# Fractional type operators on the Heisenberg group

Pablo Rocha

(e-mail<EMAIL_ADDRESS>
###### Abstract

Let $\rho(\cdot)$ be the Koranyi norm on the Heisenberg group
$\mathbb{H}^{n}\equiv(\mathbb{R}^{2n}\times\mathbb{R},\,\cdot\,)$ defined by

$\rho(x,t)=\left(|x|^{4}+16t^{2}\right)^{1/4},\,\,\,\,(x,t)\in\mathbb{H}^{n}.$

For $0\leq\alpha<Q:=2n+2$,
$m\in\mathbb{N}\cap\left(1-\frac{\alpha}{Q},\infty\right)$, and $m$ positive
constants $\alpha_{1},...,\alpha_{m}$ such that
$\alpha_{1}+\cdot\cdot\cdot+\alpha_{m}=Q-\alpha$, we consider the following
generalization of the Riesz potential on $\mathbb{H}^{n}$

$T_{\alpha,\,m}f(x,t)=\int_{\mathbb{H}^{n}}f(y,s)\prod_{j=1}^{m}\rho\left((A_{j}y,r_{j}^{-2}s)^{-1}\cdot(x,t)\right)^{-\alpha_{j}}\,dy\,ds,$

where, in the case $0<\alpha<Q$, the $A_{j}$’s are matrices belonging to
$Sp(2n,\mathbb{R})\cap SO(2n)$ and $r_{j}=1$ for every $j=1,...,m$; for
$\alpha=0$, we consider $A_{j}=r_{j}^{-1}\,I_{2n\times 2n}$ for every
$j=1,...,m$, where the $r_{j}$’s are positive constants such that
$r_{i}^{2}-r_{j}^{2}\neq 0$ if $i\neq j$. In this note we study the behavior
of these operators on variable Hardy spaces in $\mathbb{H}^{n}$.

†† 2020 Mathematics Subject Classification: 42B25, 42B30, 42B35, 43A80. †† Key
words and phrases: variable Hardy spaces, atomic decomposition, fractional
operators, Heisenberg group.

##  1 Introduction

Let $J$ be the $2n\times 2n$ skew-symmetric matrix given by

(1) $J=\frac{1}{2}\left(\begin{array}[]{cc}0&-I_{n}\\\ I_{n}&0\\\
\end{array}\right),$

where $I_{n}$ is the $n\times n$ identity matrix. The Heisenberg group
$\mathbb{H}^{n}$ is $\mathbb{R}^{2n}\times\mathbb{R}$ endowed with the
following group law (noncommutative)

(2) $(x,t)\cdot(y,s)=(x+y,\,t+s+x^{t}Jy).$

With this structure we have that $e=(0,0)$ is the neutral element and
$(x,t)^{-1}=(-x,-t)$ is the inverse of $(x,t)$.

The Koranyi norm on the Heisenberg group $\mathbb{H}^{n}$ is defined by

$\rho(x,t)=\left(|x|^{4}+16\,t^{2}\right)^{1/4},\,\,\,(x,t)\in\mathbb{H}^{n}.$

For $0<\alpha<Q:=2n+2$ and $(x,t)\in\mathbb{H}^{n}$, the Riesz potential
$\mathcal{R}_{\alpha}$ on $\mathbb{H}^{n}$ is defined by right-convolution
with the kernel $\rho(\cdot)^{\alpha-Q}$

$\mathcal{R}_{\alpha}f(x,t)=\left(f\ast\rho(\cdot)^{\alpha-Q}\right)(x,t)=\int_{\mathbb{H}^{n}}f(y,s)\rho\left((y,s)^{-1}\cdot(x,t)\right)^{\alpha-Q}dy\,ds.$

It is well known that the Heisenberg group is a homogeneous group in the sense
of G. Folland and E. Stein [4]. There the authors defined Hardy spaces on
homogeneous groups, $H^{p_{0}}(G)$ ($0<p_{0}<\infty$), and studied the
behavior of operators defined by right-convolution with kernels of type
$(\alpha,N)$ on the spaces $H^{p}(G)$ (see [4, p. 184]). In particular, when
$G=\mathbb{H}^{n}$, we have that the kernel $\rho(\cdot)^{\alpha-Q}$ is of
type $(\alpha,N)$ for $0<\alpha<Q$ and every $N\in\mathbb{N}$. Thus, from [4,
Proposition 6.2], it follows that the Reisz potential $\mathcal{R}_{\alpha}$
is bounded from $L^{p_{0}}(\mathbb{H}^{n})$ into $L^{q_{0}}(\mathbb{H}^{n})$
for $1<p_{0}<Q/\alpha$ and $1/q_{0}=1/p_{0}-\alpha/Q$ (see also [7, p. 198]);
and, by [4, Theorem 6.10], we have that $\mathcal{R}_{\alpha}$ is bounded from
$H^{p_{0}}(\mathbb{H}^{n})$ into $H^{q_{0}}(\mathbb{H}^{n})$ for $0<p_{0}\leq
1$ and $1/q_{0}=1/p_{0}-\alpha/Q$.

The variable Hardy spaces on the Heisenberg group
$H^{p(\cdot)}(\mathbb{H}^{n})$ were defined by J. Fang and J. Zhao in [2]. One
of their main goals is the atomic characterization of
$H^{p(\cdot)}(\mathbb{H}^{n})$, as an application of such characterization
they proved the boundedness on $H^{p(\cdot)}(\mathbb{H}^{n})$ of singular
integrals. Recently, with this framework, the author in [10], for
$0\leq\alpha<Q=2n+2$, $N\in\mathbb{N}$ and Hölder exponent functions
$p(\cdot):\mathbb{H}^{n}\to(0,+\infty)$ such that $\frac{Q}{Q+N}<p_{-}\leq
p(\cdot)\leq p_{+}<\frac{Q}{\alpha}$, proved the
$H^{p(\cdot)}(\mathbb{H}^{n})\to L^{q(\cdot)}(\mathbb{H}^{n})$ and
$H^{p(\cdot)}(\mathbb{H}^{n})\to H^{q(\cdot)}(\mathbb{H}^{n})$ boundedness of
right-convolution operators with kernels of type $(\alpha,N)$ on
$\mathbb{H}^{n}$, where
$\frac{1}{q(\cdot)}=\frac{1}{p(\cdot)}-\frac{\alpha}{Q}$. In particular, the
Riesz potential $\mathcal{R}_{\alpha}$ ($0<\alpha<Q$) on $\mathbb{H}^{n}$
satisfies such estimates.

In this note, for $0\leq\alpha<Q=2n+2$,
$m\in\mathbb{N}\cap\left(1-\frac{\alpha}{Q},\infty\right)$, and $m$ positive
constants $\alpha_{1},...,\alpha_{m}$ such that
$\alpha_{1}+\cdot\cdot\cdot+\alpha_{m}=Q-\alpha$, we consider the following
generalization of the Riesz potential on $\mathbb{H}^{n}$

(3)
$T_{\alpha,\,m}f(x,t)=\int_{\mathbb{H}^{n}}f(y,s)\prod_{j=1}^{m}\rho\left((A_{j}y,r_{j}^{-2}s)^{-1}\cdot(x,t)\right)^{-\alpha_{j}}\,dy\,ds,$

where, in the case $0<\alpha<Q$, the $A_{j}$’s are matrices belonging to
$Sp(2n,\mathbb{R})\cap SO(2n)$ and $r_{j}=1$ for every $j=1,...,m$; for
$\alpha=0$, we consider $A_{j}=r_{j}^{-1}\,I_{2n\times 2n}$ for every
$j=1,...,m$, where the $r_{j}$’s are positive constants such that
$r_{i}^{2}-r_{j}^{2}\neq 0$ if $i\neq j$. We observe that if $0<\alpha<Q$,
$m=1$ and $A_{1}=I_{2n\times 2n}$, then $T_{\alpha,\,1}=\mathcal{R}_{\alpha}$;
and if $0\leq\alpha<Q$ and $m\geq 2$, then the operator $T_{\alpha,m}$ is not
a convolution type operator. The operator given in (3) is inspired by the
following operator defined on $\mathbb{R}^{n}$

(4)
$U_{\alpha,\,m}f(x)=\int_{\mathbb{R}^{n}}f(y)\prod_{j=1}^{m}|x-A_{j}y|^{-\alpha_{j}}\,dy,$

where $0\leq\alpha<n$, $m\in\mathbb{N}\cap(1-\frac{\alpha}{n},\infty)$,
$|\cdot|$ is the Euclidean norm on $\mathbb{R}^{n}$, the matrices $A_{j}$’s
are invertible with $A_{i}-A_{j}$ also invertible if $i\neq j$, and the
$\alpha_{j}$’s are positive numbers such that
$\alpha_{1}+\cdot\cdot\cdot+\alpha_{m}=n-\alpha$. The operator
$U_{\alpha,\,m}$ given by (4) is a kind of generalization of the Riesz
potential on $\mathbb{R}^{n}$. In [11], the author jointly with M. Urciuolo
proved the $H^{p_{0}}(\mathbb{R}^{n})\to L^{q_{0}}(\mathbb{R}^{n})$
boundedness of $U_{\alpha,m}$ for $0<p_{0}<n/\alpha$ and
$1/q_{0}=1/p_{0}-\alpha/n$; we also showed that the $H^{p_{0}}(\mathbb{R})\to
H^{q_{0}}(\mathbb{R})$ boundedness does not hold for
$0<p_{0}\leq\frac{1}{1+\alpha}$, $\frac{1}{q_{0}}=\frac{1}{p_{0}}-\alpha$ and
$U_{\alpha,m}$ defined on $\mathbb{R}$ with $0<\alpha<1$, $m=2$, $A_{1}=1$ and
$A_{2}=-1$. This is a significant difference with respect to the case
$0<\alpha<1$, $n=m=1$ and $A_{1}=1$; indeed, in this case the operator
$U_{\alpha,m}$ coincides with the Riesz potential on $\mathbb{R}$, which is
bounded from $H^{p_{0}}(\mathbb{R})$ into $H^{q_{0}}(\mathbb{R})$ as it is
well known (see [14, 6]). In the context of variable exponents (see [8, 1]),
the author and M. Urciuolo in [12] proved the $H^{p(\cdot)}(\mathbb{R}^{n})\to
L^{q(\cdot)}(\mathbb{R}^{n})$ boundedness of the operator $U_{\alpha,m}$ and
the $H^{p(\cdot)}(\mathbb{R}^{n})\to H^{q(\cdot)}(\mathbb{R}^{n})$ boundedness
of Riesz potential via the infinite atomic and molecular decomposition
developed in [8]. In [9], the author gave another proof of the results
obtained in [12], but by using the finite atomic decomposition developed in
[1].

In this note, by using the frameworks developed in [4] and [2] joint with some
additional results obtained by the author in [10], we will prove the following
theorem.

Theorem 4.1 Let $0\leq\alpha<Q$,
$m\in\mathbb{N}\cap(1-\frac{\alpha}{Q},\infty)$ and let $T_{\alpha,\,m}$ be
the operator defined by (3). Suppose
$p(\cdot)\in\mathcal{P}^{\log}(\mathbb{H}^{n})$ such that
$p(A_{j}x,r_{j}^{-2}t)=p(x,t)$ for all $(x,t)$ and all $j=1,...,m$, and
$0<p_{-}\leq p_{+}<\frac{Q}{\alpha}$. If
$\frac{1}{q(\cdot)}=\frac{1}{p(\cdot)}-\frac{\alpha}{Q}$, then
$T_{\alpha,\,m}$ can be extended to a bounded operator from
$H^{p(\cdot)}(\mathbb{H}^{n})$ into $L^{q(\cdot)}(\mathbb{H}^{n})$.

This paper is organized as follows. Section 2 introduces the Heisenberg group
$\mathbb{H}^{n}$ and also gives some supporting results about the operator
$T_{\alpha,m}$, which are crucial to prove the Theorem 4.1. In Section 3, we
establish the basics on variable Lebesgue and Hardy spaces. Finally, in
Section 4 we prove the Theorem 4.1 and also give, for the case $0<\alpha<Q$,
non-trivial examples of variable exponents $p(\cdot)$ satisfying the condition
$p(A_{j}x,t)=p(x,t)$ for all $(x,t)$ and all $j=1,...,m$.

Notation: The symbol $A\lesssim B$ stands for the inequality $A\leq cB$ for
some positive constant $c$. The symbol $A\approx B$ stands for $B\lesssim
A\lesssim B$. For a measurable subset $E\subseteq\mathbb{H}^{n}$ we denote by
$\left|E\right|$ and $\chi_{E}$ the Haar measure of $E$ and the characteristic
function of $E$ respectively. Throughout this paper, $C$ will denote a
positive real constant not necessarily the same at each occurrence.

##  2 Preliminaries

We recall that the Heisenberg group is
$\mathbb{H}^{n}=(\mathbb{R}^{2n}\times\mathbb{R},\,\cdot\,)$ with the group
law $\cdot$ given by (2). The dilations in $\mathbb{H}^{n}$ are given by

$r\cdot(x,t)=(rx,r^{2}t),\,\,\,\,r>0,$

and satisfy

$r\cdot((x,t)\cdot(y,s))=(r\cdot(x,t))\cdot(r\cdot(y,s)).$

$M_{n}(\mathbb{R})$ denotes the algebra of all matrices of degree $n$ with
coefficients in $\mathbb{R}$. The symplectic group $Sp(2n,\mathbb{R})$ and the
special orthogonal group $SO(2n)$ are defined by

$Sp(2n,\mathbb{R})=\\{A\in M_{2n}(\mathbb{R}):A^{t}JA=J\\},$

where $J$ is the $2n\times 2n$ skew-symmetric matrix given in (1), and

$SO(2n)=\\{A\in
M_{2n}(\mathbb{R}):A^{t}=A^{-1}\,\,\text{and}\,\,\det(A)=1\\}.$

The group $Sp(2n,\mathbb{R})\cap SO(2n)$ acts on $\mathbb{H}^{n}$ by

$A\cdot(x,t)=(Ax,t),\,\,\,\,A\in Sp(2n,\mathbb{R})\cap
SO(2n)\,\,\text{and}\,\,(x,t)\in\mathbb{H}^{n};$

and satisfy

$A\cdot((x,t)\cdot(y,s))=(A\cdot(x,t))\cdot(A\cdot(y,s)).$

The topology in $\mathbb{H}^{n}$ is the induced by
$\mathbb{R}^{2n}\times\mathbb{R}$, so the borelian sets of $\mathbb{H}^{n}$
are identified with those of $\mathbb{R}^{2n}\times\mathbb{R}$. The Haar
measure in $\mathbb{H}^{n}$ is the Lebesgue measure of
$\mathbb{R}^{2n}\times\mathbb{R}$, thus $L^{p_{0}}(\mathbb{H}^{n})\equiv
L^{p_{0}}(\mathbb{R}^{2n}\times\mathbb{R})$, $0<p_{0}\leq\infty$. Moreover,
for $f\in L^{1}(\mathbb{H}^{n})$ and each $(y,s)\in\mathbb{H}^{n}$

(5)
$\int_{\mathbb{H}^{n}}f((y,s)\cdot(x,t))\,dx\,dt=\int_{\mathbb{H}^{n}}f((x,t)\cdot(y,s))\,dx\,dt=\int_{\mathbb{H}^{n}}f(x,t)\,dx\,dt,$

for $r>0$ fixed, we also have

$\int_{\mathbb{H}^{n}}f(r\cdot(x,t))\,dx\,dt=r^{-Q}\int_{\mathbb{H}^{n}}f(x,t)\,dx\,dt,$

where $Q=2n+2$. The number $2n+2$ is known as the homogeneous dimension of
$\mathbb{H}^{n}$ (we observe that the topological dimension of
$\mathbb{H}^{n}$ is $2n+1$).

