$F\in\mathfrak{C}\big{(}\mathcal{P}(\mathbb{B}),{\mathrm{C}}_{b}^{1}(\mathbb{B})\big{)}$,
then ${\mathrm{D}}_{p}F\in\left(\int_{\mathcal{P}_{p}(\mathbb{B},{\sf
d}_{\|\cdot\|})}^{\bigoplus}\mathscr{V}_{\mu}{\mathrm{d}}\mathfrak{m}(\mu)\right)_{L^{q}}$.

###### Proof.

The completeness property can be found in [10, Proposition 3.2] (see also [17]
where the notion of direct integral was introduced for the first time).
If $(\mathbb{B}^{*},\|\cdot\|_{*})$ is uniformly convex, then we can use [15,
Theorem 2.2], together with the fact that
$L^{q}(\mathcal{P}_{p}(\mathbb{B},{\sf d}_{\|\cdot\|}),\mathfrak{m})$ is
uniformly convex and that the modulus of convexity of $\mathscr{V}_{\mu}$ is
larger than the one of
$L^{p^{\prime}}(\mathbb{B},\mu;(\mathbb{B}^{*},\|\cdot\|))$, being
$\mathscr{V}_{\mu}$ just a subspace of the latter, and the fact that the
modulus of convexity of the Bochner space
$L^{p^{\prime}}(\mathbb{B},\mu;(\mathbb{B}^{*},\|\cdot\|))$ does not depend on
$\mu$ but only on $p^{\prime}$ and on the modulus of convexity of the
uniformly convex space $(\mathbb{B}^{*},\|\cdot\|_{*})$.
If $\mathbb{B}$ is reflexive, then also $\mathbb{B}^{*}$ and thus
$L^{p^{\prime}}(\mathbb{B},\mu;(\mathbb{B}^{*},\|\cdot\|_{*}))$ is reflexive
so that we can apply [17, Theorem 6.19], also noting that, obviously,
$L^{q}(\mathcal{P}_{p}(\mathbb{B},{\sf d}_{\|\cdot\|}),\mathfrak{m})$ is
reflexive.
Let now
$F=\psi\circ\mathsf{L}_{{\boldsymbol{\phi}}}\in\mathfrak{C}\big{(}\mathcal{P}(\mathbb{B}),{\mathrm{C}}_{b}^{1}(\mathbb{B})\big{)}$,
with $\psi\in{\mathrm{C}}_{b}^{1}(\mathbb{R}^{N})$ and
${\boldsymbol{\phi}}\in({\mathrm{C}}_{b}^{1}(\mathbb{B}))^{N}$, for some
$N\in\mathbb{N}$; notice that we are looking at ${\mathrm{D}}_{p}F$ as the map
sending $\mu\in\mathcal{P}_{p}(\mathbb{B},{\sf d}_{\|\cdot\|})$ to
${\mathrm{D}}_{p}[\mu]:\mathbb{B}\to\mathbb{B}^{*}$. Of course
${\mathrm{D}}_{p}F[\mu]\in\mathscr{V}_{\mu}$ for every
$\mu\in\mathcal{P}_{p}(\mathbb{B},{\sf d}_{\|\cdot\|})$ and it is uniformly
bounded. Moreover it is Bochner measurable since, in order to approximate it
with simple functions, it is enough to approximate every term of the form
$\partial\psi_{n}(\int_{\mathbb{B}}\phi_{1}{\mathrm{d}}\mu,\dots,\int_{\mathbb{B}}\phi_{N}{\mathrm{d}}\mu)$
with $\mathbb{R}$-valued simple function $(s_{k}^{n})_{k}$ on
$\mathcal{P}_{p}(\mathbb{B},{\sf d}_{\|\cdot\|})$ (which is possible since
this map is measurable) so that

$s_{k}(\mu):=\sum_{n=1}^{N}s_{k}^{n}(\mu)\nabla\phi_{n},\quad\mu\in\mathcal{P}_{p}(\mathbb{B},{\sf
d}_{\|\cdot\|}),$

is a sequence of simple functions approximating ${\mathrm{D}}_{p}F$. ∎

######  Theorem 5.2.

Let $(\mathbb{B},\|\cdot\|)$ be a separable reflexive Banach space, let
$p,q\in(1,+\infty)$ be (not necessarily conjugate) fixed exponents and let
$\mathfrak{m}$ be a positive and finite Borel measure on
$\mathcal{P}_{p}(\mathbb{B},{\sf d}_{\|\cdot\|})$. Then the Sobolev space
$H^{1,q}(\mathcal{P}_{p}(\mathbb{B},{\sf d}_{\|\cdot\|}),W_{p,{\sf
d}_{\|\cdot\|}},\mathfrak{m})$ is reflexive.

###### Proof.

It is sufficient to provide a linear isometry $\iota$ from
$H^{1,q}(\mathcal{P}_{p}(\mathbb{B},{\sf d}_{\|\cdot\|}),W_{p,{\sf
d}_{\|\cdot\|}},\mathfrak{m})$ into a reflexive Banach space. For simplicity
let us denote by

$X:=L^{q}(\mathcal{P}_{p}(\mathbb{B},{\sf d}_{\|\cdot\|}),\mathfrak{m}),\quad
Y:=\left(\int_{\mathcal{P}_{p}(\mathbb{B},{\sf
d}_{\|\cdot\|})}^{\bigoplus}\mathscr{V}_{\mu}{\mathrm{d}}\mathfrak{m}(\mu)\right)_{L^{q}}$

and let us define ${\mbox{\boldmath$G$}}\subset X\times Y$ as the closure in
$X\times Y$ (endowed with the norm $\|(x,y)\|_{X\times
Y}^{q}=\|x\|_{X}^{q}+\|y\|_{Y}^{q}$) of

$\\{(F,{\mathrm{D}}_{p}F):F\in\mathfrak{C}\big{(}\mathcal{P}(\mathbb{B}),{\mathrm{C}}_{b}^{1}(\mathbb{B})\big{)}\\}.$

Denoting by $\pi^{X}:X\times Y\to X$ the projection $\pi^{X}(x,y)=x$ for every
$x\in X$, we define the sections of $G$ by

${\mbox{\boldmath$D$}}_{\mathfrak{m}}F:=\\{G\in
Y:(F,G)\in{\mbox{\boldmath$G$}}\\},\quad F\in\pi^{X}({\mbox{\boldmath$G$}}).$

It is clear that $G$ is closed and convex so that it is also weakly closed;
let us show that $H^{1,q}(\mathcal{P}_{p}(\mathbb{B},{\sf
d}_{\|\cdot\|}),W_{p,{\sf
d}_{\|\cdot\|}},\mathfrak{m})\subset\pi^{X}({\mbox{\boldmath$G$}})$: if $F\in
H^{1,q}(\mathcal{P}_{p}(\mathbb{B},{\sf d}_{\|\cdot\|}),W_{p,{\sf
d}_{\|\cdot\|}},\mathfrak{m})$, by Corollary 4.19, we can find a sequence
$(F_{n})_{n}\subset\mathfrak{C}\big{(}\mathcal{P}(\mathbb{B}),{\mathrm{C}}_{b}^{1}(\mathbb{B})\big{)}$
such that

$F_{n}\to F,\quad\mathop{\rm lip}\nolimits_{W_{p,{\sf
d}_{\|\cdot\|}}}F_{n}\to|{\mathrm{D}}F|_{\star,q}\quad\text{ in }X.$

By Proposition 4.7, we have that $\|{\mathrm{D}}_{p}F_{n}\|_{Y}$ is uniformly
bounded so that, by the reflexivity of $Y$, we can find a subsequence of
$(F_{n_{k}})_{k}$ such that

$(F_{n_{k}},{\mathrm{D}}_{p}F_{n_{k}})\rightharpoonup(F,G)\text{ in }X\times
Y.$

This gives that $F\in\pi^{X}({\mbox{\boldmath$G$}})$ and that
$\|G\|_{Y}\leq\mathsf{C\kern-1.0ptE}_{q}^{1/q}(F)$. Moreover, if we consider
$F\in H^{1,q}(\mathcal{P}_{p}(\mathbb{B},{\sf d}_{\|\cdot\|}),W_{p,{\sf
d}_{\|\cdot\|}},\mathfrak{m})$ and
$G\in{\mbox{\boldmath$D$}}_{\mathfrak{m}}F$, then there exists
$(F_{n})_{n}\subset\mathfrak{C}\big{(}\mathcal{P}(\mathbb{B}),{\mathrm{C}}_{b}^{1}(\mathbb{B})\big{)}$
such that $(F_{n},{\mathrm{D}}_{p}F)\to(F,G)$ in $X\times Y$ so that
$\|G\|_{Y}=\lim_{n}\|{\mathrm{D}}_{p}F_{n}\|_{Y}=\lim_{n}\mathsf{pC\kern-1.0ptE}_{q}^{1/q}(F_{n})\geq\mathsf{C\kern-1.0ptE}_{q}^{1/q}(F)$
so that

$\min_{G\in{\mbox{\boldmath$D$}}_{\mathfrak{m}}F}\|G\|_{Y}=\mathsf{C\kern-1.0ptE}_{q}^{1/q}(F),\quad
F\in H^{1,q}(\mathcal{P}_{p}(\mathbb{B},{\sf d}_{\|\cdot\|}),W_{p,{\sf
d}_{\|\cdot\|}},\mathfrak{m}).$

We define
${\mbox{\boldmath$G$}}_{0}:={\mbox{\boldmath$G$}}/(\\{0\\}\times{\mbox{\boldmath$D$}}_{\mathfrak{m}}0)$
with the quotient norm $\|\cdot\|_{Y,0}$: elements of
${\mbox{\boldmath$G$}}_{0}$ are equivalence classes
$[F]=\\{(F,G):G\in{\mbox{\boldmath$D$}}_{\mathfrak{m}}F\\}$ and the quotient
norm is simply given by

$\|[F]\|_{Y,0}=\inf_{G\in{\mbox{\boldmath$D$}}_{\mathfrak{m}}F}\left(\|F\|^{q}_{X}+\|G\|_{Y}^{q})\right)^{1/q}=\left(\|F\|^{q}_{X}+\min_{G\in{\mbox{\boldmath$D$}}_{\mathfrak{m}}F}\|G\|_{Y}^{q})\right)^{1/q}=\|F\|_{H^{1,q}(\mathcal{P}_{p}(\mathbb{B},{\sf
d}_{\|\cdot\|}),W_{p,{\sf d}_{\|\cdot\|}},\mathfrak{m})}.$

The linear isometry is thus given by
$\iota:H^{1,q}(\mathcal{P}_{p}(\mathbb{B},{\sf d}_{\|\cdot\|}),W_{p,{\sf
d}_{\|\cdot\|}},\mathfrak{m})\to{\mbox{\boldmath$G$}}_{0}$ defined as

$\iota(F)=[F],\quad F\in H^{1,q}(\mathcal{P}_{p}(\mathbb{B},{\sf
d}_{\|\cdot\|}),W_{p,{\sf d}_{\|\cdot\|}},\mathfrak{m}).$

Since closed subspaces of reflexive spaces are reflexive and quotients of
reflexive spaces are reflexive, ${\mbox{\boldmath$G$}}_{0}$ is reflexive and
this concludes the proof. ∎

The next result [31, Lemma 2] provides a quantitative version of the uniform
convexity property that will pass from the pre-Cheeger energy to the Cheeger
energy.

