# Extremal of Log-Sobolev Functionals and Li-Yau Estimate on
$\text{RCD}^{*}(K,N)$ Spaces

Samuel Drapeau  Shanghai Jiao Tong University, China Academy of Financial
Research (SAIF) and School of Mathematical Sciences, Shanghai, China
<EMAIL_ADDRESS>http://www.samuel-drapeau.info and  Liming Yin
Shanghai Jiao Tong University, School of Mathematical Sciences, Shanghai,
China<EMAIL_ADDRESS>
(Date: August 28, 2024)

###### Abstract.

In this work, we study the extremal functions of log-Sobolev functional in
metric measure space satisfying $\text{RCD}^{*}(K,N)$ condition for $K>0$ and
$N$ in $(2,\infty)$. We show the existence, regularity and positivity of non-
negative extremal functions. Based on these results, we prove a Li-Yau type
estimate for the logarithmic transform of all non-negative extremal functions
of log-Sobolev functional. As applications, we show a Harnack type inequality
as well as lower and upper bounds for the non-negative extremal functions.
Keywords: Log-Sobolev functional; metric measure space; Li-Yau inequality;
Curvature-dimension condition; extremal function.

National Science Foundation of China, Grants Numbers: 11971310 and 11671257
are gratefully acknowledged.

##  1\. Introduction

In this work, we study the extremal functions of following variational problem
of Log-Sobolev functional

$\lambda(\alpha_{1},\alpha_{2}):=\inf_{f\in\mathcal{F}}\int_{X}\left(|\nabla
f|^{2}+\alpha_{1}f^{2}-\frac{\alpha_{2}}{2}f^{2}\log f^{2}\right)dm$ (1.1)

where $\alpha_{1}$ and $\alpha_{2}$ are in $\mathbb{R}$ and
$\mathcal{F}:=\\{f\in W^{1,2}:\|f\|_{L^{2}}=1\\}$. Of interest are the
existence, regularity, positivity of non negative extremal functions, and
analytic results such as Li-Yau or Harnack type estimates. While those results
are well-known in a smooth Riemannian setting, it seems natural to ask whether
they can be extended and in which form to more general non smooth metric
spaces.

While log-Sobolev inequality has vast applications in different branches of
mathematics—see Gross [20], Otto and Villani [30], Bakry et al. [5]—studying
extremal of functionals including log-Sobolev functional is important on its
own right. For example, in [38], the author shows that in the noncompact
smooth manifold setting, the geometry of manifold at infinity will affect the
existence of extremal of log-Sobolev functional and Perelman’s $W$-entropy.
Using these points, the author further shows that noncompact shrinking
breathers of Ricci flow are gradient shrinking solitons. Very recently, in
[29] the extremal function of log-Sobolev inequality is used together with
optimal transport and needle decomposition technique to show some rigidity
result of underlying weighted Riemannian manifold.

On the other hand, starting with works of Sturm [34, 35] and Lott and Villani
[27], the synthetic notion of Ricci curvature—referred to as $\text{CD}(K,N)$
condition—bounded from below by $K$ in $\mathbb{R}$ and dimension bounded from
above by $1\leq N\leq\infty$ on general metric measure space without having a
smooth structure was introduced and developed greatly in the last decade. The
key property of this notion is that it is compatible with the smooth
Riemannian manifold and stable with respect to measured Gromov-Hausdorff
convergence so that it includes the Ricci limit spaces and Alexandrov spaces.
Later, to rule out the Finsler manifold, the finer concept
$\text{RCD}(K,\infty)$ was introduced by Ambrosio and coauthors in [3] and the
finite dimensional counterpart $\text{RCD}(K,N)$ was introduced by Gigli,
Erbar and coauthor in [16, 13]. Recently, the first and second order
differential structure on $\text{RCD}(K,\infty)$ space based on the tangent
module theory was developed by Gigli in [17], and finer geometric results such
as rectifiability of $\text{RCD}(K,N)$ was also studied in [28].

With the development of analytic tools on these spaces, the geometric analysis
on metric measure space satisfying synthetic Ricci curvature condition also
developed quickly. For instance, Li-Yau-Hamilton type estimate for heat flow
[14, 22, 23] and localized gradient and Li-Yau estimate of heat equation [21,
37]. One of the breakthrough is [37], where the authors develop a Omori-Yau
type maximum principle on the $\text{RCD}^{*}(K,N)$ space and use it to show
the pointwise Li-Yau type estimate for the locally weak solution of heat
equation which may not share semigroup property.

Motivated by these works, it seems therefore natural to study extremal
functions of log-Sobolev functional (1.1) on more general metric measure
space. In particular, we are interested in whether analytic results such as
Li-Yau type estimate for non-negative extremal functions of log-Sobolev
functional holding on smooth settings can be extended to non smooth metric
measure spaces, in particular, those satisfying synthetic Ricci curvature
condition.

To do so, one of the key points is to show the existence, boundedness,
regularity and positivity of non-negative extremal functions of log-Sobolev
functional on the metric measure space. Our first main result, Theorem 3.2,
states that the log-Sobolev functional (1.1) with $\alpha_{2}>0$ on metric
measure space satisfying $\text{RCD}^{*}(K,N)$ with $K>0$ and $N$ in
$(2,\infty)$ admits non-negative extremal functions which satisfy certain
Euler-Lagrangian equation. Moreover, we show that all non-negative extremal
functions are bounded, Lipschitz continuous and bounded away from 0.

We remark that while existence and Euler-Lagrangian equation problems are
quite standard and similar to the smooth compact cases solved by Rothaus in
[33], several other problems arise on metric measure spaces. For instance, the
positivity of non-negative extremal functions in [33, pager 114] is shown by
relying heavily on the underlying smooth differential structure of Riemannian
manifold, polar coordinate and exact asymptotic ratio near the pole so that
the problem can be reduced to a one-dimensional ODE problem. However, these
smooth structures are lost in general metric measure spaces. While for
$\text{RCD}^{*}(K,N)$ spaces, the polar decomposition still works by using
“needle decomposition” generalized to $\text{CD}(K,N)$ setting by Cavalletti
and Mondino in [9], see also [10] for $\text{MCP}(K,N)$, however, similar
asymptotic ratio analysis seems to fail without further assumption on
underlying metric measure space. To overcome this difficulty, we make use of a
maximum principle type argument for the De Giorgi class on some local domain,
which was proved in [24], to show that non-negative extremal functions are
either bounded away from $0$ or vanish on the whole space. This method works
in very general metric measure spaces supporting locally doubling property and
weak Poincaré inequality, which in particular contains $\text{RCD}^{*}(K,N)$
spaces.

Our second main result is in Theorem 4.1. Based on the regularity and
positivity results obtained, we recover a Li-Yau type estimate for the
logarithmic transform of all non-negative extremal functions of (1.1). More
precisely, for any non-negative extremal functions $u$, it holds that

$|\nabla v|^{2}+(\alpha_{2}-\beta
K)v\leq\frac{N\alpha_{2}(1-\beta)}{4\beta}\left(1-\frac{\beta((2-\beta)K-\alpha_{2})}{2\alpha_{2}(1-\beta)}\right)^{2}\quad\text{$m$-a.e.,}$
(1.2)

for any $\beta$ in $(0,1)$ and $v=\log u+(\lambda-\alpha_{1})/\alpha_{2}$. The
same estimate for the smooth Riemannian cases was shown in [36], where the
argumentation relies on the pointwise Bochner formula and smoothness property
of local maximum points of the left side of (1.2). However, in the RCD
setting, neither of the functions on the left hand side of (1.2) is smooth and
pointwise defined, nor the pointwise Bochner formula is available. To overcome
the difficulty, we follow a similar argument as in [37] by using Omori-Yau
type maximum principle. Note that, to avoid the sign problem of $|\nabla
v|^{2}+(\alpha_{2}-\beta K)v$, we use different auxiliary function $\phi$ as
in [37, Theorem 1.4] which is constructed from the distance function so that
it has measure-valued Laplacian. Our construction is based on the “good” cut-
off functions from [28, Lemma 3.1], which are smoother than the one in [37] to
guarantee $L^{2}$-valued Laplacian. Furthermore, for our purpose as well as
independent interest, we prove a slightly generalized version of Omori-Yau
maximum principle holding on the whole metric space of proper
$\text{RCD}(K,\infty)$ than the one in [37], which holds for bounded domains
on $\text{RCD}^{*}(K,N)$ spaces. While most arguments are similar, our proof
follow so-called “Sobolev-to-Lip” property shared by all RCD spaces rather
than "weak maximum principle" used in [37] for $\text{RCD}^{*}(K,N)$.

Finally, we provide some applications of the regularity results and Li-Yau
type estimates. We show a Harnack type estimate as well as lower and upper
bounds of the non-negative extremal functions of (1.1) depending only on
geometry of space. These generalize results in [36] proved for Riemannian
cases. Using weak Bochner inequality, we also show that all non-negative
extremal functions are constant when $0<\alpha_{2}\leq KN/(N-1)$, which is
well-known in the smooth setting.

The paper is organized as follows: in Section 2, we introduce notations and
definitions about metric measure space and RCD condition as well as some
analytic results needed later. In Section 3, we study the variational problem
and show the existence, regularity and positivity of non-negative extremal
functions of (1.1). Section 4 is dedicated to the Li-Yau type estimate for the
non-negative extremal functions. In Section 5, we show some applications of
previous results. Finally, in an Appendix, we prove the Omori-Yau type maximum
principle for the proper $\text{RCD}(K,\infty)$ spaces.

##  2\. Preliminary and notations

We briefly introduce terminologies and notations related to calculus. For more
details, we refer readers [19, 17].

Throughout this work, we denote by $(X,d,m)$ a metric measure space where
$(X,d)$ is complete, separable and proper metric space and $m$ is a non-
negative Radon measure which is finite on each bounded set. We denote by
$B(x,r)$ and $\bar{B}(x,r)$ the open and closed metric balls centered at $x$
in $X$ with radius $r>0$, respectively. By $L^{p}:=L^{p}(X,m)$ for $1\leq
p\leq\infty$ we denote the standard $L^{p}$ spaces with $L^{p}$-norm
$\|\cdot\|_{p}$. By $L^{p}_{loc}$ we denote those measurable functions
$f:X\to\mathbb{R}$ such that $f\chi_{B}$ is in $L^{p}$ for any bounded subset
$B$ of $X$ where $\chi_{B}$ denote the indicator function of the set $B$. Let
further $\text{Lip}:=\text{Lip}(X)$, $\text{Lip}_{loc}:=\text{Lip}_{loc}(X)$,
and $\text{Lip}_{bs}:=\text{Lip}_{bs}(X)$ be the spaces of real-valued
functions $f:X\to\mathbb{R}$ which are Lipschitz, locally Lipschitz, and
Lipschitz with bounded support, respectively. For $f$ in $\text{Lip}_{loc}$,
we denote by $\text{lip}(f)$ the local Lipschitz constant, or slope, defined
for any $x$ in $X$ as

$\text{lip}f(x):=\limsup_{y\rightarrow x}\frac{|f(y)-f(x)|}{d(y,x)},$

if $x$ is not isolated and $\text{lip}f(x)=0$ if $x$ is isolated.

###  2.1. Cheeger Energy, Laplacian and Calculous Tools

The Cheeger energy is the $L^{2}$-lower-semicontinuous and convex functional
$\text{Ch}:L^{2}\rightarrow[0,\infty]$ defined as

$\text{Ch}(f):=\inf\left\\{\liminf\frac{1}{2}\int_{X}\left(\text{lip}(f_{n})\right)^{2}dm\colon(f_{n})\subseteq\text{Lip}\cap
L^{2},\|f_{n}-f\|_{2}\rightarrow 0\right\\}.$

The domain of Ch is a linear space denoted by $W^{1,2}:=W^{1.2}(X)$ and called
the Sobolev space. For $f$ in $W^{1,2}$, we identify the canonical element
$|\nabla f|$ called minimal relaxed gradient as the unique element with
minimal $L^{2}$-norm, also minimal in $m$-a.e. sense, in the set

$\left\\{G\in L^{2}:G=\lim\text{lip}f_{n}\text{ in }L^{2}\text{ for some
}(f_{n})\subseteq\text{Lip}\text{ such that }f_{n}\rightarrow f\text{ in
$L^{2}$}\right\\},$

which provides an integral representation $\text{Ch}(f)=\int_{X}|\nabla
f|^{2}dm$. The Sobolev space equipped with norm
$\|f\|^{2}_{W^{1,2}}:=\|f\|^{2}_{2}+2\text{Ch}(f)$ is a Banach space and
$\text{Lip}_{bs}$ is dense in $W^{1,2}$ with respect to $\|\cdot\|_{W^{1,2}}$.
We further denote by $W^{1,2}_{loc}:=\\{f\in L^{2}_{loc}:\eta f\in
W^{1,2}\text{ for any }\eta\in\text{Lip}_{bs}\\}$ the space of locally Sobolev
functions, and define the minimal relaxed gradient as $|\nabla
f|:=|\nabla(\eta f)|$ $m$-a.e. on $\\{\eta=1\\}$ for $f$ in $W^{1,2}_{loc}$
where $\eta$ is in $\text{Lip}_{bs}$.

We say that $(X,d,m)$ is _infinitesimally Hilbertian_ if the Cheeger energy is
a quadratic form, or equivalently, $W^{1,2}$ is a Hilbert space. Under these
assumptions, it can be proved that for any $f$ and $g$ in $W^{1,2}$, the limit

$\langle\nabla f,\nabla g\rangle:=\lim_{\varepsilon\rightarrow
0}\frac{|\nabla(f+\varepsilon g)|^{2}-|\nabla f|^{2}}{2\varepsilon}$

exists in $L^{1}$ and it is a bilinear form from $W^{1,2}\times W^{1,2}$ to
$L^{1}$, see [3].

######  Definition 2.1.

Let $(X,d,m)$ be infinitesimally Hilbertian.

  * •

Laplacian: We say that $f$ in $W^{1,2}$, is in the domain of the Laplacian,
denoted by $D(\Delta)$, provided that there exists $h$ in $L^{2}$ such that

$-\int_{X}\langle\nabla f,\nabla g\rangle dm=\int_{X}hgdm,\quad\text{for any
}g\in W^{1,2}.$ (2.1)

In this case, we denote $\Delta f=h$.

  * •

Measure-Valued Laplacian: We say that $f$ in $W^{1,2}_{loc}$ is in the domain
of measure-valued Laplacian, denoted by $D(\bm{\Delta})$, provided that there
exists a signed Radon measure $\mu$ on $X$ such that,111Recall that $X$ is
assumed to be proper, and therefore any bounded set is compact on which Radon
measures are finite.