The Koranyi norm on $\mathbb{H}^{n}$ is the function
$\rho:\mathbb{H}^{n}\to[0,\infty)$ defined by

(6)
$\rho(x,t)=\left(|x|^{4}+16\,t^{2}\right)^{1/4},\,\,\,(x,t)\in\mathbb{H}^{n},$

where $|\cdot|$ is the usual Euclidean norm on $\mathbb{R}^{2n}$. The Koranyi
norm satisfies the following properties

$\displaystyle\rho(x,t)$ $\displaystyle=$ $\displaystyle 0\,\,\,\text{if and
only if}\,\,(x,t)=(0,0),$ $\displaystyle\rho((x,t)^{-1})$ $\displaystyle=$
$\displaystyle\rho(x,t)\,\,\,\,\text{for all}\,\,(x,t)\in\mathbb{H}^{n}$
$\displaystyle\rho(r\cdot(x,t))$ $\displaystyle=$ $\displaystyle
r\rho(x,t)\,\,\,\,\text{for all}\,\,(x,t)\in\mathbb{H}^{n}\,\,\text{and
all}\,\,r>0,$ $\displaystyle\rho(A\cdot(x,t))$ $\displaystyle=$
$\displaystyle\rho(x,t)\,\,\,\,\text{for all}\,\,A\in Sp(2n,\mathbb{R})\cap
SO(2n)\,\,\text{and all}\,\,(x,t)\in\mathbb{H}^{n},$
$\displaystyle\rho((x,t)\cdot(y,s))$ $\displaystyle\leq$
$\displaystyle\rho(x,t)+\rho(y,s)\,\,\,\,\text{for
all}\,\,(x,t),\,(y,s)\in\mathbb{H}^{n},$ $\displaystyle|\rho(x,t)-\rho(y,s)|$
$\displaystyle\leq$ $\displaystyle\rho((x,t)\cdot(y,s))\,\,\,\,\text{for
all}\,\,(x,t),\,(y,s)\in\mathbb{H}^{n}.$

Moreover, $\rho$ is continuous on $\mathbb{H}^{n}$ and is smooth on
$\mathbb{H}^{n}\setminus\\{e\\}$. The $\rho$ \- ball centered at
$(x_{0},t_{0})\in\mathbb{H}^{n}$ with radius $\delta>0$ is defined by

$B_{\delta}(x_{0},t_{0}):=\\{(y,s)\in\mathbb{H}^{n}:\rho((x_{0},t_{0})^{-1}\cdot(y,s))<\delta\\}.$

######  Remark 2.1.

The topology in $\mathbb{H}^{n}$ induced by the $\rho$ \- balls coincides with
the Euclidean topology of $\mathbb{R}^{2n}\times\mathbb{R}$ (see [3,
Proposition 3.1.37]). In particular, the $\rho$ \- balls are measurable sets
of $\mathbb{H}^{n}$.

Let $|B_{\delta}(x_{0},t_{0})|$ be the Haar measure of the $\rho$ \- ball
$B_{\delta}(x_{0},t_{0})\subset\mathbb{H}^{n}$. Then,

$|B_{\delta}(x_{0},t_{0})|=c_{0}\delta^{Q},$

where $c_{0}=|B_{1}(0,0)|$ and $Q=2n+2$. Moreover, for any
$(x,t),(x_{0},t_{0})\in\mathbb{H}^{n}$ and $\delta>0$, we have $(x,t)\cdot
B_{\delta}(x_{0},t_{0})=B_{\delta}((x,t)\cdot(x_{0},t_{0}))$.

######  Remark 2.2.

If $f\in L^{1}(\mathbb{H}^{n})$, then for every $\rho$ \- ball $B$ and every
$(x_{0},t_{0})\in\mathbb{H}^{n}$, by (5), we have

$\int_{B}f(x,t)\,dxdt=\int_{(x_{0},t_{0})^{-1}\cdot
B}f((x_{0},t_{0})\cdot(x,t))\,dxdt.$

For $0<\alpha<Q$ and a locally integrable function $f$ on $\mathbb{H}^{n}$, we
define the fractional maximal function $M_{\alpha}f$ on $\mathbb{H}^{n}$ by

$M_{\alpha}f(x,t)=\sup_{B\ni(x,t)}|B|^{\frac{\alpha}{Q}-1}\int_{B}|f(y,s)|\,dy\,ds,$

where the supremum is taken over all the $\rho$ \- balls $B$ containing
$(x,t)$. For $\alpha=0$, we have that $M_{0}=M$, where $M$ is the Hardy-
Littlewood maximal operator on $\mathbb{H}^{n}$.

######  Proposition 2.3.

If $0\leq\alpha<Q$, $1<p_{0}<Q/\alpha$ and $1/q_{0}=1/p_{0}-\alpha/Q$, then

$\left(\int_{\mathbb{H}^{n}}[M_{\alpha}f(x,t)]^{q_{0}}dx\,dt\right)^{1/q_{0}}\leq
C\left(\int_{\mathbb{H}^{n}}|f(x,t)|^{p_{0}}dx\,dt\right)^{1/p_{0}}.$

Moreover, for the case $\alpha=0$, one also has that
$\|M_{0}f\|_{L^{\infty}(\mathbb{H}^{n})}\leq\|f\|_{L^{\infty}(\mathbb{H}^{n})}$
and $M_{0}$ is weak type $(1,1)$.

###### Proof..

The case $\alpha=0$ follows from [13, Note 2.5 in p. 11] and [13, Theorem 1 in
p. 13]. Now, the case $0<\alpha<Q$ follows from [10, Proposition 4.3] applied
with $\omega\equiv 1$. ∎

The following result talks about the $L^{p_{0}}(\mathbb{H}^{n})\to
L^{q_{0}}(\mathbb{H}^{n})$ boundedness of the operator $T_{\alpha,\,m}$.

######  Theorem 2.4.

If $0\leq\alpha<Q$, $m\in\mathbb{N}\cap(1-\frac{\alpha}{Q},\infty)$, and
$T_{\alpha,\,m}$ is the operator defined by (3), then $T_{\alpha,\,m}$ is
bounded from $L^{p_{0}}(\mathbb{H}^{n})$ into $L^{q_{0}}(\mathbb{H}^{n})$ for
$1<p_{0}<Q/\alpha$ and $1/q_{0}=1/p_{0}-\alpha/Q$.

###### Proof..

We study the cases $0<\alpha<Q$ and $\alpha=0$ separately. For $0<\alpha<Q$,
we have that $A_{j}\in Sp(2n,\mathbb{R})\cap SO(2n)$ for every $j=1,...,m$,
and so

$|T_{\alpha,\,m}f(x,t)|\leq\sum_{j=1}^{m}\int_{\mathbb{H}^{n}}|f(y,s)|\,\rho\left((A_{j}y,s)^{-1}\cdot(x,t)\right)^{\alpha-Q}dy\,ds=\sum_{j=1}^{m}\mathcal{R}_{\alpha}|f|(A_{j}^{-1}x,t),$

Then, the $L^{p_{0}}(\mathbb{H}^{n})\to L^{q_{0}}(\mathbb{H}^{n})$ boundedness
for $T_{\alpha,\,m}$ in the case $0<\alpha<Q$ follows from the boundedness of
the Riesz potential $\mathcal{R}_{\alpha}$.

Now we study the case $\alpha=0$. Let
$\beta=\min\\{|r_{i}-r_{j}|,\,|r_{i}^{2}-r_{j}^{2}|^{1/2}:i\neq j\\}$ and
$\gamma=\max\\{|r_{i}|:i=1,2,...,m\\}$. For $(x,t)\neq(0,0)$ and $1\leq j\leq
m$ we define the following sets

$\Omega_{j}=\left\\{(y,s)\in\mathbb{H}^{n}:\rho\left((y,s)^{-1}\cdot(r_{j}x,r_{j}^{2}t)\right)<\frac{\beta}{2}\rho(x,t)\right\\};$

we also define

$\Omega_{m+1}=\left(\bigcap_{j=1}^{m}\Omega_{j}^{c}\right)\cap\left\\{(y,s)\in\mathbb{H}^{n}:\rho(y,s)\leq(1+\gamma)\rho(x,t)\right\\}$

and

$\Omega_{m+2}=\left(\bigcap_{j=1}^{m}\Omega_{j}^{c}\right)\cap\left\\{(y,s)\in\mathbb{H}^{n}:\rho(y,s)>(1+\gamma)\rho(x,t)\right\\}$

Then, for $(x,t)\neq(0,0)$ and $C=r_{1}^{-\alpha_{1}}\cdot\cdot\cdot
r_{m}^{-\alpha_{m}}$

$T_{0,\,m}f(x,t)=C\sum_{j=1}^{m+2}\int_{\Omega_{j}}f(y,s)\prod_{k=1}^{m}\rho\left((y,s)^{-1}\cdot(r_{k}x,r_{k}^{2}t)\right)^{-\alpha_{k}}\,dy\,ds.$

Next, we fix $(x,t)\neq(0,0)$ and $1\leq j\leq m$. For $(y,s)\in\Omega_{j}$
and $i\neq j$ we have that
$\rho\left((y,s)^{-1}\cdot(r_{j}x,r_{j}^{2}t)\right)<\frac{\beta}{2}\rho(x,t)$
and
$\rho\left((r_{i}x,r_{i}^{2}t)^{-1}\cdot(r_{j}x,r_{j}^{2}t)\right)\geq\beta\rho(x,t)$;
to obtain this last inequality we use that $xJx^{t}=0$. So,

$\rho\left((y,s)^{-1}\cdot(r_{i}x,r_{i}^{2}t)\right)\geq\rho\left((r_{i}x,r_{i}^{2}t)^{-1}\cdot(r_{j}x,r_{j}^{2}t)\right)-\rho\left((y,s)^{-1}\cdot(r_{j}x,r_{j}^{2}t)\right)\geq\frac{\beta}{2}\rho(x,t)$

and

$\displaystyle
C^{-1}|T_{0,m}(\chi_{\Omega_{j}}f)(x,t)|\leq\int_{\Omega_{j}}|f(y,s)|\prod_{k=1}^{m}\rho\left((y,s)^{-1}\cdot(r_{k}x,r_{k}^{2}t)\right)^{-\alpha_{k}}\,dy\,ds$
$\displaystyle\leq$
$\displaystyle\left(\frac{\beta}{2}\rho(x,t)\right)^{\alpha_{j}-Q}\int_{\Omega_{j}}\rho\left((y,s)^{-1}\cdot(r_{j}x,r_{j}^{2}t)\right)^{-\alpha_{j}}|f(y,s)|\,dy\,ds$
$\displaystyle\leq$
$\displaystyle\left(\frac{\beta}{2}\rho(x,t)\right)^{\alpha_{j}-Q}\sum_{k=0}^{\infty}\int_{\frac{\beta\rho(x,t)}{2^{k+2}}\leq\rho\left((y,s)^{-1}\cdot(r_{j}x,r_{j}^{2}t)\right)<\frac{\beta\rho(x,t)}{2^{k+1}}}\rho\left((y,s)^{-1}\cdot(r_{j}x,r_{j}^{2}t)\right)^{-\alpha_{j}}$
$\times\,\,|f(y,s)|\,dy\,ds$ $\displaystyle\leq$
$\displaystyle\left(\frac{\beta}{2}\rho(x,t)\right)^{\alpha_{j}-Q}\sum_{k=0}^{\infty}\int_{\frac{\beta\rho(x,t)}{2^{k+2}}\leq\rho\left((y,s)^{-1}\cdot(r_{j}x,r_{j}^{2}t)\right)<\frac{\beta\rho(x,t)}{2^{k+1}}}\left(\frac{\beta\rho(x,t)}{2^{k+2}}\right)^{-\alpha_{j}}|f(y,s)|\,dy\,ds$
$\displaystyle\leq$ $\displaystyle
2^{Q}c_{0}\sum_{k=0}^{\infty}2^{-(k+1)(Q-\alpha_{j})}c_{0}^{-1}\left(\frac{\beta\rho(x,t)}{2^{k+1}}\right)^{-Q}\int_{\rho\left((y,s)^{-1}\cdot(r_{j}x,r_{j}^{2}t)\right)<\frac{\beta\rho(x,t)}{2^{k+1}}}|f(y,s)|\,dy\,ds$
$\displaystyle\leq$ $\displaystyle
2^{\alpha_{j}}c_{0}\sum_{k=0}^{\infty}2^{-k(Q-\alpha_{j})}c_{0}^{-1}\left(\frac{\beta\rho(x,t)}{2^{k+1}}\right)^{-Q}\int_{\rho\left((y,s)^{-1}\cdot(r_{j}x,r_{j}^{2}t)\right)<\frac{\beta\rho(x,t)}{2^{k+1}}}|f(y,s)|\,dy\,ds,$

where $c_{0}=|B_{1}(0,0)|$. Then

(7)
$C^{-1}|T_{0,m}(\chi_{\Omega_{j}}f)(x,t)|\leq\frac{2^{\alpha_{j}}c_{0}}{1-2^{-(Q-\alpha_{j})}}M_{0}f(r_{j}x,r_{j}^{2}t),\,\,\,\,\text{for
every}\,\,j=1,...,m.$

On the region $\Omega_{m+1}$ we have

$\displaystyle C^{-1}|T_{0,m}(\chi_{\Omega_{m+1}}f)(x,t)|$ $\displaystyle\leq$
$\displaystyle\int_{\Omega_{m+1}}|f(y,s)|\prod_{k=1}^{m}\rho\left((y,s)^{-1}\cdot(r_{k}x,r_{k}^{2}t)\right)^{-\alpha_{k}}\,dy\,ds$
$\displaystyle\leq$
$\displaystyle\left(\frac{2}{\beta}\right)^{Q}\rho(x,t)^{-Q}\int_{\rho(y,s)\leq(1+\gamma)\rho(x,t)}|f(y,s)|\,dy\,ds$
$\displaystyle\leq$
$\displaystyle\left(\frac{2}{\beta}\right)^{Q}\rho(x,t)^{-Q}\int_{\rho((x,t)^{-1}\cdot(y,s))<2(1+\gamma)\rho(x,t)}|f(y,s)|\,dy\,ds,$

so

(8)
$C^{-1}|T_{0,m}(\chi_{\Omega_{m+1}}f)(x,t)|\leq\left(\frac{4(1+\gamma)}{\beta}\right)^{Q}c\,M_{0}f(x,t).$