######  Proposition 5.3.

Let $(\mathscr{W},\|\cdot\|_{\mathscr{W}})$ be a Banach space. Then
$(\mathscr{W},\|\cdot\|_{\mathscr{W}})$ is uniformly convex if and only if for
every $t\in(1,+\infty)$ there exists a strictly increasing and continuous
function $g_{t}:[0,2]\to[0,+\infty)$ such that

$\left\|\frac{x+y}{2}\right\|_{\mathscr{W}}^{t}+\left(\|x\|_{\mathscr{W}}\vee\|y\|_{\mathscr{W}}\right)^{t}g_{t}\left(\frac{\|x-y\|_{\mathscr{W}}}{\|x\|_{\mathscr{W}}\vee\|y\|_{\mathscr{W}})}\right)\leq\frac{1}{2}\|x\|_{\mathscr{W}}^{t}+\frac{1}{2}\|y\|_{\mathscr{W}}^{t}\quad\text{
for every }x,y\in\mathscr{W}.$

######  Theorem 5.4.

Let $(\mathbb{B},\|\cdot\|)$ be a separable Banach space such that
$(\mathbb{B}^{*},\|\cdot\|_{*})$ is uniformly convex, let $p,q\in(1,+\infty)$
be (not necessarily conjugate) fixed exponents and let $\mathfrak{m}$ be a
positive and finite Borel measure on $\mathcal{P}_{p}(\mathbb{B},{\sf
d}_{\|\cdot\|})$. Then the Sobolev space
$H^{1,q}(\mathcal{P}_{p}(\mathbb{B},{\sf d}_{\|\cdot\|}),W_{p,{\sf
d}_{\|\cdot\|}},\mathfrak{m})$ is uniformly convex.

###### Proof.

By the uniform convexity of $\left(\int_{\mathcal{P}_{p}(\mathbb{B},{\sf
d}_{\|\cdot\|})}^{\bigoplus}\mathscr{V}_{\mu}{\mathrm{d}}\mathfrak{m}(\mu)\right)_{L^{q}}$
provided by Proposition 5.1 and using Proposition 5.3, we have that for every
$t\in(1,+\infty)$ there exists a continuous and strictly increasing function
$g_{t}:[0,2]\to[0,+\infty)$ such that

$\displaystyle\begin{split}\mathsf{pC\kern-1.0ptE}_{q}\left(\frac{1}{2}(F+G)\right)^{t/q}&+\left(\mathsf{pC\kern-1.0ptE}_{q}^{1/q}(F)\vee\mathsf{pC\kern-1.0ptE}_{q}^{1/q}(G)\right)^{t}g_{t}\left(\frac{\mathsf{pC\kern-1.0ptE}_{q}^{1/q}(F-G)}{\mathsf{pC\kern-1.0ptE}_{q}^{1/q}(F)\vee\mathsf{pC\kern-1.0ptE}_{q}^{1/q}(G)}\right)\\\
&\leq\frac{1}{2}\mathsf{pC\kern-1.0ptE}_{q}^{t/q}(F)+\frac{1}{2}\mathsf{pC\kern-1.0ptE}_{q}^{t/q}(G)\quad\text{
for every
}F,G\in\mathfrak{C}\big{(}\mathcal{P}(\mathbb{B}),{\mathrm{C}}_{b}^{1}(\mathbb{B})\big{)},\end{split}$
(5.1)

where we are able to identify the norm in the direct integral and the $q$-th
root of the pre-Cheeger energy thanks to Proposition 4.7. If $F,G\in
H^{1,q}(\mathcal{P}_{p}(\mathbb{B},{\sf d}_{\|\cdot\|}),W_{p,{\sf
d}_{\|\cdot}},\mathfrak{m})$, thanks to Corollary 4.19, we can find sequences
$(F_{n})_{n},(G_{n})_{n}\subset\mathfrak{C}\big{(}\mathcal{P}(\mathbb{B}),{\mathrm{C}}_{b}^{1}(\mathbb{B})\big{)}$
such that

$F_{n}\to F,\quad G_{n}\to G\quad\text{ in
}L^{q}(\mathcal{P}_{p}(\mathbb{B},{\sf
d}_{\|\cdot\|}),\mathfrak{m}),\quad\mathsf{pC\kern-1.0ptE}_{q}(F_{n})\to\mathsf{C\kern-1.0ptE}_{q}(F),\quad\mathsf{pC\kern-1.0ptE}_{q}(G_{n})\to\mathsf{C\kern-1.0ptE}_{q}(G).$

By the lower semicontinuity of the Cheeger energy w.r.t. the $L^{q}$
convergence and the continuity and monotonicity of $g_{t}$, we obtain that
(5.1) holds for $\mathsf{C\kern-1.0ptE}_{q}$ instead of
$\mathsf{pC\kern-1.0ptE}_{q}$. By Proposition 5.3 we obtain that
$\mathsf{C\kern-1.0ptE}_{q}^{1/q}$ is uniformly convex, in the sense that, for
every $\varepsilon>0$ there exists a $\delta(\varepsilon)>0$ such that,
whenever $F,G\in H^{1,q}(\mathcal{P}_{p}(\mathbb{B},{\sf
d}_{\|\cdot\|}),W_{p,{\sf d}_{\|\cdot\|}},\mathfrak{m})$ are such that
$\mathsf{C\kern-1.0ptE}_{q}^{1/q}(F)=\mathsf{C\kern-1.0ptE}_{q}^{1/q}(G)=1$
and $\mathsf{C\kern-1.0ptE}_{q}^{1/q}(F-G)\geq\varepsilon$, then
$\mathsf{C\kern-1.0ptE}_{q}^{1/q}(F+G)<2(1-\delta(\varepsilon))$. Since the
$L^{q}(\mathcal{P}_{p}(\mathbb{B},{\sf d}_{\|\cdot\|},\mathfrak{m})$-norm is
uniformly convex, the $q$-sum (i.e. the $q$-th root of the sum of the
$q$-powers) of it with $\mathsf{C\kern-1.0ptE}_{q}^{1/q}$ is uniformly convex
(see e.g. [8, Theorem 1]). This means that the Sobolev norm is uniformly
convex and concludes the proof. ∎

Given a separable Banach space $(\mathbb{B},\|\cdot\|)$ and a positive and
finite Borel measure $\mathfrak{m}$ on $\mathbb{B}$, we can consider the
measure $\mathsf{D}_{\sharp}\mathfrak{m}$ on $\mathcal{P}(\mathbb{B},{\sf
d}_{\|\cdot\|})$, where $\mathsf{D}:\mathbb{B}\to\mathcal{P}(\mathbb{B},{\sf
d}_{\|\cdot\|})$ is defined as

$\mathsf{D}(x):=\delta_{x},\quad x\in\mathbb{B}.$

It is immediate to check that, for every $q\in(1,+\infty)$, the Sobolev spaces
$H^{1,q}(\mathbb{B},{\sf d}_{\|\cdot\|},\mathfrak{m})$ and
$H^{1,q}(\mathcal{P}_{2}(\mathbb{B},{\sf d}_{\|\cdot\|}),W_{2,{\sf
d}_{\|\cdot\|}},\mathsf{D}_{\sharp}\mathfrak{m})$ are isomorphic, see also
[12, Section 5.2]. Notice that the choice $p=2$ is irrelevant, since
$\mathsf{D}(\mathbb{B})\subset\mathcal{P}_{p}(\mathbb{B},{\sf d}_{\|\cdot\|})$
for every $p\in[1,+\infty]$. As a consequence of this observation and Theorems
5.2 and 5.4 we obtain the following corollary.

######  Corollary 5.5.

Let $(\mathbb{B},\|\cdot\|)$ be a separable Banach space, let
$q\in(1,+\infty)$ be a fixed exponent, and let $\mathfrak{m}$ be a positive
and finite Borel measure on $\mathbb{B}$. If $(\mathbb{B},\|\cdot\|)$ is
reflexive (resp. if $(\mathbb{B}^{*},\|\cdot\|_{*})$ is uniformly convex),
then $H^{1,q}(\mathbb{B},{\sf d}_{\|\cdot\|},\mathfrak{m})$ is reflexive
(resp. uniformly convex).

We remark that the statement regarding reflexivity in the above corollary was
already known (see e.g. [25, Corollary 5.3.11]) while the part concerning
uniform convexity is new, at least to our knowledge.

###  5.1. Clarkson’s type inequalities

In this subsection we show how one can deduce Clarkson’s type inequalities for
the Sobolev norm starting from analogous inequalities that are satisfied for
the dual norm in the separable Banach space $\mathbb{B}$. The following
inequality was introduced for the first time by [6] and it is a generalization
of the classical Clarkson’s inequalities [8].

######  Definition 5.6.

Let $V$ be a real vector space and let $r,s\in(1,+\infty)$. We say that a
functional $\mathcal{J}:V\to[0,+\infty)$ satisfies the $(r,s)$-Boas inequality
($(r,s)$-(BI)) in $V$ if

$\left(\mathcal{J}(u+v)^{r}+\mathcal{J}(u-v)^{r}\right)^{1/r}\leq
2^{1/s^{\prime}}\left(\mathcal{J}(u)^{s}+\mathcal{J}(v)^{s}\right)^{1/s}\quad\text{
for every }u,v\in V.$

######  Lemma 5.7.

Let $V$ be a real vector space, let $q\in(1,+\infty)$ and let
$\mathcal{J}_{1},\mathcal{J}_{2}:V\to[0,+\infty)$ be two functionals
satisfying the $(r,s)$-(BI) in $V$ for some $r,s\in(1,+\infty)$ such that
$s\leq q\leq r$. Then the functional $\mathcal{J}:V\to[0,+\infty)$ defined as

$\mathcal{J}(u):=\left(\mathcal{J}_{1}(u)^{q}+\mathcal{J}_{2}(u)^{q}\right)^{1/q},\quad
u\in V,$

satisfies the $(r,s)$-(BI) in $V$ as well.

###### Proof.

Let $u,v\in V$; let us set

$\displaystyle a_{0}:=\mathcal{J}_{1}(u+v),\quad$ $\displaystyle
b_{0}:=\mathcal{J}_{1}(u-v),\quad$ $\displaystyle
c_{0}:=\mathcal{J}_{2}(u+v),\quad$ $\displaystyle
d_{0}:=\mathcal{J}_{2}(u-v),$ $\displaystyle a:=\mathcal{J}_{1}(u),\quad$
$\displaystyle b:=\mathcal{J}_{1}(v),\quad$ $\displaystyle
c:=\mathcal{J}_{2}(u),\quad$ $\displaystyle d:=\mathcal{J}_{2}(v).$

Since $\mathcal{J}_{1}$ and $\mathcal{J}_{2}$ satisfy the $(r,s)$-(BI) in $V$,
we have

$\displaystyle\left(a_{0}^{r}+b_{0}^{r}\right)^{1/r}$ $\displaystyle\leq
2^{1/s^{\prime}}\left(a^{s}+b^{s}\right)^{1/s},$ (5.2)
$\displaystyle\left(c_{0}^{r}+d_{0}^{r}\right)^{1/r}$ $\displaystyle\leq
2^{1/s^{\prime}}\left(c^{s}+d^{s}\right)^{1/s}.$ (5.3)

Then

$\displaystyle
2^{1/s^{\prime}}\left(\mathcal{J}(u)^{s}+\mathcal{J}(v)^{s}\right)^{1/s}$
$\displaystyle=2^{1/s^{\prime}}\left((a^{q}+c^{q})^{s/q}+(b^{q}+d^{q})^{s/q}\right)^{1/s}$
$\displaystyle=2^{1/s^{\prime}}\left(((a^{s})^{q/s}+(c^{s})^{q/s})^{s/q}+((b^{s})^{q/s}+(d^{s})^{q/s})^{s/q}\right)^{1/s}$
$\displaystyle\geq
2^{1/s^{\prime}}\left(\left(a^{s}+b^{s}\right)^{q/s}+\left(c^{s}+d^{s}\right)^{q/s}\right)^{1/q}$
$\displaystyle\geq\left(\left(a_{0}^{r}+b_{0}^{r}\right)^{q/r}+\left(c_{0}^{r}+d_{0}^{r}\right)^{q/r}\right)^{1/q}$
$\displaystyle=\left(\left((a_{0}^{q})^{r/q}+(b_{0}^{q})^{r/q}\right)^{q/r}+\left((c_{0}^{q})^{r/q}+(d_{0}^{q})^{r/q}\right)^{q/r}\right)^{1/q}$
$\displaystyle\geq\left((a_{0}^{q}+c_{0}^{q})^{r/q}+(b_{0}^{q}+d_{0}^{q})^{r/q}\right)^{1/r}$
$\displaystyle=\left(\mathcal{J}(u+v)^{r}+\mathcal{J}(u-v)^{r}\right)^{1/r},$

where we have used the triangular inequality for the $\alpha$-norm in
$\mathbb{R}^{2}$ first with $\alpha=q/s$ (from the second to the third line)
and then with $\alpha=r/q$ (from the fifth to the sixth line), and the
inequalities (5.2) and (5.3) to pass from the third to the fourth line. ∎

######  Proposition 5.8.

Let $(X,{\sf d},\mathfrak{m})$ be a Polish metric-measure space, let
$\mathscr{A}$ be a separating unital subalgebra of $\mathop{\rm
Lip}\nolimits_{b}(X,{\sf d})$ satisfying (2.12) and let $q,r,s\in(1,+\infty)$.
If $(\mathsf{pC\kern-1.0ptE}_{q})^{1/q}$ satisfies the $(r,s)$-(BI) in
$\mathscr{A}$, then $(\mathsf{C\kern-1.0ptE}_{q})^{1/q}$ satisfies the
$(r,s)$-(BI) in $H^{1,q}(X,{\sf d},\mathfrak{m})$. If in addition
$r^{\prime}\leq s\leq q\leq r$, then the Sobolev norm in $H^{1,q}(X,{\sf
d},\mathfrak{m})$ satisfies the $(r,s)$-(BI).