$-\int_{X}\langle\nabla f,\nabla g\rangle dm=\int_{X}gd\mu,\quad\text{for any
}g\in\text{Lip}_{bs}.$ (2.2)

In this case, we denote $\bm{\Delta}f=\mu$.

By separating property of $\text{Lip}_{bs}$ for Radon measure and
infinitesimally Hilbertian property, it’s clear that both $\Delta$ and
$\bm{\Delta}$ are well-defined, unique and linear operators. Moreover, the two
definitions are are compatible in the following sense: on the one hand, if $f$
is in $W^{1,2}$ with $\bm{\Delta}f=\rho m$ for some $\rho$ in $L^{2}$, then
$f$ is in $D(\Delta)$ and $\Delta f=\rho$. On the other hand, if $f$ is in
$W^{1,2}$ such that $\Delta f\in L^{1}$, then $f$ is in $D(\bm{\Delta})$ and
$\bm{\Delta}f=(\Delta f)m$, see [19, Proposition 6.2.13]. For $f$ in the
domain of $\bm{\Delta}$, We denote the Lebesgue decomposition with respect to
$m$

$\bm{\Delta}f=(\bm{\Delta}^{ac}f)\cdot m+\bm{\Delta}^{s}f,$ (2.3)

where $\bm{\Delta}^{ac}f$ is the Radon-Nikodym density and $\bm{\Delta}^{s}f$
is the singular part of $\bm{\Delta}f$.

For $w$ in $W^{1,2}\cap L^{\infty}$, we define weighted Laplacian $\Delta_{w}$
in a similar way but with respect to reference measure $m_{w}:=e^{w}\cdot m$
and test functions in $W^{1,2}(X,m_{w})$. For such $w$, it can be shown that
$W^{1,2}$ coincides with $W^{1,2}(X,m_{w})$, and the minimal relaxed gradient
induced by $m_{w}$ coincides with the one induced by $m$, see [2, Lemma 4.11].
Moreover, it holds that $\Delta_{w}f=\Delta f+\langle\nabla w,\nabla
f\rangle$.

The following Lemma recaps calculus rules, the proof of which can be found in
[19, 37, 18].

######  Lemma 2.2.

Let $(X,d,m)$ be infinitesimally Hilbertian. Then:

  1. (i)

Locality: $|\nabla f|=|\nabla g|$ on $\\{f-g=c\\}$ for any $f$, $g$ in
$W^{1,2}$ and constant $c$.

  2. (ii)

Chain rule: for any $f$ in $W^{1,2}$ and Lipschitz function
$\phi:\mathbb{R}\to\mathbb{R}$, it follows that

$|\nabla(\phi\circ f)|=|\phi^{\prime}(f)||\nabla f|$

In particular, if $\phi$ is a contraction, then $|\nabla(\phi\circ
f)|\leq|\nabla f|$.

  3. (iii)

Leibniz rule: for any $f$, $g$ and $h$ in $W^{1,2}\cap L^{\infty}$, it follows
that $fg$ is in $W^{1,2}$ and

$\langle\nabla(fg),\nabla h\rangle=f\langle\nabla g,\nabla
h\rangle+g\langle\nabla f,\nabla h\rangle.$

  4. (iv)

Chain rule: for any $f$ in $D(\Delta)\cap L^{\infty}$ and $C^{2}$-function
$\phi:\mathbb{R}\rightarrow\mathbb{R}$, it follows that $\phi(f)$ is in
$D(\Delta)$ and

$\Delta\phi(f)=\phi^{\prime}(f)\Delta f+\phi^{\prime\prime}(f)|\nabla f|^{2}.$
(2.4)

  5. (v)

Leibniz rule: for any $f$ and $g$ in $D(\bm{\Delta})\cap L^{\infty}$ such that
$g$ is continuous and $\bm{\Delta}g$ is absolutely continuous with respect to
$m$, then $fg$ is in $D(\bm{\Delta})$ and

$\bm{\Delta}(fg)=f\bm{\Delta}g+g\bm{\Delta}f+2\langle\nabla f,\nabla
g\rangle\cdot m.$ (2.5)

As $L^{2}$-lower semicontinuity and convexity of Cheeger energy, the heat
semigroup $P_{t}f$ is defined as the gradient flow in $L^{2}$ of Cheeger
energy starting from $f\in L^{2}$ based on classical Brezis-Komura theory,
which provides the existence and uniqueness results. Moreover, for any $f\in
L^{2}$, it holds that $t\mapsto P_{t}f$ is locally absolutely continuous on
$(0,\infty)$ and $P_{t}f$ is in $D(\Delta)$ for all $t>0$ and

$\frac{d}{dt}P_{t}f=\Delta P_{t}f,\quad\text{for almost all }t\in(0,\infty).$
(2.6)

Under the further assumption that $(X,d,m)$ is infinitesimally Hilbertian, the
heat semigroup $P_{t}$ is linear, strongly continuous, contraction and order-
preserving in $L^{2}$. Moreover, $P_{t}$ can be extended into a linear, mass
preserving and strongly continuous operator in $L^{p}$ for any $1\leq
p<\infty$, see [2] and futher results therein.

Finally we adapt the above mentioned definition and results when restricted to
balls $\Omega:=B(x,R)\subseteq X$ to which we refer to [37, Section 2.3] for a
detailed presentation. It follows the same approach up to the following subtle
fact that one needs to pay attention to the boundary as follows: We denote by
$\text{Lip}_{c}(\Omega)$ the space of Lipschitz functions on $\Omega$ with
compact support in $\Omega$. For $f$ in $\text{Lip}_{loc}(\Omega)$, we define
the $W^{1,2}(\Omega)$-norm as

$\|f\|^{2}_{W^{1,2}(\Omega)}=\|f\|^{2}_{L^{2}(\Omega)}+\|\text{lip}f\|^{2}_{L^{2}(\Omega)},$
(2.7)

and the Sobolev space $W^{1,2}(\Omega)$ as the $W^{1,2}(\Omega)$-closure of
the set $\\{f\in\text{Lip}_{loc}(\Omega):\|f\|_{W^{1,2}(\Omega)}<\infty\\}$.
By similar procedures as the global one, one can easily define $|\nabla f|$
and local inner product $\langle\nabla f,\nabla g\rangle$ in $L^{1}(\Omega)$
for any $f$ and $g$ in $W^{1,2}(\Omega)$. As for the Laplacian, it modifies as
follows: A function $f$ in $W^{1,2}(\Omega)$ belongs to $D(\Delta,\Omega)$
provided that there exists $g$ in $L^{2}(\Omega)$ such that

$-\int_{\Omega}\langle\nabla f,\nabla\phi\rangle dm=\int_{\Omega}g\phi
dm,\quad\text{for all }\phi\in\text{Lip}_{c}(\Omega),$

and we denote $\Delta_{\Omega}f:=g$. It can be easily checked that for $f$ in
$D(\Delta)$, its restriction to $\Omega$ belongs to $D(\Delta,\Omega)$.

###  2.2. RCD metric measure spaces

Gigli [16], Erbar et al. [13] introduced the notion of Riemannian curvature
dimension $\text{RCD}(K,N)$ as the finite dimensional counterpart to
$\text{RCD}(K,\infty)$, itself introduced in [3] based on curvature-dimension
condition proposed by [27, 34, 35] to rule out Finsler cases. Later the finer
notion called reduced Riemannian curvature-dimension condition
$\text{RCD}^{*}(K,N)$ was proposed in [13], which satisfies the so-called
“local-to-global” property of [4]. Very recently, Cavalletti and Milman [8]
shows that $\text{RCD}(K,N)$ and $\text{RCD}^{*}(K,N)$ are equivalent if the
reference measure is finite.

The notion of $\text{RCD}^{*}(K,N)$ condition can be introduced in several
equivalent ways, see [13]. In this work we give a definition including the
case where $N=\infty$ from an Eulerian point of view based on abstract
$\Gamma$-calculus.

######  Definition 2.3.

We say that a metric measure space $(X,d,m)$ satisfies the
$\text{RCD}^{*}(K,N)$ condition for $K$ in $\mathbb{R}$ and $N$ in
$[1,\infty]$ provided that

  1. (i)

$m(B(x,r))\leq C\exp(Cr^{2})$ for some $x$ in $X$ and $C>0$;

  2. (ii)

Sobolev-to-Lip property: Any $f$ in $W^{1,2}$ with $|\nabla f|$ in
$L^{\infty}$ admits a Lipschitz representation $\tilde{f}$ in Lip such that
$f=\tilde{f}$ $m$-a.e and $\text{Lip}(f)=\||\nabla f|\|_{\infty}$.

  3. (iii)

$(X,d,m)$ is infinitesimally Hilbertian.

  4. (iv)

Weak Bochner inequality: for any $f$ and $g$ in $D(\Delta)$ with $\Delta f$ in
$W^{1,2}$, $\Delta g$ in $L^{\infty}$ and $g\geq 0$ it holds

$\int_{X}\Delta g\frac{|\nabla
f|^{2}}{2}dm\geq\int_{X}g\left(\frac{1}{N}(\Delta f)^{2}+\langle\nabla
f,\nabla\Delta f\rangle+K|\nabla f|^{2}\right)dm$ (2.8)

for any $f\in D(\Delta)$ with $\Delta f\in W^{1,2}$ and $g\in D(\Delta)$ with
$\Delta g\in L^{\infty}$ and $g\geq 0$.

From now on, assume that $(X,d,m)$ is a $\text{RCD}^{\ast}(K,N)$ space with
$N$ in $(1,\infty)$ and $K>0$, which is the framework of the results in this
work Several important geometrical and analytical properties hold – sometimes
in more general case.

First, $X$ satisfies the Bishop-Gromov inequality, that is, for any $0<r<R$
and $x\in X$,

$\frac{m(B(x,R))}{v_{K,N}(R)}\leq\frac{m(B(x,r))}{v_{K,N}(r)},$ (2.9)

where $v_{K,N}(r)$ is the volume of ball with radius $r>0$ in the model space
with dimension $N$ and Ricci curvature $K$. In particular, $X$ is globally
doubling with constant $2^{N}$, that is,

$m(B(x,2r))\leq 2^{N}m(B(x,r)),\quad\text{for any }x\in X\text{ and }r>0.$
(2.10)

It also holds that $m(B(x,r))>0$ for any $r>0$ and $x$ in $X$, and therefore
the support of reference measure $m$ is equal to $X$.

Second, $X$ supports the weak $(1,1)$-Poincaré inequality, see [32, Theorem
1.1], that is, for any $x$ in $X$, $r>0$ and continuous function
$f:X\rightarrow\mathbb{R}$ and any upper gradient $g$ of $f$, we have

$\fint_{B(x,r)}\left|f-(f)_{x,r}\right|dm\leq Cr\fint_{B(x,2r)}gdm,$ (2.11)

where the constant $C$ only depends on $K$, $N$ and $r$, and
$\fint_{\Omega}fdm:=\frac{1}{m(\Omega)}\int_{\Omega}fdm$ and
$(f)_{x,r}:=\int_{B(x,r)}fdm$. By Hölder inequality, $X$ also supports weak
$(1,q)$-Poincaré inequality for any $1<q<2$. This implies that $X$ is
connected, see [6, Proposition 4.2]. In fact, although implicitly contained in
the definition, $X$ is geodesic space on the support of reference measure.

Third, the Monnet-Myers theorem implies that $X$ is compact, $m(X)<\infty$ and
$\text{diam}(X)\leq\pi\sqrt{(N-1)/K}$, see [34, Theorem 4.26]. Since
normalization of measure does not affect $\text{RCD}^{*}(K,N)$, it is not
restrictive to assume that $m(X)=1$ when $K>0$.

Fourth, in RCD setting, for Sobolev function $f\in W^{1,2}$, it is possible to
identify the gradient $\nabla f$, rather than modulus of the gradient $|\nabla
f|$, as the unique element in the tangent module $L^{2}(TX)$, which is a
$L^{2}(m)$-normed $L^{\infty}(m)$-module and has been introduced in [19, 17].
They also developed a second-order calculus on RCD spaces such that the notion
of Hessian $\text{Hess}f$, its pointwise norm $|\text{Hess}f|_{HS}\in L^{2}$
and covariant derivative are well-defined. For the complete theory, we refer
readers to [19, 17], we here only mention that we have the inclusion that
$D(\Delta)\subseteq H^{2,2}$. Consequently for any $f$ in
$D(\Delta)\cap\text{Lip}$, we have $|\nabla f|^{2}$ in $W^{1,2}$ and

$|\nabla|\nabla f|^{2}|\leq 2|\text{Hess}f|_{HS}|\nabla f|,$ (2.12)

where $|\text{Hess}(f)|_{HS}\in L^{2}$ is the pointwise norm of
$\text{Hess}f$, see [17, Theorem 3.3.18] or [12, Lemma 3.5].

######  Remark 2.4.

Although there exist different notions for Sobolev space such as Newtonian
spaces, see [6, 1], and different notions of weak upper gradients in the
metric measure space, all these notions are equivalent to each other in the
setting of $\text{RCD}^{*}(K,N)$ spaces. In particular, the minimal relaxed
gradient coincides with the minimal weak upper gradient, which is defined via
test plans and geodesics, see [19, Definition 2.1.8], and $W^{1,2}$ is
reflexive. For more details, we refer to [1].

Fifth, we recall regularity result of Poisson equation on the ball, see [37,
Lemma 3.4] and [25, Theorem 3.1] proced in the more general setting of Bakry-
Emery type curvature condition.

######  Lemma 2.5.

Let $(X,d,m)$ be a $\text{RCD}^{*}(K,N)$ space with $K$ in $\mathbb{R}$ and
$N$ in $[1,\infty)$. Let further $g$ in $L^{\infty}(B_{R})$ where
$B_{R}:=B(x_{0},R)$ is a geodesic ball centered at some $x_{0}$ in $X$ with
radius $R>0$. Assume that $f$ is in $W^{1,2}(B_{R})$ and $\Delta_{B_{R}}f=g$
on $B_{R}$ in the sense of distribution. Then it holds that $|\nabla f|$ is in
$L^{\infty}_{loc}(B_{R})$ and

$\||\nabla f|\|_{L^{\infty}(B_{R/2})}\leq
C(N,K,R)\left(\frac{1}{m(B_{R})}\|f\|_{L^{1}(B_{R})}+\|g\|_{L^{\infty}(B_{R})}\right).$

We also recall the following results about Sobolev inequality and compact
Sobolev embedding theorem in the $\text{RCD}(K,N)$ spaces setting with $K>0$
and $N$ in $(2,\infty)$, proved in [31, Proposition 3.3,Proposition 4.2].

######  Lemma 2.6.

Let $(X,d,m)$ be a $\text{RCD}^{*}(K,N)$ space with $K>0$ and $N$ in
$(2,\infty)$.