Now, if $(y,s)\in\Omega_{m+2}$, for every $1\leq i\leq m$, we have that
$\rho((y,s)^{-1}\cdot(r_{i}x,r_{i}^{2}t))\geq\frac{1}{1+\gamma}\rho(y,s)$.
Thus

$C^{-1}|T_{0,m}(\chi_{\Omega_{m+2}}f)(x,t)|\leq\int_{\Omega_{m+2}}|f(y,s)|\prod_{k=1}^{m}\rho\left((y,s)^{-1}\cdot(r_{k}x,r_{k}^{2}t)\right)^{-\alpha_{k}}\,dy\,ds$
$\leq(1+\gamma)^{Q}\int_{\rho(y,s)>(1+\gamma)\rho(x,t)}|f(y,s)|\rho(y,s)^{-Q}\,dy\,ds$
$\leq(1+\gamma)^{Q/p_{0}^{\prime}}\,\frac{\sigma(\\{\rho=1\\})^{1/p_{0}^{\prime}}}{((p_{0}^{\prime}-1)Q)^{1/p_{0}^{\prime}}}\,\|f\|_{L^{p_{0}}(\mathbb{H}^{n})}\,\rho(x,t)^{-Q/p_{0}},$

to obtain this last inequality we apply Hölder’s inequality and polar
coordinates on $\mathbb{H}^{n}$ (see [7, p. 211]). Then, for $\lambda>0$

(9)
$\left|\left\\{(x,t)\in\mathbb{H}^{n}:C^{-1}|T_{0,m}(\chi_{\Omega_{m+2}}f)(x,t)|>\lambda)\right\\}\right|\lesssim\left(\frac{\|f\|_{L^{p_{0}}(\mathbb{H}^{n})}}{\lambda}\right)^{p_{0}}.$

Thus, by (7), (8), Proposition 2.3, and (9) we have that $T_{0,m}$ is weak
type $(p_{0},p_{0})$ for every $1\leq p_{0}<\infty$. Finally, the
Marcinkiewicz interpolation theorem (see [5, p. 34]) allows to conclude the
theorem. ∎

For every $i=1,2,...,2n+1$, $X_{i}$ denotes the left-invariant vector field on
$\mathbb{H}^{n}$, which are given by

$X_{i}=\frac{\partial}{\partial
x_{i}}+\frac{x_{i+n}}{2}\frac{\partial}{\partial t},\,\,\,\,i=1,2,...,n;$
$X_{i+n}=\frac{\partial}{\partial
x_{i+n}}-\frac{x_{i}}{2}\frac{\partial}{\partial t},\,\,\,i=1,2,...,n;$

and

$X_{2n+1}=\frac{\partial}{\partial t}.$

Given a multiindex
$I=(i_{1},i_{2},...,i_{2n},i_{2n+1})\in(\mathbb{N}\cup\\{0\\})^{2n+1}$, we set

$|I|=i_{1}+i_{2}+\cdot\cdot\cdot+i_{2n}+i_{2n+1},\hskip
14.22636ptd(I)=i_{1}+i_{2}+\cdot\cdot\cdot+i_{2n}+2\,i_{2n+1}.$

The amount $|I|$ is called the length of $I$ and $d(I)$ the homogeneous degree
of $I$. We adopt the following multiindex notation for higher order
derivatives and for monomials on $\mathbb{H}^{n}$. If
$I=(i_{1},i_{2},...,i_{2n+1})$ is a multiindex, $X=\\{X_{i}\\}_{i=1}^{2n+1}$,
and $z=(x,t)=(x_{1},...,x_{2n},t)\in\mathbb{H}^{n}$, we put

$X^{I}:=X_{1}^{i_{1}}X_{2}^{i_{2}}\cdot\cdot\cdot
X_{2n+1}^{i_{2n+1}},\,\,\,\,\,\,\text{and}\,\,\,\,\,\,z^{I}:=x_{1}^{i_{1}}\cdot\cdot\cdot
x_{2n}^{i_{2n}}\cdot t^{i_{2n+1}}.$

A computation give

$X^{I}(f(r\cdot z))=r^{d(I)}(X^{I}f)(r\cdot
z),\,\,\,\,\,\,\text{and}\,\,\,\,\,\,(r\cdot z)^{I}=r^{d(I)}z^{I}.$

So, the operators $X^{I}$ and the monomials $z^{I}$ are homogeneous of degree
$d(I)$.

Next, we establish an estimate for higher order derivatives (in the variable
$(x,t)$) of the kernel
$\prod_{j=1}^{m}\rho\left((A_{j}y,r_{j}^{-2}s)^{-1}\cdot(x,t)\right)^{-\alpha_{j}}$.

######  Proposition 2.5.

For every $N\in\mathbb{N}$,

$\displaystyle{\left|X^{I}\left(\prod_{j=1}^{m}\rho\left((A_{j}y,r_{j}^{-2}s)^{-1}\cdot(x,t)\right)^{-\alpha_{j}}\right)\right|}$

$\leq
C\left(\prod_{j=1}^{m}\rho\left((A_{j}y,r_{j}^{-2}s)^{-1}\cdot(x,t)\right)^{-\alpha_{j}}\right)\left(\sum_{j=1}^{m}\rho\left((A_{j}y,r_{j}^{-2}s)^{-1}\cdot(x,t)\right)^{-1}\right)^{d(I)}$

holds for all multiindex $I$ such that $d(I)\leq N$, where $C$ is a positive
constant which does not depend on $(y,s)$ and $(x,t)$.

###### Proof..

We fix $N\in\mathbb{N}$. From the left-invariance and the homogeneity of
$X^{I}$ and the homogeneity of the function $\rho(\cdot)^{-\alpha_{j}}$ we
obtain that

(10)
$\left|X^{I}\left(\rho\left((A_{j}y,r_{j}^{-2}s)^{-1}\cdot(x,t)\right)^{-\alpha_{j}}\right)\right|\leq
C_{I,\,j}\,\rho\left((A_{j}y,r_{j}^{-2}s)^{-1}\cdot(x,t)\right)^{-\alpha_{j}-d(I)},$

for every multiindex $I$ such that $d(I)\leq N$ and every $1\leq j\leq m$. For
$m=2$, (10) leads to

$\left|X^{I}\left(\displaystyle{\prod_{j=1}^{2}}\rho\left((A_{j}y,r_{j}^{-2}s)^{-1}\cdot(x,t)\right)^{-\alpha_{j}}\right)\right|$

$\leq
C\left(\prod_{j=1}^{2}\rho\left((A_{j}y,r_{j}^{-2}s)^{-1}\cdot(x,t)\right)^{-\alpha_{j}}\right)\left(\sum_{j=1}^{2}\rho\left((A_{j}y,r_{j}^{-2}s)^{-1}\cdot(x,t)\right)^{-1}\right)^{d(I)},$

for every multiindex $I$ such that $d(I)\leq N$. Finally, applying induction
on $m$, the proposition follows. ∎

##  3 Basics on variable Lebesgue and Hardy spaces

Let $p(\cdot):\mathbb{H}^{n}\to(0,\infty)$ be a measurable function. Given a
measurable set $E\subset\mathbb{H}^{n}$, let

$p_{-}(E)=\mathop{\rm ess\,inf}_{(x,t)\in
E}p(x,t),\,\,\,\,\text{and}\,\,\,\,p_{+}(E)=\mathop{\rm ess\,sup}_{(x,t)\in
E}p(x,t).$

When $E=\mathbb{H}^{n}$, we will simply write $p_{-}:=p_{-}(\mathbb{H}^{n})$,
$p_{+}:=p_{+}(\mathbb{H}^{n})$ and $\underline{p}:=\min\\{p_{-},1\\}$. Such
function $p(\cdot)$ is called an exponent function.

We say that an exponent function $p(\cdot):\mathbb{H}^{n}\to(0,\infty)$ such
that $0<p_{-}\leq p_{+}<\infty$ belongs to
$\mathcal{P}^{\log}(\mathbb{H}^{n})$, if there exist positive constants $C$,
$C_{\infty}$ and $p_{\infty}$ such that $p(\cdot)$ satisfies the local log-
Hölder continuity condition, i.e.:

(11)
$|p(x,t)-p(y,s)|\leq\frac{C}{-\log(\rho((x,t)^{-1}\cdot(y,s))},\,\,\,\text{for}\,\,\rho((x,t)^{-1}\cdot(y,s))\leq\frac{1}{2},$

and is log-Hölder continuous at infinity, i.e.:

(12)
$|p(x,t)-p_{\infty}|\leq\frac{C_{\infty}}{\log(e+\rho(x,t))},\,\,\,\text{for
all}\,\,(x,t)\in\mathbb{H}^{n}.$

Here $\rho$ is the Koranyi norm given by (6).

Given an exponent function $p(\cdot)$ on $\mathbb{H}^{n}$, we define the
variable Lebesgue space $L^{p(\cdot)}=L^{p(\cdot)}(\mathbb{H}^{n})$ to be the
set of all measurable functions $f:\mathbb{H}^{n}\to\mathbb{C}$ such that for
some $\lambda>0$

$\int_{\mathbb{H}^{n}}|f(x,t)/\lambda|^{p(x,t)}dx\,dt<\infty.$

This becomes a quasi normed space when equipped with the Luxemburg norm

$\|f\|_{L^{p(\cdot)}}=\inf\left\\{\lambda>0:\int_{\mathbb{H}^{n}}|f(x,t)/\lambda|^{p(x,t)}dx\,dt\leq
1\right\\}.$

The following result follows from the definition of the $L^{p(\cdot)}$ \-
norm.

######  Lemma 3.1.

Given a measurable function $p(\cdot):\mathbb{H}^{n}\to(0,\infty)$ with
$0<p_{-}\leq p_{+}<\infty$, then
(i) $\|f\|_{L^{p(\cdot)}}\geq 0$ and $\|f\|_{L^{p(\cdot)}}=0$ if and only if
$f\equiv 0$ a.e.,
(ii) $\|cf\|_{L^{p(\cdot)}}=|c|\|f\|_{L^{p(\cdot)}}$ for all $f\in
L^{p(\cdot)}$ and all $c\in\mathbb{C}$,
(iii) $\|f+g\|_{L^{p(\cdot)}}\leq
2^{1/\underline{p}-1}(\|f\|_{L^{p(\cdot)}}+\|g\|_{L^{p(\cdot)}})$ for all
$f,g\in L^{p(\cdot)}$,
(iv) $\|f\|_{L^{p(\cdot)}}^{s}=\||f|^{s}\|_{L^{p(\cdot)/s}}$ for every $s>0$.

Next we shall define the variable Hardy spaces on $\mathbb{H}^{n}$. For them,
we introduce the Schwartz space $\mathcal{S}(\mathbb{H}^{n})$ which is defined
by

$\mathcal{S}(\mathbb{H}^{n})=\left\\{\phi\in
C^{\infty}(\mathbb{H}^{n}):\sup_{(x,t)\in\mathbb{H}^{n}}(1+\rho(x,t))^{N}|(X^{I}f)(x,t)|<\infty\,\,\,\forall\,\,N\in\mathbb{N}_{0},\,I\in(\mathbb{N}_{0})^{2n+1}\right\\}.$

We topologize the space $\mathcal{S}(\mathbb{H}^{n})$ with the following
family of seminorms

$\|f\|_{\mathcal{S}(\mathbb{H}^{n}),\,N}=\sum_{d(I)\leq
N}\sup_{(x,t)\in\mathbb{H}^{n}}(1+\rho(x,t))^{N}|(X^{I}f)(x,t)|\,\,\,\,\,\,\,(N\in\mathbb{N}_{0}),$

with $\mathcal{S}^{\prime}(\mathbb{H}^{n})$ we denote the dual space of
$\mathcal{S}(\mathbb{H}^{n})$.

Given $N\in\mathbb{N}$, define

$\mathcal{F}_{N}=\left\\{\varphi\in\mathcal{S}(\mathbb{H}^{n}):\sum\limits_{d(I)\leq
N}\sup\limits_{(x,t)\in\mathbb{H}^{n}}\left(1+\rho(x,t)\right)^{N}|(X^{I}\varphi)(x,t)|\leq
1\right\\}.$

For any $f\in\mathcal{S}^{\prime}(\mathbb{H}^{n})$, the grand maximal function
of $f$ is given by

$\mathcal{M}_{N}f(x,t)=\sup\limits_{s>0}\sup\limits_{\varphi\in\mathcal{F}_{N}}\left|\left(f\ast\varphi_{s}\right)(x,t)\right|,$

where $\varphi_{s}(x,t)=s^{-2n-2}\varphi(s^{-1}\cdot(x,t))$.

######  Definition 3.2.

Given an exponent function $p(\cdot):\mathbb{H}^{n}\to(0,\infty)$ with
$0<p_{-}\leq p_{+}<\infty$, we define the integer $\mathcal{D}_{p(\cdot)}$ by

$\mathcal{D}_{p(\cdot)}:=\min\\{k\in\mathbb{N}\cup\\{0\\}:(2n+k+3)p_{-}>2n+2\\}.$

For $N\geq\mathcal{D}_{p(\cdot)}+1$, define the variable Hardy space
$H^{p(\cdot)}(\mathbb{H}^{n})$ to be the collection of
$f\in\mathcal{S}^{\prime}(\mathbb{H}^{n})$ such that
$\|\mathcal{M}_{N}f\|_{L^{p(\cdot)}(\mathbb{H}^{n})}<\infty$. Then, the ”norm”
on the space $H^{p(\cdot)}(\mathbb{H}^{n})$ is taken to be
$\|f\|_{H^{p(\cdot)}}:=\|\mathcal{M}_{N}f\|_{L^{p(\cdot)}}$.

######  Definition 3.3.

Let $p(\cdot):\mathbb{H}^{n}\to(0,\infty)$, $0<p_{-}\leq p_{+}<\infty$, and
$p_{0}>1$. Fix an integer $D\geq\mathcal{D}_{p(\cdot)}$. A measurable function
$a(\cdot)$ on $\mathbb{H}^{n}$ is called a $(p(\cdot),p_{0},D)$ \- atom
centered at a $\rho$ \- ball $B=B_{\delta}(x_{0},t_{0})$ if it satisfies the
following conditions:

$a_{1})$ $\mathop{\rm supp}(a)\subset B$,

$a_{2})$
$\|a\|_{L^{p_{0}}(\mathbb{H}^{n})}\leq\displaystyle{\frac{|B|^{\frac{1}{p_{0}}}}{\|\chi_{B}\|_{L^{p(\cdot)}(\mathbb{H}^{n})}}}$,

$a_{3})$ $\displaystyle{\int_{\mathbb{H}^{n}}}a(z)\,z^{I}\,dz=0$ for all
multiindex $I$ such that $d(I)\leq D$ (here $z=(x,t)$).

