00footnotetext: * Corresponding author

# Commutators of multilinear singular integral operators on non-homogeneous
metric measure spaces

Rulong Xie, Huajun Gong∗ and Xiaoyao Zhou  School of Mathematical Sciences,
University of Science and Technology of China, Hefei 230026, China
xierl@mail.ustc.edu.cn;huajun84<EMAIL_ADDRESS>
###### Abstract.

Let $(X,d,\mu)$ be a metric measure space satisfying both the geometrically
doubling and the upper doubling measure conditions, which is called non-
homogeneous metric measure space. In this paper, via a sharp maximal operator,
the boundedness of commutators generated by multilinear singular integral with
$RBMO(\mu)$ function on non-homogeneous metric measure spaces in $m$-multiple
Lebesgue spaces is obtained.

###### Key words and phrases:

multilinear singular integral; commutators; non-homogeneous metric measure
spaces; $RBMO(\mu)$

###### 1991 Mathematics Subject Classification:

42B20; 42B25

##  1\. Introduction

It is well known that the standard singular integral theory is constructed
with the assumption of spaces satisfying the doubling measures condition. We
recall that $\mu$ is said to satisfy the doubling condition if there exists a
constant $C>0$ such that $\mu(B(x,2r))\leq C\mu(B(x,r))$ for all
$x\in\rm{supp}\mu$ and $r>0$. A metric measure space $(X,d,\mu)$ equipped with
a non-negative doubling measure $\mu$ is called a space of homogeneous type.
In case of non-doubling measures, a non-negative measure $\mu$ only need to
satisfy the polynomial growth condition, i.e., for all $x\in\mathbb{R}^{n}$
and $r>0$, there exist a constant $C_{0}>0$ and $k\in(0,n]$ such that,

(1.1) $\ \mu(B(x,r))\leq C_{0}r^{k},$

where $B(x,r)=\\{y\in\mathbb{R}^{n}:|y-x|<r\\}$. This breakthrough brings
rapid development in harmonic analysis (see [2,5,8-9,17-22]). And the analysis
on non-doubling measures has important applications in solving the long-
standing open Painlev$\acute{e}$’s problem (see [18]).

However, as stated by Hytönen in [11], the measure satisfying (1) does not
include the doubling measure as special cases. To solve this problem, a kind
of metric measure space $(X,d,\mu)$, which is called non-homogeneous metric
measures space, satisfying geometrically doubling and the upper doubling
measure condition (see Definition 1.1 and 1.2) is introduced by Hytönen in
[11]. The highlight of this kind of spaces is that it includes both the
homogeneous spaces and metric spaces with polynomial growth measures as
special cases. From then on, some results paralled to homogeneous spaces and
non-doubling measures space are obtained (see [1,5,11-16] and the references
therein). Hytönen et al. in [14] and Bui and Duong in [1] independently
introduced the atomic Hardy space $H^{1}(\mu)$ and proved that the dual space
of $H^{1}(\mu)$ is $RBMO(\mu)$. In [1], the authors also proved that Calderón-
Zygmund operator and commutators of Calderón-Zygmund operators and RBMO
functions are bounded in $L^{p}(\mu)$ for $1<p<\infty$. Recently, some
equivalent characterizations are established by Liu et al. in [16] for the
boundedness of Carder on-Zygmund operators on $L^{p}(\mu)$ for $1<p<\infty$.
In [4], Fu et al. established boundedness of multilinear commutators of
Calderón-Zygmund operators on Orlicz spaces on non-homogeneous spaces.

On the other hand, the theory on multilinear singular integral operators has
been considered by some researchers. In [3], Coifman and Meyers firstly
established the theory of bilinear Calderón-Zygmund operators. Later,
Gorafakos and Torres [6-7] established the boundedness of multilinear singular
integral on the product Lebesgue spaces and Hardy spaces. The properties of
multilinear singular integrals and commutators on non-doubling measures spaces
$(\mathbb{R}^{n},\mu)$ were established by Xu in [21-22]. Weighted norm
inequalities for multilinear Calderón-Zygmund operators on non-homogeneous
metric measure spaces were also constructed in [10].

In the setting of non-homogeneous metric measure spaces, it is natural to ask
whether commutators of multilinear singular integral operators is also bounded
in $m$-multiple Lebesgue spaces. This paper will give an affirmative answer to
this question. In this paper, commutators generated by multilinear singular
integrals with $RBMO(\mu)$ function on non-homogeneous metric spaces is
introduced firstly. And we will prove that it is bounded in $m$-multiple
Lebesgue spaces on non-homogeneous metric spaces, provided that multilinear
singular integrals is bounded from $m$-multiple $L^{1}(\mu)\times\ldots\times
L^{1}(\mu)$ to $L^{1/m,\infty}(\mu)$, where $L^{p}(\mu)$ and
$L^{p,\infty}(\mu)$ denote the Lebesgue space and weak Lebesgue space
respectively. This result in this paper includes the corresponding results on
both the homogeneous spaces and $(\mathbb{R}^{n},\mu)$ with non-doubling
measures space. A variant of sharp maximal operator $M^{\sharp}$, Kolmogorov’s
theorem and some good properties of the dominating function $\lambda$ (see
Definition 1.2) are the main tools for proving the results of this paper.

Before stating the main results of this paper, we firstly recall some
notations and definitions.

######  Definition 1.1.

[11]A metric space $(X,d)$ is called geometrically doubling if there exists
some $N_{0}\in\mathbf{N}$ such that, for any ball $B(x,r)\subset X$, there
exists a finite ball covering $\\{B(x_{i},r/2)\\}_{i}$ of $B(x,r)$ such that
the cardinality of this covering is at most $N_{0}$.

######  Definition 1.2.

[11]A metric measure space $(X,d,\mu)$ is said to be upper doubling if $\mu$
is a Borel measure on $X$ and there exists a dominating function
$\lambda:X\times(0,+\infty)\rightarrow(0,+\infty)$ and a constant
$C_{\lambda}>0$ such that for each $x\in X,r\longmapsto(x,r)$ is non-
decreasing, and for all $x\in X,r>0$,

(1.2) $\mu(B(x,r))\leq\lambda(x,r)\leq C_{\lambda}\lambda(x,r/2)$

######  Remark 1.3.

(i) A space of homogeneous type is a special case of upper doubling spaces,
where one can take the dominating function $\lambda(x,r)=\mu(B(x,r))$. On the
other hand, a metric space $(X,d,\mu)$ satisfying the polynomial growth
condition (1)(in particular, $(X,d,\mu)=(\mathbb{R}^{n},|\cdot|,\mu)$ with
$\mu$ satisfying (1) for some $k\in(0,n])$) is also an upper doubling measure
space if we take $\lambda(x,r)=Cr^{k}$.

(ii) Let$(X,d,\mu)$ be an upper doubling space and $\lambda$ be a dominating
function on $X\times(0,+\infty)$ as in Definition 1.2. In [14], it was showed
that there exists another dominating function $\tilde{\lambda}$ such that for
all $x,y\in X$ with $d(x,y)\leq r$,

(1.3) $\tilde{\lambda}(x,r)\leq\tilde{C}\tilde{\lambda}(y,r).$

Thus, we suppose that $\lambda$ always satisfies (1.3) in this paper.

######  Definition 1.4.

Let $\alpha,\beta\in(1,+\infty)$. A ball $B\subset X$ is called
$(\alpha,\beta)$-doubling if $\mu(\alpha B)\leq\beta\mu(B)$.

As pointed in Lemma 2.3 of [1], there exist plenties of doubling balls with
small radii and with large radii. In the rest of this paper, unless $\alpha$
and $\beta$ are specified otherwise, by an $(\alpha,\beta)$ doubling ball we
mean a $(6,\beta_{0})$-doubling with a fixed number
$\beta_{0}>\max\\{C_{\lambda}^{3\log_{2}6},6^{n}\\}$, where $n=\log_{2}N_{0}$
is viewed as a geometric dimension of the space.

######  Definition 1.5.

[1]For any two balls $B\subset Q$, define

(1.4) $K_{B,Q}=1+\int_{r_{B}\leq d(x,x_{B})\leq
r_{Q}}\frac{d\mu(x)}{\lambda(x_{B},d(x,x_{B}))}.$

And, for two balls $B\subset Q$, one define the coefficient $K^{\prime}_{B,Q}$
as follows. Let $N_{B,Q}$ be the smallest integer satisfying
$6^{N_{B,Q}}r_{B}\geq r_{Q}$, then we set

(1.5)
$K^{\prime}_{B,Q}=1+\sum_{k=1}^{N_{B,Q}}\frac{\mu(6^{k}B)}{\lambda(x_{B},6^{k}r_{B})}.$

######  Remark 1.6.

In the case that $\lambda(x,ar)=a^{t}\lambda(x,r)$ for $0<t<\infty$, $x\in X$,
and $a,r>0$, one know that $K_{B,Q}\approx K^{\prime}_{B,Q}$. However, in
general, we only have $K_{B,Q}\leq CK^{\prime}_{B,Q}$. In this paper, we
always suppose that $\lambda(x,ar)=a^{t}\lambda(x,r)$ for $0<t<\infty$, $x\in
X$, and $a,r>0$. So we don’t differentiate $K_{B,Q}$ with $K^{\prime}_{B,Q}$
and always write $K_{B,Q}$ for simplicity in this paper.

######  Definition 1.7.

A kernel $K(\cdot,\cdots,\cdot)\in
L_{loc}^{1}((X)^{m+1}\backslash\\{(x,y_{1}\cdots,y_{j},\cdots,y_{m}):x=y_{1}=\cdots=y_{j}=\cdots=y_{m}\\})$
is called an $m$-linear Calderón-Zygmund kernel if it satisfies:

(i)

(1.6) $|K(x,y_{1},\cdots,y_{j},\cdots,y_{m})|\leq
C\biggl{[}\sum_{j=1}^{m}\lambda(x,d(x,y_{j}))\biggr{]}^{-m}$

for all $(x,y_{1}\cdots,y_{j},\cdots,y_{m})\in(X)^{m+1}$ with $x\neq y_{j}$
for some $j$.

(ii) There exists $0<\delta\leq 1$ such that

(1.7)
$\begin{split}&|K(x,y_{1},\cdots,y_{j},\cdots,y_{m})-K(x^{\prime},y_{1},\cdots,y_{j},\cdots,y_{m})|\\\
\leq&\frac{Cd(x,x^{\prime})^{\delta}}{\biggl{[}\sum\limits_{j=1}^{m}d(x,y_{j})\biggr{]}^{\delta}\biggl{[}\sum\limits_{j=1}^{m}\lambda(x,d(x,y_{j}))\biggr{]}^{m}},\end{split}$

provided that $Cd(x,x^{\prime})\leq\max\limits_{1\leq j\leq m}d(x,y_{j})$ and
for each $j$,

(1.8)
$\begin{split}&|K(x,y_{1},\cdots,y_{j},\cdots,y_{m})-K(x,y_{1},\cdots,y^{\prime}_{j},\cdots,y_{m})|\\\
&\leq\frac{Cd(y_{j},y^{\prime}_{j})^{\delta}}{\biggl{[}\sum\limits_{j=1}^{m}d(x,y_{j})\biggr{]}^{\delta}\biggl{[}\sum\limits_{j=1}^{m}\lambda(x,d(x,y_{j}))\biggr{]}^{m}},\end{split}$

provided that $Cd(y_{j},y^{\prime}_{j})\leq\max\limits_{1\leq j\leq
m}d(x,y_{j})$.

A multilinear operator $T$ is called a multilinear Calderón-Zygmund singular
integral operator with the above kernel $K$ satisfying $(1.6)$, $(1.7)$ and
$(1.8)$ if, for $f_{1},\cdots f_{m}$ are $L^{\infty}$ functions with compact
support and $x\notin\bigcap_{j=1}^{m}\text{supp}f_{j}$,

(1.9) $T(f_{1},\cdots f_{m})(x)=\int_{X^{m}}K(x,y_{1},\cdots
y_{m})f_{1}(y_{1})\cdots f_{m}(y_{m})d\mu(y_{1})\cdots d\mu(y_{m}).$

######  Remark 1.8.

Because $\max\limits_{1\leq j\leq
m}d(x,y_{j})\leq\sum\limits_{j=1}^{m}d(x,y_{j})\leq m\max\limits_{1\leq j\leq
m}d(x,y_{j})$, (ii) in Definition 1.7 is equivalent to (ii’) in the following
statement.

(ii’) There exists $0<\delta\leq 1$ such that

(1.10)
$\begin{split}&|K(x,y_{1},\cdots,y_{j},\cdots,y_{m})-K(x^{\prime},y_{1},\cdots,y_{j},\cdots,y_{m})|\\\
\leq&\frac{Cd(x,x^{\prime})^{\delta}}{\biggl{[}\max\limits_{1\leq j\leq
m}d(x,y_{j})\biggr{]}^{\delta}\biggl{[}\sum\limits_{j=1}^{m}\lambda(x,d(x,y_{j}))\biggr{]}^{m}},\end{split}$

provided that $Cd(x,x^{\prime})\leq\max\limits_{1\leq j\leq m}d(x,y_{j})$ and
for each $j$,

(1.11)
$\begin{split}&|K(x,y_{1},\cdots,y_{j},\cdots,y_{m})-K(x,y_{1},\cdots,y^{\prime}_{j},\cdots,y_{m})|\\\
\leq&\frac{Cd(y_{j},y^{\prime}_{j})^{\delta}}{\biggl{[}\max\limits_{1\leq
j\leq
m}d(x,y_{j})\biggr{]}^{\delta}\biggl{[}\sum\limits_{j=1}^{m}\lambda(x,d(x,y_{j}))\biggr{]}^{m}},\end{split}$

provided that $Cd(y_{j},y^{\prime}_{j})\leq\max\limits_{1\leq j\leq
m}d(x,y_{j})$.

