields the general lower
bound

$\mathsf{c}_{E}\big{(}\mathscr{C}_{n}/\mathscr{R}\big{)}\geqslant\frac{cT^{-1}\big{\|}\varrho_{\mathscr{C}_{n}/\mathscr{R}}([\varepsilon],[\updelta])\big{\|}_{L_{p}(\mathscr{C}_{n}\times\mathscr{C}_{n})}}{\big{(}\sum_{i=1}^{n}\sigma_{n}(\partial_{i}\mathscr{R})\big{)}^{1/p}}$
(16)

for the bi-Lipschitz distortion of the quotient $\mathscr{C}_{n}/\mathscr{R}$
into a normed space of Rademacher type $p$, where $c>0$ is a universal
constant. As a consequence of the new vector-valued logarithmic Sobolev
inequalities, we will prove the following improved bound in Section 4.1 below.

######  Corollary 4.

There exists a universal constant $c>0$ such that if a normed space $E$ has
Rademacher type $p$ with constant $T$, then for any $n\in\mathbb{N}$, we have

$\mathsf{c}_{E}\big{(}\mathscr{C}_{n}/\mathscr{R}\big{)}\geqslant\frac{cT^{-1}\big{\|}\varrho_{\mathscr{C}_{n}/\mathscr{R}}([\varepsilon],[\updelta])\big{\|}_{L_{p}(\log
L)^{p/2}(\mathscr{C}_{n}\times\mathscr{C}_{n})}}{\big{(}\sum_{i=1}^{n}\sigma_{n}(\partial_{i}\mathscr{R})\big{)}^{1/p}}.$
(17)

A different (incomparable) refinement of the lower bound (16) has been
obtained by means of the vector-valued influence inequalities in [10, Section
8]. Similar lower bounds can be obtained from metric logarithmic Sobolev
inequalities in full generality including, for instance, for quotients of the
symmetric group following the result of Theorem 3.

### Acknowledgements

We are grateful to Michel Ledoux and Ramon van Handel for valuable discussions
and constructive feedback.

##  2\. Preliminary discussion

###  2.1. Historical interlude

Before demonstrating the ideas leading to Theorem 1, we digress to discuss the
existing approaches to $L_{p}$ logarithmic Sobolev inequalities in discrete
and continuous settings. The case $p=2$ of inequality (4), which is equivalent
to Bonami’s hypercontractive inequality [8], is quite singular in that it can
be proven in many distinct (and simple) ways which do not extend to other
values of $p$ so we shall purposefully not address it separately.

In [27], Ledoux initiated the study of $L_{p}$ logarithmic Sobolev
inequalities by proving the exact analogue of (4) in Gauss space
$(\mathbb{R}^{n},\upgamma_{n})$, where $\upgamma_{n}$ is the standard normal
distribution. His work thus improves upon the dimension-free $L_{p}$ Poincaré
inequality in Gauss space which had previously been proven by Maurey and
Pisier in [37]. Ledoux’s proof relies on the coarea formula and crucially uses
the Gaussian isoperimetric inequality [40, 9]. As Ledoux himself points out,
his theorem was only new in the endpoint case $p=1$. For $p\in(1,\infty)$, the
corresponding inequality also follows by a concatenation of Meyer’s Riesz
transform inequalities in Gauss space [34] (see also [38]) and a delicate
result of Bakry and Meyer [1] on the boundedness of operators of the form
$\mathscr{L}^{-\upalpha}$ in Orlicz spaces, where $\mathscr{L}$ is the
generator of a hypercontractive semigroup (see also [18]).

Moreover, Ledoux mentions in [27] that his results can be extended to a
vector-valued setting (in the spirit of [37]) but his proof never appeared in
print. His approach (see Remark 8) in fact gives a bound for
$\mathrm{Ent}(\|f\|_{E}^{2})$ which, when combined with (11), yields a
Gaussian version of Theorem 1 for $p=2$. In turn, this entropy bound is
obtained by applying the scalar logarithmic Sobolev inequality of [19] to the
function $\|f\|_{E}$ and using the chain rule to differentiate the norm.
Overcoming the use of a chain rule (which does not exist in the discrete
setting) is the main novelty of Theorem 1.

Talagrand’s technique yielding (4) on the hypercube is reminiscent of Ledoux’s
approach to the inequality’s Gaussian counterpart. Indeed, the first step of
his proof is an intricate induction which implies an isoperimetric-type
inequality [41, Theorem 1.1] for subsets of the discrete hypercube (see also
[13] for a recent refinement). Even though an immediate analogue of the coarea
formula does not exist in discrete spaces, in [41, Section 5] Talagrand uses a
discrete layer cake representation and suitable applications of his discrete
isoperimetric inequality to deduce (4). Similarly to the Gaussian case, a
different proof of the $L_{p}$ logarithmic Sobolev inequality (4) for
$p\in(1,\infty)$ can be obtained using Lust-Piquard’s dimension-free bounds
for the Riesz transform on the hypercube [29]. It is worth emphasizing that
the argument leading to (4) only requires a lower bound for the Riesz
transform which holds for every $p\in(1,\infty)$ as shown in [4, Theorem 1.1],
whereas the corresponding upper bound only holds for $p\geqslant 2$ (see [29,
p. 283]). To the best of our knowledge these are the only two known proofs to
the logarithmic Sobolev inequality (4).

We note in passing that a different refinement of the $L_{2}$ Poincaré
inequality is Talagrand’s influence inequality obtained in [42]. The original
proof of the influence inequality relied on Parseval’s identity for the
Fourier–Walsh transform along with a clever layer cake decomposition and a use
of hypercontractivity, whereas a semigroup proof was later obtained in [11].
In [10], which is a precursor to this work, we investigated vector-valued
versions of Talagrand’s influence inequality, obtained $L_{p}$ analogues and
extensions of it and studied its geometric consequences. Unlike Theorem 1, all
the vector-valued influence inequalities of [10] have either suboptimal
assumptions on the target space (e.g. some of them assume superreflexivity) or
suboptimal exponents in the Orlicz spaces.

###  2.2. About the proof

Unlike the influence inequality which has been successfully generalized to
vector-valued functions, the two available arguments for Talagrand’s
inequality (4) appear to be limited to the scalar-valued case. Indeed, the
original proof relying on a discrete isoperimetric inequality stumbles upon
the conceptual problem of understanding how to create a layer cake
representation for vector-valued functions. On the other hand, finding Banach
space-valued extensions of Lust-Piquard’s Riesz transform inequalities [29, 4]
has been a long-standing open problem in harmonic analysis which appears
intractable with current methods (see the discussion in [35, Section 4]).
Nevertheless, an argument inspired by [11, 10] can yield inequalities similar
to those of Theorem 1, alas with a suboptimal Orlicz space on the left-hand
side. This is further explained in Section 5.

Surprisingly, our proof of the optimal vector-valued logarithmic Sobolev
inequality (7) does not yield a new approach to the scalar inequality (4) but
rather uses it _as a black box_. The starting point is the observation that
the Orlicz norm on the left-hand side of (7) can be controlled by a sum of two
terms, the first being an entropy-like term which captures the oscillations of
$\|f-\mathbb{E}f\|_{E}$ and the second being further dominated by the $L_{1}$
norm of $f-\mathbb{E}f$. The latter summand can thus be controlled by the
vector-valued $L_{1}$ Poincaré inequality (6) of [22]. A direct application of
Talagrand’s inequality (4) is insufficient to control the remaining (scalar)
Orlicz term by the vector-valued gradient of $f$. Instead, we apply an
inequality which is a technical step in Talagrand’s proof of (4), which
crucially controls said Orlicz norm by an _asymmetric_ gradient of the norm of
$f$. This finally allows us to get the desired bound via a differentiation
argument, inspired by Ledoux’s approach.

##  3\. Proof of Theorem 1

The proof relies on the scalar $L_{p}$ logarithmic Sobolev inequality (4) of
[41]. In fact, rather than the inequality itself, we shall use a stronger
statement which appeared in Talagrand’s proof and is a similar inequality with
respect to an asymmetric gradient. For a function
$h:\mathscr{C}_{n}\to\mathbb{R}$, we shall consider the asymmetric gradient
$\mathsf{M}h:\mathscr{C}_{n}\to\mathbb{R}_{+}$ of $h$ that is given by

$\forall\
x\in\mathscr{C}_{n},\qquad\mathsf{M}h(x)=\left(\sum_{i=1}^{n}\partial_{i}h(x)_{+}^{2}\right)^{1/2},$
(18)

where $a_{+}=\max\\{a,0\\}$ for a real number $a\in\mathbb{R}$. Crucially for
his proof of (4), Talagrand proved the following refined technical result
which appears in [41, Proposition 5.1].