###### Proof.

The proof that $(\mathsf{C\kern-1.0ptE}_{q})^{1/q}$ satisfies the $(r,s)$-(BI)
in $H^{1,q}(X,{\sf d},\mathfrak{m})$ is the same of [12, Theorem 2.15] for the
case $r=s=q=2$. If in addition we know that $r^{\prime}\leq s\leq q\leq r$,
then the $L^{q}(X,\mathfrak{m})$-norm satisfies the $(r,s)$-(BI) in
$H^{1,q}(X,{\sf d},\mathfrak{m})$ [6, Theorem 1] so that we can apply Lemma
5.7 and conclude that the Sobolev norm satisfies the $(r,s)$-(BI) in
$H^{1,q}(X,{\sf d},\mathfrak{m})$. ∎

In the next Proposition 5.9 we show how to deduce the $(r,s)$-(BI) for the
pre-Cheeger energy from the same inequality for the
$L^{p^{\prime}}(\mathbb{B},\mu;(\mathbb{B}^{*},\|\cdot\|_{*}))$ norm. Instead
of stating our hypothesis in terms of the validity of a Clarkson’s inequality
in $(\mathbb{B},\|\cdot\|)$, we state it directly in terms of the (BI) in
$L^{p^{\prime}}(\mathbb{B},\mu;(\mathbb{B}^{*},\|\cdot\|_{*}))$: the two
notions are different in general, we refer to [16] for the treatment of the
relation between them.

######  Proposition 5.9.

Let $(\mathbb{B},\|\cdot\|)$ be a separable Banach space and let
$\mathfrak{m}$ be a positive and finite Borel measure on
$\mathcal{P}_{p}(\mathbb{B},{\sf d}_{\|\cdot\|})$; suppose that there exists
$(r,s)$ with $1<s\leq q\leq r$ such that the
$L^{p^{\prime}}(\mathbb{B},\mu;(\mathbb{B}^{*},\|\cdot\|_{*}))$-norm satisfies
the $(r,s)$-(BI) (cf. Definition 5.6) in
$L^{p^{\prime}}(\mathbb{B},\mu;(\mathbb{B}^{*},\|\cdot\|_{*}))$ for
$\mathfrak{m}$-a.e. $\mu\in\mathcal{P}_{p}(\mathbb{B},{\sf d}_{\|\cdot\|})$.
Then the functional
$\mathcal{J}:\mathfrak{C}\big{(}\mathcal{P}(\mathbb{B}),{\mathrm{C}}_{b}^{1}(\mathbb{B})\big{)}\to[0,+\infty)$
defined as

$\mathcal{J}(F):=(\mathsf{pC\kern-1.0ptE}_{q}(F))^{1/q},\quad
F\in\mathfrak{C}\big{(}\mathcal{P}(\mathbb{B}),{\mathrm{C}}_{b}^{1}(\mathbb{B})\big{)},$

satisfies the $(r,s)$-(BI) in
$\mathfrak{C}\big{(}\mathcal{P}(\mathbb{B}),{\mathrm{C}}_{b}^{1}(\mathbb{B})\big{)}$.

###### Proof.

Let
$F,G\in\mathfrak{C}\big{(}\mathcal{P}(\mathbb{B}),{\mathrm{C}}_{b}^{1}(\mathbb{B})\big{)}$.
We denote by $\|\cdot\|_{*,p^{\prime},\mu}$ the
$L^{p^{\prime}}(\mathbb{B},\mu;(\mathbb{B}^{*},\|\cdot\|_{*}))$-norm for
$\mu\in\mathcal{P}_{p}(\mathbb{B},{\sf d}_{\|\cdot\|})$ and we set

$U_{\mu}(x):={\mathrm{D}}_{p}F(\mu,x),\quad
V_{\mu}(x):={\mathrm{D}}_{p}G(\mu,x),\quad(\mu,x)\in\mathcal{P}_{p}(\mathbb{B},{\sf
d}_{\|\cdot\|})\times\mathbb{B}.$

By hypothesis we have

$\left(\|U_{\mu}+V_{\mu}\|_{*,p^{\prime},\mu}^{r}+\|U_{\mu}-V_{\mu}\|_{*,p^{\prime},\mu}^{r}\right)^{1/r}\leq
2^{1/s^{\prime}}\left(\|U_{\mu}\|_{*,p^{\prime},\mu}^{s}+\|V_{\mu}\|_{*,p^{\prime},\mu}^{s}\right)^{1/s}$

for $\mathfrak{m}$-a.e. $\mu\in\mathcal{P}_{p}(\mathbb{B},{\sf
d}_{\|\cdot\|})$. Computing the $L^{q}(\mathcal{P}_{p}(\mathbb{B},{\sf
d}_{\|\cdot\|}),\mathfrak{m})$-norm on both sides and applying Minkowski
inequalities in $L^{q/r}(\mathcal{P}_{p}(\mathbb{B},{\sf
d}_{\|\cdot\|}),\mathfrak{m})$ and in $L^{q/s}(\mathcal{P}_{p}(\mathbb{B},{\sf
d}_{\|\cdot\|}),\mathfrak{m})$ (notice that $0<q/r\leq 1$ and $q/s\geq 1$), we
get

$\displaystyle\left(\left(\int_{\mathcal{P}_{p}(\mathbb{B},{\sf
d}_{\|\cdot\|})}\|U_{\mu}+V_{\mu}\|_{*,p^{\prime},\mu}^{q}\,{\mathrm{d}}\mathfrak{m}(\mu)\right)^{r/q}+\left(\int_{\mathcal{P}_{p}(\mathbb{B},{\sf
d}_{\|\cdot\|})}\|U_{\mu}-V_{\mu}\|_{*,p^{\prime},\mu}^{q}\,{\mathrm{d}}\mathfrak{m}(\mu)\right)^{r/q}\right)^{1/r}\leq$
$\displaystyle
2^{1/s^{\prime}}\left(\left(\int_{\mathcal{P}_{p}(\mathbb{B},{\sf
d}_{\|\cdot\|})}\|U_{\mu}\|_{*,p^{\prime},\mu}^{q}\,{\mathrm{d}}\mathfrak{m}(\mu)\right)^{s/q}+\left(\int_{\mathcal{P}_{p}(\mathbb{B},{\sf
d}_{\|\cdot\|})}\|V_{\mu}\|_{*,p^{\prime},\mu}^{q}\,{\mathrm{d}}\mathfrak{m}(\mu)\right)^{s/q}\right)^{1/s}$

i.e.

$\left(\mathcal{J}(F+G)^{r}+\mathcal{J}(F-G)^{r}\right)^{1/r}\leq
2^{1/s^{\prime}}\left(\mathcal{J}(F)^{s}+\mathcal{J}(G)^{s}\right)^{1/s},$

by Proposition 4.7. ∎

######  Theorem 5.10.

In the same hypotheses of Proposition 5.9, assume in addition that
$r^{\prime}\leq s$. Then the Sobolev norm in
$H^{1,q}(\mathcal{P}_{p}(\mathbb{B},{\sf d}_{\|\cdot\|}),W_{p,{\sf
d}_{\cdot}},\mathfrak{m})$ satisfies the $(r,s)$-(BI).

###### Proof.

This follows combining Proposition 5.8, Corollary 4.19 and Proposition 5.9. ∎

######  Corollary 5.11.

If $\mathbb{B}=\mathbb{R}^{d}$ for some $d\in\mathbb{N}$ and $|\cdot|$ is the
Euclidean norm, then the Sobolev norm in
$H^{1,q}(\mathcal{P}_{p}(\mathbb{R}^{d},|\cdot|),W_{p,{\sf
d}_{|\cdot|}},\mathfrak{m})$ satisfies the $(r,s)$-(BI) for every
$1<r^{\prime}\leq s\leq q,p^{\prime}\leq r$.

###### Proof.

This follows by the fact that the
$L^{p^{\prime}}(\mathbb{R}^{d},\mu;(\mathbb{R}^{d},|\cdot|))$-norm satisfies
the $(r,s)$-(BI) for every $1<r^{\prime}\leq s\leq p^{\prime}\leq r$ and
Theorem 5.10. ∎

#### Data availability statement

Data sharing not applicable to this article as no datasets were generated or
analysed during the current study.