######  Definition 1.9.

[1] Let $\rho>1$ be some fixed constant. A function $b\in L_{loc}^{1}(\mu)$ is
said to belong to $RBMO(\mu)$ if there exists a constant $C>0$ such that for
any ball $B$

(1.12) $\frac{1}{\mu(\rho B)}\int_{B}|b(x)-m_{\widetilde{B}}b|d\mu(x)\leq C,$

and for any two doubling balls $B\subset Q$,

(1.13) $|m_{B}(b)-m_{Q}(b)|\leq CK_{B,Q},$

where $\widetilde{B}$ is the smallest $(\alpha,\beta)$-doubling ball of the
form $6^{k}B$ with $k\in{\mathbf{N}}\bigcup\\{0\\}$, and
$m_{\widetilde{B}}(b)$ is the mean value of $b$ on $\widetilde{B}$, namely,

$m_{\widetilde{B}}(b)=\frac{1}{\mu(\widetilde{B})}\int_{\widetilde{B}}b(x)d\mu(x).$

The minimal constant $C$ appearing in (1.12) and (1.13) is defined to be the
$RBMO(\mu)$ norm of $b$ and denoted by $||b||_{\ast}$.

For $1\leq i\leq k$, we denote by $C_{i}^{k}$ the family of all finite subsets
$\sigma=\\{\sigma(1),\sigma(2),\cdot\cdot\cdot,\sigma(i)\\}$ of
$\\{1,2,\cdot\cdot\cdot,k\\}$ with $i$ different elements. For any $\sigma\in
C_{i}^{k}$, the complementary sequence $\sigma^{\prime}$ is given by
$\sigma^{\prime}=\\{1,2,\cdot\cdot\cdot,k\\}\backslash\sigma$. Moreover, for
$b_{i}\in RBMO(\mu),i=1,\cdots,k$, let
$\vec{b}=(b_{1},b_{2},\cdot\cdot\cdot,b_{k})$ be a finite family of locally
integrable function. For all $1\leq i\leq k$ and
$\sigma=\\{\sigma(1),\cdot\cdot\cdot,\sigma(i)\\}\in C_{i}^{k}$, we set
$\vec{b}_{\sigma}=(b_{\sigma(1)},\cdot\cdot\cdot,b_{\sigma(i)})$ and the
product $b_{\sigma}(x)=b_{\sigma(1)}(x)\cdot\cdot\cdot b_{\sigma(i)}(x)$.
Also, we denote $\vec{f}=(f_{1},\cdots,f_{k})$,
$\vec{f}_{\sigma}=(f_{\sigma(1)},\cdots,f_{\sigma(i)})$ and
$\vec{b}_{\sigma^{\prime}}\vec{f}_{\sigma^{\prime}}=(b_{\sigma^{\prime}(i+1)}f_{\sigma^{\prime}(i+1)},\cdots,b_{\sigma^{\prime}(k)}f_{\sigma^{\prime}(k)})$.

######  Definition 1.10.

A kind of commutators generated by multilinear singular integral operator $T$
with $b_{i}\in RBMO(\mu),i=1,\cdots,k$ is defined as follows:

(1.14) $[\vec{b},T](\vec{f})(x)=\sum_{i=0}^{k}\sum_{\sigma\in
C_{i}^{k}}(-1)^{k-i}b_{\sigma}(x)T(\vec{f}_{\sigma},\vec{b}_{\sigma^{\prime}}\vec{f}_{\sigma^{\prime}})(x).$

In particular, when $k=2$, we can obtain

(1.15)
$\begin{split}[b_{1},b_{2},T](f_{1},f_{2})(x)=&b_{1}(x)b_{2}(x)T(f_{1},f_{2})(x)-b_{1}(x)T(f_{1},b_{2}f_{2})(x)\\\
&-b_{2}(x)T(b_{1}f_{1},f_{2})(x)+T(b_{1}f_{1},b_{2}f_{2})(x).\end{split}$

Also, we define $[b_{1},T]$ and $[b_{2},T]$ as follows respectively.

(1.16)
$[b_{1},T](f_{1},f_{2})(x)=b_{1}(x)T(f_{1},f_{2})(x)-T(b_{1}f_{1},f_{2})(x),$
(1.17)
$[b_{2},T](f_{1},f_{2})(x)=b_{2}(x)T(f_{1},f_{2})(x)-T(f_{1},b_{2}f_{2})(x).$

For the sake of simplicity and without loss of generality, we only consider
the case of $k=2$ in this paper. Let us state the main result as follows.

######  Theorem 1.11.

Suppose that $\mu$ is a Radon measure with $||\mu||=\infty$. Let
$[b_{1},b_{2},T]$ defined by (1.15). Let $1<p_{1},p_{2}<+\infty$, $f_{1}\in
L^{p_{1}}(\mu)$, $f_{2}\in L^{p_{2}}(\mu)$, $b_{1}\in RBMO(\mu)$ and $b_{2}\in
RBMO(\mu)$. If $T$ is bounded from $L^{1}(\mu)\times L^{1}(\mu)$ to
$L^{1/2,\infty}(\mu)$, then there exists a constant $C>0$ such that

(1.18) $||[b_{1},b_{2},T](f_{1},f_{2})||_{L^{q}(\mu)}\leq
C||f_{1}||_{L^{p_{1}}(\mu)}||f_{2}||_{L^{p_{2}}(\mu)},$

where $\dfrac{1}{q}=\dfrac{1}{p_{1}}+\dfrac{1}{p_{2}}$.

Throughout this paper, $C$ always denotes a positive constant independent of
the main parameters involved, but it may be different from line to line. And
$p^{\prime}$ is the conjugate index of $p$, namely,
$\dfrac{1}{p}+\dfrac{1}{p^{\prime}}=1.$

##  2\. Proof of Main Result

To prove the main theorem, we firstly give some notations and lemmas.

Let $f\in L_{loc}^{1}(\mu)$, the sharp maximal operator is defined by

(2.1) $M^{\sharp}f(x)=\sup_{B\ni
x}\frac{1}{\mu(6B)}\int_{B}|f(y)-m_{\widetilde{B}}(f)|d\mu(y)+\sup_{(B,Q)\in\Delta_{x}}\frac{|m_{B}(f)-m_{Q}(f)|}{K_{B,Q}},$

where $\Delta_{x}:=\\{(B,Q):x\in B\subset Q\ \text{and}\ B,\ Q\ \text{are
doubling balls}\\}$ and the non centered doubling maximal operator is denoted
by

$Nf(x)=\sup_{B\ni x,\atop B\
\text{doubling}}\frac{1}{\mu(B)}\int_{B}|f(y)|d\mu(y).$

For any $0<\delta<1$, we also define that

(2.2) $M^{\sharp}_{\delta}f(x)=\\{M^{\sharp}(|f|^{\delta})(x)\\}^{1/\delta}$

and

(2.3) $N_{\delta}f(x)=\\{N(|f|^{\delta})(x)\\}^{1/\delta}.$

We can obtain that for any $f\in L^{1}_{loc}(\mu)$,

(2.4) $|f(x)|\leq N_{\delta}f(x)$

for $\mu-a.e.\ x\in X$. Let us give an explanation for inequality (2.4). By
the Lebesgue differential theorem, we obtain that $|f(x)|\leq Nf(x)$. Hence

$|f(x)|=[|f(x)|^{\delta}]^{1/\delta}\leq\\{N(|f|^{\delta})(x)\\}^{1/\delta}=N_{\delta}f(x).$

Let $\rho>1$, $p\in(1,\infty)$ and $r\in(1,p)$, the non-centered maximal
operator $M_{r,(\rho)}f$ is defined by

(2.5) $M_{r,(\rho)}f(x)=\sup_{B\ni x}\biggl{\\{}\frac{1}{\mu(\rho
B)}\int_{B}|f(y)|^{r}d\mu(y)\biggr{\\}}^{1/r}.$

When $r=1$, we simply write $M_{1,(\rho)}f(x)$ as $M_{(\rho)}f$. If $\rho\geq
5$, then the operator $M_{(\rho)}f$ is bounded on $L^{p}(\mu)$ for $p>1$ and
$M_{r,(\rho)}$ is bounded on $L^{p}(\mu)$ for $p>r$ (see [1]).

From Theorem 4.2 in [1], it is easy to obtain that

######  Lemma 2.1.

Let $f\in L^{1}_{loc}(\mu)$ with $\int_{X}f(x)d\mu(x)=0$ if $||\mu||<\infty$.
For $1<p<\infty$ and $0<\delta<1$, if $\inf(1,N_{\delta}f)\in L^{p}(\mu)$,
then there exists a constant $C>0$ such that

(2.6) $||N_{\delta}(f)||_{L^{p}(\mu)}\leq
C||M_{\delta}^{\sharp}(f)||_{L^{p}(\mu)}.$

######  Lemma 2.2.

[4,19] Let $1\leq p<\infty$ and $1<\rho<\infty$. Then $b\in RBMO(\mu)$ if and
only if for any ball $B\subset X$,

(2.7) $\biggl{\\{}\frac{1}{\mu(\rho
B)}\int_{B}|b_{B}-m_{\tilde{B}}(b)|^{p}d\mu(x)\biggr{\\}}^{1/p}\leq
C||b||_{\ast},$

and for any two doubling balls $B\subset Q$,

(2.8) $|m_{B}(b)-m_{Q}(b)|\leq CK_{B,Q}||b||_{\ast}.$

######  Lemma 2.3.

[4]

(2.9) $|m_{\widetilde{6^{j}\frac{6}{5}B}}(b)-m_{\tilde{B}}(b)|\leq
Cj||b||_{\ast}.$

######  Lemma 2.4.

[10] Suppose that $\mu$ is a Radon measure with $||\mu||=\infty$. Let $T$ be
defined by (1.9) with $m=2$. Let $1<p_{1},p_{2}<+\infty$, $f_{1}\in
L^{p_{1}}(\mu)$ and $f_{2}\in L^{p_{2}}(\mu)$. If $T$ is bounded from
$L^{1}(\mu)\times L^{1}(\mu)$ to $L^{1/2,\infty}(\mu)$, then there exists a
constant $C>0$ such that

(2.10) $||T(f_{1},f_{2})||_{L^{q}(\mu)}\leq
C||f_{1}||_{L^{p_{1}}(\mu)}||f_{2}||_{L^{p_{2}}(\mu)},$

where $\dfrac{1}{q}=\dfrac{1}{p_{1}}+\dfrac{1}{p_{2}}$.

######  Lemma 2.5.

Suppose that $[b_{1},b_{2},T]$ is defined by (1.15), $0<\delta<1/2$,
$1<p_{1},p_{2},q<\infty$, $1<r<q$ and $b_{1},b_{2}\in RBMO(\mu)$. If $T$ is
bounded from $L^{1}(\mu)\times L^{1}(\mu)$ to $L^{1/2,\infty}(\mu)$, then
there exists a constant $C>0$ such that for any $x\in X$, $f_{1}\in
L^{p_{1}}(\mu)$ and $f_{2}\in L^{p_{2}}(\mu)$,

(2.11) $\begin{split}&M_{\delta}^{\sharp}[b_{1},b_{2},T](f_{1},f_{2})(x)\leq
C||b_{1}||_{\ast}||b_{2}||_{\ast}M_{r,(6)}(T(f_{1},f_{2}))(x)\\\ &\ \
+C||b_{1}||_{\ast}M_{r,(6)}([b_{2},T](f_{1},f_{2}))(x)+C||b_{2}||_{\ast}M_{r,(6)}([b_{1},T](f_{1},f_{2}))(x)\\\
&\ \
+C||b_{1}||_{\ast}||b_{2}||_{\ast}M_{p_{1},(5)}f_{1}(x)M_{p_{2},(5)}f_{2}(x),\end{split}$

(2.12) $\begin{split}M_{\delta}^{\sharp}[b_{1},T](f_{1},f_{2})(x)\leq
C||b_{1}||_{\ast}M_{r,(6)}(T(f_{1},f_{2}))(x)+C||b_{1}||_{\ast}M_{p_{1},(5)}f_{1}(x)M_{p_{2},(5)}f_{2}(x),\end{split}$

and

(2.13) $\begin{split}M_{\delta}^{\sharp}[b_{2},T](f_{1},f_{2})(x)\leq
C||b_{2}||_{\ast}M_{r,(6)}(T(f_{1},f_{2}))(x)+C||b_{2}||_{\ast}M_{p_{1},(5)}f_{1}(x)M_{p_{2},(5)}f_{2}(x).\end{split}$

###### Proof.

Because $L^{\infty}(\mu)$ with compact support is dense in $L^{p}(\mu)$ for
$1<p<\infty$, we only consider the situation of $f_{1},f_{2}\in
L^{\infty}(\mu)$ with compact support. Also, by Corollary 3.11 in [4], without
loss of generality, we can assume that $b_{1},b_{2}\in L^{\infty}(\mu)$.

