# Capacity in Besov and Triebel-Lizorkin spaces with generalized smoothness

Nijjwal Karak and Debarati Mondal

###### Abstract.

We prove a lower bound estimate for capacities in Hajłasz-Besov, Hajłasz-
Triebel-Lizorkin and Hajłasz-Sobolev spaces with generalized smoothness
defined on metric spaces in terms of Netrusov-Hausdorff content or Hausdorff
content. These results are improvements of the results obtained in [18].

We would like to thank the reviewer for the valuable comments and suggestions.
N.K. would like to thank DST-SERB (SRG/2021/000118) and D.M. would like to
thank BITS Pilani (ID no. 2020PHXF0038H) for the financial support.

Keywords: Hajłasz-Besov space, Hajłasz-Triebel-Lizorkin space, capacity,
Hausdorff content, Netrusov-Hausdorff content.
2010 Mathematics Subject Classification: 31E05, 31B15.

##  1\. Introduction

Function spaces with generalized smoothness have received a lot of attention
in recent years due to its applications in various fields. In [7], the authors
have studied the generalized Besov spaces
$B^{(\sigma,N)}_{p,q}(\mathbb{R}^{n})$ and Triebel-Lizorkin spaces
$F^{(\sigma,N)}_{p,q}(\mathbb{R}^{n})$ for the full range of parameters, where
$\sigma$ is a weight sequence of positive numbers and $N$ is a sequence of
strictly increasing positive numbers connected a suitable partition of unity,
and they are used to replace the classical smoothness $\\{2^{js}\\}.$
Generalized Besov and Triebel-Lizorkin spaces also have been studied where
smoothness is given by a parameter function, see [1, 5, 6, 2, 4, 3]. Recently
Li, Yang and Yuan have introduced Hajłasz-Sobolev spaces $M^{\phi,p}(X)$,
Hajłasz-Besov spaces $N^{\phi}_{p,q}(X)$ and Hajłasz-Triebel-Lizorkin spaces
$M^{\phi}_{p,q}(X)$ with generalized smoothness on metric spaces, [19]. These
spaces have been defined using Hajłasz gradients which were first introduced
by Hajłasz [8] and later defined for fractional scales independently by Hu
[11] and Yang [27] and then generalized to sequence of gradients to define
Besov and Triebel-Lizorkin spaces [17]. Please see Section 2 for the
definition of these spaces.

In this paper we study the relation between the capacity defined on Hajłasz-
Sobolev spaces $M^{\phi,p}(X)$, Hajłasz-Besov spaces $N^{\phi}_{p,q}(X)$ or
Hajłasz-Triebel-Lizorkin spaces $M^{\phi}_{p,q}(X)$ with generalized
smoothness and Netrusov-Hausdorff content (or Hausdorff content). Hajłasz-
Sobolev capacity was first studied in [16] and later it was used in [15] to
prove the existence of Lebesgue points of functions in Hajłasz-Sobolev space
$M^{1,p}(X),$ $p\in(1,Q],$ outside a set of Hajłasz-Sobolev capacity zero,
where $X$ is a doubling metric measure space and $Q$ is the doubling dimension
of $X.$ This result leads to a series of related work on other function spaces
such as fractional Hajłasz-Sobolev spaces [24], Orlicz-Sobolev spaces [20] as
well as Hajłasz-Besov and Hajłasz-Triebel-Lizorkin spaces [10]. Let us recall
here the definition of the capacities on these spaces from [18].

######  Definition 1.1.

Let $0<s<\infty,$ $0<p,q<\infty,$ $\phi\in\mathcal{A}$ and
$\mathcal{F}\in\\{M^{\phi,p},M^{\phi}_{p,q},N^{\phi}_{p,q}\\}.$ The
$\mathcal{F}$-capacity of a set $E\subset X$ is

$\operatorname{Cap}_{\mathcal{F}}(E)=\inf\\{\|u\|_{\mathcal{F}(X)}^{p}:u\in\mathcal{G_{F}}(E)\\},$

where

$\mathcal{G_{F}}(E)=\\{u\in\mathcal{F}(X):u\geq 1\,\,\text{on a neighbourhood
of}\,\,E\\}.$

Lower bound and upper bound estimates for Hajłasz-Besov capacity
($N^{s}_{p,q}$-capacity) were found in [23] in terms of generalized Netrusov-
Hausdorff content, whereas the estimates for Hajłasz-Triebel-Lizorkin capacity
($M^{s}_{p,q}$-capacity) were found in [13] in terms of generalized Hausdorff
measure. These results were also extended in [18] for spaces with generalized
smoothness and recently some of them have been improved in [14]. In this
paper, we improve the results of [18] on the relation between the capacities
and the Netrusov-Hausdorff content (or Hausdorff content) by relaxing the
assumptions on the metric space. Here is our first result:

######  Theorem 1.2.

Let $\phi\in\mathcal{A}_{0}$, $0<p<\infty$, $0<q<\infty$ and $\omega$ be any
given function of admissible growth such that, for any $L\in\mathbb{Z}_{+}$,

$\sum_{k\geq L}\frac{1}{\omega(2^{-k})}<\infty.$

Let $x_{0}\in X,$ $0<R<\infty.$ Then there are constants $C>0$ and $c>0$ such
that, for any compact set $E\subset B(x_{0},R),$

(1.1) $\mathcal{H}^{h_{\omega},q/p}_{cR}(E)\leq
C\operatorname{Cap}_{N^{\phi}_{p,q}}(E)$

where, for any $r\in(0,R],$ $h_{\omega}(r)=[\phi(r)\omega(r)]^{p}.$

Note that, this result is an improvement of [18, Theorem 7] in the sense that
here we do not require any additional assumption of the existence of some non-
empty annulus on the underlying metric space which was required for Theorem 7
of [18]. Also for Triebel-Lizorkin space, the lower bound is obtained, without
the additional assumption on the underlying metric space, in terms of
Hausdorff measure and the result turns out to be true as well for Sobolev
spaces with generalized smoothness $M^{\phi,p}=M^{\phi}_{p,\infty}.$

######  Theorem 1.3.

Let $\phi\in\mathcal{A}_{0}$, $0<p<\infty$, $0<q<\infty$ and $\omega$ be any
given function of admissible growth such that, for any $L\in\mathbb{Z}_{+}$,

$\sum_{k\geq L}\frac{1}{\omega(2^{-k})}<\infty.$

Let $x_{0}\in X,$ $0<R<\infty.$ Then there exist constants $c$ and $C$ such
that, for any compact set $E\subset B(x_{0},R),$

$H^{h_{\omega}}_{cR}(E)\leq C\operatorname{Cap}_{M^{\phi}_{p,q}}(E)$

where, for any $r\in(0,R],$ $h_{\omega}(r)=[\phi(r)\omega(r)]^{p}.$

######  Theorem 1.4.

Let $\phi\in\mathcal{A}_{0}$, $0<p<\infty$, and $\omega$ be any given function
of admissible growth such that, for any $L\in\mathbb{Z}_{+}$,

$\sum_{k\geq L}\frac{1}{\omega(2^{-k})}<\infty.$

Let $x_{0}\in X,$ $0<R<\infty.$ Then there exist constants $c$ and $C$ such
that, for any compact set $E\subset B(x_{0},R),$

$H^{h_{\omega}}_{cR}(E)\leq C\operatorname{Cap}_{M^{\phi,p}}(E)$

where, for any $r\in(0,R],$ $h_{\omega}(r)=[\phi(r)\omega(r)]^{p}.$

##  2\. Notations and Preliminaries

We assume throughout the article that $X=(X,d,\mu)$ is a doubling metric
measure space equipped with a metric $d$ and a doubling Borel regular measure
$\mu.$ Note that a Borel regular measure $\mu$ on a metric space $(X,d)$ is
said to be a doubling measure if every ball in $X$ has positive and finite
measure and there exist a constant $c_{d}\geq 1$ such that $\mu(B(x,2r))\leq
c_{d}\,\mu(B(x,r))$ holds for each $x\in X$ and $r>0.$

###  2.1. Smoothness weight sequences and functions

We first recall some tools used to describe the generalized smoothness.
Smoothness in terms of sequences can be found in [7, 12] and in terms of
sequences can be found in [18, 19].

