# Classifying Triebel–Lizorkin capacities in metric spaces

Juha Lehrbäck  Department of Mathematics and Statistics, P.O. Box 35, FI-40014
University of Jyvaskyla, Finland<EMAIL_ADDRESS>,  Kaushik Mohanta
Department of Mathematics and Statistics, P.O. Box 35, FI-40014 University of
Jyvaskyla, Finland<EMAIL_ADDRESS>and  Antti V. Vähäkangas
Department of Mathematics and Statistics, P.O. Box 35, FI-40014 University of
Jyvaskyla, Finland<EMAIL_ADDRESS>
###### Abstract.

We study non-local or fractional capacities in metric measure spaces. Our main
goal is to clarify the relations between relative Hajłasz–Triebel–Lizorkin
capacities, potentional Triebel–Lizorkin capacities, and metric space variants
of Riesz capacities. As an application of our results, we obtain a
characterization of a Hajłasz–Triebel–Lizorkin capacity density condition,
which is based on an earlier characterization of a Riesz capacity density
condition in terms of Hausdorff contents.

###### Key words and phrases:

Comparison of capacities, capacity density condition, Hajłasz–Triebel–Lizorkin
space, Riesz capacity, metric measure space

###### 2020 Mathematics Subject Classification:

31C15 (28A12, 31E05)

##  1\. Introduction

In mathematical analysis, there are several possible approaches to the concept
of _capacity of a set_ , depending, for instance, on the general setting and
the intended applications. Nevertheless, it often turns out that seemingly
different definitions still lead to comparable or otherwise closely related
concepts.

In this work, we examine and compare four different approaches to nonlocal or
fractional capacities in a metric space $X$ equipped with a doubling measure
$\mu$. More precisely, when $B\subset X$ is an open ball and
$E\subset{\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle B}\kern
0.0pt}$\hss}{B}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle B}\kern
0.0pt}$\hss}{B}}}$ is a closed set, we consider the following, for parameters
$1<p<\infty$, $1\leq q\leq\infty$, $0<\beta<1$, and $\Lambda\geq 2$:

  * •

the relative Hajłasz–Triebel–Lizorkin $(\beta,p,q)$-capacity
$\operatorname{cap}_{\dot{M}^{\beta}_{p,q}}(E,2B,\Lambda B)$,

  * •

the potentional Triebel–Lizorkin $(\beta,p,q)$-capacity
$\operatorname{Cap}^{\beta}_{p,q}(E)$,

  * •

the dual characterization of $\operatorname{Cap}^{\beta}_{p,q}(E)$ in terms of
measures supported on $E$,

  * •

the Riesz $(\beta,p)$-capacity $R_{\beta,p}(E)$.

The main purpose of this paper is to establish and clarify the relations
between these concepts. For instance, we examine when these capacities are
comparable and show that often the parameter $q$ plays no essential role. The
last part of the paper, where we characterize a capacity density condition
associated to Hajłasz–Triebel–Lizorkin capacities, can be viewed as a
continuation of [6, 7].

The Hajłasz–Triebel–Lizorkin function spaces $\dot{M}^{\beta}_{p,q}$ and the
corresponding relative capacities $\operatorname{cap}_{\dot{M}^{\beta}_{p,q}}$
are defined using fractional Hajłasz gradients. In Section 3 we introduce
these gradients, following [24], and recall in Lemma 3.2 a Poincaré type
inequality for them. Hajłasz–Triebel–Lizorkin spaces and relative capacities
$\operatorname{cap}_{\dot{M}^{\beta}_{p,q}}$ are defined in Section 4, where
we also prove a lower bound estimate for these capacities in terms of suitable
Hausdorff contents; see Theorem 4.6.

On the other hand, following [2, 3], the Triebel–Lizorkin potentional capacity
$\operatorname{Cap}^{\beta}_{p,q}(E)$ is defined in Section 6 using the
potential operator $\operatorname{\mathcal{H}}_{\beta}$,

$\operatorname{\mathcal{H}}_{\beta}f(x)=\sum_{n=n_{0}}^{\infty}2^{-\beta
n}\int_{X}\frac{\mathbf{1}_{B(y,2^{-n})}(x)}{\mu(B(y,2^{-n}))}f_{n}(y)\,d\mu(y)\,,$

for a sequence $f=(f_{n})_{n\geq n_{0}}$ of nonnegative Borel functions in
$X$. Here $n_{0}$ is the smallest integer such that $2^{-n_{0}}\leq
2\operatorname{diam}(X)$, with the interpretation that $n_{0}=-\infty$ if $X$
is unbounded.

We show that, under suitable assumptions on the space $X$ and the parameters
$p$, $q$, $\beta$ and $\Lambda$, the capacities
$\operatorname{cap}_{\dot{M}^{\beta}_{p,q}}(E,2B,\Lambda B)$ and
$\operatorname{Cap}^{\beta}_{p,q}(E)$ are comparable for all compact sets
$E\subset{\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle B}\kern
0.0pt}$\hss}{B}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle B}\kern
0.0pt}$\hss}{B}}}$. In particular, besides the doubling condition (see (3)),
we need to assume that the measure $\mu$ is also reverse doubling (see (4) and
(5)), as follows.

######  Theorem 1.1.

Let $1<p<\infty$, $1<q\leq\infty$, $0<\beta<1$ and $\Lambda\geq 41$. Assume
that the measure $\mu$ is doubling and reverse doubling, and satisfies the
quantitative reverse doubling condition (5) for some exponent $\sigma>\beta
p$. Let $B=B(x,r)\subset X$ be a ball with $x\in X$ and
$0<r<\frac{1}{80}\operatorname{diam}(X)$ and let $E\subset{\mathchoice{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle B}\kern
0.0pt}$\hss}{B}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\textstyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle B}\kern
0.0pt}$\hss}{B}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptscriptstyle B}\kern 0.0pt}$\hss}{B}}}$ be a
compact set. Then

$C^{-1}\operatorname{Cap}^{\beta}_{p,q}(E)\leq\operatorname{cap}_{\dot{M}^{\beta}_{p,q}}(E,2B,\Lambda
B)\leq C\operatorname{Cap}^{\beta}_{p,q}(E)\,,$ (1)

where the constant $C\geq 1$ is independent of $B$ and $E$.

The two inequalities in (1) hold separately under slightly different
assumptions, see Theorems 6.5 and 6.6 for precise formulations. Also recall,
as a basic example, that in ${\mathbb{R}}^{n}$ (equipped with the Euclidean
distance) the Lebesgue measure is both doubling and reverse doubling and
satisfies condition (5) with $\sigma=n$.

As a technical tool for the proof of the second inequality in (1), we
introduce a new discrete potential operator
$\operatorname{{\mathcal{Q}}}_{\beta}$, which majorizes the operator
$\operatorname{\mathcal{H}}_{\beta}$; see Lemma 6.3. The definition of
$\operatorname{{\mathcal{Q}}}_{\beta}$ (Definition 5.1) involves a partition
of unity and is inspired by the so-called discrete convolutions, see e.g. [1,
23]. Since the functions in the partition of unity are Lipschitz-continuous,
the operator $\operatorname{{\mathcal{Q}}}_{\beta}$ can be seen as a smooth
version of $\operatorname{\mathcal{H}}_{\beta}$, whose definition uses
noncontinuous characteristic functions of balls. Boundedness properties of
$\operatorname{{\mathcal{Q}}}_{\beta}$ are studied in Section 5, and in
particular we obtain for $\operatorname{{\mathcal{Q}}}_{\beta}$ a mixed norm
estimate in Lemma 5.2 and a Sobolev type inequality in Lemma 5.3.

In Section 7, we prove a dual characterization of the Triebel–Lizorkin
potentional capacity $\operatorname{Cap}^{\beta}_{p,q}(E)$ for compact sets
$E\subset X$. As in the Euclidean case in [3], this is based on the use of
(nonnegative) Radon measures $\nu$ supported on $E$. In our setting, the
sequence

$\widecheck{\mathcal{H}}_{\beta}\nu(x)=\left(2^{-\beta
n}\frac{\nu(B(x,2^{-n}))}{\mu(B(x,2^{-n}))}\right)_{n=n_{0}}^{\infty}\,,\qquad
x\in X\,,$

plays an important role. The main tool in the proof of the dual
characterization (Theorem 7.3) is a Minimax theorem, see Lemma 7.4.

In Section 8 we prove a version of the Muckenhoupt–Wheeden theorem (Theorem
8.3), which gives an estimate for the norm of
$\widecheck{\mathcal{H}}_{\beta}\nu$ in terms of the fractional maximal
function

$M_{\beta}\nu(x)=\sup_{0<r\leq
2^{-n_{0}}}r^{\beta}\frac{\nu(B(x,r))}{\mu(B(x,r))}\,,\qquad x\in X\,.$

Such results originate in [31], and a very general metric space variant of the
Muckenhoupt–Wheeden theorem can be found in [33]. Nevertheless, instead of
relying on the more abstract machinery from [33], we present an alternative
self-contained proof that is suited for our purposes.

The dual characterization of $\operatorname{Cap}^{\beta}_{p,q}$ and the
Muckenhoupt–Wheeden theorem are crucial ingredients in the proof of Theorem
9.3, which shows that for compact sets the Triebel–Lizorkin capacities
$\operatorname{Cap}^{\beta}_{p,q}$ are in fact independent of the
$q$-parameter, again under a suitable reverse doubling assumption.
Consequently, by Theorem 1.1 the same independence holds for the relative
capacity $\operatorname{cap}_{\dot{M}^{\beta}_{p,q}}$, as well.

Finally, in Sections 10 and 11 we compare the Triebel–Lizorkin capacities and
the Riesz capacities $R_{\beta,p}(E)$, which are defined in terms of a metric
space variant of the classical Riesz potential,

$I_{\beta}f(x)=\int_{X\setminus\\{x\\}}\frac{d(x,y)^{\beta}}{\mu(B(x,d(x,y)))}f(y)\,d\mu(y)\,,$
(2)

for Borel functions $f\geq 0$ in $X$. See Definitions 10.1 and 10.2 for more
details.

In Theorem 10.4 we show that $\operatorname{Cap}^{\beta}_{p,\infty}$ and
$R_{\beta,p}$ are comparable. The proof of this theorem follows by rather
straightforward applications of the doubling and quantitative reverse doubling
conditions, see Lemma 10.3. The $q$-independence of the potentional
Triebel–Lizorkin capacities yields for compact sets the same comparability
also for $\operatorname{Cap}^{\beta}_{p,q}$, for every $1<q\leq\infty$, and
thus for the relative capacity $\operatorname{cap}_{\dot{M}^{\beta}_{p,q}}$ as
well, by Theorem 1.1. In Theorem 10.5 we present an alternative direct proof,
again based on the dual characterization, for the fact that for compact sets
$\operatorname{Cap}^{\beta}_{p,q}$ is bounded from above by $R_{\beta,p}$, for
all $1<q\leq\infty$.

The main application of the above results is given in Section 11, where we
characterize, for $0<\beta<1$, a Hajłasz–Triebel–Lizorkin capacity density
condition in complete geodesic metric spaces. The proof of the
characterization is based on the following facts:

  1. (1)

The Riesz capacity $R_{\beta,p}$ dominates the corresponding
Hajłasz–Triebel–Lizorkin capacity
$\operatorname{cap}_{\dot{M}^{\beta}_{p,q}}$, for every $1<q\leq\infty$; see
Corollary 10.6.

  2. (2)

The Riesz capacity density condition is equivalent to a Hausdorff content
density condition. This is a deep and highly nontrivial result from [6,
Theorem 6.4] and [7, Theorem 9.5], and in particular implies a self-
improvement result for the Riesz capacity density condition. Here the
additional assumptions that the space $X$ is complete and geodesic are needed.

  3. (3)

The Hajłasz–Triebel–Lizorkin capacity
$\operatorname{cap}_{\dot{M}^{\beta}_{p,q}}$ can be estimated from below by
suitable Hausdorff contents, by Theorem 4.6.

In addition to these main ingredients, we also need estimates for the
capacities and contents of balls. For Hajłasz–Triebel–Lizorkin capacities such
estimates are obtained in Theorem 4.3.

The above characterization leads to a self-improvement result (Corollary 11.5)
for the Hajłasz–Triebel–Lizorkin capacity density condition, which is, to the
best of our knowledge, new even in the context of Euclidean spaces. Self-
improvement of the capacity density condition for the Riesz capacity in
${\mathbb{R}}^{n}$ was originally proven by Lewis [26]. See also [6, 7] for
corresponding results for other capacities in metric spaces.

Before going to the main parts of the paper, we begin in Section 2 by
introducing notation and recalling some useful concepts that are needed later
on.

### Acknowledgements

The authors would like to thank Lizaveta Ihnatsyeva for interesting
discussions on the topics of the paper. The second author was supported by the
Academy of Finland (project #323960) and by the Academy of Finland via Centre
of Excellence in Analysis and Dynamics Research (project #346310).

##  2\. Preliminaries

We write ${\mathbb{N}}=\\{1,2,3,\ldots\\}$ and
${\mathbb{N}}_{0}=\\{0,1,2,\ldots\\}$. We also set $p^{\prime}=\frac{p}{p-1}$
if $1<p<\infty$, $p^{\prime}=1$ if $p=\infty$, and $p^{\prime}=\infty$ if
$p=1$.

Throughout the rest of the paper we assume that $X=(X,d,\mu)$ is a metric
measure space equipped with a metric $d$ and a nonnegative complete Borel
measure $\mu$ such that $0<\mu(B)<\infty$ for all open balls $B\subset X$,
each of which is an open set of the form

$B=B(x,r)=\\{y\in X\,:\,d(y,x)<r\\}$

with $x\in X$ and $r>0$. The corresponding closed ball is

${\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle B}\kern
0.0pt}$\hss}{B}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle B}\kern
0.0pt}$\hss}{B}}}(x,r)=\\{y\in X\,:\,d(y,x)\leq r\\}.$

Under these assumptions, the space $X$ is separable, see [4, Proposition 1.6].
We also assume that $\\#X\geq 2$.

The diameter of a set $A\subset X$ is denoted by $\operatorname{diam}(A)$. We
write $n_{0}=-\infty$ if $\operatorname{diam}(X)=\infty$, and otherwise choose
$n_{0}$ to be the smallest integer such that $2^{-n_{0}}\leq
2\operatorname{diam}(X)$. In the latter case, we have
$\operatorname{diam}(X)<2^{-n_{0}}\leq 2\operatorname{diam}(X)$.

We also assume throughout the paper that the measure $\mu$ is doubling, that
is, there is a constant $c_{\mu}>1$, called the doubling constant of $\mu$,
such that

$\mu(2B)\leq c_{\mu}\,\mu(B)$ (3)

for all balls $B=B(x,r)$ in $X$. Here we use for $0<t<\infty$ the notation
$tB=B(x,tr)$.

If $X$ is connected, then the doubling measure $\mu$ also satisfies the
reverse doubling condition, that is, there is a constant
$0<c_{R}=C(c_{\mu})<1$ such that

$\mu(B(x,r/2))\leq c_{R}\,\mu(B(x,r))$ (4)

for all $x\in X$ and all $0<r<\operatorname{diam}(X)/2$; see for instance [4,
Lemma 3.7]. Iteration of (4) shows that if $X$ is connected, then there exist
an exponent $\sigma>0$ and a constant $c_{\sigma}>0$, both depending on
$c_{\mu}$ only, such that the quantitative reverse doubling condition

$\frac{\mu(B(x,r))}{\mu(B(x,R))}\leq
c_{\sigma}\Bigl{(}\frac{r}{R}\Bigr{)}^{\sigma}$ (5)

holds for all $x\in X$ and all $0<r<R\leq 2\operatorname{diam}(X)$. However,
in most of our results we will not assume $X$ to be connected, but instead we
often explicitly assume that (5) holds for some exponent $\sigma>0$.

We denote the closure of a set $A\subset X$ by ${\mathchoice{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle A}\kern
0.0pt}$\hss}{A}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\textstyle A}\kern 0.0pt}$\hss}{A}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle A}\kern
0.0pt}$\hss}{A}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptscriptstyle A}\kern 0.0pt}$\hss}{A}}}$. In
particular, if $B=B(x,r)\subset X$ is a ball, then the notation
${\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle B}\kern
0.0pt}$\hss}{B}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle B}\kern
0.0pt}$\hss}{B}}}={\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle B(x,r)}\kern 0.0pt}$\hss}{B(x,r)}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle B(x,r)}\kern
0.0pt}$\hss}{B(x,r)}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle B(x,r)}\kern 0.0pt}$\hss}{B(x,r)}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle
B(x,r)}\kern 0.0pt}$\hss}{B(x,r)}}}$ refers to the closure of the ball $B$.
The characteristic function of a set $A\subset X$ is denoted by
$\mathbf{1}_{A}$; that is, $\mathbf{1}_{A}(x)=1$ if $x\in A$ and
$\mathbf{1}_{A}(x)=0$ if $x\in X\setminus A$.