Indeed, every $(p(\cdot),p_{0},D)$ \- atom $a(\cdot)$ belongs to
$H^{p(\cdot)}(\mathbb{H}^{n})$. Moreover, there exists an universal constant
$C>0$ such that $\|a\|_{H^{p(\cdot)}}\leq C$ for all $(p(\cdot),p_{0},D)$ \-
atom $a(\cdot)$.

######  Remark 3.4.

It is easy to check that if $a(\cdot)$ is a $(p(\cdot),p_{0},D)$ \- atom
centered at the ball $B_{\delta}(x_{0},t_{0})$, then the function
$a_{(x_{0},t_{0})}(\cdot):=a((x_{0},t_{0})\cdot(\cdot))$ is a
$(p(\cdot),p_{0},D)$ \- atom centered at the ball $B_{\delta}(0,0)$.

######  Definition 3.5.

Let $p(\cdot):\mathbb{H}^{n}\to(0,\infty)$ be an exponent function such that
$0<p_{-}\leq p_{+}<\infty$. Given a sequence of nonnegative numbers
$\\{\lambda_{j}\\}_{j=1}^{\infty}$ and a family of $\rho$ \- balls
$\\{B_{j}\\}_{j=1}^{\infty}$, we define

(13)
$\mathcal{A}\left(\\{\lambda_{j}\\}_{j=1}^{\infty},\\{B_{j}\\}_{j=1}^{\infty},p(\cdot)\right):=\left\|\left\\{\sum_{j=1}^{\infty}\left(\frac{\lambda_{j}\chi_{B_{j}}}{\|\chi_{B_{j}}\|_{L^{p(\cdot)}}}\right)^{\underline{p}}\right\\}^{1/\underline{p}}\right\|_{L^{p(\cdot)}}.$

To prove our main result we will need the following additional estimate.

######  Lemma 3.6.

Let $A\in SO(2n)$, $r>0$,
$p(\cdot):\mathbb{H}^{n}\rightarrow(0,\infty)\in\mathcal{P}^{\log}(\mathbb{H}^{n})$
such that $0<p_{-}\leq p_{+}<\infty$, and let
$\\{B_{j}=B_{\delta_{j}}(x_{j}^{0},t_{j}^{0})\\}_{j=1}^{\infty}$ be a sequence
of $\rho$ \- balls of $\mathbb{H}^{n}$.

$(i)$ If $p(Ax,t)=p(x,t)$ for all $(x,t)\in\mathbb{H}^{n}$ and
$B_{j}^{\ast}=B_{\gamma\delta_{j}}(Ax_{j}^{0},t_{j}^{0})$ for each $j\geq 1$,
where $\gamma\geq 1$, then

$\mathcal{A}\left(\left\\{\lambda_{j}\right\\}_{j=1}^{\infty},\left\\{B_{j}^{\ast}\right\\}_{j=1}^{\infty},p(\cdot)\right)\lesssim\mathcal{A}\left(\left\\{\lambda_{j}\right\\}_{j=1}^{\infty},\left\\{B_{j}\right\\}_{j=1}^{\infty},p(\cdot)\right),$

holds for all sequences of nonnegative numbers
$\\{\lambda_{j}\\}_{j=1}^{\infty}$ and $\rho$ \- balls
$\\{B_{j}\\}_{j=1}^{\infty}$.

$(ii)$ If $p(rx,r^{2}t)=p(x,t)$ for all $(x,t)\in\mathbb{H}^{n}$ and
$B_{j}^{\ast}=B_{\gamma\delta_{j}}(rx_{j}^{0},r^{2}t_{j}^{0})$ for each $j\geq
1$, where $\gamma\geq 1$, then

$\mathcal{A}\left(\left\\{\lambda_{j}\right\\}_{j=1}^{\infty},\left\\{B_{j}^{\ast}\right\\}_{j=1}^{\infty},p(\cdot)\right)\lesssim\mathcal{A}\left(\left\\{\lambda_{j}\right\\}_{j=1}^{\infty},\left\\{B_{j}\right\\}_{j=1}^{\infty},p(\cdot)\right),$

holds for all sequences of nonnegative numbers
$\\{\lambda_{j}\\}_{j=1}^{\infty}$ and $\rho$ \- balls
$\\{B_{j}\\}_{j=1}^{\infty}$.

###### Proof..

We start proving $(i)$. Since $p(Ax,t)=p(x,t)$ for all
$(x,t)\in\mathbb{H}^{n}$ and $A\in SO(2n)$, a change of variable gives

$\mathcal{A}\left(\left\\{\lambda_{j}\right\\}_{j=1}^{\infty},\left\\{B_{j}^{\ast}\right\\}_{j=1}^{\infty},p(\cdot)\right)=\left\|\left\\{\sum\limits_{j=1}^{\infty}\left(\frac{\lambda_{j}\chi_{B_{j}^{\ast}}}{\left\|\chi_{B_{j}\ast}\right\|_{p(\cdot)}}\right)^{\underline{p}}\right\\}^{\frac{1}{\underline{p}}}\right\|_{p(\cdot)}$
$\approx\left\|\left\\{\sum\limits_{j=1}^{\infty}\left(\frac{\lambda_{j}\chi_{\gamma
B_{j}}}{\left\|\chi_{\gamma
B_{j}}\right\|_{p(\cdot)}}\right)^{\underline{p}}\right\\}^{\frac{1}{\underline{p}}}\right\|_{p(\cdot)}=:\mathcal{A}\left(\left\\{\lambda_{j}\right\\}_{j=1}^{\infty},\left\\{\gamma
B_{j}\right\\}_{j=1}^{\infty},p(\cdot)\right),$

where $\gamma B_{j}$ is the $\rho$ \- ball with the same center as $B_{j}$ but
whose radius is expanded by the factor $\gamma\geq 1$. It is clear that
$B_{j}\subset\gamma B_{j}$ for all $j$. Now, the argument utilized in the
proof of Lemma 4.8 in [8] works on $\mathbb{H}^{n}$ as well, but taking into
account [2, Theorem 4.2] instead of [8, Lemma 2.4]. So

$\mathcal{A}\left(\left\\{\lambda_{j}\right\\}_{j=1}^{\infty},\left\\{\gamma
B_{j}\right\\}_{j=1}^{\infty},p(\cdot)\right)\lesssim\mathcal{A}\left(\left\\{\lambda_{j}\right\\}_{j=1}^{\infty},\left\\{B_{j}\right\\}_{j=1}^{\infty},p(\cdot)\right).$

The proof of $(ii)$ is similar. ∎

To get our main result we also need the following version of the atomic
decomposition for $H^{p(\cdot)}(\mathbb{H}^{n})$ obtained in [2] (see also
[10, Theorem 5.5]).

######  Theorem 3.7.

Let $1<p_{0}<\infty$, $p(\cdot)\in\mathcal{P}^{\log}(\mathbb{H}^{n})$ with
$0<p_{-}\leq p_{+}<\infty$. Then, for every $f\in
H^{p(\cdot)}(\mathbb{H}^{n})\cap L^{p_{0}}(\mathbb{H}^{n})$ and every integer
$D\geq\mathcal{D}_{p(\cdot)}$ fixed, there exist a sequence of nonnegative
numbers $\\{\lambda_{j}\\}_{j=1}^{\infty}$, a sequence of $\rho$ \- balls
$\\{B_{j}\\}_{j=1}^{\infty}$ with the bounded intersection property and
$(p(\cdot),p_{0},D)$ \- atoms $a_{j}$ supported on $B_{j}$ such that
$f=\displaystyle{\sum_{j=1}^{\infty}\lambda_{j}a_{j}}$ converges in
$L^{p_{0}}(\mathbb{H}^{n})$ and

(14)
$\mathcal{A}\left(\\{\lambda_{j}\\}_{j=1}^{\infty},\\{B_{j}\\}_{j=1}^{\infty},p(\cdot)\right)\lesssim\|f\|_{H^{p(\cdot)}(\mathbb{H}^{n})},$

where the implicit constant in (14) is independent of
$\\{\lambda_{j}\\}_{j=1}^{\infty}$, $\\{B_{j}\\}_{j=1}^{\infty}$, and $f$.

######  Proposition 3.8.

Let $1<p_{0}<\infty$ and $p(\cdot)\in\mathcal{P}^{\log}(\mathbb{H}^{n})$ with
$0<p_{-}\leq p_{+}<\infty$. Then $H^{p(\cdot)}(\mathbb{H}^{n})\cap
L^{p_{0}}(\mathbb{H}^{n})\subset H^{p(\cdot)}(\mathbb{H}^{n})$ densely.

###### Proof..

The proof is similar to the one given in [8, see p. 3693]. ∎

##  4 Main result

In this section we will prove that the generalized Riesz operator
$T_{\alpha,m}$ on $\mathbb{H}^{n}$ can be extended to a bounded operator from
variable Hardy spaces into variable Lebesgue spaces. The main tools that we
will use are Theorem 2.4, Proposition 2.5 and the atomic decomposition of
$H^{p(\cdot)}(\mathbb{H}^{n})$ established in Theorem 3.7.

######  Theorem 4.1.

Let $0\leq\alpha<Q$, $m\in\mathbb{N}\cap(1-\frac{\alpha}{Q},\infty)$ and let
$T_{\alpha,\,m}$ be the operator defined by (3). Suppose
$p(\cdot)\in\mathcal{P}^{\log}(\mathbb{H}^{n})$ such that
$p(A_{j}x,r_{j}^{-2}t)=p(x,t)$ for every $j=1,...,m$ and $0<p_{-}\leq
p_{+}<\frac{Q}{\alpha}$. If
$\frac{1}{q(\cdot)}=\frac{1}{p(\cdot)}-\frac{\alpha}{Q}$, then
$T_{\alpha,\,m}$ can be extended to a bounded operator from
$H^{p(\cdot)}(\mathbb{H}^{n})$ into $L^{q(\cdot)}(\mathbb{H}^{n})$.

###### Proof..

We take $N\in\mathbb{N}$ such that $0<\frac{Q}{Q+N}<p_{-}$, thus
$N-1\geq\mathcal{D}_{p(\cdot)}$. Then, given $f\in
H^{p(\cdot)}(\mathbb{H}^{n})\cap L^{p_{0}}(\mathbb{H}^{n})$ (with $p_{0}>1$),
by Theorem 3.7 with $D=N-1$, there exist a sequence of nonnegative numbers
$\\{\lambda_{j}\\}_{j=1}^{\infty}$, a sequence of $\rho$ \- balls
$\\{B_{j}\\}_{j=1}^{\infty}$ and $(p(\cdot),p_{0},N-1)$ atoms $a_{j}$
supported on $B_{j}$ such that
$f=\displaystyle{\sum_{j=1}^{\infty}\lambda_{j}a_{j}}$ converges in
$L^{p_{0}}(\mathbb{H}^{n})$ and

(15)
$\mathcal{A}\left(\\{\lambda_{j}\\}_{j=1}^{\infty},\\{B_{j}\\}_{j=1}^{\infty},p(\cdot)\right)\lesssim\|f\|_{H^{p(\cdot)}(\mathbb{H}^{n})}.$

If $0<\alpha<Q$, we take $\max\\{1,p_{+}\\}<p_{0}<\frac{Q}{\alpha}$. If
$\alpha=0$, we take $p_{0}=2$. Then, by Theorem 2.4, the operator
$T_{\alpha,m}$ is bounded from $L^{p_{0}}(\mathbb{H}^{n})$ to
$L^{q_{0}}(\mathbb{H}^{n})$ for $1<p_{0}<\frac{Q}{\alpha}$ and
$\frac{1}{q_{0}}=\frac{1}{p_{0}}-\frac{\alpha}{Q}$. Since
$f=\displaystyle{\sum_{j=1}^{\infty}\lambda_{j}a_{j}}$ converges in
$L^{p_{0}}(\mathbb{H}^{n})$, we have

$|T_{\alpha,\,m}f(x,t)|\leq\sum_{j}\lambda_{j}|T_{\alpha,\,m}a_{j}(x,t)|,\,\,\,a.e.\,\,(x,t)\in\mathbb{H}^{n}.$

Let $\beta$ be the constant given in [4, Corollary 1.44], we observe that
$\beta\geq 1$ (see [4, p. 29]), and let $\gamma:=(1+\max_{1\leq i\leq
m}\\{r_{i}\\})/(\min_{1\leq i\leq m}\\{r_{i}\\})$. Given an atom $a_{j}$
supported on the $\rho$ \- ball $B_{j}=B_{\delta_{j}}(x^{0}_{j},t^{0}_{j})$,
for every $i=1,...,m$ let $B_{j\,i}^{\ast}=2\beta^{N}\gamma
B_{\delta_{j}}(A_{i}x^{0}_{j},r_{i}^{-2}t^{0}_{j})$, where $2\beta^{N}\gamma
B$ is the $\rho$ \- ball with the same center as $B$ but whose radius is
expanded by the factor $2\beta^{N}\gamma$. Now we decompose
$\mathbb{H}^{n}=\cup_{i=1}^{m}B_{j\,i}^{\ast}\cup\Omega_{j}$, where
$\Omega_{j}=\left(\cup_{i=1}^{m}B_{j\,i}^{\ast}\right)^{c}$. Then, for
$\frac{1}{q(\cdot)}=\frac{1}{p(\cdot)}-\frac{\alpha}{Q}$

(16)
$\|T_{\alpha,\,m}f\|_{L^{q(\cdot)}}\leq\sum_{i=1}^{m}\left\|\sum_{j=1}^{\infty}\lambda_{j}\chi_{B_{j\,i}^{\ast}}|T_{\alpha,\,m}a_{j}|\right\|_{L^{q(\cdot)}}+\left\|\sum_{j=1}^{\infty}\lambda_{j}\chi_{\Omega_{j}}|T_{\alpha,\,m}a_{j}|\right\|_{L^{q(\cdot)}}=:J_{1}+J_{2},$

To estimate $J_{1}$ we once again apply, for the case $0<\alpha<Q$, Theorem
2.4 with $q_{0}>\max\\{\frac{Q}{Q-\alpha},q_{+}\\}$ and
$\frac{1}{p_{0}}:=\frac{1}{q_{0}}+\frac{\alpha}{Q}$ (or with $q_{0}=p_{0}=2$,
if $\alpha=0$). So,