######  Definition 2.1.

Let $E\in\\{\mathbb{Z},\mathbb{Z_{+}}\\}.$ A sequence of positive numbers
$\\{\sigma_{j}\\}_{j\in E}$ is said to be admissible if there exist two
positive constants $d_{0}$ and $d_{1}$ such that for any $j\in E$,
$d_{0}\sigma_{j}\leq\sigma_{j+1}\leq d_{1}\sigma_{j}.$

######  Definition 2.2.

A continuous function $\phi:[0,\infty)\rightarrow[0,\infty)$ is said to be of
admissible growth if $\\{\phi(2^{j})\\}_{j\in\mathbb{Z}}$ is an admissible
sequence and $\phi(t)\sim\phi(2^{k})$ for any $k\in\mathbb{Z}$ and
$t\in[2^{k},2^{k+1})$ with the positive equivalence constants independent of
both $t$ and $k$.

Here we recall some indices from [18]. For any given sequence
$\sigma:=\\{\sigma_{k}\\}_{k\in\mathbb{Z}}$ of positive numbers and any given
function $\phi:[0,\infty)\rightarrow[0,\infty)$, let

$\alpha_{\phi}:=\max\\{\alpha^{-}_{\phi},\alpha^{+}_{\phi}\\}:=\max\left\\{\limsup_{k\rightarrow-\infty}\frac{\phi(2^{k})}{\phi(2^{k+1})},\limsup_{k\rightarrow\infty}\frac{\phi(2^{k})}{\phi(2^{k+1})}\right\\}$

and

$\beta_{\phi}:=\max\\{\beta^{-}_{\phi},\beta^{+}_{\phi}\\}:=\max\left\\{\limsup_{k\rightarrow-\infty}\frac{\phi(2^{k+1})}{\phi(2^{k})},\limsup_{k\rightarrow\infty}\frac{\phi(2^{k+1})}{\phi(2^{k})}\right\\}.$

Throughout this article, we denote by $\mathcal{A}$ the class of all
continuous and almost increasing functions
$\phi:[0,\infty)\rightarrow[0,\infty)$ such that $\phi(0)=0,\phi(1)=1,$ and
$\\{\phi(2^{k})\\}_{k\in\mathbb{Z}}$ is admissible. We also denote by
$\mathcal{A}_{\infty}$ the class of all functions $\phi\in\mathcal{A}$ so that
$\frac{\phi(t)}{t}$ almost decreases on $[0,\infty).$ Recall that a function
$f:[0,\infty)\rightarrow[0,\infty)$ is said to be almost increasing (resp.,
almost decreasing) if there exists a positive constant $C_{I}\in[1,\infty)$
such that for any $t_{1},t_{2}\in[0,\infty)$ with $t_{1}\leq
t_{2}$(resp.,$t_{1}\geq t_{2}$), $f(t_{1})\leq C_{I}f(t_{2}).$ For any
$r\in(0,\infty)$, let $\mathcal{A}_{r}$ be the set of all functions
$\phi\in\mathcal{A}_{\infty}$ so that $\phi$ is of admissible growth and there
exist an integer $i$ and two positive constants $X_{i}$ and $Y_{i}$ depending
on $i$ and $r,$ such that

(2.1) $\left\\{\sum_{j\geq i}[\phi(2^{j})]^{-r}\right\\}^{\frac{1}{r}}\leq
X_{i}\quad\text{and}\quad\left\\{\sum_{j\geq
i}2^{-jr}[\phi(2^{-j})]^{-r}\right\\}^{\frac{1}{r}}\leq Y_{i}.$

Also, let $\mathcal{A}_{0}$ be the class of all functions $\phi$ satisfying
that $\alpha_{\phi}\in(0,1)$, $\beta^{-}_{\phi}\in(0,2)$ and $\phi$ is of
admissible growth. Note that, by using the elementary inequality

(2.2)
$\sum_{i\in\mathbb{Z}}a_{i}\leq\Big{(}\sum_{i\in\mathbb{Z}}a_{i}^{\beta}\Big{)}^{\frac{1}{\beta}}$

which holds whenever $a_{i}\geq 0$ for all $i\in\mathbb{Z}$ and $0<\beta\leq
1,$ we have
$\mathcal{A}_{r_{1}}\subset\mathcal{A}_{r_{2}}\subset\mathcal{A}_{\infty}$ for
any $r_{1},r_{2}\in(0,\infty)$ with $r_{1}\leq r_{2}.$ Also, note that if
$\phi\in\mathcal{A}_{0},$ then $\phi\in\mathcal{A}_{r},$ for any
$r\in(0,\infty].$

###  2.2. Hajłasz-Sobolev, Hajłasz-Besov and Hajłasz-Triebel-Lizorkin spaces
with generalized smoothness

Here we recall the definitions of $\phi$-Hajłasz gradients and the spaces with
generalized smoothness from [19].

######  Definition 2.3.

A non-negative measurable function $g$ is called a $\phi$-Hajłasz gradient of
$u$ if there exists a set $E\subset X$ with $\mu(E)=0$ such that, for any
$x,y\in X\setminus E,$

$|u(x)-u(y)|\leq\phi(d(x,y))[g(x)+g(y)].$

We denote the collection of all $\phi$-Hajłasz gradients of $u$ by
$\mathcal{D}^{\phi}(u).$

######  Definition 2.4.

Let $\phi\in\mathcal{A}$ and $0<p\leq\infty$. The homogeneous $\phi$-Hajłasz-
Sobolev space $\dot{M}^{\phi,p}(S)$ consists of all measurable, almost
everywhere finite function $u:S\rightarrow\bar{\mathbb{R}}$ for which the semi
norm

(2.3)
$\|u\|_{\dot{M}^{\phi,p}(S)}=\inf_{g\in\mathcal{D}^{\phi}(u)}\|g\|_{L^{p}(S)}$

is finite. The non-homogeneous $\phi$-Hajłasz-Sobolev space $M^{\phi,p}(S)$ is
$\dot{M}^{\phi,p}(S)\cap L^{p}(S)$ equipped with the norm

(2.4) $\|u\|_{M^{\phi,p}(S)}=\|u\|_{L^{p}(S)}+\|u\|_{\dot{M}^{\phi,p}(S)}.$

Let $S\subset X$ be a measurable set. For $0<p,q\leq\infty$ and a sequence
$\vec{f}=(f_{k})_{k\in\mathbb{Z}}$ of measurable functions, we define

$\|(f_{k})_{k\in\mathbb{Z}}\|_{L^{p}(S,l^{q})}=\big{\|}\|(f_{k})_{k\in\mathbb{Z}}\|_{l^{q}}\big{\|}_{L^{p}(S)}$

and

$\|(f_{k})_{k\in\mathbb{Z}}\|_{l^{q}(L^{p}(S))}=\big{\|}(\|f_{k}\|_{L^{p}(S)})_{k\in\mathbb{Z}}\big{\|}_{l^{q}}$

where

$\|(f_{k})_{k\in\mathbb{Z}}\|_{l^{q}}=\begin{cases}(\sum_{k\in\mathbb{Z}}|f_{k}|^{q})^{1/q},&~{}\text{when}~{}0<q<\infty,\\\
\sup_{k\in\mathbb{Z}}|f_{k}|,&~{}\text{when}~{}q=\infty.\end{cases}$

