# A new estimate for homogeneous fractional integral operators on the weighted
Morrey space $L^{p,\kappa}$ when $\alpha p=(1-\kappa)n$

Jingliang Du and Hua Wang 111In memory of Li Xue. E-mail address:
<EMAIL_ADDRESS><EMAIL_ADDRESS>School of Mathematics and Systems
Science, Xinjiang University,
Urumqi 830046, P. R. China

###### Abstract

For any $0<\alpha<n$, the homogeneous fractional integral operator
$T_{\Omega,\alpha}$ is defined by

$T_{\Omega,\alpha}f(x)=\int_{\mathbb{R}^{n}}\frac{\Omega(x-y)}{|x-y|^{n-\alpha}}f(y)\,dy.$

In this paper, we prove that if $\Omega$ satisfies certain Dini smoothness
conditions on $\mathbf{S}^{n-1}$, then $T_{\Omega,\alpha}$ is bounded from
$L^{p,\kappa}(w^{p},w^{q})$ (weighted Morrey space) to
$\mathrm{BMO}(\mathbb{R}^{n})$.

###### keywords:

Homogeneous fractional integral operator , Dini smoothness condition ,
weighted Morrey space , BMO space

###### MSC:

[2020] Primary 42B20 , 42B25 Secondary 47G10

##  1 Introduction

One of the most significant operators in harmonic analysis is the Riesz
potential operator $I_{\alpha}$ of order $\alpha$, also known as the
fractional integral operator. Let $\mathbb{R}^{n}$ be the $n$-dimensional
Euclidean space endowed with the Lebesgue measure $dx$ and the Euclidean norm
$|\cdot|$. Given $\alpha\in(0,n)$, the Riesz potential operator $I_{\alpha}$
of order $\alpha$ is defined by

$I_{\alpha}f(x):=\frac{1}{\gamma(\alpha)}\int_{\mathbb{R}^{n}}\frac{f(y)}{|x-y|^{n-\alpha}}dy,\quad
x\in\mathbb{R}^{n},$

where
$\gamma(\alpha)=\frac{2^{\alpha}\pi^{n/2}\Gamma(\alpha/2)}{\Gamma({(n-\alpha)}/2)}$
and $\Gamma(\cdot)$ being the usual gamma function. Let
$(-\Delta)^{\alpha/2}$(under $0<\alpha<n$) denote the $\alpha/2$-th order
Laplacian. Then $u=I_{\alpha}f$ is viewed as a solution of the $\alpha/2$-th
order Laplace equation

$(-\Delta)^{\alpha/2}u=f$

in the sense of the Fourier transform; i.e., $(-\Delta)^{\alpha/2}$ exists as
the inverse of $I_{\alpha}$. It is well known that the Riesz potential
operator $I_{\alpha}$ plays an important role in harmonic analysis, PDE and
potential theory. The classical Hardy–Littlewood–Sobolev theorem states that
$I_{\alpha}$ is bounded from $L^{p}(\mathbb{R}^{n})$ to
$L^{q}(\mathbb{R}^{n})$ when $1<p<q<\infty$ and $1/q=1/p-\alpha/n$. For $p=1$,
we know that $I_{\alpha}$ is bounded from $L^{1}(\mathbb{R}^{n})$ to
$WL^{q}(\mathbb{R}^{n})$ when $0<\alpha<n$ and $q=n/{(n-\alpha)}$. Moreover,
for the critical index $p=n/{\alpha}$, it is easy to verify that $I_{\alpha}$
is not bounded from $L^{n/{\alpha}}(\mathbb{R}^{n})$ to
$L^{\infty}(\mathbb{R}^{n})$ (see [14, p.119]). In this case, we also know
that as a substitute the Riesz potential operator $I_{\alpha}$ is bounded from
$L^{n/{\alpha}}(\mathbb{R}^{n})$ to $\mathrm{BMO}(\mathbb{R}^{n})$ (see [7,
p.130]). For each locally integrable function $f$, closely related to the
Riesz potential operator is the fractional maximal operator $M_{\alpha}$ on
$\mathbb{R}^{n}$, which is defined by

$M_{\alpha}f(x):=\sup_{r>0}\frac{1}{|B(x,r)|^{1-\alpha/n}}\int_{|x-y|<r}|f(y)|\,dy,\quad
x\in\mathbb{R}^{n},$

where the supremum is taken over all $r>0$. It can be shown that $M_{\alpha}$
satisfies the same norm inequalities as $I_{\alpha}$ for $0<\alpha<n$,
$1<p<n/{\alpha}$ and $1/q=1/p-\alpha/n$. The weak type $(1,q)$ inequality can
be proved by using covering lemma arguments when $q=n/{(n-\alpha)}$ (see, for
example, [13, Theorem 2]), and $M_{\alpha}$ is clearly of strong type
$(p,\infty)$ when $p=n/{\alpha}$. Actually, by Hölder’s inequality

$\frac{1}{|B(x,r)|^{1-\alpha/n}}\int_{|x-y|<r}|f(y)|\,dy\leq\bigg{(}\int_{|x-y|<r}|f(y)|^{p}\,dy\bigg{)}^{1/p}\leq\|f\|_{L^{p}}.$

The strong type $(p,q)$ inequalities then follow by Marcinkiewicz
interpolation theorem.

In 1974, Muckenhoupt and Wheeden studied the weighted boundedness of
$I_{\alpha}$ when $0<\alpha<n$. A weight $w$ is a nonnegative, locally
integrable function on $\mathbb{R}^{n}$ that take values in $(0,\infty)$
almost everywhere. Given a weight $w$ and a measurable set
$E\subset\mathbb{R}^{n}$, we use the notation

$w(E):=\int_{E}w(x)\,dx$

to denote the $w$-measure of the set $E$. The complement of the set $E$ is
denoted by $E^{\complement}$, and the characteristic function of the set $E$
is denoted by $\chi_{E}$: $\chi_{E}(x)=1$ if $x\in E$ and 0 if $x\notin E$.
For given $x_{0}\in\mathbb{R}^{n}$ and $r>0$, $B(x_{0},r)$ denotes the open
ball centered at $x_{0}$ with radius $r$. Given a ball $B$ and $\lambda>0$,
$\lambda B$ denotes the ball with the same center as $B$ whose radius is
$\lambda$ times that of $B$. Following [13], let us give the definitions of
some weight classes.

######  Definition 1.1 ([13]).

A weight function $w$ is said to belong to the Muckenhoupt–Wheeden class
$A(p,q)$ for $1<p<q<\infty$, if there exists a constant $C>0$ such that

$\bigg{(}\frac{1}{|B|}\int_{B}w(x)^{q}\,dx\bigg{)}^{1/q}\bigg{(}\frac{1}{|B|}\int_{B}w(x)^{-p^{\prime}}\,dx\bigg{)}^{1/{p^{\prime}}}\leq
C<\infty,$

for every ball $B$ in $\mathbb{R}^{n}$, where $p^{\prime}$ denotes the
conjugate exponent of $p>1$ such that $1/{p^{\prime}}+1/p=1$, and
$p^{\prime}=1$ when $p=\infty$. When $p=1$, a weight function $w$ is in the
Muckenhoupt–Wheeden class $A(1,q)$ with $1<q<\infty$, if there exists a
constant $C>0$ such that

$\bigg{(}\frac{1}{|B|}\int_{B}w(x)^{q}\,dx\bigg{)}^{1/q}\bigg{(}\underset{x\in
B}{\mathrm{ess\,sup}}\;\frac{1}{w(x)}\bigg{)}\leq C<\infty,$

for every ball $B$ in $\mathbb{R}^{n}$. A weight function $w$ is said to
belong to the Muckenhoupt–Wheeden class $A(p,\infty)$ with $1<p<\infty$, if
there exists a constant $C>0$ such that

$\bigg{(}\underset{x\in
B}{\mathrm{ess\,sup}}\;w(x)\bigg{)}\bigg{(}\frac{1}{|B|}\int_{B}w(x)^{-p^{\prime}}\,dx\bigg{)}^{1/{p^{\prime}}}\leq
C<\infty,$

for every ball $B$ in $\mathbb{R}^{n}$.

The weighted Lebesgue spaces and weak spaces with respect to the measure
$w(x)\,dx$ are denoted by $L^{p}(w)$ and $WL^{p}(w)$ with $1\leq p<\infty$,
respectively. In 1974, Muckenhoupt and Wheeden proved the following strong
type and weak type estimates of $M_{\alpha}$ and $I_{\alpha}$ (see [13,
Theorems 2 through 5]).

######  Theorem A ([13]).

Let $0<\alpha<n$, $1<p<n/{\alpha}$ and $1/q=1/p-{\alpha}/n$. If $w\in A(p,q)$,
then the operators $M_{\alpha}$ and $I_{\alpha}$ are bounded from
$L^{p}(w^{p})$ to $L^{q}(w^{q})$. If $p=1$ and $w\in A(1,q)$ with
$q=n/{(n-\alpha)}$, then the operators $M_{\alpha}$ and $I_{\alpha}$ are
bounded from $L^{1}(w)$ to $WL^{q}(w^{q})$.

######  Definition 1.2 ([8, 13]).

A locally integrable function $\hbar$ is said to be in the bounded mean
oscillation space $\mathrm{BMO}(\mathbb{R}^{n})$, if

$\|\hbar\|_{*}:=\sup_{B}\frac{1}{|B|}\int_{B}|\hbar(x)-\hbar_{B}|\,dx<\infty,$

where $\hbar_{B}$ stands for the average of $\hbar$ on $B$; i.e.,
$\hbar_{B}:=\frac{1}{|B|}\int_{B}\hbar(y)\,dy$ and the supremum is taken over
all balls $B$ in $\mathbb{R}^{n}$. For a weight function $w$ on
$\mathbb{R}^{n}$, the weighted version of $\mathrm{BMO}$ is denoted by
$\mathrm{BMO}(w)$. We say that $\hbar$ is in the weighted space
$\mathrm{BMO}(w)$, if there is a constant $C>0$ such that for any ball $B$ in
$\mathbb{R}^{n}$,

$\bigg{(}\underset{x\in
B}{\mathrm{ess\,sup}}\;w(x)\bigg{)}\bigg{(}\frac{1}{|B|}\int_{B}|\hbar(x)-\hbar_{B}|\,dx\bigg{)}\leq
C<\infty.$

Moreover, for the critical index $p=n/{\alpha}$, Muckenhoupt and Wheeden
proved the following result (see [13, Theorem 7]).

######  Theorem B ([13]).

Let $0<\alpha<n$ and $p=n/{\alpha}$. If $w\in A(p,\infty)$, then the Riesz
potential operator $I_{\alpha}$ is bounded from $L^{p}(w^{p})$ to
$\mathrm{BMO}(w)$.

