# Pointwise Characterization of Besov and Triebel–Lizorkin Spaces on Spaces of
Homogeneous Type 00footnotetext:  2020 Mathematics Subject Classification.
Primary 46E36; Secondary 46E35, 42B25, 30L99.Key words and phrases. space of
homogeneous type, Besov space, Triebel–Lizorkin space, Hajłasz–Sobolev space,
Hajłasz–Triebel–Lizorkin space, grand Triebel–Lizorkin space, pointwise
characterization.This project is partially supported by the National Key
Research and Development Program of China (Grant No. 2020YFA0712900) and the
National Natural Science Foundation of China (Grant Nos. 11971058, 12071197
and 11871100).

Ryan Alvarado, Fan Wang, Dachun Yang111Corresponding author, E-mail:
<EMAIL_ADDRESS>21, 2021/Final version. and Wen Yuan

Abstract In this article, the authors establish the pointwise characterization
of Besov and Triebel–Lizorkin spaces on spaces of homogeneous type via
clarifying the relationship among Hajłasz–Sobolev spaces, Hajłasz–Besov and
Hajłasz–Triebel–Lizorkin spaces, grand Besov and Triebel–Lizorkin spaces, and
Besov and Triebel–Lizorkin spaces. A major novelty of this article is that all
results presented in this article get rid of both the dependence on the
reverse doubling condition of the measure and the metric condition of the
quasi-metric under consideration. Moreover, the pointwise characterization of
the inhomogeneous version is new even when the underlying space is an RD-
space.

##  1 Introduction

It is well known that Besov and Triebel–Lizorkin spaces provide a unified
frame for the study of many function spaces and indeed cover many well-known
classical concrete function spaces such as Lebesgue spaces, Sobolev spaces,
potential spaces, (local) Hardy spaces, and the space of functions with
bounded mean oscillation. We refer the reader to monographs [5, 42, 43, 44,
45] for a comprehensive treatment of these function spaces and their history.
We also refer the reader to [57] for relationships among Morrey spaces,
Campanato spaces, and Besov–Triebel–Lizorkin spaces, to [3, 40] for some new
progress of Besov and Triebel–Lizorkin spaces, and to [6, 7, 8, 9] for various
characterizations and applications of Besov and Triebel–Lizorkin spaces
associated with operators.

Particularly, fractional Sobolev spaces play an important major role in many
questions involving partial differential equations on $\mathbb{R}^{n}$. It is
known that a theory of first order Sobolev spaces on doubling metric spaces
has been established based on both upper gradients [41, 31] and pointwise
inequalities [14]; see [15, 16] for a survey on this. These different
approaches result in the same function class if the underlying space supports
a suitable Poincaré inequality [37]. In this article, we further investigate
the spaces introduced by Hajłasz [14] (see also [51] and Definition 2.10
below) which are defined via pointwise inequalities.

On another hand, as a generalization of $\mathbb{R}^{n}$, the space of
homogeneous type was introduced by Coifman and Weiss [10, 11] (see Definition
2.2 below), which provides a natural setting for the study of function spaces
and the boundedness of Calderón–Zygmund operators. Spaces of homogeneous type,
with some additional assumptions, have been extensively investigated in many
articles. For instance, the _Ahlfors $d$-regular space_ is a special space of
homogeneous type satisfying the following condition: there exists a positive
constant $C$ such that, for any ball $B(x,r)\subset{\mathcal{X}}$ with center
$x$ and radius $r\in(0,\mathop{\mathrm{diam}}{\mathcal{X}})$,

$C^{-1}r^{d}\leq\mu(B(x,r))\leq Cr^{d},$

where here, and thereafter, $\mathop{\mathrm{diam}}{\mathcal{X}}:=\sup_{x,\
y\in{\mathcal{X}}}d(x,y)$. Another case is the RD-space (see [34, 19, 20] for
instance), which is a doubling metric measure space satisfying the following
additional _reverse doubling condition_ : there exist positive constants
$\widetilde{C}_{(\mu)}\in(0,1]$ and $\kappa\in(0,\omega]$ such that, for any
ball $B(x,r)$ with $r\in(0,\mathop{\mathrm{diam}}{\mathcal{X}}/2)$ and
$\lambda\in[1,\mathop{\mathrm{diam}}{\mathcal{X}}/(2r))$,

$\widetilde{C}_{(\mu)}\lambda^{\kappa}\mu(B(x,r))\leq\mu(B(x,\lambda r)).$

Obviously, an RD-space is a generalization of an Ahlfors $d$-regular space. We
refer the reader to [56] for more equivalent characterizations of RD-spaces.

Besov and Triebel–Lizorkin spaces on spaces of homogeneous type satisfying
some additional assumptions were also studied. We refer the reader to [21, 22,
52, 54, 55] for various characterizations of Besov and Triebel–Lizorkin spaces
on Ahlfors $d$-regular spaces, and to [48, 49, 50, 53] for some applications.
We also refer the reader to [20, 38, 56] for various characterizations of
these Besov and Triebel–Lizorkin spaces on RD-spaces. Besides, Koskela et al.
[35, 36] introduced the Hajłasz–Besov and Hajłasz–Triebel–Lizorkin spaces on
RD-spaces. We refer the reader to [12, 27, 28, 29, 30] for various
characterizations and applications of Hajłasz–Besov and
Hajłasz–Triebel–Lizorkin on a metric measure space satisfying the doubling
property.

Recently, using the wavelet reproducing formulae in [18], Han et al. [17]
introduced Besov and Triebel–Lizorkin spaces on spaces of homogeneous type and
established several embedding theorems. On the other hand, Wang et al. [46]
also introduced Besov and Triebel–Lizorkin spaces on spaces of homogenous
type, based on the Calderón reproducing formulae established in [24], and
established the boundedness of Calderón–Zygmund operators on these spaces as
an application. Later, He et al. [25] obtained characterizations of Besov and
Triebel–Lizorkin spaces via wavelets, molecules, Lusin area functions, and
Littlewood–Paley $g_{\lambda}^{\ast}$-functions. Besides, He et al. [25] also
showed that those two kinds of Besov and Triebel–Lizorkin spaces studied,
respectively, in [17] and [46] coincide. Then Wang et al. [47] established the
difference characterization of Besov and Triebel–Lizorkin spaces on spaces of
homogenous type.

To complete the theory of Besov and Triebel–Lizorkin spaces on spaces of
homogenous type, it is a natural question whether or not we can also establish
a pointwise characterization of Besov and Triebel–Lizorkin spaces on space of
homogenous type. The main target of this article is to give an affirmative
answer to this question.

The organization of the remainder of this article is as follows.

In Section 2, we first recall the notions of homogenous Besov and
Triebel–Lizorkin spaces on spaces of homogeneous type introduced in [46], and
then introduce Hajłasz–Sobolev spaces and Hajłasz–Besov–Triebel–Lizorkin
spaces on spaces of homogeneous type. Then we state the main result of this
article; see Theorem 2.16 below.

In Section 3, we first introduce the homogenous grand Besov and
Triebel–Lizorkin spaces on spaces of homogeneous type. Then we investigate the
relation among homogeneous grand Besov and Triebel–Lizorkin spaces and
homogeneous Besov and Triebel–Lizorkin spaces; see Theorem 3.3 below. Later,
we establish the equivalence between homogenous Hajłasz–Besov–Triebel–Lizorkin
spaces and homogeneous Besov and Triebel–Lizorkin spaces; see Theorem 3.10
below. To this end, we first establish a Poincaré type inequality (see Lemma
3.11 below). It should be mentioned that, in the proof of Lemma 3.11, the
constant $A_{0}$ appearing in the quasi- triangle inequality (see Definition
2.1 below) also brings some difficulty. That is why we need additional
restrictions on parameters involved therein. Besides, all the proofs in
Section 3 get rid of the dependence on the reverse doubling assumption.

In Section 4, we establish the equivalence between inhomogeneous
Hajłasz–Besov–Triebel–Lizorkin spaces and inhomogeneous Besov–Triebel–Lizorkin
spaces (see Theorem 4.10 below). To this end, we first establish a new
characterization of inhomogeneous Besov–Triebel–Lizorkin spaces (see Theorem
4.11 below).

Finally, let us make some conventions on notation. For any given
$p\in(0,\infty]$, the _Lebesgue space_ $L^{p}({\mathcal{X}})$ is defined by
setting, when $p\in(0,\infty)$,

$L^{p}({\mathcal{X}}):=\left\\{f\ \text{is measurable on}\ {\mathcal{X}}:\
\|f\|_{{L^{p}(\mathcal{X})}}:=\left[\int_{{\mathcal{X}}}|f(x)|^{p}\,d\mu(x)\right]^{1/p}<\infty\right\\},$

and

$L^{\infty}({\mathcal{X}}):=\left\\{f\ \text{is measurable on}\
{\mathcal{X}}:\
\|f\|_{L^{\infty}({\mathcal{X}})}:=\displaystyle{\operatorname*{ess\
sup}_{x\in{\mathcal{X}}}|f(x)|<\infty}\right\\}.$

Throughout this article, we use $A_{0}$ to denote the positive constant
appearing in the _quasi-triangle inequality_ of $d$ (see Definition 2.1
below), the parameter $\omega$ to denote the _upper dimension_ in Definition
2.2 [see (2.2) below], and $\eta$ to denote the smoothness index of the exp-
ATI in Definition 2.5 below. Moreover, $\delta$ is a small positive number,
for instance, $\delta\leq(2A_{0})^{-10}$, coming from the construction of the
dyadic cubes on ${\mathcal{X}}$ (see Theorem 2.5). For any given
$p\in[1,\infty]$, we use $p^{\prime}$ to denote its conjugate index, that is,
$1/p+1/p^{\prime}=1$. For any $r\in\mathbb{R}$, $r_{+}$ is defined by setting
$r_{+}:=\max\\{0,r\\}$. For any $a,\ b\in\mathbb{R}$, let $a\wedge
b:=\min\\{a,b\\}$ and $a\vee b:=\max\\{a,b\\}$. The symbol $C$ denotes a
positive constant which is independent of the main parameters involved, but
may vary from line to line. We use $C_{(\alpha,\beta,\dots)}$ to denote a
positive constant depending on the indicated parameters $\alpha,\ \beta,\
\dots$. The symbol $A\lesssim B$ means that $A\leq CB$ for some positive
constant $C$, while $A\sim B$ means $A\lesssim B\lesssim A$. If $f\leq Cg$ and
$g=h$ or $g\leq h$, we then write $f\lesssim g\sim h$ or $f\lesssim g\lesssim
h$, _rather than_ $f\lesssim g=h$ or $f\lesssim g\leq h$. The set of positive
integers is denoted by $\mathbb{N}$, namely, $\mathbb{N}=\\{1,2,\dots\\}$, and
the set $\mathbb{Z}_{+}:=\\{0,1,2,\dots\\}$. For any $r\in(0,\infty)$ and $x,\
y\in{\mathcal{X}}$ with $x\neq y$, define $V(x,y):=\mu(B(x,d(x,y)))$ and
$V_{r}(x):=\mu(B(x,r))$. For any $\beta,\ \gamma\in(0,\eta)$ and
$s\in(-(\beta\wedge\gamma),\beta\wedge\gamma)$, we always let

$p(s,\beta\wedge\gamma):=\max\left\\{\frac{\omega}{\omega+(\beta\wedge\gamma)},\frac{\omega}{\omega+(\beta\wedge\gamma)+s}\right\\},$
(1.1)

where $\omega$ and $\eta$ are, respectively, as in (2.2) and Definition 2.5.
The operator $M$ always denotes the _central Hardy–Littlewood maximal
operator_ which is defined by setting, for any locally integrable function $f$
on ${\mathcal{X}}$ and any $x\in{\mathcal{X}}$,

$M(f)(x):=\sup_{r\in(0,\infty)}\frac{1}{\mu(B(x,r))}\int_{B(x,r)}|f(y)|\,d\mu(y).$
(1.2)

For any set $E\subset{\mathcal{X}}$, we use $\mathbf{1}_{E}$ to denote its
characteristic function and, for any set $J$, we use $\\#J$ to denote its
_cardinality_.

##  2 Besov and Triebel–Lizorkin spaces and Hajłasz–Sobolev spaces on spaces
of homogeneous type

In this section, we recall the notions of Besov and Triebel–Lizorkin spaces on
spaces of homogeneous type and introduce Hajłasz–Sobolev spaces on spaces of
homogeneous type. Let us begin with the notion of quasi-metric spaces.

######  Definition 2.1.

A _quasi-metric space_ $({\mathcal{X}},d)$ is a non-empty set ${\mathcal{X}}$
equipped with a _quasi-metric_ $d$, namely, a non-negative function defined on
${\mathcal{X}}\times{\mathcal{X}}$ satisfying that, for any $x,\ y,\
z\in{\mathcal{X}}$,

  1. (i)

$d(x,y)=0$ if and only if $x=y$;

  2. (ii)

$d(x,y)=d(y,x)$;

  3. (iii)

there exists a constant $A_{0}\in[1,\infty)$, independent of $x$, $y$, and
$z$, such that

$d(x,z)\leq A_{0}[d(x,y)+d(y,z)].$

The _ball_ $B$ of ${\mathcal{X}}$, centered at $x_{0}\in{\mathcal{X}}$ with
radius $r\in(0,\infty)$, is defined by setting

$B:=B(x_{0},r):=\\{x\in{\mathcal{X}}:d(x,x_{0})<r\\}.$

For any ball $B$ and $\tau\in(0,\infty)$, we denote by $\tau B$ the ball with
the same center as that of $B$ but of radius $\tau$ times that of $B$.

######  Definition 2.2.

Let $({\mathcal{X}},d)$ be a quasi-metric space and $\mu$ a non-negative
measure on ${\mathcal{X}}$. The triple $({\mathcal{X}},d,\mu)$ is called a
_space of homogeneous type_ if $\mu$ satisfies the following doubling
condition: there exists a positive constant $C\in[1,\infty)$ such that, for
any ball $B\subset{\mathcal{X}}$,

$0<\mu(2B)\leq C\mu(B)<\infty.$ (2.1)

Let $C_{(\mu)}:=\sup_{B\subset{\mathcal{X}}}\mu(2B)/\mu(B)$. Then it is easy
to show that $C_{(\mu)}$ is the smallest positive constant such that (2.1)
holds true. The above doubling condition implies that, for any ball $B$ and
any $\lambda\in[1,\infty)$,

$\mu(\lambda B)\leq C_{(\mu)}\lambda^{\omega}\mu(B),$ (2.2)

where $\omega:=\log_{2}C_{(\mu)}$ is called the _upper dimension_ of
${\mathcal{X}}$. Note that $\omega\in(0,\infty)$ (see, for instance, [2, p.
72]). If $A_{0}=1$, then $({\mathcal{X}},d,\mu)$ is called a _metric measure
space of homogeneous type_ or, simply, a _doubling metric measure space_.

Without loss of generality, we may make the following assumptions on
$({\mathcal{X}},d,\mu)$. For any point $x\in{\mathcal{X}}$, we assume that the
balls $\\{B(x,r)\\}_{r\in(0,\infty)}$ form a basis of open neighborhoods of
$x$. Moreover, we suppose that $\mu$ is _Borel regular_ which means that open
sets are measurable and every set $A\subset{\mathcal{X}}$ is contained in a
Borel set $E$ satisfying that $\mu(A)=\mu(E)$. We also assume that
$\mu(B(x,r))\in(0,\infty)$ and $\mu(\\{x\\})=0$ for any given
$x\in{\mathcal{X}}$ and $r\in(0,\infty)$.

Now, we recall the notion of test functions and distributions on
${\mathcal{X}}$, whose following versions were originally introduced in [20]
(see also [19]).

######  Definition 2.3 (test functions).

Let $x_{1}\in{\mathcal{X}}$, $r\in(0,\infty)$, $\beta\in(0,1]$, and
$\gamma\in(0,\infty)$. For any $x\in{\mathcal{X}}$, define

$D_{\gamma}(x_{1},x;r):=\frac{1}{V_{r}(x_{1})+V(x_{1},x)}\left[\frac{r}{r+d(x_{1},x)}\right]^{\gamma}.$
(2.3)

A measurable function $f$ on ${\mathcal{X}}$ is called a _test function of
type $(x_{1},r,\beta,\gamma)$_ if there exists a positive constant $C$ such
that

  1. (i)

for any $x\in{\mathcal{X}}$,

$|f(x)|\leq CD_{\gamma}(x_{1},x;r);$ (2.4)

  2. (ii)

for any $x,\ y\in{\mathcal{X}}$ satisfying
$d(x,y)\leq(2A_{0})^{-1}[r+d(x_{1},x)]$,

$|f(x)-f(y)|\leq
C\left[\frac{d(x,y)}{r+d(x_{1},x)}\right]^{\beta}D_{\gamma}(x_{1},x;r).$ (2.5)

The set of all test functions of type $(x_{1},r,\beta,\gamma)$ is denoted by
${\mathcal{G}}(x_{1},r,\beta,\gamma)$. For any
$f\in{\mathcal{G}}(x_{1},r,\beta,\gamma)$, its norm
$\|f\|_{{\mathcal{G}}(x_{1},r,\beta,\gamma)}$ in
${\mathcal{G}}(x_{1},r,\beta,\gamma)$ is defined by setting

$\|f\|_{{\mathcal{G}}(x_{1},r,\beta,\gamma)}:=\inf\\{C\in(0,\infty):\,\text{(\ref{t-size})
and (\ref{t-reg}) hold true}\\}.$

Its subspace $\mathring{{\mathcal{G}}}(x_{1},r,\beta,\gamma)$ is defined by
setting

$\mathring{{\mathcal{G}}}(x_{1},r,\beta,\gamma):=\left\\{f\in{\mathcal{G}}(x_{1},r,\beta,\gamma):\int_{{\mathcal{X}}}f(x)\,d\mu(x)=0\right\\}$

and is equipped with the norm
$\|\cdot\|_{\mathring{{\mathcal{G}}}(x_{1},r,\beta,\gamma)}:=\|\cdot\|_{{\mathcal{G}}(x_{1},r,\beta,\gamma)}$.

Note that, for any fixed $x_{1},\ x_{2}\in{\mathcal{X}}$ and $r_{1},\
r_{2}\in(0,\infty)$,
${\mathcal{G}}(x_{1},r_{1},\beta,\gamma)={\mathcal{G}}(x_{2},r_{2},\beta,\gamma)$
and
$\mathring{{\mathcal{G}}}(x_{1},r_{1},\beta,\gamma)=\mathring{{\mathcal{G}}}(x_{2},r_{2},\beta,\gamma)$
with equivalent norms, but the positive equivalence constants may depend on
$x_{1}$, $x_{2}$, $r_{1}$, and $r_{2}$. Thus, for fixed
$x_{0}\in{\mathcal{X}}$ and $r_{0}\in(0,\infty)$, we may denote
${\mathcal{G}}(x_{0},r_{0},\beta,\gamma)$ and
$\mathring{{\mathcal{G}}}(x_{0},r_{0},\beta,\gamma)$ simply, respectively, by
${\mathcal{G}}(\beta,\gamma)$ and $\mathring{{\mathcal{G}}}(\beta,\gamma)$.
Usually, the spaces ${\mathcal{G}}(\beta,\gamma)$ and
$\mathring{{\mathcal{G}}}(\beta,\gamma)$ are called the _spaces of test
functions_ on ${\mathcal{X}}$.

Fix $\epsilon\in(0,1]$ and $\beta,\ \gamma\in(0,\epsilon]$. Let
${\mathcal{G}}^{\epsilon}_{0}(\beta,\gamma)$ [resp.,
$\mathring{{\mathcal{G}}}^{\epsilon}_{0}(\beta,\gamma)$] be the completion of
the set ${\mathcal{G}}(\epsilon,\epsilon)$ [resp.,
$\mathring{{\mathcal{G}}}(\epsilon,\epsilon)$] in
${\mathcal{G}}(\beta,\gamma)$ [resp.,
$\mathring{{\mathcal{G}}}(\beta,{\gamma})$]. Furthermore, the norm of
${\mathcal{G}}^{\epsilon}_{0}(\beta,\gamma)$ [resp.,
$\mathring{{\mathcal{G}}}^{\epsilon}_{0}(\beta,\gamma)$] is defined by setting
$\|\cdot\|_{{\mathcal{G}}^{\epsilon}_{0}(\beta,\gamma)}:=\|\cdot\|_{{\mathcal{G}}(\beta,\gamma)}$
[resp.,
$\|\cdot\|_{\mathring{{\mathcal{G}}}^{\epsilon}_{0}(\beta,\gamma)}:=\|\cdot\|_{{\mathcal{G}}(\beta,\gamma)}$].
The dual space $({\mathcal{G}}^{\epsilon}_{0}(\beta,\gamma))^{\prime}$ [resp.,
$(\mathring{{\mathcal{G}}}^{\epsilon}_{0}(\beta,\gamma))^{\prime}$] is defined
to be the set of all continuous linear functionals from
${\mathcal{G}}^{\epsilon}_{0}(\beta,\gamma)$ [resp.,
$\mathring{{\mathcal{G}}}^{\epsilon}_{0}(\beta,\gamma)$] to $\mathbb{C}$,
equipped with the weak-$\ast$ topology. The spaces
$({\mathcal{G}}^{\epsilon}_{0}(\beta,\gamma))^{\prime}$ and
$(\mathring{{\mathcal{G}}}^{\epsilon}_{0}(\beta,\gamma))^{\prime}$ are called
the _spaces of distributions_ on ${\mathcal{X}}$.

The following lemma, which comes from [32, Theorem 2.2], establishes the
dyadic cube system of $({\mathcal{X}},d,\mu)$.

######  Lemma 2.4.

Let constants $0<c_{0}\leq C_{0}<\infty$ and $\delta\in(0,1)$ be such that
$12A_{0}^{3}C_{0}\delta\leq c_{0}$. Assume that a set of points,
$\\{z_{\alpha}^{k}:k\in\mathbb{Z},\alpha\in{\mathcal{A}}_{k}\\}\subset{\mathcal{X}}$
with ${\mathcal{A}}_{k}$ for any $k\in\mathbb{Z}$ being a set of indices, has
the following properties: for any $k\in\mathbb{Z}$,

$d(z^{k}_{\alpha},z^{k}_{\beta})\geq
c_{0}\delta^{k}\quad\text{if}\quad\alpha\neq\beta,\quad\text{and}\quad\min_{\alpha\in{\mathcal{A}}_{k}}d(x,z_{\alpha}^{k})<C_{0}\delta^{k}\quad\text{for
any}\quad x\in{\mathcal{X}}.$

Then there exists a family of sets,
$\\{Q_{\alpha}^{k}:k\in\mathbb{Z},\alpha\in{\mathcal{A}}_{k}\\}$, satisfying

  1. (i)

for any $k\in\mathbb{Z}$,
$\bigcup_{\alpha\in{\mathcal{A}}_{k}}Q_{\alpha}^{k}={\mathcal{X}}$ and
$\\{Q_{\alpha}^{k}:\alpha\in{\mathcal{A}}_{k}\\}$ consists of mutually
disjoint sets;

  2. (ii)

if $l,\ k\in\mathbb{Z}$ and $k\leq l$, then, for any
$\alpha\in{\mathcal{A}}_{k}$ and $\beta\in{\mathcal{A}}_{l}$, either
$Q_{\beta}^{l}\subset Q_{\alpha}^{k}$ or $Q_{\beta}^{l}\cap
Q_{\alpha}^{k}=\emptyset$;

  3. (iii)

for any $k\in\mathbb{Z}$ and $\alpha\in{\mathcal{A}}_{k}$,
$B(z^{k}_{\alpha},(3A_{0}^{2})^{-1}c_{0}\delta^{k})\subset
Q_{\alpha}^{k}\subset B(z^{k}_{\alpha},2A_{0}C_{0}\delta^{k})$.