######  Definition 2.5.

A sequence of non negative measurable functions,
$\vec{g}:=\\{g_{k}\\}_{k\in\mathbb{Z}}$ is called a $\phi$-Hajłasz gradient
sequence of $u$ if, for any $k\in\mathbb{Z},$ there exists a set $E_{k}\subset
X$ with $\mu(E_{k})=0$ such that , for any $x,y\in X\setminus E_{k},$ with
$2^{-k-1}\leq d(x,y)<2^{-k},$

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

We denote the collection of all $\phi$-Hajłasz gradient sequences of $u$ by
$\mathbb{D}^{\phi}(u).$

######  Definition 2.6.

Let $\phi\in\mathcal{A}$ and $0<p,q\leq\infty$. The homogeneous
$\phi$-Hajłasz-Besov space $\dot{N}^{\phi}_{p,q}(S)$ consists of all
measurable, almost everywhere finite function $u:S\rightarrow\bar{\mathbb{R}}$
for which the semi norm

(2.5)
$\|u\|_{\dot{N}^{\phi}_{p,q}(S)}=\inf_{(g_{k})\in\mathbb{D}^{\phi}(u)}\|(g_{k})\|_{l^{q}(L^{p}(S))}$

is finite. The non-homogeneous $\phi$-Hajłasz-Besov space $N^{\phi}_{p,q}(S)$
is $\dot{N}^{\phi}_{p,q}(S)\cap L^{p}(S)$ equipped with the norm

(2.6)
$\|u\|_{N^{\phi}_{p,q}(S)}=\|u\|_{L^{p}(S)}+\|u\|_{\dot{N}^{\phi}_{p,q}(S)}.$

######  Definition 2.7.

Let $\phi\in\mathcal{A}$ and $0<p,q\leq\infty$. The homogeneous
$\phi$-Hajłasz-Triebel-Lizorkin space $\dot{M}^{\phi}_{p,q}(S)$ consists of
all measurable, almost everywhere finite function
$u:S\rightarrow\bar{\mathbb{R}}$ for which the semi norm

(2.7)
$\|u\|_{\dot{M}^{\phi}_{p,q}(S)}=\inf_{(g_{k})\in\mathbb{D}^{\phi}(u)}\|(g_{k})\|_{L^{p}(S,l^{q})}$

is finite. The non-homogeneous $\phi$-Hajłasz-Triebel-Lizorkin space
$M^{\phi}_{p,q}(S)$ is $\dot{M}^{\phi}_{p,q}(S)\cap L^{p}(S)$ equipped with
the norm

(2.8)
$\|u\|_{M^{\phi}_{p,q}(S)}=\|u\|_{L^{p}(S)}+\|u\|_{\dot{M}^{\phi}_{p,q}(S)}.$

###  2.3. $\gamma$-median

As our functions may fail to be locally integrable, we will use the concept of
$\gamma$-median instead of integral averages. We recall here the definition
and some of its properties which from [10, 25, 23].

######  Definition 2.8.

Let $0<\gamma\leq 1/2.$ The $\gamma$-median $m_{u}^{\gamma}(A)$ of a
measurable, almost everywhere finite function $u$ over a set $A\subset X$ of
finite measure is

$m_{u}^{\gamma}(A)=\sup\\{M\in\mathbb{R}:\mu(\\{x\in
A:u(x)<M\\})\leq\gamma\mu(A)\\}.$

######  Lemma 2.9.

Let $0<\gamma\leq 1/2$ and $A\subset X.$ Then for two measurable functions
$u,v:A\rightarrow\mathbb{R},$ the $\gamma$-median has the following
properties:
$(a)$ If $\gamma\leq\gamma^{\prime},$ then $m_{u}^{\gamma^{\prime}}(A)\leq
m_{u}^{\gamma}(A).$
$(b)$ If $u\leq v$ almost everywhere, then $m_{u}^{\gamma}(A)\leq
m_{v}^{\gamma}(A).$
$(c)$ If $A\subset B$ and $\mu(B)\leq C\mu(A),$ then $m_{u}^{\gamma}(A)\leq
m_{u}^{\gamma/C}(B).$
$(d)$ If $c\in\mathbb{R},$ then $m_{u}^{\gamma}(A)+c=m_{u+c}^{\gamma}(A).$
$(e)$If $c\in\mathbb{R},$ then $m_{cu}^{\gamma}(A)=cm_{u}^{\gamma}(A).$
$(f)$ $|m_{u}^{\gamma}(A)|\leq m_{|u|}^{\gamma}(A).$
$(g)$ If $u\in L^{p}(A),$ $p>0,$ then

$m_{|u|}^{\gamma}(A)\leq\left(\gamma^{-1}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{A}|u|^{p}\,d\mu\right)^{1/p}.$

$(h)$ If $u$ is continuous, then for every $x\in X,$

$\lim_{r\rightarrow 0}m_{u}^{\gamma}(B(x,r))=u(x).$

We also recall the notion of discrete $\gamma$-median convolution introduced
in [10].

######  Definition 2.10 ([18]).

Let $\gamma\in(0,1/2]$ and $u\in L^{0}(X).$ The discrete $\gamma$\- median
convolution $u^{\gamma}_{r}$ of $u$ at scale $r\in(0,\infty)$ is defined by
setting, for any $x\in X$

$u_{r}^{\gamma}(x):=\sum_{j\in\mathcal{J}}m^{\gamma}_{u}(B_{j})\varphi_{j}(x),$

where $\mathcal{J}$ is an index set, $\\{B_{j}\\}_{j\in\mathcal{J}}$ is a
cover of $X$ with the radius $r$ such that
$\sum_{j\in\mathcal{J}}\chi_{2B_{j}}$ is finite and
$\\{\varphi_{j}\\}_{j\in\mathcal{J}}$ is a partition of unity with respect to
$\\{B_{j}\\}_{j\in\mathcal{J}}.$

###  2.4. Hausdorff content

Netrusov-Hausdorff content was first introduced by Netrusov [21, 22] and later
it was generalized using an increasing function $h$ in [23]. For the
generalized Hausdorff content (and measure), we refer to [26].

######  Definition 2.11.

Let $0<\theta<\infty,$ $0<R<\infty$ and $h:(0,\infty)\rightarrow(0,\infty)$ be
an increasing function. The generalized Netrusov-Hausdorff content of a set
$E\subset X$ is

$\mathcal{H}^{h,\theta}_{R}(E)=\inf\Big{[}\sum_{i:2^{-i}<R}\Big{(}\sum_{j\in
I_{i}}\frac{\mu(B(x_{j},r_{j}))}{h(r_{j})}\Big{)}^{\theta}\Big{]}^{\frac{1}{\theta}},$

where $I_{i}=\\{j\in\mathbb{N}:2^{-i}\leq r_{j}<2^{-i+1}\\}$ and the infimum
is taken over all coverings $\\{B(x_{j},r_{j})\\}$ of $E$ with $0<r_{j}\leq
R.$ If $R=\infty,$ then we take the infimum over all coverings of $E$ and the
first sum is over all $i\in\mathbb{Z}.$

######  Definition 2.12.

Let $0<R\leq\infty$ and $h:(0,\infty)\rightarrow(0,\infty)$ be an increasing
function. The generalized Hausdorff content of a set $E\subset X$ is

$H^{h}_{R}(E)=\inf\Big{\\{}\sum_{j\in\mathbb{N}}\frac{\mu(B(x_{j},r_{j}))}{h(r_{j})}:E\subset\bigcup
B(x_{j},r_{j}),r_{j}\leq R\Big{\\}}$

and the generalized Hausdorff measure is $H^{h}(E)=\limsup_{R\rightarrow
0}H^{h}_{R}(E).$

Note that $\mathcal{H}^{h,1}_{R}(E)=H^{h}_{R}.$ Also by using the elementary
inequality (2.2) we have $\mathcal{H}^{h,\theta}_{R}(E)\leq H^{h}_{R}(E)$ if
$\theta>1$ and $\mathcal{H}^{h,\theta}_{R}(E)\geq H^{h}_{R}(E)$ if $\theta<1.$

##  3\. Proofs

We will first prove two lemmas (Lemma 3.1 and Lemma 3.4) to restrict the set
of admissible functions for the capacities to locally Lipschitz functions. The
proof of Lemma 3.1 follows by extending the proof of [23, Theorem 4.8] and
using the convergence of approximations by discrete $\gamma$-median
convolutions, [18, Theorem 4]. We give the details here for the sake of
completeness.