We now introduce the function space $L^{\infty}(w)$, the weighted version of
$L^{\infty}(\mathbb{R}^{n})$. We say that a measurable function $\hslash$
belongs to the weighted space $L^{\infty}(w)$, if there is a constant $C>0$
such that for any ball $B$ in $\mathbb{R}^{n}$,

$\bigg{(}\underset{x\in
B}{\mathrm{ess\,sup}}\;w(x)\bigg{)}\Big{(}\big{\|}\hslash\big{\|}_{L^{\infty}(B)}\Big{)}\leq
C<\infty,$

where

$\big{\|}\hslash\big{\|}_{L^{\infty}(B)}:=\underset{x\in
B}{\mbox{ess\,sup}}\;|\hslash(x)|=\inf\big{\\{}M>0:\big{|}\\{x\in
E:|\hslash(x)|>M\\}\big{|}=0\big{\\}}.$

We prove in this paper that the strong type $(p,\infty)$ result remains valid
for $M_{\alpha}$ in our weighted spaces, under the assumptions that $w\in
A(p,\infty)$ and $p=n/{\alpha}$. Actually, by Hölder’s inequality and the
condition $w\in A(p,\infty)$, we get

$\begin{split}&\bigg{(}\underset{y\in
B(x,r)}{\mathrm{ess\,sup}}\;w(y)\bigg{)}\cdot\frac{1}{|B(x,r)|^{1-\alpha/n}}\int_{|x-y|<r}|f(y)|\,dy\\\
&\leq\bigg{(}\int_{B(x,r)}|f(y)|^{p}w(y)^{p}\,dy\bigg{)}^{1/p}\bigg{(}\underset{y\in
B(x,r)}{\mathrm{ess\,sup}}\;w(y)\bigg{)}\bigg{(}\frac{1}{|B(x,r)|}\int_{B(x,r)}w(y)^{-p^{\prime}}\,dy\bigg{)}^{1/{p^{\prime}}}\\\
&\leq C\|f\|_{L^{p}(w^{p})}.\end{split}$

Thus, the fractional maximal operator $M_{\alpha}$ is bounded from
$L^{p}(w^{p})$ to $L^{\infty}(w)$.

Let $\mathbf{S}^{n-1}=\\{x\in\mathbb{R}^{n}:|x|=1\\}$ be the unit sphere in
$\mathbb{R}^{n}$ ($n\geq 2$) equipped with the normalized Lebesgue measure
$d\sigma(x^{\prime})$. Let $\alpha\in(0,n)$ and $\Omega$ be a homogeneous
function of degree zero on $\mathbb{R}^{n}$ satisfying $\Omega\in
L^{s}(\mathbf{S}^{n-1})$ with $s\geq 1$. We study the homogeneous fractional
integral operator $T_{\Omega,\alpha}$, which is defined by

$T_{\Omega,\alpha}f(x):=\int_{\mathbb{R}^{n}}\frac{\Omega(x-y)}{|x-y|^{n-\alpha}}f(y)\,dy,\quad
x\in\mathbb{R}^{n}.$ (1.1)

Given any $\alpha\in(0,n)$ and $f\in L^{1}_{\mathrm{loc}}(\mathbb{R}^{n})$,
the fractional maximal function of $f$ with homogeneous kernel is defined by

$M_{\Omega,\alpha}f(x):=\sup_{r>0}\frac{1}{|B(x,r)|^{1-\alpha/n}}\int_{|x-y|<r}|\Omega(x-y)||f(y)|\,dy,\quad
x\in\mathbb{R}^{n},$ (1.2)

where the supremum on the right-hand side is taken over all $r>0$.

1

When $\alpha=0$ and $\Omega\equiv 1$, $M_{\Omega,\alpha}$ is the standard
Hardy–Littlewood maximal operator on $\mathbb{R}^{n}$, which is denoted by
$M$;

2

When $\Omega\equiv 1$, $M_{\Omega,\alpha}$($T_{\Omega,\alpha}$) is just the
fractional maximal (integral) operator $M_{\alpha}$($I_{\alpha}$) on
$\mathbb{R}^{n}$.

In 1971, Muckenhoupt and Wheeden [12] studied weighted norm inequalities for
$T_{\Omega,\alpha}$ with the weight $w(x)=|x|^{\beta}$. Some weak type
estimates with power weights for $M_{\Omega,\alpha}$ and $T_{\Omega,\alpha}$
were obtained by Ding in [6]. In 1998, Ding and Lu considered weighted norm
inequalities for $M_{\Omega,\alpha}$ and $T_{\Omega,\alpha}$ with more general
weights. More precisely, they proved the $(L^{p}(w^{p}),L^{q}(w^{q}))$
boundedness of the fractional maximal operator $M_{\Omega,\alpha}$ and the
fractional integral operator $T_{\Omega,\alpha}$ (see [4, Theorem 1 and
Proposition 1] and [11, Chapter 3]).

######  Theorem C ([4]).

Let $0<\alpha<n$, $1\leq s^{\prime}<p<n/{\alpha}$ and $1/q=1/p-{\alpha}/n$. If
$\Omega\in L^{s}(\mathbf{S}^{n-1})$ and $w^{s^{\prime}}\in
A(p/{s^{\prime}},q/{s^{\prime}})$, then the operators $M_{\Omega,\alpha}$ and
$T_{\Omega,\alpha}$ are all bounded from $L^{p}(w^{p})$ to $L^{q}(w^{q})$.

######  Definition 1.3 ([5]).

We say that $\Omega$ satisfies the $L^{s}$-Dini smoothness condition
$\mathfrak{D}_{s}$, if $\Omega\in L^{s}(\mathbf{S}^{n-1})$ is homogeneous of
degree zero on $\mathbb{R}^{n}$ with $1\leq s<\infty$, and

$\int_{0}^{1}\omega_{s}(\delta)\frac{d\delta}{\delta}<\infty,$

where $\omega_{s}(\delta)$ denotes the integral modulus of continuity of order
$s$, which is defined by

$\omega_{s}(\delta):=\sup_{|\rho|<\delta}\bigg{(}\int_{\mathbf{S}^{n-1}}\big{|}\Omega(\rho
x^{\prime})-\Omega(x^{\prime})\big{|}^{s}d\sigma(x^{\prime})\bigg{)}^{1/s}$

and $\rho$ is a rotation in $\mathbb{R}^{n}$ and
$|\rho|:=\|\rho-I\|=\sup_{x^{\prime}\in\mathbf{S}^{n-1}}\big{|}\rho
x^{\prime}-x^{\prime}\big{|}$. We also say that $\Omega$ satisfies the
$L^{\infty}$-Dini smoothness condition $\mathfrak{D}_{\infty}$, if $\Omega\in
L^{\infty}(\mathbf{S}^{n-1})$ is homogeneous of degree zero on
$\mathbb{R}^{n}$, and

$\int_{0}^{1}\omega_{\infty}(\delta)\frac{d\delta}{\delta}<\infty,$

where $\omega_{\infty}(\delta)$ is defined by

$\omega_{\infty}(\delta):=\sup_{|\rho|<\delta,x^{\prime}\in\mathbf{S}^{n-1}}\big{|}\Omega(\rho
x^{\prime})-\Omega(x^{\prime})\big{|},$

where $x^{\prime}=x/{|x|}$ for any $\mathbb{R}^{n}\ni x\neq 0$.

In 2002, Ding and Lu considered the critical case $p=n/{\alpha}$, and proved
the weighted boundedness of $T_{\Omega,\alpha}$ from $L^{p}(w^{p})$ to
$\mathrm{BMO}(w)$ (see [5, Theorem 1]).

######  Theorem D ([5]).

Let $0<\alpha<n$ and $p=n/{\alpha}$. If $\Omega$ satisfies the $L^{s}$-Dini
condition $\mathfrak{D}_{s}$ with $s>n/{(n-\alpha)}$ and $w^{s^{\prime}}\in
A(p/{s^{\prime}},\infty)$, then the homogeneous fractional integral operator
$T_{\Omega,\alpha}$ is bounded from $L^{p}(w^{p})$ to $\mathrm{BMO}(w)$.

Let $L^{\infty}(w)$ denote the space of all essentially bounded measurable
functions with respect to $w$ on $\mathbb{R}^{n}$. In this paper, we are able
to prove that the fractional maximal operator $M_{\Omega,\alpha}$ is bounded
from $L^{p}(w^{p})$ to $L^{\infty}(w)$ when $0<\alpha<n$ and $\Omega\in
L^{s}(\mathbf{S}^{n-1})$ with $s>n/{(n-\alpha)}$.

In 2009, Komori and Shirai [9] defined and investigated the weighted Morrey
spaces $L^{p,\kappa}(\mu,\nu)$ which could be viewed as an extension of
weighted Lebesgue spaces, and obtained the boundedness of some classical
integral operators on these weighted spaces. Let us now give the definition of
weighted Morrey spaces.

######  Definition 1.4 ([9]).

Let $1\leq p<\infty$ and $0<\kappa<1$. For two weights $\mu$ and $\nu$ on
$\mathbb{R}^{n}$, the weighted Morrey space $L^{p,\kappa}(\mu,\nu)$ is defined
by

$L^{p,\kappa}(\mu,\nu):=\Big{\\{}f\in
L^{p}_{\mathrm{loc}}(\mu):\|f\|_{L^{p,\kappa}(\mu,\nu)}<\infty\Big{\\}},$

where

$\|f\|_{L^{p,\kappa}(\mu,\nu)}:=\sup_{B}\bigg{(}\frac{1}{\nu(B)^{\kappa}}\int_{B}|f(x)|^{p}\mu(x)\,dx\bigg{)}^{1/p}$

and the supremum is taken over all balls $B$ in $\mathbb{R}^{n}$. If
$\mu=\nu$, then $L^{p,\kappa}(\mu,\nu)$ is simply denoted by
$L^{p,\kappa}(\mu)$.

It is known that $L^{p,\kappa}$ is an extension of $L^{p}$ in the sense that
$L^{p,0}=L^{p}$. Komori and Shirai obtained the strong type estimates for
$M_{\alpha}$ and $I_{\alpha}$ in weighted Morrey spaces based on Theorem A
(see [9, Theorems 3.5 and 3.6]).

######  Theorem E ([9]).

If $0<\alpha<n$, $1<p<n/{\alpha}$, $1/q=1/p-\alpha/n$, $0<\kappa<p/q$ and
$w\in A(p,q)$, then the fractional maximal operator $M_{\alpha}$ and the Riesz
potential operator $I_{\alpha}$ are bounded from $L^{p,\kappa}(w^{p},w^{q})$
to $L^{q,{\kappa q}/p}(w^{q})$.

In 2013, we established the boundedness properties of $M_{\Omega,\alpha}$ and
$T_{\Omega,\alpha}$ in weighted Morrey spaces based on Theorem C (see [15,
Theorems 1.1 and 1.2]).

######  Theorem F ([15]).

Suppose that $\Omega\in L^{s}(\mathbf{S}^{n-1})$ with $1<s\leq\infty$. If
$0<\alpha<n$, $1\leq s^{\prime}<p<n/{\alpha}$, $1/q=1/p-{\alpha}/n$,
$0<\kappa<p/q$ and $w^{s^{\prime}}\in A(p/{s^{\prime}},q/{s^{\prime}})$, then
the fractional maximal operator $M_{\Omega,\alpha}$ and the fractional
integral operator $T_{\Omega,\alpha}$ are bounded from
$L^{p,\kappa}(w^{p},w^{q})$ to $L^{q,{\kappa q}/p}(w^{q})$.