######  Proposition 5 (Talagrand).

For every $p\in[1,\infty)$, there exists $\upkappa_{p}\in(0,\infty)$ such that
the following holds for any $n\in\mathbb{N}$. If
$h:\mathscr{C}_{n}\to\mathbb{R}_{+}$ is a nonnegative function satisfying
$\sigma_{n}\\{h=0\\}\geqslant\tfrac{1}{2}$, then

$\|h\|_{L_{p}(\log
L)^{p/2}}\leqslant\upkappa_{p}\big{\|}\mathsf{M}h\big{\|}_{L_{p}}.$ (19)

Recall that any function $f:\mathscr{C}_{n}\to E$ admits a unique
Fourier–Walsh expansion of the form

$\forall\ x\in\mathscr{C}_{n},\qquad
f(x)=\sum_{S\subseteq\\{1,\ldots,n\\}}\widehat{f}(S)w_{S}(x),$ (20)

where the Fourier coefficients $\widehat{f}(S)\in E$ and the Walsh functions
$w_{S}:\mathscr{C}_{n}\to\\{-1,1\\}$ are given by $w_{S}(x)=\prod_{i\in
S}x_{i}$. By the expansion above, it is clear that any function
$f:\mathscr{C}_{n}\to E$ on the hypercube admits a unique multilinear
extension on $\mathbb{R}^{n}$, that is, there exists a unique function
$F:\mathbb{R}^{n}\to E$ such that $F|_{\mathscr{C}_{n}}=f$ and $F$ is affine
in each variable.

We shall use the following pointwise estimate which is a consequence of
convexity.

######  Lemma 6.

Let $(E,\|\cdot\|_{E})$ be a normed space. Then for any $n\in\mathbb{N}$,
every function $f:\mathscr{C}_{n}\to E$ satisfies the pointwise estimate

$\forall\
x\in\mathscr{C}_{n},\qquad\mathsf{M}\|f\|_{E}(x)\leqslant\Big{(}\rmintop\nolimits_{\mathscr{C}_{n}}\Big{\|}\sum_{i=1}^{n}\updelta_{i}\partial_{i}f(x)\Big{\|}_{E}^{2}\Big{)}^{1/2}.$
(21)

###### Proof.

Fix $x\in\mathscr{C}_{n}$. Without loss of generality, we can always assume
that $E$ is finite-dimensional as $F$ takes values in
$\mathrm{span}\\{F(x):x\in\mathbb{R}^{n}\\}=\mathrm{span}\\{f(x):x\in\mathscr{C}_{n}\\}$
which has dimension at most $2^{n}$. Therefore (by approximation), we can
assume that $f(x)\neq 0$ and that $\|\cdot\|_{E}$ is smooth on
$E\smallsetminus\\{0\\}$, that is, for every $v\in E\smallsetminus\\{0\\}$
there exists $\mathsf{D}_{v}^{\ast}\in E^{\ast}$ with
$\|\mathsf{D}_{v}^{*}\|_{E^{*}}\leqslant 1$ such that if
$\upbeta:(-\upeta,\upeta)\to E$ is a smooth function for some $\upeta>0$ with
$\upbeta(0)=v$, then

$\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}t}\Big{|}_{t=0}\big{\|}\upbeta(t)\big{\|}_{E}=\Big{\langle}\mathsf{D}_{v}^{*},\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}t}\Big{|}_{t=0}\upbeta(t)\Big{\rangle}.$
(22)

Fix also $i\in\\{1,\ldots,n\\}$ and let $F:\mathbb{R}^{n}\to E$ be the
multilinear extension of $f$. The lemma will be a consequence of the following
two simple technical claims.

Claim. The function $\phi_{i}:\mathbb{R}\to\mathbb{R}_{+}$ given by

$\forall\ s\in\mathbb{R},\qquad\phi_{i}(s)=\big{\|}F(x+se_{i})\big{\|}_{E}$
(23)

is convex.

Proof. Indeed, by the multilinear expansion of $F$, we have

$F(x+se_{i})=\sum_{\begin{subarray}{c}S\subseteq\\{1\ldots,n\\}:\\\ i\notin
S\end{subarray}}\widehat{f}(S)w_{S}(x)+(x_{i}+s)\sum_{\begin{subarray}{c}S\subseteq\\{1\ldots,n\\}:\\\
i\in S\end{subarray}}\widehat{f}(S)w_{S\smallsetminus\\{i\\}}(x),$ (24)

which is an affine function in the variable $s$. Therefore,
$s\mapsto\|F(x+se_{i})\|_{E}$ is indeed convex. $\Box$

Claim. We have the following pointwise estimate

$\partial_{i}\|f\|_{E}(x)_{+}\leqslant\Big{|}\frac{\partial\|F\|_{E}}{\partial
x_{i}}(x)\Big{|}.$ (25)

Proof. We have assumed that $\|\cdot\|_{E}$ is smooth on
$E\smallsetminus\\{0\\}$ and $f(x)\neq 0$ so the right-hand side is well-
defined. We will distinguish two cases and include all the details for
completeness.

Case 1. Suppose that $x_{i}=1$. Then, we will show that for every $h>0$,

$\partial_{i}\|f\|_{E}(x)_{+}=\left(\frac{\|F(x)\|_{E}-\|F(x-2e_{i})\|_{E}}{2}\right)_{+}\leqslant\left(\frac{\|F(x+he_{i})\|_{E}-\|F(x)\|_{E}}{h}\right)_{+}$
(26)

and the conclusion follows by taking a limit of $h\to 0^{+}$. Phrased in terms
of the function $\phi_{i}$ of (23), inequality (26) can be rewritten as

$\forall\
h>0,\qquad\left(\frac{\phi_{i}(0)-\phi_{i}(-2)}{2}\right)_{+}\leqslant\left(\frac{\phi_{i}(h)-\phi_{i}(0)}{h}\right)_{+}.$
(27)

If $\phi_{i}(0)\leqslant\phi_{i}(-2)$, then the inequality holds trivially. On
the other hand, if $\phi_{i}(0)>\phi_{i}(-2)$, we get from the convexity of
$\phi_{i}$ that

$\phi_{i}(0)\leqslant\frac{2}{2+h}\phi_{i}(h)+\frac{h}{2+h}\phi_{i}(-2),$ (28)

which can be rewritten as

$\frac{\phi_{i}(0)-\phi_{i}(-2)}{2}\leqslant\frac{\phi_{i}(h)-\phi_{i}(0)}{h}\leqslant\left(\frac{\phi_{i}(h)-\phi_{i}(0)}{h}\right)_{+}.$
(29)

Case 2. Suppose that $x_{i}=-1$. Similarly, we will show that for every $h>0$,

$\partial_{i}\|f\|_{E}(x)_{+}=\left(\frac{\|F(x)\|_{E}-\|F(x+2e_{i})\|_{E}}{2}\right)_{+}\leqslant\left(\frac{\|F(x-he_{i})\|_{E}-\|F(x)\|_{E}}{h}\right)_{+}.$
(30)

As before, we can assume that $\phi_{i}(0)>\phi_{i}(2)$ and use convexity in
the form of

$\phi_{i}(0)\leqslant\frac{2}{2+h}\phi_{i}(-h)+\frac{h}{2+h}\phi_{i}(2),$ (31)

which can be rewritten as

$\frac{\phi_{i}(0)-\phi_{i}(2)}{2}\leqslant\frac{\phi_{i}(-h)-\phi_{i}(0)}{h}\leqslant\left(\frac{\phi_{i}(-h)-\phi_{i}(0)}{h}\right)_{+}.\qed$

Combining the proven claim (25) with (22), we get that

$\begin{split}\mathsf{M}\|f\|_{E}(x)^{2}&=\sum_{i=1}^{n}\partial_{i}\|f\|_{E}(x)_{+}^{2}\stackrel{{\scriptstyle\eqref{eq:pointwise-i}}}{{\leqslant}}\sum_{i=1}^{n}\Big{(}\frac{\partial\|F\|_{E}}{\partial
x_{i}}(x)\Big{)}^{2}=\rmintop\nolimits_{\mathscr{C}_{n}}\Big{|}\sum_{i=1}^{n}\updelta_{i}\frac{\partial\|F\|_{E}}{\partial
x_{i}}(x)\Big{|}^{2}\,\mathop{}\\!\mathrm{d}\sigma_{n}(\updelta)\\\
&\stackrel{{\scriptstyle\eqref{eq:norm-
smooth}}}{{=}}\rmintop\nolimits_{\mathscr{C}_{n}}\Big{|}\sum_{i=1}^{n}\updelta_{i}\Big{\langle}\mathsf{D}_{f(x)}^{*},\frac{\partial
F}{\partial
x_{i}}(x)\Big{\rangle}\Big{|}^{2}\,\mathop{}\\!\mathrm{d}\sigma_{n}(\updelta)=\rmintop\nolimits_{\mathscr{C}_{n}}\Big{|}\Big{\langle}\mathsf{D}_{f(x)}^{*},\sum_{i=1}^{n}\updelta_{i}\frac{\partial
F}{\partial
x_{i}}(x)\Big{\rangle}\Big{|}^{2}\,\mathop{}\\!\mathrm{d}\sigma_{n}(\updelta)\\\
&\leqslant\rmintop\nolimits_{\mathscr{C}_{n}}\Big{\|}\sum_{i=1}^{n}\updelta_{i}\frac{\partial
F}{\partial
x_{i}}(x)\Big{\|}_{E}^{2}\,\mathop{}\\!\mathrm{d}\sigma_{n}(\updelta).\end{split}$
(32)

Now notice that as $x\in\mathscr{C}_{n}$, the multilinearity of $F$ easily
shows that

$\frac{\partial F}{\partial x_{i}}(x)=x_{i}\partial_{i}f(x)$ (33)

and thus we can rewrite the above bound as

$\mathsf{M}\|f\|_{E}(x)^{2}\leqslant\rmintop\nolimits_{\mathscr{C}_{n}}\Big{\|}\sum_{i=1}^{n}x_{i}\updelta_{i}\partial_{i}f(x)\Big{\|}_{E}^{2}\,\mathop{}\\!\mathrm{d}\sigma_{n}(\updelta)=\rmintop\nolimits_{\mathscr{C}_{n}}\Big{\|}\sum_{i=1}^{n}\updelta_{i}\partial_{i}f(x)\Big{\|}_{E}^{2}\,\mathop{}\\!\mathrm{d}\sigma_{n}(\updelta),$
(34)

where in the last equality we used that the random vectors $\updelta$ and
$(x_{1}\updelta_{1},\ldots,x_{n}\updelta_{n})$ have the same distribution when
$\updelta$ is distributed according to $\sigma_{n}$ and $x\in\mathscr{C}_{n}$
is fixed. $\Box$

Equipped with the above, we can prove Theorem 1.