As in the proof of Theorem 9.1 in [19], to obtain (2.11), it suffices to show
that

(2.14)
$\begin{split}&\biggl{(}\frac{1}{\mu(6B)}\int_{B}||[b_{1},b_{2},T](f_{1},f_{2})(z)|^{\delta}-|h_{B}|^{\delta}|d\mu(z)\biggr{)}^{1/\delta}\\\
\leq&C||b_{1}||_{\ast}||b_{2}||_{\ast}M_{r,(6)}(T(f_{1},f_{2}))(x)+C||b_{1}||_{\ast}M_{r,(6)}([b_{2},T](f_{1},f_{2}))(x)\\\
&+C||b_{2}||_{\ast}M_{r,(6)}([b_{1},T](f_{1},f_{2}))(x)+C||b_{1}||_{\ast}||b_{2}||_{\ast}M_{p_{1},(5)}f_{1}(x)M_{p_{2},(5)}f_{2}(x),\end{split}$

holds for any $x$ and ball $B$ with $x\in B$, and

(2.15) $\begin{split}|h_{B}-h_{Q}|&\leq
CK_{B,Q}^{2}\biggr{[}||b_{1}||_{\ast}||b_{2}||_{\ast}M_{r,(6)}(T(f_{1},f_{2}))(x)+||b_{1}||_{\ast}M_{r,(6)}([b_{2},T](f_{1},f_{2}))(x)\\\
&\ \
+||b_{2}||_{\ast}M_{r,(6)}([b_{1},T](f_{1},f_{2}))(x)+||b_{1}||_{\ast}||b_{2}||_{\ast}M_{p_{1},(5)}f_{1}(x)M_{p_{2},(5)}f_{2}(x)\biggr{]}.\end{split}$

for all balls $B\subset Q$ with $x\in B$, where $B$ is an arbitrary ball, $Q$
is a doubling ball. For any ball $B$, we denote

$h_{B}:=m_{B}(T((b_{1}-m_{\tilde{B}}(b_{1}))f_{1}\chi_{X\backslash\frac{6}{5}B},(b_{2}-m_{\tilde{B}}(b_{2}))f_{2}\chi_{X\backslash\frac{6}{5}B})),$

and

$h_{Q}:=m_{Q}(T((b_{1}-m_{Q}(b_{1}))f_{1}\chi_{X\backslash\frac{6}{5}Q},(b_{2}-m_{Q}(b_{2}))f_{2}\chi_{X\backslash\frac{6}{5}Q})).$

Write

$[b_{1},b_{2},T]=T((b_{1}-b_{1}(z))f_{1},(b_{2}-b_{2}(z))f_{2}),$

and

(2.16)
$\begin{split}&T((b_{1}-m_{\tilde{B}}(b_{1}))f_{1},(b_{2}-m_{\tilde{B}}(b_{2}))f_{2})\\\
=&T((b_{1}-b_{1}(z)+b_{1}(z)-m_{\tilde{B}}(b_{1}))f_{1},(b_{2}-b_{2}(z)+b_{2}(z)-m_{\tilde{B}}(b_{2}))f_{2})\\\
=&(b_{1}(z)-m_{\tilde{B}}(b_{1}))(b_{2}(z)-m_{\tilde{B}}(b_{2}))T(f_{1},f_{2})\\\
&\ -(b_{1}(z)-m_{\tilde{B}}(b_{1}))T(f_{1},(b_{2}-b_{2}(z))f_{2})\\\ &\
-(b_{2}(z)-m_{\tilde{B}}(b_{2}))T((b_{1}-b_{1}(z))f_{1},f_{2})+T((b_{1}-b_{1}(z))f_{1},(b_{2}-b_{2}(z))f_{2}).\end{split}$

Then

(2.17)
$\begin{split}&\biggl{(}\frac{1}{\mu(6B)}\int_{B}||[b_{1},b_{2},T](f_{1},f_{2})(z)|^{\delta}-|h_{B}|^{\delta}d\mu(z)\biggr{)}^{1/\delta}\\\
\leq&C\biggl{(}\frac{1}{\mu(6B)}\int_{B}|[b_{1},b_{2},T](f_{1},f_{2})(z)-h_{B}|^{\delta}d\mu(z)\biggr{)}^{1/\delta}\\\
\leq&C\biggl{(}\frac{1}{\mu(6B)}\int_{B}|(b_{1}(z)-m_{\tilde{B}}(b_{1}))(b_{2}(z)-m_{\tilde{B}}(b_{2}))T(f_{1},f_{2})(z)|^{\delta}d\mu(z)\biggr{)}^{1/\delta}\\\
&\ \
+C\biggl{(}\frac{1}{\mu(6B)}\int_{B}|(b_{1}(z)-m_{\tilde{B}}(b_{1}))T(f_{1},(b_{2}-b_{2}(z))f_{2})(z)|^{\delta}d\mu(z)\biggr{)}^{1/\delta}\\\
&\ \
+C\biggl{(}\frac{1}{\mu(6B)}\int_{B}|(b_{2}(z)-m_{\tilde{B}}(b_{2}))T((b_{1}-b_{1}(z))f_{1},f_{2})(z)|^{\delta}d\mu(z)\biggr{)}^{1/\delta}\\\
&\ \
+C\biggl{(}\frac{1}{\mu(6B)}\int_{B}|T((b_{1}-m_{\tilde{B}}(b_{1}))f_{1},(b_{2}-m_{\tilde{B}}(b_{2}))f_{2})(z)-h_{B}|^{\delta}d\mu(z)\biggr{)}^{1/\delta}\\\
=&:E_{1}+E_{2}+E_{3}+E_{4}.\end{split}$

We firstly estimate $E_{1}$. Let $r_{1},r_{2}>1$ such that
$\dfrac{1}{r}+\dfrac{1}{r_{1}}+\dfrac{1}{r_{2}}=\dfrac{1}{\delta}$. By
Hölder’s inequality and Lemma 2.2, we yields

(2.18) $\begin{split}E_{1}&\leq
C\biggl{(}\frac{1}{\mu(6B)}\int_{B}|b_{1}(z)-m_{\tilde{B}}b_{1}|^{r_{1}}d\mu(z)\biggr{)}^{1/r_{1}}\\\
&\ \
\times\biggl{(}\frac{1}{\mu(6B)}\int_{B}|b_{2}(z)-m_{\tilde{B}}b_{2}|^{r_{2}}d\mu(z)\biggr{)}^{1/r_{2}}\\\
&\ \
\times\biggl{(}\frac{1}{\mu(6B)}\int_{B}|T(f_{1},f_{2})|^{r}d\mu(z)\biggr{)}^{1/r}\\\
&\leq
C||b_{1}||_{\ast}||b_{2}||_{\ast}M_{r,(6)}(T(f_{1},f_{2}))(x).\end{split}$

For $E_{2}$, let $s>1$ such that
$\dfrac{1}{s}+\dfrac{1}{r}=\dfrac{1}{\delta}$, by Hölder’s inequality and
Lemma 2.2, we deduce

(2.19) $\begin{split}E_{2}&\leq
C\biggl{(}\frac{1}{\mu(6B)}\int_{B}|b_{1}(z)-m_{\tilde{B}}b_{1}|^{s}d\mu(z)\biggr{)}^{1/s}\\\
&\ \
\times\biggl{(}\frac{1}{\mu(6B)}\int_{B}|[b_{2},T](f_{1},f_{2})|^{r}d\mu(z)\biggr{)}^{1/r}\\\
&\leq C||b_{1}||_{\ast}M_{r,(6)}([b_{2},T](f_{1},f_{2}))(x).\end{split}$

Similar to estimate $E_{2}$, we immediately get

(2.20) $E_{3}\leq C||b_{2}||_{\ast}M_{r,(6)}([b_{1},T](f_{1},f_{2}))(x).$

Let us turn to estimate $E_{4}$. Denote $f_{j}^{1}=f_{j}\chi_{\frac{6}{5}B}$
and $f_{j}^{2}=f_{j}-f_{j}^{1}$ for $j=1,2$. Then

(2.21) $\begin{split}E_{4}&\leq
C\biggl{(}\frac{1}{\mu(6B)}\int_{B}|T((b_{1}-m_{\tilde{B}}b_{1})f_{1}^{1}(z),(b_{2}-m_{\tilde{B}}b_{2})f_{2}^{1})(z)|^{\delta}d\mu(z)\biggr{)}^{1/\delta}\\\
&\ \ \
+C\biggl{(}\frac{1}{\mu(6B)}\int_{B}|T((b_{1}-m_{\tilde{B}}b_{1})f_{1}^{1}(z),(b_{2}-m_{\tilde{B}}b_{2})f_{2}^{2})(z)|^{\delta}d\mu(z)\biggr{)}^{1/\delta}\\\
&\ \ \
+C\biggl{(}\frac{1}{\mu(6B)}\int_{B}|T((b_{1}-m_{\tilde{B}}b_{1})f_{1}^{2}(z),(b_{2}-m_{\tilde{B}}b_{2})f_{2}^{1})(z)|^{\delta}d\mu(z)\biggr{)}^{1/\delta}\\\
&\ \ \
+C\biggl{(}\frac{1}{\mu(6B)}\int_{B}|T((b_{1}-m_{\tilde{B}}b_{1})f_{1}^{2}(z),(b_{2}-m_{\tilde{B}}b_{2})f_{2}^{2})(z)-h_{B}|^{\delta}d\mu(z)\biggr{)}^{1/\delta}\\\
&=:E_{41}+E_{42}+E_{43}+E_{44}.\end{split}$

To estimate $E_{41}$, we need the classical Kolmogorov’s theorem: Let
$(X,\mu)$ be a probability measure space and let $0<p<q<\infty$, then there
exists a constant $C>0$, such that $||f||_{L^{p}(\mu)}\leq
C||f||_{L^{q,\infty}(\mu)}$ for any measurable function $f$. Let $p=\delta$
and $q=1/2$ such that $0<\delta<1/2$. Using Kolmogorov’s theorem, Lemma 2.2
and Hölder’s inequality, we obtain

(2.22) $\begin{split}&E_{41}\leq
C||T((b_{1}-m_{\tilde{B}}b_{1})f_{1}^{1},(b_{2}-m_{\tilde{B}}b_{2})f_{2}^{1})||_{L^{1/2,\infty}(\frac{6}{5}B,\frac{d\mu(z)}{\mu(6B)})}\\\
\leq&C\frac{1}{\mu(6B)}\int_{\frac{6}{5}B}|(b_{1}-m_{\tilde{B}}b_{1})f_{1}(z)|d\mu(z)\\\
&\ \
\times\frac{1}{\mu(6B)}\int_{\frac{6}{5}B}|(b_{2}-m_{\tilde{B}}b_{2})f_{2}(z)|d\mu(z)\\\
\leq&C(\frac{1}{\mu(6B)}\int_{\frac{6}{5}B}|(b_{1}-m_{\tilde{B}}b_{1}|^{p^{\prime}_{1}}d\mu(z))^{1/p^{\prime}_{1}}\\\
&\ \
\times(\frac{1}{\mu(6B)}\int_{\frac{6}{5}B}|f_{1}(z)|^{p_{1}}d\mu(z))^{1/p_{1}}\\\
&\ \
\times(\frac{1}{\mu(6B)}\int_{\frac{6}{5}B}|(b_{2}-m_{\tilde{Q}}b_{2})|^{p^{\prime}_{2}}d\mu(z))^{1/p^{\prime}_{2}}\\\
&\ \
\times(\frac{1}{\mu(6B)}\int_{\frac{6}{5}B}|f_{2}(z)|^{p_{2}}d\mu(z))^{1/p_{2}}\\\
\leq&C||b_{1}||_{\ast}||b_{2}||_{\ast}M_{p_{1},(5)}f_{1}(x)M_{p_{2},(5)}f_{2}(x).\end{split}$

To compute $E_{42}$, using (i) of Definition 1.7, Lemma 2.1, Lemma 2.2,
Hölder’s inequality and the properties of $\lambda$, we know

(2.23) $\begin{split}E_{42}&\leq
C\frac{1}{\mu(6B)}\int_{X}\int_{X}\int_{X\backslash\frac{6}{5}B}\frac{|b_{1}(y_{1})-m_{\tilde{B}}b_{1}||f_{1}^{1}(y_{1})|}{[\lambda(z,d(z,y_{1}))+\lambda(z,d(z,y_{2}))]^{2}}\\\
&\ \ \ \
\times|b_{2}(y_{2})-m_{\tilde{B}}b_{2}||f_{1}^{2}(y_{2})|d\mu(y_{1})d\mu(y_{2})d\mu(z)\\\
&\leq
C\frac{1}{\mu(6B)}\int_{B}\int_{\frac{6}{5}B}|b_{1}(y_{1})-m_{\tilde{B}}b_{1}||f_{1}(y_{1})|d\mu(y_{1})\\\
&\ \ \
\times\int_{X\backslash\frac{6}{5}B}\frac{|b_{2}(y_{2})-m_{\tilde{B}}b_{2}||f_{2}(y_{2})|d\mu(y_{2})}{[\lambda(z,d(z,y_{2}))]^{2}}d\mu(z)\\\
&\leq
C(\frac{1}{\mu(6B)}\int_{\frac{6}{5}B}|b_{1}(y_{1})-m_{\tilde{B}}b_{1}|^{p^{\prime}_{1}}d\mu(y_{1}))^{1/p^{\prime}_{1}}\\\
&\ \ \
\times(\frac{1}{\mu(6B)}\int_{\frac{6}{5}B}|f_{1}(y_{1})|^{p_{1}}d\mu(y_{1}))^{1/p_{1}}\\\
&\ \ \
\times\mu(B)\sum_{k=1}^{\infty}\int_{6^{k}\frac{6}{5}B}\frac{|b_{2}(y_{2})-m_{\tilde{B}}b_{2}||f_{2}(y_{2})|}{[\lambda(z,6^{k-1}\frac{6}{5}r_{B})]^{2}}d\mu(y_{2})\\\
&\leq
C||b_{1}||_{\ast}M_{p_{1},(5)}f_{1}(x)\sum_{k=1}^{\infty}6^{-km}\frac{\mu(B)}{\mu(\frac{6}{5}B)}\frac{\mu(\frac{6}{5}B)}{\lambda(z,\frac{6}{5}r_{B})}\\\
&\ \ \
\times\frac{1}{\lambda(z,6^{k-1}\frac{6}{5}r_{B})}\int_{6^{k}\frac{6}{5}B}|b_{2}(y_{2})-m_{\tilde{B}}b_{2}||f_{2}(y_{2})|d\mu(y_{2})\\\
&\leq
C||b_{1}||_{\ast}M_{p_{1},(5)}f_{1}(x)\sum_{k=1}^{\infty}6^{-km}\frac{1}{\mu(5\times
6^{k}\frac{6}{5}B)}\\\ &\ \ \
\times\int_{6^{k}\frac{6}{5}B}|b_{2}(y_{2})-m_{\widetilde{6^{k}\frac{6}{5}B}}(b_{2})+m_{\widetilde{6^{k}\frac{6}{5}B}}(b_{2})-m_{\tilde{B}}b_{2}||f_{2}(y_{2})|d\mu(y_{2})\\\
&\leq
C||b_{1}||_{\ast}M_{p_{1},(5)}f_{1}(x)\sum_{k=1}^{\infty}6^{-km}\biggl{[}\biggl{(}\frac{1}{\mu(5\times
6^{k}\frac{6}{5}B)}\\\ &\ \ \
\times\int_{6^{k+1}\frac{6}{5}B}|b_{2}(y_{2})-m_{\widetilde{6^{k}\frac{6}{5}B}}(b_{2})|^{p^{\prime}_{2}}d\mu(y_{2})\biggr{)}^{1/p^{\prime}_{2}}\\\
&\ \ \ \times\biggl{(}\frac{1}{\mu(5\times
6^{k}\frac{6}{5}B)}\int_{6^{k}\frac{6}{5}B}|f_{2}(y_{2})|^{p_{2}}d\mu(y_{2})\biggr{)}^{1/p_{2}}\\\
&\ \ \ +Ck||b_{2}||_{\ast}\frac{1}{\mu(5\times
6^{k}\frac{6}{5}B)}\int_{6^{k}\frac{6}{5}B}|f_{2}(y_{2})|d\mu(y_{2})\biggr{]}\\\
&\leq
C||b_{1}||_{\ast}||b_{2}||_{\ast}M_{p_{1},(5)}f_{1}(x)M_{p_{2},(5)}f_{2}(x).\\\
\end{split}$