When $A\subset X$ is a measurable set with $0<\mu(A)<\infty$, we write

$f_{A}=\mathchoice{\mathop{\kern 1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996ptA}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{A}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{A}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{A}}f(y)\,d\mu(y)=\frac{1}{\mu(A)}\int_{A}f(y)\,d\mu(y)$

for the integral average of $f\in L^{1}(A)$ over $A$. If $f:X\to{\mathbb{R}}$
is a measurable function, then the noncentered Hardy–Littlewood maximal
function $M^{*}f$ of $f$ is defined by

$M^{*}f(x)=\sup_{B}\mathchoice{\mathop{\kern 1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996ptB}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B}}\lvert
f(y)\rvert\,d\mu(y)\,,\qquad x\in X\,,$ (6)

where the supremum is taken over all balls $B\subset X$ such that $x\in B$.
The sublinear operator $M^{*}$ is bounded on $L^{p}(X)$ with a constant
$C(c_{\mu},p)$ for every $1<p\leq\infty$, see [4, Theorem 3.13].

Finally, we say that a function $\eta\colon X\to{\mathbb{R}}$ is
$L$-Lipschitz, if $\lvert\eta(x)-\eta(y)\rvert\leq Ld(x,y)$ for every $x,y\in
X$.

##  3\. Fractional Hajłasz gradient

We define fractional Hajłasz gradients in open sets as in [13], by adapting
[24, Definition 1.1].

######  Definition 3.1.

Let $0<\beta\leq 1$ and let $u\colon\Omega\to{\mathbb{R}}$ be a measurable
function on an open set $\Omega\subset X$. A sequence
$g=(g_{k})_{k\in{\mathbb{Z}}}$ of nonnegative measurable functions is called a
fractional $\beta$-Hajłasz gradient of $u$, if there exists a set
$E\subset\Omega$ such that $\mu(E)=0$ and for all $k\in{\mathbb{Z}}$ and
$x,y\in\Omega\setminus E$ satisfying $2^{-k-1}\leq d(x,y)<2^{-k}$, we have

$\lvert u(x)-u(y)\rvert\leq d(x,y)^{\beta}(g_{k}(x)+g_{k}(y))\,.$

We denote by $\mathbb{D}^{\beta}_{\Omega}(u)$ the collection of all fractional
$\beta$-Hajłasz gradients of $u$.

We need the following metric space variant of the Poincaré type inequality
[24, Lemma 2.1].

######  Lemma 3.2.

Let $0<\beta<1$ and assume that $\mu$ satisfies the reverse doubling condition
(4) with a constant $0<c_{R}<1$. Let $x_{0}\in X$, let $n\in{\mathbb{Z}}$ be
such that $2^{-n+3}<\operatorname{diam}(X)$, and assume that $u\in
L^{1}(B(x_{0},2^{-n+2}))$ and
$(g_{k})_{k\in{\mathbb{Z}}}\in\mathbb{D}_{B(x_{0},2^{-n+2})}^{\beta}(u)$. Then

$\mathchoice{\mathop{\kern 1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996ptB(x_{0},2^{-n})}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{0},2^{-n})}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{0},2^{-n})}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{0},2^{-n})}}|u(x)-u_{B(x_{0},2^{-n})}|\,d\mu(x)\leq
C(c_{\mu},c_{R})2^{-n\beta}\sum_{k=n-3}^{n}\;\mathchoice{\mathop{\kern
1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996ptB(x_{0},2^{-n+2})}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{0},2^{-n+2})}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{0},2^{-n+2})}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{0},2^{-n+2})}}g_{k}(x)\,d\mu(x)\,.$

###### Proof.

We estimate

$\begin{split}&\mathchoice{\mathop{\kern 1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996ptB(x_{0},2^{-n})}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{0},2^{-n})}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{0},2^{-n})}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{0},2^{-n})}}|u(x)-u_{B(x_{0},2^{-n})}|\,d\mu(x)\leq
2\mathchoice{\mathop{\kern 1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996ptB(x_{0},2^{-n})}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{0},2^{-n})}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{0},2^{-n})}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{0},2^{-n})}}|u(x)-u_{B(x_{0},2^{-n+2})\setminus
B(x_{0},2^{-n+1})}|\,d\mu(x)\\\ &\quad\leq 2\mathchoice{\mathop{\kern
1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996ptB(x_{0},2^{-n})}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{0},2^{-n})}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{0},2^{-n})}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{0},2^{-n})}}\mathchoice{\mathop{\kern
1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996ptB(x_{0},2^{-n+2})\setminus
B(x_{0},2^{-n+1})}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{0},2^{-n+2})\setminus
B(x_{0},2^{-n+1})}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{0},2^{-n+2})\setminus
B(x_{0},2^{-n+1})}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{0},2^{-n+2})\setminus
B(x_{0},2^{-n+1})}}|u(x)-u(y)|\,d\mu(y)\,d\mu(x)\,.\end{split}$ (7)

We let $E\subset B(x_{0},2^{-n+2})$ be the exceptional set for
$(g_{k})_{k\in{\mathbb{Z}}}$ as in Definition 3.1. Fix points $x\in
B(x_{0},2^{-n})$ and $y\in B(x_{0},2^{-n+2})\setminus B(x_{0},2^{-n+1})$ such
that $x,y\not\in E$. Then

$2^{-n}\leq d(x,y)<2^{-n+3}$

and therefore

$\lvert u(x)-u(y)\rvert\leq
d(x,y)^{\beta}\left(\sum_{k=n-3}^{n-1}g_{k}(x)+\sum_{k=n-3}^{n-1}g_{k}(y)\right)\,.$

Since $\mu(E)=0$, it follows that

$\begin{split}&\mathchoice{\mathop{\kern 1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996ptB(x_{0},2^{-n})}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{0},2^{-n})}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{0},2^{-n})}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{0},2^{-n})}}\mathchoice{\mathop{\kern
1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996ptB(x_{0},2^{-n+2})\setminus
B(x_{0},2^{-n+1})}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{0},2^{-n+2})\setminus
B(x_{0},2^{-n+1})}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{0},2^{-n+2})\setminus
B(x_{0},2^{-n+1})}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{0},2^{-n+2})\setminus
B(x_{0},2^{-n+1})}}|u(x)-u(y)|\,d\mu(y)\,d\mu(x)\\\ &\quad\leq
2^{-n\beta+3}\biggl{(}\mathchoice{\mathop{\kern 1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996ptB(x_{0},2^{-n})}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{0},2^{-n})}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{0},2^{-n})}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{0},2^{-n})}}\sum_{k=n-3}^{n-1}g_{k}(x)\,d\mu(x)+\mathchoice{\mathop{\kern
1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996ptB(x_{0},2^{-n+2})\setminus
B(x_{0},2^{-n+1})}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{0},2^{-n+2})\setminus
B(x_{0},2^{-n+1})}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{0},2^{-n+2})\setminus
B(x_{0},2^{-n+1})}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{0},2^{-n+2})\setminus
B(x_{0},2^{-n+1})}}\sum_{k=n-3}^{n-1}g_{k}(y)\,d\mu(y)\biggr{)}\,.\end{split}$
(8)

By the doubling condition (3), we get

$\mathchoice{\mathop{\kern 1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996ptB(x_{0},2^{-n})}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{0},2^{-n})}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{0},2^{-n})}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{0},2^{-n})}}\sum_{k=n-3}^{n-1}g_{k}(x)\,d\mu(x)\leq
c_{\mu}^{2}\sum_{k=n-3}^{n-1}\mathchoice{\mathop{\kern 1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996ptB(x_{0},2^{-n+2})}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{0},2^{-n+2})}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{0},2^{-n+2})}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{0},2^{-n+2})}}g_{k}(x)\,d\mu(x)\,.$
(9)

By the assumed reverse doubling condition (4), we have

$\mu(B(x_{0},2^{-n+1}))\leq c_{R}\,\mu(B(x_{0},2^{-n+2}))\,,$

and therefore

$\displaystyle\mu(B(x_{0},2^{-n+2})\setminus
B(x_{0},2^{-n+1}))\geq(1-c_{R})\mu(B(x_{0},2^{-n+2}))\,.$

Hence,

$\mathchoice{\mathop{\kern 1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996ptB(x_{0},2^{-n+2})\setminus
B(x_{0},2^{-n+1}))}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{0},2^{-n+2})\setminus
B(x_{0},2^{-n+1}))}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{0},2^{-n+2})\setminus
B(x_{0},2^{-n+1}))}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{0},2^{-n+2})\setminus
B(x_{0},2^{-n+1}))}}\sum_{k=n-3}^{n-1}g_{k}(y)\,d\mu(y)\leq\frac{1}{1-c_{R}}\sum_{k=n-3}^{n-1}\mathchoice{\mathop{\kern
1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996ptB(x_{0},2^{-n+2})}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{0},2^{-n+2})}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{0},2^{-n+2})}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{0},2^{-n+2})}}g_{k}(x)\,d\mu(x)\,.$
(10)

The desired Poincaré inequality follows by combining inequalities (7), (8),
(9) and (10). ∎

The following Leibniz type rule for $\beta$-Hajłasz gradients is from [13,
Lemma 3.10]. The proof is a case study.

######  Lemma 3.3.

Let $0<\beta<1$ and let
$(g_{k})_{k\in{\mathbb{Z}}}\in\mathbb{D}^{\beta}_{X}(u)$ be a fractional
$\beta$-Hajłasz gradient of a measurable function $u\colon X\to{\mathbb{R}}$.
Assume that $\eta$ is a bounded $L$-Lipschitz function in $X$ and define

$\rho_{k}=\big{(}g_{k}\|\eta\|_{\infty}+2^{k(\beta-1)}L|u|\big{)}\mathbf{1}_{\\{x\in
X\,:\,\eta(x)\not=0\\}}\,,\qquad k\in{\mathbb{Z}}\,.$

Then $(\rho_{k})_{k\in{\mathbb{Z}}}\in\mathbb{D}^{\beta}_{X}(\eta u)$.

##  4\. Triebel–Lizorkin spaces and relative capacity

Hajłasz–Triebel–Lizorkin spaces were introduced in [24]. The associated global
capacities first appeared in [15] and they have been studied, for instance, in
[21], [22] and [27]. See also [3] and the references therein for the theory of
Triebel–Lizorkin capacities in the Euclidean case. In this section, we first
recall Hajłasz–Triebel–Lizorkin spaces on open sets and then we define the
corresponding relative capacities that seem to be novel in this generality.

Fix $1\leq p<\infty$, $1\leq q\leq\infty$, and an open set $\Omega\subset X$.
For a sequence $g=(g_{k})_{k\in{\mathbb{Z}}}$ of measurable functions in
$\Omega$, we define the mixed norm

$\lVert
g\rVert_{L^{p}(\Omega;\,l^{q})}=\bigl{\|}\|g\|_{l^{q}({\mathbb{Z}})}\bigr{\|}_{L^{p}(\Omega)}\,,$

where

$\|g\|_{l^{q}({\mathbb{Z}})}=\begin{cases}\Bigl{(}\sum_{k\in{\mathbb{Z}}}|g_{k}|^{q}\Bigr{)}^{1/q},&\quad\text{
for }1\leq q<\infty,\\\ \;\sup_{k\in{\mathbb{Z}}}|g_{k}|,&\quad\text{ for
}q=\infty.\end{cases}$

We use the following Fefferman–Stein vector-valued maximal function
inequality; we refer to [11, Theorem 1.2] for a proof. If $1<p<\infty$ and
$1<q\leq\infty$, then there exists a positive constant $C>0$ such that for all
sequences of measurable functions $(g_{k})_{k\in{\mathbb{Z}}}$ in $X$ we have

$\big{\lVert}(M^{*}g_{k})_{k\in{\mathbb{Z}}}\big{\rVert}_{L^{p}(X;l^{q})}\leq
C\lVert(g_{k})_{k\in{\mathbb{Z}}}\rVert_{L^{p}(X;l^{q})}\,,$ (11)

where $M^{*}g_{k}$ is the noncentered maximal function of $g_{k}$ for each
$k\in{\mathbb{Z}}$.

######  Definition 4.1.

Let $1<p<\infty$, $1\leq q\leq\infty$, and $0<\beta<1$, and let $\Omega\subset
X$ be an open set. The homogeneous Hajłasz–Triebel–Lizorkin space
$\dot{M}^{\beta}_{p,q}(\Omega)$ is the seminormed space of all measurable
functions $u\colon\Omega\to{\mathbb{R}}$ satisfying

$\lvert
u\rvert_{\dot{M}^{\beta}_{p,q}(\Omega)}=\inf_{g\in\mathbb{D}^{\beta}_{\Omega}(u)}\lVert
g\rVert_{L^{p}(\Omega;l^{q})}<\infty\,.$

Notice that $\lvert u\rvert_{\dot{M}^{\beta}_{p,q}(\Omega)}\leq\lvert
u\rvert_{\dot{M}^{\beta}_{p,\tau}(\Omega)}$ whenever $1\leq\tau\leq
q\leq\infty$. We remark that
$\lvert\,\cdot\,\rvert_{\dot{M}^{\beta}_{p,q}(\Omega)}$ is a seminorm but not
a norm since $\lvert u\rvert_{\dot{M}^{\beta}_{p,q}(\Omega)}=0$ whenever $u$
is a constant function.

The following definition of a relative Hajłasz–Triebel–Lizorkin
$(\beta,p,q)$-capacity $\operatorname{cap}_{\dot{M}^{\beta}_{p,q}}$ is
motivated by [9, Definition 4.1]; see also [28, Definition 7.1], [10] and [29,
§11].

######  Definition 4.2.

Let $1<p<\infty$, $1\leq q\leq\infty$, $0<\beta<1$, and $\Lambda\geq 2$. Let
$B\subset X$ be a ball and let $E\subset{\mathchoice{\hbox to0.0pt{$\kern
0.0pt\kern 0.0pt\overline{\phantom{\displaystyle B}\kern
0.0pt}$\hss}{B}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\textstyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle B}\kern
0.0pt}$\hss}{B}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptscriptstyle B}\kern 0.0pt}$\hss}{B}}}$ be a
closed set. Then we write

$\operatorname{cap}_{\dot{M}^{\beta}_{p,q}}(E,2B,\Lambda
B)=\inf_{\varphi}|\varphi|_{\dot{M}^{\beta}_{p,q}(\Lambda B)}^{p}\,,$

where the infimum is taken over all continuous functions $\varphi\colon
X\to{\mathbb{R}}$ such that $\varphi(x)\geq 1$ for every $x\in E$ and
$\varphi(x)=0$ for every $x\in X\setminus 2B$.

Observe that if $F\subset E\subset{\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle B}\kern
0.0pt}$\hss}{B}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle B}\kern
0.0pt}$\hss}{B}}}$ are closed sets and $1\leq\tau\leq q\leq\infty$, then

$\operatorname{cap}_{\dot{M}^{\beta}_{p,q}}(F,2B,\Lambda
B)\leq\operatorname{cap}_{\dot{M}^{\beta}_{p,q}}(E,2B,\Lambda
B)\leq\operatorname{cap}_{\dot{M}^{\beta}_{p,\tau}}(E,2B,\Lambda B)\,.$ (12)

Next we estimate the relative capacities of balls. In part (a) we have $q=1$
and in part (b) $q=\infty$; compare to (12).

######  Theorem 4.3.

Let $1<p<\infty$, $0<\beta<1$, $x_{0}\in X$, and $r>0$.