Throughout this article, for any $k\in\mathbb{Z}$, define

${\mathcal{G}}_{k}:={\mathcal{A}}_{k+1}\setminus{\mathcal{A}}_{k}\quad\text{and}\quad{\mathcal{Y}}^{k}:=\left\\{z^{k+1}_{\alpha}\right\\}_{\alpha\in{\mathcal{G}}_{k}}=:\left\\{y_{\alpha}^{k}\right\\}_{\alpha\in{\mathcal{G}}_{k}}$

and, for any $x\in{\mathcal{X}}$, define

$d(x,{\mathcal{Y}}^{k}):=\inf_{y\in{\mathcal{Y}}^{k}}d(x,y)\quad\text{and}\quad
V_{\delta^{k}}(x):=\mu(B(x,\delta^{k})).$

Now, we recall the notion of approximations of the identity with exponential
decay from [24].

######  Definition 2.5.

A sequence $\\{Q_{k}\\}_{k\in\mathbb{Z}}$ of bounded linear integral operators
on $L^{2}({\mathcal{X}})$ is called an _approximation of the identity with
exponential decay_ (for short, exp-ATI) if there exist constants $C$,
$\nu\in(0,\infty)$, $a\in(0,1]$, and $\eta\in(0,1)$ such that, for any
$k\in\mathbb{Z}$, the kernel of the operator $Q_{k}$, a function on
${\mathcal{X}}\times{\mathcal{X}}$ , which is still denoted by $Q_{k}$,
satisfies the following conditions:

  1. (i)

(the _identity condition_) $\sum_{k=-\infty}^{\infty}Q_{k}=I$ in
$L^{2}({\mathcal{X}})$, where $I$ denotes the _identity operator_ on
$L^{2}({\mathcal{X}})$;

  2. (ii)

(the _size condition_) for any $x,\ y\in{\mathcal{X}}$,

$\displaystyle|Q_{k}(x,y)|$ $\displaystyle\leq
C\frac{1}{\sqrt{V_{\delta^{k}}(x)V_{\delta^{k}}(y)}}H_{k}(x,y),$

where here, and thereafter,

$H_{k}(x,y):=\exp\left\\{-\nu\left[\frac{d(x,y)}{\delta^{k}}\right]^{a}\right\\}\exp\left\\{-\nu\left[\frac{\max\\{d(x,{\mathcal{Y}}^{k}),d(y,{\mathcal{Y}}^{k})\\}}{\delta^{k}}\right]^{a}\right\\};$

  3. (iii)

(the _regularity condition_) for any $x,\ x^{\prime},\ y\in{\mathcal{X}}$ with
$d(x,x^{\prime})\leq\delta^{k}$,

$\displaystyle|Q_{k}(x,y)-Q_{k}(x^{\prime},y)|+|Q_{k}(y,x)-Q_{k}(y,x^{\prime})|\leq
C\left[\frac{d(x,x^{\prime})}{\delta^{k}}\right]^{\eta}\frac{1}{\sqrt{V_{\delta^{k}}(x)V_{\delta^{k}}(y)}}H_{k}(x,y);$

  4. (iv)

(the _second difference regularity condition_) for any $x,\ x^{\prime},\ y,\
y^{\prime}\in{\mathcal{X}}$ with $d(x,x^{\prime})\leq\delta^{k}$ and
$d(y,y^{\prime})\leq\delta^{k}$,

$\displaystyle|[Q_{k}(x,y)-Q_{k}(x^{\prime},y)]-[Q_{k}(x,y^{\prime})-Q_{k}(x^{\prime},y^{\prime})]|$
$\displaystyle\quad\leq
C\left[\frac{d(x,x^{\prime})}{\delta^{k}}\right]^{\eta}\left[\frac{d(y,y^{\prime})}{\delta^{k}}\right]^{\eta}\frac{1}{\sqrt{V_{\delta^{k}}(x)V_{\delta^{k}}(y)}}H_{k}(x,y);$

  5. (v)

(the _cancellation condition_) for any $x,\ y\in{\mathcal{X}}$,

$\int_{\mathcal{X}}Q_{k}(x,y^{\prime})\,d\mu(y^{\prime})=0=\int_{\mathcal{X}}Q_{k}(x^{\prime},y)\,d\mu(x^{\prime}).$

The existence of such an exp-ATI on spaces of homogeneous type is guaranteed
by [4, Theorem 7.1] with $\eta$ same as in [4, Theorem 3.1] which might be
very small (see also [24, Remark 2.8(i)]). However, if $d$ is a metric, then
$\eta$ can be taken arbitrarily close to 1 (see [33, Corollary 6.13]).

The following lemma states some basic properties of exp-ATIs. One can find
more details in [24, Remarks 2.8 and 2.9, and Proposition 2.10].

######  Lemma 2.6.

Let $\\{Q_{k}\\}_{k\in\mathbb{Z}}$ be an exp-ATI and $\eta\in(0,1)$ as in
Definition 2.5. Then, for any given $\Gamma\in(0,\infty)$, there exists a
positive constant $C$ such that, for any $k\in\mathbb{Z}$, the kernel $Q_{k}$
has the following properties:

  1. (i)

for any $x,\ y\in{\mathcal{X}}$,

$|Q_{k}(x,y)|\leq CD_{\Gamma}(x,y;\delta^{k}),$ (2.6)

where $D_{\Gamma}(x,y;\delta^{k})$ is as in (2.3);

  2. (ii)

for any $x,\ x^{\prime},\ y\in{\mathcal{X}}$ with
$d(x,x^{\prime})\leq(2A_{0})^{-1}[\delta^{k}+d(x,y)]$,

$\displaystyle|Q_{k}(x,y)-Q_{k}(x^{\prime},y)|+|Q_{k}(y,x)-Q_{k}(y,x^{\prime})|\leq
C\left[\frac{d(x,x^{\prime})}{\delta^{k}+d(x,y)}\right]^{\eta}D_{\Gamma}(x,y;\delta^{k});$
(2.7)

  3. (iii)

for any $x,\ x^{\prime},\ y,\ y^{\prime}\in{\mathcal{X}}$ with
$d(x,x^{\prime})\leq(2A_{0})^{-2}[\delta^{k}+d(x,y)]$ and
$d(y,y^{\prime})\leq(2A_{0})^{-2}[\delta^{k}+d(x,y)]$,

$\displaystyle|[Q_{k}(x,y)-Q_{k}(x^{\prime},y)]-[Q_{k}(x,y^{\prime})-Q_{k}(x^{\prime},y^{\prime})]|$
$\displaystyle\quad\leq
C\left[\frac{d(x,x^{\prime})}{\delta^{k}+d(x,y)}\right]^{\eta}\left[\frac{d(y,y^{\prime})}{\delta^{k}+d(x,y)}\right]^{\eta}D_{\Gamma}(x,y;\delta^{k}).$

Based on exp-ATIs, we now recall the notions of Besov and Triebel–Lizorkin
spaces on spaces of homogeneous type; see [46, Definitions 3.1 and 5.1].

######  Definition 2.7.

Let $\beta,\ \gamma\in(0,\eta)$ with $\eta$ as in Definition 2.5, and
$s\in(-(\beta\wedge\gamma),\beta\wedge\gamma)$. Let
$\\{Q_{k}\\}_{k\in\mathbb{Z}}$ be an exp-ATI.

  1. (i)

Let $p\in(p(s,\beta\wedge\gamma),\infty]$, with $p(s,\beta\wedge\gamma)$ as in
(1.1), and $q\in(0,\infty]$. The _homogenous Besov space_
$\dot{B}^{s}_{p,q}(\mathcal{X})$ is defined by setting

$\dot{B}^{s}_{p,q}(\mathcal{X}):=\left\\{f\in\left(\mathring{\mathcal{G}}_{0}^{\eta}(\beta,\gamma)\right)^{\prime}:\
\|f\|_{\dot{B}^{s}_{p,q}(\mathcal{X})}<\infty\right\\},$

where, for any
$f\in(\mathring{\mathcal{G}}_{0}^{\eta}(\beta,\gamma))^{\prime}$,

$\|f\|_{\dot{B}^{s}_{p,q}(\mathcal{X})}:=\left[\sum_{k=-\infty}^{\infty}\delta^{-ksq}\|Q_{k}(f)\|_{{L^{p}(\mathcal{X})}}^{q}\right]^{1/q}$

with usual modifications made when $q=\infty$.

  2. (ii)

Let $p\in(p(s,\beta\wedge\gamma),\infty)$ and
$q\in(p(s,\beta\wedge\gamma),\infty]$. The _homogenous Triebel–Lizorkin space_
$\dot{F}^{s}_{p,q}(\mathcal{X})$ is defined by setting

$\dot{F}^{s}_{p,q}(\mathcal{X}):=\left\\{f\in\left(\mathring{\mathcal{G}}_{0}^{\eta}(\beta,\gamma)\right)^{\prime}:\
\|f\|_{\dot{F}^{s}_{p,q}(\mathcal{X})}<\infty\right\\},$

where, for any
$f\in(\mathring{\mathcal{G}}_{0}^{\eta}(\beta,\gamma))^{\prime}$,

$\|f\|_{\dot{F}^{s}_{p,q}(\mathcal{X})}:=\left\|\left[\sum_{k=-\infty}^{\infty}\delta^{-ksq}|Q_{k}(f)|^{q}\right]^{1/q}\right\|_{{L^{p}(\mathcal{X})}}$

with usual modification made when $q=\infty$.

The following definition introduces the notion of Triebel–Lizorkin spaces with
$p=\infty$; see [46, Definition 5.1].

######  Definition 2.8.

Let $\beta,\ \gamma\in(0,\eta)$,
$s\in(-(\beta\wedge\gamma),\beta\wedge\gamma)$, and
$q\in(p(s,\beta\wedge\gamma),\infty]$ with $\eta$ as in Definition 2.5 and
$p(s,\beta\wedge\gamma)$ as in (1.1). Let $\\{Q_{k}\\}_{k\in\mathbb{Z}}$ be an
exp-ATI. For any $k\in\mathbb{Z}$ and $\alpha\in{\mathcal{A}}_{k}$, let
$Q_{\alpha}^{k}$ be as in Lemma 2.4. Then the _homogeneous Triebel–Lizorkin
space $\dot{F}^{s}_{\infty,q}(\mathcal{X})$_ is defined by setting

$\displaystyle\dot{F}^{s}_{\infty,q}(\mathcal{X}):=\left\\{f\in\left(\mathring{\mathcal{G}}_{0}^{\eta}(\beta,\gamma)\right)^{\prime}:\
\|f\|_{\dot{F}^{s}_{\infty,q}(\mathcal{X})}<\infty\right\\},$

where, for any
$f\in(\mathring{\mathcal{G}}_{0}^{\eta}(\beta,\gamma))^{\prime}$,

$\|f\|_{\dot{F}^{s}_{\infty,q}(\mathcal{X})}:=\sup_{l\in\mathbb{Z}}\sup_{\alpha\in{\mathcal{A}}_{l}}\left[\frac{1}{\mu(Q_{\alpha}^{l})}\int_{Q_{\alpha}^{l}}\sum_{k=l}^{\infty}\delta^{-ksq}|Q_{k}(f)(x)|^{q}\,d\mu(x)\right]^{1/q}$

with the usual modification made when $q=\infty$.

######  Remark 2.9.

  1. (i)

In Definition 2.5, we need $\mathop{\mathrm{diam}}{\mathcal{X}}=\infty$ to
guarantee (v). Observe that it was shown in [39, Lemma 5.1] (see also [4,
Lemma 8.1]) that $\mathop{\mathrm{diam}}{\mathcal{X}}=\infty$ implies
$\mu({\mathcal{X}})=\infty$. Therefore,
$\mathop{\mathrm{diam}}{\mathcal{X}}=\infty$ if and only if
$\mu({\mathcal{X}})=\infty$ under the assumptions of this article. Due to
this, we always assume that $\mu({\mathcal{X}})=\infty$ in Sections 2 and 3.

  2. (ii)

In [46], Wang et al. proved that $\dot{B}^{s}_{p,q}(\mathcal{X})$ and
$\dot{F}^{s}_{p,q}(\mathcal{X})$ in Definition 2.7 are independent of the
choices of $\beta$ and ${\gamma}$ as in in Definition 2.7, and exp-ATIs (see
[46, Propositions 3.13 and 3.16] for more details). Besides, it was also shown
that $\dot{F}^{s}_{\infty,q}(\mathcal{X})$ in Definition 2.8 is independent of
the choices of $\beta$ and ${\gamma}$, and exp-ATIs (see [46, Propositions 5.4
and 5.5] for more details).

Now, we introduce the notions of $s$-gradients and $s$-Hajłasz gradients on
spaces of homogenous type (see, for instance, [36, Definition 1.1 and (2.1)]).

######  Definition 2.10.

Let $s\in(0,\infty)$ and $u$ be a measurable function on ${\mathcal{X}}$.

  1. (i)

A nonnegative function $g$ is called an _$s$ -gradient_ of $u$ if there exists
a set $E\subset{\mathcal{X}}$ with $\mu(E)=0$ such that, for any $x,\
y\in{\mathcal{X}}\setminus E$,

$|u(x)-u(y)|\leq[d(x,y)]^{s}[g(x)+g(y)].$

Denote by ${\mathcal{D}}^{s}(u)$ the collection of all $s$-gradients of $u$.

  2. (ii)

A sequence of nonnegative functions, $\\{g_{k}\\}_{k\in\mathbb{Z}}$, is called
an _$s$ -Hajłasz gradient_ of $u$ if there exists a set
$E\subset{\mathcal{X}}$ with $\mu(E)=0$ such that, for any $k\in\mathbb{Z}$
and $x,\ y\in{\mathcal{X}}\setminus E$ with $\delta^{k+1}\leq
d(x,y)<\delta^{k}$,

$|u(x)-u(y)|\leq[d(x,y)]^{s}[g_{k}(x)+g_{k}(y)].$

Denote by $\mathbb{D}^{s}(u)$ the collection of all $s$-Hajłasz gradients of
$u$.

Next, we introduce the notions of homogeneous Hajłasz–Sobolev spaces,
Hajłasz–Triebel–Lizorkin spaces, and Hajłasz–Besov spaces (see, for instance,
[36, Definitions 1.2 and 2.1]).

######  Definition 2.11.

Let $s\in(0,\infty)$.

  1. (i)

Let $p\in(0,\infty)$. The _homogeneous Hajłasz–Sobolev space_
$\dot{M}^{s,p}({\mathcal{X}})$ is defined to be the set of all measurable
functions $u$ on ${\mathcal{X}}$ such that

$\|u\|_{\dot{M}^{s,p}({\mathcal{X}})}:=\inf_{g\in{\mathcal{D}}^{s}(u)}\|g\|_{L^{p}({\mathcal{X}})}<\infty.$

  2. (ii)

Let $p\in(0,\infty)$ and $q\in(0,\infty]$. The _homogeneous
Hajłasz–Triebel–Lizorkin space_ $\dot{M}^{s}_{p,q}({\mathcal{X}})$ is defined
to be the set of all measurable functions $u$ on ${\mathcal{X}}$ such that

$\|u\|_{\dot{M}^{s}_{p,q}({\mathcal{X}})}:=\inf_{\\{g_{k}\\}_{k\in\mathbb{Z}}\in\mathbb{D}^{s}(u)}\left\|\left(\sum_{k=-\infty}^{\infty}g_{k}^{q}\right)^{\frac{1}{q}}\right\|_{L^{p}({\mathcal{X}})}<\infty$

with the usual modification made when $q=\infty$.

  3. (iii)

Let $q\in(0,\infty)$. The _homogeneous Hajłasz–Triebel–Lizorkin space_
$\dot{M}^{s}_{\infty,q}({\mathcal{X}})$ is defined to be the set of all
measurable functions $u$ on ${\mathcal{X}}$ such that

$\|u\|_{\dot{M}^{s}_{\infty,q}({\mathcal{X}})}:=\inf_{\\{g_{k}\\}_{k\in\mathbb{Z}}\in\mathbb{D}^{s}(u)}\sup_{k\in\mathbb{Z}}\sup_{x\in{\mathcal{X}}}\left\\{\sum_{j=k}^{\infty}\frac{1}{\mu(B(x,\delta^{k}))}\int_{B(x,\delta^{k})}[g_{j}(y)]^{q}\,d\mu(y)\right\\}^{\frac{1}{q}}<\infty.$

  4. (iv)

The _homogeneous Hajłasz–Triebel–Lizorkin space_
$\dot{M}^{s}_{\infty,\infty}({\mathcal{X}})$ is defined to be the set of all
measurable functions $u$ on ${\mathcal{X}}$ such that

$\|u\|_{\dot{M}^{s}_{\infty,\infty}({\mathcal{X}})}:=\inf_{\\{g_{k}\\}_{k\in\mathbb{Z}}\in\mathbb{D}^{s}(u)}\left\|\sup_{k\in\mathbb{Z}}g_{k}\right\|_{L^{\infty}({\mathcal{X}})}<\infty.$

  5. (v)

Let $p,\ q\in(0,\infty]$. The _homogeneous Hajłasz–Besov space_
$\dot{N}^{s}_{p,q}({\mathcal{X}})$ is defined to be the set of all measurable
functions $u$ on ${\mathcal{X}}$ such that

$\|u\|_{\dot{N}^{s}_{p,q}({\mathcal{X}})}:=\inf_{\\{g_{k}\\}_{k\in\mathbb{Z}}\in\mathbb{D}^{s}(u)}\left[\sum_{k=-\infty}^{\infty}\|g_{k}\|_{L^{p}({\mathcal{X}})}^{q}\right]^{\frac{1}{q}}<\infty$

with the usual modification made when $q=\infty$.

By Definition 2.11, it is easy to see the following conclusion. We omit the
details.

######  Proposition 2.12.

Let $s\in(0,\infty)$ and $p\in(0,\infty]$. Then
$\dot{M}^{s}_{p,\infty}({\mathcal{X}})=\dot{M}^{s,p}({\mathcal{X}})$.

Next, we recall the notion of weak lower bounds (see, for instance, [17,
Definition 1.1], [47, Definition 4.4], and [1, (2) or (3)]).

######  Definition 2.13.

Let $({\mathcal{X}},d,\mu)$ be a space of homogeneous type with upper
dimension $\omega$ as in (2.2). The measure $\mu$ is said to have a _weak
lower bound_ $Q$ with $Q\in(0,\omega]$ if there exists a positive constant $C$
and a point $x_{0}\in{\mathcal{X}}$ such that, for any $r\in[1,\infty)$,

$\mu(B(x_{0},r))\geq Cr^{Q}.$

######  Remark 2.14.

We point out that, in [47, Definition 4.4], $\mu$ is said to have a lower
bound $Q$ with $Q\in(0,\omega]$ if there exists a positive constant $C$ such
that, for any $x\in{\mathcal{X}}$ and $r\in(0,\infty)$, $\mu(B(x,r))\geq
Cr^{Q}$. That is why we call it the weak lower bound in Definition 2.13.

As the next result illustrates, it follows from the doubling property of the
measure that the weak lower bound and lower bound conditions are equivalent
when $Q=\omega$, where $\omega$ is as in (2.2).

######  Proposition 2.15.

With $\omega$ as in (2.2), the measure $\mu$ has a weak lower bound $Q=\omega$
if and only if it has a lower bound $Q=\omega$.

###### Proof.

Clearly, the lower bound condition implies the weak lower bound condition.
Now, we show the converse. To this end, suppose that the measure $\mu$ has a
weak lower bound $Q=\omega$ for some fixed $x_{0}\in{\mathcal{X}}$. Fix
$x\in{\mathcal{X}}$ and $r\in(0,\infty)$. Next, choose $R\in[1,\infty)$ large
enough so that $R>r$ and $B(x,r)\subset B(x_{0},R)$. Consider the smallest
$k\in\mathbb{N}$ such that $2A_{0}R\leq(2A_{0})^{k}r$, where
$A_{0}\in[1,\infty)$ is the constant in the quasi-triangle inequality. Note
that $k\geq 1$ because $r<R$, and hence $(2A_{0})^{k}>1$. Also, this choice of
$k$ ensures that $(2A_{0})^{k}r\leq(2A_{0})^{2}R$ which further implies that
$B(x_{0},R)\subset B(x,(2A_{0})^{k}r)$. Using this, the weak lower bound
$Q=\omega$ for the ball $B(x_{0},R)$, the doubling condition in (2.1), and
$(2A_{0})^{k}r\leq(2A_{0})^{2}R$, we further conclude that

$R^{\omega}\lesssim\mu(B(x_{0},R))\lesssim\mu(B(x,(2A_{0})^{k}r))\lesssim(2A_{0})^{k\omega}\mu(B(x,r))\lesssim(2A_{0})^{2\omega}\left(\frac{R}{r}\right)^{\omega}\mu(B(x,r)),$

from which it follows that $\mu(B(x,r))\gtrsim r^{\omega}$. Thus, $\mu$ has a
lower bound $Q=\omega$, as wanted. This finishes the proof of Proposition
2.15. ∎

Now, we can state our main results of this article.

######  Theorem 2.16.

Let $\beta,\ \gamma\in(0,\eta)$ with $\eta$ as in Definition 2.5,
$s\in(0,\beta\wedge\gamma)$, $p,\ q$ be as in Definition 2.7, and $\omega$ as
in (2.2). Assume that the measure $\mu$ of ${\mathcal{X}}$ has a weak lower
bound $Q=\omega$.

  1. (i)

If $p\in(\omega/(\omega+s),\infty)$ and $q\in(\omega/(\omega+s),\infty]$, then
$\dot{M}^{s}_{p,q}({\mathcal{X}})=\dot{F}^{s}_{p,q}(\mathcal{X})$.

  2. (ii)

If $p\in(\omega/(\omega+s),\infty]$ and $q\in(0,\infty]$, then
$\dot{N}^{s}_{p,q}({\mathcal{X}})=\dot{B}^{s}_{p,q}(\mathcal{X})$.

##  3 Relations with homogeneous grand Besov and Triebel–Lizorkin spaces

Before we prove Theorem 2.16, we need to introduce the notions of another
important spaces, namely, the homogeneous grand Besov and Triebel–Lizorkin
spaces on spaces of homogenous type.

######  Definition 3.1.

Let $\eta$ be as in Definition 2.5, $s\in(-\eta,\eta)$, $\beta,\
\gamma\in(0,\eta)$, and $q\in(0,\infty]$. For any $k\in\mathbb{Z}$ and
$x\in{\mathcal{X}}$, define

${\mathcal{F}}_{k}(x):=\left\\{\phi\in\mathring{{\mathcal{G}}}_{0}^{\eta}(\beta,\gamma):\
\|\phi\|_{\mathring{{\mathcal{G}}}(x,\delta^{k},\beta,\gamma)}\leq
1\right\\}.$

  1. (i)

For any given $p\in(0,\infty]$, the _homogenous grand Besov space_
${\mathcal{A}}\dot{B}^{s}_{p,q}(\mathcal{X})$ is defined by setting

${\mathcal{A}}\dot{B}^{s}_{p,q}(\mathcal{X}):=\left\\{f\in\left(\mathring{\mathcal{G}}_{0}^{\eta}(\beta,\gamma)\right)^{\prime}:\
\|f\|_{{\mathcal{A}}\dot{B}^{s}_{p,q}(\mathcal{X})}<\infty\right\\},$

where, for any
$f\in(\mathring{\mathcal{G}}_{0}^{\eta}(\beta,\gamma))^{\prime}$,

$\|f\|_{{\mathcal{A}}\dot{B}^{s}_{p,q}(\mathcal{X})}:=\left[\sum_{k=-\infty}^{\infty}\delta^{-ksq}\left\|\sup_{\phi\in{\mathcal{F}}_{k}(\cdot)}|\langle
f,\phi\rangle|\right\|_{{L^{p}(\mathcal{X})}}^{q}\right]^{1/q}$

with usual modification made when $q=\infty$.