Similarly, we get

(2.24) $E_{43}\leq
C||b_{1}||_{\ast}||b_{2}||_{\ast}M_{p_{1},(5)}f_{1}(x)M_{p_{2},(5)}f_{2}(x).$

For $E_{44}$, by (ii) of Definition 1.7, Lemma 2.1, Lemma 2.2, Hölder’s
inequality and the properties of $\lambda$, we obtain

(2.25)
$\begin{split}&|T((b_{1}-m_{\tilde{B}}b_{1})f_{2}^{2},(b_{2}-m_{\tilde{B}}b_{2})f_{2}^{2})(z)-T((b_{1}-m_{\tilde{B}}b_{1})f_{2}^{2},(b_{2}-m_{\tilde{B}}b_{2})f_{2}^{2})(z_{0})|\\\
\leq&C\int_{X\backslash\frac{6}{5}B}\int_{X\backslash\frac{6}{5}B}|K(z,y_{1},y_{2})-K(z_{0},y_{1},y_{2})|\\\
&\ \ \
\times|\prod_{i=1}^{2}(b_{i}(y_{i})-m_{\tilde{B}}b_{i})f_{i}(y_{i})|d\mu(y_{1})d\mu(y_{2})\\\
\leq&C\int_{X\backslash\frac{6}{5}B}\int_{X\backslash\frac{6}{5}B}\frac{d(z,z_{0})^{\delta}|\prod_{i=1}^{2}(b_{i}(y_{i})-m_{\tilde{B}}b_{i})f_{i}(y_{i})|d\mu(y_{1})d\mu(y_{2})}{(d(z,y_{1})+d(z,y_{2}))^{\delta}[\sum_{j=1}^{2}\lambda(z,d(z,y_{j}))]^{2}}\\\
\leq&C\prod_{i=1}^{2}\int_{X\backslash\frac{6}{5}B}\frac{d(z,z_{0})^{\delta_{i}}|b_{i}(y_{i})-m_{\tilde{B}}b_{i}||f_{i}(y_{i})d\mu(y_{i})|}{d(z,y_{i})^{\delta_{i}}\lambda(z,d(z,y_{i}))}\\\
\leq&C\prod_{i=1}^{2}\sum_{k=1}^{\infty}\int_{6^{k}\frac{6}{5}B}6^{-k\delta_{i}}\frac{\mu(5\times
6^{k}\frac{6}{5}B)}{\lambda(z,5\times
6^{k}\frac{6}{5}r_{B})}\frac{1}{\mu(5\times
6^{k}\frac{6}{5}B)}|b_{i}(y_{i})-m_{\tilde{B}}b_{i}||f_{i}|d\mu(y_{i})\\\
\leq&C\prod_{i=1}^{2}\sum_{k=1}^{\infty}6^{-k\delta_{i}}(\frac{1}{\mu(5\times
6^{k}\frac{6}{5}B)}\int_{6^{k}\frac{6}{5}B}|b_{i}(y_{i})-m_{\tilde{B}}b_{i}|^{p^{\prime}_{i}}d\mu(y_{i}))^{1/p^{\prime}_{i}}\\\
&\ \ \ \times(\frac{1}{\mu(5\times
6^{k}\frac{6}{5}B)}\int_{6^{k}\frac{6}{5}B}|f_{i}|^{p_{i}})^{1/p_{i}}\\\
\leq&C\prod_{i=1}^{2}\sum_{k=1}^{\infty}6^{-k\delta_{i}}M_{p_{i},(6)}f_{i}(x)(\frac{1}{\mu(5\times
6^{k}\frac{6}{5}B)}\int_{6^{k}\frac{6}{5}B}|b_{i}(y_{i})-m_{\widetilde{6^{k}\frac{6}{5}B}}\\\
&\ \ \
+m_{\widetilde{6^{k}\frac{6}{5}B}}-m_{\tilde{B}}b_{i}|^{p^{\prime}_{i}}d\mu(y_{i}))^{1/p^{\prime}_{i}}\\\
\leq&C\prod_{i=1}^{2}\sum_{k=1}^{\infty}6^{-k\delta_{i}}k||b_{i}||_{\ast}M_{p_{i},(5)}f_{i}(x)\\\
\leq&C||b_{1}||_{\ast}||b_{2}||_{\ast}M_{p_{1},(5)}f_{1}(x)M_{p_{2},(5)}f_{2}(x).\end{split}$

where $\delta_{1},\delta_{2}>0$ and $\delta_{1}+\delta_{2}=\delta$.

Taking the mean over $z_{0}\in B$, we deduce

(2.26) $E_{44}\leq
C||b_{1}||_{\ast}||b_{2}||_{\ast}M_{p_{1},(5)}f_{1}(x)M_{p_{2},(5)}f_{2}(x).$

So (2.14) can be obtain from (2.17) to (2.26).

Next we prove (2.15). Consider two balls $B\subset Q$ with $x\in B$, where $B$
is an arbitrary ball and $Q$ is a doubling ball. Let $N=N_{B,Q}+1$, then we
obtain

(2.27)
$\begin{split}&\biggl{|}|m_{B}T((b_{1}-m_{\tilde{B}}b_{1})f_{1}^{2},(b_{2}-m_{\tilde{B}}b_{2})f_{2}^{2})|\\\
&\ \
-|m_{Q}T((b_{1}-m_{Q}b_{1})f_{1}^{2},(b_{2}-m_{Q}b_{2})f_{2}^{2})|\biggr{|}\\\
\leq&|m_{B}T((b_{1}-m_{\tilde{B}}b_{1})f_{1}\chi_{X\backslash
6^{N}B},(b_{2}-m_{\tilde{B}}b_{2})f_{2}\chi_{X\backslash 6^{N}B})\\\ &\ \
-m_{Q}T((b_{1}-m_{\tilde{B}}b_{1})f_{1}\chi_{X\backslash
6^{N}B},(b_{2}-m_{\tilde{B}}b_{2})f_{2}\chi_{X\backslash 6^{N}B})|\\\ &\ \
+|m_{Q}T((b_{1}-m_{Q}b_{1})f_{1}\chi_{X\backslash
6^{N}B},(b_{2}-m_{Q}b_{2})f_{2}\chi_{X\backslash 6^{N}B})\\\ &\ \
-m_{Q}T((b_{1}-m_{\tilde{B}}b_{1})f_{1}\chi_{X\backslash
6^{N}B},(b_{2}-m_{\tilde{B}}b_{2})f_{2}\chi_{X\backslash 6^{N}B})|\\\ &\ \
+|m_{B}T((b_{1}-m_{\tilde{B}}b_{1})f_{1}\chi_{6^{N}B\backslash\frac{6}{5}B},(b_{2}-m_{\tilde{B}}b_{2})f_{2}\chi_{X\backslash\frac{6}{5}B})\\\
&\ \
+|m_{B}T((b_{1}-m_{\tilde{B}}b_{1})f_{1}\chi_{X\backslash\frac{6}{5}B},(b_{2}-m_{\tilde{B}}b_{2})f_{2}\chi_{6^{N}B\backslash\frac{6}{5}B})\\\
&\ \
+|m_{Q}T((b_{1}-m_{Q}b_{1})f_{1}\chi_{6^{N}B\backslash\frac{6}{5}Q},(b_{2}-m_{Q}b_{2})f_{2}\chi_{X\backslash
6^{N}B})\\\ &\ \
+|m_{Q}T((b_{1}-m_{Q}b_{1})f_{1}\chi_{X\backslash\frac{6}{5}Q},(b_{2}-m_{Q}b_{2})f_{2}\chi_{6^{N}B\backslash\frac{6}{5}Q})\\\
=:&F_{1}+F_{2}+F_{3}+F_{4}+F_{5}+F_{6}.\end{split}$

Using the method to estimate $E_{4}$, we get

(2.28) $F_{1}\leq
CK_{B,Q}^{2}||b_{1}||_{\ast}||b_{2}||_{\ast}M_{p_{1},(5)}f_{1}(x)M_{p_{2},(5)}f_{2}(x).$

Let us estimate $F_{2}$. At first, we compute

(2.29) $\begin{split}&T((b_{1}-m_{Q}b_{1})f_{1}\chi_{X\backslash
6^{N}B},(b_{2}-m_{Q}b_{2})f_{2}\chi_{X\backslash 6^{N}B})(z)\\\ &\ \
-T((b_{1}-m_{\tilde{B}}b_{1})f_{1}\chi_{X\backslash
6^{N}B},(b_{2}-m_{\tilde{B}}b_{2})f_{2}\chi_{X\backslash 6^{N}B})(z)\\\
=&(m_{Q}b_{2}-m_{\tilde{B}}b_{2})T((b_{1}-m_{Q}b_{1})f_{1}\chi_{X\backslash
6^{N}B},f_{2}\chi_{X\backslash 6^{N}B})(z)\\\ &\ \
+(m_{Q}b_{1}-m_{\tilde{B}}b_{1})T(f_{1}\chi_{X\backslash
6^{N}B},(b_{2}-m_{Q}b_{2})f_{2}\chi_{X\backslash 6^{N}B})(z)\\\ &\ \
+(m_{Q}b_{1}-m_{\tilde{B}}b_{1})(m_{Q}b_{2}-m_{\tilde{B}}b_{2})T(f_{1}\chi_{X\backslash
6^{N}B},f_{2}\chi_{X\backslash 6^{N}B})(z).\\\ \end{split}$

Hence

(2.30)
$\begin{split}F_{2}\leq&|(m_{Q}b_{2}-m_{\tilde{B}}b_{2})\frac{1}{\mu(Q)}\int_{Q}T((b_{1}-m_{Q}b_{1})f_{1}\chi_{X\backslash
6^{N}B},f_{2}\chi_{X\backslash 6^{N}B})(z)d\mu(z)|\\\ &\
+|(m_{Q}b_{1}-m_{\tilde{B}}b_{1})\frac{1}{\mu(Q)}\int_{Q}T((f_{1}\chi_{X\backslash
6^{N}B},(b_{2}-m_{Q}b_{2})f_{2}\chi_{X\backslash 6^{N}B})(z)d\mu(z)|\\\ &\
+|(m_{Q}b_{1}-m_{\tilde{B}}b_{1})(m_{Q}b_{2}-m_{\tilde{B}}b_{2})\frac{1}{\mu(Q)}\int_{Q}T(f_{1}\chi_{X\backslash
6^{N}B},f_{2}\chi_{X\backslash 6^{N}B})(z)|\\\
=:&F_{21}+F_{22}+F_{23}.\end{split}$

To estimate $F_{21}$, we write

(2.31) $\begin{split}&T((b_{1}-m_{Q}b_{1})f_{1}\chi_{X\backslash
6^{N}Q},f_{2}\chi_{X\backslash 6^{N}Q})(z)\\\
=&T((b_{1}-m_{Q}b_{1})f_{1},f_{2})(z)-T((b_{1}-m_{Q}b_{1})f_{1}\chi_{6^{N}B}\chi_{\frac{6}{5}Q},f_{2}\chi_{\frac{6}{5}Q})(z))\\\
&\
-T((b_{1}-m_{Q}b_{1})f_{1}\chi_{\frac{6}{5}Q},f_{2}\chi_{6^{N}B}\chi_{\frac{6}{5}Q})(z))\\\
&\
+T((b_{1}-m_{Q}b_{1})f_{1}\chi_{6^{N}B}\chi_{\frac{6}{5}Q},f_{2}\chi_{6^{N}B}\chi_{\frac{6}{5}Q})(z)\\\
&\
-T((b_{1}-m_{Q}b_{1})f_{1}\chi_{X\backslash\frac{6}{5}Q},f_{2}\chi_{6^{N}B})(z)\\\
&\
-T((b_{1}-m_{Q}b_{1})f_{1}\chi_{6^{N}B},f_{2}\chi_{X\backslash\frac{6}{5}Q})(z)\\\
&\
+T((b_{1}-m_{Q}b_{1})f_{1}\chi_{6^{N}B\backslash\frac{6}{5}Q},f_{2}\chi_{6^{N}B\backslash\frac{6}{5}Q})(z)\\\
=:&H_{1}(z)+H_{2}(z)+H_{3}(z)+H_{4}(z)+H_{5}(z)+H_{6}(z)+H_{7}(z).\end{split}$