###### Proof of Theorem 1.

Let $f:\mathscr{C}_{n}\to E$ be any function, where $E$ is a space of finite
cotype, and without loss of generality assume that $\mathbb{E}f=0$. Moreover,
consider the nonnegative function
$h\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\|f\|_{E}:\mathscr{C}_{n}\to\mathbb{R}_{+}$
and denote by $\mathsf{m}\geqslant 0$ a median of $h$. Then, the function
$(h-\mathsf{m})_{+}$ satisfies
$\sigma_{n}\\{(h-\mathsf{m})_{+}=0\\}=\sigma_{n}\\{h\leqslant\mathsf{m}\\}\geqslant\tfrac{1}{2}$.
Therefore, Proposition 5 yields the estimate

$\big{\|}(h-\mathsf{m})_{+}\big{\|}_{L_{p}(\log
L)^{p/2}}\leqslant\upkappa_{p}\big{\|}\mathsf{M}(h-\mathsf{m})_{+}\big{\|}_{L_{p}}.$
(35)

Since $\mathsf{m}$ is a median of $h$, we have

$\mathsf{m}\leqslant\frac{1}{\sigma_{n}\\{h\geqslant\mathsf{m}\\}}\rmintop\nolimits_{\\{h\geqslant\mathsf{m}\\}}h\,\mathop{}\\!\mathrm{d}\sigma_{n}\leqslant
2\|h\|_{L_{1}}=2\|f\|_{L_{1}(E)},$ (36)

and thus combining the above with the inequalities $0\leqslant
h\leqslant(h-\mathsf{m})_{+}+\mathsf{m}$, we get

$\begin{split}\|f\|_{L_{p}(\log L)^{p/2}(E)}=\|h\|_{L_{p}(\log
L)^{p/2}}\leqslant\big{\|}&(h-\mathsf{m})_{+}+\mathsf{m}\big{\|}_{L_{p}(\log
L)^{p/2}}\\\ &\stackrel{{\scriptstyle\eqref{eq:use-
talagrand}\wedge\eqref{eq:use-
markov}}}{{\leqslant}}\upkappa_{p}\big{\|}\mathsf{M}(h-\mathsf{m})_{+}\big{\|}_{L_{p}}+2\|f\|_{L_{1}(E)}.\end{split}$
(37)

Claim. For any scalar function $\phi:\mathscr{C}_{n}\to\mathbb{R}$, we have
the pointwise estimate

$\forall\
x\in\mathscr{C}_{n},\qquad\mathsf{M}\phi_{+}(x)\leqslant\mathsf{M}\phi(x).$
(38)

Proof. Since $\phi_{+}(x)=\max\big{\\{}\phi(x),0\big{\\}}$, it suffices to
show that for each $x,y\in\mathscr{C}_{n}$,

$\big{(}\phi_{+}(x)-\phi_{+}(y)\big{)}_{+}\leqslant\big{(}\phi(x)-\phi(y)\big{)}_{+}.$
(39)

Denoting by $a=\phi(x)$ and $b=\phi(y)$, the inequality can be rewritten as
$(a_{+}-b_{+})_{+}\leqslant(a-b)_{+}$. If $a_{+}\leqslant b_{+}$ the
inequality is trivially true so we will assume that $a_{+}>b_{+}$, which
implies $a>0$. In this case, if $b\leqslant 0$, then it amounts to $a\leqslant
a-b$ which holds and if $b>0$ it becomes an identity. $\Box$

Combining (37) with inequality (38), we thus deduce that

$\|f\|_{L_{p}(\log
L)^{p/2}(E)}\leqslant\upkappa_{p}\big{\|}\mathsf{M}h\big{\|}_{L_{p}}+2\|f\|_{L_{1}(E)},$
(40)

since $\mathsf{M}(h-m)\equiv\mathsf{M}h$. Moreover, by the $L_{1}$ Poincaré
inequality (6) of Ivanisvili, van Handel and Volberg [22], the second term can
be further upper bounded as

$\|f\|_{L_{1}(E)}\leqslant\mathsf{C}_{1}(E)\rmintop\nolimits_{\mathscr{C}_{n}}\Big{\|}\sum_{i=1}^{n}\updelta_{i}\partial_{i}f\Big{\|}_{L_{1}(E)}\,\mathop{}\\!\mathrm{d}\sigma_{n}(\updelta)\leqslant\mathsf{C}_{1}(E)\left(\rmintop\nolimits_{\mathscr{C}_{n}}\Big{\|}\sum_{i=1}^{n}\updelta_{i}\partial_{i}f\Big{\|}_{L_{p}(E)}^{p}\,\mathop{}\\!\mathrm{d}\sigma_{n}(\updelta)\right)^{1/p},$
(41)

where we used that $E$ has finite cotype. Finally, to control the first term
in (40) we use the pointwise inequality of Lemma 6 and Kahane’s inequality
with sharp constant [23, 26] to obtain the estimate

$\big{\|}\mathsf{M}h\big{\|}_{L_{p}}\leqslant\Big{\|}\Big{(}\rmintop\nolimits_{\mathscr{C}_{n}}\Big{\|}\sum_{i=1}^{n}\updelta_{i}\partial_{i}f(x)\Big{\|}_{E}^{2}\Big{)}^{1/2}\Big{\|}_{L_{p}}\leqslant\sqrt{2}\left(\rmintop\nolimits_{\mathscr{C}_{n}}\Big{\|}\sum_{i=1}^{n}\updelta_{i}\partial_{i}f\Big{\|}_{L_{p}(E)}^{p}\,\mathop{}\\!\mathrm{d}\sigma_{n}(\updelta)\right)^{1/p}.$
(42)

This completes the proof of the theorem. ∎

The same arguments allow us to prove the refined Pisier inequality of
Corollary 2.

###### Proof of Corollary 2.

Recall that Pisier’s inequality [37, Lemma 7.3] (see also [20] for the value
of the multiplicative constant) asserts that for any $r\in[1,\infty)$, every
$f:\mathscr{C}_{n}\to E$ satisfies

$\|f-\mathbb{E}f\|_{L_{r}(E)}\leqslant(\log
n+1)\left(\rmintop\nolimits_{\mathscr{C}_{n}}\Big{\|}\sum_{i=1}^{n}\updelta_{i}\partial_{i}f\Big{\|}_{L_{r}(E)}^{r}\,\mathop{}\\!\mathrm{d}\sigma_{n}(\updelta)\right)^{1/r}$
(43)

Assuming again that $\mathbb{E}f=0$, (40) combined with (42) and Pisier’s
inequality for $r=1$ gives the desired upper bound in a general normed space.
∎

######  Remark 7.

The above argument combined with the main result of Ivanisvili, van Handel and
Volberg [22] yields a mutual refinement of (7) and (8) for general Banach
space targets. Following [22], let $\upxi_{1}(t),\ldots,\upxi_{n}(t)$ be
i.i.d. random variables satisfying

$\mathbb{P}\big{\\{}\upxi_{i}(t)=1\big{\\}}=\frac{1+e^{-t}}{2}\qquad\mbox{and}\qquad\mathbb{P}\big{\\{}\upxi_{i}(t)=-1\big{\\}}=\frac{1-e^{-t}}{2}$
(44)

and denote by
$\updelta_{i}(t)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\frac{\upxi_{i}(t)-e^{-t}}{\sqrt{1-e^{-2t}}}$.
In [22, Theorem 1.4], the authors showed the estimate

$\|f-\mathbb{E}f\|_{L_{r}(E)}\leqslant\frac{\uppi}{2}\rmintop\nolimits_{0}^{\infty}\left(\mathbb{E}\Big{\|}\sum_{i=1}^{n}\updelta_{i}(t)\partial_{i}f\Big{\|}_{L_{r}(E)}^{r}\right)^{1/r}\frac{\mathop{}\\!\mathrm{d}t}{\sqrt{e^{2t}-1}}.$
(45)

Plugging this bound for $r=1$ and (42) in (40) yields the stronger inequality

$\begin{split}\big{\|}f-\mathbb{E}f\big{\|}_{L_{p}(\log
L)^{p/2}(E)}\leqslant\upkappa_{p}&\left(\rmintop\nolimits_{\mathscr{C}_{n}}\Big{\|}\sum_{i=1}^{n}\updelta_{i}\partial_{i}f\Big{\|}_{L_{p}(E)}^{p}\,\mathop{}\\!\mathrm{d}\sigma_{n}(\updelta)\right)^{1/p}\\\
&+\uppi\rmintop\nolimits_{0}^{\infty}\mathbb{E}\Big{\|}\sum_{i=1}^{n}\updelta_{i}(t)\partial_{i}f\Big{\|}_{L_{1}(E)}\,\frac{\mathop{}\\!\mathrm{d}t}{\sqrt{e^{2t}-1}}.\end{split}$
(46)