######  Lemma 3.1.

Let $\phi\in\mathcal{A}_{0}$,
$0<p,q<\infty,\quad\mathcal{F}\in\\{M^{\phi}_{p,q}(X),N^{\phi}_{p,q}(X)\\}$
and $E\subseteq X$ be a compact set. Then

(3.1)
$\operatorname{Cap}_{\mathcal{F}}(E)\approx\inf\\{\|u\|^{p}_{\mathcal{F}}:u\in\mathcal{\tilde{G}}_{\mathcal{F}}(E)\\},$

where
$\mathcal{\tilde{G}}_{\mathcal{F}}(E)=\\{u\in\mathcal{G}_{\mathcal{F}}(E):u$
is locally Lipschitz$\\}.$

###### Proof.

Since $\mathcal{\tilde{G}_{F}}(E)\subset\mathcal{G_{F}}(E)$, it is enough to
prove the $``\gtrsim"$ part. Let $u\in\mathcal{G}_{\mathcal{F}}(E).$ Then
there is an open set $U\supset E$ such that $u\geq 1$ in $U.$ Let
$V=\\{x:d(x,E)<d(E,X\setminus U)/2\\}.$ If $x\in V$ and $r<d(E,X\setminus
U)/8,$ then $B(y,2r)\subset U$ whenever $x\in B(y,2r).$ So,
$u_{r}^{\gamma}\geq 1$ in $V$ when $r<d(E,X\setminus U)/8.$ So,
$u_{r}^{\gamma}\in\mathcal{\tilde{G}}_{\mathcal{F}}(E)$ for small $r$ and
hence by the convergence of approximations by discrete $\gamma$-median
convolutions [18, Theorem 4],

$\displaystyle\inf\\{\|v\|^{p}_{\mathcal{F}}:v\in\mathcal{\tilde{G}_{F}}(E)\\}$
$\displaystyle\leq$
$\displaystyle\lim\inf_{i\rightarrow\infty}\|u^{\gamma}_{2^{-i}}\|^{p}_{\mathcal{F}}$
$\displaystyle\leq$
$\displaystyle\lim\inf_{i\rightarrow\infty}C(\|u\|^{p}_{\mathcal{F}}+\|u^{\gamma}_{2^{-i}}-u\|^{p}_{\mathcal{F}})$
$\displaystyle\leq$ $\displaystyle C\|u\|^{p}_{\mathcal{F}}.$

The claim follows by taking the infimum over
$u\in\mathcal{G}_{\mathcal{F}}(E).$ ∎

Note that, when $q=\infty,$ the approximation by the $\gamma$-median
convolutions is not always true, even for the function $\phi(t)=t;$ see [10,
Example 3.5] for a counterexample. In this case, we use the denseness of the
$\phi$-Lipschitz class of functions, proved in [18, Theorem 2]. Recall that
for given $\phi\in\mathcal{A},$ a function $u$ on $X$ is said to be in the
$\phi$-Lipschitz class $\operatorname{Lip}_{\phi}(X)$ if there exists a
constant $C>0$ such that for any two points $x$ and $y$ in $X,$ there holds

$|u(x)-u(y)|\leq C\phi(d(x,y)).$

######  Theorem 3.2.

[18, Theorem 2] Let $\phi$ be a modulus of continuity and $p\in(0,\infty).$
Then $\operatorname{Lip}_{\phi}(X)\cap M^{\phi,p}(X)$ is a dense subset of
$M^{\phi,p}(X).$

We also need the following result to prove Lemma 3.4.

######  Lemma 3.3.

[18, Lemma 2(i)] Let $u,v\in
L^{0}(X),\\{g_{k}\\}_{k\in\mathbb{Z}}\in\mathbb{D}^{\phi}(u)$ and
$\\{h_{k}\\}_{k\in\mathbb{Z}}\in\mathbb{D}^{\phi}(v).$ Then
$(\max\\{g_{k},h_{k}\\})_{k\in\mathbb{Z}}\in\mathbb{D}^{\phi}(\min\\{u,v\\}).$

######  Lemma 3.4.

Let $\phi\in\mathcal{A},0<p<\infty$ and $E\subset X$ be a compact subset. Then

$\operatorname{Cap}_{M^{\phi,p}}(E)\approx\inf\\{\|u\|^{p}_{M^{\phi,p}(X)}:u\in\mathcal{\tilde{G}^{\prime}}_{M^{\phi,p}}(E)\\},$

where $\mathcal{\tilde{G}^{\prime}}_{M^{\phi,p}}(E)=\\{u\in
M^{\phi,p}(E):u\,\,\text{is $\phi$-Lipschitz and}\,\,$u=1$\,\,\text{on a
neighbourhood of}\,\,E\\}.$

###### Proof.

It is easy to see that
$\operatorname{Cap}_{M^{\phi,p}}(E)=\inf\\{\|u\|^{p}_{M^{\phi,p}(X)}:u\in\mathcal{G^{\prime}}_{M^{\phi,p}(E)}\\},$
where $\mathcal{G}^{\prime}_{M^{\phi,p}}(E)=\\{u\in
M^{\phi,p}(X):u=1\,\,\text{in a neighbourhood of}\,\,E\\}.$ Indeed, it is
obvious that
$\operatorname{Cap}_{M^{\phi,p}}(E)\leq\inf\\{\|w\|^{p}_{M^{\phi,p}(X)}:w\in\mathcal{G^{\prime}}_{M^{\phi,p}}(E)\\}$
and to prove the reverse inclusion, we see that for any $\epsilon>0,$ there
exists $u\in\mathcal{G}_{M^{\phi,p}}(E)$ such that

$\|u\|^{p}_{M^{\phi,p}(X)}\leq\operatorname{Cap}_{M^{\phi,p}}(E)+\epsilon.$

Hence $u^{\prime}=\min\\{1,u\\}\in\mathcal{G^{\prime}}_{M^{\phi,p}}(E)$ and by
Lemma 3.3,

$\inf\\{\|w\|^{p}_{M^{\phi,p}(X)}:w\in\mathcal{G}^{\prime}_{M^{\phi,p}}(E)\\}\leq\|u^{\prime}\|^{p}_{M^{\phi,p}(E)}\leq\|u\|^{p}_{M^{\phi,p}(E)}\leq\operatorname{Cap}_{M^{\phi,p}}(E)+\epsilon,$

which gives us the desired result by setting $\epsilon\rightarrow 0.$
Hence without loss of generality let
$u\in\mathcal{G}^{\prime}_{M^{\phi,p}}(E).$ So, we can take $u=1$ in a
neighbourhood $U$ of $E.$ Then by Lemma 3.2, there exists $u_{j}\in
Lip_{\phi}(X)\cap M^{\phi,p}(X)$ such that
$\|u_{j}-u\|_{M^{\phi,p}(X)}\rightarrow 0.$ Now choose a $\phi$-Lipschitz
function $\eta,$ $0\leq\eta\leq 1$ such that $\eta=0$ in a neighbourhood
$\tilde{U}\subset\subset U$ of $E$ and $\eta=1$ in $X\setminus U.$ Then the
sequence $v_{j}=1-\eta(1-u_{j})$ converges to $1-\eta(1-u)$ in
$M^{\phi,p}(X),$ and $1-\eta(1-u)=u.$ Hence