In the present work, we study the homogeneous fractional integral operators
and fractional maximal operators on the weighted Morrey spaces
$L^{p,\kappa}(w^{p},w^{q})$. Motivated by Theorems D and F, it is natural to
ask whether there is a corresponding estimate for the critical case
$\kappa=p/q=1-{(\alpha p)}/n$. We give a positive answer to this question. It
can be shown that the homogeneous fractional integral operator
$T_{\Omega,\alpha}$ is bounded from $L^{p,\kappa}(w^{p},w^{q})$ to
$\mathrm{BMO}(\mathbb{R}^{n})$, if $\Omega$ satisfies the Dini smoothness
condition $\mathfrak{D}_{s}$ with $1<s\leq\infty$. Moreover, the fractional
maximal operator $M_{\Omega,\alpha}$ is shown to be bounded from
$L^{p,\kappa}(w^{p},w^{q})$ to $L^{\infty}(\mathbb{R}^{n})$, if $\Omega\in
L^{s}(\mathbf{S}^{n-1})$ with $1<s\leq\infty$.

Throughout this paper, $C$ is a positive constant which is independent of the
main parameters and not necessarily the same at each occurrence.

##  2 Main results

Now let us formulate our main results as follows.

######  Theorem 2.5.

Let $0<\alpha<n$ and $p=n/{\alpha}$. If $\Omega\in L^{s}(\mathbf{S}^{n-1})$
with $s>n/{(n-\alpha)}$ and $w^{s^{\prime}}\in A(p/{s^{\prime}},\infty)$, then
the fractional maximal operator $M_{\Omega,\alpha}$ is bounded from
$L^{p}(w^{p})$ to $L^{\infty}(w)$.

######  Theorem 2.6.

Suppose that $\Omega$ satisfies the $L^{s}$-Dini condition $\mathfrak{D}_{s}$
with $1<s\leq\infty$. If $0<\alpha<n$, $s^{\prime}\leq p<n/{\alpha}$,
$1/q=1/p-{\alpha}/n$, $\kappa=p/q=1-{(\alpha p)}/n$ and $w^{s^{\prime}}\in
A(p/{s^{\prime}},q/{s^{\prime}})$, then the homogeneous fractional integral
operator $T_{\Omega,\alpha}$ is bounded from $L^{p,\kappa}(w^{p},w^{q})$ to
$\mathrm{BMO}(\mathbb{R}^{n})$.

In particular, when $\Omega\equiv 1$ and $s=\infty$, we have the following
result.

######  Corollary 2.7.

If $0<\alpha<n$, $1\leq p<n/{\alpha}$, $1/q=1/p-{\alpha}/n$,
$\kappa=p/q=1-{(\alpha p)}/n$ and $w\in A(p,q)$, then the Riesz potential
operator $I_{\alpha}$ is bounded from $L^{p,\kappa}(w^{p},w^{q})$ to
$\mathrm{BMO}(\mathbb{R}^{n})$.

######  Remark 2.8.

As a special case $w\equiv 1$, this type of estimate was shown by Adams and
implicit in [1] (see also [2, 3]).

##  3 Proofs of Theorems 2.5 and 2.6

In this section, we will prove the conclusions of Theorems 2.5 and 2.6.

###### Proof of Theorem 2.5.

Given a ball $B(x,r)$ with center $x\in\mathbb{R}^{n}$ and radius
$r\in(0,\infty)$, by using Hölder’s inequality twice (since $p>s^{\prime}$),
we get

$\begin{split}&\bigg{(}\underset{y\in
B(x,r)}{\mathrm{ess\,sup}}\;w(y)\bigg{)}\cdot\frac{1}{|B(x,r)|^{1-\alpha/n}}\int_{|x-y|<r}|\Omega(x-y)||f(y)|\,dy\\\
&\leq\bigg{(}\underset{y\in
B(x,r)}{\mathrm{ess\,sup}}\;w(y)\bigg{)}\cdot\frac{1}{|B(x,r)|^{1/{p^{\prime}}}}\bigg{(}\int_{|x-y|<r}|\Omega(x-y)|^{s}dy\bigg{)}^{1/s}\bigg{(}\int_{|x-y|<r}|f(y)|^{s^{\prime}}dy\bigg{)}^{1/{s^{\prime}}}\\\
&\leq\bigg{(}\underset{y\in
B(x,r)}{\mathrm{ess\,sup}}\;w(y)\bigg{)}\cdot\frac{1}{|B(x,r)|^{1/{p^{\prime}}}}\bigg{(}\int_{|x-y|<r}|\Omega(x-y)|^{s}dy\bigg{)}^{1/s}\\\
&\times\bigg{(}\int_{B(x,r)}|f(y)|^{p}w(y)^{p}dy\bigg{)}^{1/p}\bigg{(}\int_{B(x,r)}w^{s^{\prime}}(y)^{-\big{(}\frac{p}{s^{\prime}}\big{)}^{\prime}}dy\bigg{)}^{\frac{1}{s^{\prime}\big{(}\frac{p}{s^{\prime}}\big{)}^{\prime}}}.\end{split}$

Note that $\Omega\in L^{s}(\mathbf{S}^{n-1})$. Using polar coordinates, we
obtain

$\begin{split}&\bigg{(}\int_{|x-y|<r}|\Omega(x-y)|^{s}dy\bigg{)}^{1/s}\\\
&=\bigg{(}\int_{0}^{r}\int_{\mathbf{S}^{n-1}}|\Omega(x^{\prime})|^{s}\varrho^{n-1}d\sigma(x^{\prime})d\varrho\bigg{)}^{1/s}\\\
&=C\cdot
r^{n/s}\|\Omega\|_{L^{s}(\mathbf{S}^{n-1})}=C|B(x,r)|^{1/s}\|\Omega\|_{L^{s}(\mathbf{S}^{n-1})}.\end{split}$
(3.1)

On the other hand, a direct computation shows that

$\begin{split}\frac{1}{s^{\prime}\big{(}\frac{p}{s^{\prime}}\big{)}^{\prime}}=\frac{1}{s^{\prime}}\cdot\frac{p-s^{\prime}}{p}=\frac{1}{s^{\prime}}-\frac{1}{p}=\frac{1}{p^{\prime}}-\frac{1}{s}.\end{split}$

This, combined with (3.1), yields

$\begin{split}&\bigg{(}\underset{y\in
B(x,r)}{\mathrm{ess\,sup}}\;w(y)\bigg{)}\cdot\frac{1}{|B(x,r)|^{1-\alpha/n}}\int_{|x-y|<r}|\Omega(x-y)||f(y)|\,dy\\\
&\leq
C\|\Omega\|_{L^{s}(\mathbf{S}^{n-1})}\bigg{(}\int_{B(x,r)}|f(y)|^{p}w(y)^{p}\,dy\bigg{)}^{1/p}\\\
&\times\bigg{(}\underset{y\in
B(x,r)}{\mathrm{ess\,sup}}\;w(y)\bigg{)}\bigg{(}\frac{1}{|B(x,r)|}\int_{B(x,r)}w^{s^{\prime}}(y)^{-\big{(}\frac{p}{s^{\prime}}\big{)}^{\prime}}dy\bigg{)}^{\frac{1}{s^{\prime}\big{(}\frac{p}{s^{\prime}}\big{)}^{\prime}}}\\\
&=C\|\Omega\|_{L^{s}(\mathbf{S}^{n-1})}\bigg{(}\int_{B(x,r)}|f(y)|^{p}w(y)^{p}\,dy\bigg{)}^{1/p}\\\
&\times\bigg{[}\bigg{(}\underset{y\in
B(x,r)}{\mathrm{ess\,sup}}\;w^{s^{\prime}}(y)\bigg{)}\bigg{(}\frac{1}{|B(x,r)|}\int_{B(x,r)}w^{s^{\prime}}(y)^{-\big{(}\frac{p}{s^{\prime}}\big{)}^{\prime}}dy\bigg{)}^{\frac{1}{\big{(}\frac{p}{s^{\prime}}\big{)}^{\prime}}}\bigg{]}^{\frac{1}{s^{\prime}}}\\\
&\leq C\|f\|_{L^{p}(w^{p})},\end{split}$

where in the last step we have used the fact that $w^{s^{\prime}}\in
A(p/{s^{\prime}},\infty)$. This ends the proof of Theorem 2.5 by the
definition of $L^{\infty}(w)$. ∎

Before we proceed to prove our next theorem, we need the following estimate
which can be found in [5, Lemma 1].

######  Lemma 3.9 ([5]).

Suppose that $0<\alpha<n$ and $\Omega$ satisfies the $L^{s}$-Dini condition
$\mathfrak{D}_{s}$ with $1<s\leq\infty$. If there exists a constant
$0<\tau<1/2$ such that if $|x|<\tau R$, then we have

$\bigg{(}\int_{R\leq|z|<2R}\bigg{|}\frac{\Omega(z-x)}{|z-x|^{n-\alpha}}-\frac{\Omega(z)}{|z|^{n-\alpha}}\bigg{|}^{s}dz\bigg{)}^{1/s}\leq
C\cdot
R^{n/s-(n-\alpha)}\bigg{(}\frac{|x|}{R}+\int_{\frac{|x|}{2R}}^{\frac{|x|}{R}}\omega_{s}(\delta)\frac{d\delta}{\delta}\bigg{)},$

where the constant $C>0$ is independent of $R$ and $x$.

######  Remark 3.10.

Following the same arguments as in the proof of Lemma 5 in [10], we can also
establish the above lemma on the homogeneous kernel $\Omega(z)$.

###### Proof of Theorem 2.6.

For any ball $B=B(x_{0},r)$ with $(x_{0},r)\in\mathbb{R}^{n}\times(0,\infty)$
and $f\in L^{p,\kappa}(w^{p},w^{q})$, we are going to give the estimation of
the following expression

$\frac{1}{|B|}\int_{B}\big{|}T_{\Omega,\alpha}f(x)-(T_{\Omega,\alpha}f)_{B}\big{|}\,dx.$
(3.2)

For this purpose, we decompose $f$ as $f=f_{1}+f_{2}$, where
$f_{1}=f\cdot\chi_{4B}$, $f_{2}=f\cdot\chi_{(4B)^{\complement}}$,
$4B=B(x_{0},4r)$. By the linearity of the fractional integral operator
$T_{\Omega,\alpha}$, the expression (3.2) will be divided into two parts. That
is,

$\begin{split}&\frac{1}{|B|}\int_{B}\big{|}T_{\Omega,\alpha}f(x)-(T_{\Omega,\alpha}f)_{B}\big{|}\,dx\\\
&\leq\frac{1}{|B|}\int_{B}\big{|}T_{\Omega,\alpha}f_{1}(x)-(T_{\Omega,\alpha}f_{1})_{B}\big{|}\,dx+\frac{1}{|B|}\int_{B}\big{|}T_{\Omega,\alpha}f_{2}(x)-(T_{\Omega,\alpha}f_{2})_{B}\big{|}\,dx\\\
&:=\mathrm{I+II}.\end{split}$

Let us first consider the term $\mathrm{I}$. When $x\in B$ and $y\in 4B$, we
have $|x-y|<5r$. An application of Fubini’s theorem yields