######  Remark 8.

A Gaussian version of Theorem 1 was announced in Ledoux’s classical work [27]
yet the proof never appeared in print. In personal communications, Ledoux
confirmed to us that his claim in [27] was the following. For any normed space
$E$ and any $n\in\mathbb{N}$, every smooth function
$f:(\mathbb{R}^{n},\upgamma_{n})\to E$ satisfies the inequality

$\mathrm{Ent}\big{(}\|f\|_{E}^{2}\big{)}\leqslant
2\rmintop\nolimits_{\mathbb{R}^{n}}\Big{\|}\sum_{i=1}^{n}g_{i}\partial_{i}f\Big{\|}_{L_{2}(E)}\,\mathop{}\\!\mathrm{d}\upgamma_{n}(g),$
(47)

where the entropy and $L_{2}(E)$ norm are both with respect to the standard
Gaussian measure $\upgamma_{n}$. To see this, observe that Gross’ logarithmic
Sobolev inequality applied to the function $\|f\|_{E}$ yields

$\mathrm{Ent}\big{(}\|f\|_{E}^{2}\big{)}\leqslant
2\sum_{i=1}^{n}\big{\|}\partial_{i}\|f\|_{E}\big{\|}_{L_{2}}^{2}=2\rmintop\nolimits_{\mathbb{R}^{n}}\Big{\|}\sum_{i=1}^{n}g_{i}\partial_{i}\|f\|_{E}\Big{\|}_{L_{2}}^{2}\,\mathop{}\\!\mathrm{d}\upgamma_{n}(g).$
(48)

Arguing as in the proof of Theorem 1, we can assume without loss of generality
that $\|\cdot\|_{E}$ is smooth on $E\smallsetminus\\{0\\}$ and that $f(x)\neq
0$ almost surely, so that we have
$\partial_{i}\|f\|_{E}(x)=\langle\mathsf{D}^{\ast}_{f(x)},\partial_{i}f(x)\rangle$
for some linear functional $\mathsf{D}^{\ast}_{f(x)}$ in the unit ball of
$E^{\ast}$. Therefore,

$\mathrm{Ent}\big{(}\|f\|_{E}^{2}\big{)}\leqslant
2\rmintop\nolimits\\!\\!\\!\\!\\!\rmintop\nolimits_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\Big{\langle}\mathsf{D}^{\ast}_{f(x)},\sum_{i=1}^{n}g_{i}\partial_{i}f(x)\Big{\rangle}^{2}\,\mathop{}\\!\mathrm{d}\upgamma_{2n}(x,g)\leqslant
2\rmintop\nolimits_{\mathbb{R}^{n}}\Big{\|}\sum_{i=1}^{n}g_{i}\partial_{i}f\Big{\|}_{L_{2}(E)}^{2}\,\mathop{}\\!\mathrm{d}\upgamma_{n}(g).$
(49)

The elementary inequality (11) (see also Section 4.3) combined with Ledoux’s
inequality (47) and the vector-valued Gaussian Poincaré inequality

$\big{\|}f-\mathbb{E}f\big{\|}_{L_{2}}\leqslant\frac{\uppi}{2}\left(\rmintop\nolimits_{\mathbb{R}^{n}}\Big{\|}\sum_{i=1}^{n}g_{i}\partial_{i}f\Big{\|}_{L_{2}(E)}^{2}\,\mathop{}\\!\mathrm{d}\upgamma_{n}(g)\right)^{1/2}$
(50)

of Maurey and Pisier [37, Corollary 2.4] implies the Gaussian version of
Theorem 1 for $p=2$ and functions with values in an arbitrary normed space
$E$. An extension of this estimate to all values of $p\in[1,\infty)$ follows
from the general inequality (46) by standard considerations involving the
central limit theorem, see e.g. [22, Remark 1.3].

It is worth emphasizing that, as there is no chain rule for the discrete
derivatives on the hypercube, the main additional difficulty arising in the
proof of Theorem 1 was the need to relate discrete partial derivatives of
$\|f\|_{E}$ with classical partial derivatives of $\|F\|_{E}$. Each $i$-th
partial derivative turns out to only be controlled by a one-sided inequality
on an $(n-1)$-dimensional subcube of $\mathscr{C}_{n}$ depending on $i$ (Lemma
6) yet this is sufficient to bound the full gradient in view of Talagrand’s
asymmetric logarithmic Sobolev inequality (Proposition 5).

##  4\. Self-improvement of Poincaré inequalities

In this Section we formalize the discussion of Section 1.1, where we claimed
that metric Poincaré inequalities for norms can be boosted to logarithmic
Sobolev-type inequalities provided that the corresponding scalar-valued
inequality holds. More precisely, we prove the following proposition.

######  Proposition 9.

Let $p\geqslant 1$ and $\upalpha>0$. Fix a normed space $(E,\|\cdot\|_{E})$, a
probability space $(\Omega,\upmu)$ and a measure $\upbeta$ on
$\Omega\times\Omega$. Suppose that every bounded function $f:\Omega\to E$
satisfies

$\big{\|}f-\mathbb{E}f\big{\|}_{L_{p}(E)}^{p}\leqslant\rmintop\nolimits\\!\\!\\!\\!\\!\rmintop\nolimits_{\Omega\times\Omega}\big{\|}f(x)-f(y)\big{\|}_{E}^{p}\,\mathop{}\\!\mathrm{d}\upbeta(x,y)$
(51)

and that every scalar bounded function $h:\Omega\to\mathbb{C}$ satisfies

$\big{\|}h-\mathbb{E}h\big{\|}_{L_{p}(\log
L)^{\upalpha}}^{p}\leqslant\mathsf{C}^{p}\rmintop\nolimits\\!\\!\\!\\!\\!\rmintop\nolimits_{\Omega\times\Omega}\big{|}h(x)-h(y)\big{|}^{p}\,\mathop{}\\!\mathrm{d}\upbeta(x,y)$
(52)

for some $\mathsf{C}>0$. Then, for every bounded function $f:\Omega\to E$ we
have

$\big{\|}f-\mathbb{E}f\big{\|}_{L_{p}(\log L)^{\upalpha}(E)}^{p}\leqslant
2^{p-1}\big{(}\mathsf{C}^{p}+1)\rmintop\nolimits\\!\\!\\!\\!\\!\rmintop\nolimits_{\Omega\times\Omega}\big{\|}f(x)-f(y)\big{\|}_{E}^{p}\,\mathop{}\\!\mathrm{d}\upbeta(x,y).$
(53)

###### Proof.

Assume that $\mathbb{E}f=0$ and denote by $h=\|f\|_{E}$. Then, we have

$\|f\|_{L_{p}(\log L)^{\upalpha}(E)}^{p}=\|h\|_{L_{p}(\log
L)^{\upalpha}}^{p}\leqslant
2^{p-1}\big{(}\big{\|}h-\mathbb{E}h\big{\|}_{L_{p}(\log
L)^{\upalpha}}^{p}+\big{|}\mathbb{E}h\big{|}^{p}\big{)}$ (54)

by the triangle inequality for Orlicz norms and Hölder’s inequality
$(A+B)^{p}\leqslant 2^{p-1}(A^{p}+B^{p})$. To control the first summand in
(54), we use (52) and get

$\begin{split}\big{\|}h&-\mathbb{E}h\big{\|}_{L_{p}(\log
L)^{\upalpha}}^{p}\stackrel{{\scriptstyle\eqref{eq:sc-
lsi}}}{{\leqslant}}\mathsf{C}^{p}\rmintop\nolimits\\!\\!\\!\\!\\!\rmintop\nolimits_{\Omega\times\Omega}\big{|}h(x)-h(y)\big{|}^{p}\,\mathop{}\\!\mathrm{d}\upbeta(x,y)\\\
&=\mathsf{C}^{p}\rmintop\nolimits\\!\\!\\!\\!\\!\rmintop\nolimits_{\Omega\times\Omega}\big{|}\|f(x)\|_{E}-\|f(y)\|_{E}\big{|}^{p}\,\mathop{}\\!\mathrm{d}\upbeta(x,y)\leqslant\mathsf{C}^{p}\rmintop\nolimits\\!\\!\\!\\!\\!\rmintop\nolimits_{\Omega\times\Omega}\big{\|}f(x)-f(y)\big{\|}_{E}^{p}\,\mathop{}\\!\mathrm{d}\upbeta(x,y).\end{split}$
(55)

For the second summand, observe that (51) gives

$\big{|}\mathbb{E}h\big{|}^{p}=\big{|}\mathbb{E}\|f\|_{E}\big{|}^{p}\leqslant\mathbb{E}\|f\|_{E}^{p}\stackrel{{\scriptstyle\eqref{eq:vec-
pi}}}{{\leqslant}}\rmintop\nolimits\\!\\!\\!\\!\\!\rmintop\nolimits_{\Omega\times\Omega}\big{\|}f(x)-f(y)\big{\|}_{E}^{p}\,\mathop{}\\!\mathrm{d}\upbeta(x,y)$
(56)

and this concludes the proof. ∎

As an example, combining Proposition 9 with [8, 19, 22], we easily deduce a
metric logarithmic Sobolev inequality on the hypercube under Rademacher type
assumptions. The same estimate also follows directly from (46) but the proof
via Proposition 9 is substantially simpler.