$\displaystyle\inf_{v\in\mathcal{\tilde{G}}^{\prime}_{M^{\phi,p}}(E)}\|v\|^{p}_{M^{\phi,p}(X)}\leq\liminf_{i\rightarrow\infty}\|v_{j}\|^{p}_{M^{\phi,p}(X)}$
$\displaystyle\leq\liminf_{i\rightarrow\infty}C(\|v_{j}-u\|^{p}_{M^{\phi,p}(X)}+\|u\|^{p}_{M^{\phi,p}(X)})$
$\displaystyle=C\|u\|^{p}_{M^{\phi,p}(X)}.$

The lemma follows as the reverse inclusion is obvious. ∎

We also need the following Poincaré-type inequalities: the first one is proved
in [18, Lemma 15] and the second one can be proved by extending the proofs
available for $\phi(t)=t,$ see [23, Theorem 4.5] and also [14, Lemma 3.1].

######  Lemma 3.5.

Let $\alpha_{\phi}\in(0,1)$ and $\gamma\in(0,1/2].$
$(i)$ For any given $\lambda\in(0,\infty),$ there exists a positive constant
$C$ such that, for any $k\in\mathbb{Z},u\in
L^{0}(X),g\in\mathcal{D}^{\phi}(u)$ and $x\in X$

$\inf_{c\in\mathbb{R}}m^{\gamma}_{|u-c|}(B(x,2^{-k}))\leq
C\phi(2^{-k})\left\\{\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,2^{-k+1})}[g(y)]^{\lambda}d\mu(\gamma)\right\\}^{\frac{1}{\lambda}}.$

$(ii)$ Let $u\in\\{M^{\phi}_{p,q}(X),N^{\phi}_{p,q}(X)\\}$ and $0<p,q<\infty.$
Then there exist a positive constant $C$ and a sequence
$(g_{k})_{k\in\mathbb{Z}}\in\mathbb{D}^{\phi}(u)$ such that for any
$k\in\mathbb{Z},$

$\inf_{c\in\mathbb{R}}m^{\gamma}_{|u-c|}(B(x,2^{-k}))\leq
C\phi(2^{-k})\left\\{\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,2^{-k+1})}[g_{k}(y)]^{p}d\mu(\gamma)\right\\}^{\frac{1}{p}}.$

Proof of Theorem 1.2: For simplicity, we assume that $R=2^{-m}$ for some
$m\in\mathbb{Z}$. Without any loss of generality, we can assume that
$x_{0}\notin E.$ By Lemma 3.1, there exists a locally Lipschitz function $v\in
N^{\phi}_{p,q}(X)$ with $v\geq 1$ on a neighbourhood of $E$ which satisfies

(3.2) $\|v\|^{p}_{N^{\phi}_{p,q}(X)}\leq
C\operatorname{Cap}_{N^{\phi}_{p,q}}(E)+\delta\quad\text{for
every}\quad\delta>0.$

Now, we take a bounded $L$-Lipschitz function $\psi$ which satisfies

(3.3) $\psi=0\quad\text{on}\quad
B(x_{0},\frac{1}{4}\operatorname{dist}(x_{0},E))\quad\text{and}\quad\psi=1\quad\text{outside}\quad
B(x_{0},\frac{1}{2}\operatorname{dist}(x_{0},E)).$

Consider, $u=v\psi$. It is clear that $u\geq 1$ on $E$ and we claim that there
exists a $\phi$-Hajłasz gradient sequence $(g_{k})_{k\in\mathbb{Z}}$ of $u$
which takes $0$ in $B(x_{0},\frac{1}{4}\operatorname{dist}(x_{0},E))$ and
satisfies

(3.4) $\|(g_{k})\|_{l^{q}(L^{p}(X))}\leq C_{1}\|v\|_{N^{\phi}_{p,q}(X)}.$

Proof of the claim:
First we will prove that the sequences $(\rho_{k})_{k\in\mathbb{Z}}$,
$(\sigma_{k})_{k\in\mathbb{Z}}$ defined by

(3.5)
$\rho_{k}(t)=\begin{cases}\|\psi\|_{L^{\infty}(X)}h_{k}+\frac{2^{-k}C_{I}}{\phi(2^{-k-1})}L|v|&\text{if
$t\in X\setminus B(x_{0},\frac{1}{4}\operatorname{dist}(x_{0},E))$},\\\
0&\text{if $t\in
B(x_{0},\frac{1}{4}\operatorname{dist}(x_{0},E))$}\end{cases}$

and

(3.6)
$\sigma_{k}(t)=\begin{cases}h_{k}+\frac{2C_{I}}{\phi(2^{-k-1})}\|\psi\|_{L^{\infty}(X)}|v|&\text{if
$t\in X\setminus B(x_{0},\frac{1}{4}\operatorname{dist}(x_{0},E))$},\\\
0&\text{if $t\in
B(x_{0},\frac{1}{4}\operatorname{dist}(x_{0},E))$}\end{cases}$

are $\phi$-Hajłasz gradient sequences of $u$, where $(h_{k})_{k\in\mathbb{Z}}$
is a $\phi$-Hajłasz gradient sequence of $v$. Firstly, we will assume $y,z\in
X\setminus B(x_{0},\frac{1}{4}\operatorname{dist}(x_{0},E))$ with
$2^{-k-1}<d(y,z)\leq 2^{-k}$. The other cases follow similarly.
Now, since $\phi$ is almost increasing,

$\displaystyle|u(y)-u(z)|$ $\displaystyle=$
$\displaystyle|v(y)\psi(y)-v(z)\psi(z)|$ $\displaystyle\leq$
$\displaystyle|v(y)|\,|\psi(y)-\psi(z)|+|\psi(z)|\,|v(y)-v(z)|$
$\displaystyle\leq$
$\displaystyle|v(y)|\,|Ld(y,z)|+\|\psi\|_{L^{\infty}(X)}\phi(d(y,z))(h_{k}(y)+h_{k}(z))$
$\displaystyle\leq$
$\displaystyle\phi(d(y,z))(|v(y)|L\frac{2^{-k}C_{I}}{\phi(2^{-k-1})}+\|\psi\|_{L^{\infty}(X)}(h_{k}(y)+h_{k}(z)))$
$\displaystyle\leq$ $\displaystyle\phi(d(y,z))(\rho_{k}(y)+\rho_{k}(z)).$

Also,

$\displaystyle|u(y)-u(z)|$ $\displaystyle=$
$\displaystyle|v(y)\psi(y)-v(z)\psi(z)|$ $\displaystyle\leq$
$\displaystyle|v(y)|\,|\psi(y)-\psi(z)|+|\psi(z)|\,|v(y)-v(z)|$
$\displaystyle\leq$ $\displaystyle
2\|\psi\|_{L^{\infty}(X)}|v(y)|+\phi(d(y,z))(h_{k}(y)+h_{k}(z))$
$\displaystyle\leq$
$\displaystyle\phi(d(y,z))(2\|\psi\|_{L^{\infty}(X)}|v(y)|\frac{C_{I}}{\phi(2^{-k-1})}+h_{k}(y)+h_{k}(z))$
$\displaystyle\leq$ $\displaystyle\phi(d(y,z))(\sigma_{k}(y)+\sigma_{k}(z)).$