  2. (ii)

For any given $p\in(0,\infty)$, the _homogenous grand Triebel–Lizorkin space_
${\mathcal{A}}\dot{F}^{s}_{p,q}(\mathcal{X})$ is defined by setting

${\mathcal{A}}\dot{F}^{s}_{p,q}(\mathcal{X}):=\left\\{f\in\left(\mathring{\mathcal{G}}_{0}^{\eta}(\beta,\gamma)\right)^{\prime}:\
\|f\|_{{\mathcal{A}}\dot{F}^{s}_{p,q}(\mathcal{X})}<\infty\right\\}$

where, for any
$f\in(\mathring{\mathcal{G}}_{0}^{\eta}(\beta,\gamma))^{\prime}$,

$\|f\|_{{\mathcal{A}}\dot{F}^{s}_{p,q}(\mathcal{X})}:=\left\|\left[\sum_{k=-\infty}^{\infty}\delta^{-ksq}\sup_{\phi\in{\mathcal{F}}_{k}(\cdot)}|\langle
f,\phi\rangle|^{q}\right]^{1/q}\right\|_{{L^{p}(\mathcal{X})}}$

with usual modification made when $q=\infty$.

  3. (iii)

The _homogenous grand Triebel–Lizorkin space_
${\mathcal{A}}\dot{F}^{s}_{\infty,q}(\mathcal{X})$ is defined by setting

$\displaystyle{\mathcal{A}}\dot{F}^{s}_{\infty,q}(\mathcal{X}):=\left\\{f\in(\mathring{\mathcal{G}}_{0}^{\eta}(\beta,\gamma))^{\prime}:\
{}\|f\|_{{\mathcal{A}}\dot{F}^{s}_{\infty,q}(\mathcal{X})}<\infty\right\\},$

where, for any
$f\in(\mathring{\mathcal{G}}_{0}^{\eta}(\beta,\gamma))^{\prime}$,

$\|f\|_{{\mathcal{A}}\dot{F}^{s}_{\infty,q}(\mathcal{X})}:=\sup_{l\in\mathbb{Z}}\sup_{\alpha\in{\mathcal{A}}_{l}}\left[\frac{1}{\mu(Q_{\alpha}^{l})}\int_{Q_{\alpha}^{l}}\sum_{k=l}^{\infty}\delta^{-ksq}\sup_{\phi\in{\mathcal{F}}_{k}(x)}|\langle
f,\phi\rangle|^{q}\,d\mu(x)\right]^{1/q}$

with usual modification made when $q=\infty$.

######  Remark 3.2.

Let $\\{Q_{k}\\}_{k\in\mathbb{Z}}$ be an exp-ATI. By (2.6) and (2.7), it is
easy to see that, for any $k\in\mathbb{Z}$ and $x\in{\mathcal{X}}$,
$Q_{k}(x,\cdot)\in{\mathcal{F}}_{k}(x)$.

Now, we establish the relationship between homogeneous grand Besov and
Triebel–Lizorkin spaces and homogeneous Besov and Triebel–Lizorkin spaces.

######  Theorem 3.3.

Let $\beta,\ \gamma\in(0,\eta)$ with $\eta$ as in Definition 2.5, and
$s\in(-(\beta\wedge\gamma),\beta\wedge\gamma)$.

  1. (i)

If $p$ and $q$ are as in Definition 2.7(ii), then
$\dot{F}^{s}_{p,q}(\mathcal{X})={\mathcal{A}}\dot{F}^{s}_{p,q}(\mathcal{X})$.

  2. (ii)

If $p$ and $q$ are as in Definition 2.7(i), then
$\dot{B}^{s}_{p,q}(\mathcal{X})={\mathcal{A}}\dot{B}^{s}_{p,q}(\mathcal{X})$.

To prove Theorem 3.3, we need several lemmas. Let us begin with recalling the
following very basic inequality.

######  Lemma 3.4.

For any $\theta\in(0,1]$ and $\\{a_{j}\\}_{j\in\mathbb{N}}\subset\mathbb{C}$,
it holds true that

$\left(\sum_{j=1}^{\infty}|a_{j}|\right)^{\theta}\leq\sum_{j=1}^{\infty}|a_{j}|^{\theta}.$
(3.1)

The following lemma contains several basic and very useful estimates related
to $d$ and $\mu$ on ${\mathcal{X}}$. One can find the details in [20, Lemma
2.1] or [24, Lemma 2.4].

######  Lemma 3.5.

Let $\beta,\ \gamma\in(0,\infty)$.

  1. (i)

For any $x,\ y\in{\mathcal{X}}$ and $r\in(0,\infty)$, $V(x,y)\sim V(y,x)$ and

$V_{r}(x)+V_{r}(y)+V(x,y)\sim V_{r}(x)+V(x,y)\sim
V_{r}(y)+V(x,y)\sim\mu(B(x,r+d(x,y)))$

and, moreover, if $d(x,y)\leq r$, then $V_{r}(x)\sim V_{r}(y)$. Here the
positive equivalence constants are independent of $x$, $y$, and $r$.

  2. (ii)

There exists a positive constant $C$ such that, for any
$x_{1}\in{\mathcal{X}}$ and $r\in(0,\infty)$,

$\int_{\mathcal{X}}D_{\gamma}(x_{1},y;r)\,d\mu(y)\leq C;$

where, $D_{\gamma}(x,y;r)$ is as in (2.3).

The following homogeneous discrete Calderón reproducing formula was obtained
in [24, Theorem 5.11]. Let $j_{0}\in\mathbb{N}$ be sufficiently large such
that $\delta^{j_{0}}\leq(2A_{0})^{-3}C_{0}$. Based on Lemma 2.4, for any
$k\in\mathbb{Z}$ and $\alpha\in{\mathcal{A}}_{k}$, let

${\mathfrak{N}}(k,\alpha):=\\{\tau\in{\mathcal{A}}_{k+j_{0}}:\
Q_{\tau}^{k+j_{0}}\subset Q_{\alpha}^{k}\\}$

and $N(k,\alpha):=\\#{\mathfrak{N}}(k,\alpha)$. From Lemma 2.4, it follows
that $N(k,\alpha)\lesssim\delta^{-j_{0}\omega}$ and
$\bigcup_{\tau\in{\mathfrak{N}}(k,\alpha)}Q_{\tau}^{k+j_{0}}=Q_{\alpha}^{k}$.
We rearrange the set
$\\{Q_{\tau}^{k+j_{0}}:\tau\in{\mathfrak{N}}(k,\alpha)\\}$ as
$\\{Q_{\alpha}^{k,m}\\}_{m=1}^{N(k,\alpha)}$. Also, denote by
$y_{\alpha}^{k,m}$ an arbitrary point in $Q_{\alpha}^{k,m}$ and
$z_{\alpha}^{k,m}$ the “center” of $Q_{\alpha}^{k,m}$.

######  Lemma 3.6.

Let $\\{Q_{k}\\}_{k=-\infty}^{\infty}$ be an exp-ATI and $\beta,\
\gamma\in(0,\eta)$ with $\eta$ as in Definition 2.5. For any $k\in\mathbb{Z}$,
$\alpha\in{\mathcal{A}}_{k}$, and $m\in\\{1,\dots,N(k,\alpha)\\}$, suppose
that $y_{\alpha}^{k,m}$ is an arbitrary point in $Q_{\alpha}^{k,m}$. Then
there exists a sequence $\\{\widetilde{Q}_{k}\\}_{k=-\infty}^{\infty}$ of
bounded linear integral operators on $L^{2}(\mathcal{X})$ such that, for any
$f\in(\mathring{\mathcal{G}}_{0}^{\eta}(\beta,\gamma))^{\prime}$,

$f(\cdot)=\sum_{k=-\infty}^{\infty}\sum_{\alpha\in{\mathcal{A}}_{k}}\sum_{m=1}^{N(k,\alpha)}\mu\left(Q_{\alpha}^{k,m}\right)\widetilde{Q}_{k}(\cdot,y_{\alpha}^{k,m})Q_{k}f\left(y_{\alpha}^{k,m}\right).$

in $(\mathring{\mathcal{G}}_{0}^{\eta}(\beta,\gamma))^{\prime}$. Moreover,
there exists a positive constant $C$, independent of the choices of both
$y_{\alpha}^{k,m}$, with $k\in\mathbb{Z},\ \alpha\in{\mathcal{A}}_{k}$, and
$m\in\\{1,\dots,N(k,\alpha)\\}$, and $f$, such that, for any $k\in\mathbb{Z}$,
the kernel of $\widetilde{Q}_{k}$ satisfies

  1. (i)

for any $x,\ y\in{\mathcal{X}}$,

$\left|\widetilde{Q}_{k}(x,y)\right|\leq CD_{\gamma}(x,y;\delta^{k}),$ (3.2)

where $D_{\gamma}(x,y;\delta^{k})$ is as in (2.3);

  2. (ii)

for any $x,\ x^{\prime},\ y\in{\mathcal{X}}$ with
$d(x,x^{\prime})\leq(2A_{0})^{-1}[\delta^{k}+d(x,y)]$,

$\displaystyle\left|\widetilde{Q}_{k}(x,y)-\widetilde{Q}_{k}(x^{\prime},y)\right|\leq
C\left[\frac{d(x,x^{\prime})}{\delta^{k}+d(x,y)}\right]^{\beta}D_{\gamma}(x,y;\delta^{k});$
(3.3)

  3. (iii)

for any $x\in\mathcal{X}$,

$\int_{\mathcal{X}}\widetilde{Q}_{k}(x,y)\,d\mu(y)=0=\int_{\mathcal{X}}\widetilde{Q}_{k}(y,x)\,d\mu(y).$

We also need the following three lemmas (see, for instance, [46, Lemmas 3.5
and 3.6]).

######  Lemma 3.7.

Let $\gamma\in(0,\infty)$ and $p\in(\omega/(\omega+\gamma),1]$ with $\omega$
as in (2.2). Then there exists a constant $C\in[1,\infty)$ such that, for any
$k,\ k^{\prime}\in\mathbb{Z}$, $x\in\mathcal{X}$, and $y_{\alpha}^{k,m}\in
Q_{\alpha}^{k,m}$ with $\alpha\in{\mathcal{A}}_{k}$ and
$m\in\\{1,\dots,N(k,\alpha)\\}$,

$\displaystyle C^{-1}[V_{\delta^{k\wedge
k^{\prime}}}(x)]^{1-p}\leq\sum_{\alpha\in{\mathcal{A}}_{k}}\sum_{m=1}^{N(k,\alpha)}\mu\left(Q_{\alpha}^{k,m}\right)\left[D_{\gamma}(x,y_{\alpha}^{k,m};\delta^{k\wedge
k^{\prime}})\right]^{p}\leq C[V_{\delta^{k\wedge k^{\prime}}}(x)]^{1-p},$

where $D_{\gamma}(x,y_{\alpha}^{k,m};\delta^{k\wedge k^{\prime}})$ is as in
(2.3).

######  Lemma 3.8.

Let $\gamma\in(0,\infty)$ and $r\in(\omega/(\omega+\gamma),1]$ with $\omega$
as in (2.2). Then there exists a positive constant $C$ such that, for any $k,\
k^{\prime}\in\mathbb{Z}$, $x\in\mathcal{X}$, and
$a_{\alpha}^{k,m}\in\mathbb{C}$ and $y_{\alpha}^{k,m}\in Q_{\alpha}^{k,m}$
with $\alpha\in{\mathcal{A}}_{k}$ and $m\in\\{1,\dots,N(k,\alpha)\\}$,

$\displaystyle\sum_{\alpha\in{\mathcal{A}}_{k}}\sum_{m=1}^{N(k,\alpha)}\mu\left(Q_{\alpha}^{k,m}\right)D_{\gamma}(x,y_{\alpha}^{k,m};\delta^{k\wedge
k^{\prime}})|a_{\alpha}^{k,m}|$ $\displaystyle\quad\leq C\delta^{[k-(k\wedge
k^{\prime})]\omega(1-1/r)}\left[M\left(\sum_{\alpha\in{\mathcal{A}}_{k}}\sum_{m=1}^{N(k,\alpha)}|a_{\alpha}^{k,m}|^{r}\mathbf{1}_{Q_{\alpha}^{k,m}}\right)(x)\right]^{1/r},$

where $D_{\gamma}(x,y_{\alpha}^{k,m};\delta^{k\wedge k^{\prime}})$ is as in
(2.3) and $M$ as in (1.2).

The following lemma is the Fefferman–Stein vector-valued maximal inequality
which was established in [13, Theorem 1.2].

######  Lemma 3.9.

Let $p\in(1,\infty)$, $q\in(1,\infty]$, and $M$ be the Hardy–Littlewood
maximal operator on $\mathcal{X}$ as in (1.2). Then there exists a positive
constant $C$ such that, for any sequence $\\{f_{j}\\}_{j\in\mathbb{Z}}$ of
measurable functions on $\mathcal{X}$,

$\left\|\left\\{\sum_{j\in\mathbb{Z}}[M(f_{j})]^{q}\right\\}^{1/q}\right\|_{L^{p}(\mathcal{X})}\leq
C\left\|\left(\sum_{j\in\mathbb{Z}}|f_{j}|^{q}\right)^{1/q}\right\|_{L^{p}(\mathcal{X})}$

with the usual modification made when $q=\infty$.

Now, we prove Theorem 3.3.

###### Proof of Theorem 3.3.

We first show (i). Assume that
$f\in{\mathcal{A}}\dot{F}^{s}_{p,q}(\mathcal{X})$ and that
$\\{Q_{k}\\}_{k\in\mathbb{Z}}$ is an exp-ATI. From Remark 3.2, we deduce that
$|Q_{k}(f)|\leq\sup_{\phi\in{\mathcal{F}}_{k}(\cdot)}|\langle f,\phi\rangle|$
and hence
$\|f\|_{\dot{F}^{s}_{p,q}(\mathcal{X})}\leq\|f\|_{{\mathcal{A}}\dot{F}^{s}_{p,q}(\mathcal{X})}$.

Conversely, assume that $f\in\dot{F}^{s}_{p,q}(\mathcal{X})$. By Lemma 3.6, we
know that, for any $l\in\mathbb{Z}$, $x\in{\mathcal{X}}$, and
$\phi\in{\mathcal{F}}_{l}(x)$,

$\langle
f,\phi\rangle=\sum_{k=-\infty}^{\infty}\sum_{\alpha\in{\mathcal{A}}_{k}}\sum_{m=1}^{N(k,\alpha)}\mu\left(Q_{\alpha}^{k,m}\right)Q_{k}f\left(y_{\alpha}^{k,m}\right)\int_{{\mathcal{X}}}\widetilde{Q}_{k}(z,y_{\alpha}^{k,m})\phi(z)\,d\mu(z).$

Notice that, by an argument similar to that used in the proof of [46, Lemma
3.9], we have, for any fixed $\eta^{\prime}\in(0,\beta\wedge\gamma)$,

$\left|\int_{{\mathcal{X}}}\widetilde{Q}_{k}(z,y_{\alpha}^{k,m})\phi(z)\,d\mu(z)\right|\lesssim\delta^{|k-l|\eta^{\prime}}D_{\gamma}(x,y_{\alpha}^{k,m};\delta^{k\wedge
l}),$

where $D_{\gamma}(x,y_{\alpha}^{k,m};\delta^{k\wedge k^{\prime}})$ is as in
(2.3). Using this, Lemma 3.8, the arbitrariness of $y_{\alpha}^{k,m}$, and
choosing $r\in(\omega/(\omega+\gamma),\min\\{p,q,1\\})$, we obtain

$\displaystyle|\langle f,\phi\rangle|$
$\displaystyle\lesssim\sum_{k=-\infty}^{\infty}\delta^{|k-l|\eta^{\prime}}\sum_{\alpha\in{\mathcal{A}}_{k}}\sum_{m=1}^{N(k,\alpha)}\mu\left(Q_{\alpha}^{k,m}\right)\left|Q_{k}f\left(y_{\alpha}^{k,m}\right)\right|D_{\gamma}(x,y_{\alpha}^{k,m};\delta^{k\wedge
l})$
$\displaystyle\lesssim\sum_{k=-\infty}^{\infty}\delta^{|k-l|\eta^{\prime}}\delta^{[k-(k\wedge
l)]\omega(1-1/r)}\left[M\left(\sum_{\alpha\in{\mathcal{A}}_{k}}\sum_{m=1}^{N(k,\alpha)}\left|Q_{k}f\left(y_{\alpha}^{k,m}\right)\right|^{r}\mathbf{1}_{Q_{\alpha}^{k,m}}\right)(x)\right]^{1/r}$
$\displaystyle\lesssim\sum_{k=-\infty}^{\infty}\delta^{|k-l|\eta^{\prime}}\delta^{[k-(k\wedge
l)]\omega(1-1/r)}\left[M\left(\left|Q_{k}f\right|^{r}\right)(x)\right]^{1/r}.$

By this, we know that

$\displaystyle\|f\|_{{\mathcal{A}}\dot{F}^{s}_{p,q}(\mathcal{X})}$
$\displaystyle\lesssim\left\|\left[\sum_{l=-\infty}^{\infty}\delta^{(l-k)sq}\left\\{\sum_{k=-\infty}^{\infty}\delta^{|k-l|\eta^{\prime}}\delta^{[k-(k\wedge
l)]\omega(1-1/r)}\right.\right.\right.$
$\displaystyle\quad\left.\left.\left.\times\vphantom{\sum_{k=-\infty}^{\infty}}\left[M\left(\delta^{-ksr}\left|Q_{k}f\right|^{r}\right)\right]^{1/r}\right\\}^{q}\right]^{1/q}\right\|_{L^{p}({\mathcal{X}})},$

which, together with the Hölder inequality when $q\in(1,\infty]$, or (3.1)
when $q\in(\omega/[\omega+(\beta\wedge\gamma)],1]$, implies that

$\|f\|_{{\mathcal{A}}\dot{F}^{s}_{p,q}(\mathcal{X})}\lesssim\left\|\left\\{\sum_{k=-\infty}^{\infty}\left[M\left(\delta^{-ksr}\left|Q_{k}f\right|^{r}\right)\right]^{q/r}\right\\}^{1/q}\right\|_{L^{p}({\mathcal{X}})}.$

From this and Lemma 3.9, we deduce that

$\|f\|_{{\mathcal{A}}\dot{F}^{s}_{p,q}(\mathcal{X})}\lesssim\|f\|_{\dot{F}^{s}_{p,q}(\mathcal{X})}.$

This finishes the proof of (i).

The proof of (ii) is similar to that of (i) and we omit the details. ∎

Next, we establish the equivalence between homogenous Hajłasz–Besov spaces and
Hajłasz–Triebel–Lizorkin spaces, and homogeneous grand Besov and
Triebel–Lizorkin spaces.

######  Theorem 3.10.

Let $\beta,\ \gamma\in(0,\eta)$ with $\eta$ as in Definition 2.5, and
$s\in(0,\beta\wedge\gamma)$. Assume that the measure $\mu$ of ${\mathcal{X}}$
has a weak lower bound $Q=\omega$.

  1. (i)

If $p\in(\omega/(\omega+s),\infty]$ and $q\in(\omega/(\omega+s),\infty]$, then
${\mathcal{A}}\dot{F}^{s}_{p,q}(\mathcal{X})=\dot{M}^{s}_{p,q}({\mathcal{X}})$.

  2. (ii)

If $p\in(\omega/(\omega+s),\infty]$ and $q\in(0,\infty]$, then
${\mathcal{A}}\dot{B}^{s}_{p,q}(\mathcal{X})=\dot{N}^{s}_{p,q}({\mathcal{X}})$.

To prove Theorem 3.10, we need several lemmas. The following lemma was
originally shown in [15, Theorem 8.7] when $s=1$ and $A_{0}=1$. When
$s\in(0,1)$ and $A_{0}\in(1,\infty)$, we need more restrictions on $A_{0}$ and
$\delta$. We borrow some ideas from the proof of [15, Theorem 8.7]. In what
follows, for any measurable set $E\subset{\mathcal{X}}$ with $\mu(E)>0$, let

$\fint_{E}:=\frac{1}{\mu(E)}\int_{E}.$

######  Lemma 3.11.

Let $s\in(0,\infty)$, $p\in(0,\omega/s)$, and $p^{\ast}:=\frac{\omega
p}{\omega-sp}$ with $\omega$ as in (2.2). If $A_{0}\delta^{p/\omega}<1$, then
there exists a positive constant $C$ such that, for any
$B_{0}:=B(x_{0},r_{0})\subset{\mathcal{X}}$ with $x_{0}\in{\mathcal{X}}$ and
$r_{0}\in(0,\infty)$, $u\in\dot{M}^{s,p}(B(x_{0},\delta^{-1}r_{0}))$, and
$g\in{\mathcal{D}}^{s}(u)$, one has $u\in L^{p^{\ast}}(B_{0})$ and

$\inf_{c\in\mathbb{R}}\left[\fint_{B_{0}}|u(y)-c|^{p^{\ast}}\,d\mu(y)\right]^{\frac{1}{p^{\ast}}}\leq
Cr_{0}^{s}\left\\{\fint_{\delta^{-1}B_{0}}[g(y)]^{p}\,d\mu(y)\right\\}^{1/p}.$
(3.4)

###### Proof.

If $\int_{\delta^{-1}B_{0}}[g(y)]^{p}\,d\mu(y)=\infty$, then (3.4) holds true.
If

$\int_{\delta^{-1}B_{0}}[g(y)]^{p}\,d\mu(y)=0,$

then we know that $g(x)=0$ for almost every $x\in\delta^{-1}B_{0}$ and hence
there exists a $c\in\mathbb{R}$ such that $u(x)=c$ for almost every
$x\in\delta^{-1}B_{0}$. Thus, in this case, (3.4) holds true.