## References

  * [1] Luigi Ambrosio, Matthias Erbar, and Giuseppe Savaré.  Optimal transport, Cheeger energies and contractivity of dynamic transport distances in extended spaces.  Nonlinear Anal., 137:77–134, 2016.
  * [2] Luigi Ambrosio, Nicola Gigli, and Giuseppe Savaré.  Gradient flows in metric spaces and in the space of probability measures.  Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, second edition, 2008.
  * [3] Luigi Ambrosio, Nicola Gigli, and Giuseppe Savaré.  Density of Lipschitz functions and equivalence of weak gradients in metric measure spaces.  Rev. Mat. Iberoam., 29(3):969–996, 2013.
  * [4] Luigi Ambrosio, Nicola Gigli, and Giuseppe Savaré.  Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below.  Invent. Math., 195(2):289–391, 2014.
  * [5] Anders Björn and Jana Björn.  Nonlinear potential theory on metric spaces, volume 17 of EMS Tracts in Mathematics.  European Mathematical Society (EMS), Zürich, 2011.
  * [6] Ralph P. Boas.  Some uniformly convex spaces.  Bull. Amer. Math. Soc., 46:304–311, 1940.
  * [7] Jeff Cheeger.  Differentiability of Lipschitz functions on metric measure spaces.  Geom. Funct. Anal., 9(3):428–517, 1999.
  * [8] James A. Clarkson.  Uniformly convex spaces.  Trans. Amer. Math. Soc., 40(3):396–414, 1936.
  * [9] Jérémy Dalphin.  Some characterizations of a uniform ball property.  In Congrès SMAI 2013, volume 45 of ESAIM Proc. Surveys, pages 437–446. EDP Sci., Les Ulis, 2014.
  * [10] Marcel de Jeu and Jan Rozendaal.  Disintegration of positive isometric group representations on ${L}^{p}$-spaces.  Positivity, 21(2):673–710, 2017.
  * [11] Alessio Figalli and Nicola Gigli.  Local semiconvexity of Kantorovich potentials on non-compact manifolds.  ESAIM Control Optim. Calc. Var., 17(3):648–653, 2011.
  * [12] Massimo Fornasier, Giacomo E. Sodini, and Giuseppe Savaré.  Density of subalgebras of Lipschitz functions in metric Sobolev spaces and applications to Sobolev-Wasserstein spaces.  arXiv:2209.00974, 2022.
  * [13] Wilfrid Gangbo and Robert J. McCann.  The geometry of optimal transportation.  Acta Math., 177(2):113–161, 1996.
  * [14] Nicola Gigli.  On the differential structure of metric measure spaces and applications.  Mem. Amer. Math. Soc., 236(1113):vi+91, 2015.
  * [15] Jan-David Hardtke.  Some convexity properties in direct integrals and köthe-bochner spaces.  arXiv:1709.09532, 2017.
  * [16] Kazuo Hashimoto, Mikio Kato, and Yasuji Takahashi.  Generalized Clarkson’s inequalities for Lebesgue-Bochner spaces.  Bull. Kyushu Inst. Tech. Math. Natur. Sci., pages 15–25, 1996.
  * [17] R. Haydon, M. Levy, and Y. Raynaud.  Randomly normed spaces, volume 41 of Travaux en Cours [Works in Progress].  Hermann, Paris, 1991.
  * [18] Juha Heinonen and Pekka Koskela.  Quasiconformal maps in metric spaces with controlled geometry.  Acta Math., 181(1):1–61, 1998.
  * [19] Pekka Koskela and Paul MacManus.  Quasiconformal mappings and Sobolev spaces.  Studia Math., 131(1):1–17, 1998.
  * [20] Andreas Kriegl and Peter W. Michor.  The convenient setting of global analysis, volume 53 of Mathematical Surveys and Monographs.  American Mathematical Society, Providence, RI, 1997.
  * [21] John Lott and Cédric Villani.  Ricci curvature for metric-measure spaces via optimal transport.  ArXiv Mathematics e-prints, December 2004.
  * [22] Danka Lučić and Enrico Pasqualetto.  Gamma-convergence of Cheeger energies with respect to increasing distances.  J. Math. Anal. Appl., 515(1):Paper No. 126415, 10, 2022.
  * [23] Milica Lučić, Enrico Pasqualetto, and Ivana Vojnović.  On the reflexivity properties of Banach bundles and Banach modules.  arXiv:2205.11608, 2022.
  * [24] Filippo Santambrogio.  Optimal transport for applied mathematicians, volume 87 of Progress in Nonlinear Differential Equations and their Applications.  Birkhäuser/Springer, Cham, 2015.  Calculus of variations, PDEs, and modeling.
  * [25] Giuseppe Savaré.  Sobolev spaces in extended metric-measure spaces.  In New Trends on Analysis and Geometry in Metric Spaces, pages 117–276. Springer, 2022.
  * [26] Nageswari Shanmugalingam.  Newtonian spaces: an extension of Sobolev spaces to metric measure spaces.  Rev. Mat. Iberoamericana, 16(2):243–279, 2000.
  * [27] Karl-Theodor Sturm.  On the geometry of metric measure spaces. I.  Acta Math., 196(1):65–131, 2006.
  * [28] Karl-Theodor Sturm.  On the geometry of metric measure spaces. II.  Acta Math., 196(1):133–177, 2006.
  * [29] Cédric Villani.  Topics in optimal transportation, volume 58 of Graduate Studies in Mathematics.  American Mathematical Society, Providence, RI, 2003.
  * [30] Cédric Villani.  Optimal transport. Old and new, volume 338 of Grundlehren der Mathematischen Wissenschaften.  Springer-Verlag, Berlin, 2009.
  * [31] Zong Ben Xu and Gary F. Roach.  Characteristic inequalities of uniformly convex and uniformly smooth Banach spaces.  J. Math. Anal. Appl., 157(1):189–210, 1991.


# Phase Structure of Quantum Improved Schwarzschild-(Anti)de Sitter Black
Holes

Chiang-Mei Chen<EMAIL_ADDRESS>Department of Physics, National Central
University, Zhongli, Taoyuan 320317, Taiwan  Center for High Energy and High
Field Physics (CHiP), National Central University, Zhongli, Taoyuan 320317,
Taiwan  Yi Chen<EMAIL_ADDRESS>Department of Physics, National Central
University, Zhongli, Taoyuan 320317, Taiwan  Akihiro Ishibashi
<EMAIL_ADDRESS>Department of Physics, Kindai University, Higashi-
Osaka, Osaka 577-8502, Japan  Research Institute for Science and Technology,
Kindai University, Higashi-Osaka, Osaka 577-8502, Japan  Nobuyoshi Ohta
<EMAIL_ADDRESS>Department of Physics, National Central University, Zhongli,
Taoyuan 320317, Taiwan  Research Institute for Science and Technology, Kindai
University, Higashi-Osaka, Osaka 577-8502, Japan

###### Abstract

We study the phase structure of quantum improved Schwarzschild-(A)dS black
holes in asymptotically safe gravity. Our results confirm some of the well-
known properties of quantum black holes. For example, the quantum effect
provides a repulsive force in the core region near singularity which
stabilizes the thermodynamically unstable small black holes, and also creates
a zero temperature state with finite size. We suggest that this could be a
candidate for dark matter. We find a new second order phase transition between
small and large black holes for quantum improved Schwarzschild-Anti de Sitter
black holes. We also discuss the black holes with different spatial topologies
and find a notable duality.

##  I Introduction

The black holes are known to have thermodynamic properties including a
characteristic temperature and an intrinsic entropy [1]. The thermodynamics of
Schwarzschild black holes encounter unstable phase with negative heat
capacity. The phase structure becomes more fascinating when we take into
account a negative cosmological constant. For such Schwarzschild-Anti de
Sitter (SAdS) black holes, the temperature has a minimal value which
corresponds to the divergent point of heat capacity, giving the border line of
unstable small black holes with negative heat capacity and stable large black
holes with positive heat capacity. In particular, there is a first order
Hawking-Page (HP) transition [2] between large (stable) SAdS black holes and
thermal Anti de Sitter (AdS) at the critical temperature where both the black
hole and thermal AdS have the same free energy. Above this critical
temperature, the free energy of a large black hole is lower than the free
energy of thermal AdS of equal temperature, causing the decay of thermal AdS
into black holes. These thermodynamic aspects of black holes were originally
discovered by using quantum field theory on a classical background spacetime,
or by applying euclidean quantum gravity approach initiated in [3]. No quantum
effects of gravity are considered in these results. It is thus interesting to
revisit these issues in black hole thermodynamics, in particular the phase
structure, taking into account recent advances in quantum theory of gravity.

Quantum effects of gravity are, in general, expected to play a significant
role in the core region near the black hole singularity. This is indeed the
case for asymptotically safe quantum gravity [4, 5, 6, 7, 8]. This approach
considers the possibility of having an interacting quantum field theory of
gravitation, originated from a non-Gaussian ultraviolet (UV) fixed point in
the theory space, in the nonperturbative renormalization group (RG) framework.
The existence of such a fixed point would permit having an RG trajectory in
the theory space flowing to it and characterized by all the dimensionless
couplings remaining finite when the UV cutoff is removed. In this way, the UV
behavior of the theory becomes under control, and this provides a promising
way to quantum gravity. An important result in this case is that the coupling
“constants” depend on the energy scale along the RG flow. The essential issue
in this approach is a suitable choice of identification of the energy scale
with some length scale in the considered solutions. We call this quantum
improvement. This allows us to understand how the quantum effects manifest
themselves in the black hole geometry. Given a suitable identification, it
turns out that quantum effects provide a “repulsive” force at the core of
black hole which stabilizes the unstable phase of small black holes [9, 10].
For discussions of black holes in this context, see [11, 12, 13, 14] and
references therein.

The main purpose of this paper is to study the thermodynamic phase structure
of quantum improved static black holes in the asymptotically safe quantum
gravity. As the mass decreases, the temperature of small AdS black holes
reduces to zero, instead of rapidly increasing to infinity. Consequently, as
we will show, the quantum effects create a local maximal in all asymptotically
flat and (A)dS black holes. The small black holes become stable and, therefore
there exist finite size zero temperature remnants. We would like to suggest
that the Planck size cold remnants could be a candidate for dark matter.111
The possibility of “cold remnants” was already pointed out in Ref. [10]. In
the context of the primordial black holes (PBHs) as dark matter (see, e.g.,
Refs. [15, 16, 17] and references therein), only those with initial mass
larger than $10^{15}$g have so far been considered, since any PBHs with
smaller mass $\lesssim 10^{15}$g would have already evaporated away by now.
The asymptotic safe quantum improvement provides an interesting mechanism that
can make even much smaller PBHs free from Hawking evaporation, and thus create
stable black hole remnants. It is interesting to study the astrophysical
implications of such stable Planck size black hole remnants. In the asymptotic
safety there are a number of papers arguing that the gravitational effects
reduce the strength of interactions of fermions and scalars (not black hole
remnants), thereby making them candidates of dark matter, see, e.g. Refs. [18,
19].  Apart from that, it also raises another interesting question: Is there a
new phase transition, analogous to the HP transition, between stable quantum
improved small black holes and thermal AdS? In order to clarify this problem,
it is necessary to study the phase structure of quantum improved black holes.
We will find rich structure in the case of AdS spacetime. Note that the
importance of AdS black holes has been well recognized for some time in the
context of AdS/CFT correspondence [20], and also HP-type of phase transition,
so it is interesting to see how the results are modified by quantum effects
[21].

Here we address these questions and find that for the quantum improved SAdS
black holes, when the quantum effects are not significantly large, there is an
intermediate unstable phase between stable small and large black holes. Such
kind of sandwich structure indicates a possibility of the existence of a
second order phase transition from small to large black holes or vice versa.
The evidence of such a phase transition is the presence of a swallow-tail
pattern in free energy with respect to temperature. The phase structure of a
thermodynamical system is encoded in the associated free energy. Unfortunately
due to the lack of quantum action, we are not able to compute the free energy
via the partition function. However, by assuming the thermodynamic free energy
is still valid for quantum improved black holes, we can study the phase
structure of quantum improved SAdS black holes. We find that the analogous HP
transition can indeed occur only at unphysical negative temperature. Moreover,
a new interesting swallow-tail pattern does appear when the strength of
quantum effects are smaller than the critical value and the associated second
order phase transition can happen.

For the quantum improved asymptotically flat black holes, the heat capacity is
independent of their spatial topologies, and thus they all share similar phase
structures. When the cosmological constant is non-vanishing, we find that the
heat capacity is invariant under sign change of the cosmological constant
$\Lambda\to-\Lambda$ and the spatial topologies $\sigma\to-\sigma$. But, both
the mass and temperature flip their signs. This means, for example, the phase
structure of topological Schwarzschild-de Sitter (SdS) “black holes” is alike
those of SAdS black holes. However, for the topological SdS black holes the
mass is negative [22]. The solutions for $\sigma\neq 1$ and $\Lambda\geq 0$
correspond to cosmological solutions with positive mass parameter.

This paper is organized as follows. In the next section, we will briefly
describe how to quantum improved black hole geometries in the asymptotic
safety scenario, and discuss the first law of a quantum improved Kerr-(A)dS
black hole as an example. Then, in Sec. III, we will analyze quantum effects
on thermodynamic phase structure of asymptotically AdS static black holes with
different horizon topologies. We compute the Hawking temperature, heat
capacity, and free energy for quantum improved AdS black holes, and for each
type of horizon topology, we study in detail whether quantum improved AdS
black holes exhibit the HP type phase transition. We find that for the
spherical horizon case, a new type of second order phase transition can occur,
due to the effects of quantum improvement, whereas for the hyperbolic and the
planar horizon case, there are no phase transition analogous to the HP. We
perform similar analyses of quantum improved black holes with positive
cosmological constant in sec. IV and with vanishing cosmological constant in
sec. V. Section VI is devoted to summary and discussions. In Appendix A, we
briefly summarize how to compute free energy for SAdS black holes in general
relativity.