Therefore, the sequences $(\rho_{k})_{k\in\mathbb{Z}}$ and
$(\sigma_{k})_{k\in\mathbb{Z}}$ are $\phi$-Hajłasz gradient sequences of $u$.
Now we will prove (3.4) with the sequence $(g_{k})_{k\in\mathbb{Z}}$, where we
define, for any $i\in\mathbb{Z},$

(3.7) $g_{k}=\begin{cases}\rho_{k}&\text{if $k\geq i$}\\\ \sigma_{k}&\text{if
$k<i.$}\end{cases}$

In fact,

$\displaystyle\|(g_{k})_{k\in\mathbb{Z}}\|_{l^{q}(L^{p}(X))}$ $\displaystyle=$
$\displaystyle\left(\sum_{k\in\mathbb{Z}}\|g_{k}\|^{q}_{L^{p}(X)}\right)^{\frac{1}{q}}$
$\displaystyle\leq$ $\displaystyle c\left[\left(\sum_{k\geq
i}\|h_{k}+\frac{2^{-k}}{\phi(2^{-k-1})}|v|\,\|^{q}_{L^{p}(X)}\right)^{\frac{1}{q}}+\left(\sum_{k<i}\|h_{k}+\frac{1}{\phi(2^{-k-1})}|v|\,\|^{q}_{L^{p}(X}\right)^{\frac{1}{q}}\right]$
$\displaystyle\leq$ $\displaystyle
c\left[\left(\sum_{k\in\mathbb{Z}}\|h_{k}\|^{q}_{L^{p}(X)}\right)^{\frac{1}{q}}+\|v\|_{L^{P}(X)}\left(\sum_{k\geq
i}\frac{2^{-kq}}{\phi(2^{-k-1})^{q}}\right)^{\frac{1}{q}}+\|v\|_{L^{P}(X)}\left(\sum_{k<i}\frac{1}{\phi(2^{-k-1})^{q}}\right)^{\frac{1}{q}}\right]$
$\displaystyle\leq$ $\displaystyle
c\left[\|(h_{k})_{k\in\mathbb{Z}}\|_{l^{q}(L^{p}(X))}+\|v\|_{L^{p}(X)}\right],$

where in the last inequality we have used the inequalities in (2.1) after
choosing $i$ such that $2^{i-1}\leq L<2^{i}$. Since $(h_{k})$ is a
$\phi$-Hajłasz gradient sequence of $v$, it completes proof of (3.4).

Let $x\in E,$ $y\in B(x,2^{-m})$ and $z\in
B(x_{0},\frac{1}{4}\operatorname{dist}(x_{0},E)).$ Hence $d(y,z)<2^{-m+2}.$
Therefore there exists $k_{0}\geq m-2$ such that $2^{-k_{0}-1}\leq
d(y,z)<2^{-k_{0}}.$ So,

$\displaystyle 1\leq
u(x)\leq|u(x)-m^{\gamma}_{u}(B(x,2^{-k_{0}}))|+|m^{\gamma}_{u}(B(x,2^{-k_{0}}))|.$

Since $u$ is continuous, $x$ is a generalized Lebesgue point of $u$. By,
properties of $\gamma$-median,

$\displaystyle|u(x)-m^{\gamma}_{u}(B(x,2^{-k_{0}}))|$
$\displaystyle\leq\sum_{k\geq
k_{0}}|m^{\gamma}_{u}(B(x,2^{-k-1}))-m^{\gamma}_{u}(B(x,2^{-k}))|$
$\displaystyle\leq\sum_{k\geq
k_{0}}m^{\gamma}_{|u-m^{\gamma}_{u}(B(x,2^{-k}))|}(B(x,2^{-k-1}))$
$\displaystyle\leq\sum_{k\geq
k_{0}}m^{\gamma/c_{d}}_{|u-m^{\gamma}_{u}(B(x,2^{-k}))|}(B(x,2^{-k}))$
$\displaystyle\leq 2\sum_{k\geq
k_{0}}\inf_{c\in\mathbb{R}}\left[m^{\gamma^{\prime}}_{|u-c|}(B(x,2^{-k}))+m^{\gamma^{\prime}}_{|u-c|}(B(x,2^{-k}))\right]$

and then from Lemma 3.5 we get

(3.8) $|u(x)-m^{\gamma}_{u}(B(x,2^{-k_{0}}))|\leq c\sum_{k\geq
k_{0}}\phi(2^{-k})\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,2^{-k+1})}g^{p}_{k}d\mu\right)^{\frac{1}{p}},$

where $\gamma^{\prime}=\min\\{\gamma,\gamma/c_{d}\\}$. Using the definition of
$\phi$-Hajłasz gradient,

$\displaystyle|u(y)|=|u(y)-u(z)|\leq\phi(d(y,z))g_{k_{0}}(y).$

So, for a.e. $y,z$ with $d(y,z)<2^{-k_{0}},$ $|u(y)|\leq
c\phi(2^{-k_{0}})g_{k_{0}}(y)$, since $\phi$ is almost increasing. Hence,
using the properties of $\gamma$-median we obtain

$\displaystyle|m^{\gamma}_{u}(B(x,2^{-k_{0}}))|$ $\displaystyle\leq
m^{\gamma}_{c\phi(2^{-k_{0}})g_{k_{0}}}(B(x,2^{-k_{0}}))$
$\displaystyle=c\phi(2^{-k_{0}})m^{\gamma}_{g_{k_{0}}}(B(x,2^{-k_{0}}))$
$\displaystyle\leq
c\phi(2^{-k_{0}})\left(\gamma^{-1}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,2^{-k_{0}})}g^{p}_{k_{0}}d\mu\right)^{\frac{1}{p}}$
(3.9) $\displaystyle\leq c\sum_{k\geq
k_{0}}\phi(2^{-k})\left(\gamma^{-1}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,2^{-k})}g^{p}_{k}d\mu\right)^{\frac{1}{p}}.$

Summing (3.8) and (3)

$\displaystyle 1\leq u(x)$ $\displaystyle\leq c\sum_{k\geq
k_{0}}\phi(2^{-k})\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,2^{-k+1})}g^{p}_{k}d\mu\right)^{\frac{1}{p}}$
$\displaystyle\leq c\left(\sum_{k\geq
k_{0}}h_{\omega}(2^{-k+1})^{-\frac{1}{p}}\phi(2^{-k})\right)\sup_{k\geq
k_{0}}h_{\omega}(2^{-k+1})^{\frac{1}{p}}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,2^{-k+1})}g^{p}_{k}d\mu\right)^{\frac{1}{p}}$
$\displaystyle\leq c\left(\sum_{k\geq
k_{0}}\omega(2^{-k})^{-1}\right)\sup_{k\geq
k_{0}}h_{\omega}(2^{-k+1})^{\frac{1}{p}}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,2^{-k+1})}g^{p}_{k}d\mu\right)^{\frac{1}{p}}$
(3.10) $\displaystyle\leq c\sup_{k\geq
k_{0}}h_{\omega}(2^{-k+1})^{\frac{1}{p}}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,2^{-k+1})}g^{p}_{k}d\mu\right)^{\frac{1}{p}}.$

Now for every $x\in E$, there is a ball $B(x,2^{-k_{x}+1})$ with $k_{x}\geq
k_{0}$ such that

$\frac{\mu(B(x,2^{-k_{x}+1})}{h_{\omega}(2^{-k_{x}+1})}\leq
c\int_{B(x,2^{-k_{x}+1})}g^{p}_{k_{x}}d\mu.$