$\begin{split}\mathrm{I}&\leq\frac{2}{|B|}\int_{B}|T_{\Omega,\alpha}f_{1}(x)|\,dx\\\
&\leq\frac{2}{|B|}\int_{B}\bigg{(}\int_{4B}\frac{|\Omega(x-y)|}{|x-y|^{n-\alpha}}|f(y)|\,dy\bigg{)}dx\\\
&\leq\frac{2}{|B|}\int_{4B}|f(y)|\bigg{(}\int_{|x-y|<5r}\frac{|\Omega(x-y)|}{|x-y|^{n-\alpha}}\,dx\bigg{)}dy.\end{split}$

Since $\Omega\in L^{s}(\mathbf{S}^{n-1})$, we have $\Omega\in
L^{1}(\mathbf{S}^{n-1})$, and

$\|\Omega\|_{L^{1}(\mathbf{S}^{n-1})}\leq
C\|\Omega\|_{L^{s}(\mathbf{S}^{n-1})}.$

Note that $\alpha>0$. Using polar coordinates, we can deduce that

$\begin{split}\int_{|x-y|<5r}\frac{|\Omega(x-y)|}{|x-y|^{n-\alpha}}\,dx&=\int_{0}^{5r}\int_{\mathbf{S}^{n-1}}\frac{|\Omega(x^{\prime})|}{\varrho^{n-\alpha}}\varrho^{n-1}d\sigma(x^{\prime})d\varrho\\\
&=C\cdot r^{\alpha}\|\Omega\|_{L^{1}(\mathbf{S}^{n-1})}\leq
C|B|^{\alpha/n}\|\Omega\|_{L^{s}(\mathbf{S}^{n-1})}.\end{split}$ (3.3)

Applying the Hölder inequality, we obtain

$\begin{split}\int_{4B}|f(y)|\,dy&\leq\bigg{(}\int_{4B}|f(y)|^{p}w(y)^{p}\,dy\bigg{)}^{1/p}\bigg{(}\int_{4B}w(y)^{-p^{\prime}}dy\bigg{)}^{1/{p^{\prime}}}.\end{split}$
(3.4)

Notice that

$s^{\prime}\Big{(}\frac{p}{s^{\prime}}\Big{)}^{\prime}=\frac{s^{\prime}}{1-\frac{s^{\prime}}{p}}=\frac{ps^{\prime}}{p-s^{\prime}}.$
(3.5)

By a simple calculation, we can see that when $s^{\prime}\geq 1$,

$\frac{ps^{\prime}}{p-s^{\prime}}\geq\frac{p}{p-1}=p^{\prime}.$

Using Hölder’s inequality and the condition $w^{s^{\prime}}\in
A(p/{s^{\prime}},q/{s^{\prime}})$, we get

$\begin{split}\bigg{(}\frac{1}{|4B|}\int_{4B}w(y)^{-p^{\prime}}dy\bigg{)}^{1/{p^{\prime}}}&\leq\bigg{(}\frac{1}{|4B|}\int_{4B}w(y)^{-s^{\prime}\big{(}\frac{p}{s^{\prime}}\big{)}^{\prime}}dy\bigg{)}^{\frac{1}{s^{\prime}\big{(}\frac{p}{s^{\prime}}\big{)}^{\prime}}}\\\
&=\bigg{(}\frac{1}{|4B|}\int_{4B}w^{s^{\prime}}(y)^{-\big{(}\frac{p}{s^{\prime}}\big{)}^{\prime}}dy\bigg{)}^{\frac{1}{s^{\prime}\big{(}\frac{p}{s^{\prime}}\big{)}^{\prime}}}\\\
&\leq
C\bigg{(}\frac{1}{|4B|}\int_{4B}w^{s^{\prime}}(y)^{\frac{q}{s^{\prime}}}dy\bigg{)}^{-\frac{s^{\prime}}{q}\cdot\frac{1}{s^{\prime}}}\\\
&=C\bigg{(}\frac{1}{|4B|}\int_{4B}w(y)^{q}dy\bigg{)}^{-1/{q}}.\end{split}$
(3.6)

A direct calculation shows that

$\frac{1}{p^{\prime}}+\frac{1}{q}=1-\frac{1}{p}+\frac{1}{q}=1-\frac{\alpha}{n}.$
(3.7)

Hence, from (3.3), (3.4) and (3.6), it follows that

$\begin{split}\mathrm{I}&\leq
C\cdot\frac{1}{|B|^{1-\alpha/n}}\int_{4B}|f(y)|\,dy\\\ &\leq
C\cdot\frac{1}{|B|^{1-\alpha/n}}\cdot|4B|^{\frac{1}{p^{\prime}}+\frac{1}{q}}\bigg{(}\int_{4B}|f(y)|^{p}w(y)^{p}dy\bigg{)}^{1/p}\bigg{(}\int_{4B}w(y)^{q}dy\bigg{)}^{-1/{q}}\\\
&\leq
C\cdot\frac{w^{q}(4B)^{\kappa/p}}{w^{q}(4B)^{1/q}}\|f\|_{L^{p,\kappa}(w^{p},w^{q})}=C\|f\|_{L^{p,\kappa}(w^{p},w^{q})},\end{split}$

where the last equality holds because $\kappa=p/q$. Now we turn our attention
to the second term $\mathrm{II}$. It is easy to see that

$\begin{split}\mathrm{II}&=\frac{1}{|B|}\int_{B}\big{|}T_{\Omega,\alpha}f_{2}(x)-(T_{\Omega,\alpha}f_{2})_{B}\big{|}\,dx\\\
&\leq\frac{1}{|B|}\int_{B}\bigg{\\{}\frac{1}{|B|}\int_{B}\big{|}T_{\Omega,\alpha}f_{2}(x)-T_{\Omega,\alpha}f_{2}(y)\big{|}\,dy\bigg{\\}}dx\\\
&\leq\frac{1}{|B|}\int_{B}\bigg{\\{}\frac{1}{|B|}\int_{B}\bigg{(}\sum_{j=2}^{\infty}\int_{2^{j+1}B\backslash
2^{j}B}\Big{|}\frac{\Omega(x-z)}{|x-z|^{n-\alpha}}-\frac{\Omega(y-z)}{|y-z|^{n-\alpha}}\Big{|}|f(z)|dz\bigg{)}dy\bigg{\\}}dx.\end{split}$

Since

$\begin{split}\Big{|}\frac{\Omega(x-z)}{|x-z|^{n-\alpha}}-\frac{\Omega(y-z)}{|y-z|^{n-\alpha}}\Big{|}&\leq\Big{|}\frac{\Omega(x-z)}{|x-z|^{n-\alpha}}-\frac{\Omega(x_{0}-z)}{|x_{0}-z|^{n-\alpha}}\Big{|}\\\
&+\Big{|}\frac{\Omega(x_{0}-z)}{|x_{0}-z|^{n-\alpha}}-\frac{\Omega(y-z)}{|y-z|^{n-\alpha}}\Big{|},\end{split}$

then by using Hölder’s inequality and Minkowski’s inequality, we have

$\begin{split}&\int_{2^{j+1}B\backslash
2^{j}B}\Big{|}\frac{\Omega(x-z)}{|x-z|^{n-\alpha}}-\frac{\Omega(y-z)}{|y-z|^{n-\alpha}}\Big{|}|f(z)|dz\\\
&\leq\bigg{(}\int_{2^{j+1}B\backslash
2^{j}B}\Big{|}\frac{\Omega(x-z)}{|x-z|^{n-\alpha}}-\frac{\Omega(y-z)}{|y-z|^{n-\alpha}}\Big{|}^{s}dz\bigg{)}^{1/s}\bigg{(}\int_{2^{j+1}B\backslash
2^{j}B}|f(z)|^{s^{\prime}}dz\bigg{)}^{1/{s^{\prime}}}\\\
&\leq\bigg{(}\int_{2^{j+1}B\backslash
2^{j}B}\Big{|}\frac{\Omega(x-z)}{|x-z|^{n-\alpha}}-\frac{\Omega(x_{0}-z)}{|x_{0}-z|^{n-\alpha}}\Big{|}^{s}dz\bigg{)}^{1/s}\bigg{(}\int_{2^{j+1}B\backslash
2^{j}B}|f(z)|^{s^{\prime}}dz\bigg{)}^{1/{s^{\prime}}}\\\
&+\bigg{(}\int_{2^{j+1}B\backslash
2^{j}B}\Big{|}\frac{\Omega(x_{0}-z)}{|x_{0}-z|^{n-\alpha}}-\frac{\Omega(y-z)}{|y-z|^{n-\alpha}}\Big{|}^{s}dz\bigg{)}^{1/s}\bigg{(}\int_{2^{j+1}B\backslash
2^{j}B}|f(z)|^{s^{\prime}}dz\bigg{)}^{1/{s^{\prime}}}.\end{split}$

For any given $x\in B$ and $z\in 2^{j+1}B\backslash 2^{j}B$ with $j\geq 2$, a
trivial computation shows that

$|x-x_{0}|<\frac{\,1\,}{4}|z-x_{0}|.$

According to Lemma 3.9(if we take $R=2^{j}r$), we get

$\begin{split}&\bigg{(}\int_{2^{j+1}B\backslash
2^{j}B}\Big{|}\frac{\Omega(x-z)}{|x-z|^{n-\alpha}}-\frac{\Omega(x_{0}-z)}{|x_{0}-z|^{n-\alpha}}\Big{|}^{s}dz\bigg{)}^{1/s}\\\
&=\bigg{(}\int_{2^{j}r\leq|z-x_{0}|<2^{j+1}r}\Big{|}\frac{\Omega(z-x_{0}-(x-x_{0}))}{|z-x_{0}-(x-x_{0})|^{n-\alpha}}-\frac{\Omega(z-x_{0})}{|z-x_{0}|^{n-\alpha}}\Big{|}^{s}dz\bigg{)}^{1/s}\\\
&\leq
C(2^{j}r)^{n/s-(n-\alpha)}\bigg{[}\frac{|x-x_{0}|}{2^{j}r}+\int_{\frac{|x-x_{0}|}{2^{j+1}r}}^{\frac{|x-x_{0}|}{2^{j}r}}\omega_{s}(\delta)\frac{d\delta}{\delta}\bigg{]}\\\
&\leq
C|2^{j+1}B|^{1/s-(1-\alpha/n)}\bigg{[}\frac{1}{2^{j}}+\int_{\frac{|x-x_{0}|}{2^{j+1}r}}^{\frac{|x-x_{0}|}{2^{j}r}}\omega_{s}(\delta)\frac{d\delta}{\delta}\bigg{]}.\end{split}$

Similarly, for any given $y\in B$, we can also obtain

$\begin{split}&\bigg{(}\int_{2^{j+1}B\backslash
2^{j}B}\Big{|}\frac{\Omega(x_{0}-z)}{|x_{0}-z|^{n-\alpha}}-\frac{\Omega(y-z)}{|y-z|^{n-\alpha}}\Big{|}^{s}dz\bigg{)}^{1/s}\\\
&\leq
C(2^{j}r)^{n/s-(n-\alpha)}\bigg{[}\frac{|y-x_{0}|}{2^{j}r}+\int_{\frac{|y-x_{0}|}{2^{j+1}r}}^{\frac{|y-x_{0}|}{2^{j}r}}\omega_{s}(\delta)\frac{d\delta}{\delta}\bigg{]}\\\
&\leq
C|2^{j+1}B|^{1/s-(1-\alpha/n)}\bigg{[}\frac{1}{2^{j}}+\int_{\frac{|y-x_{0}|}{2^{j+1}r}}^{\frac{|y-x_{0}|}{2^{j}r}}\omega_{s}(\delta)\frac{d\delta}{\delta}\bigg{]}.\end{split}$

On the other hand, we now claim that the following inequality holds:

$\displaystyle\bigg{(}\int_{2^{j+1}B\backslash
2^{j}B}|f(z)|^{s^{\prime}}dz\bigg{)}^{1/{s^{\prime}}}$ (3.8)
$\displaystyle\leq
C\bigg{(}\int_{2^{j+1}B}|f(z)|^{p}w(z)^{p}dz\bigg{)}^{1/p}\bigg{(}\int_{2^{j+1}B}w(z)^{q}dz\bigg{)}^{-1/{q}}\times|2^{j+1}B|^{(1-\alpha/n)-1/s}.$

To prove (3.8), we consider the following two cases: $p=s^{\prime}$ and
$p>s^{\prime}$.