##  II Running Newton Coupling in Asymptotically Safe Gravity

Let us first summarize some relevant results here in the asymptotically safe
scenario for quantum generalization of the general relativity with
cosmological constant, in the units $c=\hbar=1,G_{0}=l_{P}^{2}=1/M_{P}^{2}$.
The action is given by

$S=-\frac{1}{16\pi G}\int d^{4}x\sqrt{-g}\left(R-2\Lambda\right).$ (1)

The asymptotically safe scenario proposes the Newton coupling $G(k)$ and
cosmological “constant” $\Lambda(k)$ are energy dependent [4]. Assuming the
cosmological constant is is sufficiently small, the associated renormalization
group equations lead to the solutions for Newton coupling [9, 24, 25, 26, 27]

$G(k)=\frac{G_{0}}{1+\omega G_{0}k^{2}},$ (2)

where $\omega>0$ is the inverse of the fixed point value. Actually the
“dimensionful” cosmological constant $\Lambda$ is given by [26, 12]

$\displaystyle\Lambda=\Lambda_{0}+\lambda_{*}k^{2},$ (3)

where $\lambda_{*}$ is the fixed point value of the “dimensionless”
cosmological constant and $\Lambda_{0}$ is the infrared value of the
cosmological constant. It might appear that $\Lambda$ increases for larger $k$
or near the singularity. However we are only concerned with the thermodynamic
behaviors of the black holes and the near-horizon area is the region we
consider. The typical value of $\lambda*$ is very small like $0.1$ [23]. So
our approximation can be justified.

How to identify the energy scale $k$ with a spacetime distance scale is an
essential issue for constructing quantum improved solutions [24]. See also
related work [28, 29, 30, 31, 32, 33]. The first law of black hole
thermodynamics, particularly for Kerr-(A)dS black holes, requires that a
consistent identification near the horizon should be a function of the horizon
area $A_{h}$ [34]. The same consequence can be derived with the relation of
entropy variation $\delta S=\delta A_{h}/4G$ [35] in which it is implicitly
presumed from the outset that the running coupling $G$ is a function of the
horizon area $A_{h}$. Therefore, a physically admissible identification for
energy scale is a function of area with given radius. According to dimensional
counting, a naturally suggested scale identification for Kerr-(A)dS black
holes at horizon is

$k(r_{h})=\xi\sqrt{\frac{1+\Lambda a^{2}/3}{r_{h}^{2}+a^{2}}},$ (4)

with a dimensionless parameter $\xi$. Strictly speaking, we may have
$r_{h}$-dependence in the cosmological constant, but we neglect this because
we are interested only in the case of small cosmological constant. This
identification is the simplest and natural one without introducing additional
dimensional parameters. The resulting running Newton coupling is

$G(r_{h})=\frac{G_{0}(r_{h}^{2}+a^{2})}{r_{h}^{2}+a^{2}+\tilde{\omega}G_{0}(1+\Lambda
a^{2}/3)},\qquad\tilde{\omega}=\xi^{2}\omega.$ (5)

The black hole entropy can be computed by integrating the first law of
thermodynamics which, up to an integrating constant $S_{0}$, gives

$S=\frac{\pi(r_{h}^{2}+a^{2})}{G_{0}(1+\Lambda
a^{2}/3)}+\pi\tilde{\omega}\ln\frac{\pi(r_{h}^{2}+a^{2})}{G_{0}(1+\Lambda
a^{2}/3)}+S_{0}.$ (6)

The temperature determined by the surface gravity $\kappa$ is

$T_{\mathrm{H}}=\frac{\kappa}{2\pi}=\frac{-\Lambda r_{h}^{4}+\left[1-\Lambda
a^{2}/3-\tilde{\omega}G_{0}(1+\Lambda
a^{2}/3)\Lambda/3\right]r_{h}^{2}-a^{2}-\tilde{\omega}G_{0}(1+\Lambda
a^{2}/3)}{4\pi r_{h}\left[r_{h}^{2}+a^{2}+\tilde{\omega}G_{0}(1+\Lambda
a^{2}/3)\right]}.$ (7)

The quantum effects enlarge the size of extremal (zero temperature) state
which can still exist in the static black holes with finite mass. The stable
extremal black holes could be a candidate of dark matter. For non-rotating
black holes $a=0$, the vanishing condition of the temperature (7) gives the
extremal limit with

$r_{\mathrm{zero}}^{2}=\frac{1-\tilde{\omega}G_{0}\Lambda/3\pm\sqrt{(1-\tilde{\omega}G_{0}\Lambda/3)^{2}-4\tilde{\omega}G_{0}\Lambda}}{2\Lambda},$
(8)

where $\pm$ corresponds to dS ($\Lambda>0$)/AdS ($\Lambda<0$) and
$r_{\mathrm{zero}}^{2}=\tilde{\omega}G_{0}$ for $\Lambda=0$.

##  III Asymptotically Anti-de Sitter Black Holes

In this section we are going to analyze the quantum effects on the
thermodynamical phase structure of the asymptotically AdS black holes for
$a=0$ with different horizon topologies, characterized by the unit sectional
curvature $\sigma$ of the horizon manifold, i.e. sphere ($\sigma=1$), flat
($\sigma=0$) and hyperbola ($\sigma=-1$):

$ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega_{\sigma}^{2},\qquad
f(r)=\sigma-\frac{2G(r)M}{r}+\frac{r^{2}}{L^{2}}.$ (9)

(We also refer to the hyperbola ($\sigma=-1$) case as the topological black
hole.) We also denote the AdS radius-squared as

$\displaystyle L^{2}=-3/\Lambda.$ (10)

The identification (4) and (5) for non-rotating black holes ($a=0$) reduces to

$k(r_{h})=\frac{\xi}{r_{h}},\qquad
G(r_{h})=\frac{G_{0}r_{h}^{2}}{r_{h}^{2}+\tilde{\omega}G_{0}}.$ (11)

Note that for the static black hole metric (9), the horizon area is given
merely in terms of the radial coordinate $r$, different from the Kerr-(A)dS
case, and therefore any scale identification of the type $k=\tilde{\xi}/r^{p}$
with $p>0$ examined in [27] (with suitable dimension for $\tilde{\xi}$) is
consistent with the black hole first-law discussed in [34]. Then, depending on
the power $p$, the quantum improved geometry near the center $r=0$ will
differ. In particular, for the identification (5) with $a=0$, for which
$k\sim\xi/r$, the metric function behaves
$f(r)\sim\sigma-(2M/\tilde{\omega})r+O(r^{2})$ near the center and therefore
the resultant quantum improved geometry admits a weak singularity at the
center. However, since the temperature and heat capacity are computed at the
horizon and also the euclidean path-integral for the free energy is performed
outside the horizon as shown in Appendix A, the behavior of the quantum
improved geometry (i.e., whether it is regular or not) near the center does
not appear to affect much the evaluation of these thermodynamic quantities.
For this reason, we will stick to the scale identification (5), which is the
simplest and natural choice without any additional dimensional parameters, in
the rest of this paper, despite possible more general dependence.

The location of the horizon $f(r_{h})=0$ gives the mass as

$M=\frac{(\sigma+r_{h}^{2}/L^{2})(r_{h}^{2}+\tilde{\omega}G_{0})}{2G_{0}r_{h}}.$
(12)

We see from (3) and (11) that there is actually some dependence on the horizon
radius $r_{h}$ in the cosmological constant, but this could be neglected for
small $\lambda_{*}$ unless we consider very small $r_{h}$ region. Moreover the
dependence on the horizon radius $r_{h}$ is smaller for larger $r_{h}$ in
which we will be interested.

The mass parameter $M$ is always positive for both $\sigma=1$ and $\sigma=0$.
However, for $\sigma=-1$ and the “horizon” radius smaller than AdS radius,
i.e. $r_{h}<L$, the mass parameter is negative. It is straightforward to
compute the Hawking temperature

$T_{\mathrm{H}}=\frac{3r_{h}^{4}/L^{2}+(\sigma+\tilde{\omega}G_{0}/L^{2})r_{h}^{2}-\sigma\tilde{\omega}G_{0}}{4\pi
r_{h}(r_{h}^{2}+\tilde{\omega}G_{0})}=\frac{\sigma+3r_{h}^{2}/L^{2}}{4\pi
r_{h}}-\frac{\tilde{\omega}G_{0}(\sigma+r_{h}^{2}/L^{2})}{2\pi
r_{h}(r_{h}^{2}+\tilde{\omega}G_{0})},$ (13)

and the heat capacity

$C=\frac{2\pi(r_{h}^{2}+\tilde{\omega}G_{0})^{2}\left[3r_{h}^{4}+(\sigma
L^{2}+\tilde{\omega}G_{0})r_{h}^{2}-\sigma
L^{2}\tilde{\omega}G_{0}\right]}{G_{0}\left[3r_{h}^{6}-(\sigma
L^{2}-8\tilde{\omega}G_{0})r_{h}^{4}+\tilde{\omega}G_{0}(4\sigma
L^{2}+\tilde{\omega}G_{0})r_{h}^{2}+\sigma
L^{2}\tilde{\omega}^{2}G_{0}^{2}\right]}.$ (14)

The heat capacity, temperature and mass parameter have an interesting duality
under transformation $L^{2}\to-L^{2}$ and $\sigma\to-\sigma$

$C\to C,\qquad T\to-T,\qquad M\to-M.$ (15)

For studying the black holes thermodynamics, it is convenient to use the
dimensionless variable $\rho$

$\rho\equiv\frac{r_{h}^{2}}{L^{2}}\geq 0,$ (16)

and dimensionless parameter $\zeta$

$\zeta\equiv\frac{\tilde{\omega}G_{0}}{L^{2}}\geq 0.$ (17)

The divergent points of heat capacity play important role in studying the
thermodynamic phase structure. This happens when the denominator of Eq. (14)
vanishes. This leads to a cubic equation

$d_{\sigma}(\rho):=3\rho^{3}-(\sigma-8\zeta)\rho^{2}+\zeta(4\sigma+\zeta)\rho+\sigma\zeta^{2}=0,$
(18)

and its discriminant is

$D_{\sigma}:=4\zeta^{2}(\zeta-\sigma)(13\zeta^{3}-303\sigma\zeta^{2}+327\sigma^{2}\zeta-5\sigma^{3}).$
(19)

In the following we will discuss the phase structure for black holes with
different spatial topology.

###  III.1 $\sigma=1$

For the typical black holes with $\sigma=1$, it is straightforward to plot the
mass (12) with respect to the horizon radius and the temperature (13) and the
heat capacity (14) with respect to the mass parameter. These are given in
Figs. 1 (a)–(c) for different choice of $\tilde{\omega}$. We note that these
figures are drawn for the fixed cosmological constant $L^{2}=10$, i.e.
$|\Lambda|=0.3$. Strictly speaking it depends on the horizon radius $r_{h}$.
However, since typically $\lambda_{*}\sim 0.1$, our approximation is valid at
least for $r_{h}$ larger than 1, and better for larger $r_{h}$. Similar
remarks apply to other figures.

We see from Fig. 1 (a) that the quantum improved black holes have two horizons
for the mass parameter greater than a lower bound which corresponds to
extremal zero temperature state $T_{\mathrm{H}}=0$. The radius of zero
temperature state is, from (8),

$r_{\mathrm{zero}}^{2}=\frac{\sqrt{L^{4}+14\tilde{\omega}G_{0}L^{2}+\tilde{\omega}^{2}G_{0}^{2}}-L^{2}-\tilde{\omega}G_{0}}{6},$
(20)

and the associated entropy is

$S_{\mathrm{BH}}=\frac{\pi
r_{\mathrm{zero}}^{2}}{G_{0}}+\pi\tilde{\omega}\ln\frac{\pi
r_{\mathrm{zero}}^{2}}{G_{0}}+\pi\tilde{\omega}S_{0}=\frac{\pi
r_{\mathrm{zero}}^{2}}{G_{0}}\left(1+\frac{\tilde{\omega}G_{0}}{r_{\mathrm{zero}}^{2}}\ln\frac{\pi
r_{\mathrm{zero}}^{2}}{G_{0}}+\frac{\tilde{\omega}G_{0}}{r_{\mathrm{zero}}^{2}}S_{0}\right).$
(21)

The logarithmic term is similar to the one obtained as quantum corrections
[36] and our result is reasonable for the quantum case. Similar result is also
obtained in loop quantum gravity [37].