$\left\|(T_{\alpha,\,m}a_{j})^{q_{*}}\right\|_{L^{q_{0}/q_{*}}(B_{j\,i}^{\ast})}=\left\|T_{\alpha,m}a_{j}\right\|_{L^{q_{0}}(B_{j\,i}^{\ast})}^{q_{*}}\lesssim\left\|a_{j}\right\|_{L^{p_{0}}}^{q_{*}}\lesssim\frac{|B_{j}|^{\frac{q_{*}}{p_{0}}}}{\left\|\chi_{B_{j}}\right\|_{L^{p(\cdot)}}^{q_{*}}}\lesssim\frac{\left|B_{j\,i}^{\ast}\right|^{\frac{q_{*}}{q_{0}}}}{\left\|\chi_{B_{j\,i}^{\ast}}\right\|_{L^{q(\cdot)/q_{*}}}},$

where $0<q_{*}<\underline{q}$ is fixed and the last inequality follows from
the estimate
$\|\chi_{B}\|_{L^{q(\cdot)}}\approx|B|^{-\alpha/Q}\|\chi_{B}\|_{L^{p(\cdot)}}$
(which is a consequence of [2, Lemma 4.1]), Lemma 3.1-(iv), and Lemma 3.2 in
[10] applied to the exponent $q(\cdot)/q_{*}$. Now, since $0<q_{*}<1$, we
apply the $q_{\ast}$-inequality and Proposition 3.3 in [10] with
$b_{j}=\left(\chi_{B_{j\,i}^{\ast}}\cdot|T_{\alpha,\,m}a_{j}|\right)^{q_{*}}$,
$A_{j}=\left\|\chi_{B_{j\,i}^{\ast}}\right\|_{L^{q(\cdot)/q_{*}}}^{-1}$ and
$s=q_{0}/q_{*}$, to obtain

$J_{1}\lesssim\sum_{i=1}^{m}\left\|\sum_{j}\left(\lambda_{j}\,\chi_{B_{j\,i}^{\ast}}\,|T_{\alpha,\,m}a_{j}|\right)^{q_{*}}\right\|^{1/q_{*}}_{L^{q(\cdot)/q_{*}}}\lesssim\sum_{i=1}^{m}\left\|\sum_{j}\left(\frac{\lambda_{j}}{\left\|\chi_{B_{j\,i}^{\ast}}\right\|_{L^{q(\cdot)}}}\right)^{q_{*}}\chi_{B_{j\,i}^{\ast}}\right\|^{1/q_{*}}_{L^{q(\cdot)/q_{*}}}.$

From this inequality, Lemma 5.7 in [10] and Lemma 3.6 applied to $q(\cdot)$
give

$J_{1}\lesssim\sum_{i=1}^{m}\mathcal{A}\left(\\{\lambda_{j}\\}_{j=1}^{\infty},\\{B_{j\,i}^{\ast}\\}_{j=1}^{\infty},q(\cdot)\right)\lesssim
m\,\mathcal{A}\left(\\{\lambda_{j}\\}_{j=1}^{\infty},\\{B_{j}\\}_{j=1}^{\infty},q(\cdot)\right).$

Then, Proposition 5.8 in [10] and (15) give

(17)
$J_{1}\lesssim\mathcal{A}\left(\\{\lambda_{j}\\}_{j=1}^{\infty},\\{B_{j}\\}_{j=1}^{\infty},q(\cdot)\right)\lesssim\mathcal{A}\left(\\{\lambda_{j}\\}_{j=1}^{\infty},\\{B_{j}\\}_{j=1}^{\infty},p(\cdot)\right)\lesssim\|f\|_{H^{p(\cdot)}}.$

Now, we proceed to estimate $J_{2}$. For them, we first consider a single
$(p(\cdot),p_{0},N-1)$ \- atom $a(\cdot)$ supported on the $\rho$ \- ball
$B=B_{\delta}(x_{0},t_{0})$ and let
$\widetilde{\Omega}=\left(\cup_{i=1}^{m}B_{i}^{\ast}\right)^{c}$, where
$B_{i}^{\ast}=2\beta^{N}\gamma B_{\delta}(A_{i}x_{0},r_{i}^{-2}t_{0})$. Then,
we decompose $\widetilde{\Omega}=\cup_{k=1}^{m}\widetilde{\Omega}_{k}$ with

$\widetilde{\Omega}_{k}=\left\\{(x,t)\in\widetilde{\Omega}:\rho\left((A_{k}^{-1}x,r_{k}^{2}t)^{-1}\cdot(x_{0},t_{0})\right)\leq\rho\left((A_{j}^{-1}x,r_{j}^{2}t)^{-1}\cdot(x_{0},t_{0})\right)\,\,\text{for
all}\,\,j\neq k\right\\}.$

Remark 2.2 and a change of variable lead to

$\displaystyle T_{\alpha,m}a(x,t)$ $\displaystyle=$
$\displaystyle\int_{B_{\delta}(x_{0},t_{0})}a(y,s)\prod_{j=1}^{m}\rho\left((A_{j}y,r_{j}^{-2}s)^{-1}\cdot(x,t)\right)^{-\alpha_{j}}\,dy\,ds$
$\displaystyle=$ $\displaystyle
C\int_{B_{\delta}(0,0)}a((x_{0},t_{0})\cdot(y,s))\prod_{j=1}^{m}\rho\left((A_{j}^{-1}x,r_{j}^{2}t)^{-1}\cdot(x_{0},t_{0})\cdot(y,s)\right)^{-\alpha_{j}}\,dy\,ds,$

for all $(x,t)\in\widetilde{\Omega}$, where
$C=r_{1}^{-\alpha_{1}}\cdot\cdot\cdot r_{m}^{-\alpha_{1}}$. By Remark 3.4, for
each $(x,t)\in\widetilde{\Omega}$, it follows that

(18) $T_{\alpha,m}a(x,t)=C\int_{B_{\delta}(0,0)}a((x_{0},t_{0})\cdot(y,s))$
$\times\left[\prod_{j=1}^{m}\rho\left((A_{j}^{-1}x,r_{j}^{2}t)^{-1}\cdot(x_{0},t_{0})\cdot(y,s)\right)^{-\alpha_{j}}-q_{N}(y,s)\right]\,dy\,ds,$

where $(y,s)\to q_{N}(y,s)$ is the left Taylor polynomial of the function

$(y,s)\to\prod_{j=1}^{m}\rho\left((A_{j}^{-1}x,r_{j}^{2}t)^{-1}\cdot(x_{0},t_{0})\cdot(y,s)\right)^{-\alpha_{j}}$

at $e=(0,0)$ of homogeneous degree $N-1$. Then by the left Taylor inequality
in [4, Corollary 1.44],

(19)
$\left|\prod_{j=1}^{m}\rho\left((A_{j}^{-1}x,r_{j}^{2}t)^{-1}\cdot(x_{0},t_{0})\cdot(y,s)\right)^{-\alpha_{j}}-q_{N}(y,s)\right|$
$\lesssim\rho(y,s)^{N}\sup_{\rho(z,u)\leq\beta^{N}\rho(y,s),\,d(I)=N}\left|X^{I}\left(\prod_{j=1}^{m}\rho\left((A_{j}^{-1}x,r_{j}^{2}t)^{-1}\cdot(x_{0},t_{0})\cdot(z,u)\right)^{-\alpha_{j}}\right)\right|.$

Now, for $(y,s)\in B_{\delta}(0,0)$,
$(x,t)^{-1}\cdot(A_{k}x_{0},r_{k}^{-2}t_{0})\notin 2\beta^{N}\gamma
B_{\delta}(0,0)$ and $\rho(z,u)\leq\beta^{N}\rho(y,s)$, we have
$\rho((A_{k}^{-1}x,r_{k}^{2}t)^{-1}\cdot(x_{0},t_{0}))\geq
2r_{k}\gamma\rho(z,u)$, where $r_{k}\gamma>1$, and hence

(20)
$\rho\left((A_{k}^{-1}x,r_{k}^{2}t)^{-1}\cdot(x_{0},t_{0})\cdot(z,u)\right)\geq\frac{2r_{k}\gamma-1}{2r_{k}\gamma}\rho((A_{k}^{-1}x,r_{k}^{2}t)^{-1}\cdot(x_{0},t_{0})).$

Then for $(x,t)\in\widetilde{\Omega}_{k}$, by (19), Proposition 2.5 applied on
the variable $(z,u)$ and (20), we get

$\left|\prod_{j=1}^{m}\rho\left((A_{j}^{-1}x,r_{j}^{2}t)^{-1}\cdot(x_{0},t_{0})\cdot(y,s)\right)^{-\alpha_{j}}-q_{N}(y,s)\right|\lesssim\delta^{N}\rho\left((A_{k}^{-1}x,r_{k}^{2}t)^{-1}\cdot(x_{0},t_{0})\right)^{\alpha-
Q-N}.$

This estimate and (18) lead to

$\displaystyle|T_{\alpha,m}a(x,t)|$ $\displaystyle\lesssim$
$\displaystyle\delta^{N}\rho\left((A_{k}^{-1}x,r_{k}^{2}t)^{-1}\cdot(x_{0},t_{0})\right)^{\alpha-
Q-N}\|a\|_{L^{1}}$ $\displaystyle\lesssim$
$\displaystyle\delta^{N}\rho\left((A_{k}^{-1}x,r_{k}^{2}t)^{-1}\cdot(x_{0},t_{0})\right)^{\alpha-
Q-N}|B|^{1-\frac{1}{p_{0}}}\|a\|_{L^{p_{0}}}$ $\displaystyle\lesssim$
$\displaystyle\frac{\delta^{N+Q}}{\|\chi_{B}\|_{L^{p(\cdot)}}}\rho\left((A_{k}^{-1}x,r_{k}^{2}t)^{-1}\cdot(x_{0},t_{0})\right)^{\alpha-
Q-N}$ $\displaystyle\lesssim$ $\displaystyle\frac{\left(M_{\frac{\alpha
Q}{Q+N}}(\chi_{B})(A_{k}^{-1}x,r_{k}^{2}t)\right)^{\frac{Q+N}{Q}}}{\|\chi_{B}\|_{L^{p(\cdot)}}},\,\,\,\,\forall\,\,(x,t)\in\widetilde{\Omega}_{k}.$

Finally, by applying the above reasoning with $a_{j}(\cdot)$ instead of
$a(\cdot)$, $\widetilde{\Omega}=\Omega_{j}$ and $j\in\mathbb{N}$, we have that
$\Omega_{j}=\cup_{k=1}^{m}\Omega_{j\,k}$ and

(21) $|T_{\alpha,m}a_{j}(x,t)|\lesssim\frac{\left(M_{\frac{\alpha
Q}{Q+N}}(\chi_{B_{j}})(A_{k}^{-1}x,r_{k}^{2}t)\right)^{\frac{Q+N}{Q}}}{\|\chi_{B_{j}}\|_{L^{p(\cdot)}}},\,\,\,\,\text{for
all}\,\,(x,t)\in\Omega_{j\,k}.$

From (21), (16) and that $q(A_{k}x,r_{k}^{-1}t)=q(x,t)$ for all $(x,t)$ and
all $k=1,...,m$, we obtain

$J_{2}\lesssim\sum_{k=1}^{m}\left\|\sum_{j}\lambda_{j}\chi_{\Omega_{j\,k}}|T_{\alpha,m}a_{j}(\cdot)|\right\|_{L^{q(\cdot)}}\lesssim\sum_{k=1}^{m}\left\|\sum_{j}\lambda_{j}\chi_{\Omega_{j\,k}}\frac{\left(M_{\frac{\alpha
Q}{Q+N}}(\chi_{B_{j}})(A_{k}^{-1}\cdot,r_{k}^{2}\cdot)\right)^{\frac{Q+N}{Q}}}{\|\chi_{B_{j}}\|_{L^{p(\cdot)}}}\right\|_{L^{q(\cdot)}}$
$\lesssim m\left\|\left\\{\sum_{j}\lambda_{j}\frac{\left(M_{\frac{\alpha
Q}{Q+N}}(\chi_{B_{j}})(\cdot)\right)^{\frac{Q+N}{Q}}}{\|\chi_{B_{j}}\|_{L^{p(\cdot)}}}\right\\}^{\frac{Q}{Q+N}}\right\|_{L^{\frac{Q+N}{Q}q(\cdot)}}^{\frac{Q+N}{Q}}.$

Since $1<\frac{Q+N}{Q}p_{-}\leq\frac{Q+N}{Q}p_{+}<\frac{Q+N}{\alpha}$, Theorem
4.4 in [10] gives

(22)
$J_{2}\lesssim\left\|\left\\{\sum_{j}\lambda_{j}\frac{\chi_{B_{j}}}{\|\chi_{B_{j}}\|_{L^{p(\cdot)}}}\right\\}^{\frac{Q}{Q+N}}\right\|_{L^{\frac{Q+N}{Q}p(\cdot)}}^{\frac{Q+N}{Q}}\lesssim\left\|\sum_{j}\lambda_{j}\frac{\chi_{B_{j}}}{\|\chi_{B_{j}}\|_{L^{p(\cdot)}}}\right\|_{L^{p(\cdot)}}$
$\lesssim\left\|\left\\{\sum_{j}\left(\lambda_{j}\frac{\chi_{B_{j}}}{\|\chi_{B_{j}}\|_{L^{p(\cdot)}}}\right)^{\underline{p}}\right\\}^{1/\underline{p}}\right\|_{L^{p(\cdot)}}=\mathcal{A}\left(\\{\lambda_{j}\\}_{j=1}^{\infty},\\{B_{j}\\}_{j=1}^{\infty},p(\cdot)\right)\lesssim\|f\|_{H^{p(\cdot)}}.$

Then, (16), (17) and (22) allow us to conclude that

$\|T_{\alpha,\,m}f\|_{L^{q(\cdot)}(\mathbb{H}^{n})}\lesssim\|f\|_{H^{p(\cdot)}(\mathbb{H}^{n})},$

for all $f\in H^{p(\cdot)}(\mathbb{H}^{n})\cap L^{p_{0}}(\mathbb{H}^{n})$. So
the theorem follows from Proposition 3.8. ∎

######  Example 4.2.

Given $0<\alpha<Q$, let $r:\mathbb{R}\to(0,\,\infty)$ be an exponent function
such that $0<r_{-}\leq r_{+}<\frac{Q}{\alpha}$ and $r(\cdot)$ is log-Hölder
continuous (locally and at infinite) on $\mathbb{R}$, i.e.:

(23)
$|r(s)-r(u)|\leq\frac{C}{-\log(|s-u|)}\,\,\,\,\text{for}\,\,|s-u|\leq\frac{1}{2},$

and

(24) $|r(s)-r_{\infty}|\leq\frac{C_{\infty}}{\log(e+|s|)}\,\,\,\,\text{for
all}\,\,s\in\mathbb{R}.$

Let $p:\mathbb{H}^{n}\to(0,\,\infty)$ be the exponent function defined by

(25) $p(x,t)=r(\rho(x,t)),\,\,\,\,\text{for all}\,\,(x,t)\in\mathbb{H}^{n},$

where $\rho$ is the Koranyi norm on $\mathbb{H}^{n}$. It is clear that
$p(Ax,t)=p(x,t)$ for all $(x,t)\in\mathbb{H}^{n}$ and all $A\in
Sp(2n,\mathbb{R})\cap SO(2n)$. Since $\rho((x,t)^{-1})=\rho(x,t)$ and
$|\rho(x,t)-\rho(y,s)|\leq\rho((x,t)\cdot(y,s))$, (23) implies (11). That (24)
implies (12) it is obvious. So, the exponent function $p(\cdot)$ is log-Hölder
continuous (locally and at infinite) on $\mathbb{H}^{n}$. Then, the exponent
$p(\cdot)$ defined by (25) satisfies the hypotheses of Theorem 4.1.