Case 1: $p=s^{\prime}$. In this case, it follows from the condition $w^{p}\in
A(1,q/{p})$ and (3.7) that

$\begin{split}&\bigg{(}\int_{2^{j+1}B\backslash
2^{j}B}|f(z)|^{s^{\prime}}dz\bigg{)}^{1/{s^{\prime}}}\\\
&\leq\bigg{(}\int_{2^{j+1}B}|f(z)|^{p}w(z)^{p}w(z)^{-p}dz\bigg{)}^{1/{p}}\\\
&\leq\bigg{(}\int_{2^{j+1}B}|f(z)|^{p}w(z)^{p}dz\bigg{)}^{1/p}\bigg{(}\underset{z\in
2^{j+1}B}{\mbox{ess\,sup}}\;\frac{1}{w^{p}(z)}\bigg{)}^{1/{p}}\\\ &\leq
C\bigg{(}\int_{2^{j+1}B}|f(z)|^{p}w(z)^{p}dz\bigg{)}^{1/p}\bigg{(}\int_{2^{j+1}B}w(z)^{q}dz\bigg{)}^{-1/{q}}\times|2^{j+1}B|^{1/q}\\\
&=C\bigg{(}\int_{2^{j+1}B}|f(z)|^{p}w(z)^{p}dz\bigg{)}^{1/p}\bigg{(}\int_{2^{j+1}B}w(z)^{q}dz\bigg{)}^{-1/{q}}\times|2^{j+1}B|^{(1-\alpha/n)-1/s}.\end{split}$

Case 2: $p>s^{\prime}$. Applying Hölder’s inequality with exponent
$t=p/{s^{\prime}}>1$ and the condition $w^{s^{\prime}}\in
A(p/{s^{\prime}},q/{s^{\prime}})$, we get

$\begin{split}&\bigg{(}\int_{2^{j+1}B\backslash
2^{j}B}|f(z)|^{s^{\prime}}dz\bigg{)}^{1/{s^{\prime}}}\leq\bigg{(}\int_{2^{j+1}B}|f(z)|^{s^{\prime}}dz\bigg{)}^{1/{s^{\prime}}}\\\
&\leq\bigg{(}\int_{2^{j+1}B}|f(z)|^{p}w(z)^{p}dz\bigg{)}^{\frac{1}{s^{\prime}}\cdot\frac{s^{\prime}}{p}}\bigg{(}\int_{2^{j+1}B}w^{s^{\prime}}(z)^{-\big{(}\frac{p}{s^{\prime}}\big{)}^{\prime}}dz\bigg{)}^{\frac{1}{s^{\prime}\big{(}\frac{p}{s^{\prime}}\big{)}^{\prime}}}\\\
&\leq
C\bigg{(}\int_{2^{j+1}B}|f(z)|^{p}w(z)^{p}dz\bigg{)}^{1/p}\bigg{(}\int_{2^{j+1}B}w(z)^{q}dz\bigg{)}^{-1/{q}}\times\big{|}2^{j+1}B\big{|}^{\frac{1}{s^{\prime}\big{(}\frac{p}{s^{\prime}}\big{)}^{\prime}}+\frac{1}{q}}.\end{split}$

It follows directly from (3.5) that

$\frac{1}{s^{\prime}\big{(}\frac{p}{s^{\prime}}\big{)}^{\prime}}+\frac{\,1\,}{q}=\frac{p-s^{\prime}}{ps^{\prime}}+\frac{\,1\,}{q}=\Big{(}1-\frac{\alpha}{n}\Big{)}-\frac{\,1\,}{s}.$

Thus, in both cases (3.8) holds. Consequently, for any $x\in B$ and $y\in B$,
we can deduce that

$\begin{split}&\sum_{j=2}^{\infty}\int_{2^{j+1}B\backslash
2^{j}B}\Big{|}\frac{\Omega(x-z)}{|x-z|^{n-\alpha}}-\frac{\Omega(y-z)}{|y-z|^{n-\alpha}}\Big{|}|f(z)|\,dz\\\
&\leq
C\sum_{j=2}^{\infty}\bigg{[}\frac{1}{2^{j-1}}+\int_{\frac{|x-x_{0}|}{2^{j+1}r}}^{\frac{|x-x_{0}|}{2^{j}r}}\omega_{s}(\delta)\frac{d\delta}{\delta}+\int_{\frac{|y-x_{0}|}{2^{j+1}r}}^{\frac{|y-x_{0}|}{2^{j}r}}\omega_{s}(\delta)\frac{d\delta}{\delta}\bigg{]}\\\
&\times\bigg{(}\int_{2^{j+1}B}|f(z)|^{p}w(z)^{p}dz\bigg{)}^{1/p}\bigg{(}\int_{2^{j+1}B}w(z)^{q}dz\bigg{)}^{-1/{q}}\\\
&\leq
C\sum_{j=2}^{\infty}\bigg{[}\frac{1}{2^{j-1}}+\int_{\frac{|x-x_{0}|}{2^{j+1}r}}^{\frac{|x-x_{0}|}{2^{j}r}}\omega_{s}(\delta)\frac{d\delta}{\delta}+\int_{\frac{|y-x_{0}|}{2^{j+1}r}}^{\frac{|y-x_{0}|}{2^{j}r}}\omega_{s}(\delta)\frac{d\delta}{\delta}\bigg{]}\\\
&\times\frac{w^{q}(2^{j+1}B)^{\kappa/p}}{w^{q}(2^{j+1}B)^{1/q}}\|f\|_{L^{p,\kappa}(w^{p},w^{q})}\\\
&=C\bigg{[}1+2\int_{0}^{1}\omega_{s}(\delta)\frac{d\delta}{\delta}\bigg{]}\|f\|_{L^{p,\kappa}(w^{p},w^{q})},\end{split}$

where the last equality follows from the assumption that $\kappa=p/q$.
Therefore, we obtain

$\begin{split}\mathrm{II}&\leq\frac{1}{|B|}\int_{B}\bigg{\\{}\frac{1}{|B|}\int_{B}\bigg{(}\sum_{j=2}^{\infty}\int_{2^{j+1}B\backslash
2^{j}B}\Big{|}\frac{\Omega(x-z)}{|x-z|^{n-\alpha}}-\frac{\Omega(y-z)}{|y-z|^{n-\alpha}}\Big{|}|f(z)|\,dz\bigg{)}dy\bigg{\\}}dx\\\
&\leq
C\bigg{[}1+2\int_{0}^{1}\omega_{s}(\delta)\frac{d\delta}{\delta}\bigg{]}\|f\|_{L^{p,\kappa}(w^{p},w^{q})}.\end{split}$

Combining the above estimates for I and II and taking the supremum over all
balls $B$, we conclude the proof of Theorem 2.6. ∎

By using the same arguments as in the proof of Theorems 2.5 and 2.6, we can
also show the following result. The proof needs appropriate but only minor
modifications and we leave this to the interested reader.

######  Theorem 3.11.

Suppose that $\Omega\in L^{s}(\mathbf{S}^{n-1})$ with $1<s\leq\infty$. If
$0<\alpha<n$, $s^{\prime}\leq p<n/{\alpha}$, $1/q=1/p-{\alpha}/n$,
$\kappa=p/q=1-{(\alpha p)}/n$ and $w^{s^{\prime}}\in
A(p/{s^{\prime}},q/{s^{\prime}})$, then the fractional maximal operator
$M_{\Omega,\alpha}$ with homogeneous kernel is bounded from
$L^{p,\kappa}(w^{p},w^{q})$ to $L^{\infty}(\mathbb{R}^{n})$.

In particular, when $\Omega\equiv 1$ and $s=\infty$, as a direct consequence
of Theorem 3.11, we have

######  Corollary 3.12.

If $0<\alpha<n$, $1\leq p<n/{\alpha}$, $1/q=1/p-{\alpha}/n$,
$\kappa=p/q=1-{(\alpha p)}/n$ and $w\in A(p,q)$, then the fractional maximal
operator $M_{\alpha}$ is bounded from $L^{p,\kappa}(w^{p},w^{q})$ to
$L^{\infty}(\mathbb{R}^{n})$.

References

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Topological classification and black hole thermodynamics
M. R. Alipour ⋆111Email<EMAIL_ADDRESS>M. A. S. Afshar ⋆222Email:
<EMAIL_ADDRESS>S. N. Gashti⋆333Email:
<EMAIL_ADDRESS>J. Sadeghi ⋆444Email<EMAIL_ADDRESS>
⋆Department of Physics, Faculty of Basic Sciences,
University of Mazandaran P. O. Box 47416-95447, Babolsar, Iran

###### Abstract

One of the new methods that can be used to study the thermodynamics critical
points of a system based on a topological approach is the study of topological
charges using Duan’s $\phi$-mapping method. In this article, we will attempt
to use this method to study three different black holes, each with different
coefficients in their metric function, in order to determine the class of
critical points these black holes have in terms of phase transition. Through
this analysis, we found that the Euler-Heisenberg black hole has two different
topological classes, and the parameter $"a"$ added to the metric function by
QED plays an important role in this classification. While a black hole with a
non-linear electrodynamic field, despite having an electromagnetic parameter,
which is added to its metric function, has only one topological class, and its
$"\alpha"$ parameter has no effect on the number of critical points and
topological class. Finally, the Young Mills black hole in massive gravity will
have a different number of critical points depending on the coefficient
$"c_{i}"$, which is related to massive gravity and leads to different
topological classes. However, this black hole exhibits the same phase
structure in all cases.