Figure 1: (a) The horizon radius, (b) Hawking temperature, (c) heat capacity
and (d) free energy of quantum improved SAdS black holes with
$G_{0}=1,L^{2}=10$. The red dash line represents the classic result without
quantum improvement. The blue solid line corresponds to
$\tilde{\omega}=0.1\,(\zeta=0.01)$ in which there is a unstable phase with
negative heat capacity. The green and purple solid lines correspond to
$\tilde{\omega}=0.2,1\,(\zeta=0.02,0.1)$ in which the phase is all stable.

In Fig. 1 (c), one can observe that the existence of $\tilde{\omega}$ creates
a thermodynamically “stable” (not run away) phase with positive heat capacity
in the small $M$ region. Consequently there are two interesting results: (i)
The quantum improved black holes can have a zero temperature state with finite
mass which could give a candidate for dark matter. (ii) Instead of the HP
transition between large black holes and AdS thermal states, quantum effects
may create new kind, either first order or second order, of phase transitions.

In GR, there is a first order HP phase transition in the SAdS black holes. It
is interesting to check how the quantum effects modify the “original” phase
transition or even generate new kinds of phase transition. The number of
divergent points of heat capacity is closely related to the phase structure of
thermodynamics. The thermodynamic phase structure is determined by the
divergent points of heat capacity (18), i.e. the roots of the following cubic
equation for dimensionless variable $\rho$:

$d_{+}(\rho):=3\rho^{3}-(1-8\zeta)\rho^{2}+\zeta(4+\zeta)\rho+\zeta^{2}=0,$
(22)

and its discriminant is

$D_{+}:=4\zeta^{2}(\zeta-1)(13\zeta^{3}-303\zeta^{2}+327\zeta-5).$ (23)

There are 5 real roots of $D_{+}=0$, numerically given as

$\zeta_{0}=0,\quad\zeta_{1}=0.01551337272,\quad\zeta_{2}=1,\quad\zeta_{3}=1.118084244,\quad\zeta_{4}=22.17409469.$
(24)

From the sign of the last term of Eq. (22), we see that the product of the 3
roots is negative, so Eq. (22) always has at least one negative root. For
$\zeta$ in the range $\zeta_{0}<\zeta<\zeta_{1},\zeta_{2}<\zeta<\zeta_{3}$ and
$\zeta_{4}<\zeta$, the discriminant (23) is positive, and we have 3 real roots
of Eq. (22). From the derivative of Eq. (22)

$d_{+}(\rho)^{\prime}=9\rho^{2}-2(1-8\zeta)\rho+\zeta(4+\zeta),$ (25)

we see that this is positive for $\zeta>\frac{1}{8}$ and positive $\rho$. This
means that $d_{+}(\rho)$ monotonically increases for positive $\rho$, and
together with $d_{+}(0)>0$, there is no positive root for $\zeta>\frac{1}{8}$.
For $\zeta_{1}<\zeta\leq\frac{1}{8}$, where the discriminant (23) is negative,
there are two complex and one negative roots. We then find that there are two
positive roots for $\rho$ only when the value of parameter $\zeta$ is in the
region $0<\zeta<\zeta_{1}$ (the region where the heat capacity is negative)
and a degenerate positive root at $\zeta=\zeta_{1}$ (region of negative heat
capacity shrinking to a point).

When the quantum effects are small ($\zeta<\zeta_{1}$), the two positive roots
of $\rho$, corresponding to two real and positive roots of $r_{h}$, are the
locations of local maximum and minimum of the temperature in accord with the
divergent points of heat capacity. In between the two divergent points, the
thermal black hole system is unstable with negative heat capacity. This case
is depicted by blue solid lines, together with other values of $\zeta$, in
Figs. 1. The physical picture of SAdS black holes is the following: In GR, the
black hole temperature has a minimum corresponding to the divergent point
separating negative (unstable small black hole states) and positive (stable
large black hole states) regions of heat capacity. Moreover, there is a first
order HP phase transition between a stable large black hole phase and thermal
AdS state. The quantum effects provide a “repulsive” force and generates
stable phases in small mass regions. This gives two possible new phase
transitions: (i) a first order phase transition (analogy of the HP transition
in classical solutions) between small black holes and AdS thermal state, (ii)
a second order phase transition in between stable small and large black holes.

However, as the quantum effects become stronger, two divergent points become
closer and degenerate at the critical value $\zeta=\zeta_{1}$, and beyond it
the heat capacity will become smooth and the corresponding black hole thermal
system is always stable (green solid lines in Fig. 1).

In order to study the details of phase transition, we have to compute the free
energy. In GR, as discussed in Appendix A, the difference in the free energies
of SAdS black holes and AdS thermal state can be computed either by saddle
approximation of partition function or directly by thermodynamic relation
$F=E-TS$. Since we do not have quantum action for asymptotically safe gravity
to compute the associated partition function, we simply use the second
relation $F=E-TS$ to calculate the difference in the free energies of quantum
improved SAdS black holes and thermal AdS state.

The result is depicted in Fig. 1 (d) as a function of the temperature. We find
a swallow-tail pattern. Such a pattern exists within the range of the
dimensionless parameter $0<\zeta<\zeta_{1}$ in which the temperature can have
both the local maxima and minima. A second order phase transition between the
small and large black holes can occur at the intersecting point. However, when
$\zeta$ is larger than the critical value $\zeta_{1}$, both the local maximal
and minimal temperatures disappear and all phases are stable. A similar
swallow-tail pattern was observed in the asymptotically flat black holes [38]
by using the identification proposed in [10].

If the first order transition of the quantum improved small black holes and
AdS thermal state were to occur, it could occur only at unphysical negative
temperature because the free energy is positive when temperature is smaller
than HP temperature and is still positive at $T=0$ so the free energy can have
only one root (“original” HP) in positive $T$ region. Indeed, if such kind of
phase transition occurred at $T>0$, then the free energy should be negative at
zero temperature corresponding to $r_{h}=r_{\mathrm{zero}}$. However, from
(12) and (20), it is obvious that
$F(r_{h}=r_{\mathrm{zero}})=M(r_{h}=r_{\mathrm{zero}})$ is positive. Thus, no
HP like first order phase transitions (in addition to the original HP
transition) could be generated for $T>0$. Mathematically, the free energy can
be negative at both small and large $r_{h}$. For small $r_{h}$ the free energy
becomes negative only at negative temperature region, i.e.
$r_{h}<r_{\mathrm{zero}}$ which is unphysical. For large $r_{h}$ the quantum
effects “reduce” the critical temperature of the “classical” HP transition,
see also Fig. 1 (d). However, it is hard to find the explicit expression for
critical temperature due to the logarithmic term in the entropy.

###  III.2 $\sigma=-1$

For topological black holes ($\sigma=-1$) in GR without quantum effects
($\tilde{\omega}=0$), we depict the mass (12) with respect to the horizon
radius and the temperature (13) and heat capacity (14) with respect to the
mass parameter in Fig. 2 by red dashed lines. We find that there is only one
horizon for positive $M$, and two horizons for negative $M$ larger than the
lower bound, as can be seen from Fig. 2 (a). The extremal limit can be found
by imposing the degenerate condition on $f(r)$ in (9) with constant
$G(r)=G_{0}$:

$r_{h}=\frac{L}{\sqrt{3}},\qquad M=-\frac{L}{3\sqrt{3}G_{0}},$ (26)

which gives the lower bound of $M$ of black hole solutions. Here the
temperature is zero.

Including the quantum effects, the allowed mass parameter is from $M=-\infty$
($r_{h}=0$) to $M=\infty$ ($r_{h}=\infty$). The temperature in Fig. 2 (b) is
plotted for full range of horizon radius $r_{h}$ in order to illustrate the
duality (15) by comparing with the plots of quantum improved de Sitter black
holes. Thus the dot parts in Fig. 2 (b) and Fig. 2 (c) are the results of the
“inner” horizon.

In GR, the topological AdS black holes are always stable. For the quantum
improved black holes, the present discriminant $D_{-}$ [obtained by setting
$\sigma=-1$ in Eq. (19)] is always positive, so there are three real roots of
$\rho$ for the equation $d_{-}(\rho)=0$. Moreover, the derivative of
$d_{-}(\rho)$ has two roots and the smaller one is always negative which
corresponds to a local maximum. Together with the fact that
$d_{-}(-\infty)<0,\,d_{-}(0)<0$ and $d_{-}(\infty)>0$, this means that the two
roots must be negative and one is positive. The heat capacity has only one
divergent point which separates the stable and unstable phases. The
temperature has only a local minimum corresponding to the divergent point of
heat capacity, as shown in Fig. 2 (b). If the quantum effect is small, the
unstable phase is located inside the horizon, but if the quantum effect is
large enough, the small quantum improved black holes, with more negative mass
parameter, can be unstable.

For the topological AdS black holes, the hyperbolic surface with unit radius,
defined by $x^{2}+y^{2}-z^{2}=-1$, has infinite area given by
$\Omega_{-}=2\pi\int_{0}^{\infty}\sinh\nu d\nu$. The total energy and entropy
are infinite large quantities. Thus we should consider their “densities” as
(the integration constant term in the entropy vanishes because it is divided
by the infinite area)

$\tilde{E}=\frac{4\pi}{\Omega_{-}}E=M=\frac{(-1+r_{h}^{2}/L^{2})(r_{h}^{2}+\tilde{\omega}G_{0})}{2G_{0}r_{h}},\qquad\tilde{S}=\frac{4\pi}{\Omega_{-}}S=\frac{\pi
r_{h}^{2}}{G_{0}}+\pi\tilde{\omega}\ln\frac{r_{h}^{2}}{4G_{0}}.$ (27)

The last term is the logarithmic corrections from the quantum effects and our
result is again reasonable for the quantum case. Then the free energy density
is

$\tilde{F}=\frac{4\pi}{\Omega_{-}}F=\tilde{E}-T\tilde{S}.$ (28)

From the free energy density plotted in Fig. 2 (d), we find that there is no
analogous HP transition.

Figure 2: (a) The horizon radius, (b) Hawking temperature, (c) heat capacity
and (d) free energy density for quantum improved “topological SAdS black
holes” ($\sigma=-1$) with $G_{0}=1,L^{2}=10$. The red dash line represents
classic result without quantum improvement. The blue, green and purple solid
lines correspond to $\tilde{\omega}=0.1,0.2,1\,(\zeta=0.01,0.02,0.1$).

###  III.3 $\sigma=0$

For the planar black holes ($\sigma=0$), we plot the mass, temperature and
heat capacity in Fig. 3. They are all positive, and the heat capacity is never
divergent. Thus the quantum improved planar AdS black holes do not change the
stable nature in GR. The phase structure does not have an obvious change due
to the quantum effects.

Figure 3: (a) The Hawking temperature and (b) heat capacity for quantum
improved “planar SAdS black holes” ($\sigma=0$) with $G_{0}=1,L^{2}=10$. The
red dash line represents classic result without quantum improvement. The blue,
green and purple solid lines correspond to
$\tilde{\omega}=0.1,0.2,1\,(\zeta=0.01,0.02,0.1$).