## References

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# A Possible Influence on Standard Model of Quasars and Active Galactic Nuclei
in Strong Magnetic Field

Qiu-He Peng 11affiliationmark:  33affiliationmark:  , Jing-Jing, Liu
22affiliationmark:  , and Chi-Kang Chou 44affiliationmark<EMAIL_ADDRESS>and<EMAIL_ADDRESS>
###### Abstract

Recent observational evidence indicates that the center of our Milky Way
galaxy harbors a super-massive object with ultra-strong radial magnetic field
(Eatough et al., 2013). Here we demonstrate that the radiations observed in
the vicinity of the Galactic Center (GC) (Falcke & Marko, 2013) cannot be
emitted by the gas of the accretion disk since the accreting plasma is
prevented from approaching to the GC by the abnormally strong radial magnetic
field. These fields obstruct the infalling accretion flow from the inner
region of the disk and the central massive black hole in the standard model.
It is expected that the observed radiations near the GC can not be generated
by the central black hole. We also demonstrate that the observed ultra-strong
radial magnetic field near the GC (Eatough et al., 2013) can not be generated
by the generalized $\alpha$-turbulence type dynamo mechanism since preliminary
qualitative estimate in terms of this mechanism gives a magnetic field
strength six orders of magnitude smaller than the observed field strength at
$r=0.12$ pc. However, both these difficulties or the dilemma of the standard
model can be overcome if the central black hole in the standard model is
replaced by a model of a supper-massive star with magnetic monopoles ( SMSMM)
(Peng & Chou, 2001). Five predictions about the GC have been proposed in the
SMSMM model. Especially, three of them are quantitatively consistent with the
observations. They are: 1) Plenty of positrons are produced, the production
rate is $6\times 10^{42}e^{+}\rm{s}^{-1}$ or so, this prediction is confirmed
by the observation (Knödlseder et al., 2003); 2) The lower limit of the
observed ultra-strong radial magnetic field near the GC (Eatough et al.,
2013), is just good agreement with the predicted estimated radial magnetic
field from the SMSMM model, which really is an exclusive and a key prediction;
3) The observed power peaking of the thermal radiation is essentially the same
as the theoretical prediction from the SMSMM model. Furthermore, the observed
ultra-strong radial magnetic field in the vicinity of the GC may be considered
as the astronomical evidence for the existence of magnetic monopoles as
predicted by the particle physics.

††slugcomment: Not to appear in Nonlearned J., 45.00footnotetext:
Corresponding to Jing-Jing, Liu00footnotetext: Department of Astronomy,
Nanjing University, Nanjing, Jiangshu 210000, China.00footnotetext: College of
Marine Science and Technology, Hainan Tropical Ocean University, Sanya, Hainan
572022, China.00footnotetext: School of Physics and Space Science, China West
Normal University, Nanchong, 637009,China.00footnotetext: National
Astronomical Observatory, Chinese Academy of Sciences, Beijing, 100000, China.

Keywords quasars: general — galaxies: magnetic fields — Physical Date and
Processes: black hole physics.

##  1 Introduction

It is now generally believed that bright quasars observed at large red-shift
are supermassive and rapidly spinning black holes formed in the primordial
universe. The spectacularly huge luminosity is supplied by the black hole and
the surrounding accretion disk. In such models, magnetic fields play a very
important role. More specifically, the magnetic coupling between the central
black hole (or some supermassive stellar object) and the accretion disk enable
effective transport of energy and angular momentum between them. If the black
hole spins faster than the disk, energy and angular momentum can be extracted
from the black hole and transferred to the disk vie the pointing flux
(Blandford & Payne, 1982; Blandford & Znajek, 1977; Yuan & Narayan, 2014). It
is now well established that the transfer of energy and angular momentum in
such sping black hole and accretion disk system with magnetic coupling can
generate relativistic jet by the mechanism of Blandford & Znajek (1977) if the
energy source is the spinning black hole.

The accretion model of GC is hot accretion flow. One of the most important
progress in this field is the finding of strong wind from black hole hot
accretion flow. For example, reformulated the adiabatic inflow and outflow
solution (ADIOS) model for radiatively inefficient accretion flows, treating
the inflow and outflow zones on an equal footing. Narayan et al. (2012)
presented the results from two long-duration general relativistic magneto-
hydrodynamic (GRMHD) simulations of advection-dominated accretion around a
non-spinning black hole. However, accretion onto a super-massive black hole of
a rotating inflow is a particularly difficult problem to study because of the
wide range of length scales involved. By using the ZEUS code, Li et al. (2013)
run hydrodynamical simulations of rotating, axisymmetric accretion flows with
Bremsstrahlung cooling, considering solutions for which the centrifugal
balance radius significantly exceeds the Schwarzschild radius, with and
without viscous angular momentum transport.

On the other hand, Hydrodynamical (HD) and magnetohydrodynamics (MHD)
numerical simulations of hot accretion flows have indicated that the inflow
accretion rate decreases inward. Two models have been proposed to explain this
result. In the adiabatic inflow-outflow solution (ADIOS), this is because of
the loss of gas in the outflow. In the alternative convection-dominated
accretion flow model, it is thought that the flow is convectively unstable and
gas is locked in convective eddies. Some authors have discuss detaily these
problems (e.g.,Yuan (2015); Bu (2013); Yuan (2012)). Based on the no-outflow
assumption, Gu (2015) investigated steady-state, axisymmetric, optically thin
accretion flows in spherical coordinates. By comparing the vertically
integrated advective cooling rate with the viscous heating rate, They found
that the former is generally less than 30% of the latter, which indicates that
the advective cooling itself cannot balance the viscous heating.

Some researches show that the quasi-relativistic jets and nonrelativistic wind
may be produced by the joint action of Blandford & Payne (1982) and other
mechanisms and the energy source in these cases is rotational energy of the
accretion flow originated from the inner region of the disk (Yuan & Narayan,
2014). Moreover, in magnetically arrested disk models, the accreting plasma
drags in a strong poloidal magnetic field to the center such that the
resulting accumulated magnetic flux disrupts the axisymmetric accretion flow
at a relatively large magnetospheric radius (Narayan et al., 2003).
Furthermore, magnetic flux threading the black hole, rather than black hole
spin is the dominant factor in launching powerful jets and determining the
radio loudness of active galactic nuclei (AGN) (Sikora & Begelman, 2013).
Thus, in the current models just briefly delineated above, magnetic fields
always play a key role. In the absence of these fields, it is almost
impossible to construct such useful, elegant and realistic models for quasars
and AGN.

More latest observational evidence indicates that the center of our Milky Way
harbours the closest candidate for a supermassive black hole with strong
magnetic fields. Using multi-frequency measurements with several radio
telescopes, Eatough et al. (2013) showed that there is a dynamically relevant
magnetic field near the black hole. If this field is accreted down to the
event horizon, it provides enough magnetic flux to explain the observed
emission from the black hole, from radio to X-rays. In addition, Zamaninasab
et al. (2014) reported that jet magnetic field and accretion disk luminosity
are tightly correlated over seven orders of magnitude for a sample of 76
radio-loud active galaxies. They concluded that the jet-launching regions of
these radio-loud galaxies are threaded by dynamically important magnetic
fields, which will affect the disk properties. In this paper, we investigate
the plausible modifications of the standard models of quasars and AGN in light
of the very recent observational evidence for the important discovery of a
dynamically relevant magnetic field near the GC. In particular, we focus on
the possible origin of the strong magnetic field in the galactic nucleus and
study some of the important effects of the ultra-strong magnetic field. The
very recent astronomical observations concerning the strongly magnetized
super-massive central black hole are depicted in considerable detail in
Section 2. The key roles played by such observed ultrastrong radial magnetic
fields in the standard models due to the effects of these fields are
elaborated in Section 3. The possible origin of these strong magnetic fields
near the GC will be considered in detail in section 4. Our model of super-
massive star with magnetic monopoles (SMSMM)is given in Section 5. We show
explicitly there the generally accepted $\alpha$-turbulence dynamo mechanism
of Parker can not be used to generate the observed strong radial magnetic
field by a preliminary estimate in terms of the observed W51 data. However,
good agreement with observations may be achieved if the central black hole of
the standard model is replaced by a supermassive stellar object containing
magnetic monopoles (Peng & Chou, 2001). In these model, the production of the
strong radial magnetic fields can be naturally explained. We also discuss
other evidences against the black hole model of quasars and AGN in Section 6\.
Finally, in Section 7, we briefly summarize and emphasize our results.

##  2 Recent Astronomical Observations

New progress and some discoveries of radio astronomical observations near the
GC have been reported in recent years. The main performances of these are as
follows:

(i) The measurement of an abnormally strong radial magnetic field near the GC
has been reported by Eatough et al. (2013). The very important results are as
follows, in particular, at $r=0.12$ pc, the lower limit of the outward radial
magnetic field near the GC is:

$B\geq
8[\frac{RM}{66.960\rm{m^{-2}}}][\frac{n_{e}}{26\rm{cm^{-3}}}]^{-1}\rm{mG},$
(1)

where $n_{e}$ is the number density of electrons. At $r=0.12$ pc by Chandra
X-ray observation, $n_{e}\approx 26\rm{cm^{-3}}$(Baganoff et al., 2003). A
theoretically calculated electron density of radioactively inefficient
accretion flows for the accretion disk around the GC) at is about $20\sim
100\rm{cm^{-3}}$, near the observational value(Yuan et al., 2003).

It is well known that the interstellar magnetic field in the galaxy is usually
along the galactic spiral arms, and the average strength of the magnetic field
is about $1\rm{\mu}$Gauss. So the magnetic field shown in Eq.(1) is abnormally
strong.

(ii)Some radiation have been detected in the region near the central region of
the GC. We can see the review article written by Falcke & Marko (2013). First,
some radiations from the radio to the sub-mm wavelength band have been
detected in the region $(10-50)R_{g}$ around the central black hole with mass
$4.3\times 10^{6}\rm{M}_{\odot}$ for the GC. Second, Sgr A* is identified as a
surprisingly weak X-ray source by Chandra and it is inferred as radiated from
the region $\leq 10R_{g}$ due to the hour long timescale for some detected
weak X-ray flare and small NIR flare. Finally, The radio flux density shows a
flat-to-inverted spectrum, i.e., it rises slowly with frequency with the power
peaking around $10^{12}$Hz in the sub mm band.

$L_{\nu}=4\pi D^{2}_{\rm{SgrA^{*}}}S_{\nu},$ (2)

At GHz frequencies, $S_{\nu}\propto\nu^{\alpha},\alpha\sim 0.3\pm 0.1$, The
spectrum continues towards low frequencies ( 300MHz) with no sigh of
absorption. At higher frequencies the spectrum extends into Sub-mm wavelength
regime, where the spectrum peaks and then suddenly cuts-off (Falcke & Marko,
2013).

##  3 The Effect of the Observed Strong Radial Magnetic Field on the Standard
Model of Quasars and AGN

The most important effect of the observed ultra-strong radial magnetic field
near the GC is that the assumption made in standard models of the GC is
invalid. This is because in the presence of the strong radial magnetic field
in the vicinity of the GC will prevent material in the accretion disk from
approaching to the GC due to the magnetic freeze effect, when the kinematic
energy density of the material is less than the energy density of the magnetic
field, or the magnetic field is stronger than the Alphen critical value:

$B>B_{\rm{Alphen}}=(4\pi\rho v^{2}_{\rm{rot}})^{\frac{1}{2}},$ (3)

where $v_{rot}$ is the rotation velocity of the accretion disk around the GC
and $\rho$ is the mass density. Making use of the relation
$v_{\rm{rot}}/c=\sqrt{(R_{S}/r)}$, we then have

$\displaystyle B_{\rm{Alphen}}$ $\displaystyle=(4\pi\rho
v^{2}_{\rm{rot}})^{\frac{1}{2}}=[\frac{4\pi
nc^{2}}{N_{A}}]^{\frac{1}{2}}(\frac{R_{S}}{r})^{\frac{1}{2}}$ (4)
$\displaystyle\sim
13(\frac{n}{10^{4}\rm{cm^{-3}}})^{\frac{1}{2}}(\frac{R_{S}}{r})^{\frac{1}{2}}\rm{G}$

Here we note that magnetic field lines may be brought by a intense turbulent
plasma fluid. Turbulent Richardson advection brings field lines implosively
together from distances far apart to separations of the order of gyro radii
(Eyink et al., 2013). This effect of advection in rough velocity fields, which
appear non-differentiable in space, leads to line motions that are completely
indeterministic or ’spontaneously stochastic’. The turbulent breakdown of
standard flux freezing at scales greater than the ion gyro radius can explain
fast reconnection of very large-scale flux structures, both observed (solar
flares and coronal mass ejections) and predicted (the inner heliosheath,
accretion disks, $\gamma$-ray bursts and so on). For laminar plasma flows with
smooth velocity fields or for low turbulence intensity, stochastic flux
freezing reduces to the usual frozen-in condition (Eyink et al., 2013)

No violent active phenomena such as explosion or mass ejection has been
detected in the region near $r\approx 0.12$ pc , so the foregoing usual
frozen-in condition is valid. However, the lower limit of the outward radial
magnetic field at $r=0.12$ pc, from the GC is $B\geq
8[RM/(66.960\rm{m^{-2}})][n_{e}/26\rm{cm^{-3}}]^{-1}\rm{mG}$.

As is well known in the popular black hole model of the GC, the radiation from
the vicinity of the black hole originates from the inflowing material of the
accretion disk (Yuan & Narayan, 2014). However, the accretion plasma is
clearly prevented from approaching to the GC by the radial magnetic field in
the region near $r\sim 0.12$pc around the GC at least as explicitly
demonstrated above. Thus, the accretion material can’t reach the region near
the central black hole. Consequently, the radiations observed near the GC
cannot be emitted by the gas of the accretion disk. This is a dilemma of the
standard accretion disk model of black hole at the GC.

##  4 The Origin of the Radial Magnetic Field and the $\alpha$-turbulence
Dynamo Mechanism

Now another important question is how to generate the strong magnetic field
near the GC by use of the known common conventional physics.

A rotating magnetic instabilities (RMI) is proposed as a angular momentum
transfer mechanism in an ionized plasma stream for producing a thin hot
coronal plasma jet in the polar region out of the accretion disk (Balbus &
Hawley, 1991, 1998). The magnetic field will be enhanced due to increases of
RMI. Under normal circumstances, the magnetic field may be enhanced to that
the magnetic pressure reaches at about $1/10$ of the hot gas pressure in
general. But in a special case, the magnetic pressure may reach at near the
hot gas pressure.