  1. (i)

Sobolev inequality: There exist constants $A\geq 1$ and $B>0$ depending only
on $K$ and $N$, such that for every $f$ in $W^{1,2}$, it holds

$\|f\|^{2}_{2^{*}}\leq A\|f\|^{2}_{2}+B\text{Ch}(f),$

where $2^{*}=2N/(N-2)$ is the Sobolev conjugate of $2$.

  2. (ii)

Rellich-Kondrachov: Let $(f_{n})$ be a sequence in $W^{1,2}$ with
$\sup_{n}\|f_{n}\|_{W^{1,2}}<\infty$. Then there exists $f$ in $W^{1,2}$ and a
subsequence $(f_{n_{k}})$ such that for every $1\leq q<2^{*}$, it holds that

$f_{n_{k}}\rightarrow f\quad\text{in }L^{q}(X,m).$

Finally, we mention a key result about the heat semigroup and resolvent of
Laplacian. Recall that for the heat semigroup $P_{t}$, we say $P_{t}$ is
ultracontractive if for $1\leq p<q\leq\infty$, there exists a constant
$C(t)>0$ such that for any $f$ in $L^{p}$, it holds that

$\|P_{t}f\|_{q}\leq C(t)\|f\|_{p},\quad t>0,$

and we denote $\|P_{t}\|_{(p,q)}:=\sup_{\|f\|_{p}\leq 1}\|P_{t}f\|_{q}$. The
ultracontractive property is equivalent to the Sobolev inequality on the
underlying metric measure space, see [5, Theorem 6.3.1]. For
$\text{RCD}^{*}(K,N)$ with $K>0$ and $N$ in $(2,\infty)$, since Sobolev
inequality holds, it follows that $P_{t}$ has the ultracontractive property.
More precisely, for $1\leq p<q\leq\infty$ and $0<t\leq 1$, it holds that

$\|P_{t}\|_{(p,q)}\leq Ct^{-\frac{N}{2}(\frac{1}{p}-\frac{1}{q})},$

where $C>0$ is a constant depending on $K$ and $N$. As a consequence, the
ultracontractive property of heat semigroup provides the following useful
boundedness result for the resolvent operator of Laplacian
$R_{\lambda}:=(\lambda I-\Delta)^{-1}$ for $\lambda>0$, which is needed later,
the proof of which can be found in [31, Lemma 4.1] and [5, Corollary 6.3].

######  Lemma 2.7.

In the context of an $\text{RCD}(K,N)$ space with $K>0$ and $N$ in
$(2,\infty)$, let $\lambda>0$. If $1\leq p\leq N/2$, the resolvent operator
$R_{\lambda}:L^{p}\rightarrow L^{q}$ is bounded for each $1\leq q<pN/(N-2p)$.
If $p>N/2$, then the resolvent operator $R_{\lambda}:L^{p}\rightarrow
L^{\infty}$ is bounded.

##  3\. Existence of positive extremal functions

From now on, we always assume that $(X,d,m)$ is a $\text{RCD}^{*}(K,N)$ space
with $K>0$ and $N$ in $(2,\infty)$, and $m$ is a Borel probability measure. We
consider the following variational problem

$\inf\left\\{\int_{X}\left(|\nabla
f|^{2}+\alpha_{1}f^{2}-\frac{\alpha_{2}}{2}f^{2}\log f^{2}\right)dm:f\in
W^{1,2},\|f\|_{2}=1\right\\},$ (3.1)

where $\alpha_{1}$ and $\alpha_{2}$ are in $\mathbb{R}$. The infimum quantity
of variational problem (3.1) is called the log Sobolev constant
$\lambda:=\lambda(\alpha_{1},\alpha_{2})$ on $X$ with parameters $\alpha_{1}$
and $\alpha_{2}$, and the functional in (3.1) is called log Sobolev
functional.

######  Definition 3.1.

Provided that $\lambda=\lambda(\alpha_{1},\alpha_{2})$ is a finite number, we
call a function $u$ in $W^{1,2}$ the extremal of variational problem (3.1) if
$\|u\|_{2}=1$ and

$\lambda=\int_{X}\left(|\nabla
u|^{2}+\alpha_{1}u^{2}-\frac{\alpha_{2}}{2}u^{2}\log u^{2}\right)dm.$ (3.2)

In the following, we provide our main result of this section. It states the
existence of non-negative extremal functions of variational problem (3.1).
Moreover, we show that all non-negative extremal functions are actually
Lipschitz continuous and bounded away from zero on $X$. As a corollary, we
show that the logarithmic transform of all non-negative extremal functions are
Lipschitz and in the domain of Laplacian and satisfies some Euler-Lagrangian
equation. Under the further assumption that $\lambda(\alpha_{1},\alpha_{2})$
the solutions thereof are non-constant.

######  Theorem 3.2.

Let $\alpha_{1}$ and $\alpha_{2}$ be given constants with $\alpha_{2}>0$. Then
the log Sobolev constant $\lambda=\lambda(\alpha_{1},\alpha_{2})$ has finite
value and the variational problem (3.1) admits non-negative extremal
functions. Moreover, any non-negative extremal function $u$ satisfies the
following properties:

  1. (i)

$u$ is in $D(\Delta)$ and satisfies

$-\Delta u=\alpha_{2}u\log u+(\lambda-\alpha_{1})u.$ (3.3)

Furthermore if $\lambda\neq\alpha_{1}$, then $u$ is non-constant.

  2. (ii)

$u$ is Lipschitz continuous;

  3. (iii)

$u$ is positive.222Since $X$ is compact, $u>\delta>0$ for some $\delta$.

######  Corollary 3.3.

Let $\alpha_{1}$ and $\alpha_{2}$ be given constants with $\alpha_{2}>0$ and
$u$ an arbitrary non-negative extremal function of (3.1). Then $v:=\log u$ is
Lipschitz and in $D(\Delta)$ and satisfies

$-\Delta v=|\nabla v|^{2}+\alpha_{2}v+\lambda-\alpha_{1}$ (3.4)

In particular, the equation $-\Delta f=|\nabla f|^{2}+\alpha_{2}f$ admits
Lipschitz weak solutions which are non-constant whenever
$\lambda\neq\alpha_{1}$.

######  Remark 3.4.

A classical example of the above variational problem is the weak log-Sobolev
inequalities, that is, the variational problems

$\lambda_{\varepsilon}=\inf\left\\{\frac{2}{\alpha_{\varepsilon}}\int_{X}|\nabla
f|^{2}dm+\int_{X}\left(\varepsilon f^{2}-f^{2}\log f^{2}\right)dm:f\in
W^{1,2},\|f\|_{2}=1\right\\},$ (3.5)

where $\varepsilon\geq 0$ and $\alpha_{\varepsilon}$ is defined as the
supremum of those $C>0$ such that

$\int_{X}f^{2}\log f^{2}dm\leq\int_{X}\left(\frac{2}{C}|\nabla
f|^{2}+\varepsilon f^{2}\right)dm$

for all $f$ in $W^{1,2}$ and $\|f\|_{2}=1$.

The log-Sobolev inequality in $\text{RCD}(K,N)$, see [9], implies that
$\alpha_{\varepsilon}\geq KN/(N-1)$ for all $\varepsilon\geq 0$, and
$\alpha_{\varepsilon}<\infty$ when $\varepsilon$ is small enough.
Straightforward inspection shows that if $\alpha_{\varepsilon}$ has finite
value, then $\lambda_{\varepsilon}=0$. In this case, by using Theorem 3.2 and
Corollary 3.3, we obtain that the weak Sobolev inequality admits Lipschitz and
positive extremal functions which are further non-constant if $\varepsilon>0$.
Furthermore, we derive that the PDE $-\Delta f=|\nabla
f|^{2}+\alpha_{\varepsilon}f$ admits non-constant, Lipschitz and positive weak
solutions if $\varepsilon>0$.

We first prove all assertions of Theorem 3.2 but positivity. While the
existence result follows classical methods in [33] using variational
techniques and compact Sobolev embedding, the boundedness and Lipschitz
regularity results are different. Our methods are based on ultracontractive
property of resolvent operator and local regularity result of Poisson equation
on a ball in Lemma 2.5. In the following proofs, we denote by $C>0$ a
universal constant which may vary from one step to the next.

###### First part of the proof of Theorem 3.2.

  1. Step 1:

We show the existence of non-negative extremal functions of variational
problem (3.1). Let $\mathcal{F}:=\\{f\in W^{1,2}:\|f\|_{2}=1\\}$ and
$F:\mathcal{F}\rightarrow\mathbb{R}$ be the log Sobolev functional in (3.1),
that is,

$F(f):=F_{\alpha_{1},\alpha_{2}}(f)=\int_{X}\left(|\nabla
f|^{2}+\alpha_{1}f^{2}-\frac{\alpha_{2}}{2}f^{2}\log
f^{2}\right)dm,\quad\text{for }f\in\mathcal{F}.$ (3.6)

We claim that $F$ is coercive on $\mathcal{F}$. Indeed, choosing
$0<\delta<N/(N-2)$, since $\|f\|_{2}=1$, by Jensen’s inequality, it follows
that

$\int f^{2}\log
f^{2}dm=\frac{1}{\delta}\int\log|f|^{2\delta}(f^{2}dm)\leq\frac{1+\delta}{\delta}\log\|f\|^{2}_{2+2\delta}.$
(3.7)

Note that the Sobolev inequality, see Lemma 2.6, implies that

$\|f\|^{2}_{2+2\delta}\leq\|f\|^{2}_{2^{*}}\leq
A\|f\|^{2}_{2}+B\text{Ch}(f)\leq C(1+\text{Ch}(f)),$ (3.8)

and therefore

$F(f)\geq 2\text{Ch}(f)+\alpha_{1}-\frac{(1+\delta)\alpha_{2}}{2\delta}\log
C\left(1+\text{Ch}(f)\right),$ (3.9)

which implies that $F\rightarrow\infty$ for $f$ in $\mathcal{F}$ with
$\|f\|_{W^{1,2}}\rightarrow\infty$. Let $(f_{n})\subseteq\mathcal{F}$ be a
minimizing sequence which can be assumed non-negative since
$\text{Ch}(|f|)\leq\text{Ch}(f)$. By the coercivity of $F$, it follows that
$(f_{n})$ is bounded in $W^{1,2}$. The compact Sobolev embedding implies the
existence of a non-negative $u$ in $W^{1,2}\cap\mathcal{F}$ and a subsequence
of $(f_{n})$, relabelled as $(f_{n})$, such that $f_{n}\rightarrow u$ strongly
in $L^{q}$ for all $q$ in $[1,2^{*})$. By the $L^{2}$-semicontinuity of
Cheeger energy as well as the $L^{2+\delta}$-continuity of $\int f^{2}\log
f^{2}$ for small $\delta>0$,333Which can be shown from mean value theorem and
equality $f^{2}\log f^{2}-g^{2}\log g^{2}=2(|f|-|g|)(1+\log\theta^{2})\theta$
where $\theta$ is between $|f|$ and $|g|$, see [33, page 112] it follows that
$F$ reaches its minimum at $u$. Furthermore, if $\alpha_{1}\neq\lambda$,
straightforward inspection shows that $u\equiv 1$ cannot be the extremal
function since otherwise $\lambda=\alpha_{1}$.

  2. Step 2:

Given any non-negative extremal function $u$, we show that $u\in D(\Delta)$
and satisfies (3.3) by using classical Euler-Lagrangian method. Let $\phi$ in
$\text{Lip}_{bs}$ be an arbitrary Lipschitz function and define the functional
$G$ as follows

$G(t,\beta)=F(u+t\phi)+\beta\left(\|u+t\phi\|_{2}^{2}-1\right).$ (3.10)

The mean value theorem yields

$\left|u^{2}\log u^{2}-(u+t\phi)^{2}\log(u+t\phi)^{2}\right|\leq
2t|\phi|\left|(1+\log\theta^{2})\theta\right|$ (3.11)

with a function $\theta$ taking values between $|u|$ and $|u+t\phi|$. From the
inequality $|\theta\log\theta|\leq\max(1/e,C(\delta)\theta^{1+\delta})$ for
some small $\delta>0$ together with the dominated convergence of Lebesgue it
follows that

$0=\frac{1}{2}\frac{dG(t,\beta)}{dt}\Big{|}_{t=0}=\int\langle\nabla
u,\nabla\phi\rangle+\alpha_{1}u\phi-\frac{\alpha_{2}}{2}u\phi\log
u^{2}-\alpha_{2}u\phi+\beta u\phi dm.$ (3.12)

Since $\text{Lip}_{bs}$ is dense in $W^{1,2}$ and $u\log u$ is in $L^{2}$ by
Sobolev inequality, it follows that

$\int\langle\nabla u,\nabla\phi\rangle dm=\int\left(\frac{\alpha_{2}}{2}\phi
u\log u^{2}+(\alpha_{2}-\alpha_{1}-\beta)\phi u\right)dm,\quad\text{for all
}\phi\in W^{1,2}.$ (3.13)

Plugging $\phi=u$ into (3.13) and using the real value of the log Sobolev
constant $\lambda$, it follows that $\alpha_{2}-\beta=\lambda$. By definition
of the Laplacian we deduce that $u$ is in $D(\Delta)$ and that $-\Delta
u=\alpha_{2}u\log u+(\lambda-\alpha_{1})u$.

  3. Step 3:

We show that $u$ is in Lip. We start by showing that $u$ is in $L^{\infty}$.
Let $\beta>0$. Note that using the resolvent of the Laplacian, (3.3) can be
rewritten as

$u=\left(\beta I-\Delta\right)^{-1}\left(\alpha_{2}u\log
u+(\beta+\lambda-\alpha_{1})u\right)=R_{\beta}\left(\alpha_{2}u\log
u+\alpha_{3}u\right),$ (3.14)

where $I$ be identity map and $R_{\beta}$ is the resolvent operator and
$\alpha_{3}=\beta+\lambda-\alpha_{1}$. Note that in order to prove that $u$ is
in $L^{\infty}$, it is sufficient to show that $g:=\alpha_{2}u\log
u+\alpha_{3}u$ is in $L^{r}$ for some $r>N/2$, according to Lemma 2.7. By the
Sobolev inequality, we know that $u$ is in $L^{2^{*}}$, which implies that $g$
is in $L^{r}$ for all $r\in(2,2^{*})$, (recall that $2^{\ast}=2N/(N-2)$). Fix
$\delta\geq 2N$. It implies that $2<r_{1}<2^{*}$ for
$1/r_{1}=1/2^{*}+1/\delta$. If $r_{1}>N/2$, then Lemma 2.7 and identity (3.14)
implies that $u\in L^{\infty}$. If $r_{1}\leq N/2$, then Lemma 2.7 implies
that $u\in L^{r}$ for all $r<r_{1}N/(N-2r_{1})$. Repeating the previous step,
we can obtain that $g\in L^{r_{2}}$ for some $r_{2}<r_{1}N/(N-2r_{1})$ such
that

$\frac{1}{r_{2}}=\frac{1}{r_{1}}-\frac{2}{N}+\frac{1}{\delta}=\frac{1}{2^{*}}-\frac{2}{N}+\frac{2}{\delta}.$
(3.15)

Define by induction $1/r_{k}:=1/2^{*}-2(k-1)/N+k/\delta$ if $r_{k-1}\leq N/2$
which by definition of $\delta$ implies that after finitely many iterations
$g$ is in $L^{r_{k}}$ for some $r_{k}>N/2$ and deduce that $u$ is in
$L^{\infty}$.