Keywords: Critical Point, Topological Charge, Thermodynamics of Black holes

###### Contents

  1. 1 Introduction
  2. 2  Thermodynamics of black holes
    1. 2.1 Case I: Euler-Heisenberg-AdS black hole
    2. 2.2 Case II: Black hole with non-linear electrodynamic field
    3. 2.3 Case III: Five-dimensional Yang–Mills black holes in massive gravity
  3. 3 Topology of black hole thermodynamics
  4. 4 The models
    1. 4.1 Case I
    2. 4.2 Case II
    3. 4.3 Case III
  5. 5 Concluding remarks
  6. 6 Acknowledgments

##  1 Introduction

A new method for studying the thermodynamics and phase transitions of black
holes in Anti-de Sitter space has been created. The method demonstrates a
precise connection between the phase transition of charged AdS black holes
from small to large size and the phase transition of ordinary fluids from
liquid to gas [1]. This also involves a critical area where the phase
transition shifts from being first-order to second-order [1, 2, 3]. This
critical region is particularly fascinating since it displays certain
universal characteristics that are dependent upon the scaling of the
thermodynamic quantities. A recent proposal has suggested utilizing a
topological approach, based on Duan’s $\phi$-mapping theory, to determine the
characteristics of critical points for certain types of black holes that
possess electric charges and nonlinear electromagnetic fields [4, 5, 6]. It is
important to determine if this approach can be applied to more complex
scenarios with multiple critical points, not just limited to the liquid-gas
phase transition [7]. Apart from the transition of black holes from small to
large, there are other fascinating phenomena that bear resemblance to
conventional thermodynamic systems. These include phase transitions that occur
multiple times in the same direction as a parameter is changed, or those that
involve more than two phases, such as solid-liquid-gas transitions. These
phenomena occur in various types of black holes, particularly in higher
dimensions, and involve additional terms in the gravitational equations that
rely on higher powers or derivatives of the curvature [8, 9, 10, 11]. The
thermodynamics in extended phase space pertains to a framework utilized for
analyzing the thermodynamics and phase transitions of black holes in Anti-de
Sitter (AdS) space. In this framework, the cosmological constant is regarded
as a dynamic variable and interpreted as thermal pressure. In this framework,
the mass parameter of the black hole is viewed as enthalpy instead of internal
energy. This allows for more intricate and diverse phenomena, such as Van der
Waals-like phase transitions, reentrant phase transitions, superfluidity,
polymer-like phase transitions, triple points, and new dual relationships. The
formula for pressure in this extended phase space is as follows:
$P=-\Lambda/8\pi$, where $\Lambda$ represents the cosmological constant [12,
13, 14, 15]. An application of this framework involves examining the
holographic dualities that exist between black holes in AdS space and
conformal field theories on the boundary. For instance, the Hawking-Page phase
transition between Schwarzschild AdS black holes and thermal AdS space is
linked to the confinement/deconfinement transition in the boundary gauge
theory [16, 17, 18]. Another use is the creation of holographic heat engines
via the utilization of black holes in AdS space as working substances. These
processes are cyclical and can convert heat into work or vice versa, making
them suitable for assessing the efficiency and effectiveness of different
thermodynamic cycles. A third application involves examining the
microstructure of black holes in AdS space using thermodynamic geometry
techniques [19, 20]. This approach enables the investigation of fluctuations
and correlations of thermodynamic variables near critical points and can
unveil universal characteristics of black hole phase transitions.
Wei and Liu state that it is possible to use a combination of classical
thermodynamic functions and Duan’s $\phi$-mapping method to achieve a new
approach to studying the thermodynamic structures of black holes [5]. This
method can also easily determine the possibility of a phase transition at the
critical points of the system being studied. Based on this, it is claimed that
critical points with a topological charge of +1 $(Q_{t}=+1)$ do not exhibit a
first-order phase transition, which is considered ”novel”. On the other hand,
points with a topological charge of -1 $(Q_{t}=-1)$ can be considered to have
a first-order phase transition, which is referred to as ”conventional”. But in
[5], this proposal was doubted by examining the study of charged Gauss-Bonnet
black holes in AdS spacetime. They showed that there are conventional critical
points that do not cause first-order phase transition, so condition Wei cannot
be considered a necessary and sufficient condition for classifying points for
the phase transition structure. Therefore, to solve this problem, the critical
points were reclassified in another way. In this way, as pressure increases,
the novel critical point will be the point that results in a new phase
(whether stable or unstable). In contrast, the conventional point will be
equivalent to the point where the change of phases disappears [21, 22]. It is
mentioned in [5] that changes in topological charge class can be caused by
either gravity or gauge corrections. Of course, it is suggested in the article
[21] that gravitational corrections have no effect on changing the topological
class of a black hole (Gauss-Bonnet black hole), while gauge corrections can
change the topological class (Born-Infeld black hole) [22]. However, when both
factors are present, they will not affect the topological charge class. On the
other hand, the application of the ”wei” method to the Lovelock black hole has
also yielded other results, which demonstrate that for a charged Lovelock
black hole with arbitrary dimensions, a certain class of topological charge
can exhibit a different phase structure [23]. Furthermore, the uncharged
Lovelock black hole in dimensions $d=7$ and $d\geq 8$ can have different phase
structures depending on the class of topological charge. Therefore, it can be
concluded that a change in the topological charge class may indicate the
presence of changes in the phase structure, but the reverse is not necessarily
true [23]. According to the aforementioned experiences, we selected black
holes that had gravitational and gauge corrections, as well as different
dimensions. We attempted to investigate the claims and discovered some
intriguing points. The Euler-Heisenberg black hole, which has electromagnetic
gauge corrections due to the coefficient $"a"$ in its metric function,
confirms the previous results regarding the effect of gauge corrections on the
topological charge class in the study of the Born-Infeld black hole [22].
However, black hole with a non-linear electrodynamic field does not exhibit
any effects despite having gauge corrections that appear with the $"\alpha"$
coefficient in its metric function. Another interesting point is that the
Young Mills black hole in massive gravity, which has both gravitational and
gauge corrections, behaves differently because it is the gravitational
corrections that change the topological class. Also, the Young Mills black
hole in massive gravity challenges the proposal presented in [23] because our
results show that despite belonging to a different topological charge class,
it displays the same phase structure. Finally, we can state that although the
”Wei” method can be a simple and quick way to make an initial comment about
the existence of a first-order phase transition at critical points. But this
method still cannot be considered a reliable way to make a final statement,
and the conventional method of thermodynamics should still be used to verify
the first-order phase transition at the critical point. Therefore, the above
content motivated us to organize this article in the following form.
In section 2, we reviewed the thermodynamics of the considered black holes. In
section 3, we investigated the relationship between the critical points of the
black hole and its topological charge. In section 4, we calculated the
critical points and topological charges related to the considered black holes
and stated the related phase structure. Finally, we expressed our conclusion
in Section 5.

##  2  Thermodynamics of black holes

Black hole thermodynamics is the study area that aims to reconcile the laws of
thermodynamics with the existence of black hole event horizons. Researchers
consider various approaches, including the study of black holes and their
thermodynamic characteristics, as promising tool to gain insight into quantum
gravity. A considerable amount of research has been conducted on the
thermodynamics of black holes, and for further review, refer to [24, 25, 26,
27, 28, 29]. We first review the thermodynamic quantities of the black hole in
the extended space (considering the cosmological constant $\Lambda$ as the
pressure $P$ and the thermodynamic volume $V$ as its conjugate quantity). In
the following, we will examine the thermodynamic quantities of black holes in
Euler-Heisenberg, nonlinear electrodynamic, and Yang-Mills’ five-dimensional
massive gravity, respectively.

###  2.1 Case I: Euler-Heisenberg-AdS black hole

Euler Heisenberg’s black hole, which has spherical symmetry, its metric
function is obtained [30],

$\begin{split}f(r)=1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}-\frac{\Lambda
r^{2}}{3}-\frac{aQ^{4}}{20r^{6}}\end{split}$ (1)

Where $M$ and $Q$ are the mass and electric charge of the black hole,
respectively. Additionally, $\Lambda$ is a cosmological constant whose
expanded phase space is interpreted as pressure $P=-\frac{\Lambda}{8\pi}$, and
$a$ is related to the QED parameter. When $a=0$, it is reduced to a charged
AdS black hole. The thermodynamic quantities for this black hole can be
obtained from [30],

$\begin{split}T=\frac{1}{4\pi r_{h}}-\frac{Q^{2}}{4\pi
r_{h}^{3}}+\frac{aQ^{4}}{16\pi r_{h}^{7}}+2Pr_{h}\end{split}$ (2)
$\begin{split}S=\pi r_{h}^{2}\end{split}$ (3)
$\begin{split}\Phi_{em}=\frac{Q}{r_{h}}\left(1-\frac{aQ^{2}}{10r_{h}^{4}}\right)\end{split}$
(4) $\begin{split}M=\frac{r_{h}}{2}\left(1+\frac{Q^{2}}{r_{h}^{2}}+\frac{8\pi
P}{3}r_{h}^{2}-\frac{aQ^{4}}{20r_{h}^{6}}\right)\end{split}$ (5)

In the above relations, $T$, $S$, $\Phi_{em}$, and $r_{h}$ are the Hawking
temperature, entropy, electromagnetic potential, and the radius of the black
hole event horizon, respectively.

###  2.2 Case II: Black hole with non-linear electrodynamic field

When we add a non-linear electrodynamic field correction term to Einstein-
Maxwell’s theory, the black hole metric is obtained as follows [31],

$\begin{split}f(r)=1+2\sqrt{\alpha}Q-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}+\frac{r^{2}}{3}\left(\frac{8\pi}{3}P-\alpha\right)\end{split}$
(6)

where $\alpha$ is the coupling constant. When $\alpha=0$, it reduces to
Einstein-Maxwell theory. Furthermore, we can find the thermodynamic properties
of this black hole in [31],

$\begin{split}T=\frac{1}{4\pi r_{h}}(1+2\alpha Q)-\frac{Q^{2}}{4\pi
r_{h}^{3}}+\frac{r_{h}}{4\pi}(8\pi P-\alpha)\end{split}$ (7)
$\begin{split}S=\pi r_{h}^{2}\end{split}$ (8)
$\begin{split}\Phi_{em}=\frac{Q}{r_{h}}\left(1+\sqrt{\alpha}r_{h}^{2}\right)\end{split}$
(9) $\begin{split}M=\frac{r_{h}}{2}\left(1+\frac{Q^{2}}{r_{h}^{2}}+\frac{8\pi
P}{3}r_{h}^{2}-\frac{\alpha}{3}r_{h}^{2}+2\sqrt{\alpha}Q\right)\end{split}$
(10)

We can also obtain the Gibbs free energy ($G$) by using the relations above
and $G=M-TS$.

###  2.3 Case III: Five-dimensional Yang–Mills black holes in massive gravity

In this section, we introduce the five-dimensional Young Mills black hole in
massive gravity. Its metric is given by [32],

$\begin{split}&f(r)=\frac{4\pi
Pr^{2}}{3}\left[1-\left(\frac{r_{h}}{r}\right)^{4}\right]-\frac{2e^{2}}{r^{2}}\ln\frac{r}{r_{h}}\\\
&+\frac{m^{2}c_{0}c_{1}r}{3}\left[1-\left(\frac{r_{h}}{r}\right)^{3}\right]+m^{2}c_{0}^{2}c_{2}\left[1-\left(\frac{r_{h}}{r}\right)^{2}\right]+\frac{2m^{2}c_{0}^{3}c_{3}}{r}\left[1-\left(\frac{r_{h}}{r}\right)\right]\end{split}$
(11)

where $e$, $m$, and $c_{\nu}$ $(\nu=0,1,2,3)$ are coupling constants of the
Yang-Mills theory, the mass, and constants term of massive gravity,
respectively. Also, its thermodynamic quantities are as follows [31],

$\begin{split}T=\frac{1}{2\pi r_{h}^{3}}\left(\frac{8\pi
Pr_{h}^{4}}{3}-e\right)+\frac{m^{2}c_{0}}{2\pi
r_{h}^{2}}\left(\frac{c_{1}r_{h}^{2}}{2}+c_{0}c_{2}r_{h}+c_{0}^{2}c_{3}\right)\end{split}$
(12) $\begin{split}M=\frac{3}{16\pi}\left(\frac{4\pi
Pr_{h}^{4}}{3}-2e^{2}\ln\frac{r_{h}}{L}\right)+\frac{3m^{2}c_{0}r_{h}}{16\pi}\left(\frac{c_{1}r_{h}^{2}}{3}+c_{0}c_{2}r_{h}+2c_{0}^{2}c_{3}\right)\end{split}$
(13) $\begin{split}S=\frac{r_{h}^{3}}{4}\end{split}$ (14)

where $L$ is an arbitrary length constant that is considered to have a
dimensionless argument of a logarithmic term.