In what follows, we assume that

$0<\int_{\delta^{-1}B_{0}}[g(y)]^{p}\,d\mu(y)<\infty.$

Note that this implies $g>0$ almost everywhere in ${\mathcal{X}}$. Moreover,
we may also assume that, for every $x\in\delta^{-1}B_{0}$,

$g(x)\geq\delta^{1+1/p}\left\\{\fint_{\delta^{-1}B_{0}}[g(y)]^{p}\,d\mu(y)\right\\}^{1/p},$
(3.5)

as otherwise we may replace $g$ by
$\widetilde{g}(x):=g(x)+\\{\fint_{\delta^{-1}B_{0}}[g(y)]^{p}\,d\mu(y)\\}^{1/p}$
for any $x\in\delta^{-1}B_{0}$, because

$\left\\{\fint_{\delta^{-1}B_{0}}[\widetilde{g}(y)]^{p}\,d\mu(y)\right\\}^{1/p}\lesssim\left\\{\fint_{\delta^{-1}B_{0}}[g(y)]^{p}\,d\mu(y)\right\\}^{1/p}.$

For any $k\in\mathbb{Z}$, define

$E_{k}:=\left\\{x\in\delta^{-1}B_{0}:\ g(x)\leq\delta^{-k}\right\\}.$

It is easy to see that, for any $k\in\mathbb{Z}$, $E_{k-1}\subset E_{k}$ and

$\lim_{k\to\infty}\mu(E_{k})=\mu(\delta^{-1}B_{0}).$ (3.6)

Since $g>0$ almost everywhere in ${\mathcal{X}}$, we also have

$\mu\left(\delta^{-1}B_{0}\setminus\bigcup_{k\in\mathbb{Z}}(E_{k}\setminus
E_{k-1})\right)=0,$

which allows us to write

$\int_{\delta^{-1}B_{0}}[g(y)]^{p}\,d\mu(y)\sim\sum_{k=-\infty}^{\infty}\delta^{-kp}\mu(E_{k}\setminus
E_{k-1}).$ (3.7)

For any $c\in\mathbb{R}$, if we let $a_{k}:=\sup_{y\in B_{0}\cap
E_{k}}|u(y)-c|$, then $a_{k}$ is nondecreasing and

$\int_{B_{0}}|u(y)-c|^{p^{\ast}}\,d\mu(y)\leq\sum_{k=-\infty}^{\infty}a_{k}^{p^{\ast}}\mu(B_{0}\cap[E_{k}\setminus
E_{k-1}]).$ (3.8)

Note that, if $\mu(\delta^{-1}B_{0}\setminus E_{k-1})=0$, then
$\mu(E_{k}\setminus E_{k-1})=0$. Thus, to estimate (3.7) and (3.8), we only
need to consider $k\in\mathbb{Z}$ such that $\mu(\delta^{-1}B_{0}\setminus
E_{k-1})>0$, which is always assumed in what follows. Let

$b:=(4A_{0})^{-\omega}\delta^{\omega}r_{0}^{-\omega}\mu(\delta^{-1}B_{0})$
(3.9)

and

$r_{k}:=2b^{-\frac{1}{\omega}}[\mu(\delta^{-1}B_{0}\setminus
E_{k-1})]^{\frac{1}{\omega}}.$

Then we know that $r_{k}\in(0,\infty)$. Moreover, by the Chebyshev inequality,
we know that

$\mu(\delta^{-1}B_{0}\setminus E_{k})=\mu(\\{x\in\delta^{-1}B_{0}:\
g(x)>\delta^{-k}\\})\leq\delta^{kp}\int_{\delta^{-1}B_{0}}[g(y)]^{p}\,d\mu(y),$
(3.10)

which implies that $\lim_{k\to\infty}r_{k}=0$. Thus, there exists a
$k_{0}\in\mathbb{Z}$, which will be determined later, such that, for any
$k>k_{0}$, we can find an $x_{k}\in B_{0}$ satisfying
$B(x_{k},r_{k})\subset\delta^{-1}B_{0}$, where $r_{k}\leq\delta^{-1}r_{0}$.
Observe that, by the doubling condition of ${\mathcal{X}}$, we can conclude
that, for any $k>k_{0}$, $\mu(B(x_{k},r_{k}))\geq br_{k}^{\omega}$. Combining
this and the definition of $r_{k}$, we find that

$\mu(B(x_{k},r_{k}))\geq br_{k}^{\omega}>\mu(\delta^{-1}B_{0}\setminus
E_{k-1})=\mu(\delta^{-1}B_{0})-\mu(E_{k-1}).$

From this, we deduce that $B(x_{k},r_{k})\cap E_{k-1}\neq\emptyset$, that is,
there exists an $x_{k-1}\in B(x_{k},r_{k})\cap E_{k-1}$. Now, if
$B(x_{k-1},r_{k-1})\subset\delta^{-1}B_{0}$, then we can repeat the above
procedure to find an $x_{k-2}$ such that $x_{k-2}\in B(x_{k-1},r_{k-1})\cap
E_{k-2}$. As a summary, for any $i\in\\{1,\dots,k-k_{0}+1\\}$, if
$B(x_{k-i},r_{k-i})\subset\delta^{-1}B_{0}$, then we can find an $x_{k-i-1}$
such that $x_{k-i-1}\in B(x_{k-i},r_{k-i})\cap E_{k-i-1}$. We now want to
determine $k_{0}$. Note that, by (3.10), $x_{k}\in B_{0}$, and the assumption
that $A_{0}\delta^{p/\omega}<1$, we know that, for any $y\in
B(x_{k_{0}},r_{k_{0}})$,

$\displaystyle d(y,x_{0})$ $\displaystyle\leq
A_{0}[d(y,x_{k})+d(x_{k},x_{0})]<A_{0}d(y,x_{k})+A_{0}r_{0}$
$\displaystyle\leq A_{0}^{2}[d(y,x_{k_{0}})+d(x_{k_{0}},x_{k})]+A_{0}r_{0}$
$\displaystyle\leq
A_{0}^{2}r_{k_{0}}+A_{0}^{3}[d(x_{k_{0}},x_{k_{0}+1})+d(x_{k_{0}+1},x_{k})]+A_{0}r_{0}$
$\displaystyle\leq
A_{0}^{2}r_{k_{0}}+A_{0}^{3}r_{k_{0}+1}+\cdots+A_{0}^{k-k_{0}+1}r_{k-1}+A_{0}^{k-k_{0}+1}r_{k}+A_{0}r_{0}$
$\displaystyle\leq
2A_{0}^{-k_{0}+3}b^{-1/\omega}\left\\{\int_{\delta^{-1}B_{0}}[g(z)]^{p}\,d\mu(z)\right\\}^{1/\omega}\sum_{i=k_{0}-1}^{k-1}(A_{0}\delta^{p/\omega})^{i}+A_{0}r_{0}$
$\displaystyle\leq
A_{0}^{2}\delta^{(k_{0}-1)p/\omega}\frac{2b^{-1/\omega}}{1-A_{0}\delta^{p/\omega}}\left\\{\int_{\delta^{-1}B_{0}}[g(z)]^{p}\,d\mu(z)\right\\}^{1/\omega}+A_{0}r_{0}.$

If

$A_{0}^{2}\delta^{(k_{0}-1)p/\omega}\frac{2b^{-1/\omega}}{1-A_{0}\delta^{p/\omega}}\left\\{\int_{\delta^{-1}B_{0}}[g(z)]^{p}\,d\mu(z)\right\\}^{1/\omega}+A_{0}r_{0}\leq\delta^{-1}r_{0},$
(3.11)

then we know that, for any $i\in\\{1,\dots,k-k_{0}\\}$,
$B(x_{k-i},r_{k-i})\subset\delta^{-1}B_{0}$. Observe that (3.11) is equivalent
to

$\delta^{1-k_{0}}\geq\left[\frac{2A_{0}^{2}}{(1-A_{0}\delta^{p/\omega})(\delta^{-1}-A_{0})}\right]^{\omega/p}(br_{0}^{\omega})^{-1/p}\left\\{\int_{\delta^{-1}B_{0}}[g(z)]^{p}\,d\mu(z)\right\\}^{1/p}.$
(3.12)

We claim that, if (3.12) holds true, then, for any $k\geq k_{0}$,
$r_{k}\leq\delta^{-1}r$. Indeed, by the definition of $r_{k}$, (3.10), (3.12),
and the fact that $\delta$ is very small, we conclude that

$\displaystyle r_{k}$ $\displaystyle\leq
2b^{-\frac{1}{\omega}}\left\\{\delta^{(k-1)p}\int_{\delta^{-1}B_{0}}[g(y)]^{p}\,d\mu(y)\right\\}^{\frac{1}{\omega}}$
$\displaystyle\leq
2b^{-\frac{1}{\omega}}\delta^{\frac{(k_{0}-1)p}{\omega}}\left\\{\int_{\delta^{-1}B_{0}}[g(y)]^{p}\,d\mu(y)\right\\}^{\frac{1}{\omega}}$
$\displaystyle\leq
2b^{-\frac{1}{\omega}}\left[\frac{2A_{0}^{2}}{(1-A_{0}\delta^{p/\omega})(\delta^{-1}-A_{0})}\right]^{-1}b^{\frac{1}{\omega}}r_{0}\left\\{\int_{\delta^{-1}B_{0}}[g(y)]^{p}\,d\mu(y)\right\\}^{-\frac{1}{\omega}}$
$\displaystyle\leq\frac{(1-A_{0}\delta^{p/\omega})(\delta^{-1}-A_{0})}{A_{0}^{2}}r_{0}$
$\displaystyle\leq\delta^{-1}r_{0}.$

Thus, the above claim holds true. Observe that (3.5) implies $E_{k}=\emptyset$
for $k\in\mathbb{Z}$ small enough. By this and (3.6), we conclude that there
exists a $\widetilde{k}_{0}\in\mathbb{Z}$ such that

$\mu(E_{\widetilde{k}_{0}-1})<\delta\mu(\delta^{-1}B_{0})\leq\mu(E_{\widetilde{k}_{0}}).$
(3.13)

From this, we deduce that $E_{\widetilde{k}_{0}}\neq\emptyset$ and, by (3.5),
we have, for any $x\in E_{\widetilde{k}_{0}}$,

$\delta^{1+1/p}\left\\{\fint_{\delta^{-1}B_{0}}[g(y)]^{p}\,d\mu(y)\right\\}^{1/p}\leq
g(x)\leq\delta^{-\widetilde{k}_{0}}.$

On the other hand, since $\delta$ is very small, we may assume that
$\delta<1/2$. Then, by (3.13) and (3.10), we know that

$\delta\mu(\delta^{-1}B_{0})<(1-\delta)\mu(\delta^{-1}B_{0})<\mu(\delta^{-1}B_{0}\setminus
E_{\widetilde{k}_{0}-1})\leq\delta^{(\widetilde{k}_{0}-1)p}\int_{\delta^{-1}B_{0}}[g(y)]^{p}\,d\mu(y).$

Combining the above two estimates, we find that

$\delta^{1+1/p}\left\\{\fint_{\delta^{-1}B_{0}}[g(y)]^{p}\,d\mu(y)\right\\}^{1/p}\leq\delta^{-\widetilde{k}_{0}}\leq\delta^{-1-1/p}\left\\{\fint_{\delta^{-1}B_{0}}[g(y)]^{p}\,d\mu(y)\right\\}^{1/p}.$

Let $l_{0}$ be the smallest integer such that

$\delta^{-l_{0}}>\max\left\\{\delta^{-2-1/p}\left[\frac{2A_{0}^{2}}{(1-A_{0}\delta^{p/\omega})(\delta^{-1}-A_{0})}\right]^{\omega/p},1\right\\}\left[\frac{\mu(\delta^{-1}B_{0})}{br_{0}^{\omega}}\right]^{1/p}$

and let $k_{0}:=\widetilde{k}_{0}+l_{0}$. Then we conclude that (3.12) holds
true and

$\displaystyle\delta^{-k_{0}}$
$\displaystyle=\delta^{-1}\delta^{-\widetilde{k}_{0}}\delta^{-(l_{0}-1)}$
$\displaystyle\leq\delta^{-1}\delta^{-1-1/p}\left\\{\fint_{\delta^{-1}B_{0}}[g(y)]^{p}\,d\mu(y)\right\\}^{1/p}$
$\displaystyle\quad\times\max\left\\{\delta^{-2-1/p}\left[\frac{2A_{0}^{2}}{(1-A_{0}\delta^{p/\omega})(\delta^{-1}-A_{0})}\right]^{\omega/p},1\right\\}\left[\frac{\mu(\delta^{-1}B_{0})}{br_{0}^{\omega}}\right]^{1/p}$
$\displaystyle\lesssim(br_{0}^{\omega})^{-1/p}\left\\{\int_{\delta^{-1}B_{0}}[g(z)]^{p}\,d\mu(z)\right\\}^{1/p},$

which, together with (3.12), implies that

$\delta^{-k_{0}}\sim(br_{0}^{\omega})^{-1/p}\left\\{\int_{\delta^{-1}B_{0}}[g(z)]^{p}\,d\mu(z)\right\\}^{1/p}.$
(3.14)

Now, we estimate $a_{k}$. We consider two cases on $k$.

Case 1) $k>k_{0}$. In this case, it suffices to consider $k>k_{0}$ such that
$E_{k}\cap B_{0}\neq\emptyset$. For any $x_{k}\in E_{k}\cap B_{0}$, choose
$\\{x_{k-1},\dots,x_{k_{0}}\\}$ as above. Then, by $g\in{\mathcal{D}}^{s}(u)$,
the definition of $r_{k}$, (3.10), and $p\in(0,\omega/s)$, we find that, for
any $c\in\mathbb{R}$,

$\displaystyle|u(x_{k})-c|$
$\displaystyle\leq\sum_{i=0}^{k-k_{0}-1}|u(x_{k-i})-u(x_{k-i-1})|+|u(x_{k_{0}})-c|$
$\displaystyle\leq\sum_{i=0}^{k-k_{0}-1}[d(x_{k-i},x_{k-i-1})]^{s}[g(x_{k-i})+g(x_{k-i-1})]+|u(x_{k_{0}})-c|$
$\displaystyle\lesssim\sum_{i=0}^{k-k_{0}-1}\delta^{-k+i}r_{k-i}^{s}+|u(x_{k_{0}})-c|$
$\displaystyle\lesssim
b^{-s/\omega}\left\\{\int_{\delta^{-1}B_{0}}[g(z)]^{p}\,d\mu(z)\right\\}^{s/\omega}\sum_{i=0}^{k-k_{0}-1}\delta^{-k+i}\delta^{(k-i-1)ps/\omega}+|u(x_{k_{0}})-c|$
$\displaystyle\lesssim
b^{-s/\omega}\left\\{\int_{\delta^{-1}B_{0}}[g(z)]^{p}\,d\mu(z)\right\\}^{s/\omega}\delta^{-(k-1)(1-ps/\omega)}+|u(x_{k_{0}})-c|,$

which implies that

$a_{k}\lesssim
b^{-s/\omega}\left\\{\int_{\delta^{-1}B_{0}}[g(z)]^{p}\,d\mu(z)\right\\}^{s/\omega}\delta^{-k(1-ps/\omega)}+\sup_{x\in
E_{k_{0}}}|u(x)-c|.$

Choose $\widetilde{c}\in\mathbb{R}$ such that $\operatorname*{ess\ inf}_{x\in
E_{k_{0}}}|u(x)-\widetilde{c}|=0$. Then we can find
$\\{y_{j}\\}_{j\in\mathbb{N}}\subset E_{k_{0}}$ such that
$\lim_{j\to\infty}|u(y_{j})-\widetilde{c}|=0$. By $g\in{\mathcal{D}}^{s}(u)$
and the definition of $E_{k_{0}}$, we have, for any $x\in E_{k_{0}}$,

$\displaystyle|u(x)-\widetilde{c}|$
$\displaystyle=\lim_{j\to\infty}|[u(x)-\widetilde{c}]-[u(y_{j})-\widetilde{c}]|=\lim_{j\to\infty}|u(x)-u(y_{j})|$
$\displaystyle\leq\varlimsup_{j\to\infty}[d(x,y_{j})]^{s}[g(x)+g(y_{j})]\leq
2^{s+1}A_{0}^{s}r_{0}^{s}\delta^{-k_{0}-s},$ (3.15)

which further implies that, for any $k>k_{0}$,

$a_{k}\lesssim
b^{-s/\omega}\left\\{\int_{\delta^{-1}B_{0}}[g(z)]^{p}\,d\mu(z)\right\\}^{s/\omega}\delta^{-k(1-ps/\omega)}+r_{0}^{s}\delta^{-k_{0}}.$
(3.16)

Case 2) $k\leq k_{0}$. In this case, by (3) and the fact that $E_{k}$ is
increasing, we know that

$a_{k}=\sup_{y\in B_{0}\cap E_{k}}|u(y)-\widetilde{c}|\leq\sup_{y\in B_{0}\cap
E_{k_{0}}}|u(y)-\widetilde{c}|\leq\sup_{y\in
E_{k_{0}}}|u(y)-\widetilde{c}|\lesssim r_{0}^{s}\delta^{-k_{0}},$ (3.17)

where we let $a_{k}:=0$ if $B_{0}\cap E_{k}=\emptyset$.

From (3.16), (3.17), (3.8), (3.7), (3.14), and (3.9), we deduce that

$\displaystyle\int_{B_{0}}|u(y)-\widetilde{c}|^{p^{\ast}}\,d\mu(y)$
$\displaystyle\leq\sum_{k=-\infty}^{\infty}a_{k}^{p^{\ast}}\mu(B_{0}\cap[E_{k}\setminus
E_{k-1}])$ $\displaystyle\lesssim
b^{-sp^{\ast}/\omega}\left\\{\int_{\delta^{-1}B_{0}}[g(z)]^{p}\,d\mu(z)\right\\}^{sp^{\ast}/\omega}\sum_{k=-\infty}^{\infty}\delta^{-k(1-ps/\omega)p^{\ast}}\mu(E_{k}\setminus
E_{k-1})$
$\displaystyle\quad+r_{0}^{sp^{\ast}}\delta^{-k_{0}p^{\ast}}\mu(B_{0})$
$\displaystyle\lesssim
b^{-sp^{\ast}/\omega}\left\\{\int_{\delta^{-1}B_{0}}[g(z)]^{p}\,d\mu(z)\right\\}^{sp^{\ast}/\omega}\left\\{\int_{\delta^{-1}B_{0}}[g(z)]^{p}\,d\mu(z)\right\\}$
$\displaystyle\quad+\mu(B_{0})r_{0}^{sp^{\ast}}(br_{0}^{\omega})^{-p^{\ast}/p}\left\\{\int_{\delta^{-1}B_{0}}[g(z)]^{p}\,d\mu(z)\right\\}^{p^{\ast}/p}$
$\displaystyle\lesssim\left[1+\frac{\mu(B_{0})}{br_{0}^{\omega}}\right]b^{-sp^{\ast}/\omega}\left\\{\int_{\delta^{-1}B_{0}}[g(z)]^{p}\,d\mu(z)\right\\}^{p^{\ast}/p}$
$\displaystyle\lesssim\frac{\mu(B_{0})}{br_{0}^{\omega}}b^{-sp^{\ast}/\omega}\left\\{\int_{\delta^{-1}B_{0}}[g(z)]^{p}\,d\mu(z)\right\\}^{p^{\ast}/p},$

which implies that

$\left[\fint_{B_{0}}|u(y)-\widetilde{c}|^{p^{\ast}}\,d\mu(y)\right]^{\frac{1}{p^{\ast}}}\lesssim(br_{0}^{\omega})^{-1/p^{\ast}}b^{-s/\omega}\left\\{\int_{\delta^{-1}B_{0}}[g(y)]^{p}\,d\mu(y)\right\\}^{1/p}.$

Recalling that
$b=(4A_{0})^{-\omega}\delta^{\omega}r_{0}^{-\omega}\mu(\delta^{-1}B_{0})$,
then we conclude that

$(br_{0}^{\omega})^{-1/p^{\ast}}b^{-s/\omega}\lesssim
r_{0}^{s}[\mu(\delta^{-1}B_{0})]^{-1/p}.$

This finishes the proof of Lemma 3.11. ∎

######  Remark 3.12.

Let $p^{\ast}$, $u$, and $B_{0}$ be as in Lemma 3.11. If
$p^{\ast}\in[1,\infty)$, then $u\in L^{1}(B_{0})$ and, moreover, the left hand
side of (3.4) can be replaced by

$\left[\fint_{B_{0}}|u(y)-u_{B_{0}}|^{p^{\ast}}\,d\mu(y)\right]^{\frac{1}{p^{\ast}}},$

where

$u_{B_{0}}:=\fint_{B_{0}}u(y)\,d\mu(y).$

Indeed, we can find a $c_{0}\in\mathbb{R}$ such that

$\left[\fint_{B_{0}}|u(y)-c_{0}|^{p^{\ast}}\,d\mu(y)\right]^{\frac{1}{p^{\ast}}}\leq
2\inf_{c\in\mathbb{R}}\left[\fint_{B_{0}}|u(y)-c|^{p^{\ast}}\,d\mu(y)\right]^{\frac{1}{p^{\ast}}}.$

By the Hölder inequality, we have

$\displaystyle\left[\fint_{B_{0}}|u(y)-u_{B_{0}}|^{p^{\ast}}\,d\mu(y)\right]^{\frac{1}{p^{\ast}}}$
$\displaystyle=\left[\fint_{B_{0}}\left|u(y)-\fint_{B_{0}}u(z)\,d\mu(z)\right|^{p^{\ast}}\,d\mu(y)\right]^{\frac{1}{p^{\ast}}}$
$\displaystyle\leq\left[\fint_{B_{0}}\fint_{B_{0}}|u(y)-u(z)|^{p^{\ast}}\,d\mu(z)\,d\mu(y)\right]^{\frac{1}{p^{\ast}}}$
$\displaystyle\lesssim\left[\fint_{B_{0}}|u(y)-c_{0}|^{p^{\ast}}\,d\mu(y)\right]^{\frac{1}{p^{\ast}}}.$

This finishes the proof of the above claim.

The next result is a consequence of Lemma 3.11 and highlights the fact that
functions in $\dot{M}^{s}_{p,q}({\mathcal{X}})$ are actually locally
integrable whenever $p\in(\omega/(\omega+s),\infty)$.

######  Corollary 3.13.

Let $s\in(0,\infty)$, $q\in(0,\infty]$, and $p\in(\omega/(\omega+s),\infty)$,
where $\omega$ as in (2.2). Then every function in
$\dot{M}^{s}_{p,q}({\mathcal{X}})$ is locally integrable on ${\mathcal{X}}$.

###### Proof.

Fix $u\in\dot{M}^{s}_{p,q}({\mathcal{X}})$ and observe that, if
$\\{g_{k}\\}_{k\in\mathbb{Z}}\in\mathbb{D}^{s}(u)$, then
$g\in{\mathcal{D}}^{s}(u)$, where

$g:=\left(\sum\limits_{k=-\infty}^{\infty}g_{k}^{q}\right)^{1/q}$

with the usual modification when $q=\infty$. Thus,
$u\in\dot{M}^{s,p}({\mathcal{X}})$. Consider any ball
$B_{0}\subset{\mathcal{X}}$ and suppose that $A_{0}\delta^{p/\omega}<1$. If we
choose $t\in(\omega/(\omega+s),p\wedge(\omega/s))$, then
$u\in\dot{M}^{s,t}(\delta^{-1}B_{0})$ and Lemma 3.11 implies $u\in
L^{t^{\ast}}(B_{0})$, where $t^{\ast}=\frac{\omega t}{\omega-st}>1$. Thus,
$u\in L^{1}(B_{0})$, which completes the proof of Corollary 3.13. ∎

The following result also follows from Lemma 3.11.

######  Corollary 3.14.

Let $s\in(0,\infty)$, $p\in(\omega/(\omega+s),\omega/s)$, and
$p^{\ast}:=\frac{\omega p}{\omega-sp}$ with $\omega$ as in (2.2). Assume that
${\mathcal{X}}$ has a weak lower bound $Q=\omega$. Then there exists a
constant $C\in\mathbb{R}$ such that, for any
$u\in\dot{M}^{s,p}({\mathcal{X}})$, $u-C\in L^{p^{\ast}}({\mathcal{X}})$ and

$\|u-C\|_{L^{p\ast}({\mathcal{X}})}\leq\widetilde{C}\|u\|_{\dot{M}^{s,p}({\mathcal{X}})},$
(3.18)

where $\widetilde{C}$ is a positive constant independent of $u$.