  * (a)

If $\Lambda\geq 2$, then

$\operatorname{cap}_{\dot{M}^{\beta}_{p,1}}\bigl{(}\overline{B(x_{0},r)},B(x_{0},2r),B(x_{0},\Lambda
r)\bigr{)}\leq C(c_{\mu},\beta,p,\Lambda)r^{-\beta p}\mu(B(x_{0},r))\,.$

  * (b)

Assume in addition that $X$ is connected and $r<(1/8)\operatorname{diam}(X)$.
If $\Lambda>2$, then

$r^{-\beta p}\mu(B(x_{0},r))\leq
C(\beta,p,c_{\mu},\Lambda)\operatorname{cap}_{\dot{M}^{\beta}_{p,\infty}}\bigl{(}\overline{B(x_{0},r)},B(x_{0},2r),B(x_{0},\Lambda
r)\bigr{)}\,.$

###### Proof.

(a) Write $B=B(x_{0},r)$ and let

$\varphi(x)=\max\bigl{\\{}0,1-r^{-1}\operatorname{dist}(x,B)\bigr{\\}}\,$

for every $x\in X$. Then $0\leq\varphi\leq 1$ in $X$, $\varphi=1$ in
${\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle B}\kern
0.0pt}$\hss}{B}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle B}\kern
0.0pt}$\hss}{B}}}$, $\varphi=0$ in $X\setminus 2B$, and $\varphi$ is an
$r^{-1}$-Lipschitz function in $X$. Since $\varphi$ is admissible for the
relative capacity, we have

$\operatorname{cap}_{\dot{M}^{\beta}_{p,1}}({\mathchoice{\hbox to0.0pt{$\kern
0.0pt\kern 0.0pt\overline{\phantom{\displaystyle B}\kern
0.0pt}$\hss}{B}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\textstyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle B}\kern
0.0pt}$\hss}{B}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptscriptstyle B}\kern
0.0pt}$\hss}{B}}},2B,\Lambda
B)\leq\lvert\varphi\rvert_{\dot{M}^{\beta}_{p,1}(\Lambda B)}^{p}\,.$

Lemma 3.3, with $u=1$ and $g_{k}=0$ for all $k\in{\mathbb{Z}}$, implies that
$(\rho_{k})_{k\in{\mathbb{Z}}}\in\mathbb{D}^{\beta}_{X}(\varphi)$, where

$\rho_{k}=2^{k(\beta-1)}r^{-1}\mathbf{1}_{\\{x\in
X\,:\,\varphi(x)\not=0\\}}\,,\qquad\text{ for all }k\in{\mathbb{Z}}\,.$

Let $m$ be the smallest integer such that $2^{-m-1}\leq 2\Lambda r$. Observe
that $(\mathbf{1}_{k\geq m}\rho_{k})_{k\in{\mathbb{Z}}}\in\mathbb{D}_{\Lambda
B}^{\beta}(\varphi)$. Thus

$\displaystyle\lvert\varphi\rvert_{\dot{M}^{\beta}_{p,1}(\Lambda B)}^{p}$
$\displaystyle\leq\big{\lVert}(\mathbf{1}_{k\geq
m}\rho_{k})_{k\in{\mathbb{Z}}}\big{\rVert}_{L^{p}(\Lambda
B;l^{1})}^{p}=\int_{\Lambda
B}\left(\sum_{k=m}^{\infty}\rho_{k}(x)\right)^{p}\,d\mu(x)$ $\displaystyle\leq
r^{-p}\int_{2B}\left(\sum_{k=m}^{\infty}2^{k(\beta-1)}\right)^{p}\,d\mu(x)$
$\displaystyle\leq C(\beta,p)r^{-p}2^{mp(\beta-1)}\mu(2B)\leq
C(c_{\mu},\beta,p,\Lambda)r^{-\beta p}\mu(B)\,,$

and the claim (a) follows from this.

(b) The proof is similar to the proof of Lemma 3.2. Write $B=B(x_{0},r)$ and
choose $\lambda=\min\\{3,\Lambda\\}>2$. Let $\varphi\colon X\to{\mathbb{R}}$
be a continuous function such that $\varphi(x)\geq 1$ for every
$x\in{\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle B}\kern
0.0pt}$\hss}{B}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle B}\kern
0.0pt}$\hss}{B}}}$ and $\varphi(x)=0$ for every $x\in X\setminus 2B$, and let
$g=(g_{k})_{k\in{\mathbb{Z}}}\in\mathbb{D}^{\beta}_{\Lambda B}(\varphi)$.

Fix $m\in{\mathbb{Z}}$ such that $2^{-m-1}\leq r<2^{-m}$. Then

$2^{-m-1}\leq r\leq d(x,y)<(1+\lambda)r\leq 4r<2^{-m+2}$

for all $x\in B$ and $y\in\lambda B\setminus 2B$. Since $\varphi\geq 1$ in $B$
and $\varphi=0$ in $\lambda B\setminus 2B$, we can proceed as in (7) and
obtain

$\begin{split}1&\leq\mathchoice{\mathop{\kern 1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996ptB}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B}}\varphi(x)\,d\mu(x)=\mathchoice{\mathop{\kern
1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996ptB}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B}}\lvert\varphi(x)-\varphi_{\lambda
B\setminus 2B}\rvert\,d\mu(x)\\\ &\leq\mathchoice{\mathop{\kern
1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996ptB}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B}}\mathchoice{\mathop{\kern
1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996pt\lambda
B\setminus 2B}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{\lambda
B\setminus 2B}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{\lambda
B\setminus 2B}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{\lambda
B\setminus 2B}}\lvert\varphi(x)-\varphi(y)\rvert\,d\mu(y)\,d\mu(x)\\\ &\leq
4r^{\beta}\sum_{k=m-2}^{m}\mathchoice{\mathop{\kern 1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996ptB}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B}}g_{k}(x)\,d\mu(x)+4r^{\beta}\sum_{k=m-2}^{m}\mathchoice{\mathop{\kern
1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996pt\lambda
B\setminus 2B}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{\lambda
B\setminus 2B}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{\lambda
B\setminus 2B}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{\lambda
B\setminus 2B}}g_{k}(y)\,d\mu(y)\,.\end{split}$ (13)

Here

$\begin{split}\sum_{k=m-2}^{m}\mathchoice{\mathop{\kern 1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996ptB}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B}}g_{k}(x)\,d\mu(x)&\leq\sum_{k=m-2}^{m}\left(\mathchoice{\mathop{\kern
1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996ptB}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B}}g_{k}(x)^{p}\,d\mu(x)\right)^{1/p}\\\
&\leq 3\left(\mathchoice{\mathop{\kern 1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996ptB}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B}}\left(\sup_{k\in{\mathbb{Z}}}g_{k}(y)\right)^{p}\,d\mu(x)\right)^{1/p}\,.\end{split}$
(14)

Since $X$ is connected and $\lambda r<4r<\operatorname{diam}(X)/2$, by [4,
Lemma 3.7] there exists a constant $0<c_{R}=C(c_{\mu},\Lambda)<1$ such that

$\mu(2B)=\mu(B(x_{0},2r))\leq c_{R}\,\mu(B(x_{0},\lambda
r))=c_{R}\,\mu(\lambda B)\,,$

and therefore

$\displaystyle\mu(\lambda B\setminus 2B)\geq(1-c_{R})\mu(\lambda B)\,.$

Hence,

$\begin{split}\sum_{k=m-2}^{m}\mathchoice{\mathop{\kern 1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996pt\lambda
B\setminus 2B}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{\lambda
B\setminus 2B}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{\lambda
B\setminus 2B}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{\lambda
B\setminus
2B}}g_{k}(y)\,d\mu(y)&\leq\frac{1}{1-c_{R}}\sum_{k=m-2}^{m}\mathchoice{\mathop{\kern
1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996pt\lambda
B}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{\lambda
B}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{\lambda
B}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{\lambda
B}}g_{k}(x)\,d\mu(x)\\\
&\leq\frac{1}{1-c_{R}}\sum_{k=m-2}^{m}\left(\mathchoice{\mathop{\kern
1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996pt\lambda
B}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{\lambda
B}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{\lambda
B}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{\lambda
B}}g_{k}(x)^{p}\,d\mu(x)\right)^{1/p}\\\
&\leq\frac{3}{1-c_{R}}\left(\mathchoice{\mathop{\kern 1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996pt\lambda
B}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{\lambda
B}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{\lambda
B}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{\lambda
B}}\left(\sup_{k\in{\mathbb{Z}}}g_{k}(y)\right)^{p}\,d\mu(x)\right)^{1/p}\,.\end{split}$
(15)

By combining inequalities (13), (14) and (15), we obtain

$\displaystyle 1$ $\displaystyle\leq\left(\mathchoice{\mathop{\kern
1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996ptB}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B}}\mathchoice{\mathop{\kern
1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996pt\lambda
B\setminus 2B}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{\lambda
B\setminus 2B}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{\lambda
B\setminus 2B}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{\lambda
B\setminus 2B}}\lvert\varphi(x)-\varphi(y)\rvert\,d\mu(y)\,d\mu(x)\right)^{p}$
$\displaystyle\leq C(p,c_{R})r^{\beta p}\left(\mathchoice{\mathop{\kern
1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996ptB}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B}}\left(\sup_{k\in{\mathbb{Z}}}g_{k}(y)\right)^{p}\,d\mu(x)+\mathchoice{\mathop{\kern
1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996pt\lambda
B}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{\lambda
B}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{\lambda
B}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{\lambda
B}}\left(\sup_{k\in{\mathbb{Z}}}g_{k}(y)\right)^{p}\,d\mu(x)\right)\,,$

and thus

$r^{-\beta p}\mu(B)\leq C(p,c_{R},\beta)\int_{\Lambda
B}\left(\sup_{k\in{\mathbb{Z}}}g_{k}(y)\right)^{p}\,d\mu(x)\,.$

Since $c_{R}=C(c_{\mu},\Lambda)$, the desired inequality follows by taking
infimum over all functions $\varphi$ and $g$ as above. ∎

Observe the strict lower bound $\Lambda>2$ in part (b) of Theorem 4.3. The
following Example 4.4, together with inequality (12), shows that we cannot
relax this lower bound to $\Lambda\geq 2$.

######  Example 4.4.

Let $X={\mathbb{R}}^{n}$ equipped with the Euclidean distance and the Lebesgue
measure. Fix $1<p<\infty$ and $0<\beta<1$ such that $\beta p<1$. We show that

$\operatorname{cap}_{\dot{M}^{\beta}_{p,1}}\bigl{(}\overline{B(0,1)},B(0,2),B(0,2)\bigr{)}=0\,.$
(16)

Fix $j\in{\mathbb{N}}$ and let

$\varphi_{j}(x)=\min\Bigl{\\{}1,2^{j}\operatorname{dist}(x,{\mathbb{R}}^{n}\setminus
B(0,2))\Bigr{\\}}$

for every $x\in{\mathbb{R}}^{n}$. Observe that $0\leq\varphi_{j}\leq 1$ in
${\mathbb{R}}^{n}$, $\varphi_{j}(x)=1$ for every $x\in\overline{B(0,1)}$ and
$\varphi_{j}(x)=0$ for every ${\mathbb{R}}^{n}\setminus B(0,2)$. It follows
that $\varphi_{j}$ is an admissible test function for the capacity on the
left-hand side of (16). Write $B_{j}=B(0,2-2^{-j})$ and $A_{j}=B(0,2)\setminus
B_{j}$. For every $k\in{\mathbb{Z}}$, we define

$g_{k}(x)=\begin{cases}0\,,\quad&\text{ if }k<-2\,,\\\
2^{j\beta}\mathbf{1}_{A_{j}}\,,\quad&\text{ if }-2\leq k<j\,,\\\
2^{j-k(1-\beta)}\mathbf{1}_{A_{j}}\,,\quad&\text{ if }j\leq k\,.\end{cases}$

Since $\varphi_{j}=1$ in $B_{j}$, a straightforward case study shows that
$(g_{k})_{k\in\mathbb{Z}}\in\mathbb{D}^{\beta}_{B(0,2)}(\varphi_{j})$. Hence,

$\displaystyle\operatorname{cap}_{\dot{M}^{\beta}_{p,1}}\bigl{(}\overline{B(0,1)},B(0,2),B(0,2)\bigr{)}\leq\lvert\varphi_{j}\rvert_{\dot{M}^{\beta}_{p,1}(B(0,2))}^{p}\leq\int_{B(0,2)}\left(\sum_{k\in{\mathbb{Z}}}g_{k}(x)\right)^{p}\,d\mu(x)$
$\displaystyle\qquad\leq
2^{p-1}\int_{A_{j}}\left(\sum_{k=-2}^{j-1}2^{j\beta}\right)^{p}\,d\mu(x)+2^{p-1}\int_{A_{j}}\left(\sum_{k=j}^{\infty}2^{j-k(1-\beta)}\right)^{p}\,d\mu(x)\,.$

Here

$\displaystyle\int_{A_{j}}\left(\sum_{k=-2}^{j-1}2^{j\beta}\right)^{p}\,d\mu(x)\leq\lvert
A_{j}\rvert(j+2)^{p}2^{j\beta p}\leq C(n)2^{-j(1-\beta p)}(j+2)^{p}$

and

$\displaystyle\int_{A_{j}}\left(\sum_{k=j}^{\infty}2^{j-k(1-\beta)}\right)^{p}\,d\mu(x)$
$\displaystyle=\lvert A_{j}\rvert 2^{j\beta
p}\left(\sum_{k=j}^{\infty}2^{(j-k)(1-\beta)}\right)^{p}$ $\displaystyle\leq
C(p,n,\beta)2^{-j(1-\beta p)}\,.$

Since $\beta p<1$, by taking $j\to\infty$ we see that (16) holds.

We need suitable Hausdorff contents to obtain more elaborate lower bounds for
relative capacities of general sets, see Theorem 4.6 below.

######  Definition 4.5.

Let $0<\rho\leq\infty$ and $d\geq 0$. The ($\rho$-restricted) Hausdorff
content of codimension $d$ of a set $F\subset X$ is defined by

${\mathcal{H}}^{\mu,d}_{\rho}(F)=\inf\Biggl{\\{}\sum_{k}\mu(B(x_{k},r_{k}))\,r_{k}^{-d}\,:\,F\subset\bigcup_{k}B(x_{k},r_{k})\text{
and }0<r_{k}\leq\rho\Biggr{\\}}\,.$

For $q=\infty$ we have the following lower bound estimate for relative
capacities in terms of Hausdorff contents. Observe that by (12) the
corresponding result holds for all $1\leq q<\infty$ as well. We refer to [22]
for recent advances on related lower bound estimates.

######  Theorem 4.6.

Let $1<p<\infty$, $0\leq\eta<p$, $0<\beta<1$, and $\Lambda\geq 41$, and assume
that $\mu$ satisfies the reverse doubling condition (4) with a constant
$0<c_{R}<1$. Let $B=B(x_{0},r)$, where $x_{0}\in X$ and
$0<r<(1/80)\operatorname{diam}(X)$, and let $E\subset{\mathchoice{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle B}\kern
0.0pt}$\hss}{B}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\textstyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle B}\kern
0.0pt}$\hss}{B}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptscriptstyle B}\kern 0.0pt}$\hss}{B}}}$ be a
closed set. Then there exists a constant
$C=C(\beta,p,\eta,c_{R},c_{\mu},\Lambda)>0$ such that

$\mathcal{H}^{\mu,\beta\eta}_{5\Lambda r}(E)\leq
Cr^{\beta(p-\eta)}\operatorname{cap}_{\dot{M}^{\beta}_{p,\infty}}(E,2B,\Lambda
B).$

###### Proof.

Let $\varphi\colon X\to{\mathbb{R}}$ be a continuous function such that
$\varphi(x)\geq 1$ for every $x\in E$ and $\varphi(x)=0$ for every $x\in
X\setminus 2B$ and let
$g=(g_{k})_{k\in{\mathbb{Z}}}\in\mathbb{D}^{\beta}_{\Lambda B}(\varphi)$. By
replacing the function $\varphi$ with $\max\\{0,\min\\{\varphi,1\\}\\}$, if
necessary, we may assume that $0\leq\varphi\leq 1$. Thus $\varphi$ is
continuous on $X$, $\varphi=1$ on $E$, and $\varphi=0$ on $X\setminus 2B$. Fix
$m\in{\mathbb{Z}}$ such that $2^{-m-1}\leq 5r<2^{-m}$. Let $x\in E$ and write
$B_{0}=B(x,2^{-m})$.