The distribution of the enhanced magnetic field in the region $100R_{g}$
($R_{g}=R_{S}/2$ ) around the central black hole at the GC has been shown in
the Fig.4 of a review paper written by Yuan & Narayan (2014). You may see that
the enhance of the magnetic field is in the polar thin hot coronal plasma
region perpendicular to the main accretion disk. However, it is impossible to
produce the observed strong radial magnetic field ($\geq 8$mG ) at a distance
$r=0.12$ pc from the GC by RMI mechanism. The reasons are given as follows: 1)
The plasma with electron density, $n_{e}\approx 26\rm{cm^{-3}}$, is one on the
main accretion disk rather than one of the polar thin hot coronal plasma
outside the main disk. RMI mechanism is negligible for this region; 2) The
enhanced magnetic field is less than $1\mu$G , even though RMI is taken into
account.

The most famous dynamo known up to now is a type of $\alpha$-turbulence dynamo
mechanism firstly proposed by Parker in 1953 (Mestel, 1999; Chatteriee et al.,
2011) in the solar convection zone. The key idea of the $\alpha$-turbulence
dynamo mechanism is that the induced electro-dynamic potential of turbulence
is parallel to the magnetic field (Mestel, 1999)

$\vec{\varepsilon}=\alpha\vec{B},$ (5)

$\alpha\equiv\alpha(\sigma
t_{c},\overline{\vec{v}\cdot\vec{\omega}})=-\frac{\sigma
t_{c}}{3c}\overline{\vec{v}\cdot\bigtriangledown\times\vec{v}}=-\frac{\sigma
t_{c}}{3c}\overline{\vec{v}\cdot\vec{w}},$ (6)

where $\vec{\omega}=\bigtriangledown\times\vec{v}$ is the curl of the
turbulent velocity of the fluid, and it is approximately equivalent to the
large-scale vortex rotational angular velocity. $\sigma$ is the electrical
conductivity of the fluid and $t_{c}$ is the typical timescale of the
turbulence. Eq.(5) shows that the electro-dynamic potential of turbulence is
proportional to the scalar product of the turbulent velocity with vorticity.

In order to appreciate the important physical significance of the -turbulence
dynamo mechanism, it is appropriate at this point to elaborate more clearly
about the novel idea of the mechanism. It is easy to visualize the production
of the toroidal magnetic field by the stretching poloidal field lines due to
differential rotation, if an astronomical body has some internal differential
rotation and a poloidal field that can be stretched. However, if the poloidal
field cannot be sustained, it will eventually decay and the production of the
toroidal field will also stop. In a famous paper, Parker argued that in a
rotating stellar object, turbulent convective motions would be able to
complete the cycle by generating a poloidal field from a toroidal field. If
the convection takes place in a rotating stellar object, as a rising blob of
plasma expands it feels a Coriolis force so that the fluid motion become
helical in nature. The nearly frozen-in toroidal field is thus twisted so as
to yield poloidal field. The small scale poloidal loops so formed are
coalesced through reconnection to yield a large-scale poloidal magnetic field
because turbulent diffusion in partially ionized plasma can smoothen out the
magnetic fields of the loops. The poloidal and toroidal field can sustain each
other through a cyclic feedback process. Thus, briefly, the poloidal field can
be stretched by the differential rotation to generate the toroidal field
whereas the helical turbulence associated with convection in a rotating frame
to give back a field in the poloidal plane.

Historically, the original treatment of Parker was very much based on
intuitive arguments. A formal and systematic approach was developed later by
Steenbeck, Krause and Rädler, known as mean field magneto-hydrodynamics. The
most important physical quantity in this mathematical theory is the mean
e.m.f. (or electrical potential) $\vec{\varepsilon}$ induced by the
fluctuating flow $\vec{v}$ and magnetic fields $\vec{B}$, namely,
$\vec{\varepsilon}=\overline{\vec{v}\times\vec{B}}$, where the overine denotes
the ensemble average. For a homogeneous, weakly anisotropic turbulence, the
mean electromotive force in the $\alpha$-turbulence dynamo mechanism is given
by equations (5) and (6). In particular, we note that the helical turbulent
motions can twist the toroidal field lines to produce the poloidal field as
mentioned before. It is the -coefficient which encapsulates this effect of
helical motions in the mathematical theory.

Therefore, the principle of the $\alpha$-turbulent dynamo may be briefly
sketched as follows: A toroidal magnetic field $\Longrightarrow$ A toroidal
electro-dynamic potential$\Longrightarrow$ A toroidal current
$\Longrightarrow$ A poloidal magnetic field $\Longrightarrow$ A poloidal
electro dynamic potential $\Longrightarrow$ A poloidal current
$\Longrightarrow$ A toroidal magnetic field (Priest, 1984).

In 1980s, the simulation showed that the intensity of the magnetic flux tube
on the surface of the sun is about $10^{5}$ G. Such a strong magnetic tube
should be formed at the bottom of the troposphere rather than in the
troposphere. The traditional $\alpha$-turbulence generator cannot operate in
such an intense circular magnetic field. To give a more convincible
explanation for the periodicity of the sunspot, lots of similar $\alpha$
effect dynamo theories are developed (Charbonneau, 2010; Dikpati & Gilman,
2001; Ferriz-Mas et al., 1994). For example, the instability of rotational
magnetic field may magnify the magnetic field’. That means the toroidal
magnetic field is transformed to the poloidal field due to the magnetofluid
instability that results from the interaction of the Coriolis force and the
differential rotation (in different latitude) in the sun. Generally speaking,
the magnification of the magnetic field is driven by the differential rotation
that interacts with the Coriolis force.

In fact, the Babcock-Leighton mechanism that is developed at the same time as
the turbulent dynamo is recently regarded as the most prospective explanation
for the periodicity of the magnetic field activity in the sun (Cameron &
Schassler, 2015). In accordance with the turbulent dynamo, the poloidal
magnetic field is also generated by the Coriolis force. But the Babcock-
Leighton mechanism is the Coriolis force acting on the magnetic flux tube in a
large scale, leading to the tilt angle that is observed on the surface of the
sun in the activation region. So the toroidal magnetic flux tube has a certain
poloidal component when it emerges on the surface of the sun. The poloidal
component is the result of the attenuation of the active region. The authors
in the paper (Ferriz-Mas et al., 1994) indicated that the nonaxisymmetric
instability of the toroidal magnetic flux tube in a rotating star can provide
a dynamo effect. This instability occurs in the form of spiral waves. The
increase in their amplitude causes a phase shift between the disturbed
magnetic field and the disturbed flow field, which leads to the generation of
an electric field in a direction parallel to the undisturbed field. Coupled
with the differential rotation, this effect will produce a type of dynamo. The
difference between the traditional turbulence dynamo and it is the traditional
turbulence dynamo cannot be applied to the magnetic field that is quite strong
while this new type of dynamo qualifies in this case.

Dikpati & Gilman (2001) propose an $\alpha\Omega$ flux-transport dynamo for
the Sun that is driven by a tachocline effect. This $\alpha$-effect comes from
the global hydrodynamic instability of latitudinal differential rotation in
the tachocline, as calculated using a shallow-water model. Growing, unstable
shallow-water modes propagating longitudinally in the tachocline create
vortices that correlate with radial motion in the layer to produce a
longitude-averaged net kinetic helicity and, hence, an $\alpha$-effect. It is
shown that such a dynamo is equally successful as a Babcock-Leighton-type
flux-transport dynamo in reproducing many large-scale solar cycle
features(Dikpati & Gilman, 2001). The success of both dynamo types depends on
the inclusion of meridional circulation of a sign and magnitude similar to
that seen on the Sun. Both $\alpha$-effects (the Babcock-Leighton-type and
tachocline $\alpha$-effect) are likely to exist in the Sun, but it is hard to
estimate their relative magnitudes. By extending the simulation to a full
spherical shell, It is shown that the flux-transport dynamo driven by the
tachocline $\alpha$-effect selects a toroidal field that is antisymmetric
about the equator, while the Babcock-Leighton flux-transport dynamo selects a
symmetric toroidal field (Dikpati & Gilman, 2001). Since our present Sun
selects antisymmetric fields, the tachocline $\alpha$-effect must be more
important than the Babcock-Leighton $\alpha$-effect.

These theories are still under further research and discussion. The average
electric potential generated by them (e.g., see the discussions by Ferriz-Mas
et al. (1994)) is just proportional to the magnetic field strength, and its
direction is parallel to the magnetic field direction. However, the
proportional coefficient $\alpha$ in Eq.(5) differs a lot in different
theories (Charbonneau, 2010). We start the discussion from Eq.(5). They are
called by a joint name $\alpha$-turbulent dynamo’ while the proportional
coefficient $\alpha$ may have an uncertainty in 1-2 orders of magnitude. In
the theory of $\alpha$-turbulent dynamo, the energy of magnetic field per unit
volume is equal to the energy of induced electric current per unit volume,
thus

In the theory of $\alpha$-turbulent dynamo, the energy of magnetic field per
unit volume is equal to the energy of induced electric current per unit
volume, thus,

$\frac{B^{2}}{4\pi}=ne\varepsilon=yne\alpha(\sigma
t_{c},\overline{\vec{v}\cdot\vec{\omega}})B$ (7)

so we have

$B=8\pi eyn\alpha(\sigma t_{c},\overline{\vec{v}\cdot\vec{\omega}}),$ (8)

where $n$ is the number density of the plasma particles, $y$ is the degree of
ionization.

Some relevant data in the Sun are given as follows. The mass density in the
solar convection zone where dynamo mechanism is valid, is $\rho\approx
8\rm{gcm^{-3}}$, or the number density of particles $n=5\times
10^{24}\rm{cm^{3}}$; The maximum magnetic field in the solar convection zone
is $B_{\rm{max}}\sim 10^{5}$G (The details of the differential rotation in
solar interior can be found from the following webpage of
http$://$www.aip.de/en/press/images/

It is expected that the $\alpha$-turbulent dynamo model originally developed
by Parker valid in the solar convection region can be applied also in
accretion disks surrounding massive black holes. The origin of the magnetic
fields in the inner region of the accretion disks may be explained in terms of
the -turbulent dynamo model. To estimate the value of in the inner regions
near the GC we may compare the value of relevant parameters involved in the
solar convection zone with those in the star forming region in the
interstellar clouds. Using this method, it is plausible to believe that the
uncertainties of the turbulent velocity $\vec{v}$, he electrical conductivity
$\sigma$, the time scale for turbulence $t_{c}$, and the vorticity
$\vec{\omega}$ of the fluid in the solar convection region (see Charbonneau
(2010)) would not seriously affect the accuracy of our estimation for the
magnetic field strength in the inner region of accretion disk near the GC.

Assuming the validity of the $\alpha$-turbulence dynamo mechanism similar to
that in the solar convection zone, and comparing the interstellar magnetic
field strength with that of the sunspot, we may estimate the uncertainty of
the value of $\alpha$ in terms of the recent observation for the collapsing
core W51e2 of the star forming region (Koch et al., 2012).

We deduce from Eq.(8)

$B=B_{\odot\rm{max}}\frac{n}{n_{\odot}}\frac{y}{y_{\odot}}\frac{\alpha(\sigma
t_{c},\overline{\vec{v}\cdot\vec{\omega}})}{\alpha(\sigma
t_{c},\overline{\vec{v}\cdot\vec{\omega}})_{\odot}}$ (9)

$B\sim 10^{-19}\frac{n}{5\rm{cm^{-3}}}r(\vec{v}_{\rm{turb}},\sigma
t_{c})\rm{G},$ (10)

where $r(\vec{v}_{\rm{turb}},\sigma
t_{c})=\frac{n}{n_{\odot}}\frac{y}{y_{\odot}}\frac{\alpha(\sigma
t_{c},\overline{\vec{v}\cdot\vec{\omega}})}{\alpha(\sigma
t_{c},\overline{\vec{v}\cdot\vec{\omega}})_{\odot}}$.

Though the turbulent velocity in an interstellar cloud nay reach
$\vec{v}_{turb}\sim 10\rm{kms^{-1}}$, the curl of the turbulence velocity in
the interstellar cloud is far smaller than that in the Sun, but both the
typical timescale of turbulence, $t_{c}$ and the electric conductivity
($\sigma$) may be much larger than those of the sun. Thus, the value of the
factor $\alpha(\sigma t_{c},\overline{\vec{v}\cdot\vec{\omega}})$ is rather
uncertain.

However, we may estimate it as follows. In some interstellar cloud, the
strongest magnetic field may reach 20mG and the number density is $n=2.7\times
10^{7}\rm{cm^{-3}}$ near the collapsing core W51e2. Using Eq.(10), the
uncertain factor, $r(\vec{v}_{turb},\sigma t_{c})$, may be determined in terms
of the observed data for W51e2 just delineated, thus

$r(\vec{v}_{\rm{turb}},\sigma t_{c})\sim 3.7\times 10^{10}$ (11)

In recent work by Qiu et al. (2014), Submillimeter Array Observations of
magnetic fields in a $\rm{H}_{2}$ molecular cloud has been made. the magnetic
field is estimated about 1 mG, and the corresponding number density is about
$2.7\times 10^{5}\rm{cm^{-3}}$. Thus

$r(\vec{v}_{\rm{turb}},\sigma t_{c})\sim 1.85\times 10^{11}$ (12)

We can find that the difference of Eq.(11) with Eq.(12) is a factor of 5 and
it is in the uncertainty region of the $\alpha$-coefficient (it is may be to
reach to (1-2) orders of magnitude, see the discussions from Chatteriee et al.
(2011)).

Using Eq.(9) in eq.(11), with the observed electron number density $n\sim
26{\rm{cm}}^{-3}$ at the distance of 0.12pc from GC (Eatough et al., 2013),
the resulting magnetic field is given by $B\leq 0.1$G which is five order of
magnitude smaller than the observed lower limit for the field strength, 8mG
(Eatough et al., 2013). At this point, we would like to mention that such
strong magnetic field at the distance of 0.12pc from GC can not be generated
by the recent Magnetically arrested accretion disk model(Yuan & Narayan,
2014), although a strong vertical bipolar magnetic field is pushed into the
central black hole by the thermal and ram pressure of the accreting gas and
the maximum magnetic field strength at the horizon ($R_{S}\approx
10^{12}\rm{cm}$) is roughly $10^{3}$ G.

A broad class of astronomical accretion disks is shown to be dynamically
unstable to axisymmetric disturbances in the presence of a weak magnetic field
by Balbus & Hawley (1991, 1998), an insight with consequently broad
applicability to gaseous, differentially-rotating systems. However, we have
shown that the RMI (Balbus & Hawley, 1991) can not generate the observed
strong magnetic field. Also, the magnetic field is not generated by
$\alpha$-dynamo. In addition to the two mechanisms of magnetic field
amplification, Cao (2011) proposed that the radial inward advection can also
generate strong magnetic field. Cao (2011) calculate the advection/diffusion
of the large-scale magnetic field threading an advection-dominated accretion
flow (ADAF) and find that the magnetic field can be dragged inward by the
accretion flow. The measurement of an abnormally strong radial magnetic field
near the GC has been reported in 2013 by Eatough et al. (2013). The lower
limit of the outward radial magnetic field near the GC is: B $\geq$ 8mG (at
r=0.12 pc). In our paper, we have discussed and demonstrated that the
kinematic energy density of the material is less than the energy density of
the magnetic field at r=0.12 pc.