###### Proof.

Let $s$ and $p$ be as in this corollary. Let
$u\in\dot{M}^{s,p}({\mathcal{X}})$ and fix a point $x_{0}\in{\mathcal{X}}$.
For any $k\in\mathbb{N}$, let $B_{k}:=B(x_{0},k)$. Choose a
$g\in\mathcal{D}^{s}(u)$ such that $\|g\|_{L^{p}({\mathcal{X}})}\leq
2\|u\|_{\dot{M}^{s,p}({\mathcal{X}})}$. By Lemma 3.11, we find that, for any
$k\in\mathbb{N}$,

$\displaystyle\left[\fint_{B_{k}}|u(y)-u_{B_{k}}|^{p^{\ast}}\,d\mu(y)\right]^{\frac{1}{p^{\ast}}}\lesssim
k^{s}\left\\{\fint_{\delta^{-1}B_{k}}[g(y)]^{p}\,d\mu(y)\right\\}^{1/p}\lesssim
k^{s}[\mu(B_{k})]^{-1/p}\|u\|_{\dot{M}^{s,p}({\mathcal{X}})}.$

This, together with the assumption that ${\mathcal{X}}$ has a weak lower bound
$Q=\omega$, and Proposition 2.15, implies that

$\displaystyle\left[\int_{B_{k}}|u(y)-u_{B_{k}}|^{p^{\ast}}\,d\mu(y)\right]^{\frac{1}{p^{\ast}}}$
$\displaystyle\lesssim
k^{s}[\mu(B_{k})]^{1/p^{\ast}-1/p}\|u\|_{\dot{M}^{s,p}({\mathcal{X}})}$
$\displaystyle\lesssim
k^{s(1-Q/\omega)}\|u\|_{\dot{M}^{s,p}({\mathcal{X}})}\lesssim\|u\|_{\dot{M}^{s,p}({\mathcal{X}})}.$
(3.19)

By this, we find that, for any $k\in\mathbb{N}$,

$\displaystyle\left|u_{B_{k}}-u_{B_{1}}\right|$
$\displaystyle\leq\frac{1}{\mu(B_{1})}\int_{B_{1}}\left|u(y)-u_{B_{k}}\right|\,d\mu(y)$
$\displaystyle\leq\frac{1}{[\mu(B_{1})]^{1/p^{\ast}}}\left[\int_{B_{1}}\left|u(y)-u_{B_{k}}\right|^{p^{\ast}}\,d\mu(y)\right]^{\frac{1}{p^{\ast}}}\leq\frac{1}{[\mu(B_{1})]^{1/p^{\ast}}}\|u\|_{\dot{M}^{s,p}({\mathcal{X}})},$

which implies that $\\{u_{B_{k}}\\}_{k\in\mathbb{N}}\subset\mathbb{R}$ is a
bounded sequence. From this, we deduce that there exist a subsequence
$\\{u_{B_{k_{j}}}\\}_{j\in\mathbb{N}}$ and a constant $C\in\mathbb{R}$ such
that $C=\lim_{j\to\infty}u_{B_{k_{j}}}$. Moreover, by (3) and the Fatou lemma,
we further conclude that

$\displaystyle\left[\int_{{\mathcal{X}}}|u(x)-C|^{p^{\ast}}\,d\mu(x)\right]^{\frac{1}{p^{\ast}}}$
$\displaystyle=\left[\int_{{\mathcal{X}}}\lim_{j\to\infty}\left|[u(x)-u_{B_{k}}]\mathbf{1}_{B_{k_{j}}}(x)\right|^{p^{\ast}}\,d\mu(x)\right]^{\frac{1}{p^{\ast}}}$
$\displaystyle\leq\varliminf_{j\to\infty}\left[\int_{B_{k_{j}}}\left|u(x)-u_{B_{k_{j}}}\right|^{p^{\ast}}\,d\mu(x)\right]^{\frac{1}{p^{\ast}}}\lesssim\|u\|_{\dot{M}^{s,p}({\mathcal{X}})}.$

This finishes the proof of Corollary 3.14. ∎

######  Remark 3.15.

Let $\omega$ be as in (2.2) and $p\in(0,\omega)$. In [1, Theorem 22], Alvarado
et al. proved that, if ${\mathcal{X}}$ is uniformly perfect (see [1, (39)]),
then that (3.18) holds true with $s=1$ is equivalent to that ${\mathcal{X}}$
has a lower bound.

The following lemma is a Poincaré type inequality for $\mathbb{D}^{s}(u)$ (see
also [36, Lemma 2.1]).

######  Lemma 3.16.

Let $s\in(0,\infty)$. Then there exists a positive constant $C$ such that, for
any $k\in\mathbb{Z}$, any measurable function $u$ on ${\mathcal{X}}$,
$x\in{\mathcal{X}}$, and $\\{g_{j}\\}_{j\in\mathbb{Z}}\in\mathbb{D}^{s}(u)$,

$\inf_{c\in\mathbb{R}}\fint_{B(x,\delta^{k})}|u(y)-c|\,d\mu(y)\leq
C\delta^{ks}\sum_{j=k-3}^{k-1}\fint_{B(x,\delta^{k-2})}g_{j}(y)\,d\mu(y).$
(3.20)

###### Proof.

Let $s$, $u$, and $\\{g_{j}\\}_{j\in\mathbb{Z}}$ be as in this lemma. Observe
that, for any $k\in\mathbb{Z}$ and $x\in{\mathcal{X}}$,

$\displaystyle\inf_{c\in\mathbb{R}}\fint_{B(x,\delta^{k})}|u(y)-c|\,d\mu(y)$
$\displaystyle\leq\fint_{B(x,\delta^{k})}|u(y)-u_{B(x,\delta^{k-2})\setminus
B(x,A_{0}\delta^{k-1})}|\,d\mu(y)$
$\displaystyle\leq\fint_{B(x,\delta^{k})}\fint_{B(x,\delta^{k-2})\setminus
B(x,A_{0}\delta^{k-1})}|u(y)-u(z)|\,d\mu(y)\mu(z).$ (3.21)

Note that, due to the fact that $\delta$ is very small, we have, for any $y\in
B(x,\delta^{k})$ and $z\in B(x,\delta^{k-2})\setminus B(x,A_{0}\delta^{k-1})$,

$d(y,z)\leq A_{0}[d(y,x)+d(x,z)]<2A_{0}\delta^{k-2}\leq\delta^{k-3}$

and

$d(x,z)\leq A_{0}[d(x,y)+d(y,z)]<A_{0}\delta^{k}+A_{0}d(y,z),$

which implies that

$\delta^{k}\leq d(y,z)<\delta^{k-3}.$

From this, we deduce that there exists a unique $j_{0}\in\\{k-1,k-2,k-3\\}$
such that

$\delta^{j_{0}+1}\leq d(y,z)<\delta^{j_{0}}$

and hence

$|u(y)-u(z)|\leq[d(y,z)]^{s}[g_{j_{0}}(y)+g_{j_{0}}(z)]\leq\delta^{(k-3)s}\sum_{j=k-3}^{k-1}[g_{j}(y)+g_{j}(z)].$

Therefore, by this and (3), we conclude that

$\displaystyle\inf_{c\in\mathbb{R}}\fint_{B(x,\delta^{k})}|u(y)-c|\,d\mu(y)$
$\displaystyle\quad\lesssim\delta^{ks}\sum_{j=k-3}^{k-1}\fint_{B(x,\delta^{k})}\fint_{B(x,\delta^{k-2})\setminus
B(x,A_{0}\delta^{k-1})}[g_{j}(y)+g_{j}(z)]\,d\mu(y)\mu(z)$
$\displaystyle\quad\lesssim\delta^{ks}\sum_{j=k-3}^{k-1}\fint_{B(x,\delta^{k-2})}g_{j}(y)\,d\mu(y),$

which completes the proof of Lemma 3.16. ∎

######  Remark 3.17.

Similarly to Remark 3.12, under the assumptions same as in Lemma 3.16, the
left hand side of (3.20) can be replaced by

$\fint_{B(x,\delta^{k})}|u(y)-u_{B(x,\delta^{k})}|\,d\mu(y).$

Using Lemma 3.11, we can show the following Poincaré type inequality, which is
very useful in the case when $p\in(0,1]$.

######  Lemma 3.18.

Let $s\in(0,\infty)$, $p\in(0,1]$, and $\varepsilon,\
\varepsilon^{\prime}\in(0,s)$ with $\varepsilon<\varepsilon^{\prime}$. If
$A_{0}\delta^{p/\omega}<1$, then there exists a positive constant $C$ such
that, for any $k\in\mathbb{Z}$, $x\in{\mathcal{X}}$, any measurable function
$u$, and $\\{g_{j}\\}_{j\in\mathbb{Z}}\in\mathbb{D}^{s}(u)$,

$\displaystyle{}\inf_{c\in\mathbb{R}}\left[\fint_{B(x,\delta^{k})}|u(y)-c|^{\frac{\omega
p}{\omega-\varepsilon p}}\,d\mu(y)\right]^{\frac{\omega-\varepsilon p}{\omega
p}}$ $\displaystyle\quad\leq
C\delta^{k\varepsilon^{\prime}}\sum_{j=k-2}^{\infty}\delta^{j(s-\varepsilon^{\prime})}\left\\{\fint_{B(x,\delta^{k-1})}[g_{j}(y)]^{p}\,d\mu(y)\right\\}^{\frac{1}{p}}.$
(3.22)

###### Proof.

Let all the notation be the same as in this lemma. Without loss of generality,
we may assume that the right hand side of (3.18) is less than infinity. For
any $k\in\mathbb{Z}$ and $x\in{\mathcal{X}}$, let

$g(x):=\left\\{\sum_{j=k-2}^{\infty}\delta^{j(s-\varepsilon)p}[g_{j}(x)]^{p}\right\\}^{\frac{1}{p}}.$

We claim that $g\in{\mathcal{D}}^{s}(u)$ and
$u\in\dot{M}^{\varepsilon,p}(B(x,\delta^{k-1}))$. Indeed, for any $y,\ z\in
B(x,\delta^{k-1})$, we have

$d(y,z)\leq A_{0}[d(y,x)+d(x,z)]<2A_{0}\delta^{k-1}<\delta^{k-2}.$

Therefore, there exists a unique integer $j_{0}\geq k-2$ such that

$\delta^{j_{0}+1}\leq d(y,z)<\delta^{j_{0}}.$

Then, since $\\{g_{j}\\}_{j\in\mathbb{Z}}\in\mathbb{D}^{s}(u)$, it follows
that there exists an $E\subset{\mathcal{X}}$ with $\mu(E)=0$ such that, for
any $y,\ z\in B(x,\delta^{k-1})\setminus E$,

$\displaystyle|u(y)-u(z)|$
$\displaystyle\leq[d(y,z)]^{s}[g_{j_{0}}(y)+g_{j_{0}}(z)]\leq[d(y,z)]^{\varepsilon}\delta^{j(s-\varepsilon)}[g_{j_{0}}(y)+g_{j_{0}}(z)]$
$\displaystyle\leq[d(y,z)]^{\varepsilon}[g(y)+g(z)],$

which implies that $g\in{\mathcal{D}}^{\varepsilon}(u)$. On the other hand, by
$p\in(0,1]$, the Hölder inequality with exponent $1/p\geq 1$, and
$0<\varepsilon<\varepsilon^{\prime}<s$, we know that

$\displaystyle\|g\|_{L^{p}(B(x,\delta^{k-1}))}$
$\displaystyle=\left\\{\int_{B(x,\delta^{k-1})}\sum_{j=k-2}^{\infty}\delta^{j(s-\varepsilon)p}[g_{j}(y)]^{p}\,d\mu(y)\right\\}^{\frac{1}{p}}$
$\displaystyle=\left\\{\sum_{j=k-2}^{\infty}\delta^{j(\varepsilon^{\prime}-\varepsilon)p}\delta^{j(s-\varepsilon^{\prime})p}\int_{B(x,\delta^{k-1})}[g_{j}(y)]^{p}\,d\mu(y)\right\\}^{\frac{1}{p}}$
$\displaystyle\leq\left[\sum_{j=k-2}^{\infty}\delta^{j(\varepsilon^{\prime}-\varepsilon)p(1/p)^{\prime}}\right]^{\frac{1}{(1/p)^{\prime}p}}\sum_{j=k-2}^{\infty}\delta^{j(s-\varepsilon^{\prime})}\left\\{\int_{B(x,\delta^{k-1})}[g_{j}(y)]^{p}\,d\mu(y)\right\\}^{\frac{1}{p}}$
$\displaystyle\lesssim\delta^{k(\varepsilon^{\prime}-\varepsilon)}\left[V_{\delta^{k-1}}(x)\right]^{1/p}\sum_{j=k-2}^{\infty}\delta^{j(s-\varepsilon^{\prime})}\left\\{\fint_{B(x,\delta^{k-1})}[g_{j}(y)]^{p}\,d\mu(y)\right\\}^{\frac{1}{p}}<\infty.$

From this, we deduce that $u\in\dot{M}^{\varepsilon,p}(B(x,\delta^{k-1}))$.
Combining the above claim and Lemma 3.11, and using the Hölder inequality with
exponent $1/p\geq 1$, we find that

$\displaystyle\inf_{c\in\mathbb{R}}\left[\fint_{B(x,\delta^{k})}|u(y)-c|^{\frac{\omega
p}{\omega-\varepsilon p}}\,d\mu(y)\right]^{\frac{\omega-\varepsilon p}{\omega
p}}$
$\displaystyle\quad\lesssim\delta^{k\varepsilon}\left\\{\fint_{B(x,\delta^{k-1})}[g(y)]^{p}\,d\mu(y)\right\\}^{1/p}$
$\displaystyle\quad\sim\delta^{k\varepsilon}\left\\{\sum_{j=k-2}^{\infty}\delta^{j(s-\varepsilon)p}\fint_{B(x,\delta^{k-1})}[g_{j}(y)]^{p}\,d\mu(y)\right\\}^{1/p}$
$\displaystyle\quad\lesssim\delta^{k\varepsilon}\left\\{\sum_{j=k-2}^{\infty}\delta^{j(\varepsilon^{\prime}-\varepsilon)p(1/p)^{\prime}}\right\\}^{\frac{1}{(1/p)^{\prime}p}}\sum_{j=k-2}^{\infty}\delta^{j(s-\varepsilon^{\prime})}\left\\{\fint_{B(x,\delta^{k-1})}[g_{j}(y)]^{p}\,d\mu(y)\right\\}^{\frac{1}{p}}$
$\displaystyle\quad\lesssim\delta^{k\varepsilon^{\prime}}\sum_{j=k-2}^{\infty}\delta^{j(s-\varepsilon^{\prime})}\left\\{\fint_{B(x\delta^{k-1})}[g_{j}(y)]^{p}\,d\mu(y)\right\\}^{\frac{1}{p}}.$

This finishes the proof of Lemma 3.18. ∎

The following lemma illustrates that any element of
${\mathcal{A}}\dot{F}^{s}_{p,\infty}({\mathcal{X}})$ is a locally integrable
function.

######  Lemma 3.19.

Let $\beta,\ \gamma\in(0,\eta)$ with $\eta$ as in Definition 2.5,
$s\in(0,\beta\wedge\gamma)$, and $p\in(\omega/(\omega+s),\infty)$ with
$\omega$ as in (2.2). Assume that the measure $\mu$ of ${\mathcal{X}}$ has a
weak lower bound $Q=\omega$. Then, for any
$f\in{\mathcal{A}}\dot{F}^{s}_{p,\infty}({\mathcal{X}})$, there exists an
$\widetilde{f}\in L^{1}_{{\mathop{\mathrm{missing}}{\,loc\,}}}({\mathcal{X}})$
such that $f=\widetilde{f}$ in
$(\mathring{\mathcal{G}}_{0}^{\eta}(\beta,\gamma))^{\prime}.$

To prove Lemma 3.19, we need the notion of approximations of the identity with
exponential decay and integration 1; see [23, Definition 2.8] for more
details.

######  Definition 3.20.

Let $\eta\in(0,1)$ be as in Definition 2.5. A sequence
$\\{Q_{k}\\}_{k\in\mathbb{Z}}$ of bounded linear integral operators on
$L^{2}({\mathcal{X}})$ is called an _approximation of the identity with
exponential decay and integration 1_ (for short, 1-exp-ATI) if
$\\{Q_{k}\\}_{k\in\mathbb{Z}}$ has the following properties:

  1. (i)

for any $k\in\mathbb{Z}$, $Q_{k}$ satisfies (ii), (iii), and (iv) of
Definition 2.5 but without the decay factor

$\exp\left\\{-\nu\left[\frac{\max\\{d(x,{\mathcal{Y}}^{k}),d(y,{\mathcal{Y}}^{k})\\}}{\delta^{k}}\right]^{a}\right\\};$

  2. (ii)

for any $k\in\mathbb{Z}$ and $x\in{\mathcal{X}}$,

$\int_{\mathcal{X}}Q_{k}(x,y)\,d\mu(y)=1=\int_{\mathcal{X}}Q_{k}(y,x)\,d\mu(y);$

  3. (iii)

letting, for any $k\in\mathbb{Z}$, $P_{k}:=Q_{k}-Q_{k-1}$, then
$\\{P_{k}\\}_{k\in\mathbb{Z}}$ is an exp-ATI.

######  Remark 3.21.

As was pointed out in [23, Remark 2.9], the existence of the 1-exp-ATI is
guaranteed by [4, Lemma 10.1]. Moreover, by the proofs of [24, Proposition
2.9] and [20, Proposition 2.7(iv)], we know that, if
$\\{Q_{k}\\}_{k\in\mathbb{Z}}$ is a 1-exp-ATI, then, for any $f\in
L^{2}({\mathcal{X}})$, $\lim_{k\to\infty}Q_{k}f=f$ in $L^{2}({\mathcal{X}})$.

By an argument similar to that used in the proof of [24, Proposition 2.10], we
have the following conclusion; we omit the details.

######  Lemma 3.22.

Let $\beta,\ \gamma\in(0,\eta)$ with $\eta$ as in Definition 2.5,
$s\in(0,\beta\wedge\gamma)$, $k\in\mathbb{Z}$, $x$, $y\in{\mathcal{X}}$, and
$\\{Q_{k}\\}_{k\in\mathbb{Z}}$ be a 1-exp-ATI. For any $z\in{\mathcal{X}}$,
let

$\phi(z):=\delta^{ks}[d(x,y)]^{-s}[Q_{k}(x,z)-Q_{k}(y,z)].$

If $d(x,y)\in(0,\delta^{k}]$, then $\phi\in{\mathcal{F}}_{k}(x)$, where
${\mathcal{F}}_{k}(x)$ is as in Definition 3.1.

Next, we prove Lemma 3.19.

###### Proof of Lemma 3.19.

Assume that $f\in{\mathcal{A}}\dot{F}^{s}_{p,\infty}({\mathcal{X}})$ and
$\\{Q_{k}\\}_{k\in\mathbb{Z}}$ is an 1-exp-ATI. For any $x\in{\mathcal{X}}$,
let

$g(x):=\sup_{k\in\mathbb{Z}}\delta^{-ks}\sup_{\phi\in{\mathcal{F}}_{k}(x)}|\langle
f,\phi\rangle|.$

First, for any $k\in\mathbb{Z}$, $i\in\mathbb{N}$, and $x\in{\mathcal{X}}$, we
have

$\displaystyle|Q_{k}f(x)-Q_{k+i}f(x)|\leq\sum_{j=0}^{i-1}|Q_{k+j}f(x)-Q_{k+j+1}f(x)|=\sum_{j=0}^{i-1}|\langle
f,Q_{k+j}(x,\cdot)-Q_{k+j+1}(x,\cdot)\rangle|.$

Note that, by (2.6) and (2.7), we obtain, for any $k\in\mathbb{Z}$, and
$x\in{\mathcal{X}}$,

$Q_{k+j}(x,\cdot)-Q_{k+j+1}(x,\cdot)\in{\mathcal{F}}_{k+j+1}(x).$

From this, we deduce that

$\displaystyle|Q_{k}f(x)-Q_{k+i}f(x)|\leq\sum_{j=0}^{i-1}\delta^{(k+j+1)s}g(x)\lesssim\delta^{ks}g(x).$
(3.23)

We now consider the case $p\in(1,\infty)$ first. To this end, note that (3.23)
implies that $\\{Q_{k}f-Q_{k+i}f\\}_{i\in\mathbb{N}}$ is a Cauchy sequence in
$L^{p}({\mathcal{X}})$. By the completeness of $L^{p}({\mathcal{X}})$, we find
that there exists an $f_{k}\in L^{p}({\mathcal{X}})$ such that

$\lim_{i\to\infty}(Q_{k}f-Q_{k+i}f)=f_{k}$

both in $L^{p}({\mathcal{X}})$ and pointwise. Observe that, for any $k,\
k^{\prime}\in\mathbb{Z}$, we have

$\displaystyle f_{k}$
$\displaystyle=\lim_{i\to\infty}(Q_{k}f-Q_{k+i}f)=Q_{k}f-Q_{k^{\prime}}f+\lim_{i\to\infty}(Q_{k^{\prime}}f-Q_{k+i}f)$
$\displaystyle=Q_{k}f-Q_{k^{\prime}}f+f_{k^{\prime}}$

both in $L^{p}({\mathcal{X}})$ and pointwise. Let
$\widetilde{f}:=Q_{0}f-f_{0}$. Then, by this, we conclude that
$\widetilde{f}\in L^{1}_{{\mathop{\mathrm{missing}}{\,loc\,}}}({\mathcal{X}})$
and, for any $k\in\mathbb{Z}$, $\widetilde{f}=Q_{k}f-f_{k}$. On the other
hand, since $Q_{k}f\to f$ in
$(\mathring{\mathcal{G}}_{0}^{\eta}(\beta,\gamma))^{\prime}$ as $k\to\infty$,
it follows that, for any
$\psi\in\mathring{\mathcal{G}}_{0}^{\eta}(\beta,\gamma)$,

$\displaystyle\langle\widetilde{f},\psi\rangle$
$\displaystyle=\int_{{\mathcal{X}}}\widetilde{f}(x)\psi(x)\,d\mu(x)$
$\displaystyle=\int_{{\mathcal{X}}}\left\\{Q_{0}f(x)-\lim_{i\to\infty}[Q_{0}f(x)-Q_{i}f(x)]\right\\}\psi(x)\,d\mu(x)$
$\displaystyle=\int_{{\mathcal{X}}}Q_{0}f(x)\psi(x)\,d\mu(x)-\lim_{i\to\infty}\int_{{\mathcal{X}}}[Q_{0}f(x)-Q_{i}f(x)]\psi(x)\,d\mu(x)$
$\displaystyle=\lim_{i\to\infty}\int_{{\mathcal{X}}}Q_{i}f(x)\psi(x)\,d\mu(x)=\lim_{i\to\infty}\left\langle
Q_{i}f,\psi\right\rangle=\langle f,\psi\rangle,$

which completes the proof of this lemma in the case when $p\in(1,\infty)$.