######  Corollary 10.

For every $p\in[1,2]$ and every normed space $(E,\|\cdot\|_{E})$ of Rademacher
type $p$, there exists $\uptau_{p}(E)\in(0,\infty)$ such that for any
$n\in\mathbb{N}$, every $f:\mathscr{C}_{n}\to E$ satisfies

$\big{\|}f-\mathbb{E}f\big{\|}_{L_{p}(\log
L)^{p/2}(E)}\leqslant\uptau_{p}(E)\left(\sum_{i=1}^{n}\big{\|}\partial_{i}f\big{\|}_{L_{p}(E)}^{p}\right)^{1/p}.$
(57)

Moreover, $\uptau_{p}(E)$ is proportional to the Rademacher type $p$ constant
of $E$.

###  4.1. Distortion lower bounds for quotients

Using Corollary 10, we will prove Corollary 4.

###### Proof of Corollary 4.

Let $\mathscr{R}\subseteq\mathscr{C}_{n}\times\mathscr{C}_{n}$ be an
equivalence relation and consider an embedding
$f:\mathscr{C}_{n}/\mathscr{R}\to E$ satisfying the bi-Lipschitz condition

$\forall\
[\varepsilon],[\updelta]\in\mathscr{C}_{n}/\mathscr{R},\qquad\sigma\varrho_{\mathscr{C}_{n}/\mathscr{R}}\big{(}[\varepsilon],[\updelta]\big{)}\leqslant\big{\|}f\big{(}[\varepsilon]\big{)}-f\big{(}[\updelta]\big{)}\big{\|}_{E}\leqslant\sigma
D\varrho_{\mathscr{C}_{n}/\mathscr{R}}\big{(}[\varepsilon],[\updelta]\big{)}$
(58)

for some $\sigma>0$ and $D\geqslant 1$. Also, consider the lift
$F:\mathscr{C}_{n}\to E$ of $f$ given by $F(\varepsilon)=f([\varepsilon])$ for
$\varepsilon\in\mathscr{C}_{n}$. Then, the lower Lipschitz condition on $f$,
yields

$\begin{split}\big{\|}F(\varepsilon)-F(\updelta)\big{\|}_{L_{p}(\log
L)^{p/2}(\mathscr{C}_{n}\times\mathscr{C}_{n};E)}&=\big{\|}f\big{(}[\varepsilon]\big{)}-f\big{(}[\updelta]\big{)}\big{\|}_{L_{p}(\log
L)^{p/2}(\mathscr{C}_{n}\times\mathscr{C}_{n};E)}\\\
&\stackrel{{\scriptstyle\eqref{lip}}}{{\geqslant}}\sigma\big{\|}\varrho_{\mathscr{C}_{n}/\mathscr{R}}\big{(}[\varepsilon],[\updelta]\big{)}\big{\|}_{L_{p}(\log
L)^{p/2}(\mathscr{C}_{n}\times\mathscr{C}_{n})}.\end{split}$ (59)

On the other hand, by the triangle inequality and Corollary 10, we have

$\begin{split}\big{\|}F(\varepsilon)-F(\updelta)\big{\|}_{L_{p}(\log
L)^{p/2}(\mathscr{C}_{n}\times\mathscr{C}_{n};E)}\leqslant
2\big{\|}F-\mathbb{E}F\big{\|}_{L_{p}(\log L)^{p/2}(E)}\leqslant
2\uptau_{p}(E)\left(\sum_{i=1}^{n}\big{\|}\partial_{i}F\big{\|}_{L_{p}(E)}^{p}\right)^{1/p}.\end{split}$

Now notice that for each $i$,

$\begin{split}\big{\|}\partial_{i}F\big{\|}_{L_{p}(E)}^{p}=\frac{1}{2^{p}}\rmintop\nolimits_{\mathscr{C}_{n}}\big{\|}f\big{(}[\varepsilon]\big{)}-f\big{(}[(\varepsilon_{1},\ldots,-\varepsilon_{i},\ldots,\varepsilon_{n})]\big{)}\big{\|}_{E}^{p}\,\mathop{}\\!\mathrm{d}\sigma_{n}(\varepsilon)\stackrel{{\scriptstyle\eqref{lip}}}{{\leqslant}}\sigma^{p}D^{p}\sigma_{n}\big{(}\partial_{i}\mathscr{R}\big{)},\end{split}$
(60)

as
$\varrho_{\mathscr{C}_{n}/\mathscr{R}}\big{(}[\varepsilon],[(\varepsilon_{1},\ldots,-\varepsilon_{i},\ldots,\varepsilon_{n})]\big{)}=2$
if $\varepsilon\in\partial_{i}\mathscr{R}$ and otherwise it is 0. Combining
all the above readily gives the desired lower bound on $D$. ∎

###  4.2. Proof of Theorem 3

Recall that a normed space $(E,\|\cdot\|_{E})$ has martingale type 2 with
constant $M\in(0,\infty)$ if for any $n\in\mathbb{N}$, any probability space
$(\Omega,\mathscr{F},\upmu)$ and any filtration
$\\{\mathscr{F}_{i}\\}_{i=0}^{n}$ of sub-$\sigma$-algebras of $\mathscr{F}$,
every $E$-valued martingale $\\{\mathscr{M}_{i}:\Omega\to E\\}_{i=0}^{n}$
adapted to $\\{\mathscr{F}_{i}\\}_{i=0}^{n}$ satisfies

$\big{\|}\mathscr{M}_{n}-\mathscr{M}_{0}\big{\|}_{L_{2}(\upmu;E)}^{2}\leqslant
M^{2}\sum_{i=1}^{n}\big{\|}\mathscr{M}_{i}-\mathscr{M}_{i-1}\big{\|}^{2}_{L_{2}(\upmu;E)}.$
(61)

The least constant $M>0$ for which (61) is satisfied shall be denoted by
$\mathsf{m}_{2}(E)$.

In [25, Section 6], Khot and Naor proved lower bounds for the bi-Lipschitz
distortion of metric spaces with prescribed length in the sense of Schechtman
[39] into normed spaces of given martingale type. An important example of such
a metric space is the Cayley graph of the symmetric group $\mathscr{S}_{n}$
equipped with generating set $\mathscr{T}_{n}$ consisting of all
transpositions, as was proven in influential earlier work of Maurey [31]. For
their proof of the distortion lower bounds, Khot and Naor use a Poincaré type
inequality in such spaces. In the case of the symmetric group, the following
inequality can be extracted from the proof of [25, Theorem 7].

######  Theorem 11 (Khot–Naor).

If $(E,\|\cdot\|_{E})$ is a Banach space of martingale type 2, then for any
$n\in\mathbb{N}$, every function $f:\mathscr{S}_{n}\to E$ satisfies

$\big{\|}f-\mathbb{E}f\big{\|}_{L_{2}(E)}^{2}\leqslant\frac{2\mathsf{m}_{2}(E)^{2}}{n-1}\sum_{\uptau\in\mathscr{T}_{n}}\rmintop\nolimits_{\mathscr{S}_{n}}\big{\|}f(\uppi)-f(\uptau\circ\uppi)\big{\|}_{E}^{2}\,\mathop{}\\!\mathrm{d}\upnu_{n}(\uppi).$
(62)

To derive the vector-valued logarithmic Sobolev inequality of Theorem 3, we
shall also need the following scalar-valued result of Diaconis and Saloff-
Coste [12, Section 4.3].

######  Theorem 12 (Diaconis–Saloff-Coste).

For any $n\in\mathbb{N}$, every function $f:\mathscr{S}_{n}\to\mathbb{R}$
satisfies

$\mathrm{Ent}\big{(}f^{2}\big{)}\leqslant\frac{6\log
n}{n-1}\sum_{\uptau\in\mathscr{T}_{n}}\rmintop\nolimits_{\mathscr{S}_{n}}\big{|}f(\uppi)-f(\uptau\circ\uppi)\big{|}^{2}\,\mathop{}\\!\mathrm{d}\upnu_{n}(\uppi).$
(63)

###### Proof of Theorem 3.

In view of the inequality (11) (see also the more general Lemma 13) and the
scalar-valued case of inequality (62), Theorem 12 implies that every function
$f:\mathscr{S}_{n}\to\mathbb{R}$ satisfies

$\big{\|}f-\mathbb{E}f\big{\|}_{L_{2}(\log L)}^{2}\leqslant\frac{C\log
n}{n-1}\sum_{\uptau\in\mathscr{T}_{n}}\rmintop\nolimits_{\mathscr{S}_{n}}\big{|}f(\uppi)-f(\uptau\circ\uppi)\big{|}^{2}\,\mathop{}\\!\mathrm{d}\upnu_{n}(\uppi),$
(64)

where $C>0$ is a universal constant. Combining this with the vector-valued
inequality of Theorem 11 and the general implication of Proposition 9, we
deduce Theorem 3. ∎

###  4.3. Orlicz norms and $\upalpha$-entropies

In order to use Proposition 9 in concrete examples (including Theorem 3), we
needed the elementary estimate (11) relating the $L_{2}(\log L)$ norm with
entropy. In this section, we extend this two-sided bound to more general
Orlicz norms. If $x\in(0,\infty)$ and $\upalpha>0$, we use the ad hoc notation

$\log^{\upalpha}x=\mathrm{sign}(\log x)|\log x|^{\upalpha},$ (65)

so that $x\mapsto\log^{\upalpha}x$ is increasing on $(0,\infty)$. Finally, for
a nonnegative function $h:(\Omega,\upmu)\to[0,\infty)$, and $\upalpha>0$ we
denote its _$\upalpha$ -entropy_ by

$\mathrm{Ent}^{\upalpha}(h)=\rmintop\nolimits_{\Omega}h\log^{\upalpha}\Big{(}\frac{h}{\rmintop\nolimits_{\Omega}h\,\mathop{}\\!\mathrm{d}\upmu}\Big{)}\,\mathop{}\\!\mathrm{d}\upmu,$
(66)

where $0\log^{\upalpha}0$ will always be understood to be 0.