Let us estimate $H_{1}(z)$ firstly. Since

$\frac{1}{\mu(Q)}\int_{Q}|T(b_{1}-b_{1}(z)f_{1},f_{2})(z)|d\mu(z)\leq
CM_{r,(6)}([b_{1},T]f_{1},f_{2})(x)$

and by Hölder’s inequality, we have

$\frac{1}{\mu(Q)}\int_{Q}|(b_{1}(z)-m_{Q}(b_{1}))T(f_{1},f_{2})(z)|d\mu(z)\leq
C||b_{1}||_{\ast}M_{r,(6)}(T(f_{1},f_{2}))(x),$

then we obtain

(2.32)
$\begin{split}|m_{Q}(H_{1})|&\leq|m_{Q}(T(b_{1}-b_{1}(z)f_{1},f_{2}))|+|m_{Q}((b_{1}(z)-m_{Q}(b_{1}))T(f_{1},f_{2}))|\\\
&\leq
CM_{r,(6)}([b_{1},T]f_{1},f_{2})(x)+||b_{1}||_{\ast}M_{r,(6)}(T(f_{1},f_{2}))(x).\end{split}$

For $H_{2}(z)$, let $r>1$ and $1<s_{1}<p_{1}$ such that
$\dfrac{1}{r}=\dfrac{1}{s_{1}}+\dfrac{1}{p_{2}}$. Denote
$\dfrac{1}{s_{2}}=\dfrac{1}{s_{1}}-\dfrac{1}{p_{1}}$, using the fact of $Q$ is
a doubling balls, Kolmogorov’s inequality, Hölder’s inequality and Lemma 2.4,
we yield

(2.33)
$\begin{split}|m_{Q}(H_{2})|\leq&C||H_{2}||_{L^{r,\infty}(Q,\frac{d\mu(z)}{\mu(Q)})}\\\
\leq&C(\frac{1}{\mu(Q)}\int_{\frac{6}{5}Q}|b_{1}-m_{Q}b_{1}|^{s_{1}}d\mu(z))^{1/s_{1}}(\frac{1}{\mu(Q)}\int_{\frac{6}{5}Q}|f_{2}|^{p_{2}}d\mu(z))^{1/p_{2}}\\\
\leq&C(\frac{1}{\mu(6Q)}\int_{\frac{6}{5}Q}|b_{1}-m_{Q}b_{1}|^{s_{2}}d\mu(z))^{1/s_{2}}(\frac{1}{\mu(6Q)}\int_{\frac{6}{5}Q}|f_{2}|^{p_{1}}d\mu(z))^{1/p_{1}}\\\
&\ \
\times(\frac{1}{\mu(6Q)}\int_{\frac{6}{5}Q}|f_{2}|^{p_{2}}d\mu(z))^{1/p_{2}}\\\
\leq&||b_{1}||_{\ast}M_{p_{1},(5)}f_{1}(x)M_{p_{2},(5)}f_{2}(x).\end{split}$

We also can obtain

(2.34) $|m_{Q}(H_{3})|+|m_{Q}(H_{4})|\leq
C||b_{1}||_{\ast}M_{p_{1},(5)}f_{1}(x)M_{p_{2},(5)}f_{2}(x).$

For $H_{5}$, since $z\in Q$, by (i) of Definition 1.7, Lemma 2.1, Lemma 2.2,
Hölder’s inequality and $Q$ is a doubling ball, we deduce

(2.35) $\begin{split}|H_{5}(z)|&\leq
C\int_{6^{N}B}\int_{X\backslash\frac{6}{5}Q}\frac{|b_{1}(y_{1})-m_{Q}b_{1}||f_{1}(y_{1})||f_{2}(y_{2})|d\mu(y_{1})d\mu(y_{2})}{[\sum_{j=1}^{2}\lambda(z,d(x,y_{j}))]^{2}}\\\
&\leq
C\int_{6^{N}B}|f_{2}(y_{2})|d\mu(y_{2})\sum_{k=1}^{\infty}\int_{6^{k}\frac{6}{5}Q}\frac{|b_{1}(y_{1})-m_{Q}b_{1}||f_{1}(y_{1})|}{(\lambda(z,6^{k-1}\frac{6}{5}r_{Q}))^{2}}d\mu(y_{1})\\\
&\leq C\int_{6^{N}B}|f_{2}(y_{2})|d\mu(y_{2})\sum_{k=1}^{\infty}6^{-km}\\\ &\
\ \
\times\int_{6^{k}\frac{6}{5}Q}\frac{1}{\lambda(z,\frac{6}{5}r_{Q})}\frac{|b_{1}(y_{1})-m_{Q}b_{1}||f_{1}(y_{1})|d\mu(y_{1})}{\lambda(z,6^{k-1}\frac{6}{5}r_{Q})}\\\
&\leq
C\frac{1}{\lambda(z,6r_{Q})}\int_{6^{N}B}|f_{2}(y_{2})|d\mu(y_{2})\sum_{k=1}^{\infty}6^{-km}\frac{1}{\lambda(z,5\times
6^{k}\frac{6}{5}r_{Q})}\\\ &\ \ \
\times\biggl{[}\int_{6^{k}\frac{6}{5}Q}|b_{1}(y_{1})-m_{6^{k}\frac{6}{5}Q}(b_{1})||f_{1}(y_{1})|d\mu(y_{1})\\\
&\ \ \
+\int_{6^{k}\frac{6}{5}Q}|m_{6^{k}\frac{6}{5}Q}(b_{1})-m_{Q}b_{1}||f_{1}(y_{1})|d\mu(y_{1})\biggr{]}\\\
&\leq
C\frac{1}{\lambda(z,6r_{Q})}\int_{6^{N}B}|f_{2}(y_{2})|d\mu(y_{2})\sum_{k=1}^{\infty}6^{-km}\\\
&\ \ \ \times\biggl{[}\biggl{(}\frac{1}{\lambda(z,5\times
6^{k}\frac{6}{5}r_{Q})}\int_{6^{k}\frac{6}{5}Q}|b_{1}(y_{1})-m_{6^{k}\frac{6}{5}Q}(b_{1})|^{p^{\prime}_{1}}d\mu(y_{1})\biggr{)}^{1/p^{\prime}_{1}}\\\
&\ \ \ \ \times(\frac{1}{\lambda(z,5\times
6^{k}\frac{6}{5}r_{Q})}\int_{6^{k}\frac{6}{5}Q}|f_{1}(y_{1})|^{p_{1}}d\mu(y_{1}))^{1/p_{1}}\\\
&\ \ \ \ +k||b_{1}||_{\ast}\frac{1}{\lambda(z,5\times
6^{k}\frac{6}{5}r_{Q})}\int_{6^{k}\frac{6}{5}Q}|f_{1}(y_{1})|d\mu(y_{1})\biggr{]}\\\
&\leq
C\sum_{k=1}^{N}\frac{1}{\lambda(z,6r_{Q})}\int_{6^{k}B}|f_{2}(y_{2})|d\mu(y_{2})||b_{1}||_{\ast}M_{p_{1},(5)}f_{1}(x)\\\
&\leq C\sum_{k=1}^{N}\frac{\mu(5\times 6^{k}B)}{\lambda(z,5\times
6^{k}r_{B})}\frac{\lambda(z,5\times 6^{k}r_{B})}{\lambda(z,6r_{Q})}\\\ &\ \ \
\ \times\frac{1}{\mu(5\times
6^{k}B)}\int_{6^{k}B}|f_{2}(y_{2})|d\mu(y_{2})||b_{1}||_{\ast}M_{p_{1},(5)}f_{1}(x)\\\
&\leq
CK_{B,Q}||b_{1}||_{\ast}M_{p_{1},(5)}f_{1}(x)M_{p_{2},(5)}f_{2}(x).\end{split}$

Then it yields

(2.36) $|m_{Q}(H_{5})|\leq
CK_{B,Q}||b_{1}||_{\ast}M_{p_{1},(5)}f_{1}(x)M_{p_{2},(5)}f_{2}(x).$

In the similar way to estimate $m_{Q}(H_{5})$, we also obtain

(2.37) $|m_{Q}(H_{6})|+|m_{Q}(H_{7})|\leq
CK_{B,Q}||b_{1}||_{\ast}M_{p_{1},(5)}f_{1}(x)M_{p_{2},(5)}f_{2}(x).$

From (2.8) in Lemma 2.2, we deduce

(2.38)
$\begin{split}F_{21}\leq&CK_{B,Q}^{2}\\{||b_{1}||_{\ast}||b_{2}||_{\ast}M_{r,(6)}(T(f_{1},f_{2}))(x)\\\
&\ \ \ +||b_{1}||_{\ast}M_{r,(6)}([b_{2},T](f_{1},f_{2}))(x)\\\ &\ \ \
+||b_{2}||_{\ast}M_{r,(6)}([b_{1},T](f_{1},f_{2}))(x)\\\ &\ \ \
+||b_{1}||_{\ast}||b_{2}||_{\ast}M_{p_{1},(5)}f_{1}(x)M_{p_{2},(5)}f_{2}(x)\\}.\\\
\end{split}$

$F_{22}$ and $F_{23}$ also have similar estimate of $F_{21}$, therefore,

(2.39)
$\begin{split}F_{2}\leq&CK_{B,Q}^{2}\biggr{\\{}||b_{1}||_{\ast}||b_{2}||_{\ast}M_{r,(6)}(T(f_{1},f_{2}))(x)\\\
&\ \ \ +||b_{1}||_{\ast}M_{r,(6)}([b_{2},T](f_{1},f_{2}))(x)\\\ &\ \ \
+||b_{2}||_{\ast}M_{r,(6)}([b_{1},T](f_{1},f_{2}))(x)\\\ &\ \ \
+||b_{1}||_{\ast}||b_{2}||_{\ast}M_{p_{1},(5)}f_{1}(x)M_{p_{2},(5)}f_{2}(x)\biggr{\\}}.\end{split}$

From $F_{3}$ to $F_{6}$, using the similar method to estimate $I_{4}$, we
conclude

(2.40) $F_{3}+F_{4}+F_{5}+F_{6}\leq
C||b_{1}||_{\ast}||b_{2}||_{\ast}M_{p_{1},(5)}f_{1}(x)M_{p_{2},(5)}f_{2}(x).$

Thus (2.15) holds by from (2.27) to (2.40) and hence (2.11) is proved. With
the same method to prove (2.11), we can obtain that (2.12) and (2.13) are also
hold. Here we omit the details. Thus the Lemma 2.5 is proved. ∎

###### Proof of Theorem 1.11.

Let $0<\delta<1/2$, $1<p_{1},p_{2},q<\infty$,
$\dfrac{1}{q}=\dfrac{1}{p_{1}}+\dfrac{1}{p_{2}}$, $1<r<q$, $f_{1}\in
L^{p_{1}}(\mu)$, $f_{2}\in L^{p_{2}}(\mu)$, $b_{1}\in RBMO(\mu)$ and $b_{2}\in
RBMO(\mu)$. By $|f(x)|\leq N_{\delta}f(x)$, Lemma 2.1, Lemma 2.3, Lemma 2.4,
Hörder’s inequality and the boundedness of $M_{(\rho)}$ and $M_{r,(\rho)}$ for
$\rho\geq 5$ and $q>r$, we obtain

(2.41)
$\begin{split}&||[b_{1},b_{2},T](f_{1},f_{2})||_{L^{q}(\mu)}\leq||N_{\delta}([b_{1},b_{2},T](f_{1},f_{2}))||_{L^{q}(\mu)}\\\
\leq&C||M^{\sharp}_{\delta}([b_{1},b_{2},T](f_{1},f_{2}))||_{L^{q}(\mu)}\\\
\leq&C||b_{1}||_{\ast}||b_{2}||_{\ast}||M_{r,(6)}(T(f_{1},f_{2}))||_{L^{q}(\mu)}\\\
&+C||b_{1}||_{\ast}||M_{r,(6)}([b_{2},T](f_{1},f_{2}))||_{L^{q}(\mu)}\\\
&+C||b_{2}||_{\ast}||M_{r,(6)}([b_{1},T](f_{1},f_{2}))||_{L^{q}(\mu)}\\\
&+C||b_{1}||_{\ast}||b_{2}||_{\ast}||M_{p_{1},(5)}f_{1}(x)M_{p_{2},(5)}f_{2}(x)||_{L^{q}(\mu)}\\\
\leq&C||b_{1}||_{\ast}||b_{2}||_{\ast}||f_{1}(x)||_{L^{p_{1}}(\mu)}||f_{2}(x)||_{L^{p_{2}}(\mu)}\\\
&+C||b_{1}||_{\ast}||([b_{2},T](f_{1},f_{2}))||_{L^{q}(\mu)}\\\
&+C||b_{2}||_{\ast}||([b_{1},T](f_{1},f_{2}))||_{L^{q}(\mu)}\\\
\leq&C||b_{1}||_{\ast}||b_{2}||_{\ast}||f_{1}(x)||_{L^{p_{1}}(\mu)}||f_{2}(x)||_{L^{p_{2}}(\mu)}\\\
&+C||b_{1}||_{\ast}||M^{\sharp}_{\delta}([b_{2},T](f_{1},f_{2}))||_{L^{q}(\mu)}\\\
&+C||b_{2}||_{\ast}||M^{\sharp}_{\delta}([b_{1},T](f_{1},f_{2}))||_{L^{q}(\mu)}\\\
\leq&||b_{1}||_{\ast}||b_{2}||_{\ast}||f_{1}(x)||_{L^{p_{1}}(\mu)}||f_{2}(x)||_{L^{p_{2}}(\mu)}\\\
&+C||b_{1}||_{\ast}||M_{r,(6)}(T(f_{1},f_{2}))(x)||_{L^{q}(\mu)}\\\
&+C||b_{1}||_{\ast}||M_{p_{1},(5)}f_{1}(x)M_{p_{2},(5)}f_{2}(x)||_{L^{q}(\mu)}\\\
&+C||b_{2}||_{\ast}||M_{r,(6)}(T(f_{1},f_{2}))(x)||_{L^{q}(\mu)}\\\
&+C||b_{2}||_{\ast}||M_{p_{1},(5)}f_{1}(x)M_{p_{2},(5)}f_{2}(x)||_{L^{q}(\mu)}\\\
\leq&C||b_{1}||_{\ast}||b_{2}||_{\ast}||f_{1}(x)||_{L^{p_{1}}(\mu)}||f_{2}(x)||_{L^{p_{2}}(\mu)}.\end{split}$

Thus the proof of Theorem 1.11 is completed. ∎

Acknowledgements This work was supported by National Natural Science
Foundation of China (Grant No. 10371087) and Excellent Young Talent Foundation
of Anhui Province (Grant No.2013SQRL080ZD).