We now show that $u$ is actually in Lip. Let $x_{0}$ be any point in $X$ and
$B_{R}=B(x_{0},R)$ be an open ball with $0<R<\text{diam}(X)/3$ and
$g:=\alpha_{2}u\log u+(\lambda-\alpha_{1})u$. Since $u$ is in $D(\Delta)$ and
identity (3.3), it follows by definition that $u$ belongs to $W^{1,2}(B_{R})$
as well as $D(\Delta,B_{R})$ and that $\Delta_{B_{R}}u=-g$ holds on $B_{R}$ in
the distributional sense. Hence, by Lemma 2.5, it follows that $|\nabla u|\in
L^{\infty}_{loc}(B_{R})$ and

$\||\nabla u|\|_{L^{\infty}(B_{R/2})}\leq
C(N,K,R)\left(\frac{1}{m(B_{R})}\|u\|_{L^{2}}+\|g\|_{L^{\infty}}\right).$
(3.16)

However, $X$ is compact and $m(B_{R})>0$ by doubling property, implying that
$|\nabla u|$ belongs to $L^{\infty}$. Sobolev-to-Lipschitz property thus
implies that $u$ has a Lipschitz conditions representation ending the proof of
Theorem 3.2 but positivity.

∎

As for the last assertion of Theorem 3.2, the positivity of non-negative
extremal functions, we address the following auxiliary lemma stating that any
non-negative extremal function vanishing at one point must also vanish in a
neighborhood of that point. Our approach is based on a maximum principle type
of result for the De Giorgi class proved in [24].

######  Lemma 3.5.

Suppose that the hypothesis of Theorem 3.2 holds. Let $u$ be any non-negative
extremal function of variational problem (3.1). Assume that $u(x_{0})=0$ for
some $x_{0}$ in $X$, then $u\equiv 0$ on a neighborhood of $x_{0}$.

###### Proof.

From the first part of the Proof of Theorem 3.2, any non-negative extremal
function of 3.2 is Lipschitz continuous. Furthermore, since $\alpha_{2}>0$ and
$\lambda$ is finite, the function $g(t):=\alpha_{2}t\log
t+(\lambda-\alpha_{1})t\leq 0$ for all small enough $t>0$. So we can find
$r_{0}>0$ such that $g(u)\leq 0$ on $B(x_{0},r_{0})$. We first claim that $-u$
is of De Giorgi class $\text{DG}_{2}(B(x_{0},r_{0}))$. In other terms, there
exists $C>0$ such that for all $k$ in $\mathbb{R}$ and $z$ in $B(x_{0},r_{0})$
and all $0<\rho<R\leq\text{diam}(X)/3$ with $B(z,R)\subseteq B(x_{0},r_{0})$,
it holds that

$\int_{A_{z}(k,\rho)}|\nabla
u|^{2}dm\leq\frac{C}{(R-\rho)^{2}}\int_{A_{z}(k,R)}(-u-k)^{2}dm,$ (3.17)

where $A_{z}(k,r)=\\{x\in B(z,r):-u(x)>k\\}$. Indeed, let $\eta$ be a
Lipschitz cut-off function such that $\eta=1$ on $B(z,\rho)$ and
$\text{supp}(\eta)\subseteq B(z,R)$ with $|\nabla\eta|\leq C/(R-\rho)$ for
some $C>0$. Taking $\phi=\eta(-u-k)_{+}$ as test function for (3.3), it
follows that

$\int\langle\nabla(\eta(-u-k)_{+}),\nabla u\rangle
dm=\int\eta(-u-k)_{+}\left(\alpha_{2}u\log u+(\lambda-\alpha_{1})u\right)dm.$
(3.18)

As for the left hand side of (3.18), Lebniz rule yields

$\int\langle\nabla(\eta(-u-k)_{+}),\nabla u\rangle
dm\geq\int_{\\{-u>k\\}}\eta|\nabla
u|^{2}dm-\int_{\\{-u>k\\}}(-u-k)_{+}|\nabla\eta||\nabla u|dm\\\
\geq\int_{A_{z}(k,\rho)}|\nabla
u|^{2}dm-\frac{C}{(R-\rho)^{2}}\int_{A_{z}(k,R)}(-u-k)^{2}_{+}dm-\int_{A_{z}(k,R)\setminus
A_{z}(k,\rho)}C|\nabla u|^{2}dm,$ (3.19)

where we use the locality of minimal weak upper gradient
$|\nabla(-u-k)|=|\nabla u|$ for the first inequality, Young’s inequality,
$|\nabla\eta|\leq C/(R-\rho)$ and $|\nabla\eta|=0$ on $B(z,\rho)$ for the
second inequality. As for the right hand side of (3.18), by the very choice of
$B(x_{0},r_{0})$ and non-negativity of $u$, it follows that $\alpha_{2}u\log
u+(\lambda-\alpha_{1})u\leq 0$. Hence

$\int\eta(-u-k)_{+}\left(\alpha_{2}u\log u+(\lambda-\alpha_{1})u\right)dm\leq
0.$ (3.20)

Taking equations (3.19) and (3.20), we obtain

$\int_{A_{z}(k,\rho)}|\nabla
u|^{2}dm\leq\frac{C}{(R-\rho)^{2}}\int_{A_{z}(k,R)}(-u-k)^{2}_{+}dm+C\int_{A_{z}(k,R)\setminus
A_{z}(k,\rho)}|\nabla u|^{2}dm.$ (3.21)

Adding $C\int_{A_{z}(k,\rho)}|\nabla u|^{2}dm$ and then dividing by $(1+C)$ on
both sides, it follows that

$\int_{A_{z}(k,\rho)}|\nabla
u|^{2}dm\leq\frac{C}{(R-\rho)^{2}}\int_{A_{z}(k,R)}(-u-k)^{2}_{+}dm+\theta\int_{A_{z}(k,R)}|\nabla
u|^{2}dm,$ (3.22)

where $\theta=C/(1+C)<1$. Hence, for all $0<\rho<r\leq R$, it holds

$\int_{A_{z}(k,\rho)}|\nabla
u|^{2}dm\leq\frac{C}{(r-\rho)^{2}}\int_{A_{z}(k,R)}(-u-k)^{2}_{+}dm+\theta\int_{A_{z}(k,r)}|\nabla
u|^{2}dm.$ (3.23)

Taking $f(r):=\int_{A_{z}(k,r)}|\nabla u|^{2}dm$. Using [24, Lemma 3.2], see
also [15, Lemma 3.1], we obtain that

$\int_{A_{z}(k,\rho)}|\nabla
u|^{2}dm\leq\frac{C}{(R-\rho)^{2}}\int_{A_{z}(k,R)}(-u-k)^{2}_{+}dm,$ (3.24)

which implies that $-u$ is of De Giorgi class $\text{DG}_{2}(B(x_{0},r_{0}))$.

Note that by the assumption that $X$ is a $\text{RCD}^{*}(K,N)$ space with
$K>0$ and $N\in(2,\infty)$, it follows that $X$ is compact, global doubling,
and supports global weak $(1,1)$-Poincaré inequality, see [32]. Together with
Hölder’s inequality, $X$ supports global weak $(1,q)$-Poincare inequality for
any $q$ in $(1,2)$. Since the minimal weak upper gradient in RCD setting
coincides with the minimal weak upper gradient in Newtonian setting, see [1],
it follows that $\text{RCD}^{*}(K,N)$ space together with that $u\geq 0$ and
$-u\in\text{DG}_{2}(B(x_{0},r_{0}))$ satisfies the assumptions of [24, Lemma
6.1 and Lemma 6.2].

By contradiction, suppose that the assertion of our Lemma does not hold. Let
then $0<R<r_{0}$ and $x$ in $B(x_{0},R)$ and $\tau:=u(x)>0$ be fixed. By
Lipschitz continuity of $u$, we can find $r>0$ with $B(x,r)\subseteq
B(x_{0},R)$ such that $u\geq\tau/2$ on $B(x,r)$. Doubling property yields

$\frac{m(B(x,r))}{m(B(x_{0},R))}\geq\left(\frac{r}{R}\right)^{N}.$ (3.25)

Hence it follows that

$m\left(\left\\{z\in B(x_{0},R):u(z)\geq\tau/2\right\\}\right)\geq
m\left(B(x,r)\right)\geq\left(\frac{r}{R}\right)^{N}m(B(x_{0},R)).$ (3.26)

Taking $\gamma:=1-(r/R)^{N}$, the inequality (3.26) implies that

$m\left(\left\\{x\in B(x_{0},R):u<\tau/2\right\\}\right)\leq\gamma
m(B(x_{0},R)).$ (3.27)

Since $0<\gamma<1$, [24, Lemma 6.2] yields the existence of
$\bar{\lambda}=\bar{\lambda}(\gamma)>0$ such that

$u(x_{0})=\inf_{B(x_{0},R/2)}u\geq\frac{\bar{\lambda}\tau}{2}>0,$ (3.28)

which is a contradiction. ∎

We can now address the positivity assertion of Theorem 3.2.

###### Final part of the proof of Theorem 3.2.

  1. Step 4:

We show the positivity of non-negative extremal function. From the continuity
of $u$, the set $A=\\{x\colon u(x)=0\\}$ is closed. By contradiction, suppose
that $A$ is non empty, by Lemma 3.5 if follows that $A$ is also open. Since
$X$ is connected, it follows that $A=X$ and therefore $u\equiv 0$. This
however contradicts the fact that $\|u\|_{2}=1$.

∎

Finally we address the proof of Corollary 3.3.

###### Proof of Corollary 3.3.

Let $\phi(t)=\log t,t>0$ and $u$ be an arbitrary non-negative extremal
function of variational problem (3.1). By Theorem 3.2, we know that $0<c\leq
u\leq C$ for some some positive constants $c$ and $C$ and that $u$ is
Lipschitz. Since $\phi$ is $C^{2}$-function with bounded first and second
derivative on $[c,C]$, it follows by chain rule that $v:=\phi(u)$ is
Lipschitz, $v$ is in $D(\Delta)$ and

$\Delta(\phi(u))=\phi^{\prime}(u)\Delta u+\phi^{\prime\prime}(u)|\nabla
u|^{2}=\frac{1}{u}\Delta u-\frac{1}{u^{2}}|\nabla u|^{2}.$ (3.29)

By (3.3) we get

$-\Delta v=|\nabla v|^{2}+\alpha_{2}v+(\lambda-\alpha_{1}).$ (3.30)

Taking $\tilde{v}=v+(\lambda-\alpha_{1})/\alpha_{2}$. By locality of minimal
weak upper gradient and Laplacian, we obtain that $\tilde{v}\in D(\Delta)$
satisfies $-\Delta\tilde{v}=|\nabla\tilde{v}|^{2}+\alpha_{2}\tilde{v}$. If
$\lambda\neq\alpha_{1}$, then Theorem 3.2 implies that $u$ is non-constant,
which also implies that $\tilde{v}$ is non-constant. Since $\tilde{v}$
satisfies (3.4), we obtain the result. ∎

##  4\. Li-Yau type inequality for logarithmic extremal functions

In this section, we derive a Li-Yau type estimate for the Lipschitz solutions
of equation $-\Delta v=|\nabla v|^{2}+\alpha v$, whose existence is guaranteed
by Corollary 3.3. In particular, based on the regularity and positivity
results obtained in previous section, this Li-Yau estimate holds for all
logarithmic transform of non-negative extremal functions of (3.1).

######  Theorem 4.1.

Let $(X,d,m)$ be a $\text{RCD}^{*}(K,N)$ space with $K>0$ and $N$ in
$(2,\infty)$. Let $v\in\text{Lip}\cap D(\Delta)$ such that $-\Delta v=|\nabla
v|^{2}+\alpha v$ for some $\alpha>0$. Then, for all $0<\beta<1$ it holds

$|\nabla v|^{2}+(\alpha-\beta
K)v\leq\frac{N\alpha(1-\beta)}{4\beta}\left(1-\frac{\beta((2-\beta)K-\alpha)}{2\alpha(1-\beta)}\right)^{2},\quad\text{$m$-a.e.}$
(4.1)

######  Corollary 4.2.

Let $u$ be any non-negative extremal function of (3.1) with log-Sobolev
constant $\lambda(\alpha_{1},\alpha_{2})$ and $\alpha_{2}>0$. Then $v:=\log
u+(\lambda-\alpha_{1})/\alpha_{2}$ satisfies Li-Yau type estimate (4.1) with
$\alpha=\alpha_{2}$. Moreover, if $0<\alpha_{2}\leq K$, then any non-negative
extremal function is constant.

The proof of Theorem 4.1 is divided into three parts: In a first step, we show
regularity results for $|\nabla v|^{2}+(\alpha-\beta Kv)$ using the weak
Bochner inequality 2.8. In a second step, following similar computation
arguments as in [36] we derive a lower bound of absolutely continuous part of
$\bm{\Delta}(|\nabla v|^{2}+(\alpha-\beta Kv))$, where all inequalities are
understood in $m$-a.e. sense. In a last step, we make use of a slightly
generalized Omori-Yau type maximum principle proved in Appendix A together
with a “good” cut-off function inspired by [28] to derive the desired Li-Yau
estimate.

###### Proof of Theorem 4.1.

Recall that $\text{Test}^{\infty}:=\\{f\in\text{Lip}\cap D(\Delta)\cap
L^{\infty}:\Delta f\in W^{1,2}\cap L^{\infty}\\}$ and
$\text{Test}^{\infty}_{+}:=\\{f\in\text{Test}^{\infty}:f\geq 0\text{ $m$-a.e.
on $X$}\\}$.