We shall need a two-sided estimate for Orlicz norms. The special case $p=2$
and $\upalpha=1$ is implicit in the work [7, Section 4] of Bobkov and Götze.
We have been generous with the dependence of the constants on the various
parameters involved.

######  Lemma 13.

Fix $p\geqslant 1$ and $\upalpha>0$. If $(\Omega,\upmu)$ is a probability
space, $(E,\|\cdot\|_{E})$ is a normed space and $f:\Omega\to E$ is a
measurable function, then we have

$c_{\upalpha}^{-1}\max\big{\\{}\|f\|_{L_{p}(E)}^{p},\mathrm{Ent}^{\upalpha}\big{(}\|f\|_{E}^{p}\big{)}\big{\\}}\leqslant\|f\|_{L_{p}(\log
L)^{\upalpha}(E)}^{p}\leqslant
C_{\upalpha}\max\big{\\{}\|f\|_{L_{p}(E)}^{p},\mathrm{Ent}^{\upalpha}\big{(}\|f\|_{E}^{p}\big{)}\big{\\}},$
(67)

where
$c_{\upalpha}=2^{(\upalpha-1)_{+}}\big{(}1+(\upalpha/e)^{\upalpha}\big{)}+(\upalpha/e)^{\upalpha}$
and

$C_{\upalpha}=\max\Big{\\{}\log^{\upalpha}(e+1)+2(\upalpha/e)^{\upalpha},\tfrac{2^{1/\upalpha}+e-1}{2^{1/\upalpha}-1}\log^{\upalpha}\Big{(}e+\tfrac{2^{1/\upalpha}+e-1}{2^{1/\upalpha}-1}\Big{)}\Big{\\}}+2.$
(68)

###### Proof.

Since all quantities involved depend only on $\|f\|_{E}$, we can assume
without loss of generality that $f$ is scalar-valued and nonnegative. We start
by proving the inequality on the left. Assume that $\|f\|_{L_{p}(\log
L)^{\upalpha}}^{p}\leqslant 1$, that is,

$\rmintop\nolimits_{\Omega}f^{p}\log^{\upalpha}(e+f^{p})\,\mathop{}\\!\mathrm{d}\upmu\leqslant
1.$ (69)

The inequality $\log(e+x)\geqslant 1$ for $x>0$ implies that
$\rmintop\nolimits_{\Omega}f^{p}\,\mathop{}\\!\mathrm{d}\upmu\leqslant 1$. To
control the term $\mathrm{Ent}^{\upalpha}(f^{p})$, observe that if
$A=\big{\\{}f^{p}>\rmintop\nolimits_{\Omega}f^{p}\,\mathop{}\\!\mathrm{d}\upmu\big{\\}}$,
we have

$\mathrm{Ent}^{\upalpha}(f^{p})\leqslant\rmintop\nolimits_{A}f^{p}\Big{(}\log
f^{p}-\log\rmintop\nolimits_{\Omega}f^{p}\,\mathop{}\\!\mathrm{d}\upmu\Big{)}^{\upalpha}\,\mathop{}\\!\mathrm{d}\upmu.$
(70)

We will break this integral in two parts. First, notice that

$\rmintop\nolimits_{A\cap\\{f\leqslant 1\\}}f^{p}\Big{(}\log
f^{p}-\log\rmintop\nolimits_{\Omega}f^{p}\,\mathop{}\\!\mathrm{d}\upmu\Big{)}^{\upalpha}\,\mathop{}\\!\mathrm{d}\upmu\leqslant\Big{(}\rmintop\nolimits_{\Omega}f^{p}\,\mathop{}\\!\mathrm{d}\upmu\Big{)}\Big{(}-\log\rmintop\nolimits_{\Omega}f^{p}\,\mathop{}\\!\mathrm{d}\upmu\Big{)}^{\upalpha}\leqslant\Big{(}\frac{\upalpha}{e}\Big{)}^{\upalpha},$
(71)

as $x\log^{\upalpha}(1/x)\leqslant\big{(}\upalpha/e\big{)}^{\upalpha}$ for
$x\in(0,1)$. For the remaining integral we write

$\begin{split}\rmintop\nolimits_{\\{f>1\\}}&f^{p}\Big{(}\log
f^{p}-\log\rmintop\nolimits_{\Omega}f^{p}\,\mathop{}\\!\mathrm{d}\upmu\Big{)}^{\upalpha}\,\mathop{}\\!\mathrm{d}\upmu\\\
&\leqslant
2^{(\upalpha-1)_{+}}\rmintop\nolimits_{\Omega}f^{p}\Big{(}\log^{\upalpha}f^{p}-\log^{\upalpha}\Big{(}\rmintop\nolimits_{\Omega}f^{p}\,\mathop{}\\!\mathrm{d}\upmu\Big{)}\Big{)}\,\mathop{}\\!\mathrm{d}\upmu\leqslant
2^{(\upalpha-1)_{+}}\big{(}1+(\upalpha/e)^{\upalpha}\big{)},\end{split}$ (72)

where $\upgamma_{+}=\max\\{\upgamma,0\\}$ and we used the inequality
$(A+B)^{\upalpha}\leqslant 2^{(\upalpha-1)_{+}}(A^{\upalpha}+B^{\upalpha})$
which holds for every $A,B\geqslant 0$. This proves the leftmost inequality of
(67).

For the inequality on the right, assume that

$\max\big{\\{}\|f\|_{L_{p}}^{p},\mathrm{Ent}^{\upalpha}\big{(}f^{p}\big{)}\big{\\}}\leqslant
1.$ (73)

Then, since
$\rmintop\nolimits_{\Omega}f^{p}\,\mathop{}\\!\mathrm{d}\upmu\leqslant 1$, the
monotonicity of $x\mapsto\log^{\upalpha}x$ on $(0,\infty)$ gives

$\rmintop\nolimits_{\Omega}f^{p}\log^{\upalpha}f^{p}\,\mathop{}\\!\mathrm{d}\upmu\leqslant\rmintop\nolimits_{\Omega}f^{p}\log^{\upalpha}\Big{(}\frac{f^{p}}{\rmintop\nolimits
f^{p}\,\mathop{}\\!\mathrm{d}\upmu}\Big{)}\,\mathop{}\\!\mathrm{d}\upmu=\mathrm{Ent}^{\upalpha}\big{(}f^{p}\big{)}\leqslant
1.$ (74)

At this point we shall use the elementary inequality

$y\log^{\upalpha}(e+y)\leqslant\underbrace{\max\Big{\\{}\log^{\upalpha}(e+1)+2(\upalpha/e)^{\upalpha},\tfrac{2^{1/\upalpha}+e-1}{2^{1/\upalpha}-1}\log^{\upalpha}\Big{(}e+\tfrac{2^{1/\upalpha}+e-1}{2^{1/\upalpha}-1}\Big{)}\Big{\\}}}_{K_{\upalpha}}+2y\log^{\upalpha}y$
(75)

which is valid for all $y>0$. Indeed, assuming (75), we have

$\rmintop\nolimits_{\Omega}f^{p}\log^{\upalpha}(e+f^{p})\,\mathop{}\\!\mathrm{d}\upmu\stackrel{{\scriptstyle\eqref{eq:eleme}}}{{\leqslant}}K_{\upalpha}+2\rmintop\nolimits_{\Omega}f^{p}\log^{\upalpha}f^{p}\,\mathop{}\\!\mathrm{d}\upmu\stackrel{{\scriptstyle\eqref{eq:upper-
for-Orlicz}}}{{\leqslant}}K_{\upalpha}+2$ (76)

and therefore

$\rmintop\nolimits_{\Omega}\frac{f^{p}}{K_{\upalpha}+2}\log^{\upalpha}\Big{(}e+\frac{f^{p}}{K_{\upalpha}+2}\Big{)}\,\mathop{}\\!\mathrm{d}\upmu\leqslant\frac{1}{K_{\upalpha}+2}\rmintop\nolimits_{\Omega}f^{p}\log^{\upalpha}(e+f^{p})\,\mathop{}\\!\mathrm{d}\upmu\leqslant
1.$ (77)

This implies that $\|f\|_{L_{p}(\log
L)^{\upalpha}}\leqslant(K_{\upalpha}+2)^{1/p}$ which is the desired estimate.