##  3 Topology of black hole thermodynamics

In the extended thermodynamic framework, in order to study the critical points
of a thermodynamic system such as a black hole, the temperature is rewritten
as a function of pressure, entropy, and $x_{i}$, i.e., $T=T(P,S,x_{i})$. Then,
we use the following relationship to obtain the critical points [5, 21, 22,
23],

$\begin{split}\left(\frac{\partial T}{\partial
S}\right)_{P,x_{i}}=0,\qquad\left(\frac{\partial^{2}T}{\partial
S^{2}}\right)_{P,x_{i}}=0.\end{split}$ (15)

Recently, it has been shown in [5] that each critical point can have a
topological charge, which can be divided into two types: conventional and
novel. In the following, we can eliminate the parameter $P$ using the relation
$\left(\frac{\partial T}{\partial S}\right)_{P,x_{i}}=0$. Additionally, to
check the topological charge of critical points, we define the thermodynamic
function as follows [5, 21, 22],

$\begin{split}\Phi=\frac{1}{\sin\theta}T(S,x_{i})\end{split}$ (16)

Whereas $\frac{1}{\sin\theta}$ is considered an auxiliary factor to simplify
the analysis of topological calculations, we can then define a new vector
field $\phi=(\phi^{S},\phi^{\theta})$ using Doan’s $\phi$-mapping theory [5,
21, 22] and we have,

$\begin{split}\phi^{S}=\left(\frac{\partial\Phi}{\partial
S}\right)_{\theta,x_{i}},\qquad\phi^{\theta}=\left(\frac{\partial\Phi}{\partial\theta}\right)_{S,x_{i}}.\end{split}$
(17)

The vector field $\phi$ is always zero at the point $\theta=\frac{\pi}{2}$,
allowing for easy identification of the critical points in the thermodynamic
system. Additionally, the points $\theta=0$ and $\theta=\pi$, where the field
vector is perpendicular to the horizontal lines, act as boundaries in the
parameter space. The proposed structure has an important feature: when the
vector field $\phi^{a}$ becomes zero (i.e., $\phi^{a}(x^{i})=0$), its
topological current $j^{\mu}$ becomes non-zero. As a result, we can calculate
its topological charge in the parameter region $\Sigma$ as follows [33, 34],

$\begin{split}Q_{t}=\int_{\Sigma}j^{0}d^{2}x=\Sigma_{i=1}^{N}\omega_{i}\end{split}$
(18)

where $\omega_{i}$ is the winding number for $\phi^{a}(x^{i})=0$. Since the
winding number can be either $+1$ or $-1$, $Q_{t}$ can be positive or negative
at critical points. This is suggested in [5], which divides them into two
categories: conventional charge when $Q_{t}=-1$ and novel charge when
$Q_{t}=+1$. Also, to obtain the topological charge, we can use the contour $C$
around the critical points in the $\sqrt{S}-\theta$ or $r-\theta$ plane where
the $n$ vector field is drawn, and obtain the normalization of the vector
field,

$\begin{split}n=(\frac{\phi^{S}}{\|\phi\|},\frac{\phi^{\theta}}{\|\phi\|})\end{split}$
(19)

On the other hand, contour $C$ should be parametrized by
$\vartheta\in(0,2\pi)$,

$\begin{split}r=a\cos\vartheta+r_{0},\qquad\theta=b\sin\vartheta+\frac{\pi}{2}\end{split}$
(20)

Which $(r_{0},\pi/2)$ is the center of the counter. Now, according to (19) and
(2) , we define a new quantity that measures the deflection of the vector
field along the contour. So, we will have,

$\begin{split}\Omega(\vartheta)=\int_{0}^{\vartheta}\epsilon_{ij}n^{i}\partial_{\vartheta}n^{j}d\vartheta\end{split}$
(21)

where $i,j=S,\theta$. Then the topological charge must be
$Q_{t}=\frac{\Omega(2\pi)}{2\pi}$.

##  4 The models

First, we attempt to find the critical points of black holes using equations
(15), (16), and (17). Next, we determine the corresponding topological charge
for these critical points using equations (18), (19), (19), and (2). Finally,
we analyze their properties by creating diagrams.

###  4.1 Case I

At first, we use relations (2) and (15) to eliminate the parameter $P$,
resulting in,

$\begin{split}T=\frac{a\,Q^{4}\pi^{3}-2\pi
Q^{2}S^{2}+S^{3}}{2\sqrt{\pi}\,S^{\frac{7}{2}}}\end{split}$ (22)

Then, by substituting equation (22) into (16), the thermodynamic function
$\Phi$ for the Euler-Heisenberg black hole can be determined as follows,

$\begin{split}\Phi=\frac{a\,Q^{4}\pi^{3}-2\pi
Q^{2}S^{2}+S^{3}}{2\sin\\!\left(\theta\right)S^{\frac{7}{2}}\sqrt{\pi}}\end{split}$
(23)

Now, with respect to equations (17), and (23) we investigate the vector fields
as,

$\begin{split}&\phi^{S}=-\frac{7a\,Q^{4}\pi^{3}-6\pi
Q^{2}S^{2}+S^{3}}{4\sin\\!\left(\theta\right)S^{\frac{9}{2}}\sqrt{\pi}}\\\
&\phi^{\theta}=-\frac{\left(a\,Q^{4}\pi^{3}-2\pi
Q^{2}S^{2}+S^{3}\right)\cos\\!\left(\theta\right)}{2\sin\\!\left(\theta\right)^{2}S^{\frac{7}{2}}\sqrt{\pi}}\end{split}$
(24)

According to relation (24), we can obtain the critical points. Of course, the
parameter ”a” plays an important role in determining the number of critical
points shown in Fig. 1.

Figure 1: The arrows represent the vector field n on the $\sqrt{S}-\theta$
plane for the Euler-Heisenberg black hole with the charge $Q=1$ and different
values for parameter $a$. So that, (a) a= 3, (b) a= -1, (c) a= 5.

As shown in Fig. 1, the parameter $"a"$ affects the number of critical points,
and can be divided into the following three categories: (1) When $0\leq
a\leq\frac{32Q^{2}}{7}$, we have two critical points as shown in Figure (1a).
The topological charge corresponding to each critical point is $Q_{CP_{2}}=-1$
and $Q_{CP_{1}}=+1$. In this case, the total topological charge is
$Q_{total}=Q_{CP_{1}}+Q_{CP_{2}}=0$. (2) When $a<0$, we have a critical point
as shown in Fig. (1b), and the topological charge corresponding to it is -1,
i.e. $Q_{CP_{1}}=-1$. So, the total topological charge is
$Q_{total}=Q_{CP_{1}}=-1$. (3) When $a>\frac{32Q^{2}}{7}$, we do not have a
critical point and topological charge for it.

We can see in Fig. 2 that the novel critical point is at the minimum point of
a part of the spinodal curve, while the conventional critical point is the
maximum point of the spinodal curve. As shown in Fig. 2, at the novel critical
point, the black hole has an unstable state, but when the pressure is
$P>P_{c1}$, a new phase of stability and instability appears in the black
hole. Also, the black hole is in a stable state at the conventional critical
point, and the stability of the black hole is maintained by increasing the
pressure. In this case, we will not have a new phase state. This result is in
agreement with proposal in [21], where the novel critical point of the novel
phase of stability and instability appears and disappears at the conventional
critical point of the phase.

Figure 2: Isobaric curves (blue=stable, red=unstable) and spinodal curve (gray
line) for the Euler-Heisenberg black hole. We have set Q = 1 and (a) $a=3$,
(b) $a=-1$.

We have found that the Euler-Heisenberg black hole falls into two distinct
topological classes, resulting in two different phase structures similar to
those of uncharged Lovelock AdS black holes with $d>7$ [23]. As depicted in
Fig. 3(a), there is a reentrant phase transition comprising a first-order
phase transition (blue curve) and a zeroth-order phase transition (green
curve). However, in Fig. 3(b), only the first-order phase transition is
present.

Figure 3: Phase diagram showing for the Euler-Heisenberg black hole. (a)
Reentrant phase structure for the $a=3$ and $Q=1$. (b) The first-order phase
transitions near the conventional critical point $CP_{1}$ for the $a=-1$ and
$Q=1$.

###  4.2 Case II

Regarding equations (7) and (15), we will have,

$\begin{split}T=\frac{-2q^{2}\pi+2\alpha
qS+S}{2S^{\frac{3}{2}}\sqrt{\pi}}\end{split}$ (25)

Also, by placing (25) in (16), we get the thermodynamic function,

$\begin{split}\Phi=\frac{-2q^{2}\pi+2\alpha
qS+S}{2\sin\\!\left(\theta\right)S^{\frac{3}{2}}\sqrt{\pi}}\end{split}$ (26)

In the following, we have the vector field components,

$\begin{split}&\phi^{S}=-\frac{\alpha
qS-3q^{2}\pi+\frac{1}{2}S}{2S^{\frac{5}{2}}\sqrt{\pi}\,\sin\\!\left(\theta\right)}\\\
&\phi^{\theta}=-\frac{\left(-2q^{2}\pi+2\alpha
qS+S\right)\cos\\!\left(\theta\right)}{2\sin\\!\left(\theta\right)^{2}S^{\frac{3}{2}}\sqrt{\pi}}\end{split}$
(27)

Now, we can obtain the critical points using equation (27), which are
illustrated in Fig. 4 for various values of the parameter $"\alpha"$.

Figure 4: The arrows represent the vector field n on the $\sqrt{S}-\theta$
plane for the nonlinear electrodynamic black hole with the charge $q=1$ and
different values for parameter $\alpha$. So that, (a) $\alpha$= 0.1, (b)
$\alpha$= 2.

Since the critical point at $(\sqrt{S},\theta)=(\sqrt{\frac{6\pi
q^{2}}{1+2\alpha q}},\pi/2)$, we observe that critical points occur at smaller
entropy or event horizons with increasing $\alpha$, as shown in Fig. 4.
However, this does not alter the topological charge. Specifically, we have a
conventional critical point for all values of $\alpha$, denoted by $e$, i.e.
$Q_{total}=Q_{CP_{1}}=-1$, and this result is consistent with the charged AdS
black hole in [5]. The nonlinear electrodynamics black hole is in the same
topological class as charged Gauss-Bonnet AdS black holes [21], charged
Lovelock AdS black holes [23], and charged AdS black holes [5]. As shown in
Fig. 5, when $P<P_{c1}$, the black hole can exist in both stable and unstable
states. With increasing pressure, the stability of the black hole increases,
such that at $P\geq P_{c1}$, the black hole becomes completely stable.