  1. Step 1:

We claim that $|\nabla v|^{2}$ is in $W^{1,2}\cap L^{\infty}$ and $|\nabla
v|^{2}$ is in $D(\bm{\Delta})$ with $\bm{\Delta}^{s}(|\nabla v|^{2})\geq 0$.
Indeed, by assumption $v$ is in $D(\Delta)$, hence $v$ belongs to $H^{2,2}$
and

$|\nabla|\nabla v|^{2}|\leq 2|\text{Hess}v|_{HS}|\nabla v|,$ (4.2)

with $|\text{Hess}v|_{HS}\in L^{2}$. By the fact that $|\nabla v|$ is in
$L^{\infty}$, we obtain that $|\nabla v|^{2}$ is also in $W^{1,2}$. We now
show that $|\nabla v|^{2}\in D(\bm{\Delta})$. For any
$\phi\in\text{Test}^{\infty}_{+}$, by weak Bochner inequality (2.8), it
follows that

$\int_{X}|\nabla v|^{2}\Delta\phi dm\geq 2\int_{X}\phi\left(\frac{(\Delta
v)^{2}}{N}+\langle\nabla v,\nabla\Delta v\rangle+K|\nabla
v|^{2}\right)dm=\int_{X}\phi d\mu,$ (4.3)

where $\mu=2((\Delta v)^{2}/N+\langle\nabla v,\nabla\Delta v\rangle+K|\nabla
v|^{2})m$. By standard regularization via mollified heat flow, see [19,
Corollary 6.2.17], the inequality (4.3) holds for all
$\phi\in\text{Lip}_{bs}^{+}$. Then by [19, Proposition 6.2.16], it follows
that $|\nabla v|^{2}\in D(\bm{\Delta})$ and that

$\bm{\Delta}\left(|\nabla v|^{2}\right)\geq 2\left(\frac{(\Delta
v)^{2}}{N}+\langle\nabla v,\nabla\Delta v\rangle+K|\nabla v|^{2}\right)\cdot
m.$ (4.4)

In particular we obtain that $\bm{\Delta}^{s}(|\nabla v|^{2})\geq 0$.

  2. Step 2:

We provide lower bounds for $(\bm{\Delta}^{ac}g)$ based on inequality (4.4)
where $g:=|\nabla v|^{2}+(\alpha-\beta K)v$ for some $0<\beta<1$ to be
determined later. First note that since $-\Delta v=|\nabla v|^{2}+\alpha v$,
it follows that $-\Delta v=g+\beta Kv$. Hence, from the first step, we get
that $g$ is in $D(\bm{\Delta})$ and $\bm{\Delta}^{s}g=\bm{\Delta}^{s}(|\nabla
v|^{2})\geq 0$. Then inequality (4.4) implies that

$\bm{\Delta}^{ac}g=\bm{\Delta}^{ac}(|\nabla v|^{2}+(\alpha-\beta K)v)\\\ \geq
2\frac{(\Delta v)^{2}}{N}+2\langle\nabla v,\nabla\Delta v\rangle+2K|\nabla
v|^{2}+(\alpha-\beta K)\Delta v.$ (4.5)

Plugging $\Delta v=-g-\beta Kv$ into (4.5), it follows that

$\bm{\Delta}^{ac}g\geq 2\frac{(g+\beta Kv)^{2}}{N}-2\langle\nabla
v,\nabla(g+\beta Kv)\rangle+2K|\nabla v|^{2}-(\alpha-\beta K)(g+\beta Kv)\\\
=\frac{2}{N}\left(g^{2}+2\beta Kgv+\beta^{2}K^{2}v^{2}\right)-2\beta K|\nabla
v|^{2}-2\langle\nabla v,\nabla g\rangle+2K|\nabla v|^{2}\\\ -(\alpha-\beta
K)g-\beta K(\alpha-\beta K)v.$ (4.6)

Plugging identity $|\nabla v|^{2}=g-(\alpha-\beta K)v$ into the right hand of
(4.6), it follows that

$\bm{\Delta}^{ac}g\geq\frac{2}{N}g^{2}+\left(\frac{4\beta
K}{N}v+(2-\beta)K-\alpha\right)g+\frac{2\beta^{2}K^{2}}{N}v^{2}\\\
-K(2-\beta)(\alpha-\beta K)v-2\langle\nabla v,\nabla g\rangle\\\
=\frac{2}{N}\left[g+\left(\beta
Kv+\frac{N[(2-\beta)K-\alpha]}{4}\right)\right]^{2}-\frac{2}{N}\left(\beta
Kv+\frac{N[(2-\beta)K-\alpha]}{4}\right)^{2}\\\
+\frac{2\beta^{2}K^{2}}{N}v^{2}-K(2-\beta)(\alpha-\beta K)v-2\langle\nabla
v,\nabla g\rangle.$ (4.7)

For $a=N((2-\beta)K-\alpha)$ and $b=N\alpha(1-\beta)/\beta$, inequality (4.7)
simplifies to

$\bm{\Delta}^{ac}g\geq\frac{2}{N}(g+\beta
Kv+\frac{a}{4})^{2}-\frac{2}{N}(Kabv)-\frac{2}{N}\frac{a^{2}}{16}-2\langle\nabla
v,\nabla g\rangle.$ (4.8)

  3. Step 3:

We show that the assumptions of Lemma A.2 are valid for $g$ based on estimate
(4.8). Let $D:=\text{diam}(X)$ and fix $0<R<D/4$. Since $X$ is compact and
$g\in L^{\infty}$, we can find $x_{0}$ in $X$ such that:

$M_{1}:=\operatorname*{ess\,sup}_{B(x_{0},R)}g=\operatorname*{ess\,sup}_{X}g$
(4.9)

Define

$M_{2}:=\operatorname*{ess\,sup}_{X\setminus B(x_{0},R)}g\leq M_{1}$

By our choice of $R$ and doubling property of $X$, we have $m(B(x_{0},R))>0$
and $m(X\setminus B(x_{0},R))>0$. Without loss of generality, we assume that
$M_{1}>0$, otherwise nothing needs to be shown. We consider different cases
for $M_{1}$ and $M_{2}$.

    1. Case 1:

$M_{1}>M_{2}$. By regularity result in Step 1, we can apply Lemma A.2 to $g$.
So taking $w=2v$ which is in $W^{1,2}\cap\text{Lip}_{b}$, it follows from
Lemma A.2 that we can find a sequence $(x_{j})\subseteq X$ such that
$g(x_{j})>M_{1}-1/j$ and

$\bm{\Delta}^{ac}g(x_{j})+\langle\nabla g,\nabla w\rangle(x_{j})\leq 1/j.$
(4.10)

Plugging (4.8) into (4.10) and letting $h_{v}=(K\beta bv+a^{2}/16)^{1/2}$, it
follows that

$g(x_{j})\leq-\beta Kv(x_{j})-\frac{a}{4}+h_{v}(x_{j})+\sqrt{\frac{N}{2j}}\\\
=-\frac{1}{b}h_{v}(x_{j})^{2}+h_{v}(x_{j})+\frac{a^{2}}{16b^{2}}-\frac{a}{4}+\sqrt{\frac{N}{2j}}\\\
=-\frac{1}{b}\left(h_{v}(x_{j})-\frac{b}{2}\right)^{2}+\frac{b}{4}\left(1-\frac{a}{2b}\right)^{2}+\sqrt{\frac{N}{2j}}.$
(4.11)

For $j\rightarrow\infty$ together with $g(x_{j})>M_{1}-1/j$, we obtain that

$\operatorname*{ess\,sup}_{X}g\leq\frac{b}{4}\left(1-\frac{a}{2b}\right)^{2}.$
(4.12)

    2. Case 2:

$M_{1}=M_{2}$. Let $\varepsilon\in(0,1/2)$. By [28, Lemma 3.1], we can find a
Lipschitz cut-off function $\psi:X\rightarrow\mathbb{R}$ with $\psi\in
D(\Delta)$ such that $0\leq\psi\leq 1$ and $\psi\equiv 1$ on $B(x_{0},R)$ and
$\text{supp}(\psi)\subseteq B(x_{0},2R)$, and

$R^{2}|\Delta\psi|+R|\nabla\psi|\leq C,$ (4.13)

where $C>0$ is a constant depending only on $K,N,D$. Let
$\phi_{\varepsilon}:X\rightarrow\mathbb{R}$ be defined as
$\phi_{\varepsilon}=1-\varepsilon+\varepsilon\psi$ and
$G_{\varepsilon}:=\phi_{\varepsilon}\cdot g$. Note that since
$\phi_{\varepsilon}\in D(\Delta)$, it holds that
$\bm{\Delta}\phi_{\varepsilon}=\Delta\phi_{\varepsilon}\cdot m$. Together with
the fact that $\phi_{\varepsilon}$ is continuous, by the Leibniz rule for
measure-valued Laplacian, it follows that

$\bm{\Delta}(G_{\varepsilon})=\phi_{\varepsilon}\bm{\Delta}g+g\Delta\phi_{\varepsilon}\cdot
m+2\langle\nabla\phi_{\varepsilon},\nabla g\rangle\cdot m.$ (4.14)

Together with the result in the first step of this proof that
$\bm{\Delta}^{s}g\geq 0$, we deduce that

$\bm{\Delta}^{s}(G_{\varepsilon})=\phi_{\varepsilon}\bm{\Delta}^{s}g\geq 0.$
(4.15)

Furthermore, from (4.13), it follows that

$\frac{|\nabla\phi_{\varepsilon}|^{2}}{\phi_{\varepsilon}}\leq\varepsilon^{2}\frac{C}{R^{2}}\quad\text{and}\quad|\Delta\phi_{\varepsilon}|\leq\varepsilon\frac{C}{R^{2}}.$
(4.16)

Denote by $H_{v}$ the right side of (4.8) without last term $-2\langle\nabla
v,\nabla g\rangle$. Then, by using $g=G_{\varepsilon}/\phi_{\varepsilon}$ and
inequality (4.8), it follows that

$\bm{\Delta}^{ac}(G_{\varepsilon})=\phi_{\varepsilon}\bm{\Delta}^{ac}g+\frac{G_{\varepsilon}}{\phi_{\varepsilon}}\Delta\phi_{\varepsilon}+2\left\langle\nabla\phi_{\varepsilon},\nabla\left(\frac{G_{\varepsilon}}{\phi_{\varepsilon}}\right)\right\rangle\\\
\geq\phi_{\varepsilon}\left(H_{v}-2\left\langle\nabla v,\nabla
g\right\rangle\right)+2\left\langle\nabla\phi_{\varepsilon},\nabla
G_{\varepsilon}\right\rangle/\phi_{\varepsilon}+\frac{G_{\varepsilon}}{\phi_{\varepsilon}}\left(\Delta\phi_{\varepsilon}-2\frac{|\nabla\phi_{\varepsilon}|^{2}}{\phi_{\varepsilon}}\right)$
(4.17)

Using estimate (4.16) with $\varepsilon^{2}<\varepsilon$, it follows that

$\bm{\Delta}^{ac}(G_{\varepsilon})\geq\phi_{\varepsilon}H_{v}-2\phi_{\varepsilon}\left(\left\langle\nabla
v,\frac{\nabla
G_{\varepsilon}}{\phi_{\varepsilon}}\right\rangle-\left\langle\nabla
v,\frac{G_{\varepsilon}\nabla\phi_{\varepsilon}}{\phi_{\varepsilon}^{2}}\right\rangle\right)\\\
+2\left\langle\nabla\phi_{\varepsilon},\nabla
G_{\varepsilon}\right\rangle/\phi_{\varepsilon}-\varepsilon\|g\|_{\infty}\frac{C}{R^{2}}\\\
\geq\phi_{\varepsilon}H_{v}-2\left\langle\nabla(v-\log\phi_{\varepsilon}),\nabla
G_{\varepsilon}\right\rangle+2\frac{G_{\varepsilon}}{\phi_{\varepsilon}}\langle\nabla
v,\nabla\phi_{\varepsilon}\rangle-\varepsilon\|g\|_{\infty}\frac{C}{R^{2}}\\\
\geq\phi_{\varepsilon}H_{v}-2\langle\nabla(v-\log\phi_{\varepsilon}),\nabla
G_{\varepsilon}\rangle-\varepsilon\|g\|_{\infty}\||\nabla
v|\|_{\infty}\frac{C}{R}-\varepsilon\|g\|_{\infty}\frac{C}{R^{2}}.$ (4.18)

Since $\operatorname*{ess\,sup}_{X\setminus
B(x_{0},2R)}g\leq\operatorname*{ess\,sup}_{X\setminus B(x_{0},R)}g<M_{1}$, by
definition of $\phi_{\varepsilon}$, it follows that

$\operatorname*{ess\,sup}_{X\setminus
B(x_{0},2R)}G_{\varepsilon}\leq(1-\varepsilon)M_{1}<M_{1}=\operatorname*{ess\,sup}_{B(x_{0},R)}G_{\varepsilon}=\operatorname*{ess\,sup}_{B(x_{0},2R)}G_{\varepsilon}.$
(4.19)

Doubling property and $R<D/4$ implies that $m(B(x_{0},2R))>0$ as well as
$m(X\setminus B(x_{0},2R))>0$ which together with the fact that
$\bm{\Delta}^{s}(G_{\varepsilon})\geq 0$, allows us to apply Lemma A.2 to
$G_{\varepsilon}$. Taking $w_{\varepsilon}=2v-2\log\phi_{\varepsilon}\in
W^{1,2}\cap\text{Lip}_{b}$, by Lemma A.2, it follows that there exists a
sequence $(x_{j})\subseteq X$ such that
$G_{\varepsilon}(x_{j})>\operatorname*{ess\,sup}_{X}G_{\varepsilon}-1/j$ and

$\bm{\Delta}^{ac}G_{\varepsilon}(x_{j})+\langle\nabla G_{\varepsilon},\nabla
w_{\varepsilon}\rangle(x_{j})\leq 1/j.$ (4.20)

Plugging (4.18), we obtain that

$(1-\varepsilon)H_{v}(x_{j})\leq\phi_{\varepsilon}(x_{j})H_{v}(x_{j})\leq\frac{1}{j}+\varepsilon
C_{1},$ (4.21)

where $C_{1}=C(K,N,D)\|g\|_{\infty}\left(\||\nabla
f|\|_{\infty}/R+1/R^{2}\right)$. Following the similar argument in Case 1, we
obtain that

$g(x_{j})\leq\frac{b}{4}\left(1-\frac{a}{2b}\right)^{2}+\sqrt{\frac{N(j^{-1}+\varepsilon
C_{1})}{2(1-\varepsilon)}}.$ (4.22)

Since
$\operatorname*{ess\,sup}_{X}G_{\varepsilon}=\operatorname*{ess\,sup}_{X}g=M_{1}$,
so multiplying $\phi(x_{j})$ on both side of (4.22) and letting
$j\rightarrow\infty$, it follows that

$\operatorname*{ess\,sup}_{X}g\leq\frac{b}{4}\left(1-\frac{a}{2b}\right)^{2}+\sqrt{\frac{\varepsilon
NC_{1}}{2(1-\varepsilon)}}.$ (4.23)

Inequality (4.23) holding for any $0<\varepsilon<1/2$, we send $\varepsilon$
to $0$ to obtain

$\operatorname*{ess\,sup}_{X}g\leq\frac{b}{4}\left(1-\frac{a}{2b}\right)^{2}.$
(4.24)

With Case 1 and Case 2, together with the definition of $a$ and $b$, we obtain
inequality (4.1).