To prove the elementary inequality (75) we consider several cases. If $0<y<1$,

$y\log^{\upalpha}(e+y)+2y\log^{\upalpha}(1/y)\leqslant\log^{\upalpha}(e+1)+2(\upalpha/e)^{\upalpha}$
(78)

since the first summand is increasing and the second is bounded by
$2(\upalpha/e)^{\upalpha}$. One the other hand, if
$y\geqslant\tfrac{2^{1/\upalpha}+e-1}{2^{1/\upalpha}-1}$ and we denote by
$\upbeta=2^{1/\upalpha}>1$, then we have

$2\log^{\upalpha}y=\log^{\upalpha}y^{\upbeta}\geqslant\log^{\upalpha}\big{(}\upbeta(y-1)+1\big{)}>\log^{\upalpha}(y+e)$
(79)

and thus (75) holds without the term $K_{\upalpha}$. Finally, if
$1<y<\tfrac{2^{1/\upalpha}+e-1}{2^{1/\upalpha}-1}$, then we just use that

$y\log^{\upalpha}(e+y)\leqslant\frac{2^{1/\upalpha}+e-1}{2^{1/\upalpha}-1}\log^{\upalpha}\Big{(}e+\frac{2^{1/\upalpha}+e-1}{2^{1/\upalpha}-1}\Big{)}$
(80)

and (75) holds without the term $2y\log^{\upalpha}y$. This completes the proof
of the lemma. ∎

##  5\. Further remarks

###  5.1. Limitations of semigroup methods

We include here, mainly for pedagogical reasons, an argument inspired by [22,
10] which leads to versions of Theorem 1 with a suboptimal Orlicz space on the
left-hand side. This material was obtained jointly with Ramon van Handel, whom
we thank for allowing us to include it here. We shall need the following
technical result which is due to [17, Lemma 44 and Remark 48]. Observe that
the case $p=2$ can be proven easily by an expansion in the Walsh basis and
Parseval’s identity.

######  Lemma 14.

For every $p\in(1,\infty]$, there exists $A_{p}\in(0,\infty)$ such that for
any $n\in\mathbb{N}$, every function $h:\mathscr{C}_{n}\to\mathbb{C}$
satisfies

$\forall\ t>0,\qquad\big{\|}\nabla
P_{t}h\big{\|}_{L_{p}}\leqslant\frac{A_{p}}{\big{(}e^{p^{*}t}-1\big{)}^{1/p^{*}}}\|h\|_{L_{p}},$
(81)

where $p^{\ast}=\min\\{p,2\\}$.

By Talagrand’s logarithmic Sobolev inequality [41] and interpolation, we
deduce the following Orlicz space version of hypercontractivity.

######  Lemma 15.

For every $p\in(1,\infty)$, there exists $\mathsf{L}_{p}\in(0,\infty)$ such
that the following holds. For any normed space $(E,\|\cdot\|_{E})$, any
$\upalpha\in[0,p/2]$ and any $n\in\mathbb{N}$, every function
$f:\mathscr{C}_{n}\to E$ satisfies

$\forall\ t\in(0,1),\qquad\big{\|}P_{t}f\big{\|}_{L_{p}(\log
L)^{\upalpha}(E)}\leqslant\frac{\mathsf{L}_{p}}{t^{2\upalpha/pp^{\ast}}}\|f\|_{L_{p}(E)},$
(82)

where $p^{\ast}=\min\\{p,2\\}$.

###### Proof.

The desired inequality is obvious for $\upalpha=0$ since $P_{t}$ is a
contraction. Therefore, the general claim would follow from the endpoint case
$\upalpha=p/2$ by complex interpolation of $L_{p}(\log L)^{\upalpha}$ Orlicz
spaces (which appears explicitly, e.g., in [33]). As $P_{t}$ has a positive
kernel, Jensen’s inequality implies the pointwise bound
$\|P_{t}f\|_{E}\leqslant P_{t}h$, where $h=\|f\|_{E}$ and thus we have

$\big{\|}P_{t}f\big{\|}_{L_{p}(\log
L)^{p/2}(E)}\leqslant\big{\|}P_{t}h\big{\|}_{L_{p}(\log
L)^{p/2}}\leqslant\big{\|}P_{t}h-\mathbb{E}h\big{\|}_{L_{p}(\log
L)^{p/2}}+\mathbb{E}h.$ (83)

As $\mathbb{E}P_{t}h=\mathbb{E}h$, Talagrand’s inequality (4) gives

$\big{\|}P_{t}h-\mathbb{E}h\big{\|}_{L_{p}(\log
L)^{p/2}}\stackrel{{\scriptstyle\eqref{eq:talagrand}}}{{\leqslant}}\mathsf{C}_{p}\big{\|}\nabla
P_{t}h\big{\|}_{L_{p}}\stackrel{{\scriptstyle\eqref{eq:ei}}}{{\leqslant}}\frac{\mathsf{C}_{p}A_{p}}{\big{(}e^{p^{*}t}-1\big{)}^{1/p^{*}}}\|f\|_{L_{p}(E)}.$
(84)

The proof follows by combining the above since additionally
$\mathbb{E}h=\|f\|_{L_{1}(E)}\leqslant\|f\|_{L_{p}(E)}$. ∎

Equipped with these lemmas, we can now try to prove Theorem 1 by the methods
of [22, 10].

###### Attempted proof of Theorem 1.

Consider a function $f:\mathscr{C}_{n}\to E$ and without loss of generality
assume that $\mathbb{E}f=0$. Then,

$\begin{split}\|f\|_{L_{p}(\log
L)^{\upalpha}(E)}=2\Big{\|}\rmintop\nolimits_{0}^{\infty}P_{t}\Delta
P_{t}f\,\mathop{}\\!\mathrm{d}t&\Big{\|}_{L_{p}(\log
L)^{\upalpha}(E)}\leqslant 2\rmintop\nolimits_{0}^{\infty}\big{\|}P_{t}\Delta
P_{t}f\big{\|}_{L_{p}(\log L)^{\upalpha}(E)}\,\mathop{}\\!\mathrm{d}t\\\
&\stackrel{{\scriptstyle\eqref{eq:orl-
hyp}}}{{\leqslant}}\widetilde{\mathsf{L}}_{p}\rmintop\nolimits_{0}^{\infty}\big{\|}\Delta
P_{t}f\big{\|}_{L_{p}(E)}\max\big{\\{}t^{-2\upalpha/pp^{\ast}},1\big{\\}}\,\mathop{}\\!\mathrm{d}t.\end{split}$
(85)

The last estimate follows by combining (82) for $t\in(0,1)$ and the usual
hypercontractive inequality

$\forall\ t\geqslant 1,\qquad\big{\|}P_{t}\Delta P_{t}f\big{\|}_{L_{p}(\log
L)^{\upalpha}(E)}\leqslant\mathsf{K}_{p,\upalpha}\big{\|}P_{t}\Delta
P_{t}f\big{\|}_{L_{1+e^{2t}(p-1)}(E)}\leqslant\mathsf{K}_{p,\upalpha}\big{\|}\Delta
P_{t}f\big{\|}_{L_{p}(E)},$ (86)

where in the first inequality we used a bound between Orlicz norms which is
valid for every function. By the main technical bound of [22] as expressed in
[10, Proposition 18], we deduce that

$\|f\|_{L_{p}(\log
L)^{\upalpha}(E)}\leqslant\widetilde{\mathsf{L}}_{p}\rmintop\nolimits_{0}^{\infty}\left(\mathbb{E}\Big{\|}\sum_{i=1}^{n}\updelta_{i}(t)\partial_{i}f\Big{\|}_{L_{p}(E)}^{p}\right)^{1/p}\max\big{\\{}t^{-2\upalpha/pp^{\ast}},1\big{\\}}\,\frac{\mathop{}\\!\mathrm{d}t}{\sqrt{e^{2t}-1}},$
(87)

where the random variables $\updelta_{i}(t)$ are defined in Remark 7. Assuming
that the space has cotype $q\in[2,\infty)$, it was shown in [22, Section 4]
that

$\left(\mathbb{E}\Big{\|}\sum_{i=1}^{n}\updelta_{i}(t)\partial_{i}f\Big{\|}_{L_{p}(E)}^{p}\right)^{1/p}\leqslant\mathsf{C}_{p,q}t^{\frac{1}{\max\\{p,q\\}}-\frac{1}{2}}\left(\mathbb{E}\Big{\|}\sum_{i=1}^{n}\updelta_{i}\partial_{i}f\Big{\|}_{L_{p}(E)}^{p}\right)^{1/p}$
(88)

and combining the above we finally get the estimate

$\|f\|_{L_{p}(\log
L)^{\upalpha}(E)}\leqslant\widetilde{\mathsf{L}}_{p}\mathsf{C}_{p,q}\left(\rmintop\nolimits_{0}^{\infty}t^{\frac{1}{\max\\{p,q\\}}-\frac{1}{2}}\max\big{\\{}t^{-2\upalpha/pp^{\ast}},1\big{\\}}\,\frac{\mathop{}\\!\mathrm{d}t}{\sqrt{e^{2t}-1}}\right)\left(\mathbb{E}\Big{\|}\sum_{i=1}^{n}\updelta_{i}\partial_{i}f\Big{\|}_{L_{p}(E)}^{p}\right)^{1/p}.$

Observe however that this integral converges if and only if
$\upalpha<\frac{pp^{\ast}}{2\max\\{p,q\\}}\leqslant\frac{p}{2}$. Therefore,
this approach cannot recover the optimal Orlicz space attained in Theorem 1. ∎

###  5.2. Reductions for vector-valued influence inequalities

Conversely, the arguments used in this paper may shed light to some of the
questions left open in [10]. As a matter of fact, they readily lead to two
potentially useful reductions.