Next, we consider the case $p\in(\omega/(\omega+s),1]$. For any $x,\
y\in{\mathcal{X}}$, let $k_{0}\in\mathbb{Z}$ be such that
$\delta^{k_{0}+1}<d(x,y)\leq\delta^{k_{0}}$. Then, by Lemma 3.22 and (3.23),
we know that, for any $k\in\mathbb{Z}$ with $k>k_{0}$,

$\displaystyle|Q_{k}f(x)-Q_{k}f(y)|$
$\displaystyle\leq|Q_{k}f(x)-Q_{k_{0}}f(x)|+|Q_{k_{0}}f(x)-Q_{k_{0}}f(y)|{}$
$\displaystyle\quad+|Q_{k_{0}}f(y)-Q_{k}f(y)|{}$
$\displaystyle\lesssim\delta^{k_{0}s}[g(x)+g(y)]\lesssim[d(x,y)]^{s}[g(x)+g(y)].$
(3.24)

On the other hand, we find that, for any $k\in\mathbb{Z}$ with $k\leq k_{0}$,

$|Q_{k}f(x)-Q_{k}f(y)|=[d(x,y)]^{s}\delta^{-ks}\left\langle
f,\delta^{ks}[d(x,y)]^{-s}[Q_{k}(x,\cdot)-Q_{k}(y,\cdot)]\right\rangle.$

Note that, due to $k\leq k_{0}$, then $d(x,y)\leq\delta^{k}$. From this and
Lemma 3.22, we deduce that

$\delta^{ks}[d(x,y)]^{-s}[Q_{k}(x,\cdot)-Q_{k}(y,\cdot)]\in{\mathcal{F}}_{k}(x),$

which implies that

$|Q_{k}f(x)-Q_{k}f(y)|\lesssim[d(x,y)]^{s}[g(x)+g(y)].$ (3.25)

Combining (3) and (3.25), we know that, for any $k\in\mathbb{Z}$,

$|Q_{k}f(x)-Q_{k}f(y)|\lesssim[d(x,y)]^{s}[g(x)+g(y)].$ (3.26)

Moreover, by $f\in{\mathcal{A}}\dot{F}^{s}_{p,\infty}({\mathcal{X}})$, we know
that $g\in L^{p}({\mathcal{X}})$. From this and (3.26), we further deduce
that, for any $k\in\mathbb{Z}$,

$Q_{k}f\in\dot{M}^{s,p}({\mathcal{X}})\quad\text{and}\quad\|Q_{k}f\|_{\dot{M}^{s,p}({\mathcal{X}})}\lesssim\|g\|_{L^{p}({\mathcal{X}})}\sim\|f\|_{{\mathcal{A}}\dot{F}^{s}_{p,\infty}({\mathcal{X}})}.$

By this and Corollary 3.14, we find that, for any $k\in\mathbb{Z}$, there
exists a constant $C_{k}$ such that

$Q_{k}f-C_{k}\in
L^{p^{\ast}}({\mathcal{X}})\quad\text{and}\quad\|Q_{k}f-C_{k}\|_{L^{p^{\ast}}({\mathcal{X}})}\lesssim\|Q_{k}f\|_{\dot{M}^{s,p}({\mathcal{X}})}\lesssim\|f\|_{{\mathcal{A}}\dot{F}^{s}_{p,\infty}({\mathcal{X}})}.$

From this and the weak compactness property of $L^{p^{\ast}}({\mathcal{X}})$
(recall that $p^{\ast}>1$ in this case), we deduce that there exist a
subsequence $\\{Q_{k_{j}}f-C_{k_{j}}\\}_{j\in\mathbb{N}}$ and a function
$\widetilde{f}\in L^{p^{\ast}}({\mathcal{X}})$ such that

$\widetilde{f}=\lim_{j\to\infty}[Q_{k_{j}}f-C_{k_{j}}]$

both weakly in $L^{p^{\ast}}({\mathcal{X}})$ and also in
$(\mathring{\mathcal{G}}_{0}^{\eta}(\beta,\gamma))^{\prime}$. Besides, by
(3.23), we find that, for any $j,\ j^{\prime}\in\mathbb{N}$,
$Q_{k_{j}}f-Q_{k_{j^{\prime}}}f\in L^{p}({\mathcal{X}})$ and
$[Q_{k_{j}}f-C_{k_{j}}]-[Q_{k_{j^{\prime}}}f-C_{k_{j^{\prime}}}]\in
L^{p^{\ast}}({\mathcal{X}})$, which further implies that
$C_{k_{j}}=C_{k_{j^{\prime}}}$. By the fact that $Q_{k_{j}}f\to f$ in
$(\mathring{\mathcal{G}}_{0}^{\eta}(\beta,\gamma))^{\prime}$ as $j\to\infty$,
we conclude that

$f=f-C_{k_{0}}=\lim_{j\to\infty}[Q_{k_{j}}f-C_{k_{j}}]=\widetilde{f}$

in $(\mathring{\mathcal{G}}_{0}^{\eta}(\beta,\gamma))^{\prime}$. This finishes
the proof of Lemma 3.19. ∎

We now present the proof of Theorem 3.10.

###### Proof of Theorem 3.10.

By similarity, we only prove (i). We first show
$\dot{M}^{s}_{p,q}({\mathcal{X}})\subset{\mathcal{A}}\dot{F}^{s}_{p,q}(\mathcal{X})$.
To this end, fix $u\in\dot{M}^{s}_{p,q}({\mathcal{X}})$ and recall that $u$ is
locally integrable on ${\mathcal{X}}$ by Corollary 3.13. With this in mind, we
consider five cases on $p$ and $q$.

Case 1) $p\in(1,\infty)$ and $q\in(1,\infty]$. In this case, we only consider
the case $q\in(1,\infty)$, because the proof of the case $q=\infty$ is similar
to that of $q\in(1,\infty)$. Choose
$\\{g_{k}\\}_{k\in\mathbb{Z}}\in\mathbb{D}^{s}(u)$ such that

$\left\|\left(\sum_{k=-\infty}^{\infty}g_{k}^{q}\right)^{\frac{1}{q}}\right\|_{L^{p}({\mathcal{X}})}\lesssim\|u\|_{\dot{M}^{s}_{p,q}({\mathcal{X}})}.$

Then we have, for any $k\in\mathbb{Z}$, $x\in{\mathcal{X}}$, and
$\phi\in{\mathcal{F}}_{k}(x)$,

$\displaystyle|\langle u,\phi\rangle|$
$\displaystyle=\left|\int_{{\mathcal{X}}}\phi(y)u(y)\,d\mu(y)\right|=\left|\int_{{\mathcal{X}}}\phi(y)\left[u(y)-u_{B(x,\delta^{k})}\right]\,d\mu(y)\right|$
$\displaystyle\leq\int_{{\mathcal{X}}}D_{\gamma}\left(x,y;\delta^{k}\right)\left|u(y)-u_{B(x,\delta^{k})}\right|\,d\mu(y)$
$\displaystyle\lesssim\sum_{j=0}^{\infty}\delta^{j\gamma}\fint_{B(x,\delta^{k-j})}\left|u(y)-u_{B(x,\delta^{k})}\right|\,d\mu(y),$

where $D_{\gamma}(x,y;\delta^{k})$ is as in (2.3). Notice that, for any
$k\in\mathbb{Z}$, $j\in\mathbb{Z}_{+}$, and $x\in{\mathcal{X}}$,

$\displaystyle\fint_{B(x,\delta^{k-j})}\left|u(y)-u_{B(x,\delta^{k})}\right|\,d\mu(y)$
$\displaystyle\quad=\fint_{B(x,\delta^{k-j})}\left|u(y)-u_{B(x,\delta^{k-1})}+u_{B(x,\delta^{k-1})}-u_{B(x,\delta^{k})}\right|\,d\mu(y)$
$\displaystyle\quad\leq\fint_{B(x,\delta^{k-j})}\left|u(y)-u_{B(x,\delta^{k-1})}\right|\,d\mu(y)+\frac{1}{V_{\delta^{k}}(x)}\int_{B(x,\delta^{k})}\left|u(y)-u_{B(x,\delta^{k-1})}\right|\,d\mu(y)$
$\displaystyle\quad\lesssim\fint_{B(x,\delta^{k-j})}\left|u(y)-u_{B(x,\delta^{k-1})}\right|\,d\mu(y)+\fint_{B(x,\delta^{k-1})}\left|u(y)-u_{B(x,\delta^{k-1})}\right|\,d\mu(y)$
$\displaystyle\quad\lesssim\sum_{i=0}^{j}\fint_{B(x,\delta^{k-i})}\left|u(y)-u_{B(x,\delta^{k-i})}\right|\,d\mu(y).$

Combining the above two inequalities, we obtain, for any $k\in\mathbb{Z}$,

$\displaystyle|\langle u,\phi\rangle|$
$\displaystyle\lesssim\sum_{j=0}^{\infty}\delta^{j\gamma}\sum_{i=0}^{j}\fint_{B(x,\delta^{k-i})}\left|u(y)-u_{B(x,\delta^{k-i})}\right|\,d\mu(y)$
$\displaystyle\sim\sum_{i=0}^{\infty}\sum_{j=i}^{\infty}\delta^{j\gamma}\fint_{B(x,\delta^{k-i})}\left|u(y)-u_{B(x,\delta^{k-i})}\right|\,d\mu(y)$
$\displaystyle\lesssim\sum_{i=0}^{\infty}\delta^{i\gamma}\fint_{B(x,\delta^{k-i})}\left|u(y)-u_{B(x,\delta^{k-i})}\right|\,d\mu(y).$
(3.27)

From this and Lemma 3.16 (see also Remark 3.17), we deduce that, for any
$k\in\mathbb{Z}$,

$\displaystyle|\langle u,\phi\rangle|$
$\displaystyle\lesssim\sum_{i=0}^{\infty}\delta^{i\gamma}\delta^{(k-i)s}\sum_{j=k-i-3}^{k-i-1}\fint_{B(x,\delta^{k-i-2})}g_{j}(y)\,d\mu(y)$
$\displaystyle\lesssim\delta^{ks}\sum_{j=-\infty}^{k}\sum_{i=k-j-3}^{k-j-1}\delta^{i(\gamma-s)}\fint_{B(x,\delta^{k-i-2})}g_{j}(y)\,d\mu(y)$
$\displaystyle\lesssim\delta^{k\gamma}\sum_{j=-\infty}^{k}\delta^{-j(\gamma-s)}\fint_{B(x,\delta^{j-2})}g_{j}(y)\,d\mu(y)\lesssim\delta^{k\gamma}\sum_{j=-\infty}^{k}\delta^{-j(\gamma-s)}M(g_{j})(x).$
(3.28)

By this, the Hölder inequality, and Lemma 3.9, we conclude that

$\displaystyle\|u\|_{{\mathcal{A}}\dot{F}^{s}_{p,q}(\mathcal{X})}$
$\displaystyle\lesssim\left\|\left\\{\sum_{k=-\infty}^{\infty}\delta^{-ksq}\left[\delta^{k\gamma}\sum_{j=-\infty}^{k}\delta^{-j(\gamma-s)}M(g_{j})\right]^{q}\right\\}^{1/q}\right\|_{L^{p}({\mathcal{X}})}$
$\displaystyle\lesssim\left\|\left\\{\sum_{k=-\infty}^{\infty}\delta^{k(\gamma-s)}\sum_{j=-\infty}^{k}\delta^{-j(\gamma-s)}[M(g_{j})]^{q}\right\\}^{1/q}\right\|_{L^{p}({\mathcal{X}})}$
$\displaystyle\lesssim\left\|\left\\{\sum_{j=-\infty}^{\infty}[M(g_{j})]^{q}\right\\}^{1/q}\right\|_{L^{p}({\mathcal{X}})}\lesssim\left\|\left\\{\sum_{j=-\infty}^{\infty}g_{j}^{q}\right\\}^{1/q}\right\|_{L^{p}({\mathcal{X}})}\lesssim\|u\|_{\dot{M}^{s}_{p,q}({\mathcal{X}})},$

which is the desired estimate in this case.

Case 2) $p\in(1,\infty)$ and $q\in(\omega/(\omega+s),1]$. In this case, choose
$\\{g_{k}\\}_{k\in\mathbb{Z}}\in\mathbb{D}^{s}(u)$ as in Case 1). Recall that
we may assume $A_{0}\delta^{p/\omega}<1$. As such, combining (3) and Lemma
3.18 [with $p=\omega/(\omega+\varepsilon)<1$], we find that for any fixed
$\varepsilon,\ \varepsilon^{\prime}\in(0,s)$ with
$\varepsilon<\varepsilon^{\prime}$, there holds true

$\displaystyle|\langle u,\phi\rangle|$
$\displaystyle\lesssim\sum_{i=0}^{\infty}\delta^{i\gamma}\delta^{(k-i)\varepsilon^{\prime}}\sum_{j=k-i-2}^{\infty}\delta^{j(s-\varepsilon^{\prime})}\left\\{\fint_{B(x,\delta^{k-i-1})}[g_{j}(y)]^{\frac{\omega}{\omega+\varepsilon}}\,d\mu(y)\right\\}^{\frac{\omega+\varepsilon}{\omega}}$
$\displaystyle\lesssim\delta^{k\varepsilon^{\prime}}\sum_{j=-\infty}^{k-2}\delta^{j(s-\varepsilon^{\prime})}\left[M\left([g_{j}(x)]^{\frac{\omega}{\omega+\varepsilon}}\right)\right]^{\frac{\omega+\varepsilon}{\omega}}\sum_{i=k-j-2}^{\infty}\delta^{i(\gamma-\varepsilon^{\prime})}$
$\displaystyle\quad+\delta^{k\varepsilon^{\prime}}\sum_{j=k-1}^{\infty}\delta^{j(s-\varepsilon^{\prime})}\left[M\left([g_{j}(x)]^{\frac{\omega}{\omega+\varepsilon}}\right)\right]^{\frac{\omega+\varepsilon}{\omega}}\sum_{i=0}^{\infty}\delta^{i(\gamma-\varepsilon^{\prime})}$
$\displaystyle\lesssim\delta^{k\gamma}\sum_{j=-\infty}^{k-2}\delta^{j(s-\gamma)}\left[M\left([g_{j}(x)]^{\frac{\omega}{\omega+\varepsilon}}\right)\right]^{\frac{\omega+\varepsilon}{\omega}}+\delta^{k\varepsilon^{\prime}}\sum_{j=k-1}^{\infty}\delta^{j(s-\varepsilon^{\prime})}\left[M\left([g_{j}(x)]^{\frac{\omega}{\omega+\varepsilon}}\right)\right]^{\frac{\omega+\varepsilon}{\omega}}.$
(3.29)

Choosing $\varepsilon\in(0,s)$ such that
$\frac{\omega}{\omega+s}<\frac{\omega}{\omega+\varepsilon}<q$, and using (3),
(3.1) (with $\theta=q$), and Lemma 3.9, we conclude that

$\displaystyle\|u\|_{{\mathcal{A}}\dot{F}^{s}_{p,q}(\mathcal{X})}$
$\displaystyle\lesssim\left\|\left(\sum_{k=-\infty}^{\infty}\delta^{-ksq}\left\\{\delta^{k\gamma}\sum_{j=-\infty}^{k-2}\delta^{j(s-\gamma)}\left[M\left([g_{j}]^{\frac{\omega}{\omega+\varepsilon}}\right)\right]^{\frac{\omega+\varepsilon}{\omega}}\right.\right.\right.$
$\displaystyle\quad\left.\left.\left.+\delta^{k\varepsilon^{\prime}}\sum_{j=k-1}^{\infty}\delta^{j(s-\varepsilon^{\prime})}\left[M\left([g_{j}]^{\frac{\omega}{\omega+\varepsilon}}\right)\right]^{\frac{\omega+\varepsilon}{\omega}}\right\\}^{q}\right)^{1/q}\right\|_{L^{p}({\mathcal{X}})}$
$\displaystyle\lesssim\left\|\left\\{\sum_{k=-\infty}^{\infty}\delta^{-ksq}\delta^{k\gamma
q}\sum_{j=-\infty}^{k-2}\delta^{j(s-\gamma)q}\left[M\left([g_{j}]^{\frac{\omega}{\omega+\varepsilon}}\right)\right]^{\frac{(\omega+\varepsilon)q}{\omega}}\right\\}^{1/q}\right\|_{L^{p}({\mathcal{X}})}$
$\displaystyle\quad+\left\|\left\\{\sum_{k=-\infty}^{\infty}\delta^{-ksq}\delta^{k\varepsilon^{\prime}q}\sum_{j=k-1}^{\infty}\delta^{j(s-\varepsilon^{\prime})q}\left[M\left([g_{j}]^{\frac{\omega}{\omega+\varepsilon}}\right)\right]^{\frac{(\omega+\varepsilon)q}{\omega}}\right\\}^{1/q}\right\|_{L^{p}({\mathcal{X}})}$
$\displaystyle\lesssim\left\|\left\\{\sum_{j=-\infty}^{\infty}\left[M\left([g_{j}]^{\frac{\omega}{\omega+\varepsilon}}\right)\right]^{\frac{(\omega+\varepsilon)q}{\omega}}\right\\}^{1/q}\right\|_{L^{p}({\mathcal{X}})}\lesssim\left\|\left\\{\sum_{j=-\infty}^{\infty}g_{j}^{q}\right\\}^{1/q}\right\|_{L^{p}({\mathcal{X}})}$
$\displaystyle\lesssim\|u\|_{\dot{M}^{s}_{p,q}({\mathcal{X}})}.$

This is the desired estimate in this case.

Case 3) $p\in(\omega/(\omega+s),1]$ and $q\in(\omega/(\omega+s),\infty]$. In
this case, the proof of Theorem 3.10 is similar to that of Case 2); the
details are omitted.

Case 4) $p=\infty$ and $q\in(1,\infty]$. In this case, we only consider the
case $q\in(1,\infty)$ because the proof of the case $q=\infty$ is similar to
that of $q\in(1,\infty)$. Choose
$\\{g_{k}\\}_{k\in\mathbb{Z}}\in\mathbb{D}^{s}(u)$ such that

$\sup_{k\in\mathbb{Z}}\sup_{x\in{\mathcal{X}}}\left\\{\sum_{j=k}^{\infty}\fint_{B(x,\delta^{k})}[g_{j}(y)]^{q}\,d\mu(y)\right\\}^{\frac{1}{q}}\lesssim\|u\|_{\dot{M}^{s}_{\infty,q}({\mathcal{X}})}.$
(3.30)

By (3) and the Hölder inequality, we know that, for any $l\in\mathbb{Z}$ and
$x\in{\mathcal{X}}$,

$\displaystyle\fint_{B(x,2A_{0}C_{0}\delta^{l})}\sum_{k=l}^{\infty}\delta^{-ksq}\sup_{\phi\in{\mathcal{F}}_{k}(z)}|\langle
u,\phi\rangle|^{q}\,d\mu(z)$
$\displaystyle\quad\lesssim\fint_{B(x,2A_{0}C_{0}\delta^{l})}\sum_{k=l}^{\infty}\delta^{-ksq}\left[\delta^{k\gamma}\sum_{j=-\infty}^{k}\delta^{-j(\gamma-s)}\fint_{B(z,\delta^{j-2})}g_{j}(y)\,d\mu(y)\right]^{q}\,d\mu(z)$
$\displaystyle\quad\lesssim\fint_{B(x,2A_{0}C_{0}\delta^{l})}\sum_{k=l}^{\infty}\delta^{k(\gamma-s)}\sum_{j=-\infty}^{k}\delta^{-j(\gamma-s)}\left[\fint_{B(z,\delta^{j-2})}g_{j}(y)\,d\mu(y)\right]^{q}\,d\mu(z)$
$\displaystyle\quad\lesssim\fint_{B(x,2A_{0}C_{0}\delta^{l})}\sum_{k=l}^{\infty}\delta^{k(\gamma-s)}\sum_{j=-\infty}^{l-1}\delta^{-j(\gamma-s)}\left[\fint_{B(z,\delta^{j-2})}g_{j}(y)\,d\mu(y)\right]^{q}\,d\mu(z)$
$\displaystyle\quad\quad+\fint_{B(x,2A_{0}C_{0}\delta^{l})}\sum_{k=l}^{\infty}\delta^{k(\gamma-s)}\sum_{j=l}^{k}\delta^{-j(\gamma-s)}\left[\fint_{B(z,\delta^{j-2})}g_{j}(y)\,d\mu(y)\right]^{q}\,d\mu(z)$
$\displaystyle\quad=:\mathrm{Y}_{1}+\mathrm{Y}_{2}.$

To estimate $\mathrm{Y}_{1}$, by the Hölder inequality with exponent $q>1$,
and (3.30), we find that, for any $j\in\mathbb{Z}$ and $z\in{\mathcal{X}}$,

$\left[\fint_{B(z,\delta^{j-2})}g_{j}(y)\,d\mu(y)\right]^{q}\leq\fint_{B(z,\delta^{j-2})}[g_{j}(y)]^{q}\,d\mu(y)\lesssim\|u\|_{\dot{M}^{s}_{\infty,q}({\mathcal{X}})}^{q}.$

From this, we further deduce that

$\displaystyle\mathrm{Y}_{1}\lesssim\|u\|_{\dot{M}^{s}_{\infty,q}({\mathcal{X}})}^{q}\fint_{B(x,2A_{0}C_{0}\delta^{l})}\sum_{k=l}^{\infty}\delta^{k(\gamma-s)}\sum_{j=-\infty}^{l-1}\delta^{-j(\gamma-s)}\,d\mu(z)\lesssim\|u\|_{\dot{M}^{s}_{\infty,q}({\mathcal{X}})}^{q}.$

To estimate $\mathrm{Y}_{2}$, note that, for any $j\in\mathbb{Z}$ with $j\geq
l$, $z\in B(x,2A_{0}C_{0}\delta^{l})$, and $y\in B(z,\delta^{j-2})$, we have

$d(y,x)\leq
A_{0}[d(y,z)+d(z,x)]<A_{0}\delta^{j-2}+2A_{0}^{2}C_{0}\delta^{l}\leq(A_{0}\delta+2A_{0}^{2}C_{0}\delta^{3})\delta^{l-3}\leq\delta^{l-3},$

which further implies that

$\displaystyle\mathrm{Y}_{2}$
$\displaystyle\lesssim\fint_{B(x,2A_{0}C_{0}\delta^{l})}\sum_{k=l}^{\infty}\delta^{k(\gamma-s)}\sum_{j=l}^{k}\delta^{-j(\gamma-s)}\left[\fint_{B(z,\delta^{j-2})}g_{j}(y)\mathbf{1}_{B(x,{\delta^{l-3}})}(y)\,d\mu(y)\right]^{q}\,d\mu(z)$
$\displaystyle\lesssim\fint_{B(x,2A_{0}C_{0}\delta^{l})}\sum_{k=l}^{\infty}\delta^{k(\gamma-s)}\sum_{j=l}^{k}\delta^{-j(\gamma-s)}\left[M\left(g_{j}\mathbf{1}_{B(x,\delta^{l-3})}\right)(z)\right]^{q}\,d\mu(z)$
$\displaystyle\lesssim\sum_{j=l}^{\infty}\fint_{B(x,2A_{0}C_{0}\delta^{l})}\left[M\left(g_{j}\mathbf{1}_{B(x,\delta^{l-3})}\right)(z)\right]^{q}\,d\mu(z)$
$\displaystyle\lesssim\sum_{j=l}^{\infty}\fint_{B(x,2A_{0}C_{0}\delta^{l})}\left[g_{j}(z)\mathbf{1}_{B(x,\delta^{l-3})}(z)\right]^{q}\,d\mu(z)$
$\displaystyle\lesssim\sum_{j=l-3}^{\infty}\fint_{B(x,\delta^{l-3})}\left[g_{j}(z)\right]^{q}\,d\mu(z)\lesssim\|u\|_{\dot{M}^{s}_{\infty,q}({\mathcal{X}})}^{q}.$

Combining the estimates of $\mathrm{Y}_{1}$ and $\mathrm{Y}_{2}$, we conclude
that, for any $l\in\mathbb{Z}$ and $x\in{\mathcal{X}}$,

$\fint_{B(x,2A_{0}C_{0}\delta^{l})}\sum_{k=l}^{\infty}\delta^{-ksq}\sup_{\phi\in{\mathcal{F}}_{k}(z)}|\langle
u,\phi\rangle|^{q}\,d\mu(z)\lesssim\|u\|_{\dot{M}^{s}_{\infty,q}({\mathcal{X}})}^{q},$

which, together with Lemma 2.4(iii), implies the desired estimate in this
case.