∎

###### Proof of Corollary 4.2.

By Theorem 3.2 and Corollary 3.3, we know that $v\in\text{Lip}\cap D(\Delta)$
satisfies $-\Delta v=|\nabla v|^{2}+\alpha_{2}v$. Hence inequality (4.1) holds
for $\tilde{v}$. If $\alpha_{2}\in(0,K]$, then taking $0<\beta<\alpha_{2}/K$
and letting $\beta\nearrow\alpha_{2}/K$. Then it follows from (4.1) that
$|\nabla\log u|^{2}=0$. By the fact that $|\nabla\log u|^{2}\in W^{1,2}$ and
Sobolev-to-Lip property, it follows that $u$ is constant. ∎

##  5\. Applications

In this section, we present applications of the regularity result in Theorem
3.2 and Li-Yau type estimate in Theorem 4.1, which generalize results in [36]
in smooth Riemannian setting to the $\text{RCD}^{*}(K,N)$ setting.

Since the Lipschitz solution of $-\Delta v=|\nabla v|^{2}+\alpha v$ is
constant when $\alpha\in(0,K]$ as shown in Theorem 4.1, we only focus on the
case where $\alpha>K$. For notational simplicity, we define the non-negative
constants $C_{1}$ and $C_{2}$ as follows

$C_{1}(\beta):=\alpha-\beta K\quad\text{and}\quad
C_{2}(\beta):=\frac{N\alpha(1-\beta)}{4\beta}\left(1-\frac{\beta((2-\beta)K-\alpha)}{2\alpha(1-\beta)}\right)^{2},$
(5.1)

where $0<\beta<1$. That that $C_{1}$ is positive.

The first direct consequence is a Harnack-type inequality for the non-negative
extremal functions.

######  Corollary 5.1.

Let $(X,d,m)$ be a $\text{RCD}^{*}(K,N)$ space with $K>0$ and $N$ in
$(2,\infty)$. Suppose that $v\in Lip$ satisfies $-\Delta v=|\nabla
v|^{2}+\alpha v$ for some $\alpha>K$. Then for any $x,y\in X$, it holds that

$e^{v(x)}\leq
e^{(1-\varepsilon)v(y)}\exp\left(\frac{C_{1}(\beta)d^{2}(x,y)}{4\varepsilon}+\frac{\varepsilon
C_{2}(\beta)}{C_{1}(\beta)}\right),$ (5.2)

for any $0<\varepsilon<1$.

###### Proof.

First note that by Theorem 4.1, we have

$|\nabla v|\leq\sqrt{C_{2}(\beta)-C_{1}(\beta)v},\quad m\text{-a.e.}$ (5.3)

Let $x_{0},y_{0}\in X$ be arbitrary points in $X$ and let
$\mu_{0}=\frac{1}{m(B(x_{0},r))}m|_{B(x_{0},r)}$ and
$\mu_{1}=\frac{1}{m(B(y_{0},r))}m|_{B(y_{0},r)}$ for $0<r<\text{diam}(X)/3$.
By [7, Proposition 1.5] and the references therein, $\text{RCD}^{*}(K,N)$ has
the essentially non-branching property. As a consequence of which there exists
a unique $W_{2}$-geodesic $(\mu_{t})_{\\{t\in[0,1]\\}}$ joining $\mu_{0}$ and
$\mu_{1}$ with $\mu_{t}\leq Cm$ for any $t$ in $[0,1]$ for some $C>0$, and
test plan $\pi$ in $\mathcal{P}(C([0,1];X))$ such that
$(e_{t})_{\sharp}\pi=\mu_{t}$ for any $t$ in $[0,1]$ and $\pi$ is concentrated
on the set of geodesics. Let $\gamma\in\text{supp}(\pi)$ be an arbitrary
geodesic. By the continuity of $v$, let $s_{1}$ and $s_{2}$ in $[0,1]$ be the
maximum and minimum respectively, that is

$v(\gamma_{s_{1}})=\max_{s\in[0,1]}v(\gamma_{s})\quad\text{and}\quad
v(\gamma_{s_{2}})=\min_{s\in[0,1]}v(\gamma_{s}).$ (5.4)

Then by the definition of minimal weak upper gradient, it follows that

$\left|v(\gamma_{s_{1}})-v(\gamma_{s_{2}})\right|\leq\int_{\min(s_{1},s_{2})}^{\max(s_{1},s_{2})}|\nabla
v|(\gamma_{t})|\dot{\gamma}_{t}|dt\leq
d(\gamma_{0},\gamma_{1})\sqrt{C_{2}(\beta)-C_{1}(\beta)v(\gamma_{s_{2}})}.$
(5.5)

For $0<\varepsilon<1$, together with $v(\gamma_{0})\leq v(\gamma_{s_{1}})$, it
follows that

$v(\gamma_{0})\leq(1-\varepsilon)v(\gamma_{s_{2}})+\varepsilon
v(\gamma_{s_{2}})+d(\gamma_{0},\gamma_{1})\sqrt{C_{2}(\beta)-C_{1}(\beta)v(\gamma_{s_{2}})}.$
(5.6)

For $a=\sqrt{C_{2}(\beta)-C_{1}(\beta)v(\gamma_{s_{2}})}$, we get

$v(\gamma_{0})\leq(1-\varepsilon)v(\gamma_{s_{2}})-\frac{\varepsilon}{C_{1}(\beta)}a^{2}+d(\gamma_{0},\gamma_{1})a+\frac{\varepsilon
C_{2}(\beta)}{C_{1}(\beta)}\\\
\leq(1-\varepsilon)v(\gamma_{1})+\frac{C_{1}(\beta)d^{2}(\gamma_{0},\gamma_{1})}{4\varepsilon}+\frac{\varepsilon
C_{2}(\beta)}{C_{1}(\beta)}.$ (5.7)

Integrating (5.7) with respect to $\pi$ on both side and noting that
$(e_{t})_{\sharp}\pi=\mu_{t}$, it follows that

$\int
v(x)d\mu_{0}\leq\int(1-\varepsilon)v(y)+\frac{C_{1}(\beta)d^{2}(x,y)}{4\varepsilon}+\frac{\varepsilon
C_{2}(\beta)}{C_{1}(\beta)}d\tilde{\pi}(x,y),$ (5.8)

where $\tilde{\pi}=(e_{0},e_{1})_{\sharp}\pi$ is the optimal transport plan
between $\mu_{0}$ and $\mu_{1}$. Since $\mu_{0}$ and $\mu_{1}$ weakly
converges to $\delta_{x_{0}}$ and $\delta_{y_{0}}$ in duality of $C_{b}$ as
$r\rightarrow 0$, it follows that, up to a subsequence, $\tilde{\pi}$ weakly
converges to $\delta_{x_{0}}\times\delta_{y_{0}}$. Using $u=\exp(v)$, we
obtain that

$u(x_{0})\leq
u(y_{0})^{1-\varepsilon}\exp\left(\frac{C_{1}(\beta)d^{2}(x_{0},y_{0})}{4\varepsilon}+\frac{\varepsilon
C_{2}(\beta)}{C_{1}(\beta)}\right).$ (5.9)

∎

######  Remark 5.2.

Note that using the deep result proved in [11] that the minimal weak upper
gradient of Lipschitz function coincides with its local Lipschitz slope in the
complete doubling metric space supporting weak $(1,p)$-Poincaré inequality for
$p>1$, Corollary 5.1 can be shown directly following the lines of [36,
Corollary 2.2] instead of using optimal transport methods.

Next corollary is the estimates on the upper bound and lower bound of $v$
based on the dimension-free Harnack inequality in $\text{RCD}(K,\infty)$ in
[26]. The proof of which is essentially the same one as in the Riemannian case
from [36, Corollary 2.4]. For the sake of completeness, we provide the proof.

######  Corollary 5.3.

Let $(X,d,m)$ be a $\text{RCD}^{*}(K,N)$ space with $K>0$ and $N$ in
$(2,\infty)$. Then for any non-negative extremal function $u$ of (3.1) with
log Sobolev constant $\lambda(\alpha_{1},\alpha_{2})$ with $\alpha_{2}>K$, it
holds that

$\displaystyle\sup_{X}\log
u\leq\frac{\lambda-\alpha_{1}}{\alpha_{2}}+\alpha_{2}\text{diam}(X)^{2},$
(5.10) $\displaystyle\inf_{X}\log
u\geq-\frac{27}{16}\alpha_{2}\text{diam}(X)^{2}-\frac{\lambda-\alpha_{1}}{\alpha_{2}}.$
(5.11)

In particular, it holds that $|\nabla\log u|^{2}\leq
C_{2}(\beta)+\frac{27}{16}C_{1}(\beta)\alpha_{2}\text{diam}(X)^{2}$ $m$-a.e.
for any $0<\beta<1$.

###### Proof.

Let $D:=\text{diam}(X)$ and $x$ and $y$ be the maximum and minimum point of
$u$ on $X$, respectively, existence of which is guaranteed by the regularity
of extremal functions proved in Theorem 3.2.

As for the upper bound, firstly, by the dimension-free Harnack inequality in
$\text{RCD}(K,\infty)$ shown in [26, Theorem 3.1] for $p>1$, it follows that

$\left(P_{t}u(x)\right)^{p}\leq
P_{t}(u^{p})(z)\exp\left(\frac{pKd^{2}(x,z)}{2(p-1)(e^{2Kt}-1)}\right),\quad\forall
z\in X.$ (5.12)

Since $\|P_{t}u^{2}\|_{1}=\|u^{2}\|_{1}=1$ by the mass-preserving property of
heat semigroup, we deduce from (5.12) by taking $p=2$ that

$\left(P_{t}u(x)\right)^{2}\leq\exp\left(\frac{KD^{2}}{e^{2Kt}-1}\right).$
(5.13)

Secondly, using heat equation, equation (3.3) and the commutation between
$P_{t}$ and $\Delta$, it follows that

$P_{t}u(x)-P_{s}u(x)=\int_{s}^{t}P_{\tau}\Delta
u(x)d\tau=-\int_{s}^{t}P_{\tau}\left(\alpha_{2}u\log
u+(\lambda-\alpha_{1})u\right)(x)d\tau$ (5.14)

for any $0<s<t$. Since $u$ is positive, it follows that $(u\log u)(z)\leq
u(z)\log u(x)$ for any $z$ in $X$ which by the comparison principle for heat
semigroup, yields

$P_{t}u(x)-P_{s}u(x)\geq-\alpha_{2}\log
u(x)\int_{s}^{t}P_{\tau}u(x)d\tau-(\lambda-\alpha_{1})\int_{s}^{t}P_{\tau}u(x)d\tau.$
(5.15)

Gronwall’s inequality further implies that

$P_{t}u(x)\geq u(x)\exp\left(-\alpha_{2}t\log
u(x)-(\lambda-\alpha_{1})t\right)=e^{-(\lambda-\alpha_{1})t}u(x)^{1-\alpha_{2}t}.$
(5.16)

So, together with (5.13), it follows that for any $0<t<1/\alpha_{2}$:

$\log
u(x)\leq\frac{(\lambda-\alpha_{1})t}{1-\alpha_{2}t}+\frac{KD^{2}}{2(1-\alpha_{2}t)(e^{2Kt}-1)}\leq\frac{(\lambda-\alpha_{1})t}{1-\alpha_{2}t}+\frac{D^{2}}{4(1-\alpha_{2}t)t},$
(5.17)

where we use $K/(e^{2Kt}-1)\leq 1/(2t)$ in the last inequality which for
$t=1/(2\alpha_{2})$ yields

$\log u(x)\leq\frac{\lambda-\alpha_{1}}{\alpha_{2}}+\alpha_{2}D^{2}.$ (5.18)

As for the lower bound, from $(u\log u)(z)\geq u(z)\log u(y)$ and similar
arguments as above, it follows that

$P_{t}u(y)\leq e^{-(\lambda-\alpha_{1})t}u(y)^{1-\alpha_{2}t}.$ (5.19)

Taking $z=y$ in dimension-free inequality (5.12) and plugging (5.16) and
(5.19), it follows that

$e^{-p(\lambda-\alpha_{1})t}u(x)^{p-p\alpha_{2}t}\leq\left(P_{t}u(x)\right)^{p}\leq
P_{t}(u^{p})(y)\exp\left(\frac{pKD^{2}}{2(p-1)(e^{2Kt}-1)}\right)\\\ \leq
u(x)^{p-1}(P_{t}u)(y)\exp\left(\frac{pKD^{2}}{2(p-1)(e^{2Kt}-1)}\right)\\\
\leq
u(x)^{p-1}e^{-(\lambda-\alpha_{1})t}u(y)^{1-\alpha_{2}t}\exp\left(\frac{pKD^{2}}{2(p-1)(e^{2Kt}-1)}\right),$
(5.20)

which implies that

$u(x)^{1-p\alpha_{2}t}\leq
e^{(p-1)(\lambda-\alpha_{1})t}u(y)^{1-\alpha_{2}t}\exp\left(\frac{pKD^{2}}{2(p-1)(e^{2Kt}-1)}\right).$
(5.21)

Taking $p=1/(\alpha_{2}t)$ and $t=1/(3\alpha_{2})$, we obtain that

$\log
u(y)\geq-\frac{27}{16}\alpha_{2}D^{2}-\frac{\lambda-\alpha_{1}}{\alpha_{2}}.$
(5.22)

∎

As a final application, following similar methods as in [31, Lemma 5.3]
together with Theorem 3.2, we recover the classical result from [5, Theorem
5.7.4] that any non-negative extremal functions with log-Sobolev constant
$\lambda(\alpha_{1},\alpha_{2})$ is constant when $0<\alpha_{2}\leq KN/(N-1)$.

######  Corollary 5.4.

Let $(X,d,m)$ be a $\text{RCD}^{*}(K,N)$ space with $K>0$ and
$N\in(2,\infty)$. Then any non-negative extremal function of (3.1) with
positive log-Sobolev constant $\lambda(\alpha_{1},\alpha_{2})$ is constant
whenever $0<\alpha_{2}\leq KN/(N-1)$.