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# Holomorphic reflexivity for locally finite and profinite groups: the Abelian
and general cases

O. Yu. Aristov<EMAIL_ADDRESS>
###### Abstract.

Akbarov’s theory of holomorphic reflexivity for topological Hopf algebras has
been developed in two directions, namely, by the complication of definitions
when expanding the scope and by their simplification when restricting. In the
framework of the latter approach, we establish the holomorphic reflexivity for
topological Hopf algebras associated with locally finite countable groups and
second-countable profinite groups. In the Abelian case, the reflexivity is
described in terms close to the classical ones.

In the paper [1], Akbarov presented a holomorphic version of the Pontryagin
duality and showed that an analogue of the classical result on reflexivity is
valid for compactly generated Abelian Stein groups. He also expanded the
context by considering the holomorphic duality for some class of topological
Hopf algebras, which includes algebras of holomorphic functions on general
(not only Abelian) compactly generated Stein groups, and also convolution
algebras of exponential analytic functionals. The fact that there is no need
to either prove or assume the existence of the Haar measure or its
noncommutative analogues is an accomplishment of this theory. However, there
are obstacles to reflexivity (for example, in some form, the linearity of a
group is a necessary condition; see [2]). Thus, the question of which
topological Hopf algebras are holomorphically reflexive arises.

The holomorphic duality theories that we are discussing can be described with
the help of _reflexivity diagrams_ of the form

$\textstyle{H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(H^{\bullet})^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{env}$$\textstyle{H^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{env}$$\textstyle{H^{\bullet}\,,\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

where $H$ is an object of some category of topological Hopf algebras, the
vertical arrows correspond to some duality functor for topological vector
spaces, and the horizontal ones correspond to some envelope functor (left
adjoint to some forgetful functor between categories of topological algebras).
If the Hopf algebra structure is preserved under all these operations, it can
be said that $H^{\bullet}$ is _dual_ to $H$. If, in addition, the diagram is
commutative (up to functor isomorphism), i.e., $H^{\bullet\bullet}\to H$ is a
functor isomorphism, then we can say what $H$ is _reflexive_.

We take for $H$ the following topological Hopf algebras: the function algebras
$\mathbb{C}^{\Gamma}$ and $\mathcal{O}(G)$ and also the group algebra
$\mathbb{C}\Gamma$ and the convolution algebra $\mathcal{O}(G)^{\prime}$,
where $\Gamma$ is a locally finite group and $G$ is a profinite group (here
$\mathcal{O}(G)$ stands for the set of locally constant functions on $G$; we
call them ‘‘holomorphic,’’ see Lemma 1.4, and the prime denotes the strong
dual space). In this paper, we prove the reflexivity of all these Hopf
algebras within the framework of the approach of [2].

The main difficulty in the scheme described above is the requirement that both
the operations involved in the definition of $H^{\bullet}$ must be functors
between some categories of topological Hopf algebras. In this paper, as well
as in [2], we consider Hopf algebras in the symmetric monoidal category of
complete locally convex spaces with the bifunctor
$(-)\mathbin{\widehat{\otimes}}(-)$ of complete projective tensor product
(Hopf $\mathbin{\widehat{\otimes}}$-algebras) and the functor of strong dual
to a locally convex space. At the same time, in [1], the spaces, their duals,
and tensor products are considered in the framework of the theory of
stereotype spaces. The other approach to definitions chosen in [2] is
motivated by the fact that Akbarov’s construction is rather complicated
technically and can be simplified under appropriate conditions on the topology
of the underlying spaces. Moreover, both approaches give the same result for
connected Lie groups considered in [2]; see, e.g., [3, Remark 2.9] (this is
apparently not the case in the general situation).

Consider first the simplest case in which $G$ is a compactly generated Abelian
Lie group. The statement about the _holomorphic reflexivity_ can be described
as the commutativity of the diagram

$\textstyle{\mathcal{O}(G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}(G^{\bullet})^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Fourier}$$\textstyle{\mathcal{O}(G)^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Fourier}$$\textstyle{\mathcal{O}(G^{\bullet})\,,\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

where $\mathcal{O}(G)$ is the set of holomorphic functions on $G$, the
horizontal arrows are the Fourier transform, and $G^{\bullet}$ stands for the
group of holomorphic characters of $G$ (see Sec. 1). A remarkable observation
made in [1] is that, in this case, the Fourier transform coincides with the
Arens–Michael envelope functor (the completion with respect to the topology
given by all continuous submultiplicative prenorms). Thus, we can write the
diagram in a form suitable not only for Abelian groups:

$\textstyle{\mathcal{O}(G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{{\mathscr{A}}}(G)^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{AM\,env}$$\textstyle{{\mathscr{A}}(G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{AM\,env}$$\textstyle{\widehat{{\mathscr{A}}}(G)\,,\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

where ${\mathscr{A}}(G)=\mathcal{O}(G)^{\prime}$, and the ‘‘hat’’ denotes the
Arens–Michael envelope. Studying the commutativity question for this diagram,
in other words, the question of the _holomorphic reflexivity_ of
$\mathcal{O}(G)$ in the case of compactly generated Stein group (not
necessarily Abelian) was started in [1]. For connected $G$, the following
answer was obtained in [2]: _$\mathcal{O}(G)$ is holomorphically reflexive if
and only if $G$ is linear_.

Note that, for the class of Hopf $\mathbin{\widehat{\otimes}}$-algebras and in
the general situation, there are no problems when choosing the Arens–Michael
envelope as the second functor in the definition, since, by [4, Proposition
6.7], we cannot leave this class by using the envelope functor.

In the present paper, we discuss the case of a zero-dimensional, i.e.,
discrete group $G$. The algebra $\mathcal{O}(G)$ can be identified with the
algebra $\mathbb{C}^{G}$ of all functions, and the reflexivity diagram takes
the form

$\textstyle{\mathbb{C}^{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\widehat{\mathbb{C}G})^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{AM\,env}$$\textstyle{\mathbb{C}G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{AM\,env}$$\textstyle{\widehat{\mathbb{C}G}\,.\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

If $G$ is finitely generated (equivalently, compactly generated), then
$\mathbb{C}^{G}$ is holomorphically reflexive (the assertion is implicitly
contained in [1]). This is connected with the fact that, in this case, both
the algebras $\mathbb{C}^{G}$ and $\widehat{\mathbb{C}G}$ are nuclear Fréchet
spaces; this is a situation in which the functor $H\mapsto H^{\prime}$ is
compatible with the structure of Hopf $\mathbin{\widehat{\otimes}}$-algebra.
It was discovered in [5] that $\widehat{\mathbb{C}G}$ need not be a Fréchet
space; moreover, the condition that G is finitely generated is necessary and
sufficient. There are grounds to assume that, in the general case, the
topology on $\widehat{\mathbb{C}G}$ may have pathologies that can violate the
compatibility of the functor $H\mapsto H^{\prime}$ with the projective tensor
product. Since we are interested in arbitrary countable groups, it is
important to understand how our duality scheme works for $H$ that are not
Fréchet spaces.

There are at least two possible ways of generalization here. The first way is
in the modification of the scheme itself within the framework of the theory of
stereotype spaces (as in [1]). This version was developed in [6]; see below.
In this paper, an alternative approach is chosen; we consider a wider class of
Hopf $\mathbin{\widehat{\otimes}}$-algebras for which the scheme works
practically without modifications. This uses the fact that a Hopf
$\mathbin{\widehat{\otimes}}$-algebra whose underlying space is a nuclear
Fréchet space belongs to the class of _well-behaved_ Hopf
$\mathbin{\widehat{\otimes}}$-algebras, which, by definition, also includes
the algebras with the underlying nuclear (DF)-space. This concept was
introduced in [7]; see the details in Sec. 2 and a more detailed discussion in
[2]. The ground for distinguishing this class is the following property: if
$H$ is a well-behaved Hopf $\mathbin{\widehat{\otimes}}$-algebra, then
$H^{\prime}$ is also a well-behaved Hopf $\mathbin{\widehat{\otimes}}$-algebra
(with respect to operations defined in a dual way). Thus, the general scheme
of [2] admits an extension to the class of underlying spaces opposite in
properties to Fréchet spaces, namely, to (DF)-spaces. Moreover, as is proved
in [8], the Arens–Michael envelopes of group algebras of locally finite groups
(which are opposite in properties to finitely generated ones) are (DF)-spaces.
This justifies the interest in locally finite groups in this context. On the
other hand, as is known, in the framework of the classical Pontryagin theory,
the group dual to a locally finite Abelian group is profinite (see, e.g., [9,
Theorem 2.9.6]). Therefore, the natural next step is to include also profinite
groups into consideration.

In the opinion of the author, it is expedient to consider the Abelian case
separately, in view of the fact that this part of the theory can be developed
without involving Hopf algebras. For this reason, we consider the holomorphic
duality between locally finite and profinite groups in the Abelian case in
terms of characters and verify that it has exactly the same form as in the
classical case in Sec. 1. The main results are contained in Sec. 2. We prove
the holomorphic duality of function algebras and Hopf group algebras for the
same classes of groups, but already without the assumption of commutativity
(Theorems 2.2 and 2.10). Note that it is reasonable to regard both
${\mathbb{C}\Gamma}$ and $\mathcal{O}(G)$ (where $\Gamma$ is locally finite
and G is profinite) as Hopf algebras of the same type and similarly combine
$\mathbb{C}^{\Gamma}$ and $\mathcal{O}(G)^{\prime}$.

Note that, in [6], Akbarov proposed a generalization of his theory suitable,
in particular, for group algebras and function algebras on arbitrary (not only
finitely generated, as before, or locally finite, as here) countable discrete
groups by showing that these algebras are holomorphically reflexive in this
broader sense (within the framework of the theory of stereotype spaces).
However, the result was achieved at the price of further complication of the
definitions. In particular, along with Arens–Michael envelopes, two additional
operations are applied: the pseudo-saturation, which, in general, modifies the
topology of the locally convex space, and the passage to an immediate
stereotype subspace (see [6], formulas (100)–(105), and also (81) and (86)).
For function algebras and group algebras associated with discrete groups, the
second operation is excessive [6, Theorems 4.3 and 4.4], but the first one (or
a similar one) is apparently inevitable. It is possible that one of the main
results of the present paper, Theorem 2.2, can be derived as a special case of
[6, Theorem 4.8]. However, the Hopf algebras treated here have good
topological properties, and both the additional operations are not necessary.
Moreover, the direct proof of Theorem 2.2 is undoubtedly shorter. Therefore,
we use a simpler definition from [2], to which the author of the present paper
prefers to refer when possible. Establishing the limits of its applicability
is one of the goals of further research.

##  1\. Abelian case

In this section, we show that the Abelian version of holomorphic duality can
be extended to locally finite and profinite groups.

By a _holomorphic character_ of a complex Lie group $G$ we mean a holomorphic
homomorphism from $G$ to $\mathbb{C}^{\times}$ (the group of invertible
complex numbers) [1, Sec. 7.1]. Just as this is done in Pontryagin’s theory,
one can define the _holomorphically dual group_ $G^{\bullet}$ as the set of
all holomorphic characters of $G$ endowed with the pointwise multiplication
and the topology of uniform convergence on compacta. It was Akbarov who showed
that, if $G$ is a compactly generated Stein group, then $G^{\bullet}$ can be
endowed with the structure of a complex Lie group and is also a compactly
generated Stein group (obviously Abelian) [1, Theorem 7.1]. The direct proof
uses structural theory, but does not provide a canonical construction.
However, the passage to Hopf algebras shows that such a construction does
exist, see Remark 2.11.

Let us turn to the case in which $G$ is zero-dimensional, i.e., discrete, and,
in particular, every character is holomorphic. Certainly, the term
‘‘holomorphic’’ in the context of discrete groups seems excessive, but it is
justified by the fact that the dual group to discrete may turn out to be a
nondiscrete Lie group (for example,
$\mathbb{Z}^{\bullet}\cong\mathbb{C}^{\times}$) or even a pro-Lie group; see
Remark 1.3.

The next assertion is a particular case of the aforementioned theorem, Theorem
7.1 of [1].

######  Proposition 1.1.

Let $G$ be a finitely generated (discrete) Abelian group. Then its dual group
$G^{\bullet}$ is a compactly generated Abelian Stein group. Here $G$ and
$G^{\bullet}$ are holomorphically dual to each other.

We now turn to locally finite Abelian groups.

######  Proposition 1.2.

Let $G$ be a locally finite Abelian group. Then $G^{\bullet}$ coincides as a
topological group with the Pontryagin dual and is profinite.

###### Proof.

It can readily be seen that $G$ is locally finite if and only if it is
periodic. In this case, all characters of $G$ take values in $\mathbb{T}$.
Thus, $G^{\bullet}$ coincides with the Pontryagin dual group, which is
profinite in the case under consideration (see, e.g., [9, Theorem 2.9.6]). ∎

Certainly, a profinite group need not be a Lie group. Therefore, before
introducing the group holomorphically dual to it, it is necessary to clarify
which functions on it are regarded as holomorphic ones. To do this, we expand
the class of groups under consideration. We call a topological group a
_complex pro-Lie group_ if it is a limit (in the category of topological
groups) of a projective system in the category of finite-dimensional complex
Lie groups (cf. the real case in [10, 11]). In particular, profinite groups
are complex pro-Lie groups. Also, a complex-valued function on a complex pro-
Lie group is said to be _holomorphic_ if it is a composition of a continuous
homomorphism into a finite-dimensional complex Lie group and a holomorphic
function. Accepting this definition, we can consider the holomorphically dual
group $G^{\bullet}$ for a profinite Abelian group $G$.