$\bullet$ In [10, Theorem 6] it was shown that when $E$ is a Banach space of
Rademacher type $2$, then for any $\varepsilon\in(0,1)$, we have the
inequality

$\big{\|}f-\mathbb{E}f\big{\|}_{L_{2}(E)}^{2}\leqslant\frac{\mathsf{C}(E)}{\varepsilon}\sum_{i=1}^{n}\big{\|}\partial_{i}f\big{\|}_{L_{2}(\log
L)^{-1+\varepsilon}(E)}^{2}$ (89)

and it remains unknown whether one is allowed to take $\varepsilon=0$ as in
the scalar case treated in [42] without additional assumptions on the space
$E$. Assuming as usual that $\mathbb{E}f=0$ and denoting $h=\|f\|_{E}$, we can
first use the triangle inequality

$\begin{split}\|f\|_{L_{2}(E)}^{2}=\|h\|_{L_{2}}^{2}\leqslant
2\big{(}\|h-\mathbb{E}h\|_{L_{2}}^{2}+|\mathbb{E}h|^{2}\big{)}.\end{split}$
(90)

Then, we observe that by Talagrand’s scalar influence inequality the first
summand satisfies

$\|h-\mathbb{E}h\|_{L_{2}}^{2}\leqslant\mathsf{C}\sum_{i=1}^{n}\big{\|}\partial_{i}h\big{\|}_{L_{2}(\log
L)^{-1}}^{2}\leqslant\mathsf{C}\sum_{i=1}^{n}\big{\|}\partial_{i}f\big{\|}_{L_{2}(\log
L)^{-1}(E)}^{2},$ (91)

where in the last inequality we used the pointwise bound
$|\partial_{i}h|\leqslant\|\partial_{i}f\|_{E}$. Therefore, in order to prove
(89) with $\varepsilon=0$ it is both necessary and sufficient to prove the
strictly weaker estimate

$\big{\|}f-\mathbb{E}f\big{\|}_{L_{1}(E)}^{2}\leqslant\mathsf{C}(E)\sum_{i=1}^{n}\big{\|}\partial_{i}f\big{\|}_{L_{2}(\log
L)^{-1}(E)}^{2}$ (92)

since the left-hand side is just $|\mathbb{E}h|^{2}$.

$\bullet$ In [10] we also investigated the so-called $L_{1}-L_{p}$
inequalities which are morally dual estimates to Talagrand’s inequality (4).
In the case of scalar-valued functions, [10, Theorem 33] asserts that for any
$p\in(1,\infty)$ there exists $\mathsf{C}_{p}>0$ such that every
$f:\mathscr{C}_{n}\to\mathbb{C}$ satisfies

$\big{\|}f-\mathbb{E}f\big{\|}_{L_{p}}\leqslant\mathsf{C}_{p}\big{\|}\nabla
f\big{\|}_{L_{p}(\log L)^{-p/2}}.$ (93)

Moreover, a semigroup argument similar to that of Section 5.1 proved an
analogue of (93) with a suboptimal Orlicz space on the right-hand side for
functions with values in normed spaces of finite cotype, see [10, Theorem 29].
An inspection of the proof of Theorem 1 reveals that if one could show a
strengthening of (93) for real-valued functions with the asymmetric gradient
$\mathsf{M}f$ on the right-hand side, this would also sharpen the vector-
valued results of [10].

###  5.3. Beckner inequalities

In [2], Beckner put forth a family of functional inequalities in Gauss space
interpolating between the Poincaré and logarithmic Sobolev inequalities. More
specifically, he proved that for any $n\in\mathbb{N}$ and $q\in[1,2]$, every
function $f:\mathbb{R}^{n}\to\mathbb{R}$ satisfies

$\|f\|_{L_{2}}^{2}-\|f\|_{L_{q}}^{2}\leqslant(2-q)\big{\|}\nabla
f\big{\|}_{L_{2}}^{2}.$ (94)

Observe that dividing both sides by $2-q$ and taking a limit as $q\to 2^{-}$,
one recovers Gross’ logarithmic Sobolev inequality [19]. In fact, Beckner-type
inequalities are equivalent (up to the value of the constants) to logarithmic
Sobolev inequalities. Indeed, the convexity of $r\mapsto\log\|f\|_{L_{1/r}}$
for $r\in(0,1]$ implies that the quantity

$\frac{1}{\frac{1}{q}-\frac{1}{2}}\Big{(}\|f\|_{L_{2}}^{2}-\|f\|_{L_{q}}^{2}\Big{)}$
(95)

is increasing in $q\in[1,2)$. Combining this observation with Theorem 1, we
deduce the following.

######  Corollary 16.

Let $(E,\|\cdot\|_{E})$ be a Banach space of finite cotype. There exists a
constant $\mathsf{B}(E)\in(0,\infty)$ such that for any $n\in\mathbb{N}$ and
$q\in[1,2]$, every function $f:\mathscr{C}_{n}\to E$ with $\mathbb{E}f=0$
satisfies

$\big{\|}f\big{\|}_{L_{2}(E)}^{2}-\big{\|}f\big{\|}_{L_{q}(E)}^{2}\leqslant\mathsf{B}(E)(2-q)\rmintop\nolimits_{\mathscr{C}_{n}}\Big{\|}\sum_{i=1}^{n}\updelta_{i}\partial_{i}f\Big{\|}_{L_{2}(E)}^{2}\,\mathop{}\\!\mathrm{d}\sigma_{n}(\updelta).$
(96)

One can obtain similar results in Gauss space or the symmetric group, using
Theorem 3.

###  5.4. Functional isoperimetric-type inequalities

Talagrand’s isoperimetric inequality from [41] asserts that any Boolean
function $f:\\{-1,1\\}^{n}\to\\{-1,1\\}$ satisfies

$\|\nabla
f\|_{L_{2}}\geqslant\mathsf{c}\mathrm{Var}(f)\sqrt{\log\big{(}1/\mathrm{Var}(f)\big{)}},$
(97)

where $\mathsf{c}>0$ is a universal constant. In the recent work [3], Beltran,
Ivanisvili and Madrid put forth some extensions of (97) for functions with
values in spaces of finite cotype. Their proofs build on the method of [22]
and, similarly to the results of Section 5.1, the cotype of the target space
appears as an exponent in some of their estimates. We present here a result in
this spirit which is a direct consequence of Theorem 1 (observe also the
analogy with the influence inequality [42, 10]).

######  Proposition 17.

Let $(E,\|\cdot\|_{E})$ be a Banach space of finite cotype and
$p\in[1,\infty)$. There exists a constant
$\mathsf{\upeta}_{p}(E)\in(0,\infty)$ such that for any $n\in\mathbb{N}$,
every function $f:\mathscr{C}_{n}\to E$ with $\mathbb{E}f=0$ satisfies

$\left(\rmintop\nolimits_{\mathscr{C}_{n}}\Big{\|}\sum_{i=1}^{n}\updelta_{i}\partial_{i}f\Big{\|}_{L_{p}(E)}^{p}\,\mathop{}\\!\mathrm{d}\sigma_{n}(\updelta)\right)^{1/p}\geqslant\upeta_{p}(E)\cdot\|f\|_{L_{p}(E)}\Big{(}1+\sqrt{\log\big{(}\|f\|_{L_{p}(E)}/\|f\|_{L_{1}(E)}\big{)}}\Big{)}.$
(98)

###### Proof.

In view of Theorem 1 it suffices to show that $\|f\|_{L_{p}(\log L)^{p/2}(E)}$
is bounded below by the right-hand side of (98). The more general estimate

$\|f\|_{L_{p}(\log
L)^{\upalpha}(E)}\geqslant\mathsf{c}_{p,\upalpha}\|f\|_{L_{p}(E)}\Big{(}1+\log^{\upalpha/p}\big{(}\|f\|_{L_{p}(E)}/\|f\|_{L_{1}(E)}\big{)}\Big{)}$
(99)

follows by concatenating [10, Lemma 17] with the Hölder inequality

$\|FG\|_{L_{p/2}}\leqslant\mathsf{C}_{\upalpha}\|F\|_{L_{p}(\log
L)^{\upalpha}}\|G\|_{L_{p}(\log L)^{-\upalpha}}$ (100)

for Orlicz spaces. As we were unable to locate a reference for (100), we
mention that it follows immediately from the two-point inequality

$(xy)^{p/2}\leqslant\tfrac{\mathsf{C}_{\upalpha}}{2}\big{(}x^{p}\log^{\upalpha}(e+x^{p})+y^{p}\log^{-\upalpha}(e+y^{p})\big{)},$
(101)

which is valid for every $x,y\geqslant 0$ and some fixed constant
$\mathsf{C}_{\upalpha}>0$. ∎

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