###### Proof.

Let $u$ be an arbitrary non-negative extremal function of (3.1) with log-
Sobolev constant $\lambda(\alpha_{1},\alpha_{2})$ with $0<\alpha_{2}\leq
KN/(N-1)$, and define $v=\log u$. Let further $a$, $b$, and $d$ be reals to be
determined later. We first estimate $\int e^{bv}(\Delta v)^{2}dm$ from both
PDE equation (3.3) and Bochner inequality, and then derive the desired result.

  1. Step 1:

We first derive estimate from PDE equation (3.3). Let
$\alpha_{3}:=\lambda-\alpha_{1}$ in (3.3). By the Lipschitz regularity of $v$
and regularity of $|\nabla v|^{2}$ proved in Step 1 in Theorem 4.1, it follows
that $\phi:=e^{(b-1)v}\Delta v\in W^{1,2}$ and $\psi:=e^{(b-1)v}|\nabla
v|^{2}\in W^{1,2}$. On the one hand, applying $\phi$ and then $\psi$ to the
right hand side of (3.3) it follows that

$I:=\int(\alpha_{2}v+\alpha_{3})e^{v}\phi
dm=\int(\alpha_{2}v+\alpha_{3})e^{v}e^{(b-1)v}\Delta vdm\\\
=-\alpha_{2}\int|\nabla v|^{2}e^{bv}dm-b\int(\alpha_{2}v+\alpha_{3})|\nabla
v|^{2}e^{bv}dm\\\ =-\alpha_{2}\int|\nabla
v|^{2}e^{bv}dm-b\int(\alpha_{2}v+\alpha_{3})e^{v}\psi dm\\\
=-\alpha_{2}\int|\nabla v|^{2}e^{bv}dm+b\int\Delta(e^{v})e^{(b-1)v}|\nabla
v|^{2}dm\\\ =-\alpha_{2}\int|\nabla v|^{2}e^{bv}dm+b\int|\nabla
v|^{4}e^{bv}dm+b\int(\Delta v)|\nabla v|^{2}e^{bv}dm.$ (5.23)

On the other hand, applying $\phi$ to the left hand side of (3.3), we get

$I:=\int-\Delta(e^{v})e^{(b-1)v}\Delta vdm=-\int(\Delta v)|\nabla
v|^{2}e^{bv}dm-\int(\Delta v)^{2}e^{bv}dm.$ (5.24)

Showing that

$\int(\Delta v)^{2}e^{bv}dm=\alpha_{2}\int|\nabla v|^{2}e^{bv}dm-b\int|\nabla
v|^{4}e^{bv}dm-(b+1)\int(\Delta v)|\nabla v|^{2}e^{bv}dm.$ (5.25)

  2. Step 2:

We derive the estimate from Bochner inequality (2.8). Note that $f:=e^{av}$
and $g:=e^{dv}$ satisfies the regularity requirement in (2.8). So plugging $f$
and $g$ into (2.8), it follows that the left side of (2.8) can be expressed as

$\frac{1}{2}\int\Delta(g)|\nabla f|^{2}dm=\frac{a^{2}d}{2}\int
e^{(2a+d)v}(\Delta v)|\nabla v|^{2}dm+\frac{a^{2}d^{2}}{2}\int
e^{(2a+d)v}|\nabla v|^{4}dm,$ (5.26)

and right hand of (2.8) can be expressed as

$-\frac{N-1}{N}\int g(\Delta f)^{2}-\int\Delta f\langle\nabla g,\nabla
f\rangle dm-K\int g|\nabla f|^{2}dm\\\ =-a^{2}\frac{N-1}{N}\int
e^{(2a+d)}(\Delta v)^{2}dm-a^{2}\left(2a\frac{N-1}{N}+d\right)\int
e^{(2a+d)v}(\Delta v)|\nabla v|^{2}dm\\\
-a^{2}\left(a^{2}\frac{N-1}{N}+ad\right)\int e^{(2a+d)v}|\nabla
v|^{4}+a^{2}K\int e^{(2a+d)v}|\nabla v|^{2}dm.$ (5.27)

From (5.26) and (5.27) it follows that (2.8) reads as follows

$\int e^{(2a+d)v}(\Delta v)^{2}dm\geq\frac{KN}{N-1}\int e^{(2a+d)v}|\nabla
v|^{2}\\\ -\left(a^{2}+\frac{N}{N-1}ad+\frac{N}{2(N-1)}d^{2}\right)\int
e^{(2a+d)v}|\nabla v|^{4}dm\\\ -\left(2a+\frac{3N}{2(N-1)}d\right)\int
e^{(2a+d)v}(\Delta v)|\nabla v|^{2}dm.$ (5.28)

  3. Step 3:

We conclude by comparing the coefficients of (5.25) and (5.28) and choosing
particular value for $b,a,d$. Let $\varepsilon>0$ be determined later. Let
$b,a,d$ and $\varepsilon>0$ satisfies the following system of equations:

$\left\\{\begin{aligned} &b=2a+d\\\
&b-\varepsilon=a^{2}+\frac{N}{N-1}ad+\frac{N}{2(N-1)}d^{2}\\\
&b+1=2a+\frac{3N}{2(N-1)}d.\end{aligned}\right.$ (5.29)

This system (5.29) admits real-valued solutions if and only if
$0<\varepsilon\leq 4N/(N+2)^{2}$. Hence, choosing arbitrary
$\varepsilon\in(0,4N/(N+2)^{2}]$ and

$\displaystyle d$ $\displaystyle=\frac{2(N-1)}{N+2},$ $\displaystyle a$
$\displaystyle=\frac{2}{N+2}+\sqrt{\frac{4N}{(N+2)^{2}}-\varepsilon},$
$\displaystyle b$ $\displaystyle=2a+d,$

by comparing (5.25) and (5.28), it follows that

$\left(\alpha_{2}-\frac{KN}{N-1}\right)\int|\nabla
v|^{2}e^{bv}dm\geq\varepsilon\int|\nabla v|^{4}e^{bv}dm.$ (5.30)

If $0<\alpha_{2}\leq KN/(N-1)$, then it follows that $|\nabla v|$ has to be
$0$, implying that $u$ is constant.

∎

##  Appendix A Omori-Yau Maximum Principle

In the appendix, we provide a slightly generalized version of Omori-Yau type
maximum principle for the whole metric space of $\text{RCD}(K,\infty)$ spaces
with $K$ in $\mathbb{R}$ which may not support doubling property. To show it,
we will first show the Kato’s inequality in the proper $\text{RCD}(K,\infty)$
setting whose proof follows a similar argument as in [37]. For the sake of
completeness, we provide the complete argumentation.

Beforehand, recall that the definition of weak Laplacian. We say an operator
$L$ on $W^{1,2}_{loc}$ is the weak Laplacian provided that for each $f\in
W^{1,2}_{loc}$, $Lf$ is a linear functional acting on $W^{1,2}\cap L^{\infty}$
with bounded support given by:

$Lf(g):=-\int\langle\nabla f,\nabla g\rangle dm,\quad\forall g\in W^{1,2}\cap
L^{\infty}\text{ with bounded support}.$ (A.1)

For each $h$ in $W^{1,2}_{loc}\cap L^{\infty}$, $h\cdot Lf$ is the linear
functional given by $h\cdot Lf(g):=Lf(hg)$ for each $g$ in $W^{1,2}\cap
L^{\infty}$ with bounded support. We say that $Lf$ is a signed Radon measure
provided that there exists a signed Radon measure $\mu$ such that $Lf(g)=\int
gd\mu$ for all $g\in W^{1,2}\cap L^{\infty}$ with bounded support. It’s clear
that in this case, we have $f\in D(\bm{\Delta})$ and $Lf=\bm{\Delta}f$. For
$w\in W^{1,2}\cap L^{\infty}$ and $m_{w}:=e^{w}\cdot m$, $L_{w}$ denotes the
weighted weak Laplacian on $W^{1,2}_{loc}$ given by

$L_{w}f(g):=-\int\langle\nabla f,\nabla g\rangle dm_{w},\quad\text{for all
}g\in W^{1,2}\cap L^{\infty}\text{ with bounded support}.$ (A.2)

It’s easy to check that $L_{w}f=e^{w}\cdot(Lf+\langle\nabla w,\nabla f\rangle
m)$. When $L_{w}f$ is a signed Radon measure, we denote by
$L_{w}f=(L^{ac}_{w}f)\cdot m_{w}+L_{w}^{s}f$ its Lebesgue decomposition with
respect to $m_{w}$. Finally, we remark that using similar argument as in [37,
Lemma 3.2], we have following chain rule: for $f\in W^{1,2}_{loc}\cap
L^{\infty}$ and $\phi\in C^{2}(\mathbb{R})$, it holds that $\phi(f)\in
W^{1,2}_{loc}\cap L^{\infty}$ and that

$L\left(\phi(f)\right)=\phi^{\prime}(f)\cdot Lf+\phi^{\prime\prime}(f)|\nabla
f|^{2}\cdot m.$ (A.3)

######  Lemma A.1.

(Kato’s inequality) Let $(X,d,m)$ be a proper $\text{RCD}(K,\infty)$ space and
$w$ in $W^{1,2}\cap L^{\infty}$. Suppose $f$ in $W^{1,2}_{loc}\cap L^{\infty}$
is such that $L_{w}f$ is a signed Radon measure and that $L^{s}_{w}f\geq 0$.
Then $L_{w}(f_{+})$ is a signed Radon measure such that

$L_{w}(f_{+})\geq\chi_{\\{f>0\\}}L_{w}^{ac}f\cdot m_{w},$ (A.4)

where $f_{+}:=\max\\{f,0\\}$ and $f_{-}:=\max\\{-f,0\\}$.

###### Proof.

It suffices to prove that $L_{w}(|f|)$ is a signed Radon measure and that

$L_{w}(|f|)\geq\text{sgn}(f)\cdot L_{w}f,$ (A.5)

where $\text{sgn}(t)=1$ if $t>0$ and $\text{sgn}(t)=-1$ if $t<0$ and
$\text{sgn}(t)=0$ if $t=0$. Indeed, if inequality (A.5) holds, then it follows
that both $L_{w}(f_{+})$ and $L_{w}(f_{-})$ are signed Radon measure and (A.5)
implies that

$L_{w}(f_{+})+L_{w}(f_{-})\geq\chi_{\\{f>0\\}}\cdot
L_{w}f-\chi_{\\{f<0\\}}\cdot L_{w}f.$ (A.6)

By the locality of minimal weak upper gradient and inner regularity of Radon
measure, it is immediate to check that $L_{w}(f_{+})$ is concentrated on set
$\\{f\geq 0\\}$. Then inequality (A.6) with the assumption that
$L^{s}_{w}f\geq 0$ implies that

$L_{w}(f_{+})\geq\chi_{\\{f>0\\}}L_{w}f=\chi_{\\{f>0\\}}\left(L^{ac}_{w}f\cdot
m_{w}+L^{s}_{w}f\right)\geq\chi_{\\{f>0\\}}L^{ac}_{w}f\cdot m_{w}.$ (A.7)

We are left to show (A.5). Let $\varepsilon>0$ and
$\phi_{\varepsilon}(t):=\sqrt{t^{2}+\varepsilon^{2}}-\varepsilon$ and
$f_{\varepsilon}:=\phi_{\varepsilon}(f)$. Since $\phi_{\varepsilon}$ is in
$C^{2}(\mathbb{R})$, it follows that $f_{\varepsilon}\leq|f|$ and that

$|\nabla f_{\varepsilon}|=|\phi^{\prime}_{\varepsilon}(f)||\nabla
f|=\frac{|f|}{\sqrt{f^{2}+\varepsilon^{2}}}|\nabla f|\leq|\nabla f|.$ (A.8)

Further, the chain rule of $L_{w}$ for $f\in W^{1,2}_{loc}\cap L^{\infty}$ and
$\psi(t)=t^{2}$ yields

$2f\cdot L_{w}f+2|\nabla f|^{2}\cdot
m_{w}=L_{w}f^{2}=L_{w}\left((f_{\varepsilon}+\varepsilon)^{2}-\varepsilon^{2}\right)\\\
=2(f_{\varepsilon}+\varepsilon)L_{w}f_{\varepsilon}+2|\nabla
f_{\varepsilon}|^{2}\cdot m_{w}.$ (A.9)

So by (A.8), it follows that

$L_{w}f_{\varepsilon}\geq\frac{f}{f_{\varepsilon}+\varepsilon}\cdot L_{w}f.$
(A.10)

Now, let $\Omega\subseteq X$ be any arbitrary bounded open subset. Clearly,
$(f_{\varepsilon})$ and $f$ are in $W^{1,2}(\Omega,m_{w})$. Since $|\nabla
f_{\varepsilon}|\leq|\nabla f|$ and $0\leq f_{\varepsilon}\leq|f|$, it follows
that $(f_{\varepsilon})$ is bounded in $W^{1,2}(\Omega,m_{w})$. Since
$W^{1,2}(\Omega,m_{w})$ is reflexive, there exists a subsequence
$(f_{\varepsilon_{j}})$ with $\varepsilon_{j}\searrow 0$ converging weakly in
$W^{1,2}(\Omega,m_{w})$ to some $g$. As $f_{\varepsilon_{j}}\rightarrow|f|$ in
$L^{2}(\Omega,m_{w})$, we obtain that $g=|f|$ $m_{w}$-a.e. on $\Omega$. Since
$f_{\varepsilon_{j}}(x)+\varepsilon\rightarrow|f|(x)$ for all $x$ in $X$ and
that $|f|/(f_{\varepsilon_{j}}+\varepsilon_{j})\leq 1$, for any non-negative
Lipschitz $\phi$ with $\text{supp}(\phi)\subset\Omega$, we obtain that

$L_{\Omega}(\phi):=-\int_{\Omega}\langle\nabla\phi,\nabla|f|\rangle
dm_{w}=-\lim\int_{\Omega}\langle\nabla\phi,\nabla f_{\varepsilon_{j}}\rangle
dm_{w}\\\ =\lim L_{w}f_{\varepsilon_{j}}(\phi)\geq\lim\int_{\Omega}\frac{\phi
f}{f_{\varepsilon_{j}}+\varepsilon_{j}}dL_{w}f=\int_{\Omega}\phi\text{sgn}(f)dL_{w}f.$
(A.11)

Following a similar argument as in [10, (4-28),(4-29) in Theorem 4.14], there
exists constant $C_{\Omega}>0$ such that $|L_{\Omega}(\phi)|\leq
C_{\Omega}\max|\phi|$ for any Lipschitz function $\phi$ with
$\text{supp}(\phi)\subset\Omega$. Since $X$ is proper, by Riesz representation
theorem, there exists a signed Radon measure $\mu_{\Omega}$ on $\Omega$ such
that $L_{\Omega}(\phi)=\int_{\Omega}\phi d\mu_{\Omega}$ for each $\phi$ in
$\text{Lip}_{c}(\Omega)$ and that $\mu_{\Omega}\geq\text{sgn}(f)L_{w}f$ on
$\Omega$, see the remark before Theorem 1.2 in [37] or [19, Proposition
6.2.16] for finite signed Radon measures. Clearly $\mu_{\Omega_{1}}$ and
$\mu_{\Omega_{2}}$ coincide on $\Omega_{1}\cap\Omega_{2}$ for any bounded open
subsets $\Omega_{1}$ and $\Omega_{2}$. Henceforth, there exists a unique
signed Radon measure $\nu$ on $X$ such that $\nu|_{\Omega}=\mu_{\Omega}$ for
all bounded open domain $\Omega$, and thus, we obtain that $L_{w}|f|$ is a
signed Radon measure with $L_{w}|f|\geq\text{sgn}(f)\cdot L_{w}f$. ∎

We now show the Omori-Yau type maximum principle in $\text{RCD}(K,\infty)$
based on the previous Kato’s inequality. While the proof in [37] relies on
doubling property and weak Poincaré inequality of underlying metric measure
space and weak maximum principle, our proof is based on the "Sobolev-to-Lip"
property of $\text{RCD}(K,\infty)$ space which in general does not support
doubling property.