##  IV Asymptotically de Sitter Black Holes

The thermodynamic quantities for de Sitter black holes can be obtained simply
from the corresponding results in Anti-de Sitter cases by replacing
$L^{2}\to-L^{2}$, namely

$\displaystyle M=\frac{(\sigma-
r_{h}^{2}/L^{2})(r_{h}^{2}+\tilde{\omega}G_{0})}{2G_{0}r_{h}},\qquad
T_{\mathrm{H}}=\frac{-3r_{h}^{4}/L^{2}+(\sigma-\tilde{\omega}G_{0}/L^{2})r_{h}^{2}-\sigma\tilde{\omega}G_{0}}{4\pi
r_{h}(r_{h}^{2}+\tilde{\omega}G_{0})},$ $\displaystyle
C=\frac{2\pi(r_{h}^{2}+\tilde{\omega}G_{0})^{2}\left[3r_{h}^{4}-(\sigma
L^{2}-\tilde{\omega}G_{0})r_{h}^{2}+\sigma
L^{2}\tilde{\omega}G_{0}\right]}{G_{0}\left[3r_{h}^{6}+(\sigma
L^{2}+8\tilde{\omega}G_{0})r_{h}^{4}-\tilde{\omega}G_{0}(4\sigma
L^{2}-\tilde{\omega}G_{0})r_{h}^{2}-\sigma
L^{2}\tilde{\omega}^{2}G_{0}^{2}\right]}.$ (29)

Let us discuss the thermodynamic properties for three different $\sigma$
separately.

###  IV.1 $\sigma=1$

According to the duality, it is not a surprise that the figures of
thermodynamic quantities in Figs. 4 are a kind of “reflection” of the related
quantities in Figs. 2 for topological AdS black holes.

In GR, the mass of the SdS black holes has an upper bound corresponding to the
degeneracy of the black hole and cosmological horizons. Only when the black
holes horizon radius is not larger than the radius of cosmological horizon,
the Hawking temperature is positive and physical.

With quantum effects, there is another zero temperature state whose radius is
given by (20) with $L^{2}$ replaced by $-L^{2}$. Only the black hole mass
radius is in between this zero temperature radius and cosmological radius, the
Hawking temperature is positive, as shown by the solid lines in Fig. 4 (b). If
quantum effects are large enough, there are no solutions with positive Hawking
temperature.

In GR, the de Sitter black holes is unstable with negative heat capacity. With
the quantum effects, the small black holes can be stable, see Fig. 4 (c).
There are no phase transitions in de Sitter black holes, as can be seen from
Fig. 4 (d).

###  IV.2 $\sigma=-1$ and $\sigma=0$

For the cases $\sigma=-1$ and $\sigma=0$ of asymptotically dS, there is no
“horizon” within positive value of $M$. Moreover, the lapse function

$f(r)=\sigma-\frac{2G(r)M}{r}-\frac{r^{2}}{L^{2}},$ (30)

is always negative. This means that the $r$ is actually the time coordinate in
these cases which would give cosmological solutions. Therefore, the black
holes can exist if the mass parameter is negative. However, $\sigma=-1$
solution is “dual” to anti-de Sitter black holes with $\sigma=1$. From Fig. 4,
according to the duality (15), we see that the solutions with negative mass
have negative Hawking temperature.

Figure 4: (a) The horizon radius, (b) Hawking temperature, (c) heat capacity
and (d) free energy for quantum improved SdS black holes with
$G_{0}=1,L^{2}=10$. The red dash line represents classic result without
quantum improvement. The blue, green and purple solid lines correspond to
$\tilde{\omega}=0.1,0.2,1\,(\zeta=0.01,0.02,0.1$).

##  V Asymptotically Flat Black Holes

For this case, the heat capacity reduces to

$C=-\frac{2\pi(r_{h}^{2}+\tilde{\omega}G_{0})^{2}(r_{h}^{2}-\tilde{\omega}G_{0})}{G_{0}(r_{h}^{4}-4\tilde{\omega}G_{0}r_{h}^{2}-\tilde{\omega}^{2}G_{0}^{2})},$
(31)

which does not depend on the sign of $\sigma$. The divergent point of the heat
capacity is then determined by the equation:

$r_{h}^{4}-4\tilde{\omega}G_{0}r_{h}^{2}-\tilde{\omega}^{2}G_{0}^{2}=0.$ (32)

The only positive root is $r_{h}^{2}=(2+\sqrt{5})\tilde{\omega}G_{0}$.

The Schwarzschild black holes ($\sigma=1$) are always unstable in GR, and Fig.
5 shows that the quantum effects stabilize the small black holes. For
$\sigma=-1$, the black holes solutions may exist for negative mass parameter.
However, according to the duality, and the Fig. 5 (a), the associated Hawking
temperature is negative. For $\sigma=0$, there are no black hole solutions.

This exhausts the quantum gravity effects in all possible cases.

Figure 5:  (a) The Hawking temperature and (b) heat capacity for quantum
improved Schwarzschild black holes with $G_{0}=1$. The red dash line
represents classic result without quantum improvement. The blue, green and
purple solid lines correspond to
$\tilde{\omega}=0.1,0.2,1\,(\zeta=0.01,0.02,0.1$).

##  VI Conclusion

We have studied the quantum effects of gravity on various black holes with and
without cosmological constant. Our findings can be summarized as follows.

  * •

For SAdS black holes, there always exists one and only one divergent point in
GR (boundary of stable and unstable phases). Quantum improvement, with small
quantum-effect parameter $\omega$, increases one more divergent point and
creates a new stable phase in the small mass region (consequently generates a
new zero temperature state with finite size). If $\omega$ increases, the
“distance” (in mass coordinate) between two divergent points reduces, and when
$\omega$ is larger than the critical value, both divergent points and unstable
phase disappear, see Fig. 1 (b). If $\omega$ is not too big, the black holes
can have a second order phase transition.

  * •

For the topological AdS black holes, the mass parameter can be negative (black
holes with negative parameter had been discussed in [22, 39]). The system is
thermodynamically stable if the quantum effects are small. Otherwise, the
small black holes become unstable when the quantum effects are big enough.

  * •

For the planar AdS black holes ($\sigma=0$), since the black holes is always
stable in GR, the “repulsive” effects does not give an obvious change.

  * •

For the SdS black holes, there are no divergent point in GR (only unstable
phase exists). The quantum improvement again creates a new stable phase in
small mass region which then also generates a divergent point. If the quantum
improvement is big enough, there are no black hole solution with positive
Hawking temperature, see Fig. 4 (b).

  * •

There is a duality in black hole thermodynamics. The heat capacity is
invariant under the transformation $L^{2}\to-L^{2}$ and $\sigma\to-\sigma$,
but the mass and temperature change sign. The topological SdS black holes
share various properties with SAdS black holes.

As listed above, we found that quantum improved Schwarzschild black holes with
negative cosmological constant have a richer phase structure than their
classical counter-parts, as well as those with positive and vanishing
cosmological constant. For this, we have however assumed that the
thermodynamic free energy is valid for the black holes quantum improved at the
solution level. It would be interesting to investigate whether one can also
compute the free energy by considering quantum improvement of the reduced
action for euclidean black holes. In this respect, once such an action for the
quantum improvement is formulated, it would also be interesting to study the
phase structure of quantum improved AdS black holes from the perspective of
the AdS/CFT correspondence.

In the cosmological context, we are more concerned with quantum improved
Schwarzschild black holes with positive or vanishing cosmological constant. In
general, primordial black holes of mass $\sim 10^{14}$g produced in the early
universe are expected to be undergoing now their final stages of the Hawking
evaporation. However, once the quantum effects are turned on, the evaporating
process would stop at some point where their horizon size is still finite, but
the Hawking temperature is vanishing as can be seen in Fig. 4 and 5. Thus,
such primordial black holes eventually settle down to thermodynamically
stable, zero-temperature remnants, rather than being completely evaporated
away. Although they may typically be Planck-size objects, such remnants might
possibly be a candidate of dark matter. It is worth pursuing such an
intriguing possibility.

In this paper, we have chosen the simplest identification (4) to study the
phase structure of the black holes. In this case, we find that the singularity
at the center of the black holes may not be resolved. Unfortunately, in the
Kerr black holes, the consistency of the thermodynamics requires that the
scale $k$ should be a function of the horizon area, and naive extension of
this away from the horizon does not allow such singularity resolution [34].
However, it is also possible to consider the possibility to include the mass
parameter of the black hole in the identification. Indeed, with such
dependence, it is possible that the singularity may be resolved [9]. It would
be interesting to investigate what restriction emerges when we allow the mass
parameter in the identification, and whether we may get the possibility that
the singularity may be resolved.

###### Acknowledgements.

The work of C.M.C. was supported by the National Science and Technology
Council of the R.O.C. (Taiwan) under the grants NSTC 111-2112-M-008-012. The
work of A.I. was supported in part by JSPS KAKENHI Grants No. 21H05182,
21H05186, 20K03938, 20K03975, 17K05451, and 15K05092. The work of N.O. was
supported in part by the Grant-in-Aid for Scientific Research Fund of the JSPS
(C) No. 16K05331, 20K03980, and Taiwan NSTC 111-2811-M-008-024.

##  Appendix A Free Energy and Hawking-Page Transition

For the classical SAdS black holes, the free energy can be computed from the
action. In $(d+1)$-dimensional spacetimes, the action is
($\Lambda=-d(d-1)/2L^{2}$)

$S=-\frac{1}{16\pi
G}\int_{\mathcal{M}}d^{d+1}x\sqrt{-g}\left(R+\frac{d(d-1)}{L^{2}}\right)+\frac{1}{8\pi
G}\int_{\partial\mathcal{M}}d^{d}x\sqrt{-\gamma}\left(-K+\frac{d-1}{L}+\cdots\right),$
(33)

where $K$ is the trace of the extrinsic curvature of the boundary
$\partial\mathcal{M}$. For the SAdS black holes

$ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega_{d-1}^{2},\qquad
f=1-\frac{2G_{0}M}{r^{d-2}}+\frac{r^{2}}{L^{2}},$ (34)

the horizon of black holes is located at $2GM=r_{h}^{d-2}(1+r_{h}^{2}/L^{2})$
and the associated temperature is

$T=\frac{f^{\prime}(r_{h})}{4\pi}=\frac{d-2}{4\pi r_{h}}+\frac{dr_{h}}{4\pi
L^{2}}=\frac{dr_{h}^{2}+(d-2)L^{2}}{4\pi r_{h}L^{2}}.$ (35)

There is a minimal value of temperature when
$r_{h}=r_{\mathrm{min}}=\sqrt{\frac{d-2}{d}}\,L$:

$T_{\mathrm{min}}=\frac{\sqrt{d(d-2)}}{2\pi L}.$ (36)

The free energy in the saddle point approximation is
$F=-T\ln\mathcal{Z}\approx T\,S_{\textrm{on shell}}^{E}$. The Euclidean on-
shell action, $S_{\textrm{on shell}}^{E}$ for SAdS black hole and thermal AdS
are obtained by the integration outside the horizon in the full spacetime with
$R=\frac{2(d+1)}{d-1}\Lambda$:

$\displaystyle S_{\mathrm{SAdS}}$ $\displaystyle=$ $\displaystyle\frac{d}{8\pi
GL^{2}}\int_{0}^{\beta}d\tau\int
d\Omega_{d-1}\int_{r_{h}}^{r_{\infty}}r^{d-1}dr=\frac{\beta\Omega_{d-1}(r_{\infty}^{d}-r_{h}^{d})}{8\pi
GL^{2}},$ (37) $\displaystyle S_{\mathrm{AdS}}$ $\displaystyle=$
$\displaystyle\frac{d}{8\pi GL^{2}}\int_{0}^{\beta_{0}}d\tau\int
d\Omega_{d-1}\int_{0}^{r_{\infty}}r^{d-1}dr=\frac{\beta_{0}\Omega_{d-1}r_{\infty}^{d}}{8\pi
GL^{2}}.$ (38)

By matching the proper temperature $T(r)=T/\sqrt{-g_{tt}}$ on the horizon, we
have

$\beta\sqrt{f}=\beta_{0}\sqrt{f_{0}}\quad\rightarrow\quad\beta_{0}=\beta\sqrt{f/f_{0}}.$
(39)

Using this relation, we can obtain the difference in the free energies:

$\displaystyle\Delta F$ $\displaystyle=$
$\displaystyle\frac{S_{\mathrm{SAdS}}-S_{\mathrm{AdS}}}{\beta}=\frac{\Omega_{d-1}}{8\pi
GL^{2}}\left(r_{\infty}^{d}-r_{h}^{d}-r_{\infty}^{d}\sqrt{\frac{f(r_{\infty})}{f_{0}(r_{\infty})}}\right)$
(40) $\displaystyle\approx$ $\displaystyle\frac{\Omega_{d-1}}{8\pi
GL^{2}}\left[r_{\infty}^{d}-r_{h}^{d}-r_{\infty}^{d}\left(1-\frac{GML^{2}}{r_{\infty}^{d}}\right)\right]=-\frac{\Omega_{d-1}r_{h}^{d-2}(r_{h}^{2}-L^{2})}{16\pi
GL^{2}}.$

The critical temperature for the HP transition corresponding to the point with
zero free energy difference $r_{h}=r_{\mathrm{HP}}=L>r_{\mathrm{min}}$ is

$T_{\mathrm{HP}}=\frac{d-1}{2\pi L}.$ (41)

There is a first order phase transition, namely HP transition, indicating a
phase change from AdS thermal spacetime (higher free energy) to large SAdS
black hole (lower free energy) for $T>T_{\mathrm{HP}}$.