Since $8r<80r<\operatorname{diam}(X)$, by inequality (4) we have

$\displaystyle 0\leq\varphi_{B_{0}}=\mathchoice{\mathop{\kern 1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996ptB_{0}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B_{0}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B_{0}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B_{0}}}\varphi(y)\,dy\leq\frac{\mu(2B)}{\mu(B_{0})}\leq\frac{\mu(B(x_{0},2r))}{\mu(B(x_{0},4r))}\leq
c_{R}<1.$

As a consequence, since $x\in E$, we find that

$\lvert\varphi(x)-\varphi_{B_{0}}\rvert\geq 1-c_{R}>0.$

Define $B_{n}=2^{-n}B_{0}=B(x,2^{-(m+n)})$ for each $n\in{\mathbb{N}}$. By the
doubling property of $\mu$,

$\displaystyle 0<1-c_{R}$
$\displaystyle\leq\lvert\varphi(x)-\varphi_{B_{0}}\rvert=\lim_{k\to\infty}\lvert\varphi_{B_{k}}-\varphi_{B_{0}}\rvert=\lim_{k\to\infty}\left\lvert\sum_{n=0}^{k-1}(\varphi_{B_{n+1}}-\varphi_{B_{n}})\right\rvert$
$\displaystyle\leq\sum_{n=0}^{\infty}\mathchoice{\mathop{\kern 1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996ptB_{n+1}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B_{n+1}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B_{n+1}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B_{n+1}}}\lvert\varphi(y)-\varphi_{B_{n}}\rvert\,d\mu(y)\leq
c_{\mu}\sum_{n=0}^{\infty}\mathchoice{\mathop{\kern 1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996ptB_{n}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B_{n}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B_{n}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B_{n}}}\lvert\varphi(y)-\varphi_{B_{n}}\rvert\,d\mu(y)\,.$

Fix $n\in{\mathbb{N}}_{0}$. Observe that $2^{-(m+n)+3}\leq
80r<\operatorname{diam}(X)$ and that $B(x,2^{-(m+n)+2})\subset\Lambda B$.
Hence, Lemma 3.2 implies that

$\displaystyle\mathchoice{\mathop{\kern 1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996ptB_{n}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B_{n}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B_{n}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B_{n}}}\lvert\varphi(y)-\varphi_{B_{n}}\rvert\,d\mu(y)$
$\displaystyle\leq
C(c_{\mu},c_{R})2^{-(m+n)\beta}\sum_{k=n+m-3}^{n+m}\;\mathchoice{\mathop{\kern
1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996ptB(x,2^{-(m+n)+2})}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x,2^{-(m+n)+2})}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x,2^{-(m+n)+2})}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x,2^{-(m+n)+2})}}g_{k}(y)\,d\mu(y)\,.$

By combining the above estimates, we get

$\displaystyle 0$ $\displaystyle<1-c_{R}\leq
c_{\mu}\sum_{n=0}^{\infty}\mathchoice{\mathop{\kern 1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996ptB_{n}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B_{n}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B_{n}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B_{n}}}\lvert\varphi(y)-\varphi_{B_{n}}\rvert\,d\mu(y)$
$\displaystyle\leq
C(c_{\mu},c_{R})\sum_{n=0}^{\infty}2^{-(m+n)\beta}\sum_{k=n+m-3}^{n+m}\mathchoice{\mathop{\kern
1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996ptB(x,2^{-(m+n)+2})}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x,2^{-(m+n)+2})}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x,2^{-(m+n)+2})}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x,2^{-(m+n)+2})}}g_{k}(y)\,d\mu(y)$
$\displaystyle\leq
C(c_{\mu},c_{R})\sum_{n=m-2}^{\infty}2^{-n\beta}\;\sum_{k=n-1}^{n+2}\mathchoice{\mathop{\kern
1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996ptB(x,2^{-n})}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x,2^{-n})}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x,2^{-n})}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x,2^{-n})}}g_{k}(y)\,d\mu(y)$
$\displaystyle\leq
C(c_{\mu},c_{R})\sum_{n=m-2}^{\infty}2^{-n\beta}\;\sum_{k=n-1}^{n+2}\left(\mathchoice{\mathop{\kern
1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996ptB(x,2^{-n})}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x,2^{-n})}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x,2^{-n})}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x,2^{-n})}}g_{k}(y)^{p}\,d\mu(y)\right)^{1/p}$
$\displaystyle\leq
C(c_{\mu},c_{R})\sum_{n=m-2}^{\infty}2^{-n\beta}\;\left(\mathchoice{\mathop{\kern
1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996ptB(x,2^{-n})}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x,2^{-n})}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x,2^{-n})}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x,2^{-n})}}\left(\sup_{k\in{\mathbb{Z}}}g_{k}(y)\right)^{p}\,d\mu(y)\right)^{1/p}\,.$

Let $\delta=\beta(p-\eta)/p>0$. Then

$\displaystyle
2^{(m-2)\delta}\sum_{n=m-2}^{\infty}2^{-n\delta}=C(\beta,p,\eta,c_{R})(1-c_{R})$
$\displaystyle\qquad\leq
C(\beta,p,\eta,c_{R},c_{\mu})\sum_{n=m-2}^{\infty}2^{-n\beta}\;\left(\mathchoice{\mathop{\kern
1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996ptB(x,2^{-n})}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x,2^{-n})}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x,2^{-n})}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x,2^{-n})}}\left(\sup_{k\in{\mathbb{Z}}}g_{k}(y)\right)^{p}\,d\mu(y)\right)^{1/p}\,.$

In particular, by comparing the above sums it follows that there exists $n\geq
m-2$, depending on $x$, such that

$\mu(B(x,2^{-n}))(2^{-n})^{-\beta\eta}\leq
C(\beta,p,\eta,c_{R},c_{\mu})2^{-\beta
m(p-\eta)}\;\int_{B(x,2^{-n})}\left(\sup_{k\in{\mathbb{Z}}}g_{k}(y)\right)^{p}\,d\mu(y).$

Write $r_{x}=2^{-n}$ and $B_{x}=B(x,r_{x})=B(x,2^{-n})$. Then the previous
estimate gives

$\mu(B_{x})r_{x}^{-\beta\eta}\leq
C(\beta,p,\eta,c_{R},c_{\mu})r^{\beta(p-\eta)}\;\int_{B_{x}}\left(\sup_{k\in{\mathbb{Z}}}g_{k}(y)\right)^{p}\,d\mu(y).$

By the $5r$-covering lemma [4, Lemma 1.7], we obtain points $x_{\ell}\in E$,
$\ell\in{\mathbb{N}}$, such that the balls $B_{x_{\ell}}\subset\Lambda B$ with
radii $r_{x_{\ell}}\leq\Lambda r$ are pairwise disjoint and
$E\subset\bigcup_{\ell=1}^{\infty}5B_{x_{\ell}}$. Hence,

$\displaystyle\mathcal{H}^{\mu,\beta\eta}_{5\Lambda r}(E)$
$\displaystyle\leq\sum_{\ell=1}^{\infty}\mu(5B_{x_{\ell}})(5r_{x_{\ell}})^{-\beta\eta}\leq
Cr^{\beta(p-\eta)}\;\sum_{\ell=1}^{\infty}\int_{B_{x_{\ell}}}\left(\sup_{k\in{\mathbb{Z}}}g_{k}(y)\right)^{p}\,d\mu(y)$
$\displaystyle\leq Cr^{\beta(p-\eta)}\;\int_{\Lambda
B}\left(\sup_{k\in{\mathbb{Z}}}g_{k}(y)\right)^{p}\,d\mu(y)\,.$

The desired inequality follows by taking infimum over all functions $g$ and
$\varphi$ as above. ∎

##  5\. Discrete potential operator

In this section we introduce and study a discrete potential operator
$\operatorname{{\mathcal{Q}}}_{\beta}$. The main importance of this operator
is that it majorizes the potential $\operatorname{\mathcal{H}}_{\beta}$ that
is used in the next section to define the Triebel–Lizorkin
$(\beta,p,q)$-capacity $\operatorname{Cap}^{\beta}_{p,q}$; see Definitions 6.1
and 6.2. Moreover, due to the Lipschitz-continuity of the bump functions
$\psi_{n,i}$ that are used in the definition of
$\operatorname{{\mathcal{Q}}}_{\beta}$, this operator has better smoothing
properties than $\operatorname{\mathcal{H}}_{\beta}$ whose definition is based
on characteristic functions of balls. In particular, Lemma 5.2 seem not to be
available for $\operatorname{\mathcal{H}}_{\beta}$.

We begin with a construction of a partition of unity, following the ideas from
[23] and [1]. Let $n\in{\mathbb{Z}}$ be such that $n\geq n_{0}$; recall that
$2^{-n_{0}}$ is essentially $\operatorname{diam}(X)$. By a maximal packing
argument [16, p. 101] and using the separability of $X$, we may select a
countable number of points $x_{n,i}\in X$, $i\in P_{n}$, such that the balls
$B(x_{n,i},2^{-n-1})$, $i\in P_{n}$, are pairwise disjoint, while

$X=\bigcup_{i\in P_{n}}B(x_{n,i},2^{-n}).$ (17)

Since $X$ is equipped with a doubling measure $\mu$, the balls
$B(x_{n,i},6\cdot 2^{-n})$ have a finite overlapping property, that is, there
exists a constant $N=C(c_{\mu})$ such that

$\sum_{i\in P_{n}}\mathbf{1}_{B(x_{n,i},6\cdot 2^{-n})}\leq N.$ (18)

In addition, there are constants $\kappa=C(c_{\mu})$ and $L=C(c_{\mu})$ and a
family of functions $\psi_{n,i}$, $i\in P_{n}$, with the following properties:
$0\leq\psi_{n,i}\leq 1$, $\psi_{n,i}=0$ in $X\setminus B(x_{n,i},6\cdot
2^{-n})$, $\psi_{n,i}\geq\kappa$ in $B(x_{n,i},3\cdot 2^{-n})$, $\psi_{n,i}$
is $2^{n}L$-Lipschitz in $X$, and

$\sum_{i\in P_{n}}\psi_{n,i}(x)=1$

for every $x\in X$.

The family $\psi_{n,i}$, $i\in P_{n}$, is called a partition of unity
subordinate to the cover $B(x_{n,i},2^{-n})$. We keep these families, for
$n\geq n_{0}$, fixed throughout the rest of the paper, starting from the
following definition.

######  Definition 5.1.

Let $\beta>0$ and let $f=(f_{n})_{n=n_{0}}^{\infty}$ be a sequence of
nonnegative Borel functions in $X$. The discrete potential
$\operatorname{{\mathcal{Q}}}_{\beta}f$ is defined by

$\operatorname{{\mathcal{Q}}}_{\beta}f(x)=\sum_{n=n_{0}}^{\infty}2^{-\beta
n}\sum_{i\in P_{n}}\psi_{n,i}(x)\mathchoice{\mathop{\kern 1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996ptB(x_{n,i},3\cdot
2^{-n})}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{n,i},3\cdot
2^{-n})}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{n,i},3\cdot
2^{-n})}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{n,i},3\cdot
2^{-n})}}f_{n}(y)\,d\mu(y)\,,\qquad x\in X\,.$

The closely related discrete maximal operator is studied for instance in [1,
14]. We also mention that a median type approach to discrete potential
operators appears in [15], where approximation properties of
Hajłasz–Triebel–Lizorkin functions are studied in the full range
$0<p,q<\infty$.

The following lemma allows us to control the Hajłasz–Triebel–Lizorkin seminorm
of the discrete potential $\operatorname{{\mathcal{Q}}}_{\beta}f$ in terms of
a mixed norm of $f$.

######  Lemma 5.2.

Let $1<p<\infty$, $1<q\leq\infty$, and $0<\beta<1$, and assume that
$f=(f_{n})_{n=n_{0}}^{\infty}$ is a sequence of nonnegative Borel functions in
$X$. Then there exists a constant $C=C(c_{\mu},\beta,p,q)$ such that

$\lvert\operatorname{{\mathcal{Q}}}_{\beta}f\rvert_{\dot{M}^{\beta}_{p,q}(X)}\leq
C\lVert f\rVert_{L^{p}(X;l^{q})}\,.$

###### Proof.

We may clearly assume that $\lVert f\rVert_{L^{p}(X;l^{q})}<\infty$. If $X$ is
bounded, then we extend $f$ as a sequence $(f_{n})_{n\in{\mathbb{Z}}}$ by
setting $f_{n}=0$ if $n<n_{0}$. Define $g=(g_{k})_{k\in{\mathbb{Z}}}$, where

$g_{k}(x)=Nc_{\mu}2^{\beta}\sum_{n=k}^{\infty}2^{\beta(k-n)}M^{*}f_{n}(x)+LNc_{\mu}\sum_{n=-\infty}^{k-1}2^{(\beta-1)(k-n)}M^{*}f_{n}(x)$

for every $x\in X$ and $k\in{\mathbb{Z}}$, and $N=C(c_{\mu})$ and
$L=C(c_{\mu})$ are the constants from the definition of the partitions of
unity $\psi_{n,i}$. We aim to show that
$g\in\mathbb{D}_{X}^{\beta}(\operatorname{{\mathcal{Q}}}_{\beta}f)$. For this
purpose we fix $k\in{\mathbb{Z}}$ and $x,y\in X$ such that $2^{-k-1}\leq
d(x,y)<2^{-k}$. Observe that $k\geq n_{0}$.

Write $B_{n,i}=B(x_{n,i},2^{-n})$. From the definition of
$\operatorname{{\mathcal{Q}}}_{\beta}$ we obtain

$\lvert\operatorname{{\mathcal{Q}}}_{\beta}f(x)-\operatorname{{\mathcal{Q}}}_{\beta}f(y)\rvert\leq\sum_{n=n_{0}}^{\infty}2^{-\beta
n}\sum_{i\in
P_{n}}\lvert\psi_{n,i}(x)-\psi_{n,i}(y)\rvert\mathchoice{\mathop{\kern
1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996pt3B_{n,i}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{3B_{n,i}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{3B_{n,i}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{3B_{n,i}}}f_{n}(z)\,d\mu(z)\,.$
(19)

We split the outer summation in two parts. First,

$\begin{split}&\sum_{n=k}^{\infty}2^{-\beta n}\sum_{i\in
P_{n}}\lvert\psi_{n,i}(x)-\psi_{n,i}(y)\rvert\mathchoice{\mathop{\kern
1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996pt3B_{n,i}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{3B_{n,i}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{3B_{n,i}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{3B_{n,i}}}f_{n}(z)\,d\mu(z)\\\
&\qquad\leq 2^{-k\beta}\sum_{n=k}^{\infty}2^{\beta(k-n)}\sum_{i\in
P_{n}}(\psi_{n,i}(x)+\psi_{n,i}(y))\mathchoice{\mathop{\kern 1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996pt3B_{n,i}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{3B_{n,i}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{3B_{n,i}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{3B_{n,i}}}f_{n}(z)\,d\mu(z)\\\
&\qquad\leq
c_{\mu}2^{\beta}d(x,y)^{\beta}\sum_{n=k}^{\infty}2^{\beta(k-n)}\sum_{i\in
P_{n}}\bigl{(}\mathbf{1}_{6B_{n,i}}(x)+\mathbf{1}_{6B_{n,i}}(y)\bigr{)}\mathchoice{\mathop{\kern
1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996pt6B_{n,i}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{6B_{n,i}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{6B_{n,i}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{6B_{n,i}}}f_{n}(z)\,d\mu(z)\\\
&\qquad\leq
c_{\mu}2^{\beta}d(x,y)^{\beta}\sum_{n=k}^{\infty}2^{\beta(k-n)}\sum_{i\in
P_{n}}\bigl{(}\mathbf{1}_{6B_{n,i}}(x)M^{*}f_{n}(x)+\mathbf{1}_{6B_{n,i}}(y)M^{*}f_{n}(y)\bigr{)}\\\
&\qquad\leq
Nc_{\mu}2^{\beta}d(x,y)^{\beta}\sum_{n=k}^{\infty}2^{\beta(k-n)}(M^{*}f_{n}(x)+M^{*}f_{n}(y))\,.\end{split}$
(20)

Next we assume that $n_{0}\leq n\leq k-1$ and $i\in P_{n}$. If
$\psi_{n,i}(x)\not=0$, then we have

$\displaystyle\lvert\psi_{n,i}(x)-\psi_{n,i}(y)\rvert\mathchoice{\mathop{\kern
1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996pt3B_{n,i}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{3B_{n,i}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{3B_{n,i}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{3B_{n,i}}}f_{n}(z)\,d\mu(z)$
$\displaystyle\qquad\leq
2^{n}Ld(x,y)\mathbf{1}_{6B_{n,i}}(x)c_{\mu}\mathchoice{\mathop{\kern
1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996pt6B_{n,i}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{6B_{n,i}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{6B_{n,i}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{6B_{n,i}}}f_{n}(z)\,d\mu(z)$
$\displaystyle\qquad\leq
2^{n}Lc_{\mu}d(x,y)^{\beta}2^{-k(1-\beta)}\mathbf{1}_{6B_{n,i}}(x)M^{*}f_{n}(x)\,.$

On the other hand, if $\psi_{n,i}(y)\not=0$, then a similar computation shows
that

$\displaystyle\lvert\psi_{n,i}(x)-\psi_{n,i}(y)\rvert\mathchoice{\mathop{\kern
1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996pt3B_{n,i}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{3B_{n,i}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{3B_{n,i}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{3B_{n,i}}}f_{n}(z)\,d\mu(z)\leq
2^{n}Lc_{\mu}d(x,y)^{\beta}2^{-k(1-\beta)}\mathbf{1}_{6B_{n,i}}(y)M^{*}f_{n}(y)\,.$

Thus in any case we have

$\begin{split}&\sum_{n=n_{0}}^{k-1}2^{-\beta n}\sum_{i\in
P_{n}}\lvert\psi_{n,i}(x)-\psi_{n,i}(y)\rvert\mathchoice{\mathop{\kern
1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996pt3B_{n,i}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{3B_{n,i}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{3B_{n,i}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{3B_{n,i}}}f_{n}(z)\,d\mu(z)\\\
&\quad\leq Lc_{\mu}d(x,y)^{\beta}\sum_{n=n_{0}}^{k-1}2^{-\beta
n+n-k(1-\beta)}\sum_{i\in
P_{n}}(\mathbf{1}_{6B_{n,i}}(x)M^{*}f_{n}(x)+\mathbf{1}_{6B_{n,i}}(y)M^{*}f_{n}(y))\\\
&\quad\leq
LNc_{\mu}d(x,y)^{\beta}\sum_{n=-\infty}^{k-1}2^{(\beta-1)(k-n)}(M^{*}f_{n}(x)+M^{*}f_{n}(y))\,.\end{split}$
(21)

By combining inequalities (19), (20) and (21), it follows that

$\lvert\operatorname{{\mathcal{Q}}}_{\beta}f(x)-\operatorname{{\mathcal{Q}}}_{\beta}f(y)\rvert\leq
d(x,y)^{\beta}(g_{k}(x)+g_{k}(y))\,,$

and therefore
$g\in\mathbb{D}_{X}^{\beta}(\operatorname{{\mathcal{Q}}}_{\beta}f)$.