By $5B$-covering lemma, there exists a countable family of disjoint balls
$B_{j}=B(x_{j},2^{-k_{x_{j}}+1})$ of radii $r_{j}=2^{-k_{x_{j}}+1}\leq
2^{-m+2}$, such that the dilated balls $5B_{j}$ cover the set $E$. Now we set
$I_{i}=\\{j:2^{-i}\leq 5r_{j}\leq 2^{-i+1}\\}$. Then $k_{x_{j}}=i+3$ for $j\in
I_{j}$. Since $\phi$ is increasing, $\omega$ is admissible, $\mu$ is doubling
and the balls are disjoint, we obtain

$\displaystyle\sum_{j\in I_{i}}\frac{\mu(5B_{j})}{h_{\omega}(5r_{j})}$
$\displaystyle\leq c\sum_{j\in I_{i}}\frac{\mu(B_{j})}{h_{\omega}(r_{j})}$
$\displaystyle\leq c\sum_{j\in I_{i}}\int_{B_{j}}g^{p}_{i+3}d\mu$
$\displaystyle\leq c\|g_{i+3}\|^{p}_{L^{p}(X)},$

where in the penultimate inequality we have used the disjointness of the balls
$B_{j}.$ Summing over $i,$ we get

$\displaystyle\sum_{2^{-i}<5.2^{-m+2}}\left(\sum_{j\in
I_{i}}\frac{\mu(5B_{j})}{h_{\omega}(5r_{j})}\right)^{\frac{q}{p}}$
$\displaystyle\leq c\sum_{i\in\mathbb{Z}}\|g_{i+3}\|^{q}_{L^{p}(X)}$

and hence $\mathcal{H}^{h_{\omega},q/p}_{20R}(E)\leq
c\left(\sum_{i\in\mathbb{Z}}\|g_{i+3}\|^{q}_{L^{p}(X)}\right)^{p/q}.$ Now,
taking $\delta\rightarrow 0$ in (3.2) and using (3.4) we get the desired
result. ∎

Proof of Theorem 1.3: For simplicity, we assume that $R=2^{-m}$ for some
$m\in\mathbb{Z}$. Let $E\subset B(x_{0},2^{-m})$ be a compact set. Without any
loss of generality, we can assume that $x_{0}\notin E.$ By Lemma 3.1 there
exists a locally Lipschitz function $v\in M^{\phi}_{p,q}(X)$ with $v\geq 1$ in
a neighbourhood of $E$ which satisfies

(3.11) $\|v\|^{p}_{M^{\phi}_{p,q}(X)}\leq
C\operatorname{Cap}_{M^{\phi}_{p,q}}(E)+\delta\quad\text{for
every}\quad\delta>0.$

Now, we take a bounded $L$-Lipschitz function $\psi$ which satisfies

(3.12) $\psi=0\quad\text{on}\quad
B(x_{0},\frac{1}{4}\operatorname{dist}(x_{0},E))\quad\text{and}\quad\psi=1\quad\text{outside}\quad
B(x_{0},\frac{1}{2}\operatorname{dist}(x_{0},E)).$

Consider, $u=v\psi$. It is clear that $u\geq 1$ on $E$ and we claim that there
exists a $\phi$-Hajłasz gradient sequence $(g_{k})_{k\in\mathbb{Z}}$ of $u$
which takes $0$ in $B(x_{0},\frac{1}{4}\operatorname{dist}(x_{0},E))$ and
satisfies

(3.13) $\|(g_{k})\|_{L^{p}(X,l^{q})}\leq C_{1}\|v\|_{M^{\phi}_{p,q}(X)}.$

In fact, the sequence $\\{g_{k}\\}$ defined in (3.7) is the $\phi$-Hajłasz
gradient satisfies (3.13):

$\displaystyle\left(\sum_{k\in\mathbb{Z}}g_{k}^{q}\right)^{1/q}$
$\displaystyle\leq$ $\displaystyle c\left[\left(\sum_{k\geq
i}(h_{k}+\frac{2^{-k}}{\phi(2^{-k-1})}|v|\,)^{q}\right)^{\frac{1}{q}}+\left(\sum_{k<i}(h_{k}+\frac{1}{\phi(2^{-k-1})}|v|\,)^{q}\right)^{\frac{1}{q}}\right]$
$\displaystyle\leq$ $\displaystyle
c\left[\left(\sum_{k\in\mathbb{Z}}h_{k}^{q}\right)^{\frac{1}{q}}+|v|\left(\sum_{k\geq
i}\frac{2^{-kq}}{\phi(2^{-k-1})^{q}}\right)^{\frac{1}{q}}+|v|\left(\sum_{k<i}\frac{1}{\phi(2^{-k-1})^{q}}\right)^{\frac{1}{q}}\right]$

and hence

$\|g_{k}\|_{L^{p}(X,l^{q})}\leq
c\left[\|(h_{k})_{k\in\mathbb{Z}}\|_{L^{p}(X,l^{q})}+\|v\|_{L^{p}(X)}\right],$

where in the last inequality we have used the inequalities in (2.1) after
choosing $i$ such that $2^{i-1}\leq L<2^{i}$.

Let $x\in E,$ $y\in B(x,2^{-m})$ and $z\in
B(x_{0},\frac{1}{4}\operatorname{dist}(x_{0},E)).$ Hence $d(y,z)<2^{-m+2}.$
Therefore there exists $k_{0}\geq m-2$ such that $2^{-k_{0}-1}\leq
d(y,z)<2^{-k_{0}}.$ So,

$\displaystyle 1\leq
u(x)\leq|u(x)-m^{\gamma}_{u}(B(x,2^{-k_{0}}))|+|m^{\gamma}_{u}(B(x,2^{-k_{0}}))|.$

Since $u$ is continuous, $x$ is a generalized Lebesgue point of $u$. By,
properties of $\gamma$-median,

$\displaystyle|u(x)-m^{\gamma}_{u}(B(x,2^{-k_{0}}))|$
$\displaystyle\leq\sum_{k\geq
k_{0}}|m^{\gamma}_{u}(B(x,2^{-k-1}))-m^{\gamma}_{u}(B(x,2^{-k}))|$
$\displaystyle\leq\sum_{k\geq
k_{0}}m^{\gamma}_{|u-m^{\gamma}_{u}(B(x,2^{-k}))|}(B(x,2^{-k-1}))$
$\displaystyle\leq\sum_{k\geq
k_{0}}m^{\gamma/c_{d}}_{|u-m^{\gamma}_{u}(B(x,2^{-k}))|}(B(x,2^{-k}))$
$\displaystyle\leq 2\sum_{k\geq
k_{0}}\inf_{c\in\mathbb{R}}m^{\gamma^{\prime}}_{|u-c|}(B(x,2^{-k}))$

and then from Lemma 3.5 we get

(3.14) $|u(x)-m^{\gamma}_{u}(B(x,2^{-k_{0}}))|\leq c\sum_{k\geq
k_{0}}\phi(2^{-k})\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,2^{-k+1})}g^{p}_{k}d\mu\right)^{\frac{1}{p}},$

where $\gamma^{\prime}=\min\\{\gamma,\gamma/c_{d}\\}$. Using the definition of
$\phi$-Hajłasz gradient,

$\displaystyle|u(y)|=|u(y)-u(z)|\leq\phi(d(y,z))g_{k_{0}}(y).$

So, for a.e. $y,z$ with $d(y,z)<2^{-k_{0}},$ $|u(y)|\leq
c\phi(2^{-k_{0}})g_{k_{0}}(y)$, since $\phi$ is almost increasing. Use
inequality (2.2) when $q\leq 1$ and Hölder’s inequality when $q\geq 1$ to
obtain, for a.e. $y\in B(x,2^{-m}),$