######  Remarks 1.3.

(A) The choice of the class of complex pro-Lie groups can be motivated by the
following observation. Just like every Abelian discrete group is the union of
the inductive system of its finitely generated subgroups, every Abelian
finite-dimensional complex Lie group is the union of the inductive system of
its compactly generated open subgroups. In turn, to latter system, there
corresponds projective system of groups of holomorphic characters each of
which is also an (Abelian finite-dimensional compactly generated) complex Lie
group. For example, the holomorphic characters of $\mathbb{Q}$ form a group
which is the complexification of a solenoid, namely, the projective limit of
the coverings of the form $\mathbb{C}^{\times}\to\mathbb{C}^{\times}$. As we
see, when considering the duality, the complex pro-Lie groups arise naturally.

(B) The differences from the case of real pro-Lie groups are as follows.
First, a continuous homomorphism of complex Lie groups need not be
holomorphic, while continuous homomorphisms of real Lie groups are always
smooth, which reduces real pro-Lie groups to projective limits in the category
of topological groups. This means that there should be noticeably fewer
complex pro-Lie groups than real ones, which is confirmed by the famous Yamabe
theorem that every almost connected locally compact group is a real pro-Lie
group (see the discussion in [10]).

######  Lemma 1.4.

A function on a profinite group (not necessarily Abelian) is holomorphic if
and only if it is locally constant.

###### Proof.

We claim first that it follows from the definition we have adopted that the
function on a profinite group $G$ is holomorphic if and only if it is a
composition of a continuous homomorphism into a finite group and of some
function. Indeed, let $\phi$ be a continuous homomorphism of $G$ into a Lie
group. Since $G$ is profinite, it has a neighborhood base of identity
consisting of open subgroups [9, Lemma 2.1.1]; however, the kernel of $\phi$
is open, because any Lie group does not contain small subgroups. Since $G$ is
compact, it follows that the image of $\phi$ is finite (for details, see,
e.g., [12]).

Thus, a function $f$ on $G$ is holomorphic if and only if there is a normal
subgroup $H$ of finite index such that $f$ is constant on every coset of $H$
(which is open according to [9, Lemma 2.1.1]). On the other hand, the last
condition is equivalent to the fact that $f$ is locally constant (see, e.g.,
[13, Lemma I.6.2]). ∎

Note that our definition of a holomorphic function on a profinite group is a
particular case of the definition of Bruhat-smooth functions formulated in
[14, Sec. 1]. Therefore, in the theory of profinite groups and spaces, locally
constant functions are sometimes called ‘‘smooth’’; see, e.g., [13, Chapter I,
Part II, Sec. 5].

######  Lemma 1.5.

Every locally constant function on a profinite Abelian group $G$ is a linear
combination of holomorphic characters.

###### Proof.

Let $f$ be a locally constant function. It follows from [9, Theorem 2.1.3]
that the topology has a base consisting of cosets of subgroups of finite
index. Since the group is compact, there is a finite cover by subsets in this
base on each of which $f$ is constant. Then $f$ is a linear combination of the
characteristic functions of these subsets.

It can readily be seen that every complex-valued function on a finite Abelian
group is a linear combination of characters of the group. Therefore, the
function on $G$ which is constant on every coset of some subgroup of finite
index is a linear combination of holomorphic characters of $G$. In particular,
this concerns the characteristic function of the coset. The rest is clear. ∎

######  Proposition 1.6.

Let $G$ be a profinite Abelian group. Then $G^{\bullet}$ coincides with the
Pontryagin dual group and is a locally finite discrete group.

###### Proof.

By Lemma 1.4, the holomorphic characters of $G$ are exactly its locally
constant characters. Since every locally constant function is continuous, we
see that any holomorphic character of $G$ is a continuous homomorphism into
$\mathbb{C}^{\times}$. Since the profinite group is compact, it follows that
the image of any continuous homomorphism from the group to
$\mathbb{C}^{\times}$ is contained in $\mathbb{T}$. On the other hand, the
image of a continuous homomorphism in $\mathbb{T}$ is a finite group [9, Lemma
2.9.2], and thus this homomorphism is locally constant [13, Lemma I.6.2].
Thus, the set of holomorphic characters of $G$ coincides with the set of
continuous characters in the classical sense. Consequently, $G^{\bullet}$
coincides with the Pontryagin dual group. The latter is locally finite by [9,
Theorem 2.9.6]. ∎

Obviously, for every $g\in G$, the formula $\iota(g)\\!:\chi\mapsto\chi(g)$,
where $\chi\in G^{\bullet}$, defines a homomorphism
$G^{\bullet}\to\mathbb{C}^{\times}$. It follows from Propositions 1.2 and 1.6
that this character is holomorphic in both the cases, and thus we obtain the
following assertion concerning the reflexivity.

######  Corollary 1.7.

If an Abelian group $G$ is locally finite or profinite, then the homomorphism
$\iota\\!:G\to G^{\bullet\bullet}$ is well defined and is a topological
isomorphism.

######  Remark 1.8.

Note that here we do not claim that the isomorphism in Corollary 1.7 is
canonical, since the specific definition of holomorphic functions on a
profinite group is used. We still have no definition suitable for both classes
of groups (not to mention the consistency with the definition for Lie groups).
The search for such a definition is an urgent problem, and it seems that
complex pro-Lie groups can be useful here. A possible approach is to consider
the duality at the level of Hopf algebras as the primary concept; see Remark
2.11.

##  2\. General case

The construction of holomorphic reflexivity for some topological Hopf algebras
and the term itself were introduced by Akbarov in the context of rigid
stereotype Hopf algebras [1, Russian p. 128]. Akbarov studied the holomorphic
reflexivity for Hopf algebras associated with compactly generated complex Lie
groups as the first problem. Below we follow the formulation of the notion of
holomorphic reflexivity given in [2]. This narrower definition, although
restricting the class of objects under consideration, allows one to solve the
problem without any reference to the theory of stereotype spaces, which is
used in [1]. (The definition of holomorphic reflexivity in a broader sense,
within the framework of this theory, can be found in [6].)

Recall that a Hopf algebra in the symmetric monoidal category of complete
locally convex spaces with the bifunctor $(-)\mathbin{\widehat{\otimes}}(-)$
of complete projective tensor product is called a _Hopf
$\mathbin{\widehat{\otimes}}$-algebra_, see [15] or [4]. Following [7], we say
that a Hopf $\mathbin{\widehat{\otimes}}$-algebra is _well-behaved_ if it is
either a nuclear Fréchet space or a (complete) nuclear (DF)-space.

The following definitions are formulated in [2] for the case in which $H$ is a
nuclear Fréchet space. However, they can be carried over unchanged to the case
in which $H$ is a nuclear (DF)–space, and hence also to the case of an
arbitrary well-behaved $H$. Recall that the completion of a topological
associative $\mathbb{C}$-algebra $A$ with respect to the topology given by all
continuous submultiplicative prenorms is called the Arens–Michael envelope and
is denoted by $\widehat{A}$. If $H$ is well-behaved, then
$H^{\bullet}\\!:=(H^{\prime})^{^}\,\,$ is also a Hopf
$\mathbin{\widehat{\otimes}}$-algebra by [4, Proposition 6.7] (not necessarily
well-behaved) and an Arens–Michael algebra. We say that $H^{\bullet}$
_holomorphically dual_ to $H$ (cf. [2, Definition 1.1]). If $H^{\bullet}$ is
nevertheless well-behaved, then the homomorphism of Hopf
$\mathbin{\widehat{\otimes}}$-algebras $H^{\bullet\bullet}\to H$ is well
defined, cf. [2, formula (1.4)]. If, moreover, this is a topological
isomorphism, then we say that $H$ is _holomorphically reflexive_ (cf. [2,
Definition 1.2]).

Let $G$ be a countable discrete group. We are interested in two Hopf
$\mathbin{\widehat{\otimes}}$-algebras associated with it, namely,
$\widehat{\mathbb{C}G}$ (the Arens–Michael envelope of the group algebra) and
$\mathbb{C}^{G}$ (the algebra of all functions on $G$). Note that
$\widehat{\mathbb{C}G}$ is a nuclear space [8, Theorem 2]. Moreover,
$\widehat{\mathbb{C}G}$ is a Fréchet space if and only if $G$ is finitely
generated [5, Proposition 3.15] and, on the other hand,
$\widehat{\mathbb{C}G}$ is a (DF)–space if and only if $G$ is locally finite
[8, Theorem 6]. Thus, these two cases exhaust all opportunities for
$\widehat{\mathbb{C}G}$ to be well-behaved. On the contrary, $\mathbb{C}^{G}$
is always a nuclear Fréchet space, and thus a well-behaved Hopf
$\mathbin{\widehat{\otimes}}$–algebra.

The following assertion is implicitly contained in [1].

######  Theorem 2.1.

Let $G$ be a finitely generated group. Then $\widehat{\mathbb{C}G}$ and
$\mathbb{C}^{G}$ are holomorphically dual to each other, and thus are
holomorphically reflexive.

###### Proof.

We see directly from the definition that $\widehat{\mathbb{C}G}$ is
holomorphically dual to $\mathbb{C}^{G}$. The other assertion follows from the
fact that $\bigl{(}\widehat{\mathbb{C}G}\bigr{)}^{\prime}\to\mathbb{C}^{G}$ is
the ArensMichael envelope, see [2, Proposition 3.7]. ∎

We now prove the first of our main results, namely, claiming that an analogous
assertion holds also for locally finite groups.

######  Theorem 2.2.

Let $G$ be a locally finite countable group. Then the Hopf
$\mathbin{\widehat{\otimes}}$-algebras $\widehat{\mathbb{C}G}$ and
$\mathbb{C}^{G}$ are holomorphically dual to each other and thus are
holomorphically reflexive.

(Since the topological properties of Hopf
$\mathbin{\widehat{\otimes}}$-algebras are substantial in the definition of
reflexivity, here we have to additionally assume that the group is countable,
in contrast to the Abelian case, which was described in terms of characters.)

###### Proof.

Since $G$ is locally finite, the homomorphism
$\mathbb{C}G\to\widehat{\mathbb{C}G}$ is a topological isomorphism according
to [8, Proposition 5] (Here $\mathbb{C}G$ is equipped with the strongest
locally convex topology). Then
$\bigl{(}\widehat{\mathbb{C}G}\bigr{)}^{\prime}\cong(\mathbb{C}G)^{\prime}\cong\mathbb{C}^{G}$
(as locally convex spaces). Since $\mathbb{C}^{G}$ is an Arens–Michael
algebra, it follows that
$(\widehat{\mathbb{C}G})^{\bullet}\cong\mathbb{C}^{G}$ (as a Hopf
$\mathbin{\widehat{\otimes}}$-algebra). Similarly, it follows from
$(\mathbb{C}^{G})^{\prime}\cong\mathbb{C}G$ that
$(\mathbb{C}^{G})^{\bullet}\cong\widehat{\mathbb{C}G}$. ∎

Thus, if $G$ is a locally finite countable group, then the reflexivity diagram
becomes

$\textstyle{\mathbb{C}^{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{C}^{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{AM\,env}$$\textstyle{\mathbb{C}G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{AM\,env}$$\textstyle{\mathbb{C}G\,.\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Since the locally finite and profinite groups are dual to each other in the
Abelian case, it is natural to assume that an assertion similar to Theorem 2.2
holds also for profinite groups (not necessarily Abelian). This is really the
case, but auxiliary results are needed for the verification.

Let $G$ be a profinite group. As above, denote by $\mathcal{O}(G)$ the set of
holomorphic functions on $G$. Since $\mathcal{O}(G)$ is an algebra of locally
constant functions by Lemma 1.4, it follows that the topology on this algebra
can be introduced in the standard way (see, e.g., [16, Sec. 5]). Namely, let
$I$ be the family of all possible partitions of $G$ into finitely many
disjoint open subsets. For a given $\alpha\in I$, denote by
$\mathcal{O}(G)_{\alpha}$ the algebra of functions that are constant on each
of the open subsets of $\alpha$. Since $\mathcal{O}(G)_{\alpha}$ is finite
dimensional, it can be regarded with the standard topology. Then
$\mathcal{O}(G)$ is the union of the directed family
$(\mathcal{O}(G)_{\alpha})$ and it can be equipped with the topology of the
inductive limit. In addition, one can give a more sparing description of the
topology by presenting $G$ as the projective limit of a directed family of
finite groups $(G_{\alpha})$. Then
$\mathcal{O}(G)_{\alpha}\cong\mathbb{C}^{G_{\alpha}}$ for every $\alpha$, and
$\mathcal{O}(G)$ is the inductive limit of the sequence of finite-dimensional
linear spaces $(\mathbb{C}^{G_{\alpha}})$.

By [1, Theorem 7.2], $\mathcal{O}(G^{\bullet})\cong\mathcal{O}(G)^{\bullet}$
for every compactly generated Abelian Stein group $G$. We now prove a similar
result concerning the consistency of the notions of holomorphic reflexivity
for the groups and Hopf algebras in our case. First, Lemmas 1.4 and 1.5
immediately imply the following assertion.

######  Lemma 2.3.

Let $G$ be a profinite Abelian group. Then
$\mathcal{O}(G)\cong\mathbb{C}(G^{\bullet})$.

To provide the desired topological properties of Hopf
$\mathbin{\widehat{\otimes}}$-algebras, we also impose the countability
condition below. Just like a locally finite group is countable if and only if
it is a countable inductive limit of finite groups, a profinite group has a
countable topology base if and only if it is a countable projective limit of
finite groups (this is easy to show; cf., e.g., [9, p. 11, Corollary 1.1.13]).
Moreover, a locally finite Abelian group is countable if and only if its
holomorphically dual profinite group has a countable topology base (this
follows from [9, p. 60, Lemma 2.9.3]). Below, for the sake of unity of
notation, we also sometimes denote the set $\mathbb{C}^{G}$ of all functions
on a discrete group $G$ by $\mathcal{O}(G)$.