Case 5) $p=\infty$ and $q\in(\omega/(\omega+s),1]$. In this case, choose
$\\{g_{k}\\}_{k\in\mathbb{Z}}\in\mathbb{D}^{s}(u)$ as in Case 4). Arguing as
in (3), we have, for any fixed $\varepsilon,\ \varepsilon^{\prime}\in(0,s)$
with $\varepsilon<\varepsilon^{\prime}$,

$\displaystyle\fint_{B(x,2A_{0}C_{0}\delta^{l})}\sum_{k=l}^{\infty}\delta^{-ksq}\sup_{\phi\in{\mathcal{F}}_{k}(z)}|\langle
u,\phi\rangle|^{q}\,d\mu(z)$
$\displaystyle\quad\lesssim\fint_{B(x,2A_{0}C_{0}\delta^{l})}\sum_{k=l}^{\infty}\delta^{-ksq}\sum_{i=0}^{\infty}\delta^{i\gamma
q}\delta^{(k-i)\varepsilon^{\prime}q}$
$\displaystyle\quad\quad\times\sum_{j=k-i-2}^{\infty}\delta^{j(s-\varepsilon^{\prime})q}\left\\{\fint_{B(z,\delta^{k-i-1})}[g_{j}(y)]^{\frac{\omega}{\omega+\varepsilon}}\,d\mu(y)\right\\}^{\frac{(\omega+\varepsilon)q}{\omega}}\,d\mu(z)$
$\displaystyle\quad\sim\fint_{B(x,2A_{0}C_{0}\delta^{l})}\sum_{k=l}^{\infty}\delta^{-ksq}\sum_{m=-\infty}^{k}\delta^{(k-m)\gamma
q}\delta^{m\varepsilon^{\prime}q}$
$\displaystyle\quad\quad\times\sum_{j=m-2}^{\infty}\delta^{j(s-\varepsilon^{\prime})q}\left\\{\fint_{B(z,\delta^{m-1})}[g_{j}(y)]^{\frac{\omega}{\omega+\varepsilon}}\,d\mu(y)\right\\}^{\frac{(\omega+\varepsilon)q}{\omega}}\,d\mu(z)$
$\displaystyle\quad\lesssim\fint_{B(x,2A_{0}C_{0}\delta^{l})}\sum_{m=-\infty}^{\infty}\delta^{(l\vee
m)(\gamma-s)q}\delta^{m(\varepsilon^{\prime}-\gamma)q}$
$\displaystyle\quad\quad\times\sum_{j=m-2}^{\infty}\delta^{j(s-\varepsilon^{\prime})q}\left\\{\fint_{B(z,\delta^{m-1})}[g_{j}(y)]^{\frac{\omega}{\omega+\varepsilon}}\,d\mu(y)\right\\}^{\frac{(\omega+\varepsilon)q}{\omega}}\,d\mu(z)$
$\displaystyle\quad\lesssim\fint_{B(x,2A_{0}C_{0}\delta^{l})}\sum_{m=-\infty}^{l}\delta^{l(\gamma-s)q}\delta^{m(\varepsilon^{\prime}-\gamma)q}$
$\displaystyle\quad\quad\times\sum_{j=m-2}^{\infty}\delta^{j(s-\varepsilon^{\prime})q}\left\\{\fint_{B(z,\delta^{m-1})}[g_{j}(y)]^{\frac{\omega}{\omega+\varepsilon}}\,d\mu(y)\right\\}^{\frac{(\omega+\varepsilon)q}{\omega}}\,d\mu(z)$
$\displaystyle\quad\quad+\fint_{B(x,2A_{0}C_{0}\delta^{l})}\sum_{m=l+1}^{\infty}\delta^{m(\varepsilon^{\prime}-s)q}\cdots$
$\displaystyle\quad=:\mathrm{Y}_{3}+\mathrm{Y}_{4}.$

To estimate $\mathrm{Y}_{3}$, choosing $\varepsilon\in(0,s)$ such that
$\omega/(\omega+s)<\omega/(\omega+\varepsilon)<q$, and using the Hölder
inequality and (3.30), we find that, for any $m\in\mathbb{Z}$ and
$z\in{\mathcal{X}}$,

$\left\\{\fint_{B(z,\delta^{m-1})}[g_{j}(y)]^{\frac{\omega}{\omega+\varepsilon}}\,d\mu(y)\right\\}^{\frac{(\omega+\varepsilon)q}{\omega}}\leq\fint_{B(z,\delta^{m-1})}[g_{j}(y)]^{q}\,d\mu(y)\lesssim\|u\|_{\dot{M}^{s}_{\infty,q}({\mathcal{X}})}^{q},$

which implies that

$\mathrm{Y}_{3}\lesssim\|u\|_{\dot{M}^{s}_{\infty,q}({\mathcal{X}})}^{q}\sum_{m=-\infty}^{l}\delta^{l(\gamma-s)q}\delta^{m(\varepsilon^{\prime}-\gamma)q}\sum_{j=m-2}^{\infty}\delta^{j(s-\varepsilon^{\prime})q}\lesssim\|u\|_{\dot{M}^{s}_{\infty,q}({\mathcal{X}})}^{q}.$

To estimate $\mathrm{Y}_{4}$, observe that, for any $m\in\mathbb{Z}$ with
$m\geq l+1$, $z\in B(x,2A_{0}C_{0}\delta^{l})$, and $y\in B(z,\delta^{m-1})$,
we have

$d(y,x)\leq
A_{0}[d(y,z)+d(z,x)]<A_{0}\delta^{m-2}+2A_{0}^{2}C_{0}\delta^{l}\leq(A_{0}\delta+2A_{0}^{2}C_{0}\delta^{2})\delta^{l-2}\leq\delta^{l-2},$

which further implies that

$\displaystyle\mathrm{Y}_{4}$
$\displaystyle\lesssim\fint_{B(x,2A_{0}C_{0}\delta^{l})}\sum_{m=l+1}^{\infty}\delta^{m(\varepsilon^{\prime}-s)q}\sum_{j=m-2}^{\infty}\delta^{j(s-\varepsilon^{\prime})q}$
$\displaystyle\quad\times\left\\{\fint_{B(z,\delta^{m-1})}[g_{j}(y)\mathbf{1}_{B(x,\delta^{l-2})}(y)]^{\frac{\omega}{\omega+\varepsilon}}\,d\mu(y)\right\\}^{\frac{(\omega+\varepsilon)q}{\omega}}\,d\mu(z)$
$\displaystyle\lesssim\fint_{B(x,2A_{0}C_{0}\delta^{l})}\sum_{m=l+1}^{\infty}\delta^{m(\varepsilon^{\prime}-s)q}\sum_{j=m-2}^{\infty}\delta^{j(s-\varepsilon^{\prime})q}\left\\{M\left([g_{j}\mathbf{1}_{B(x,\delta^{l-2})}]^{\frac{\omega}{\omega+\varepsilon}}\right)(z)\right\\}^{\frac{(\omega+\varepsilon)q}{\omega}}\,d\mu(z)$
$\displaystyle\lesssim\fint_{B(x,2A_{0}C_{0}\delta^{l})}\sum_{j=l-1}^{\infty}\left\\{M\left([g_{j}\mathbf{1}_{B(x,\delta^{l-2})}]^{\frac{\omega}{\omega+\varepsilon}}\right)(z)\right\\}^{\frac{(\omega+\varepsilon)q}{\omega}}\,d\mu(z)$
$\displaystyle\lesssim\sum_{j=l-2}^{\infty}\fint_{B(x,\delta^{l-2})}\left[g_{j}(z)\right]^{q}\,d\mu(z)\lesssim\|u\|_{\dot{M}^{s}_{\infty,q}({\mathcal{X}})}^{q}.$

Combining the estimates of $\mathrm{Y}_{3}$ and $\mathrm{Y}_{4}$, we conclude
that, for any $l\in\mathbb{Z}$ and $x\in{\mathcal{X}}$,

$\fint_{B(x,2A_{0}C_{0}\delta^{l})}\sum_{k=l}^{\infty}\delta^{-ksq}\sup_{\phi\in{\mathcal{F}}_{k}(z)}|\langle
u,\phi\rangle|^{q}\,d\mu(z)\lesssim\|u\|_{\dot{M}^{s}_{\infty,q}({\mathcal{X}})}^{q},$

which, together with Lemma 2.4(iii), implies the desired estimate in this
case.

Thus, we have
$\dot{M}^{s}_{p,q}({\mathcal{X}})\subset{\mathcal{A}}\dot{F}^{s}_{p,q}(\mathcal{X})$.

We finally show
${\mathcal{A}}\dot{F}^{s}_{p,q}(\mathcal{X})\subset\dot{M}^{s}_{p,q}({\mathcal{X}})$.
Assume that $f\in{\mathcal{A}}\dot{F}^{s}_{p,q}(\mathcal{X})$. By Lemma 3.19
and its proof, we know that there exist a subsequence
$\\{Q_{k_{j}}f\\}_{j\in\mathbb{N}}$ and a constant $C\in\mathbb{R}$ such that,
for almost every $x\in{\mathcal{X}}$, $\lim_{j\to\infty}Q_{k_{j}}f(x)=f(x)-C$.
For any $k\in\mathbb{Z}$ and $x\in{\mathcal{X}}$, let

$g_{k}(x):=\delta^{-ks}\sup_{\phi\in{\mathcal{F}}_{k}(x)}|\langle
f,\phi\rangle|.$

For almost every $x,\ y\in{\mathcal{X}}$ and $k_{0}\in\mathbb{Z}$ satisfying
$\delta^{k_{0}+1}\leq d(x,y)<\delta^{k_{0}}$, we can find a
$j_{0}\in\mathbb{N}$ satisfying $k_{0}\leq k_{j_{0}}$ and we can estimate

$\displaystyle|f(x)-f(y)|$
$\displaystyle=\left|f(x)-Q_{k_{j_{0}}}f(x)+Q_{k_{j_{0}}}f(x)-Q_{k_{j_{0}}}f(y)+Q_{k_{j_{0}}}f(y)-f(y)\right|$
$\displaystyle\leq\left|Q_{k_{j_{0}}}f(x)-Q_{k_{j_{0}}}f(y)\right|+\sum_{j=j_{0}}^{\infty}\left[\left|Q_{k_{j+1}}f(x)-Q_{k_{j}}f(x)\right|+\left|Q_{k_{j+1}}f(x)-Q_{k_{j}}f(x)\right|\right]$
$\displaystyle\leq\delta^{k_{j_{0}}s}[g_{k_{j_{0}}}(x)+g_{k_{j_{0}}}(y)]+\sum_{k=k_{0}}^{\infty}\left[\left|Q_{k+1}f(x)-Q_{k}f(x)\right|+\left|Q_{k+1}f(x)-Q_{k}f(x)\right|\right]$
$\displaystyle\leq 2\sum_{k=k_{0}}^{\infty}\delta^{ks}[g_{k}(x)+g_{k}(y)].$

For any $k\in\mathbb{Z}$, define

$h_{k}:=2\sum_{j=k}^{\infty}\delta^{(-k+j-1)s}g_{j}.$

We then have, for almost every $x,\ y\in{\mathcal{X}}$ with $\delta^{k+1}\leq
d(x,y)<\delta^{k}$,

$\displaystyle|f(x)-f(y)|$ $\displaystyle\leq
2\sum_{j=k}^{\infty}\delta^{js}[g_{j}(x)+g_{j}(y)]\leq\delta^{(k+1)s}[h_{k}(x)+h_{k}(y)]$
$\displaystyle\leq[d(x,y)]^{s}[h_{k}(x)+h_{k}(y)],$

which implies that $\\{h_{k}\\}_{k\in\mathbb{Z}}\in\mathbb{D}^{s}(f)$. Note
that, by the Hölder inequality when $q\in(1,\infty]$, or (3.1) when
$q\in(\omega/(\omega+s),1]$, we obtain

$\displaystyle\sum_{k=-\infty}^{\infty}h_{k}^{q}\lesssim\sum_{k=-\infty}^{\infty}\left[\sum_{j=k}^{\infty}\delta^{(-k+j-1)s}g_{j}\right]^{q}\lesssim\sum_{k=-\infty}^{\infty}\sum_{j=k}^{\infty}\delta^{(-k+j-1)sq/2}g_{j}^{q}\lesssim\sum_{j=-\infty}^{\infty}g_{j}^{q},$

which implies that, for any given $p\in(\omega/(\omega+s),\infty)$ and
$q\in(\omega/(\omega+s),\infty]$,

$\displaystyle\|f\|_{\dot{M}^{s}_{p,q}({\mathcal{X}})}\leq\left\|\left(\sum_{k=\infty}^{\infty}h_{k}^{q}\right)^{1/q}\right\|_{L^{p}({\mathcal{X}})}\lesssim\left\|\left(\sum_{j=\infty}^{\infty}g_{j}^{q}\right)^{1/q}\right\|_{L^{p}({\mathcal{X}})}\sim\|f\|_{{\mathcal{A}}\dot{F}^{s}_{p,q}(\mathcal{X})}.$

When $p=\infty$, then, by the Hölder inequality when $q\in(1,\infty]$, or
(3.1) when $q\in(\omega/(\omega+s),1)$, we have, for any $l\in\mathbb{Z}$,

$\displaystyle\sum_{k=l}^{\infty}h_{k}^{q}\lesssim\sum_{k=l}^{\infty}\left[\sum_{j=k}^{\infty}\delta^{(-k+j-1)s}g_{j}\right]^{q}\lesssim\sum_{k=l}^{\infty}\sum_{j=k}^{\infty}\delta^{(-k+j-1)sq/2}g_{j}^{q}\lesssim\sum_{j=l}^{\infty}g_{j}^{q},$

which implies that, for any $x\in{\mathcal{X}}$,

$\sum_{k=l}^{\infty}\fint_{B(x,\delta^{l})}[h_{k}(y)]^{q}\,d\mu(y)\lesssim\sum_{j=l}^{\infty}\fint_{B(x,\delta^{l})}[g_{j}(y)]^{q}\,d\mu(y).$

From this, we deduce that
$\|f\|_{\dot{M}^{s}_{\infty,q}({\mathcal{X}})}\lesssim\|f\|_{{\mathcal{A}}\dot{F}^{s}_{\infty,q}(\mathcal{X})}$,
which completes the proof of (i) and hence of Theorem 3.10. ∎

Now, we can show Theorem 2.16.

###### Proof of Theorem 2.16.

The Theorem 2.16 is a direct corollary of Theorems 3.3 and 3.10; we omit the
details. ∎

##  4 Pointwise characterization of inhomogeneous Besov and Triebel–Lizorkin
spaces

In this section, we establish the inhomogeneous version of Theorem 2.16. Let
us begin with the notion of inhomogeneous approximations of the identity with
exponential decay (see [24, Definition 6.1]).

######  Definition 4.1.

Let $\eta\in(0,1)$ be as in Definition 2.5. A sequence
$\\{Q_{k}\\}_{k\in\mathbb{Z}_{+}}$ of bounded linear integral operators on
$L^{2}({\mathcal{X}})$ is called an _inhomogeneous approximation of the
identity with exponential decay_ (for short, exp-IATI) if
$\\{Q_{k}\\}_{k\in\mathbb{Z}_{+}}$ has the following properties:

  1. (i)

$\sum_{k=0}^{\infty}Q_{k}=I$ in $L^{2}({\mathcal{X}})$;

  2. (ii)

for any $k\in\mathbb{N}$, $Q_{k}$ satisfies (ii) through (v) of Definition
2.5;

  3. (iii)

$Q_{0}$ satisfies (ii), (iii), and (iv) of Definition 2.5 with $k=0$ but
without the decay factor

$\exp\left\\{-\nu\left[\max\left\\{d(x,{\mathcal{Y}}^{0}),d(y,{\mathcal{Y}}^{0})\right\\}\right]^{a}\right\\};$

moreover, for any $x\in{\mathcal{X}}$,

$\int_{\mathcal{X}}Q_{0}(x,y)\,d\mu(y)=1=\int_{\mathcal{X}}Q_{0}(y,x)\,d\mu(y).$
(4.1)

######  Remark 4.2.

As was pointed out in [24, Remark 6.2], the existence of an exp-IATI on
${\mathcal{X}}$ is guaranteed by the main results from [4]. In Definition 4.1,
due to (4.1), we do not need $\mathop{\mathrm{diam}}{\mathcal{X}}=\infty$ to
guarantee the existence of an exp-IATI on ${\mathcal{X}}$. In other words,
$\mathop{\mathrm{diam}}{\mathcal{X}}$ can be finite or infinite.

Based on the notion of exp-IATIs, He et al. established the following
inhomogeneous discrete Calderón reproducing formulae in [24, Theorems 6.10 and
6.13].

######  Lemma 4.3.

Let $\\{Q_{k}\\}_{k\in\mathbb{Z}_{+}}$ be an exp-IATI and $\beta,\
\gamma\in(0,\eta)$ with $\eta$ as in Definition 2.5. For any
$k\in\mathbb{Z}_{+},$ $\alpha\in{\mathcal{A}}_{k}$, and
$m\in\\{1,\dots,N(k,\alpha)\\}$, suppose that $y_{\alpha}^{k,m}$ is an
arbitrary point in $Q_{\alpha}^{k,m}$. Then there exist an $N\in\mathbb{N}$
and a sequence $\\{\widetilde{Q}_{k}\\}_{k\in\mathbb{Z}_{+}}$ of bounded
linear integral operators on $L^{2}(\mathcal{X})$ such that, for any
$f\in(\mathcal{G}_{0}^{\eta}(\beta,\gamma))^{\prime}$,

$\displaystyle f(\cdot)$
$\displaystyle=\sum_{\alpha\in{\mathcal{A}}_{0}}\sum_{m=1}^{N(0,\alpha)}\int_{Q_{\alpha}^{0,m}}\widetilde{Q}_{0}(\cdot,y)\,d\mu(y)Q^{0,m}_{\alpha,1}(f)$
$\displaystyle\quad+\sum_{k=1}^{N}\sum_{\alpha\in{\mathcal{A}}_{k}}\sum_{m=1}^{N(k,\alpha)}\mu\left(Q_{\alpha}^{k,m}\right)\widetilde{Q}_{k}(\cdot,y_{\alpha}^{k,m})Q^{k,m}_{\alpha,1}(f)$
$\displaystyle\quad+\sum_{k=N+1}^{\infty}\sum_{\alpha\in{\mathcal{A}}_{k}}\sum_{m=1}^{N(k,\alpha)}\mu\left(Q_{\alpha}^{k,m}\right)\widetilde{Q}_{k}(\cdot,y_{\alpha}^{k,m})Q_{k}f\left(y_{\alpha}^{k,m}\right)$

in $(\mathcal{G}_{0}^{\eta}(\beta,\gamma))^{\prime},$ where, for any
$k\in\\{0,\ldots,N\\}$, $\alpha\in{\mathcal{A}}_{k}$,
$m\in\\{1,\dots,N(k,\alpha)\\}$, and $x\in{\mathcal{X}}$,

$Q^{k,m}_{\alpha,1}(x):=\frac{1}{\mu(Q_{\alpha}^{k,m})}\int_{Q_{\alpha}^{k,m}}Q_{k}(y,x)\,d\mu(y),$

and $Q^{k,m}_{\alpha,1}(f):=\langle f,Q^{k,m}_{\alpha,1}\rangle$. Moreover,
for any $k\in\mathbb{Z}_{+}$, the kernel of $\widetilde{Q}_{k}$, still denoted
by $\widetilde{Q}_{k}$, satisfies (3.2), (3.3), and the following integral
condition: for any $x\in\mathcal{X}$,

$\int_{\mathcal{X}}\widetilde{Q}_{k}(x,y)\,d\mu(y)=\int_{\mathcal{X}}\widetilde{Q}_{k}(y,x)\,d\mu(y)=\begin{cases}1&\text{if
}k\in\\{0,\dots,N\\},\\\ 0&\text{if }k\in\\{N+1,N+2,\ldots\\}.\end{cases}$

Now, we recall the notions of inhomogeneous spaces $B^{s}_{p,q}(\mathcal{X})$
and $F^{s}_{p,q}(\mathcal{X})$ introduced in [46]. To this end, for any dyadic
cube $Q$ and any non-negative measurable function $f$ on $X$, let

$m_{Q}(f):=\frac{1}{\mu(Q)}\int_{Q}f(y)\,d\mu(y).$

######  Definition 4.4.

Let $\beta,\gamma\in(0,\eta)$ with $\eta$ as in Definition 2.5, and
$s\in(-(\beta\wedge\gamma),\beta\wedge\gamma)$. Let
$\\{Q_{k}\\}_{k\in\mathbb{Z}_{+}}$ be an exp-IATI and $N\in\mathbb{N}$ as in
Lemma 4.3.

  1. (i)

If $p\in(p(s,\beta\wedge\gamma),\infty]$ with $p(s,\beta\wedge\gamma)$ as in
(1.1), and $q\in(0,\infty]$, then the _inhomogeneous Besov space
$B^{s}_{p,q}({\mathcal{X}})$_ is defined to be the set of all
$f\in(\mathcal{G}_{0}^{\eta}(\beta,\gamma))^{\prime}$ such that

$\displaystyle\|f\|_{B^{s}_{p,q}({\mathcal{X}})}:=$
$\displaystyle\,\left\\{\sum_{k=0}^{N}\sum_{\alpha\in{\mathcal{A}}_{k}}\sum_{m=1}^{N(k,\alpha)}\mu\left(Q_{\alpha}^{k,m}\right)\left[m_{Q_{\alpha}^{k,m}}(|Q_{k}f|)\right]^{p}\right\\}^{1/p}$
$\displaystyle\quad+\left[\sum_{k=N+1}^{\infty}\delta^{-ksq}\|Q_{k}f\|_{L^{p}({\mathcal{X}})}^{q}\right]^{1/q}$
$\displaystyle<$ $\displaystyle\,\infty$

with the usual modifications made when $p=\infty$ or $q=\infty$.

  2. (ii)

If $p\in(p(s,\beta\wedge{\gamma}),\infty)$ and
$q\in(p(s,\beta\wedge{\gamma}),\infty]$, then the _inhomogeneous
Triebel–Lizorkin space $F^{s}_{p,q}({\mathcal{X}})$_ is defined to be the set
of all $f\in(\mathcal{G}_{0}^{\eta}(\beta,\gamma))^{\prime}$ such that

$\displaystyle\|f\|_{F^{s}_{p,q}({\mathcal{X}})}:=$
$\displaystyle\left\\{\sum_{k=0}^{N}\sum_{\alpha\in{\mathcal{A}}_{k}}\sum_{m=1}^{N(k,\alpha)}\mu\left(Q_{\alpha}^{k,m}\right)\left[m_{Q_{\alpha}^{k,m}}(|Q_{k}f|)\right]^{p}\right\\}^{1/p}$
$\displaystyle\quad+\left\|\left(\sum_{k=N+1}^{\infty}\delta^{-ksq}|Q_{k}f|^{q}\right)^{1/q}\right\|_{L^{p}({\mathcal{X}})}$
$\displaystyle<$ $\displaystyle\,\infty$

with the usual modification made when $q=\infty$.

######  Remark 4.5.

  1. (i)

We point out that we _do not_ need the assumption $\mu({\mathcal{X}})=\infty$
in Definitions 4.1 and 4.4.

  2. (ii)

It was proved in [46, Propositions 4.3 and 4.4] that, when $\beta$, $\gamma$,
$s$, $p$, and $q$ are as in Definition 4.4, the inhomogeneous Besov and
Triebel–Lizorkin spaces are independent of the choices of both exp-IATIs and
the spaces of distributions.

We next recall the notions of 1-exp-IATIs (see, for instance, [26, Definition
3.1]) and the local Hardy space $h^{p}({\mathcal{X}})$ (see, for instance,
[26, Section 3]).

######  Definition 4.6.

Let $\eta\in(0,1)$ be as in Definition 2.5. A sequence
$\\{P_{k}\\}_{k\in\mathbb{Z}_{+}}$ of bounded linear integral operators on
$L^{2}({\mathcal{X}})$ is called an _inhomogeneous approximation of the
identity with exponential decay and integration 1_ (for short, 1-exp-IATI) if
$\\{P_{k}\\}_{k\in\mathbb{Z}_{+}}$ has the following properties:

  1. (i)

for any $k\in\mathbb{Z}_{+}$, $P_{k}$ satisfies (ii) and (iii) of Definition
2.5 but without the term

$\exp\left\\{-\nu\left[\max\\{d(x,{\mathcal{Y}}^{k}),d(y,{\mathcal{Y}}^{k})\\}\right]^{a}\right\\};$

  2. (ii)

for any $k\in\mathbb{Z}_{+}$ and $x\in{\mathcal{X}}$,

$\int_{\mathcal{X}}P_{k}(x,y)\,d\mu(y)=1=\int_{\mathcal{X}}P_{k}(y,x)\,d\mu(y);$

  3. (iii)

letting $Q_{0}:=P_{0}$ and, for any $k\in\mathbb{N}$, $Q_{k}:=P_{k}-P_{k-1}$,
then $\\{Q_{k}\\}_{k\in\mathbb{Z}_{+}}$ is an exp-IATI.