Next, define $a=(a_{k})_{k\in{\mathbb{Z}}}$, where
$a_{k}=Nc_{\mu}2^{\beta}\cdot 2^{\beta k}$ if $k\leq 0$ and
$a_{k}=LNc_{\mu}2^{(\beta-1)k}$ if $k>0$. Observe that
$(g_{k}(x))_{k\in{\mathbb{Z}}}=a\star(M^{*}f_{n}(x))_{n\in{\mathbb{Z}}}$,
where $\star$ designates the discrete convolution

$\bigl{(}a\star(M^{*}f_{n}(x))_{n\in{\mathbb{Z}}}\bigr{)}_{k}=\sum_{n\in{\mathbb{Z}}}a_{k-n}M^{*}f_{n}(x)\,,\qquad
k\in{\mathbb{Z}}\,.$

Young’s convolution inequality [18, Corollary 20.14] implies for every $x\in
X$ that

$\displaystyle\big{\lVert}(g_{k}(x))_{k\in{\mathbb{Z}}}\big{\rVert}_{l^{q}({\mathbb{Z}})}$
$\displaystyle=\big{\lVert}a\star(M^{*}f_{n}(x))_{n\in{\mathbb{Z}}}\big{\rVert}_{l^{q}({\mathbb{Z}})}$
$\displaystyle\leq\lVert
a\rVert_{l^{1}({\mathbb{Z}})}\big{\lVert}(M^{*}f_{n}(x))_{n\in{\mathbb{Z}}}\big{\rVert}_{l^{q}({\mathbb{Z}})}$
$\displaystyle\leq
C(L,N,c_{\mu},\beta)\big{\lVert}(M^{*}f_{n}(x))_{n\in{\mathbb{Z}}}\big{\rVert}_{l^{q}({\mathbb{Z}})}\,.$

Hence, we see that

$\lvert\operatorname{{\mathcal{Q}}}_{\beta}f\rvert_{\dot{M}^{\beta}_{p,q}(X)}\leq\lVert(g_{k})_{k\in{\mathbb{Z}}}\rVert_{L^{p}(X;l^{q}({\mathbb{Z}}))}\leq
C(L,N,c_{\mu},\beta)\big{\lVert}(M^{*}f_{n})_{n\in{\mathbb{Z}}}\big{\rVert}_{L^{p}(X;l^{q}({\mathbb{Z}}))}\,.$

By the Fefferman–Stein vector-valued maximal function inequality (11), we
obtain

$\displaystyle\big{\lVert}(M^{*}f_{n})_{n\in{\mathbb{Z}}}\big{\rVert}_{L^{p}(X;l^{q}({\mathbb{Z}}))}$
$\displaystyle=\Biggl{(}\int_{X}\Biggl{(}\sum_{n\in{\mathbb{Z}}}\left(M^{*}f_{n}(x)\right)^{q}\Biggr{)}^{p/q}\,d\mu(x)\Biggl{)}^{1/p}$
$\displaystyle\leq
C(c_{\mu},p,q)\Biggl{(}\int_{X}\Biggl{(}\sum_{n\in{\mathbb{Z}}}f_{n}(x)^{q}\Biggl{)}^{p/q}\,d\mu(x)\Biggl{)}^{1/p}$
$\displaystyle=C(c_{\mu},p,q)\lVert(f_{n})_{n\in{\mathbb{Z}}}\rVert_{L^{p}(X;l^{q}({\mathbb{Z}}))}$

(with the usual modification if $q=\infty$). This concludes the proof, since
the constants $L$ and $N$ only depend on $c_{\mu}$. ∎

The following lemma gives for the discrete potential operator
$\operatorname{{\mathcal{Q}}}_{\beta}$ a local embedding, which can be viewed
as a Sobolev type inequality.

######  Lemma 5.3.

Let $1<p<\infty$, $1<q\leq\infty$, and $\beta>0$. Assume that $\mu$ satisfies
the quantitative reverse doubling condition (5) for some exponent
$\sigma>\beta p$, and let $f=(f_{n})_{n=n_{0}}^{\infty}$ be a sequence of
nonnegative Borel functions in $X$. Then there exists a constant
$C=C(c_{\sigma},p,q,\sigma,\beta,c_{\mu})$ such that

$\left(\int_{B(x_{0},r)}(\operatorname{{\mathcal{Q}}}_{\beta}f(x))^{p}\,d\mu(x)\right)^{1/p}\leq
Cr^{\beta}\lVert f\rVert_{L^{p}(X;l^{q})}$

for every $x_{0}\in X$ and $r>0$.

###### Proof.

Fix $x_{0}\in X$ and $r>0$. If $X$ is bounded, then we may clearly assume that
$0<r\leq 2^{-n_{0}}$. Let $k\in\mathbb{Z}$ be such that $2^{-k-1}<r\leq
2^{-k}$ and let $\psi_{n,i}$ be as in the definition of
$\operatorname{{\mathcal{Q}}}_{\beta}$. We write $B_{n,i}=B(x_{n,i},2^{-n})$
and $\operatorname{{\mathcal{Q}}}_{\beta}f=F+G$, where

$F(x)=\sum_{n=k}^{\infty}2^{-\beta n}\sum_{i\in
P_{n}}\psi_{n,i}(x)\mathchoice{\mathop{\kern 1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996pt3B_{n,i}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{3B_{n,i}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{3B_{n,i}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{3B_{n,i}}}f_{n}(y)\,d\mu(y)$

and

$G(x)=\sum_{n=n_{0}}^{k-1}2^{-\beta n}\sum_{i\in
P_{n}}\psi_{n,i}(x)\mathchoice{\mathop{\kern 1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996pt3B_{n,i}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{3B_{n,i}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{3B_{n,i}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{3B_{n,i}}}f_{n}(y)\,d\mu(y)$

for every $x\in X$.

Fix a point $x\in B(x_{0},r)$. We estimate $F(x)$ by using Hölder’s
inequality, with exponents $q$ and $q^{\prime}=\frac{q}{q-1}$,

$\displaystyle F(x)$ $\displaystyle=2^{-\beta
k}\sum_{n=k}^{\infty}2^{-\beta(n-k)}\sum_{i\in
P_{n}}\psi_{n,i}(x)\mathchoice{\mathop{\kern 1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996pt3B_{n,i}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{3B_{n,i}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{3B_{n,i}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{3B_{n,i}}}f_{n}(y)\,d\mu(y)$
$\displaystyle\leq 2^{-\beta
k}\left(\sum_{n=k}^{\infty}2^{-\beta(n-k)q^{\prime}}\right)^{1/q^{\prime}}\left(\sum_{n=k}^{\infty}\left(\sum_{i\in
P_{n}}\psi_{n,i}(x)\mathchoice{\mathop{\kern 1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996pt3B_{n,i}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{3B_{n,i}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{3B_{n,i}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{3B_{n,i}}}f_{n}(y)\,d\mu(y)\right)^{q}\right)^{1/q}$
$\displaystyle\leq C(q,\beta,c_{\mu})2^{-\beta
k}\left(\sum_{n=k}^{\infty}\left(\sum_{i\in
P_{n}}\mathbf{1}_{6B_{n,i}}(x)\mathchoice{\mathop{\kern 1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996pt6B_{n,i}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{6B_{n,i}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{6B_{n,i}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{6B_{n,i}}}f_{n}(y)\,d\mu(y)\right)^{q}\right)^{1/q}$
$\displaystyle\leq C(q,\beta,c_{\mu},N)2^{-\beta
k}\left(\sum_{n=k}^{\infty}\left(M^{*}f_{n}(x)\right)^{q}\right)^{1/q}\,,$

where $N=C(c_{\mu})$ is the constant from (18). By the Fefferman–Stein vector-
valued maximal function inequality (11), we obtain

$\begin{split}\int_{B(x_{0},r)}(F(x))^{p}\,d\mu(x)&\leq
C(p,q,\beta,c_{\mu},N)2^{-\beta
pk}\int_{X}\left(\sum_{n=k}^{\infty}\left(M^{*}f_{n}(x)\right)^{q}\right)^{p/q}\,d\mu(x)\\\
&\leq C(p,q,\beta,c_{\mu},N)2^{-\beta
pk}\int_{X}\left(\sum_{n=k}^{\infty}f_{n}(x)^{q}\right)^{p/q}\,d\mu(x)\end{split}$
(22)

(with an obvious modification if $q=\infty$).

On the other hand, from the quantitative reverse doubling condition (5) with
exponent $\sigma>\beta p$, it follows that

$1\leq
c_{\sigma}\frac{\mu(B(x_{0},2^{-n}))}{\mu(B(x_{0},2^{-k}))}\left(\frac{2^{-k}}{2^{-n}}\right)^{\sigma}=c_{\sigma}2^{\sigma(n-k)}\frac{\mu(B(x_{0},2^{-n}))}{\mu(B(x_{0},2^{-k}))}$

whenever $n_{0}\leq n\leq k-1$. Fix $x\in B(x_{0},r)$. We estimate $G(x)$ as
follows

$\displaystyle G(x)$
$\displaystyle=2^{-k\beta}\sum_{n=n_{0}}^{k-1}2^{-\beta(n-k)}\sum_{i\in
P_{n}}\psi_{n,i}(x)\mathchoice{\mathop{\kern 1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996ptB(x_{n,i},3\cdot
2^{-n})}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{n,i},3\cdot
2^{-n})}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{n,i},3\cdot
2^{-n})}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{n,i},3\cdot
2^{-n})}}f_{n}(y)\,d\mu(y)$ $\displaystyle\leq
c_{\sigma}^{1/p}2^{-k\beta}\sum_{n=n_{0}}^{k-1}2^{(\sigma/p-\beta)(n-k)}\left(\frac{\mu(B(x_{0},2^{-n}))}{\mu(B(x_{0},2^{-k}))}\right)^{1/p}\sum_{i\in
P_{n}}\psi_{n,i}(x)\mathchoice{\mathop{\kern 1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996pt3B_{n,i}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{3B_{n,i}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{3B_{n,i}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{3B_{n,i}}}f_{n}(y)\,d\mu(y)\,.$

Writing
$2^{(\sigma/p-\beta)(n-k)}=2^{(\sigma/p-\beta)(n-k)/p^{\prime}}2^{(\sigma/p-\beta)(n-k)/p}$
and using Hölder’s inequality, we have

$\begin{split}G(x)&\leq
C(c_{\sigma},p)2^{-k\beta}\left(\sum_{n=n_{0}}^{k-1}2^{(\sigma/p-\beta)(n-k)}\right)^{1/p^{\prime}}\\\
&\qquad\times\left(\sum_{n=n_{0}}^{k-1}2^{(\sigma/p-\beta)(n-k)}\frac{\mu(B(x_{0},2^{-n}))}{\mu(B(x_{0},2^{-k}))}\left(\sum_{i\in
P_{n}}\psi_{n,i}(x)\mathchoice{\mathop{\kern 1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996pt3B_{n,i}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{3B_{n,i}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{3B_{n,i}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{3B_{n,i}}}f_{n}(y)\,d\mu(y)\right)^{p}\right)^{1/p}\,.\end{split}$
(23)

Since $n_{0}\leq n\leq k-1$ and $x\in B(x_{0},r)$ with $r\leq 2^{-k}$, we
obtain

$\begin{split}&\left(\sum_{i\in P_{n}}\psi_{n,i}(x)\mathchoice{\mathop{\kern
1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996pt3B_{n,i}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{3B_{n,i}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{3B_{n,i}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{3B_{n,i}}}f_{n}(y)\,d\mu(y)\right)^{p}\\\
&\qquad\qquad\leq\left(\sum_{i\in
P_{n}}\mathbf{1}_{6B_{n,i}}(x)\mathchoice{\mathop{\kern 1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996pt3B_{n,i}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{3B_{n,i}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{3B_{n,i}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{3B_{n,i}}}f_{n}(y)\,d\mu(y)\right)^{p}\\\
&\qquad\qquad\leq C(c_{\mu},p)\left(\sum_{i\in
P_{n}}\mathbf{1}_{6B_{n,i}}(x)\mathchoice{\mathop{\kern 1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996ptB(x_{0},10\cdot
2^{-n})}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{0},10\cdot
2^{-n})}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{0},10\cdot
2^{-n})}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{0},10\cdot
2^{-n})}}f_{n}(y)\,d\mu(y)\right)^{p}\\\ &\qquad\qquad\leq
C(c_{\mu},p,N)\left(\mathchoice{\mathop{\kern 1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996ptB(x_{0},10\cdot
2^{-n})}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{0},10\cdot
2^{-n})}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{0},10\cdot
2^{-n})}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{0},10\cdot
2^{-n})}}f_{n}(y)\,d\mu(y)\right)^{p}\\\ &\qquad\qquad\leq
C(c_{\mu},p,N)\mathchoice{\mathop{\kern 1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996ptB(x_{0},10\cdot
2^{-n})}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{0},10\cdot
2^{-n})}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{0},10\cdot
2^{-n})}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{0},10\cdot
2^{-n})}}f_{n}(y)^{p}\,d\mu(y)\,.\end{split}$ (24)

By combining the estimates (23) and (24), we see that

$\displaystyle G(x)\leq
C(c_{\sigma},p,\sigma,\beta,c_{\mu},N)2^{-k\beta}\left(\sum_{n=n_{0}}^{k-1}\frac{2^{(\sigma/p-\beta)(n-k)}}{\mu(B(x_{0},2^{-k}))}\int_{B(x_{0},10\cdot
2^{-n})}f_{n}(y)^{p}\,d\mu(y)\right)^{1/p}\,.$

It follows that

$\begin{split}\int_{B(x_{0},r)}&G(x)^{p}\,d\mu(x)\\\ &\leq
C(c_{\sigma},p,\sigma,\beta,c_{\mu},N)2^{-kp\beta}\sum_{n=n_{0}}^{k-1}2^{(\sigma/p-\beta)(n-k)}\int_{B(x_{0},10\cdot
2^{-n})}f_{n}(y)^{p}\,d\mu(y)\\\ &\leq
C(c_{\sigma},p,\sigma,\beta,c_{\mu},N)2^{-kp\beta}\sum_{n=n_{0}}^{k-1}2^{(\sigma/p-\beta)(n-k)}\int_{X}\left(\sum_{m=n_{0}}^{k-1}f_{m}(y)^{q}\right)^{p/q}\,d\mu(y)\\\
&\leq
C(c_{\sigma},p,\sigma,\beta,c_{\mu},N)2^{-kp\beta}\int_{X}\left(\sum_{n=n_{0}}^{k-1}f_{n}(y)^{q}\right)^{p/q}\,d\mu(y)\end{split}$
(25)

(with an obvious modification if $q=\infty$).

By combining the estimates (22) and (25), we get

$\displaystyle\int_{B(x_{0},r)}(\operatorname{{\mathcal{Q}}}_{\beta}f(x))^{p}\,d\mu(x)$
$\displaystyle\leq
2^{p-1}\int_{B(x_{0},r)}(F(x))^{p}\,d\mu(x)+2^{p-1}\int_{B(x_{0},r)}(G(x))^{p}\,d\mu(x)$
$\displaystyle\leq
C(c_{\sigma},p,q,\sigma,\beta,c_{\mu},N)2^{-kp\beta}\int_{X}\left(\sum_{n=n_{0}}^{\infty}f_{n}(y)^{q}\right)^{p/q}\,d\mu(y)$
$\displaystyle\leq C(c_{\sigma},p,q,\sigma,\beta,c_{\mu},N)r^{\beta p}\lVert
f\rVert_{L^{p}(X;l^{q})}^{p}\,.$

This concludes the proof since $N=C(c_{\mu})$. ∎

##  6\. Capacity with respect to a potential

Next we define the potential $\operatorname{\mathcal{H}}_{\beta}$ and the
associated Triebel–Lizorkin capacity $\operatorname{Cap}^{\beta}_{p,q}$. The
main goal in this section is to show that the capacities
$\operatorname{Cap}^{\beta}_{p,q}$ and
$\operatorname{cap}_{\dot{M}^{\beta}_{p,q}}$ are equivalent, as was stated in
Theorem 1.1 in the introduction. The proof of Theorem 1.1 follows by combining
Theorems 6.5 and 6.6 that are presented at the end of this section.