######  Proposition 2.4.

Let $G$ be a locally finite countable Abelian group or a profinite Abelian
group with countable base. Then
$\mathcal{O}(G^{\bullet})\cong\mathcal{O}(G)^{\bullet}$.

###### Proof.

Assume that G is locally finite and countable. Then $G^{\bullet}$ is profinite
according to Proposition 1.2. Lemma 2.3 implies that
$\mathcal{O}(G^{\bullet})$ coincides with $\mathbb{C}(G^{\bullet\bullet})$. It
follows from Corollary 1.7 that $G^{\bullet\bullet}\cong G$, and thus
$\mathcal{O}(G^{\bullet})\cong\mathbb{C}G$. Since $G$ is discrete, we obtain
$\mathcal{O}(G)=\mathbb{C}^{G}$, and, since $G$ is countable, it follows from
Theorem 2.2 that
$\mathbb{C}G\cong(\mathbb{C}^{G})^{\bullet}=\mathcal{O}(G)^{\bullet}$. Thus,
$\mathcal{O}(G^{\bullet})\cong\mathcal{O}(G)^{\bullet}$.

Assume that $G$ is profinite and has a countable base. Then $G^{\bullet}$ is
locally finite by Proposition 1.6, and thus
$\mathcal{O}(G^{\bullet})=\mathbb{C}^{G^{\bullet}}$. On the other hand,
$\mathbb{C}(G^{\bullet})\cong\mathcal{O}(G)$ by Lemma 2.3. Since $G$ has a
countable base, it follows that $G^{\bullet}$ is countable. By Theorem 2.2 we
have $(\mathbb{C}(G^{\bullet}))^{\bullet}\cong\mathbb{C}^{G^{\bullet}}$. Thus,
$\mathcal{O}(G^{\bullet})\cong\mathcal{O}(G)^{\bullet}$. ∎

The key point needed to be verified to prove an analogue of Theorem 2.2 for a
profinite group $G$ is that $\mathcal{O}(G)$ and $\mathcal{O}(G)^{\prime}$ are
Arens–Michael algebras, and thus the Arens–Michael envelope functor acts
trivially both on $\mathcal{O}(G)$ and on $\mathcal{O}(G)^{\prime}$.

######  Lemma 2.5.

Let G be a profinite group with countable base. Then the topology on
$\mathcal{O}(G)$ coincides with the strongest locally convex topology.

###### Proof.

Presenting $G$ as the projective limit of a sequence $(G_{n})$ of finite
groups, we see that $\mathcal{O}(G)$ is the inductive limit of a sequence
$(\mathbb{C}^{G_{n}})$ of finite-dimensional linear spaces, which can be
regarded as subspaces of $\mathcal{O}(G)$. Choose in $\mathcal{O}(G)$ a
countable linear basis consistent with the sequence $(\mathbb{C}^{G_{n}})$ and
take the corresponding decomposition of $\mathcal{O}(G)$ into a direct sum of
one-dimensional subspaces. Then the inductive topology on $\mathcal{O}(G)$
coincides with the direct sum topology, which, in turn, is the strongest
locally convex one; see [17], Sec. II.6, Russian pp. 74–75, (6.2) and an
example after it. ∎

Note that the question of when an algebra endowed with the inductive topology
has continuous multiplication and, in particular, is an Arens–Michael algebra
turned out to be rather subtle (see a discussion in [18]). However, there are
no problems in our special case, namely, the following assertion holds.

######  Proposition 2.6.

Let $G$ be a profinite group with countable base. Then $\mathcal{O}(G)$ is an
Arens–Michael algebra

###### Proof.

By [19, Theorem 2.1], the inductive limit topology (in the category of locally
convex spaces) in the case of a sequence of normed algebras can be given by a
family of submultiplicative prenorms. To ensure that an Arens–Michael algebra
is obtained, it remains to verify the completeness. By Lemma 2.5, the topology
on $\mathcal{O}(G)$ is the strongest locally convex one, whence the
completeness follows. ∎

We now show that $\mathcal{O}(G)$ is a Hopf
$\mathbin{\widehat{\otimes}}$-algebra.

######  Lemma 2.7.

Let $G$ be a profinite group with countable base. Then
$\mathcal{O}(G)\otimes\mathcal{O}(G)\cong\mathcal{O}(G\times G)$ as a linear
space, and
$\mathcal{O}(G)\mathbin{\widehat{\otimes}}\mathcal{O}(G)\cong\mathcal{O}(G\times
G)$ as a locally convex space.

###### Proof.

We use Lemma 1.4. First, we claim that the mapping
$\psi\\!:\mathcal{O}(G)\otimes\mathcal{O}(G)\to\mathcal{O}(G\times G)$ is well
defined. Indeed, any element of the tensor product is a finite sum of
elementary tensors $f_{i}\otimes h_{i}$. Since all $f_{i}$ and $h_{i}$ are
locally constant, it follows that the function on $G\times G$ corresponding to
this sum is locally constant.

Note that $\psi$ is injective (this follows from the fact that the analogous
mapping for functions on finite sets is injective). On the other hand,
represent $G$ as the projective limit of a sequence of finite groups
$(G_{n})$. Then $G\times G$ is the projective limit of the sequence
$(G_{n}\times G_{n})$. Therefore, every locally constant function on $G\times
G$ is constant on cosets of the kernel of the homomorphism $G\times G\to
G_{n}\times G_{n}$ for some $n$. This readily implies that $\psi$ is
surjective. Thus, $\psi$ is an isomorphism of linear spaces.

By Lemma 2.5, the topology on the space $\mathcal{O}(G)$ coincides with the
strongest locally convex topology, which means that this space can be
represented as a direct sum of one-dimensional subspaces. Further, we use the
fact that there is an isomorphism

$\mathcal{O}(G)\mathbin{\widehat{\otimes}}\mathcal{O}(G)\cong(\mathcal{O}(G)^{\prime}\mathbin{\widehat{\otimes}}\mathcal{O}(G)^{\prime})^{\prime}.$

Since the strong dual of a sum is the product and conversely [20, Sec. 22.5,
p. 287], and the projective tensor product commutes with direct products [21,
Sec. 41.6, p. 194, (5)], it follows that
$\mathcal{O}(G)\mathbin{\widehat{\otimes}}\mathcal{O}(G)$ is also a direct sum
of one-dimensional subspaces, and thus its topology is the strongest locally
convex one. Since $\mathcal{O}(G\times G)$ is also equipped with the strongest
locally convex topology, we have a topological isomorphism
$\mathcal{O}(G)\mathbin{\widehat{\otimes}}\mathcal{O}(G)\cong\mathcal{O}(G\times
G)$. ∎

######  Proposition 2.8.

Let $G$ be a profinite group with countable base. Then $\mathcal{O}(G)$ is a
commutative well-behaved Hopf $\mathbin{\widehat{\otimes}}$-algebra with
respect to standard operations for a function algebra on a group.

###### Proof.

It follows from Proposition 2.6 that $\mathcal{O}(G)$ is a
$\mathbin{\widehat{\otimes}}$-algebra. By Lemma 2.7, the comultiplication is
well defined. Since $\mathcal{O}(G)$ is the set of locally constant functions
(Lemma 1.4), it is invariant with respect to the antipode. The continuity of
the operations follows from the fact that $\mathcal{O}(G)$ is regarded with
the strongest locally convex topology (Lemma 2.5). The axioms of Hopf
$\mathbin{\widehat{\otimes}}$-algebra are verified directly. Since the
topology is the strongest locally convex one, it follows that $\mathcal{O}(G)$
is a nuclear (DF)-space, which means that $\mathcal{O}(G)$ is a well-behaved
Hopf $\mathbin{\widehat{\otimes}}$-algebra. ∎

######  Proposition 2.9.

Let $G$ be a profinite group with countable base. Then
$\mathcal{O}(G)^{\prime}$ is an Arens–Michael algebra.

###### Proof.

Just as in the proof of Lemma 2.5, we represent $G$ as the projective limit of
a sequence $(G_{n})$ of finite groups and, correspondingly, represent
$\mathcal{O}(G)$ as the inductive limit of the sequence $(\mathbb{C}^{G_{n}})$
of finite-dimensional linear spaces. It can readily be seen that the linking
mappings are homomorphisms of Hopf $\mathbin{\widehat{\otimes}}$-algebras. In
particular, the family $(\mathbb{C}^{G_{n}}\to\mathcal{O}(G))$ is a cone in
the category of $\mathbin{\widehat{\otimes}}$-coalgebras. Applying the functor
of strong dual, we obtain the cone
$(\mathcal{O}(G)^{\prime}\to\mathbb{C}G_{n})$ in the category of
$\mathbin{\widehat{\otimes}}$-algebras (taking into account that
$(\mathbb{C}^{G_{n}})^{\prime}\cong\mathbb{C}G_{n}$). Being finite
dimensional, $\mathbb{C}G_{n}$ can be regarded as Banach algebras (for
example, by providing them with $\ell_{1}$-norms). Denote by $A$ the
Arens–Michael algebra which is the projective limit of the sequence
$(\mathbb{C}G_{n})$. The above cone induces a homomorphism of
$\mathbin{\widehat{\otimes}}$-algebras $\mathcal{O}(G)^{\prime}\to A$. It
remains to show that this homomorphism is an isomorphism of
$\mathbin{\widehat{\otimes}}$-algebras. It can readily be seen that, to this
end, it suffices to establish that this homomorphism is a topological
isomorphism of locally convex spaces.

The inductive system $(\mathbb{C}^{G_{n}})$ is regular in the following sense:
every bounded subset of $\mathcal{O}(G)$ is bounded in some
$(\mathbb{C}^{G_{n}})$ (see the definition in [22, Sec. 1, p. 46, Definition
2] or [23, Sec. 23, Sec. 5.2, p. 123]). Indeed, since all spaces are finite-
dimensional, it follows that the system is strict, and one can apply [23, Sec.
24, p. 127, Satz 2.2]. The regularity implies that there is a topological
isomorphism between the strong dual to the inductive limit and the projective
limit of strong duals [22, Sec. 2, p. 57, Proposition 1] (see, e.g., a proof
in [23, Sec. 25, p. 145, Satz 2.1]). Since the first space is
$\mathcal{O}(G)^{\prime}$ and the other is $A$, this means that
$\mathcal{O}(G)^{\prime}\to A$ is a topological isomorphism, as required. ∎

In conclusion, we prove our second main result, which is similar to Theorem
2.2.

######  Theorem 2.10.

Let $G$ be a profinite group with countable base. Then the
$\mathbin{\widehat{\otimes}}$-algebras $\mathcal{O}(G)$ and
$\mathcal{O}(G)^{\prime}$ are holomorphically dual to each other and thus are
holomorphically reflexive.

###### Proof.

Since $\mathcal{O}(G)^{\prime}$ is an Arens–Michael algebra (according to
Proposition 2.9), we see that
$\mathcal{O}(G)^{\bullet}=\mathcal{O}(G)^{\prime}$. Then
$\mathcal{O}(G)^{\prime}$ is a nuclear Fréchet space being the strong dual to
a nuclear (DF)-space, and hence a well-behaved Hopf
$\mathbin{\widehat{\otimes}}$-algebra. Thus, the homomorphism of Hopf
$\mathbin{\widehat{\otimes}}$-algebras

$(\mathcal{O}(G)^{\prime})^{\bullet}\to\mathcal{O}(G)$

is well defined. Further, since any space with the strongest locally convex
topology is reflexive in the sense of the strong dual, it follows that
$\mathcal{O}(G)^{\prime\prime}\cong\mathcal{O}(G)$. Finally, $\mathcal{O}(G)$
is an Arens–Michael algebra by Proposition 2.6, and thus
$(\mathcal{O}(G)^{\prime})^{\bullet}\to\mathcal{O}(G)$ is an isomorphism. ∎

Thus, if $G$ is a profinite group with countable base, then the reflexivity
diagram becomes

$\textstyle{\mathcal{O}(G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}(G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{AM\,env}$$\textstyle{\mathcal{O}(G)^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{AM\,env}$$\textstyle{\mathcal{O}(G)^{\prime}\,.\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

######  Remark 2.11.

If $G$ is an Abelian group, then $\widehat{\mathbb{C}G}$ is a commutative
ArensMichael algebra. Applying the Gelfand transformation, we can identify
this algebra with the algebra of functions on the spectrum, i.e., on the set
of continuous algebraic characters (the Gelfand transformation is injective,
since $\widehat{\mathbb{C}G}$ is a dense subalgebra in the semisimple Banach
algebra $L^{1}(G)$). It can readily be seen that the resulting group is just
$G^{\bullet}$. If, in addition, G is finitely generated, then
$\widehat{\mathbb{C}G}$ is a holomorphically finitely generated Hopf algebra
in the sense of [5]. It follows from the commutativity of
$\widehat{\mathbb{C}G}$ that $G^{\bullet}$ has a canonical Lie group structure
[5, Theorem 2.2]. A similar assertion holds also in the more general case of
compactly generated Lie groups [5, Theorem 3.1], which allows one to give a
canonical form to Akbarov’s construction in [1, 7.1].

If $G$ is Abelian and locally finite, then, in a similar way, the Gelfand
transformation gives us not only a group $G^{\bullet}$, but also the algebra
of functions on $G^{\bullet}$. Thus, transposing $G$ and $G^{\bullet}$, we can
use the formula of Lemma 2.3 as the definition of the algebra of holomorphic
functions on a profinite (Abelian) group and obtain the definition accepted
above as its consequence (recall that $\widehat{\mathbb{C}G}\cong\mathbb{C}G$
according to Proposition 2.6).

The author is grateful to S. S. Akbarov for useful remarks that improved the
quality of the presentation.

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