######  Definition 4.7.

Let ${\mathcal{X}}$ be a space of homogeneous type. Let
$\\{P_{k}\\}_{k\in\mathbb{Z}}$ be a 1-exp-IATI. The _local radial maximal
function_ ${\mathcal{M}}^{+}_{0}(f)$ of $f$ is defined by setting, for any
$x\in{\mathcal{X}}$,

${\mathcal{M}}^{+}_{0}(f)(x):=\max\left\\{\max_{k\in\\{0,\dots,N\\}}\left\\{\sum_{\alpha\in{\mathcal{A}}_{k}}\sum_{m=1}^{N(k,\alpha)}\sup_{z\in
Q_{\alpha}^{k,m}}|P_{k}f(z)|\mathbf{1}_{Q_{\alpha}^{k,m}}(x)\right\\},\sup_{k\in\\{N+1,N+2,\dots\\}}|P_{k}f(x)|\right\\},$

where $N\in\mathbb{N}$ is as in Lemma 4.3. For any $p\in(0,\infty)$, the
_local Hardy space_ $h^{p}({\mathcal{X}})$ is defined by setting

$h^{p}({\mathcal{X}}):=\left\\{f\in(\mathcal{G}_{0}^{\eta}(\beta,\gamma))^{\prime}:\
\|f\|_{h^{p}({\mathcal{X}})}:=\|{\mathcal{M}}^{+}_{0}(f)\|_{L^{p}({\mathcal{X}})}<\infty\right\\}.$

######  Remark 4.8.

In [26, Theorem 3.3], it was shown that, when $p\in(1,\infty]$, then
$h^{p}({\mathcal{X}})=L^{p}({\mathcal{X}})$. Besides, in [46, Theorem 6.13],
it was proved that, when $p\in(1,\infty)$, $F^{0}_{p,2}=L^{p}({\mathcal{X}})$.
Moreover, the Littlewood–Paley $g$-function characterization of
$h^{p}({\mathcal{X}})$ in [26, Theorem 5.7] implies that, for any given
$p\in(\omega/(\omega+\eta),1]$,
$F_{p,2}^{0}({\mathcal{X}})=h^{p}({\mathcal{X}})$.

Now, we introduce the notions of inhomogeneous Hajłasz–Sobolev spaces,
Hajłasz–Triebel–Lizorkin spaces, and Hajłasz–Besov spaces.

######  Definition 4.9.

Let $s\in(0,\infty)$.

  1. (i)

Let $p\in(0,\infty)$. The _inhomogeneous Hajłasz–Sobolev space_
$M^{s,p}({\mathcal{X}})$ is defined to be the set of all measurable functions
$u$ on ${\mathcal{X}}$ such that

$\|u\|_{M^{s,p}({\mathcal{X}})}:=\|u\|_{h^{p}({\mathcal{X}})}+\|u\|_{\dot{M}^{s,p}({\mathcal{X}})}<\infty.$

  2. (ii)

Let $p,\ q\in(0,\infty]$. The _inhomogeneous Hajłasz–Triebel–Lizorkin space_
$M^{s}_{p,q}({\mathcal{X}})$ is defined to be the set of all measurable
functions $u$ on ${\mathcal{X}}$ such that

$\|u\|_{\dot{M}^{s}_{p,q}({\mathcal{X}})}:=\|u\|_{h^{p}({\mathcal{X}})}+\|u\|_{\dot{M}^{s}_{p,q}({\mathcal{X}})}<\infty.$

  3. (iii)

Let $p,\ q\in(0,\infty]$. The _inhomogeneous Hajłasz–Besov space_
$N^{s}_{p,q}({\mathcal{X}})$ is defined to be the set of all measurable
functions $u$ on ${\mathcal{X}}$ such that

$\|u\|_{N^{s}_{p,q}({\mathcal{X}})}:=\|u\|_{h^{p}({\mathcal{X}})}+\|u\|_{\dot{N}^{s}_{p,q}({\mathcal{X}})}.$

The following theorem states the inhomogeneous version of Theorem 2.16.

######  Theorem 4.10.

Let $\omega$ and $\eta$ be, respectively, as in (2.2) and Definition 2.5,
$\beta,\ \gamma\in(0,\eta)$, $s\in(0,\beta\wedge\gamma)$, and $p,\ q$ be as in
Definition 4.4. Assume that $\mu({\mathcal{X}})=\infty$ and the measure $\mu$
of ${\mathcal{X}}$ has a weak lower bound $Q=\omega$.

  1. (i)

If $p\in(\omega/(\omega+s),\infty)$ and $q\in(\omega/(\omega+s),\infty]$, then
$M^{s}_{p,q}({\mathcal{X}})=F^{s}_{p,q}(\mathcal{X})$.

  2. (ii)

If $p\in(\omega/(\omega+s),\infty]$ and $q\in(0,\infty]$, then
$N^{s}_{p,q}({\mathcal{X}})=B^{s}_{p,q}(\mathcal{X})$.

To prove Theorem 4.10, we first establish a relationship between local Hardy
spaces and inhomogeneous Besov and Triebel–Lizorkin spaces, whose RD-space
version was obtained in [56, Theorem 1.2]. We point out that the proof of [56,
Theorem 1.2] just depends on the size, the regularity, and the cancellation
conditions of the approximation of the identity (for short, ATI), but _does
not_ involve the reverse doubling condition of the underlying space and the
bounded support of ATI. This results in that the proof of [56, Theorem 1.2] is
still valid in any space of homogeneous type due to Definition 2.5(v) and
Lemma 2.6; we omit the details.

######  Theorem 4.11.

Let $\beta,\ \gamma\in(0,\eta)$ with $\eta$ as in Definition 2.5,
$s\in(0,\beta\wedge\gamma)$, and $p\in(\omega/(\omega+s),\infty)$ with
$\omega$ as in (2.2). Assume $\mu({\mathcal{X}})=\infty$. Let
$\\{Q_{k}\\}_{k\in\mathbb{Z}}$ be an exp-ATI.

  1. (i)

If $q\in(0,\infty]$, then $f\in B^{s}_{p,q}(\mathcal{X})$ if and only if $f\in
h^{p}({\mathcal{X}})$ and

${\rm
J}_{1}:=\left[\sum_{k=-\infty}^{\infty}\delta^{-ksq}\|Q_{k}f\|_{L^{p}(X)}^{q}\right]^{1/q}<\infty.$

Moreover, $\|f\|_{B^{s}_{p,q}(\mathcal{X})}$ is equivalent to
$\|f\|_{h^{p}({\mathcal{X}})}+{\rm J}_{1}$ with positive equivalence constants
independent of $f$.

  2. (ii)

If $q\in(\omega/(\omega+s),\infty]$, then $f\in F^{s}_{p,q}(\mathcal{X})$ if
and only if $f\in h^{p}({\mathcal{X}})$ and

${\rm
J}_{2}:=\left\|\left(\sum_{k=-\infty}^{\infty}\delta^{-ksq}|Q_{k}f|^{q}\right)^{1/q}\right\|_{L^{p}(X)}<\infty.$

Moreover, $\|f\|_{F^{s}_{p,q}(\mathcal{X})}$ is equivalent to
$\|f\|_{h^{p}({\mathcal{X}})}+{\rm J}_{2}$ with positive equivalence constants
independent of $f$.

######  Remark 4.12.

Usually, it makes no sense to write the conclusions of Theorem 4.11 as
$B^{s}_{p,q}(\mathcal{X})=h^{p}({\mathcal{X}})\cap\dot{B}^{s}_{p,q}(\mathcal{X})$
and
$F^{s}_{p,q}(\mathcal{X})=h^{p}({\mathcal{X}})\cap\dot{F}^{s}_{p,q}(\mathcal{X})$
because homogeneous and inhomogeneous spaces are defined via different kinds
of spaces of distributions (see [56, Remark 1.1(iv)]).

Now, we show Theorem 4.10.

###### Proof of Theorem 4.10.

We only prove (i) because the proof of (ii) is similar to that of (i). We
first show $M^{s}_{p,q}({\mathcal{X}})\subset F^{s}_{p,q}(\mathcal{X})$. To
this end, assume that $u\in M^{s}_{p,q}({\mathcal{X}})$. By Definition 4.9, we
know that $u\in h^{p}({\mathcal{X}})$ and
$u\in\dot{M}^{s}_{p,q}({\mathcal{X}})$. We consider two cases on $p$.

Case 1) $p\in(1,\infty)$. In this case, since $p\in(1,\infty)$, from [26,
Theorem 3.3], it follows that $u\in L^{p}({\mathcal{X}})$. Moreover, by
Theorem 2.16, we know that $u\in\dot{F}^{s}_{p,q}(\mathcal{X})$. These,
together with [46, Theorem 6.12], imply that $u\in F^{s}_{p,q}(\mathcal{X})$
and
$\|u\|_{F^{s}_{p,q}(\mathcal{X})}\lesssim\|u\|_{M^{s}_{p,q}({\mathcal{X}})}$.

Case 2) $p\in(\omega/(\omega+s),1]$. In this case, by Theorem 2.16, we know
that $u\in\dot{F}^{s}_{p,q}(\mathcal{X})$ and
$\|u\|_{\dot{F}^{s}_{p,q}(\mathcal{X})}\lesssim\|u\|_{\dot{M}^{s}_{p,q}({\mathcal{X}})}$.
Then, using Theorem 4.11(ii), we conclude that $u\in F^{s}_{p,q}(\mathcal{X})$
and
$\|u\|_{F^{s}_{p,q}(\mathcal{X})}\lesssim\|u\|_{M^{s}_{p,q}({\mathcal{X}})}$.

Next, we show $F^{s}_{p,q}(\mathcal{X})\subset M^{s}_{p,q}({\mathcal{X}})$. To
this end, assume that $u\in F^{s}_{p,q}(\mathcal{X})$. We also consider two
cases on $p$.

Case 1) $p\in(1,\infty)$. In this case, since $p\in(1,\infty)$, from [46,
Theorem 6.12], it follows that $u\in
L^{p}({\mathcal{X}})\cap\dot{F}^{s}_{p,q}(\mathcal{X})$ and

$\|u\|_{F^{s}_{p,q}(\mathcal{X})}\sim\|u\|_{L^{p}({\mathcal{X}})}+\|u\|_{\dot{F}^{s}_{p,q}(\mathcal{X})}.$

By [26, Theorem 3.3] again, we know that $u\in h^{p}({\mathcal{X}})$ and
$\|u\|_{h^{p}({\mathcal{X}})}\sim\|u\|_{L^{p}({\mathcal{X}})}$. Besides, from
Theorem 2.16, we deduce that $u\in\dot{M}^{s}_{p,q}({\mathcal{X}})$ and
$\|u\|_{\dot{M}^{s}_{p,q}({\mathcal{X}})}\sim\|u\|_{\dot{F}^{s}_{p,q}(\mathcal{X})}$,
which further implies that $u\in M^{s}_{p,q}({\mathcal{X}})$ and
$\|u\|_{M^{s}_{p,q}({\mathcal{X}})}\lesssim\|u\|_{F^{s}_{p,q}({\mathcal{X}})}$.

Case 2) $p\in(\omega/(\omega+s),1]$. In this case, since $u\in
F^{s}_{p,q}(\mathcal{X})$, from [46, Proposition 4.4], it follows that
$u\in(\mathcal{G}_{0}^{\eta}(\beta,\gamma))^{\prime}$ with $\beta$ and
$\gamma$ as in Definition 4.4. By $p\in(\omega/(\omega+s),1]$, we know that
$s>\omega(1/p-1)$. Choosing $\beta_{0}\in(0,\eta)$ and
$\gamma_{0}\in(s,\eta)\subset(\omega[1/p-1],\eta)$, then we find that
$u\in({\mathcal{G}}_{0}^{\eta}(\beta_{0},\gamma_{0}))^{\prime}\subset(\mathring{{\mathcal{G}}}_{0}^{\eta}(\beta_{0},\gamma_{0}))^{\prime}$.
From this, Theorem 4.11(ii), and [46, Proposition 3.15], we deduce that
$u\in\dot{F}^{s}_{p,q}(\mathcal{X})$ and

$\|u\|_{F^{s}_{p,q}(\mathcal{X})}\sim\|u\|_{L^{p}({\mathcal{X}})}+\|u\|_{\dot{F}^{s}_{p,q}(\mathcal{X})},$

which implies that $u\in M^{s}_{p,q}({\mathcal{X}})$ and
$\|u\|_{M^{s}_{p,q}({\mathcal{X}})}\lesssim\|u\|_{F^{s}_{p,q}({\mathcal{X}})}$.
This finishes the proof of (i) and hence of Theorem 4.10. ∎

######  Remark 4.13.

We point out that it is not clear whether or not Theorem 4.10 still holds true
when $\mu({\mathcal{X}})\neq\infty$, due to that the existence of exp-ATIs in
Theorem 4.11 needs $\mu({\mathcal{X}})=\infty$ [see Remark 2.9(i)].

### Acknowledgments.

The second author would like to thank Ziyi He for his valuable discussions and
suggestions for this article.

## References

  * [1] R. Alvarado, H. Górka and P. Hajłasz, Sobolev embedding for $M^{1,p}$ spaces is equivalent to a lower bound of the measure, J. Funct. Anal. 279 (2020), 108628, 39 pp.
  * [2] R. Alvarado and M. Mitrea, Hardy Spaces on Ahlfors-Regular Quasi Metric Spaces, Lecture Notes in Mathematics 2142, Springer, New York, 2015.
  * [3] K. Asami and Y. Sawano, Non-smooth decomposition of homogeneous Triebel–Lizorkin–Morrey spaces, Comment. Math. 58 (2018), 37–56.
  * [4] P. Auscher and T. Hytönen, Orthonormal bases of regular wavelets in spaces of homogeneous type, Appl. Comput. Harmon. Anal. 34 (2013), 266–296.
  * [5] O. V. Besov, V. P. Il’in and S. M. Nikol’skiĭ, Integralnye Predstavleniya Funktsii i Teoremy Vlozheniya, Second edition, Fizmatlit “Nauka”, Moscow, 1996.
  * [6] T. A. Bui, Besov and Triebel–Lizorkin spaces for Schrödinger operators with inverse-square potentials and applications, J. Differential Equations 269 (2020), 641–688.
  * [7] T. A. Bui, Hermite pseudo-multipliers on new Besov and Triebel–Lizorkin spaces, J. Approx. Theory 252 (2020), 105348, 16 pp.
  * [8] T. A. Bui and X. T. Duong, Inhomogeneous Besov spaces associated to operators with off-diagonal semigroup estimates, Adv. Differential Equations 22 (2017), 191–234.
  * [9] H.-Q. Bui, T. A. Bui and X. T. Duong, Weighted Besov and Triebel–Lizorkin spaces associated to operators and applications, Forum Math. Sigma 8 (2020), e11, 95 pp.
  * [10] R. R. Coifman and G. Weiss, Analyse Harmonique Non-Commutative sur Certains Espaces Homogènes, (French) Étude de Certaines Intégrales Singulières, Lecture Notes in Mathematics 242, Springer-Verlag, Berlin–New York, 1971.
  * [11] R. R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), 569–645.
  * [12] A. Gogatishvili, P. Koskela and Y. Zhou, Characterizations of Besov and Triebel–Lizorkin spaces on metric measure spaces. Forum Math. 25 (2013), 787–819.
  * [13] L. Grafakos, L. Liu and D. Yang, Vector-valued singular integrals and maximal functions on spaces of homogeneous type, Math. Scand. 104 (2009), 296–310.
  * [14] P. Hajłasz, Sobolev spaces on an arbitrary metric spaces, Potential Anal. 5 (1996), 403–415.
  * [15] P. Hajłasz, Sobolev spaces on metric-measure spaces, in: Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces, Paris, 2002, in: Contemp. Math., vol. 338, Amer. Math. Soc., Providence, RI, 2003, pp. 173–218.
  * [16] P. Hajłasz and P. Koskela, Sobolev met Poincaré, Mem. Amer. Math. Soc. 145 (688) (2000), 1–101.
  * [17] Ya. Han, Yo. Han, Z. He, J. Li and C. Pereyra, Geometric characterizations of embedding theorems: For Sobolev, Besov, and Triebel–Lizorkin spaces on spaces of homogeneous type — via orthonormal wavelets, J. Geom. Anal. 31 (2021), 8947–8978.
  * [18] Y. Han, J. Li and L. A. Ward, Hardy space theory on spaces of homogeneous type via orthonormal wavelet bases, Appl. Comput. Harmon. Anal. 45 (2018), 120–169.
  * [19] Y. Han, D. Müller and D. Yang, Littlewood–Paley characterizations for Hardy spaces on spaces of homogeneous type, Math. Nachr. 279 (2006), 1505–1537.
  * [20] Y. Han, D. Müller and D. Yang, A theory of Besov and Triebel–Lizorkin spaces on metric measure spaces modeled on Carnot–Carathéodory spaces, Abstr. Appl. Anal. 2008, Art. ID 893409, 1–250.
  * [21] Y. Han and D. Yang, New characterizations and applications of inhomogeneous Besov and Triebel–Lizorkin spaces on homogeneous type spaces and fractals, Dissertationes Math. (Rozprawy Mat.) 403 (2002), 1–102.
  * [22] Y. Han and D. Yang, Some new spaces of Besov and Triebel–Lizorkin type on homogeneous spaces, Studia Math. 156 (2003), 67–97.
  * [23] Z. He, Y. Han, J. Li, L. Liu, D. Yang and W. Yuan, A complete real-variable theory of Hardy spaces on spaces of homogeneous type, J. Fourier Anal. Appl. 25 (2019), 2197–2267.
  * [24] Z. He, L. Liu, D. Yang and W. Yuan, New Calderón reproducing formulae with exponential decay on spaces of homogeneous type, Sci. China Math. 62 (2019), 283–350.
  * [25] Z. He, F. Wang, D. Yang and W. Yuan, Wavelet characterizations of Besov and Triebel–Lizorkin spaces on spaces of homogeneous type and its applications, Appl. Comput. Harmon. Anal. 54 (2021), 176–226.
  * [26] Z. He, D. Yang and W. Yuan, Real-variable characterizations of local Hardy spaces on spaces of homogeneous type, Math. Nachr. 294 (2021), 900–955.
  * [27] T. Heikkinen, L. Ihnatsyeva and H. Tuominen, Measure density and extension of Besov and Triebel–Lizorkin functions, J. Fourier Anal. Appl. 22 (2016), 334–382.
  * [28] T. Heikkinen, P. Koskela and H. Tuominen, Approximation and quasicontinuity of Besov and Triebel–Lizorkin functions, Trans. Amer. Math. Soc. 369 (2017), 3547–3573.
  * [29] T. Heikkinen and H. Tuominen, Approximation by Hölder functions in Besov and Triebel–Lizorkin spaces, Constr. Approx. 44 (2016), 455–482.
  * [30] T. Heikkinen and H. Tuominen, Smoothing properties of the discrete fractional maximal operator on Besov and Triebel–Lizorkin spaces, Publ. Mat. 58 (2014), 379–399.
  * [31] J. Heinonen and P. Koskela, Quasiconformal maps in metric spaces with controlled geometry, Acta Math. 181 (1998), 1–61.
  * [32] T. Hytönen and A. Kairema, Systems of dyadic cubes in a doubling metric space, Colloq. Math. 126 (2012), 1–33.
  * [33] T. Hytönen and O. Tapiola, Almost Lipschitz-continuous wavelets in metric spaces via a new randomization of dyadic cubes, J. Approx. Theory 185 (2014), 12–30.
  * [34] A. Jonsson, Besov spaces on closed subsets of $\mathbb{R}^{n}$, Trans. Amer. Math. Soc. 341 (1994), 355–370.
  * [35] P. Koskela, D. Yang and Y. Zhou, A characterization of Hajłasz–Sobolev and Triebel–Lizorkin spaces via grand Littlewood–Paley functions, J. Funct. Anal. 258 (2010), 2637–2661.
  * [36] P. Koskela, D. Yang and Y. Zhou, Pointwise characterizations of Besov and Triebel–Lizorkin spaces and quasiconformal mappings, Adv. Math. 226 (2011), 3579–3621.
  * [37] S. Keith and X. Zhong, The Poincaré inequality is an open ended condition, Ann. of Math. (2) 167 (2008), 575–599.
  * [38] D. Müller and D. Yang, A difference characterization of Besov and Triebel–Lizorkin spaces on RD-spaces, Forum Math. 21 (2009), 259–298.
  * [39] E. Nakai and K. Yabuta, Pointwise multipliers for functions of weighted bounded mean oscillation on spaces of homogeneous type, Math. Japon. 46 (1997), 15–28.
  * [40] Y. Sawano, Theory of Besov Spaces, Developments in Mathematics 56, Springer, Singapore, 2018.
  * [41] N. Shanmugalingam, Newtonian spaces: An extension of Sobolev spaces to metric measure spaces, Rev. Mat. Iberoam. 16 (2000), 243–279.
  * [42] H. Triebel, Theory of Function Spaces. I, Monographs in Mathematics 78, Birkhäuser Verlag, Basel, 1983.
  * [43] H. Triebel, Theory of Function Spaces. II, Monographs in Mathematics 84, Birkhäuser Verlag, Basel, 1992.
  * [44] H. Triebel, Theory of Function Spaces. III, Monographs in Mathematics 100, Birkhäuser Verlag, Basel, 2006.
  * [45] H. Wallin, New and old function spaces, in: Function Spaces and Applications (Lund, 1986), 99–114, Lecture Notes in Mathematics 1302, Springer, Berlin, 1988.
  * [46] F. Wang, Y. Han, Z. He and D. Yang, Besov and Triebel–Lizorkin spaces on spaces of homogeneous type with applications to boundedness of Calderón–Zygmund operators, Dissertationes Math. 565 (2021), 1–113.
  * [47] F. Wang, Z. He, D. Yang and W. Yuan, Difference characterization of Besov and Triebel–Lizorkin spaces on spaces of homogeneous type, Commun. Math. Stat. (2021), https://doi. org/10.1007/s40304-021-00243-w.
  * [48] D. Yang, $T1$ theorems on Besov and Triebel–Lizorkin spaces on spaces of homogeneous type and their applications, Z. Anal. Anwendungen 22 (2003), 53–72.
  * [49] D. Yang, Riesz potentials in Besov and Triebel–Lizorkin spaces over spaces of homogeneous type, Potential Anal. 19 (2003), 193–210.
  * [50] D. Yang, Embedding theorems of Besov and Triebel–Lizorkin spaces on spaces of homogeneous type, Sci. China Ser. A 46 (2003), 187–199.
  * [51] D. Yang, New characterizations of Hajłasz–Sobolev spaces on metric spaces, Sci. China Ser. A 46 (2003), 675–689.
  * [52] D. Yang, Localization principle of Triebel–Lizorkin spaces on spaces of homogeneous type, Rev. Mat. Complut. 17 (2004), 229–249.
  * [53] D. Yang, Real interpolations for Besov and Triebel–Lizorkin spaces on spaces of homogeneous type, Math. Nachr. 273 (2004), 96–113.
  * [54] D. Yang, Some new inhomogeneous Triebel–Lizorkin spaces on metric measure spaces and their various characterizations, Studia Math. 167 (2005), 63–98.
  * [55] D. Yang, Some new Triebel–Lizorkin spaces on spaces of homogeneous type and their frame characterizations, Sci. China Ser. A 48 (2005), 12–39.
  * [56] D. Yang and Y. Zhou, New properties of Besov and Triebel–Lizorkin spaces on RD-spaces, Manuscripta Math 134 (2011), 59–90.
  * [57] W. Yuan, W. Sickel and D. Yang, Morrey and Campanato Meet Besov, Lizorkin and Triebel, Lecture Notes in Mathematics 2005, Springer-Verlag, Berlin, 2010.

Ryan Alvarado

Department of Mathematics and Statistics, Amherst College, 303 Seeley Mudd,
Amherst, MA 01002, United States of America

E-mail<EMAIL_ADDRESS>

Fan Wang, Dachun Yang (Corresponding author) and Wen Yuan

Laboratory of Mathematics and Complex Systems (Ministry of Education of
China), School of Mathematical Sciences, Beijing Normal University, Beijing
100875, People’s Republic of China

E-mails<EMAIL_ADDRESS>(F. Wang)

E-mails<EMAIL_ADDRESS>(D. Yang)

E-mails<EMAIL_ADDRESS>(W. Yuan)