The discrete potential operator $\operatorname{{\mathcal{Q}}}_{\beta}$ serves
as an important technical tool in the proof of Theorem 6.5. More precisely, in
Lemmas 6.3 and 6.4 we connect $\operatorname{{\mathcal{Q}}}_{\beta}$ with the
potential $\operatorname{\mathcal{H}}_{\beta}$ and the capacity
$\operatorname{cap}_{\dot{M}^{\beta}_{p,q}}$, respectively, and these two
elements are then brought together in the proof of Theorem 6.5. On the other
hand, Theorem 6.6 follows by a much more direct argument.

######  Definition 6.1.

Let $\beta>0$ and let $f=(f_{n})_{n=n_{0}}^{\infty}$ be a sequence of
nonnegative Borel functions in $X$. The potential
$\operatorname{\mathcal{H}}_{\beta}f$ is defined by

$\operatorname{\mathcal{H}}_{\beta}f(x)=\sum_{n=n_{0}}^{\infty}2^{-\beta
n}\int_{X}\frac{\mathbf{1}_{B(y,2^{-n})}(x)}{\mu(B(y,2^{-n}))}f_{n}(y)\,d\mu(y)\,,\qquad
x\in X\,.$

We use potentials $\operatorname{\mathcal{H}}_{\beta}$ to define the capacity
$\operatorname{Cap}^{\beta}_{p,q}$, by adapting the treatment in [3, Section
4.4] to the metric space setting. This approach is a Triebel–Lizorkin variant
of the so-called Meyer’s theory of $L^{p}$-capacities, that was initially
developed in [30]. See also [35, Section 3.2] for a clear presentation of the
$L^{p}$-theory in (weighted) Euclidean spaces.

######  Definition 6.2.

Let $1<p<\infty$, $1\leq q\leq\infty$, and $\beta>0$. The
$(\beta,p,q)$-capacity of a set $E\subset X$ is defined by

$\operatorname{Cap}^{\beta}_{p,q}(E)=\inf\bigl{\\{}\lVert
f\rVert_{L^{p}(X;l^{q})}^{p}\,:\,f=(f_{n})_{n=n_{0}}^{\infty}\geq 0\text{ and
}\operatorname{\mathcal{H}}_{\beta}f(x)\geq 1\text{ for all }x\in
E\bigr{\\}}\,,$

where $(f_{n})_{n=n_{0}}^{\infty}\geq 0$ in the infimum means that
$(f_{n})_{n=n_{0}}^{\infty}$ is a sequence of nonnegative Borel functions in
$X$.

Observe that if $F\subset E\subset X$ and $1\leq\tau\leq q\leq\infty$, then

$\operatorname{Cap}^{\beta}_{p,q}(F)\leq\operatorname{Cap}^{\beta}_{p,q}(E)\leq\operatorname{Cap}^{\beta}_{p,\tau}(E).$
(26)

We have the following pointwise comparison between the potentials
$\operatorname{\mathcal{H}}_{\beta}f$ and
$\operatorname{{\mathcal{Q}}}_{\beta}f$.

######  Lemma 6.3.

Let $\beta>0$ and let $f=(f_{n})_{n=n_{0}}^{\infty}$ be a sequence of
nonnegative Borel functions in $X$. Then there exists a constant
$C=C(c_{\mu})>0$ such that

$\operatorname{\mathcal{H}}_{\beta}f(x)\leq
C\operatorname{{\mathcal{Q}}}_{\beta}f(x)$

for every $x\in X$.

###### Proof.

Let $x\in X$, let $n\in{\mathbb{Z}}$ be such that $n\geq n_{0}$, and let
functions $\psi_{n,i}$, $i\in P_{n}$, be as in the definition of
$\operatorname{{\mathcal{Q}}}_{\beta}f$, see Definition 5.1. By the covering
property (17), there exists $j\in P_{n}$ such that $x\in B(x_{n,j},2^{-n})$.
Then $B(x,2^{-n})\subset B(x_{n,j},3\cdot 2^{-n})\subset B(y,5\cdot 2^{-n})$
for every $y\in B(x,2^{-n})$. Observe also that
$\mathbf{1}_{B(y,2^{-n})}(x)=\mathbf{1}_{B(x,2^{-n})}(y)$ for every $y\in X$.
It follows that

$\displaystyle 2^{-\beta
n}\int_{X}\frac{\mathbf{1}_{B(y,2^{-n})}(x)}{\mu(B(y,2^{-n}))}f_{n}(y)\,d\mu(y)$
$\displaystyle\leq c_{\mu}^{3}2^{-\beta n}\mathchoice{\mathop{\kern
1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996ptB(x_{n,j},3\cdot
2^{-n})}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{n,j},3\cdot
2^{-n})}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{n,j},3\cdot
2^{-n})}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{n,j},3\cdot
2^{-n})}}f_{n}(y)\,d\mu(y)$ $\displaystyle\leq\kappa^{-1}c_{\mu}^{3}2^{-\beta
n}\psi_{n,j}(x)\mathchoice{\mathop{\kern 1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996ptB(x_{n,j},3\cdot
2^{-n})}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{n,j},3\cdot
2^{-n})}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{n,j},3\cdot
2^{-n})}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{n,j},3\cdot
2^{-n})}}f_{n}(y)\,d\mu(y)$ $\displaystyle\leq\kappa^{-1}c_{\mu}^{3}2^{-\beta
n}\sum_{i\in P_{n}}\psi_{n,i}(x)\mathchoice{\mathop{\kern 1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996ptB(x_{n,i},3\cdot
2^{-n})}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{n,i},3\cdot
2^{-n})}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{n,i},3\cdot
2^{-n})}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{n,i},3\cdot
2^{-n})}}f_{n}(y)\,d\mu(y)\,,$

where $\kappa=C(c_{\mu})$ is as in the definition of the functions
$\psi_{n,i}$. By summing over all $n\geq n_{0}$, we find that

$\displaystyle\mathcal{H}_{\beta}f(x)$
$\displaystyle=\sum_{n=n_{0}}^{\infty}2^{-\beta
n}\int_{X}\frac{\mathbf{1}_{B(y,2^{-n})}(x)}{\mu(B(y,2^{-n}))}f_{n}(y)\,d\mu(y)$
$\displaystyle\leq\kappa^{-1}c_{\mu}^{3}\sum_{n=n_{0}}^{\infty}2^{-\beta
n}\sum_{i\in P_{n}}\psi_{n,i}(x)\mathchoice{\mathop{\kern 1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996ptB(x_{n,i},3\cdot
2^{-n})}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{n,i},3\cdot
2^{-n})}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{n,i},3\cdot
2^{-n})}}{\mathop{\kern 1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x_{n,i},3\cdot
2^{-n})}}f_{n}(y)\,d\mu(y)=\kappa^{-1}c_{\mu}^{3}\operatorname{{\mathcal{Q}}}_{\beta}f(x)\,,$

and this concludes the proof. ∎

Next we connect the relative Hajłasz–Triebel–Lizorkin capacity to the
$\operatorname{{\mathcal{Q}}}_{\beta}$ potentials.

######  Lemma 6.4.

Let $1<p<\infty$, $1<q\leq\infty$, $0<\beta<1$, and $\Lambda\geq 2$. Assume
that $\mu$ satisfies the quantitative reverse doubling condition (5) for some
exponent $\sigma>\beta p$. Let $B\subset X$ be a ball, let
$E\subset{\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle B}\kern
0.0pt}$\hss}{B}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle B}\kern
0.0pt}$\hss}{B}}}$ be a compact set, and let $f=(f_{n})_{n=n_{0}}^{\infty}$ is
sequence of nonnegative Borel functions in $X$ such that
$\operatorname{{\mathcal{Q}}}_{\beta}f(x)\geq 1$ for every $x\in E$. Then
there exists a constant $C=C(c_{\sigma},p,q,\sigma,\beta,c_{\mu},\Lambda)>0$
such that

$\operatorname{cap}_{\dot{M}^{\beta}_{p,q}}(E,2B,\Lambda B)\leq C\lVert
f\rVert_{L^{p}(X;l^{q})}^{p}\,.$ (27)

###### Proof.

Clearly, we may assume that the right-hand side of (27) is finite. First we
consider the case where $\operatorname{{\mathcal{Q}}}_{\beta}f$ is continuous.
We truncate $\operatorname{{\mathcal{Q}}}_{\beta}f$ and then use the
fractional type Leibniz rule, as follows. Write $B=B(x_{0},r)\subset X$, with
$x_{0}\in X$ and $r>0$, and let

$\eta(x)=\max\biggl{\\{}0,1-\frac{\operatorname{dist}(x,B)}{r}\biggr{\\}}$

for every $x\in X$. Then

$\lvert\eta(x)-\eta(y)\rvert\leq\frac{d(x,y)}{r},\qquad\text{for every }x,y\in
X\,,$

$0\leq\eta\leq 1$ in $X$, $\eta=1$ in the closure ${\mathchoice{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle B}\kern
0.0pt}$\hss}{B}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\textstyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle B}\kern
0.0pt}$\hss}{B}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptscriptstyle B}\kern 0.0pt}$\hss}{B}}}$, and
$\eta=0$ in $X\setminus 2B$. The function
$\varphi=\eta\operatorname{{\mathcal{Q}}}_{\beta}f$ is continuous in $X$.
Since $E\subset{\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle B}\kern
0.0pt}$\hss}{B}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle B}\kern
0.0pt}$\hss}{B}}}$ and $\operatorname{{\mathcal{Q}}}_{\beta}f\geq 1$ in $E$,
we have $\varphi\geq 1$ in $E$, and clearly $\varphi=0$ in $X\setminus 2B$.
Let
$g=(g_{k})_{k\in{\mathbb{N}}}\in\mathbb{D}^{\beta}_{X}(\operatorname{{\mathcal{Q}}}_{\beta}f)$.
Lemma 3.3 implies that
$(\rho_{k})_{k\in{\mathbb{Z}}}\in\mathbb{D}_{X}^{\beta}(\varphi)$, where

$\rho_{k}=\big{(}g_{k}\|\eta\|_{\infty}+2^{k(\beta-1)}r^{-1}\operatorname{{\mathcal{Q}}}_{\beta}f\big{)}\mathbf{1}_{\\{x\in
X\,:\,\eta(x)\not=0\\}}\,,\quad\text{ for every }k\in{\mathbb{Z}}\,.$

Let $m$ be the smallest integer such that $2^{-m-1}\leq 2\Lambda r$. Observe
that $(\mathbf{1}_{k\geq m}\rho_{k})_{k\in{\mathbb{Z}}}\in\mathbb{D}_{\Lambda
B}^{\beta}(\varphi)$.

By Definition 4.2 of the capacity
$\operatorname{cap}_{\dot{M}^{\beta}_{p,q}}$, we have

$\begin{split}&\operatorname{cap}_{\dot{M}^{\beta}_{p,q}}(E,2B,\Lambda
B)\leq\lvert\varphi\rvert_{\dot{M}^{\beta}_{p,q}(\Lambda
B)}^{p}\leq\int_{{\Lambda B}}\left(\sum_{k\geq
m}\rho_{k}(x))^{q}\right)^{p/q}\,d\mu(x)\\\ &\quad\leq
C(p)\int_{2B}\left(\sum_{k\geq
m}g_{k}(x))^{q}\right)^{p/q}\,d\mu(x)+C(p)\int_{2B}\left(\sum_{k\geq
m}\left(2^{k(\beta-1)}r^{-1}\operatorname{{\mathcal{Q}}}_{\beta}f(x)\right)^{q}\right)^{p/q}\,d\mu(x)\\\
&\quad\leq
C(p)\int_{X}\left(\sum_{k\in{\mathbb{Z}}}g_{k}(x))^{q}\right)^{p/q}\,d\mu(x)+C(p)r^{-p}\left(\sum_{k\geq
m}2^{kq(\beta-1)}\right)^{p/q}\int_{2B}\left(\operatorname{{\mathcal{Q}}}_{\beta}f(x)\right)^{p}\,d\mu(x)\\\
&\quad\leq
C(p)\int_{X}\left(\sum_{k\in{\mathbb{Z}}}g_{k}(x))^{q}\right)^{p/q}\,d\mu(x)+C(p,q,\beta)r^{-p}2^{mp(\beta-1)}\int_{2B}\left(\operatorname{{\mathcal{Q}}}_{\beta}f(x)\right)^{p}\,d\mu(x)\\\
&\quad\leq
C(p)\int_{X}\left(\sum_{k\in{\mathbb{Z}}}g_{k}(x))^{q}\right)^{p/q}\,d\mu(x)+C(p,q,\beta,\Lambda)r^{-\beta
p}\int_{2B}\left(\operatorname{{\mathcal{Q}}}_{\beta}f(x)\right)^{p}\,d\mu(x)\,.\end{split}$

By taking infimum over all $g$ as above, we obtain

$\operatorname{cap}_{\dot{M}^{\beta}_{p,q}}(E,2B,\Lambda B)\leq
C(p)\lvert\operatorname{{\mathcal{Q}}}_{\beta}f\rvert_{\dot{M}^{\beta}_{p,q}(X)}^{p}+C(p,q,\beta,\Lambda)r^{-\beta
p}\int_{2B}\left(\operatorname{{\mathcal{Q}}}_{\beta}f(x)\right)^{p}\,d\mu(x)\,,$

and the claim in the present case follows from Lemmas 5.2 and 5.3.

Next we consider the case where the function
$\operatorname{{\mathcal{Q}}}_{\beta}f$ is not continuous. For every
$k\in{\mathbb{N}}$, we let $h_{k}=(h_{k,n})_{n=n_{0}}^{\infty}$, where
$h_{k,n}=\mathbf{1}_{\lvert n\rvert\leq k}f_{n}$ for every $n\geq n_{0}$. Then
$(\operatorname{{\mathcal{Q}}}_{\beta}h_{k})_{k\in{\mathbb{N}}}$ is an
increasing sequence of continuous real-valued functions in $X$ that converges
to $\operatorname{{\mathcal{Q}}}_{\beta}f$ pointwise in $X$. We consider the
open sets

$E_{k}=\left\\{x\in
X\,:\,\operatorname{{\mathcal{Q}}}_{\beta}h_{k}(x)>\frac{1}{2}\right\\}\,,\qquad
k\in{\mathbb{N}}\,.$

Since $\operatorname{{\mathcal{Q}}}_{\beta}f(x)\geq 1$ for every $x\in E$, we
see that $E\subset\bigcup_{k\in{\mathbb{N}}}E_{k}$. By the compactness of $E$,
there exists a finite subcover, and since $E_{n}\subset E_{n+1}$ for every
$n\in{\mathbb{N}}$, there exists $k\in{\mathbb{N}}$ such that $E\subset
E_{k}$. Thus we have $\operatorname{{\mathcal{Q}}}_{\beta}(2h_{k})>1$ in $E$,
and by the first part of the proof

$\operatorname{cap}_{\dot{M}^{\beta}_{p,q}}(E,2B,\Lambda B)\leq C\lVert
2h_{k}\rVert_{L^{p}(X;l^{q})}^{p}\leq 2^{p}C\lVert
f\rVert_{L^{p}(X;l^{q})}^{p}\,.$

This shows that inequality (27) holds also in the non-continuous case, and the
proof is complete. ∎

We now have all the tools needed for the comparison of the two capacities. We
begin with the second inequality in (1).