On the other hand, the problems discussed and the results obtained by Cao
(2011) shall not apply to the content of our paper discussed according to the
astronomical observation. We know that for magnetic field, the magnetic flux
is conserved. For the radial magnetic field, if the field is adverted inward
by gas, Strength of B is proportional to reciprocal of radius square.
Therefore, the magnetic field can be easily amplified by inward motion of gas.
At this point, we would like to mention that such strong magnetic field at the
distance of 0.12pc from the GC cannot be generated by the recent Magnetically
arrested accretion disk model (Yuan & Narayan, 2014) due to the strength of B
is proportional to reciprocal of radius square, although a strong vertical
bipolar magnetic field is pushed into the central black hole by the thermal
and ram pressure of the accreting gas and the maximum magnetic field strength
at the horizon ($R_{S}\approx 10^{12}$ cm) is roughly $10^{3}$Gauss. Thus the
inward advection of magnetic field by gas from larger radii (0.12pc) can
produce magnetic field about 0.1 G much less than observed $B\geq 8$mG.

Even in strongly magnetized magnetically arrested disk (MAD) state, the black
hole can accrete gas through non-axisymmetric spiral streams. Therefore, there
should be radiation from the black hole hot accretion flow. Igumenshchev
(2008) investigated the dynamics and structure of accretion disks, which
accumulate a vertical magnetic field in their centers by using two- and three-
dimensional MHD simulations. The central field can be built up to the
equipartition level, where it disrupts a nearly axisymmetric outer accretion
disk inside a magnetospheric radius, forming a magnetically arrested disk
(MAD). On the other hand, the black hole (BH) accretion flows and jets are
qualitatively affected by the presence of ordered magnetic fields. McKinney et
al. (2012) discuss fully 3D global general relativistic MHD simulations of
radially extended and thick (height H-to-cylindrical radius R ratio of
$|H/R|\sim 0.2-1$) accretion flows around BHs with various dimensionless spins
(a/M, with BH mass M) and with initially toroidally dominated
($\varphi$-directed) and poloidally dominated (R-z directed) magnetic fields.
However, the above opinions expressed have nothing to do with the problem of
radial magnetic field (i.e., $B\geq 8$mG) at the distance of 0.12pc from the
GC in our paper.

In addition, someone once put forward whether the plasma or gas of accretion
disks can flow into a black hole along the radial magnetic field at the center
of the Milky Way galaxy. It is simple answer that if there are the material
flow along the radial magnetic field to the center of the galaxy, the speed of
the radial free fall material will be close to the speed of light. Therefore,
the strong powerful radiation will be produced. But this kind of abnormal
quite powerful radiation has always been not observed. So this kind of
circumstance should be ruled out.

##  5 Our Model of Super-massive Star with Magnetic Monopoles (SMSMM)

We note that the important discovery of very strong radial magnetic field in
the Vicinity of the GC is consistent with the prediction from our model of
SMSMM (Peng, 2002). Thus, it is plausible to believe that this is just the
astronomical evidence needed for the existence of magnetic monopoles as
predicted by the grand unified theory of particle physics. In addition, the
observed radiation from radio to sub-mm wavelength band with power peaking
around $10^{12}$ Hz in the sub mm band and the x-ray radiation near the GC are
also essentially in agreement with the prediction of our paper(Peng & Chou,
2001). In other words, the dilemma of the standard accretion disk model with
supermassive black holes at the GC would disappear.

We have investigated the model of SMSMM in a series of papers since 1985 (Peng
et al., 1985; Peng & Wang, 1985; Peng et al., 1985; Peng, 1989; Peng & Chou,
1998, 2001; Peng, 2002; Wang & Peng, 1986), and the main ideas of our model
are as follows:

1) The fact that magnetic monopoles (M) may catalyze nucleons to decay (the
Rubakov-Callan (RC) effect, $pM\rightarrow e^{+}\pi^{0}M(85\%)$ or
$pM\rightarrow e^{+}\mu^{+}\mu^{-}M(15\%)$, with the number of baryons being
non-conserved) as predicted by the grand unified theory of particle physics is
invoked as the main energy source of quasars and AGN. The supermassive central
black hole in the standard model is replaced by a supermassive object
containing magnetic monopoles. And the accretion disk acts only as a minor
energy supply.

2) The gravitational effect around the SMSMM in the galactic center is similar
to that around a massive black hole. However, the supermassive object
containing sufficient magnetic monopoles has neither the horizon nor the
central singularity. This is because the reaction rate of the nucleon decay
catalyzed by magnetic monopoles is proportional to the square of mass density.
Both the leptons and photons from the decay are emitted outward, and the
central density cannot approach infinity. Combined with the RC effect from
particle physics, our model can avoid the central singularity problem in the
standard model of black hole theory.

On the other hand, some predictions about the GC in our model are as
follows(Peng & Chou, 2001):

1) Plenty of positrons are generated and are emitted from the GC. The
producing rate is about $6.0\times 10^{42}e^{+}\rm{s^{-1}}$. This prediction
is quantitatively confirmed by observation of high energy astrophysics
quantitatively ($3.4-6.30\times 10^{42}e^{+}\rm{s^{-1}}$).

2) Some higher energy radiation above 0.5MeV may be emitted. The integral
energy of the high energy radiation is much higher than both the total energy
of the spectra of electron and positron annihilation, and the total thermal
luminosity of the central object. This prediction is also consistent with
observations.

3) The magnetic monopole condensed in the core region of the supermassive
object can generate radial magnetic field. The magnetic field strength at the
surface of the object is about 20-100 Gauss (the radius of the object is about
$8.1\times 10^{15}$cm or $1.1\times 10^{4}R_{S}$ ($R_{S}$ is the Schwarzschild
radius). We declared previously in our article that this prediction is the
most crucial one, which can be testified by future radio observations(Peng &
Chou, 2001). Because the decrease of the magnetic field strength is
proportional to the inverse square of the distance from the source, so we have
$B\approx(10-50)$ mG at $r=0.12$ pc. This prediction is in agreement with the
lower limit of the observed magnetic field (the detailed discussions can be
found from the article of Eatough et al. (2013)).

4) The super-massive objects containing saturated magnetic monopoles in the
centers of all the AGN in the region $D\leq 50$ Mpc from the earth may be the
sources of observed ultra-high energy cosmic rays
($E_{\gamma}\sim(10^{18}-10^{21})$ eV), which can not be explained up to date
except our model (Peng & Chou, 2001).

5) The surface temperature of the super-massive object in the galactic center
is about 120 K and the corresponding spectrum peak of the thermal radiation is
at $10^{12}$ Hz in the sub-mm wavelength regime. This prediction is basically
consistent with the recent observation (Falcke & Marko, 2013).

The non- thermal radiation such as synchrotron radiation, may be emitted due
to the motion of the relativistic electrons in the magnetic field. However,
quantitative comparison of observations with theory is rather difficult now,
because the power indexes of both the thermal radiation and the non- thermal
radiation for the radio wavelength band have not been well determined yet up
to now. The predictions 1), 3), 5) have been confirmed by the astronomical or
by the astrophysical observations in quantitative. Especially, the third one
is an exclusive prediction. It is hardly a coincidence.

##  6 Other Evidences Against the Black Hole Model of Quasars and AGN

We now briefly mention some other relevant evidences against the black hole
model of quasars and active galactic nuclei (AGN).

1) On strong radial magnetic fields in the galactic center for 76 radio-loud
active galaxies

Zamaninasab et al. (2014) did a statistical analysis on 76 radio-loud active
galaxies and concluded that there are very strong radial magnetic fields in
the galactic center preventing material in the accretion disk from falling in,
i.e. the accretion disk is not near the central black holes in AGN. This could
invalidate the standard accretion disk model of black holes in AGN.

2) On two hot dust-free quasars

Using the Spitzer Space Telescope, Jiang et al. (2006, 2010) discovered two
quasars without hot-dust emission in a sample of 21 ($z\approx 6$) quasars by
deep infrared photometry.

It is generally believed that quasars are powered by mass accretion onto the
central black holes and hot dust is directly heated by quasar activity. So the
discovery of the two hot-dust-free quasars becomes a puzzle. The puzzle may be
explained in terms of the standard model for quasars with central black holes
as follows.

We note that the two hot dust-free quasars with the lowest hot-dust abundances
have the smallest black hole masses ($(2-3)\times 10^{8}\rm{M_{\odot}}$) and
highest Eddington luminosity ratios ($\sim 2$) in the $z\approx 6$ sample,
thus they are in an early stage of quasar evolution with rapid mass accretion,
but are too young to have formed a detectable amount of hot dust around them.

Since the accretion disk is not near the central black holes in the quasars
and AGN due to the presence of the observed strong radial magnetic field near
the GC, no material can reach the central region of the quasars and AGN and
the ultra -luminous radiation cannot be emitted by the accretion disk model of
black holes. The observations of the two hot dust-free quasars may be
considered as the important evidence that the ultra -luminous radiation of the
very young quasars in the early stage of the universe cannot be emitted by the
accretion material flow from the accretion disk around, but it may be produced
by the RC effect in our AGN model containing magnetic monopoles.

3) On the mass of the quasars

It is now generally believed by most astronomers that bright quasars observed
at large redshift (for example, $z>1$ or even $z>5$) are supermassive black
holes ($m>10^{10}\rm{M_{\odot}}$) formed in the primordial universe. The
spectacularly huge luminosity is supplied by the accretion of matter outside
these black holes. As a result, the mass of nearby galactic nuclei and quasars
must be greater than that of the remote quasars with larger redshift. This is
because the mass of the black holes must continuously increase due to
accretion. But the deduction is just contrary to the observation that no
supermassive black hole with mass $m>10^{9}\rm{M_{\odot}}$. Indeed, this is
the dilemma of the black hole model of quasars and active galactic nuclei
(AGN), although some proposals had been suggested such as: (i) the merger of
two galactic nuclei may also form a larger quasar or an AGN; (ii) the mass of
the supermassive black holes at the center of AGN (and quasars) in the high
red shift region are much larger than those in the low red shift region is
only a select effect in observation because the supermassive quasars are easy
to observe due to their huge luminosities. But, these proposals cannot explain
the fact that why no supermassive black holes ($m>10^{9}\rm{M_{\odot}}$) have
been found near the Milky Way galaxy ( $D<1$ Gpc).

However, it is naturally explained by our AGN model containing magnetic
monopoles. The mass of the supermassive object must decrease gradually due to
the baryons decay catalyzed by the magnetic monopoles and the decaying
products (including pions , muons , positrons and the radiation) are lost from
the central massive stellar object continuously. So the conclusion from our
AGN model containing magnetic monopoles is that the mass of the quasars and
AGN would decrease with the redshift (z). This is consistent with
observations.

4) On Jets of AGNs

Sell et al. (2014)investigated the process of rapid star formation quenching
in a sample of 12 massive galaxies at intermediate redshift ($z\sim 0.6$) that
host high-velocity ionized gas outflows ($v>1000$ km s-1).They conclude that
these fast outflows are most likely driven by feedback from star formation
rather than active galactic nuclei (AGN). They use multiwavelength survey and
targeted observations of the galaxies to assess their star formation, AGN
activity, and morphology. Common attributes include diffuse tidal features
indicative of recent mergers accompanied by bright, unresolved cores with
effective radii less than a few hundred parsecs. The galaxies are
extraordinarily compact for their stellar mass, even when compared with
galaxies at $z\sim 2-3$. For $9/12$ galaxies, we rule out an AGN contribution
to the nuclear light and hypothesize that the unresolved core comes from a
compact central starburst triggered by the dissipative collapse of very gas-
rich progenitor merging disks. They find evidence of AGN activity in half the
sample but we argue that it accounts for only a small fraction ($\sim 10\%$)
of the total bolometric luminosity. They find no correlation between AGN
activity and outflow velocity and they conclude that the fast outflows in our
galaxies are not powered by on-going AGN activity, but rather by recent,
extremely compact star-bursts.

Numerical simulations of hot accretion flows around black holes have shown the
existence of strong wind Bu et al. (2016a, b). They performed hydrodynamic
(HD) simulations and the MHD simulations by taking into account the
gravitational potential of both the black hole and the nuclear star cluster.
they found that, just as for the accretion flow at small radii, the mass
inflow rate decreases inward, and the flow is convectively unstable. However,
a trajectory analysis shows that there is very little wind launched from the
flow. They concluded the result that wind cannot be produced in the region
$R>R_{A}$ (here $R_{A}$ is similar to the Bondi radius). The winds are mainly
determined by the galactic nuclei clusters of gravitational potential. Near
the event horizon surface accretion flow is determined by the black hole’s
gravitational potential. Their results shown that Jets of active galactic
nuclei (i.e., Jet, stellar winds) is actually dominated by the gravitational
potential of the galactic nuclei clusters, and has nothing to do with the
central black hole. This is very obvious that they do not support the popular
view, which the active galactic nuclei jets are strong evidence of black holes
model.

##  7 Conclusions

In conclusion, we have demonstrated that the radiations observed in the
contiguous region of the central black hole cannot be emitted by the gas of
the disk since the accreting plasma is prevented from approaching to the GC by
the ultrastrong magnetic fields. In addition, we have also shown that the
observed strong radial magnetic fields near the GC by Eatough et al. (2013),
cannot be generated by the -turbulence dynamo mechanism of Parker because
qualitative estimate gives a magnetic field strength six orders of magnitude
smaller than the observed field strength at $r=0.12$ pc. The dilemma of the
standard model for quasars and AGN can be avoided if the central black hole in
the standard model is replaced by a supermassive stellar object containing
magnetic monopoles. The radiations emitted from the inner region of the
galactic nucleus and the discovery of the strong radial magnetic field near
the GC can all be naturally explained by our model(Peng & Chou, 2001).
Moreover, the observed ultra-strong radial magnetic field in the vicinity of
the GC may be considered as important astronomical evidence for the existence
of magnetic monopoles as predicted by the Grand Unified Theory of particle
physics.

Acknowledgements We would like to thank Prof. Daniel Wang, Prof. Y.F. Huang,
Prof. P.F. Chen and Prof. J. L. Han for their help to inform us some new
information of observations. This work was supported in part by the National
Natural Science Foundation of China under grants 10773005, 11273020, 11565020,
and the Counterpart Foundation of Sanya under grant 2016PT43, the Special
Foundation of Science and Technology Cooperation for Advanced Academy and
Regional of Sanya under grant 2016YD28, and the Natural Science Foundation of
Hainan province under grant 114012.

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