Figure 5: Isobaric curves (blue=stable, red=unstable) and spinodal curve (gray
line) for nonlinear electrodynamic black hole. We have set q = 1 and (a)
$\alpha=0.1$, (b) $\alpha=2$.

In Figure 6, we can observe a phase structure for this black hole for
different values of $\alpha$. As $\alpha$ increases, the pressure and
temperature values at critical points also increase.

Figure 6: Phase diagram showing the nonlinear electrodynamic black hole. The
first-order phase transitions occur near the conventional critical point
($CP_{1}$) for $q=1$ and (a) $\alpha=0.1$, (b) $\alpha=2$.

###  4.3 Case III

We use equations (12) and (15) to obtain the temperature and eliminate the
variable $P$, which results in,

$\begin{split}T=\frac{4\sqrt[3]{2}c_{0}^{2}c_{2}m^{2}S^{2/3}+3\
2^{2/3}c_{0}^{3}c_{3}m^{2}\sqrt[3]{S}+2c_{0}c_{1}m^{2}S-4e^{2}}{8\pi
S}\end{split}$ (28)

Then, we obtained the thermodynamic function by inserting (28) into (16),

$\begin{split}\Phi=\frac{4\sqrt[3]{2}c_{0}^{2}c_{2}m^{2}S^{2/3}+3\
2^{2/3}c_{0}^{3}c_{3}m^{2}\sqrt[3]{S}+2c_{0}c_{1}m^{2}S-4e^{2}}{8\pi
S\sin(\theta)}\end{split}$ (29)

Also, the vector field of the five-dimensional Yang-Mills black holes in
massive gravity, according to relations (17) and (29), is as follows,

$\begin{split}&\phi^{S}=-\frac{\csc(\theta)\left(2\sqrt[3]{2}c_{0}^{2}c_{2}m^{2}S^{2/3}+3\
2^{2/3}c_{0}^{3}c_{3}m^{2}\sqrt[3]{S}-6e^{2}\right)}{12\pi S^{2}}\\\
&\phi^{\theta}=-\frac{\cot(\theta)\csc(\theta)\left(4\sqrt[3]{2}c_{0}^{2}c_{2}m^{2}S^{2/3}+3\
2^{2/3}c_{0}^{3}c_{3}m^{2}\sqrt[3]{S}+2c_{0}c_{1}m^{2}S-4e^{2}\right)}{8\pi
S}\end{split}$ (30)

Therefore, we can calculate the critical points with respect to equation (30),

$\begin{split}&S_{1}=-\frac{3\left(9c_{3}c_{0}^{2}m\left(2c_{2}e^{2}+c_{3}^{2}c_{0}^{4}m^{2}\right)+\sqrt{24c_{2}e^{2}+9c_{3}^{2}c_{0}^{4}m^{2}}\left(2c_{2}e^{2}+3c_{3}^{2}c_{0}^{4}m^{2}\right)\right)}{8c_{0}^{3}c_{2}^{3}m^{3}}\\\
&S_{2}=-\frac{3\left(9c_{0}^{2}c_{3}m\left(2c_{2}e^{2}+c_{3}^{2}c_{0}^{4}m^{2}\right)-\left(2c_{2}e^{2}+3c_{3}^{2}c_{0}^{4}m^{2}\right)\sqrt{24c_{2}e^{2}+9c_{3}^{2}c_{0}^{4}m^{2}}\right)}{8c_{0}^{3}c_{2}^{3}m^{3}}\end{split}$
(31)

According to the above relationship, we can check the number of critical
points in two cases. When $-\frac{3c_{0}^{4}c_{3}^{2}m^{2}}{8e^{2}}<c_{2}<0$
and $c_{3}>0$, we have two critical points, otherwise we will have only one or
nothing critical point.

Figure 7: The arrows represent the vector field n on the $\sqrt{S}-\theta$
plane for the five-dimensional Yang–Mills black holes in massive gravity. So
that, (a) $e=2$ and $m=c_{0}=c_{1}=c_{2}=c_{3}=1$, (b) $c_{2}=-0.3$ and
$m=e=c_{0}=c_{1}=c_{3}=1$.

As shown in Fig. 7, the values of $c_{i}$ (where $i=1,2,3$) determine the
topological charges of the black hole. Therefore, massive gravity can affect
these charges, as seen in Figure 7(a) where we have a conventional topological
charge and $Q_{total}=Q_{CP1}=-1$. In Figure 7(b), where the values of $c_{i}$
have changed, we have two conventional and novel topological charges, and
$Q_{total}=Q_{CP1}+Q_{CP}=0$.

Figure 8: Isobaric curves (blue=stable, red=unstable) and spinodal curve (gray
line) for five-dimensional Yang–Mills black holes in massive gravity. We have
set (a) $e=2$ and $m=c_{0}=c_{1}=c_{2}=c_{3}=1$, (b) $c_{2}=-0.3$ and
$m=e=c_{0}=c_{1}=c_{3}=1$.

In Fig. 8, we can observe that the black hole is stable on both sides of the
conventional critical point $CP_{1}$. However, in Figure 8(b), for the novel
critical point $CP_{2}$, the black hole is unstable on both sides of this
point. Therefore, we can conclude that phases (stability or instability)
appear for critical points that have a topological charge of +1 ($Q_{t}=+1$),
while phases disappear for critical points that have a topological charge of
-1 ($Q_{t}=-1$).

Figure 9: Phase diagram showing for the five-dimensional Yang–Mills black
holes in massive gravity. The first-order phase transitions occur near the
conventional critical point for (a) $e=2$ and $m=c_{0}=c_{1}=c_{2}=c_{3}=1$,
(b) $c_{2}=-0.3$ and $m=e=c_{0}=c_{1}=c_{3}=1$ .

We have found that the five-dimensional Yang-Mills black hole in massive
gravity belongs to two different topology classes. However, as shown in Fig.
9, this black hole exhibits the same phase structure behavior in both
topological classes. In contrast, it has been suggested in [23] that a black
hole with a different topological class would have a different phase
structure. An important finding is that massive gravity can alter the number
of critical points of a Yang-Mills black hole, resulting in a different
topology class. Nevertheless, it does not affect the phase structure of the
Yang-Mills black hole. In Table 1, we summarize the topological charges
associated with the critical points of the black hole under consideration.
Additionally, the critical values can be found in Table 2.

Case |  Conditions |  Topological Charge |  Total Topological Charge
---|---|---|---
Euler-Heisenberg-AdS
black hole |  $0\leq a\leq\frac{32a^{2}}{7}$ |  $Q_{t|CP_{1}}=+1$,
$Q_{t|CP_{2}}=-1$ |  $0$
$a<0$ |  $Q_{t|CP_{1}}=-1$ |  $-1$
Nonlinear
electrodynamic
black hole |  $\alpha\geq 0$ |  $Q_{t|CP_{1}}=-1$ |  $-1$
Five-dimensional
Yang–Mills black holes
in massive gravity |  $c_{2}\geq 0$,
$m,e,c_{0},c_{1},c_{3}>0$ |  $Q_{t|CP_{1}}=-1$ |  $-1$
|  $-\frac{3c_{0}^{4}c_{3}^{2}m^{2}}{8e^{2}}<c_{2}<0$,
$m,e,c_{0},c_{1},c_{3}>0$ |  $Q_{t|CP_{1}}=-1$ ,
$Q_{t|CP_{2}}=+1$ |  $0$
Table 1: Summary of the results.

Case |  Conditions |  Critical Point (CP) |  $r_{c}$ |  $T_{c}$ |  $P_{c}$
---|---|---|---|---|---
Euler-Heisenberg-AdS
black hole |  $a=3$ |  $CP_{1}$
$CP_{2}$ |  1.5565
2.2876 |  0.0394
0.0444 |  0.0021
0.0035
$a=-1$ |  $CP_{1}$ |  2.4865 |  0.0430 |  0.0032
Nonlinear
electrodynamic
black hole |  $\alpha=0.1$ |  $CP_{1}$ |  2.2361 |  0.0569 |  0.0087
$\alpha=2$ |  $CP_{1}$ |  1.0954 |  0.4842 |  0.1624
Five-dimensional
Yang–Mills black holes
in massive gravity |  $c_{2}=1,e=2$,
$m=c_{0}=c_{1}=c_{3}=1$ |  $CP_{1}$ |  3.6235 |  0.1502 |  0.0058
|  $c_{2}=-0.3,e=1$,
$m=c_{0}=c_{1}=c_{3}=1$ |  $CP_{1}$
$CP_{2}$ |  2.7639
7.2361 |  0.0774
0.0738 |  0.00048
-0.00018
Table 2: Critical values.

##  5 Concluding remarks

In this paper, we investigated a topological charge that is associated with
critical points of Euler-Heisenberg, nonlinear, and Young-Mills black holes in
massive gravity. We found that the number of critical points in the Euler-
Heisenberg black hole is dependent on variable $a$. This leads to the
classification of the black hole into two distinct topological classes, each
with its own unique phase structure. The $Q_{total}=-1$ exhibits a small/large
black hole phase structure, while the $Q_{total}=0$ displays a reentrant phase
structure. Also, if $a\rightarrow 0$, we get the same results as the charged
AdS black hole [5]. On the other hand, for the nonlinear electrodynamic black
hole, we have only one conventional critical point, which has a small/large
black hole phase structure and similar behavior to the charged AdS black hole.
Additionally, we found that the massive gravity coefficient, $c_{2}$, can
impact the number of critical points of the Young Mills black hole in massive
gravity, resulting in the black hole being placed in two distinct topological
classes. However, the Young Mills term does not affect the topological class.
We also found that this black hole only has small/large phase structure for
different total topological charge. Here, the Euler-Heisenberg black hole and
the Young Mills black hole in massive gravity have different topological
classes, such as Born-Infeld AdS black holes in 4D novel Einstein-Gauss-Bonnet
gravity [22] and uncharged Lovelock AdS black holes [23]. While the Young
Mills black hole in heavy gravity is different from these black holes in terms
of phase structure because it has a phase structure despite the different
topology class. The obtained result is in conflict with the proposition [23],
which states that black holes with different topological charges can have
different phase structures. On the other hand, the novel and conventional
critical points for the considered black holes correspond to the new
classification [21]. The way they categorize critical points is as follows:
the novel critical point is where new phases (stable or unstable) appear as
pressure increases, while the conventional point is where the phases
disappear, as shown in Figs. 2, 5 and 8. We also found that, besides gauge
corrections [22], massive gravity can alter the topological class. In
addition, our findings align with the critical point parity conjecture
presented in [21], which states that the total topological charge is odd
(even) for an odd (even) number of critical points. Finally, based on our
results and [21, 22, 23] we conclude that the proposal presented in [5], which
states the conventional critical points of the first phase transition, is a
_sufficient condition for black holes that have at most two critical points.
However, if a black hole has more than two critical points, this proposition
is only a necessary condition._

##  6 Acknowledgments

This work is dedicated to the memory of Zohreh Sadeghi of the University of
Washington, Tacoma, USA, (daughter of Prof. J. Sadeghi). (For her diligent
correction of the English texts of our papers).

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