######  Theorem 6.5.

Let $1<p<\infty$, $1<q\leq\infty$, $0<\beta<1$, and $\Lambda\geq 2$, and
assume that $\mu$ satisfies the quantitative reverse doubling condition (5)
for some exponent $\sigma>\beta p$. Let $B\subset X$ be a ball and
$E\subset{\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle B}\kern
0.0pt}$\hss}{B}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle B}\kern
0.0pt}$\hss}{B}}}$ be a compact set. Then there exists a constant
$C=C(c_{\sigma},p,q,\sigma,\beta,c_{\mu},\Lambda)>0$ such that

$\operatorname{cap}_{\dot{M}^{\beta}_{p,q}}(E,2B,\Lambda B)\leq
C\operatorname{Cap}^{\beta}_{p,q}(E)\,.$ (28)

###### Proof.

We may assume that $\operatorname{Cap}^{\beta}_{p,q}(E)<\infty$. Let
$f=(f_{n})_{n=n_{0}}^{\infty}$ be a sequence of nonnegative Borel functions in
$X$ such that $\operatorname{\mathcal{H}}_{\beta}f\geq 1$ in $E$. By Lemma
6.3, there is a constant $C=C(c_{\mu})>0$ such that
$C\operatorname{{\mathcal{Q}}}_{\beta}f(x)\geq 1$ for every $x\in E$. Hence
$\operatorname{{\mathcal{Q}}}_{\beta}(Cf)(x)\geq 1$ for every $x\in E$ and
therefore Lemma 6.4 implies that

$\operatorname{cap}_{\dot{M}^{\beta}_{p,q}}(E,2B,\Lambda B)\leq
C(c_{\sigma},p,q,\sigma,\beta,c_{\mu},\Lambda)\lVert
Cf\rVert_{L^{p}(X;l^{q})}^{p}\leq
C(c_{\sigma},p,q,\sigma,\beta,c_{\mu},\Lambda)\lVert
f\rVert_{L^{p}(X;l^{q})}^{p}\,.$

Inequality (28) follows by taking infimum over all sequences $f$ as above. ∎

The converse inequality, that is, the first inequality in (1), requires
further restrictions on $\Lambda$ and $r$, but otherwise the assumptions are
less restrictive than in Theorem 6.5.

######  Theorem 6.6.

Let $1<p<\infty$, $1\leq q\leq\infty$, $0<\beta<1$, and $\Lambda\geq 41$, and
assume that $\mu$ satisfies the reverse doubling condition (4), with a
constant $0<c_{R}<1$. Assume that $B=B(x_{0},r)$, with $x_{0}\in X$ and
$0<r<(1/80)\operatorname{diam}(X)$, and let $E\subset{\mathchoice{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle B}\kern
0.0pt}$\hss}{B}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\textstyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle B}\kern
0.0pt}$\hss}{B}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptscriptstyle B}\kern 0.0pt}$\hss}{B}}}$ be a
closed set. Then there exists a constant $C=C(p,c_{\mu},c_{R})>0$ such that

$\operatorname{Cap}^{\beta}_{p,q}(E)\leq
C\operatorname{cap}_{\dot{M}^{\beta}_{p,q}}(E,2B,\Lambda B)\,.$

###### Proof.

The proof is similar to that of Theorem 4.6. Let $\varphi\colon
X\to{\mathbb{R}}$ be a continuous function such that $\varphi(x)\geq 1$ for
every $x\in E$ and $\varphi(x)=0$ for every $x\in X\setminus 2B$, and let
$g=(g_{k})_{k\in{\mathbb{Z}}}\in\mathbb{D}^{\beta}_{\Lambda B}(\varphi)$. By
replacing $\varphi$ with $\max\\{0,\min\\{\varphi,1\\}\\}$, if necessary, we
may assume that $0\leq\varphi\leq 1$.

Let $x\in E$ and let $m\in{\mathbb{Z}}$ be such that $2^{-m-1}\leq 5r<2^{-m}$.
Arguing as in the proof of Theorem 4.6 and using the inclusion
$B(x,2^{-n})\subset B(x_{0},\Lambda r)$ if $n\geq m-2$, we obtain

$\displaystyle 0<1-c_{R}$ $\displaystyle\leq
C(c_{\mu},c_{R})\sum_{n=m-2}^{\infty}2^{-n\beta}\;\sum_{k=n-1}^{n+2}\mathchoice{\mathop{\kern
1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996ptB(x,2^{-n})}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x,2^{-n})}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x,2^{-n})}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x,2^{-n})}}\mathbf{1}_{\Lambda
B}(y)g_{k}(y)\,d\mu(y)\,.$

Define $f=(f_{n})_{n\geq n_{0}}$, where $f_{n}=\mathbf{1}_{\Lambda
B}\sum_{k=n-1}^{n+2}g_{k}$ for each $n\geq n_{0}$. Since $m-2>n_{0}$, we have

$\displaystyle 0<1-c_{R}$ $\displaystyle\leq
C(c_{\mu},c_{R})\sum_{n=n_{0}}^{\infty}2^{-n\beta}\;\mathchoice{\mathop{\kern
1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996ptB(x,2^{-n})}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x,2^{-n})}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x,2^{-n})}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{B(x,2^{-n})}}f_{n}(y)\,d\mu(y)$
$\displaystyle\leq
C(c_{\mu},c_{R})\sum_{n=n_{0}}^{\infty}2^{-n\beta}\;\int_{X}\frac{\mathbf{1}_{B(y,2^{-n})}(x)}{\mu(B(y,2^{-n}))}f_{n}(y)\,d\mu(y)$
$\displaystyle=C(c_{\mu},c_{R})\operatorname{\mathcal{H}}_{\beta}f(x)\,.$

Hence, there exists a constant $C=C(c_{\mu},c_{R})>0$ such that
$\operatorname{\mathcal{H}}_{\beta}(Cf)(x)\geq 1$ for every $x\in E$. Thus

$\operatorname{Cap}^{\beta}_{p,q}(E)\leq\lVert
Cf\rVert_{L^{p}(X;l^{q})}^{p}=C^{p}\lVert
f\rVert_{L^{p}(X;l^{q})}^{p}\leq(4C)^{p}\lVert g\rVert_{L^{p}(\Lambda
B;l^{q})}^{p}\,,$

and the claim follows by taking infimum first over all $g$ and the over all
$\varphi$ as above. ∎

##  7\. Dual characterization of capacity

We now turn to the dual characterization of the capacity
$\operatorname{Cap}^{\beta}_{p,q}$ given in terms of measures, see Theorem 7.3
below. In Euclidean spaces, this type of results have a long history, see for
instance [2, 3, 30] and the references therein. We prove the dual
characterization by adapting the proof of equality (4.4.7) in [3] to the
setting of metric measure spaces. A fundamental tool in the proof is the
Minimax theorem [3, Theorem 2.4.1]. The somewhat technical task of verifying
the assumptions of the Minimax theorem in our setting is carried out
separately in Lemma 7.4.

######  Definition 7.1.

Let $\beta>0$ and let $\nu$ be a nonnegative Borel measure in $X$. We define

$\widecheck{\mathcal{H}}_{\beta}\nu(x)=\left(2^{-\beta
n}\frac{\nu(B(x,2^{-n}))}{\mu(B(x,2^{-n}))}\right)_{n=n_{0}}^{\infty}\,,\qquad
x\in X.$ (29)

Of particular interest to us are the nonnegative Radon measures on a compact
set $E\subset X$.

######  Definition 7.2.

Let $E\subset X$ be a compact set. We say that $\nu$ is a nonnegative Radon
measure on $E$, if $\nu$ is a nonnegative Borel measure on $E$ that satisfies
the following properties:

  * •

$\nu(E)<\infty$.

  * •

For every Borel set $F$ in $E$, we have

$\nu(F)=\inf\\{\nu(U)\,:\,F\subset U\text{ and }U\text{ is open in }E\\}\,.$

  * •

For every Borel set $F$ in $E$, we have

$\nu(F)=\sup\\{\nu(K)\,:\,K\subset F\text{ and }K\text{ is compact in
}E\\}\,.$

We denote by $\mathcal{M}^{+}(E)$ the set of all nonnegative Radon measures on
$E$. If $f$ is a Borel function on $X$ and $\nu\in\mathcal{M}^{+}(E)$, we
define

$\int_{X}f\,d\nu=\int_{E}f|_{E}\,d\nu\,.$

We also define $\nu(A)=\nu(A\cap E)$ if $A\subset X$ is a Borel set, and this
extends $\nu$ as a nonnegative Borel measure in $X$.

We are ready to state and prove the dual characterization of the capacity.

######  Theorem 7.3.

Let $1<p<\infty$, $1\leq q\leq\infty$, and $\beta>0$, and let $E\subset X$ be
a compact set. Then

$\operatorname{Cap}^{\beta}_{p,q}(E)^{1/p}=\sup\Biggl{\\{}\frac{\nu(E)}{\big{\lVert}\widecheck{\mathcal{H}}_{\beta}\nu\big{\rVert}_{L^{p^{\prime}}(X;l^{q^{\prime}})}}\,:\,\nu\in\mathcal{M}^{+}(E)\Biggr{\\}}\,.$

###### Proof.

Let $\nu\in\mathcal{M}^{+}(E)$, let $f=(f_{n})_{n=n_{0}}^{\infty}$ be a
sequence of nonnegative Borel functions, and write

$\mathcal{E}_{\beta}(\nu,f)=\int_{X}\operatorname{\mathcal{H}}_{\beta}f(x)\,d\nu(x)\,.$

Using Definition 6.1 and Fubini’s theorem, we have

$\begin{split}\mathcal{E}_{\beta}(\nu,f)&=\sum_{n=n_{0}}^{\infty}2^{-\beta
n}\int_{X}\int_{X}\frac{\mathbf{1}_{B(y,2^{-n})}(x)}{\mu(B(y,2^{-n}))}f_{n}(y)\,d\mu(y)\,d\nu(x)\\\
&=\sum_{n=n_{0}}^{\infty}2^{-\beta
n}\int_{X}\frac{f_{n}(y)}{\mu(B(y,2^{-n}))}\int_{X}\mathbf{1}_{B(y,2^{-n})}(x)\,d\nu(x)\,d\mu(y)\\\
&=\int_{X}\sum_{n=n_{0}}^{\infty}2^{-\beta
n}\frac{\nu(B(y,2^{-n}))}{\mu(B(y,2^{-n}))}\,f_{n}(y)\,d\mu(y)\\\
&=\int_{X}\langle\widecheck{\mathcal{H}}_{\beta}\nu(y),f(y)\rangle\,d\mu(y)\,.\end{split}$
(30)

Now, assume that $\operatorname{\mathcal{H}}_{\beta}f(x)\geq 1$ for all $x\in
E$. Then, by using also Hölder’s inequality twice, we get

$\nu(E)\leq\mathcal{E}_{\beta}(\nu,f)\leq\big{\lVert}\widecheck{\mathcal{H}}_{\beta}\nu\big{\rVert}_{L^{p^{\prime}}(X;l^{q^{\prime}})}\lVert
f\rVert_{L^{p}(X;l^{q})}\,.$

By taking infimum over all sequences $f$ with
$\operatorname{\mathcal{H}}_{\beta}f\geq 1$ in $E$, we thus obtain

$\frac{\nu(E)}{\big{\lVert}\widecheck{\mathcal{H}}_{\beta}\nu\big{\rVert}_{L^{p^{\prime}}(X;l^{q^{\prime}})}}\leq\operatorname{Cap}^{\beta}_{p,q}(E)^{1/p}\,.$

Taking supremum over all $\nu\in\mathcal{M}^{+}(E)$ gives

$\sup\Biggl{\\{}\frac{\nu(E)}{\big{\lVert}\widecheck{\mathcal{H}}_{\beta}\nu\big{\rVert}_{L^{p^{\prime}}(X;l^{q^{\prime}})}}\,:\,\nu\in\mathcal{M}^{+}(E)\Biggr{\\}}\leq\operatorname{Cap}^{\beta}_{p,q}(E)^{1/p}\,.$

For the converse inequality, we will use the Minimax theorem [3, Theorem
2.4.1]; we refer to Lemma 7.4 for further details. Define

$Z=\\{\nu\in\mathcal{M}^{+}(E)\,:\,\nu(E)=1\\}$

and

$W=\big{\\{}f\in L^{p}(X;l^{q})\,:\,f=(f_{n})_{n=n_{0}}^{\infty}\geq 0\text{
and }\lVert f\rVert_{L^{p}(X;l^{q})}\leq 1\big{\\}}\,.$

Observe from [11, Lemma 2.3] and (30) that for $\nu\in Z$ we have

$\frac{\big{\lVert}\widecheck{\mathcal{H}}_{\beta}\nu\big{\rVert}_{L^{p^{\prime}}(X;l^{q^{\prime}})}}{\nu(E)}=\sup_{f\in
W}\int_{X}\langle\widecheck{\mathcal{H}}_{\beta}\nu(y),f(y)\rangle\,d\mu(y)=\sup_{f\in
W}\mathcal{E}_{\beta}(\nu,f)\,.$

By taking infimum over all $\nu\in Z$, we get

$\inf_{\nu\in
Z}\frac{\big{\lVert}\widecheck{\mathcal{H}}_{\beta}\nu\big{\rVert}_{L^{p^{\prime}}(X;l^{q^{\prime}})}}{\nu(E)}=\inf_{\nu\in
Z}\sup_{f\in W}\mathcal{E}_{\beta}(\nu,f)\,.$ (31)

On the other hand, since the Dirac delta measure $\delta_{x}$ belongs to $Z$
for any $x\in E$, we have

$\inf_{\nu\in Z}\mathcal{E}_{\beta}(\nu,f)=\inf_{\nu\in
Z}\int_{X}\operatorname{\mathcal{H}}_{\beta}f(x)\,d\nu(x)\leq\inf_{x\in
E}\operatorname{\mathcal{H}}_{\beta}f(x)\leq\frac{\inf_{x\in
E}\operatorname{\mathcal{H}}_{\beta}f(x)}{\lVert f\rVert_{L^{p}(X;l^{q})}}$

for every $f\in W$. By scaling and Definition 6.2, it follows that

$\begin{split}\sup_{f\in W}\inf_{\nu\in
Z}\mathcal{E}_{\beta}(\nu,f)&\leq\sup_{f\in W}\frac{\inf_{x\in
E}\operatorname{\mathcal{H}}_{\beta}f(x)}{\lVert
f\rVert_{L^{p}(X;l^{q})}}=\sup_{0\leq f\in L^{p}(X;l^{q})}\frac{\inf_{x\in
E}\operatorname{\mathcal{H}}_{\beta}f(x)}{\lVert f\rVert_{L^{p}(X;l^{q})}}\\\
&=\sup_{0\leq f\in L^{p}(X;l^{q})}\Biggl{(}\left\lVert\frac{f}{\inf_{x\in
E}\operatorname{\mathcal{H}}_{\beta}f(x)}\right\rVert_{L^{p}(X;l^{q})}^{p}\Biggr{)}^{-1/p}\\\
&=\Biggl{(}\inf_{0\leq f\in L^{p}(X;l^{q})}\left\lVert\frac{f}{\inf_{x\in
E}\operatorname{\mathcal{H}}_{\beta}f(x)}\right\rVert_{L^{p}(X;l^{q})}^{p}\Biggr{)}^{-1/p}\leq\operatorname{Cap}^{\beta}_{p,q}(E)^{-1/p}\,.\end{split}$
(32)

By (31), the Minimax theorem in the following Lemma 7.4, and (32) we obtain

$\displaystyle\inf_{\nu\in
Z}\frac{\big{\lVert}\widecheck{\mathcal{H}}_{\beta}\nu\big{\rVert}_{L^{p^{\prime}}(X;l^{q^{\prime}})}}{\nu(E)}$
$\displaystyle=\inf_{\nu\in Z}\sup_{f\in
W}\mathcal{E}_{\beta}(\nu,f)=\sup_{f\in W}\inf_{\nu\in
Z}\mathcal{E}_{\beta}(\nu,f)$ $\displaystyle\leq\sup_{f\in W}\inf_{\nu\in
Z}\mathcal{E}_{\beta}(\nu,f)\leq\operatorname{Cap}^{\beta}_{p,q}(E)^{-1/p}\,.$

Hence

$\operatorname{Cap}^{\beta}_{p,q}(E)^{1/p}\leq\sup_{\nu\in
Z}\frac{\nu(E)}{\big{\lVert}\widecheck{\mathcal{H}}_{\beta}\nu\big{\rVert}_{L^{p^{\prime}}(X;l^{q^{\prime}})}}\leq\sup\Biggl{\\{}\frac{\nu(E)}{\big{\lVert}\widecheck{\mathcal{H}}_{\beta}\nu\big{\rVert}_{L^{p^{\prime}}(X;l^{q^{\prime}})}}\,:\,\nu\in\mathcal{M}^{+}(E)\Biggr{\\}}\,,$

as claimed. ∎

To complete the proof of Theorem 7.3, we still need to verify that the Minimax
theorem is valid in our setting. The underlying ideas are well known, but we
present the details for the sake of completeness.