Lets consider another approach to deriving the free energy via the black hole
thermodynamics. The related thermodynamic quantities, namely energy and
entropy are given by

$E=\frac{(d-1)\Omega_{d-1}}{8\pi}M,\qquad
S=\frac{\Omega_{d-1}r_{h}^{d-1}}{4G}.$ (42)

According to the first law

$\delta E=T\delta S,$ (43)

the “enthalpy” [40] $E$ is identical with the internal energy $U=E$. Therefore
the Helmholtz free energy (identical with the Gibbs function) for SAdS black
holes is

$F=U-TS=E-TS=-\frac{\Omega_{d-1}r_{h}^{d-2}}{16\pi GL^{2}}(r_{h}^{2}-L^{2}).$
(44)

This is exactly the $\Delta F$ in (40).

## References

  * [1] R. M. Wald, “The thermodynamics of black holes,” Living Rev. Rel. 4 (2001), 6 doi:10.12942/lrr-2001-6 [arXiv:gr-qc/9912119 [gr-qc]].
  * [2] S. W. Hawking and D. N. Page, “Thermodynamics of Black Holes in anti-De Sitter Space,” Commun. Math. Phys. 87 (1983), 577 doi:10.1007/BF01208266
  * [3] G. W. Gibbons and S. W. Hawking, “Action Integrals and Partition Functions in Quantum Gravity,” Phys. Rev. D 15 (1977), 2752 doi:10.1103/PhysRevD.15.2752
  * [4] M. Reuter, “Nonperturbative evolution equation for quantum gravity,” Phys. Rev. D 57 (1998), 971 doi:10.1103/PhysRevD.57.971 [arXiv:hep-th/9605030 [hep-th]].
  * [5] R. Percacci, “An introduction to covariant quantum gravity and asymptotic safety”, World Scientific, Singapore (2017).
  * [6] M. Reuter and F. Saueressig, “Quantum Gravity and the Functional Renormalization Group : The Road towards Asymptotic Safety,” CUP, Cambridge (2019)
  * [7] M. Niedermaier and M. Reuter, “The Asymptotic Safety Scenario in Quantum Gravity,” Living Rev. Rel. 9 (2006), 5 doi:10.12942/lrr-2006-5
  * [8] M. Niedermaier, “The Asymptotic safety scenario in quantum gravity: An Introduction,” Class. Quant. Grav. 24 (2007), R171 doi:10.1088/0264-9381/24/18/R01 [arXiv:gr-qc/0610018 [gr-qc]].
  * [9] A. Bonanno and M. Reuter, “Renormalization group improved black hole space-times,” Phys. Rev. D 62 (2000), 043008 doi:10.1103/PhysRevD.62.043008 [arXiv:hep-th/0002196 [hep-th]].
  * [10] A. Bonanno and M. Reuter, “Spacetime structure of an evaporating black hole in quantum gravity,” Phys. Rev. D 73 (2006), 083005 doi:10.1103/PhysRevD.73.083005 [arXiv:hep-th/0602159 [hep-th]].
  * [11] K. Falls, D. F. Litim and A. Raghuraman, “Black Holes and Asymptotically Safe Gravity,” Int. J. Mod. Phys. A 27 (2012), 1250019 doi:10.1142/S0217751X12500194 [arXiv:1002.0260 [hep-th]].
  * [12] B. Koch and F. Saueressig, “Black holes within Asymptotic Safety,” Int. J. Mod. Phys. A 29 (2014) no.8, 1430011 doi:10.1142/S0217751X14300117 [arXiv:1401.4452 [hep-th]].
  * [13] A. Eichhorn and A. Held, “Black holes in asymptotically safe gravity and beyond,” [arXiv:2212.09495 [gr-qc]].
  * [14] A. Platania, “Black Holes in Asymptotically Safe Gravity,” [arXiv:2302.04272 [gr-qc]].
  * [15] B. Carr and F. Kuhnel, “Primordial black holes as dark matter candidates,” SciPost Phys. Lect. Notes 48, 1 (2022) doi:10.21468/SciPostPhysLectNotes.48 [arXiv:2110.02821 [astro-ph.CO]].
  * [16] A. M. Green and B. J. Kavanagh, “Primordial Black Holes as a dark matter candidate,” J. Phys. G 48, no.4, 043001 (2021) doi:10.1088/1361-6471/abc534 [arXiv:2007.10722 [astro-ph.CO]].
  * [17] T. Harada, C. M. Yoo and K. Kohri, “Threshold of primordial black hole formation,” Phys. Rev. D 88, no.8, 084051 (2013) [erratum: Phys. Rev. D 89, no.2, 029903 (2014)] doi:10.1103/PhysRevD.88.084051 [arXiv:1309.4201 [astro-ph.CO]].
  * [18] F. Sannino and I. M. Shoemaker, “Asymptotically Safe Dark Matter,” Phys. Rev. D 92 (2015) no.4, 043518 doi:10.1103/PhysRevD.92.043518 [arXiv:1412.8034 [hep-ph]].
  * [19] A. Eichhorn, A. Held and P. Vander Griend, “Asymptotic safety in the dark,” JHEP 08 (2018), 147 doi:10.1007/JHEP08(2018)147 [arXiv:1802.08589 [hep-ph]].
  * [20] J. M. Maldacena, “The Large N limit of superconformal field theories and supergravity,” Adv. Theor. Math. Phys. 2 (1998), 231 doi:10.4310/ATMP.1998.v2.n2.a1 [arXiv:hep-th/9711200 [hep-th]].
  * [21] R. Ferrero and M. Reuter, “On the possibility of a novel (A)dS/CFT relationship emerging in Asymptotic Safety,” JHEP 12 (2022), 118 doi:10.1007/JHEP12(2022)118 [arXiv:2205.12030 [hep-th]].
  * [22] R. B. Mann, “Black holes of negative mass,” Class. Quant. Grav. 14 (1997), 2927 doi:10.1088/0264-9381/14/10/018 [arXiv:gr-qc/9705007 [gr-qc]].
  * [23] A. Codello, R. Percacci and C. Rahmede, “Investigating the Ultraviolet Properties of Gravity with a Wilsonian Renormalization Group Equation,” Annals Phys. 324 (2009), 414 doi:10.1016/j.aop.2008.08.008 [arXiv:0805.2909 [hep-th]].
  * [24] M. Reuter and E. Tuiran, “Quantum Gravity Effects in the Kerr Spacetime,” Phys. Rev. D 83 (2011), 044041 doi:10.1103/PhysRevD.83.044041 [arXiv:1009.3528 [hep-th]].
  * [25] U. Harst and M. Reuter, “QED coupled to QEG,” JHEP 05 (2011), 119 doi:10.1007/JHEP05(2011)119 [arXiv:1101.6007 [hep-th]].
  * [26] J. M. Pawlowski and D. Stock, “Quantum-improved Schwarzschild-(A)dS and Kerr-(A)dS spacetimes,” Phys. Rev. D 98 (2018) no.10, 106008 doi:10.1103/PhysRevD.98.106008 [arXiv:1807.10512 [hep-th]].
  * [27] A. Ishibashi, N. Ohta and D. Yamaguchi, “Quantum improved charged black holes,” Phys. Rev. D 104 (2021) no.6, 066016 doi:10.1103/PhysRevD.104.066016 [arXiv:2106.05015 [hep-th]].
  * [28] B. Koch, P. Rioseco and C. Contreras, “Scale Setting for Self-consistent Backgrounds,” Phys. Rev. D 91 (2015) no.2, 025009 doi:10.1103/PhysRevD.91.025009 [arXiv:1409.4443 [hep-th]].
  * [29] B. Koch and I. Ramirez, “Exact renormalization group with optimal scale and its application to cosmology,” Class. Quant. Grav. 28 (2011), 055008 doi:10.1088/0264-9381/28/5/055008 [arXiv:1010.2799 [gr-qc]].
  * [30] B. Koch and F. Saueressig, “Structural aspects of asymptotically safe black holes,” Class. Quant. Grav. 31 (2014), 015006 doi:10.1088/0264-9381/31/1/015006 [arXiv:1306.1546 [hep-th]].
  * [31] B. Koch and P. Rioseco, “Black Hole Solutions for Scale Dependent Couplings: The de Sitter and the Reissner-Nordström Case,” Class. Quant. Grav. 33 (2016), 035002 doi:10.1088/0264-9381/33/3/035002 [arXiv:1501.00904 [gr-qc]].
  * [32] C. González and B. Koch, “Improved Reissner-Nordström-(A)dS black hole in asymptotic safety,” Int. J. Mod. Phys. A 31 (2016) no.26, 1650141 doi:10.1142/S0217751X16501414 [arXiv:1508.01502 [hep-th]].
  * [33] J. N. Borissova and A. Platania, “Formation and evaporation of quantum black holes from the decoupling mechanism in quantum gravity,” JHEP 03 (2023), 046 doi:10.1007/JHEP03(2023)046 [arXiv:2210.01138 [gr-qc]].
  * [34] C.-M. Chen, Y. Chen, A. Ishibashi, N. Ohta and D. Yamaguchi, “Running Newton Coupling, Scale Identification and Black Hole Thermodynamics,” Phys. Rev. D 105 (2022) no.10, 106026 doi:10.1103/PhysRevD.105.106026 [arXiv:2204.09892 [hep-th]].
  * [35] K. Falls and D. F. Litim, “Black hole thermodynamics under the microscope,” Phys. Rev. D 89 (2014), 084002 doi:10.1103/PhysRevD.89.084002 [arXiv:1212.1821 [gr-qc]].
  * [36] R. K. Kaul and P. Majumdar, “Logarithmic correction to the Bekenstein-Hawking entropy,” Phys. Rev. Lett. 84 (2000), 5255 [arXiv:gr-qc/0002040 [gr-qc]].
  * [37] K. A. Meissner, “Black hole entropy in loop quantum gravity,” Class. Quant. Grav. 21 (2004), 5245 [arXiv:gr-qc/0407052 [gr-qc]].
  * [38] R. Mandal and S. Gangopadhyay, “Black hole thermodynamics in asymptotically safe gravity,” Gen. Rel. Grav. 54 (2022) no.12, 159 doi:10.1007/s10714-022-03045-9 [arXiv:2204.11616 [gr-qc]].
  * [39] B. R. Hull and R. B. Mann, “Negative mass black holes in de Sitter space,” Phys. Rev. D 107 (2023) no.6, 064027 doi:10.1103/PhysRevD.107.064027 [arXiv:2207.03507 [gr-qc]].
  * [40] S. Q. Hu, Y. C. Ong and D. N. Page, “No evidence for violation of the second law in extended black hole thermodynamics,” Phys. Rev. D 100 (2019) no.10, 104022 doi:10.1103/PhysRevD.100.104022 [arXiv:1906.05870 [gr-qc]].