######  Lemma A.2.

(Omori-Yau type maximum principle) Let $(X,d,m)$ be a proper
$\text{RCD}(K,\infty)$ space with $K$ in $\mathbb{R}$. Let further $f$ in
$W^{1,2}\cap L^{\infty}\cap D(\bm{\Delta})$ such that $\bm{\Delta}^{s}f\geq
0$. Suppose that $f$ achieves one of its strict maximum in $X$ in the sense
that there exits a bounded subset $U\subset X$ satisfying $m(U)>0$ and
$m(X\setminus U)>0$ with

$\operatorname*{ess\,sup}_{U}f>\operatorname*{ess\,sup}_{X\setminus U}f.$
(A.12)

Then, given any $w$ in $W^{1,2}\cap\text{Lip}_{b}$, for any $\varepsilon>0$,
we have

$m\left(\left\\{x\in X\colon
f(x)\geq\operatorname*{ess\,sup}_{X}f-\varepsilon\text{ and
}(\bm{\Delta}^{ac}f)(x)+\langle\nabla f,\nabla
w\rangle(x)\leq\varepsilon\right\\}\right)>0.$ (A.13)

In particular, there exists a sequence $(x_{j})$ in $X$ such that
$f(x_{j})\geq\operatorname*{ess\,sup}f(x_{j})-1/j$ and
$(\bm{\Delta}^{ac}f)(x_{j})+\langle\nabla f,\nabla w\rangle(x_{j})\leq 1/j$.

###### Proof.

We adapt the proof in [37]. Let $M:=\operatorname*{ess\,sup}_{X}f$. Suppose by
contradiction that there exists $\varepsilon_{0}>0$ and $w$ in
$W^{1,2}\cap\text{Lip}_{b}$ such that (A.13) fails. Then for possibly smaller
$\varepsilon_{0}>0$ such that
$M-\varepsilon_{0}>\operatorname*{ess\,sup}_{X\setminus U}f$, it follows that
$g:=(f-(M-\varepsilon_{0}))_{+}$ is in $W^{1,2}$ with $g=0$ $m$-a.e. in
$X\setminus U$, and that

$m\left(\left\\{x\in X:f(x)>M-\varepsilon_{0}\text{ and
}\bm{\Delta}^{ac}f(x)+\langle\nabla f,\nabla
w\rangle(x)\leq\varepsilon_{0}\right\\}\right)=0.$ (A.14)

Then it follows that for $m$-a.e. $x$ in $\\{y\in
X:f(y)>M-\varepsilon_{0}\\}$, we have

$\bm{\Delta}^{ac}f(x)+\langle\nabla f,\nabla w\rangle(x)>\varepsilon_{0}.$
(A.15)

Note that since $w\in W^{1,2}\cap\text{Lip}_{b}$ and $\bm{\Delta}f$ is a
signed Radon measure, it follows that $e^{w}(\bm{\Delta}f+\langle\nabla
w,\nabla f\rangle m)$ is a well-defined signed Radon measure. By the identity
of weighted weak Laplacian $L_{w}f=e^{w}(Lf+\langle\nabla w,\nabla f\rangle
m)$ and $Lf=\bm{\Delta}f$, it follows that

$L^{ac}_{w}f\cdot m_{w}=e^{w}\left(L^{ac}f+\langle\nabla w,\nabla
f\rangle\right)\cdot m\geq e^{-\|w\|_{L^{\infty}}}\varepsilon_{0}\cdot m>0$
(A.16)

on $\\{y\in X:f(y)>M-\varepsilon_{0}\\}$. Moreover, the identity of weighted
weak Laplacian together with $\bm{\Delta}^{s}f\geq 0$ implies $L^{s}_{w}f\geq
0$, and that $f-(M-\varepsilon_{0})$ is in $W^{1,2}_{loc}\cap L^{\infty}$,
applying Lemma A.1 to $f-(M-\varepsilon_{0})$, yields

$L_{w}g=L_{w}\left(f-(M-\varepsilon_{0})\right)_{+}\geq\chi_{\\{f>M-\varepsilon_{0}\\}}L^{ac}_{w}f\cdot
m_{w}\geq 0.$ (A.17)

By definition of $L_{w}$, it follows from (A.17) that

$-\int_{X}\langle\nabla g,\nabla g\rangle dm_{w}=L_{w}g(g)\geq 0,$ (A.18)

which implies that $|\nabla g|=0$ $m_{w}$-a.e. From $m$ being equivalent to
$m_{w}$ since $w\in L^{\infty}$, we get that $|\nabla g|=0$ $m$-a.e. Together
with $g$ being in $W^{1,2}$ and $|\nabla g|$ in $L^{\infty}$, by Sobolev-to-
Lip property of $\text{RCD}(K,\infty)$ spaces, it follows that $g$ admits a
Lipschitz version $\tilde{g}$ with $\text{Lip}(\tilde{g})\leq\||\nabla
g|\|_{\infty}$. Then, from $|\nabla g|=0$ $m$-a.e. as well as $g=0$ $m$-a.e on
$X\setminus U$, it follows that $\tilde{g}$ is constant and $\tilde{g}\equiv
0$. Hence, $f\leq M-\varepsilon_{0}$ $m$-a.e., which is a contradiction. ∎

## References

  * Ambrosio et al. [2013] Luigi Ambrosio, Nicola Gigli, and Giuseppe Savaré.  Density of lipschitz functions and equivalence of weak gradients in metric measure spaces.  _Revista Matemática Iberoamericana_ , 29(3):969–996, 2013.
  * Ambrosio et al. [2014a] Luigi Ambrosio, Nicola Gigli, and Giuseppe Savaré.  Calculus and heat flow in metric measure spaces and applications to spaces with ricci bounds from below.  _Inventiones mathematicae_ , 195(2):289–391, 2014a.
  * Ambrosio et al. [2014b] Luigi Ambrosio, Nicola Gigli, and Giuseppe Savaré.  Metric measure spaces with riemannian ricci curvature bounded from below.  _Duke Mathematical Journal_ , 163(7):1405–1490, 2014b.
  * Bacher and Sturm [2010] Kathrin Bacher and Karl-Theodor Sturm.  Localization and tensorization properties of the curvature-dimension condition for metric measure spaces.  _Journal of Functional Analysis_ , 259(1):28–56, 2010.
  * Bakry et al. [2014] Dominique Bakry, Ivan Gentil, Michel Ledoux, et al.  _Analysis and geometry of Markov diffusion operators_ , volume 103\.  Springer, 2014.
  * Björn and Björn [2011] Anders Björn and Jana Björn.  _Nonlinear potential theory on metric spaces_ , volume 17.  European Mathematical Society, 2011.
  * Brué and Semola [2020] Elia Brué and Daniele Semola.  Constancy of the dimension for RCD(K, N) spaces via regularity of Lagrangian flows.  _Communications on Pure and Applied Mathematics_ , 73(6):1141–1204, 2020.
  * Cavalletti and Milman [2021] Fabio Cavalletti and Emanuel Milman.  The globalization theorem for the curvature-dimension condition.  _Inventiones mathematicae_ , 226(1):1–137, 2021\.
  * Cavalletti and Mondino [2017] Fabio Cavalletti and Andrea Mondino.  Sharp geometric and functional inequalities in metric measure spaces with lower ricci curvature bounds.  _Geometry & Topology_, 21(1):603–645, 2017\.
  * Cavalletti and Mondino [2020] Fabio Cavalletti and Andrea Mondino.  New formulas for the laplacian of distance functions and applications.  _Analysis & PDE_, 13(7):2091–2147, 2020.
  * Cheeger [1999] Jeff Cheeger.  Differentiability of lipschitz functions on metric measure spaces.  _Geometric & Functional Analysis GAFA_, 9(3):428–517, 1999.
  * Debin et al. [2021] Clément Debin, Nicola Gigli, and Enrico Pasqualetto.  Quasi-continuous vector fields on rcd spaces.  _Potential Analysis_ , 54(1):183–211, 2021.
  * Erbar et al. [2015] Matthias Erbar, Kazumasa Kuwada, and Karl-Theodor Sturm.  On the equivalence of the entropic curvature-dimension condition and bochner’s inequality on metric measure spaces.  _Inventiones mathematicae_ , 201(3):993–1071, 2015.
  * Garofalo and Mondino [2014] Nicola Garofalo and Andrea Mondino.  Li–Yau and Harnack type inequalities in $\text{RCD}^{*}(K,N)$ metric measure spaces.  _Nonlinear Analysis: Theory, Methods & Applications_, 95:721–734, 2014.
  * Giaquinta [1983] Mariano Giaquinta.  _Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems_.  Annals of Mathematics Studies 105. Princeton University Press, 1983.
  * Gigli [2015] Nicola Gigli.  _On the differential structure of metric measure spaces and applications_.  American Mathematical Soc., 2015.
  * Gigli [2018] Nicola Gigli.  _Nonsmooth differential geometry–An approach tailored for spaces with Ricci curvature bounded from below_ , volume 251.  American Mathematical Society, 2018.
  * Gigli and Mondino [2013] Nicola Gigli and Andrea Mondino.  A PDE approach to nonlinear potential theory in metric measure spaces.  _Journal de Mathématiques Pures et Appliquées_ , 100(4):505–534, 2013.
  * Gigli et al. [2020] Nicola Gigli, Enrico Pasqualetto, et al.  _Lectures on nonsmooth differential geometry_ , volume 2.  Springer, 2020.
  * Gross [1975] Leonard Gross.  Logarithmic sobolev inequalities.  _American Journal of Mathematics_ , 97(4):1061–1083, 1975.
  * Huang and Zhang [2020] Jia-Cheng Huang and Hui-Chun Zhang.  Localized elliptic gradient estimate for solutions of the heat equation on $\text{RCD}^{*}(K,N)$ metric measure spaces.  _manuscripta mathematica_ , 161(3):303–324, 2020\.
  * Jiang [2015] Renjin Jiang.  The Li–Yau inequality and heat kernels on metric measure spaces.  _Journal de Mathématiques Pures et Appliquées_ , 104(1):29–57, 2015.
  * Jiang and Zhang [2016] Renjin Jiang and Huichun Zhang.  Hamilton’s gradient estimates and a monotonicity formula for heat flows on metric measure spaces.  _Nonlinear Analysis_ , 131:32–47, 2016.
  * Kinnunen and Shanmugalingam [2001] Juha Kinnunen and Nageswari Shanmugalingam.  Regularity of quasi-minimizers on metric spaces.  _manuscripta mathematica_ , 105(3):401–423, 2001\.
  * Koskela et al. [2003] Pekka Koskela, Kai Rajala, and Nageswari Shanmugalingam.  Lipschitz continuity of cheeger-harmonic functions in metric measure spaces.  _Journal of Functional Analysis_ , 202(1):147–173, 2003.
  * Li [2016] Huaiqian Li.  Dimension-Free Harnack Inequalities on $RCD(K,\infty)$ Spaces.  _Journal of Theoretical Probability_ , 29(4):1280–1297, 2016.
  * Lott and Villani [2009] John Lott and Cédric Villani.  Ricci curvature for metric-measure spaces via optimal transport.  _Annals of Mathematics_ , pages 903–991, 2009.
  * Mondino and Naber [2019] Andrea Mondino and Aaron Naber.  Structure theory of metric measure spaces with lower ricci curvature bounds.  _Journal of the European Mathematical Society_ , 21(6):1809–1854, 2019.
  * Ohta and Takatsu [2020] Shin-ichi Ohta and Asuka Takatsu.  Equality in the logarithmic sobolev inequality.  _manuscripta mathematica_ , 162(1):271–282, 2020\.
  * Otto and Villani [2000] Felix Otto and Cédric Villani.  Generalization of an inequality by talagrand and links with the logarithmic sobolev inequality.  _Journal of Functional Analysis_ , 173(2):361–400, 2000.
  * Profeta [2015] Angelo Profeta.  The sharp sobolev inequality on metric measure spaces with lower ricci curvature bounds.  _Potential Analysis_ , 43(3):513–529, 2015.
  * Rajala [2012] Tapio Rajala.  Interpolated measures with bounded density in metric spaces satisfying the curvature-dimension conditions of sturm.  _Journal of Functional Analysis_ , 263(4):896–924, 2012.
  * Rothaus [1981] Oskar S Rothaus.  Logarithmic sobolev inequalities and the spectrum of schrödinger operators.  _Journal of functional analysis_ , 42(1):110–120, 1981.
  * Sturm [2006a] Karl-Theodor Sturm.  On the geometry of metric measure spaces.  _Acta mathematica_ , 196(1):65–131, 2006a.
  * Sturm [2006b] Karl-Theodor Sturm.  On the geometry of metric measure spaces. II.  _Acta Mathematica_ , 196(1):133 – 177, 2006b.  doi: 10.1007/s11511-006-0003-7.
  * Wang [1999] Feng-Yu Wang.  Harnack inequalities for log-sobolev functions and estimates of log-sobolev constants.  _The Annals of Probability_ , 27(2):653–663, 1999\.
  * Zhang and Zhu [2016] Hui-Chun Zhang and Xi-Ping Zhu.  Local Li–Yau’s estimates on $\text{RCD}^{*}(K,N)$ metric measure spaces.  _Calculus of Variations and Partial Differential Equations_ , 55(4):1–30, 2016.
  * Zhang [2012] Qi S Zhang.  Extremal of Log Sobolev inequality and W entropy on noncompact manifolds.  _Journal of Functional Analysis_ , 263(7):2051–2101, 2012.