######  Lemma 7.4.

Under the notation of the proof of Theorem 7.3, we have

$\inf_{\nu\in Z}\sup_{f\in W}\mathcal{E}_{\beta}(\nu,f)=\sup_{f\in
W}\inf_{\nu\in Z}\mathcal{E}_{\beta}(\nu,f)\,.$

###### Proof.

Since $E$ is compact, the Riesz representation theorem [34, Theorem 6.19]
implies that every $T\in(C(E),\lVert\cdot\rVert_{\infty})^{*}$ is represented
by a unique regular Borel measure $\nu$, in the sense that
$T(g)=\int_{E}g\,d\nu$ for every $f\in C(E)$. The operator norm
$\lVert\nu\rVert$ of $\nu$ is the total variation of $\nu$. Hence, we can
regard $Z$ as a subset of the dual $(C(E),\|\cdot\|_{\infty})^{*}$, and endow
$Z$ with the relative weak* topology.

We will apply the Minimax theorem [3, Theorem 2.4.1] to the functional
$\mathcal{E}_{\beta}$ defined in $Z\times W$. The sets $Z$ and $W$ are convex
and $\mathcal{E}_{\beta}$ is convex in $Z$ and concave in $W$, see [3, p. 30].
Hence, we need to show that

  * (a)

$Z$ is a Hausdorff space,

  * (b)

$Z$ is compact, and

  * (c)

for any $f\in W$, the functional $\mathcal{E}_{\beta}(\cdot,f)$ is lower
semicontinuous in $Z$.

We first prove (a). Let $\nu_{1},\nu_{2}\in
Z=\\{\nu\in\mathcal{M}^{+}(E):\nu(E)=1\\}$ be two distinct measures. By the
uniqueness part of the Riesz representation theorem, there exist a real-valued
$g\in C(E)$ and $\alpha\in{\mathbb{R}}$ such that
$\int_{E}g\,d\nu_{1}<\alpha<\int_{E}g\,d\nu_{2}$. Then

$U=\left\\{\nu\in Z:\int_{E}g\,d\nu<\alpha\right\\}\quad\text{ and }\quad
V=\left\\{\nu\in Z:\int_{E}g\,d\nu>\alpha\right\\}$

are open and disjoint subsets of $Z$, and $\nu_{1}\in U$ and $\nu_{2}\in V$.
Thus $Z$ is a Hausdorff space.

Next we prove (b). Given $\nu\in Z$, the operator norm of $\nu$ on
$(C(E),\|\cdot\|_{\infty})$ is $\lVert\nu\rVert=\nu(E)=1$, and therefore

$Z\subset{\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle B}\kern
0.0pt}$\hss}{B}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle B}\kern
0.0pt}$\hss}{B}}}(0,1)=\\{\nu\in(C(E),\lVert\cdot\rVert_{\infty})^{*}\,:\,\lVert\nu\rVert\leq
1\\}\,.$

Recall that $Z$ is equipped with the relative weak* topology. By the
Banach–Alaoglu theorem, the closed unit ball ${\mathchoice{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle B}\kern
0.0pt}$\hss}{B}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\textstyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle B}\kern
0.0pt}$\hss}{B}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptscriptstyle B}\kern 0.0pt}$\hss}{B}}}(0,1)$ is
compact in the weak* topology. Hence, in order to show that $Z$ is compact, it
suffices to verify that $Z$ is closed in the weak* topology of
$(C(E),\lVert\cdot\rVert_{\infty})^{*}$. Write

$G=\\{g\in C(E)\,:\,g(E)\subset[0,\infty)\\}\,.$

For every $g\in G$ we define
$\Lambda_{g}\colon(C(E),\lVert\cdot\rVert_{\infty})^{*}\to\mathbb{R}$,
$\Lambda_{g}(\nu)=\int_{E}g\,d\nu$. Then the functionals $\Lambda_{g}$ are
continuous for every $g\in G$ and

$Z=\Lambda_{1}^{-1}(\\{1\\})\cap\bigcap_{g\in
G}\Lambda_{g}^{-1}([0,\infty))\,.$

Indeed, above the inclusion from left to right is immediate. Conversely, if
$\nu\in(C(E),\lVert\cdot\rVert_{\infty})^{*}$ belongs to the set on the right-
hand side, then $\nu(E)=1$ and $0\leq\int_{E}g\,d\nu<\infty$ for all $g\in
C(E)$ satisfying $0\leq g<\infty$ in $E$. Hence $\nu$ is a nonnegative linear
functional on $C(E)$, and [34, Theorem 2.14] implies that
$\nu\in\mathcal{M}^{+}(E)$. Thus $\nu\in Z$, and it follows that $Z$ is closed
in $(C(E),\lVert\cdot\rVert_{\infty})^{*}$ since it is an intersection of
closed sets.

It remains to prove condition (c). Since $E\subset X$ is compact, by
Stone–Weierstrass theorem and Urysohn’s lemma, the Banach space
$(C(E),\lVert\cdot\rVert_{\infty})$ is separable. Since
$Z\subset(C(E),\lVert\cdot\rVert_{\infty})^{*}$ is weak* compact, the weak*
topology relative to $Z$ is metrizable. Hence, it suffices to show sequential
lower semicontinuity. To this end, we fix $f\in W$. Recall that

$\operatorname{\mathcal{H}}_{\beta}f(x)=\sum_{n=n_{0}}^{\infty}2^{-\beta
n}\int_{X}\frac{\mathbf{1}_{B(y,2^{-n})}(x)}{\mu(B(y,2^{-n}))}f_{n}(y)\,d\mu(y)\,,\qquad
x\in X.$

For each fixed $n\geq n_{0}$, we define a function $g_{n}\colon
E\to[0,\infty]$,

$g_{n}(x)=2^{-\beta
n}\int_{X}\frac{\mathbf{1}_{B(y,2^{-n})}(x)}{\mu(B(y,2^{-n}))}f_{n}(y)\,d\mu(y)\,,\qquad
x\in E\,.$

Let $x\in E$ and let $(x_{j})_{j\in{\mathbb{N}}}$ be a sequence in $E$ such
that $x_{j}\to x$ in $E$. Then

$\mathbf{1}_{B(y,2^{-n})}(x)\leq\liminf_{j\to\infty}\mathbf{1}_{B(y,2^{-n})}(x_{j})$

for all $y\in X$. Thus, by Fatou’s lemma,

$\displaystyle g_{n}(x)$ $\displaystyle\leq 2^{-\beta
n}\int_{X}\liminf_{j\to\infty}\left(\frac{\mathbf{1}_{B(y,2^{-n})}(x_{j})}{\mu(B(y,2^{-n}))}f_{n}(y)\right)\,d\mu(y)$
$\displaystyle\leq\liminf_{j\to\infty}2^{-\beta
n}\int_{X}\frac{\mathbf{1}_{B(y,2^{-n})}(x_{j})}{\mu(B(y,2^{-n}))}f_{n}(y)\,d\mu(y)=\liminf_{j\to\infty}g_{n}(x_{j})\,.$

Hence, the function $g_{n}$ is lower semicontinuous in $E$, and therefore also
$\mathcal{H}_{\beta}f=\sum_{n=n_{0}}^{\infty}g_{n}$ is lower semicontinuous in
$E$. By [17, Proposition 4.2.2], there exists a sequence
$(h_{k})_{k\in{\mathbb{N}}}$ of continuous functions in $E$ such that $0\leq
h_{1}(x)\leq h_{2}(x)\leq\dotsb\leq\mathcal{H}_{\beta}f(x)$ and
$\lim_{k\to\infty}h_{k}(x)=\mathcal{H}_{\beta}f(x)$ for every $x\in E$. Now,
let $\nu\in Z$ and let $(\nu_{i})_{i\in{\mathbb{N}}}$ be a sequence in $Z$
such that $\nu_{i}\to\nu$ in $Z$. Then, for all $k\in{\mathbb{N}}$,

$\displaystyle\int_{E}h_{k}(x)\,d\nu(x)$
$\displaystyle=\lim_{i\to\infty}\int_{E}h_{k}(x)\,d\nu_{i}(x)\leq\liminf_{i\to\infty}\int_{E}\mathcal{H}_{\beta}f(x)\,d\nu_{i}(x)$
$\displaystyle=\liminf_{i\to\infty}\int_{X}\mathcal{H}_{\beta}f(x)\,d\nu_{i}(x)\,.$

By using also the monotone convergence theorem, we obtain

$\displaystyle\mathcal{E}_{\beta}(\nu,f)$
$\displaystyle=\int_{X}\mathcal{H}_{\beta}f(x)\,d\nu(x)=\int_{E}\mathcal{H}_{\beta}f(x)\,d\nu(x)=\lim_{k\to\infty}\int_{E}h_{k}(x)\,d\nu(x)$
$\displaystyle\leq\liminf_{i\to\infty}\int_{X}\mathcal{H}_{\beta}f(x)\,d\nu_{i}(x)=\liminf_{i\to\infty}\mathcal{E}_{\beta}(\nu_{i},f)\,.$

This shows that $\mathcal{E}_{\beta}(\,\cdot\,,f)$ is sequentially lower
semicontinuous, and the proof is complete. ∎

##  8\. Muckenhoupt–Wheeden theorem

The original formulation of the celebrated Muckenhoupt–Wheeden Theorem in [31]
bounds the $L^{p}$-norm of the Riesz potential by the $L^{p}$-norm of a
fractional maximal function in the (suitably weighted) Euclidean space. In
Theorem 8.3 we present a variant of this result for the mixed
$L^{p}(X,l^{1})$-norm of the operator $\widecheck{\mathcal{H}}_{\beta}$.
Theorem 8.3 could actually be obtained as a consequence of a very general
metric space formulation of the Muckenhoupt–Wheeden Theorem in [33], but for
the sake of completeness and transparency we give below a more organic and
self-contained proof. In fact, the application of the general result from [33]
would still require rather tedious verification of the various assumptions;
compare for instance to [8, Theorem 4.2]. Our proof of Theorem 8.3 follows the
general approach in [3, Theorem 3.6.1], based on a good $\lambda$-inequality,
but the present setting requires several modifications.

For the formulation of Theorem 8.3, we need fractional maximal functions.

######  Definition 8.1.

Let $\beta\geq 0$ and let $\nu$ be a nonnegative Borel measure in $X$. The
fractional maximal function $M_{\beta}\nu$ is defined by

$M_{\beta}\nu(x)=\sup_{0<r\leq
2^{-n_{0}}}r^{\beta}\frac{\nu(B(x,r))}{\mu(B(x,r))}\,,\qquad x\in X\,.$ (33)

When $X$ is unbounded, the supremum is taken over all radii $0<r<\infty$. When
$\beta=0$ we get the centered maximal function which is denoted by
$M\nu=M_{0}\nu$.

We need the following technical lemma concerning lower semicontinuity. Recall
that here ${\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle B}\kern
0.0pt}$\hss}{B}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle B}\kern
0.0pt}$\hss}{B}}}(x,2^{-n})$ is the closed ball centered at $x$.

######  Lemma 8.2.

Let $\beta>0$ and let $\nu$ be a nonnegative Borel measure in $X$. Then the
function

$\sum_{n=n_{0}}^{\infty}2^{-\beta
n}\frac{\nu(B(x,2^{-n}))}{\mu({\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle B}\kern
0.0pt}$\hss}{B}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle B}\kern
0.0pt}$\hss}{B}}}(x,2^{-n}))}\,,\qquad x\in X\,,$

is lower semicontinuous in $X$.

###### Proof.

Fix $n\in{\mathbb{Z}}$ such that $n\geq n_{0}$. By the properties of
semicontinuous functions, see [5, p. 362], it suffices to show that the
nonnegative functions $f(x)=\nu(B(x,2^{-n}))$ and $g(x)=\mu({\mathchoice{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle B}\kern
0.0pt}$\hss}{B}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\textstyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle B}\kern
0.0pt}$\hss}{B}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptscriptstyle B}\kern
0.0pt}$\hss}{B}}}(x,2^{-n}))^{-1}$ are lower semicontinuous in $X$. Let $x\in
X$ and let $(x_{j})_{n\in\mathbb{N}}$ be a sequence in $X$ such that $x_{j}\to
x$ in $X$. By Fatou’s lemma,

$\displaystyle f(x)$
$\displaystyle=\int_{X}\mathbf{1}_{B(x,2^{-n})}(y)\,d\nu(y)\leq\int_{X}\liminf_{j\to\infty}\mathbf{1}_{B(x_{j},2^{-n})}(y)\,d\nu(y)$
$\displaystyle\leq\liminf_{j\to\infty}\int_{X}\mathbf{1}_{B(x_{j},2^{-n})}(y)\,d\nu(y)=\liminf_{j\to\infty}f(x_{j})\,,$

showing that $f$ is lower semicontinuous in $X$.

Define $h(x)=\mu({\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle B}\kern
0.0pt}$\hss}{B}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle B}\kern
0.0pt}$\hss}{B}}}(x,2^{-n}))$, $x\in X$. Then, if $(x_{j})_{j\in{\mathbb{N}}}$
is a sequence converging to $x\in X$ as above, we have

$\displaystyle h(x)=\int_{X}\mathbf{1}_{{\mathchoice{\hbox to0.0pt{$\kern
0.0pt\kern 0.0pt\overline{\phantom{\displaystyle B}\kern
0.0pt}$\hss}{B}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\textstyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle B}\kern
0.0pt}$\hss}{B}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptscriptstyle B}\kern
0.0pt}$\hss}{B}}}(x,2^{-n})}(y)\,d\mu(y)\geq\int_{X}\limsup_{j\to\infty}\mathbf{1}_{{\mathchoice{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle B}\kern
0.0pt}$\hss}{B}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\textstyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle B}\kern
0.0pt}$\hss}{B}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptscriptstyle B}\kern
0.0pt}$\hss}{B}}}(x_{j},2^{-n})}(y)\,d\mu(y)\,.$

Write ${\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle B}\kern
0.0pt}$\hss}{B}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle B}\kern
0.0pt}$\hss}{B}}}_{j}={\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle B}\kern
0.0pt}$\hss}{B}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle B}\kern
0.0pt}$\hss}{B}}}(x_{j},2^{-n})$ and $B=B(x,M)$, where $M>0$ is chosen so that
${\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle B}\kern
0.0pt}$\hss}{B}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle B}\kern
0.0pt}$\hss}{B}}}_{j}\subset B$ for all $j\in{\mathbb{N}}$. Then
$\mathbf{1}_{{\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle B}\kern
0.0pt}$\hss}{B}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle B}\kern
0.0pt}$\hss}{B}}}_{j}}\leq\mathbf{1}_{B}\in L^{1}(d\mu)$ for all
$j\in{\mathbb{N}}$. By applying Fatou’s lemma to the nonnegative functions
$\mathbf{1}_{B}-\mathbf{1}_{{\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle B}\kern
0.0pt}$\hss}{B}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle B}\kern
0.0pt}$\hss}{B}}}_{j}}$, we obtain

$\displaystyle\mu(B)-\int_{X}\limsup_{j\to\infty}\mathbf{1}_{{\mathchoice{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle B}\kern
0.0pt}$\hss}{B}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\textstyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle B}\kern
0.0pt}$\hss}{B}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptscriptstyle B}\kern
0.0pt}$\hss}{B}}}_{j}}(y)\,d\mu(y)$
$\displaystyle=\int_{X}\liminf_{j\to\infty}\bigl{(}\mathbf{1}_{B}-\mathbf{1}_{{\mathchoice{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle B}\kern
0.0pt}$\hss}{B}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\textstyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle B}\kern
0.0pt}$\hss}{B}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptscriptstyle B}\kern
0.0pt}$\hss}{B}}}_{j}}\bigr{)}(y)\,d\mu(y)$
$\displaystyle\leq\liminf_{j\to\infty}\int_{X}\bigl{(}\mathbf{1}_{B}-\mathbf{1}_{{\mathchoice{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle B}\kern
0.0pt}$\hss}{B}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\textstyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle B}\kern
0.0pt}$\hss}{B}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptscriptstyle B}\kern
0.0pt}$\hss}{B}}}_{j}}\bigr{)}(y)\,d\mu(y)$
$\displaystyle=\mu(B)-\limsup_{j\to\infty}\int_{X}\mathbf{1}_{{\mathchoice{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle B}\kern
0.0pt}$\hss}{B}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\textstyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle B}\kern
0.0pt}$\hss}{B}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptscriptstyle B}\kern
0.0pt}$\hss}{B}}}_{j}}(y)\,d\mu(y)\,.$

Consequently, we see that

$h(x)\geq\int_{X}\limsup_{j\to\infty}\mathbf{1}_{{\mathchoice{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle B}\kern
0.0pt}$\hss}{B}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\textstyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle B}\kern
0.0pt}$\hss}{B}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptscriptstyle B}\kern
0.0pt}$\hss}{B}}}_{j}}(y)\,d\mu(y)\geq\limsup_{j\to\infty}\int_{X}\mathbf{1}_{{\mathchoice{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle B}\kern