$|u(y)|\leq C\phi(2^{-k_{0}})\big{(}\sum_{j\geq
k_{0}}g_{j}(y)^{q}\big{)}^{1/q}$

Hence, using the properties of $\gamma$-median we obtain

$\displaystyle|m^{\gamma}_{u}(B(x,2^{-k_{0}}))|$ $\displaystyle\leq
m^{\gamma}_{c\phi(2^{-k_{0}})\|(g_{j})\|_{l^{q}}}(B(x,2^{-k_{0}}))$
$\displaystyle=c\phi(2^{-k_{0}})m^{\gamma}_{\|(g_{j})\|_{l^{q}}}(B(x,2^{-k_{0}}))$
$\displaystyle\leq
c\phi(2^{-k_{0}})\left(\gamma^{-1}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,2^{-k_{0}})}\|(g_{j})\|_{l^{q}}^{p}d\mu\right)^{\frac{1}{p}}$
(3.15) $\displaystyle\leq c\sum_{k\geq
k_{0}}\phi(2^{-k})\left(\gamma^{-1}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,2^{-k})}\|(g_{j})\|_{l^{q}}^{p}d\mu\right)^{\frac{1}{p}}.$

Summing (3.14) and (3)

$\displaystyle 1\leq u(x)$ $\displaystyle\leq c\sum_{k\geq
k_{0}}\phi(2^{-k})\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,2^{-k+1})}\|(g_{j})\|_{l^{q}}^{p}d\mu\right)^{\frac{1}{p}}$
$\displaystyle\leq c\left(\sum_{k\geq
k_{0}}h_{\omega}(2^{-k+1})^{-\frac{1}{p}}\phi(2^{-k})\right)\sup_{k\geq
k_{0}}h_{\omega}(2^{-k+1})^{\frac{1}{p}}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,2^{-k+1})}\|(g_{j})\|_{l^{q}}^{p}d\mu\right)^{\frac{1}{p}}$
$\displaystyle\leq c\left(\sum_{k\geq
k_{0}}\omega(2^{-k})^{-1}\right)\sup_{k\geq
k_{0}}h_{\omega}(2^{-k+1})^{\frac{1}{p}}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,2^{-k+1})}\|(g_{j})\|_{l^{q}}^{p}d\mu\right)^{\frac{1}{p}}$
(3.16) $\displaystyle\leq c\sup_{k\geq
k_{0}}h_{\omega}(2^{-k+1})^{\frac{1}{p}}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,2^{-k+1})}\|(g_{j})\|_{l^{q}}^{p}d\mu\right)^{\frac{1}{p}}.$

Now for every $x\in E$, there is a ball $B(x,2^{-k_{x}+1})$ with $k_{x}\geq
k_{0}$ such that

$\frac{\mu(B(x,2^{-k_{x}+1})}{h_{\omega}(2^{-k_{x}+1})}\leq
c\int_{B(x,2^{-k_{x}+1})}\|(g_{j})\|_{l^{q}}^{p}d\mu.$

By $5B$-covering lemma, there exists a countable family of disjoint balls
$B_{j}=B(x_{j},2^{-k_{x_{j}}+1})$ of radii $r_{j}=2^{-k_{x_{j}}+1}$, such that
the dilated balls $5B_{j}$ cover the set $E$. Now, summing over $i$ and using
the disjointness of the balls, doubling condition, increasing property of
$\phi$ as well as the admissible property of $\omega,$ we obtain

$\displaystyle\sum_{j}\frac{\mu(5B_{j})}{h_{\omega}(5r_{j})}$
$\displaystyle\leq c\sum_{j}\frac{\mu(B_{j})}{h_{\omega}(r_{j})}$
$\displaystyle\leq\|g_{j}\|^{p}_{L^{p}(X,l^{q})}.$

and hence $H^{\phi}_{c2^{-m}}(E)\leq C\|g_{j}\|^{p}_{L^{p}(X,l^{q})}.$ Then by
taking $\delta\rightarrow 0$ in (3.11) and applying (3.13) we obtain the
desired result. ∎

Proof of Theorem 1.4: For simplicity, let us assume that $R=2^{-m}$ for some
$m\in\mathbb{Z}.$ Without loss of generality, we can assume that $x_{0}\notin
E.$ By Lemma 3.4, for any $\delta>0,$ there exists a $\phi$-Lipschitz function
$v\in M^{\phi,p}(X)$ such that $v=1$ on a neighbourhood of $E$ and

(3.17)
$\|v\|^{p}_{M^{\phi,p}(X)}<\operatorname{Cap}_{M^{\phi}_{p,\infty}}(E)+\delta.$

Now, take a bounded $L$-Lipschitz function $\psi$ which satisfies

(3.18) $\psi=0\quad\text{on}\quad
B(x_{0},\frac{1}{4}\operatorname{dist}(x_{0},E))\quad\text{and}\quad\psi=1\quad\text{outside}\quad
B(x_{0},\frac{1}{2}\operatorname{dist}(x_{0},E)).$

Consider, $u=v\psi$.
So, $u=1$ on $E$ and we claim that there exists a $\phi$-Hajłasz gradient of
$u$ which takes $0$ in $B(x_{0},\frac{1}{4}\operatorname{dist}(x_{0},E))$ and
satisfies

(3.19) $\|g\|_{L^{p}(X)}\leq c\|v\|_{M^{\phi,p}(X)}.$

In fact, for any $\phi$-Hajłasz gradient of $h$ of $v,$ the function

(3.20)
$g(t)=\begin{cases}\|\psi\|_{L^{\infty}(X)}h+[\|\psi\|_{L^{\infty}(X)}+1][\phi(L^{-1})]^{-1}|v|&\text{if
$t\in X\setminus B(x_{0},\frac{1}{4}\operatorname{dist}(x_{0},E))$},\\\
0&\text{if $t\in
B(x_{0},\frac{1}{4}\operatorname{dist}(x_{0},E))$}\end{cases}$

is a $\phi$-Hajłasz gradient of $u.$ This can be proved by defining $g_{k}$ as
in (3.7) with $h_{k}$ replaced by $h$ therein, choosing $i$ with $2^{i-1}\leq
L<2^{i}$ and by taking $g=\sup_{k}g_{k}.$
Let $x\in E,$ $y\in B(x,2^{-m})$ and $z\in
B(x_{0},\frac{1}{4}\operatorname{dist}(x_{0},E)).$ So, $d(y,z)\leq 2^{-m+1}.$
Since $u$ is continuous, $x$ is a Lebesgue point of $u.$ Using properties of
$\gamma$-median, doubling property of $\mu$ and Lemma 3.5 in a similar fashion
to the estimate (3.14), we get

(3.21) $\displaystyle|u(x)-m^{\gamma}_{u}(B(x,2^{-m+1}))|$ $\displaystyle\leq
c\sum_{k\geq
m-1}\phi(2^{-k})\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,2^{-k+1})}g^{p}d\mu\right)^{\frac{1}{p}}.$

Using the definition of $\phi$-Hajłasz gradient,

$\displaystyle|u(y)|=|u(y)-u(z)|\leq\phi(d(y,z))g(y)$

and hence similar to (3), we have

(3.22) $\displaystyle|m^{\gamma}_{u}(B(x,2^{-m+1}))|\leq c\sum_{k\geq
m-1}\phi(2^{-k})\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,2^{-k+1})}g^{p}d\mu\right)^{\frac{1}{p}}.$

Summing (3.21) and (3.22)

$\displaystyle 1\leq u(x)$ $\displaystyle\leq c\sum_{k\geq
m-1}\phi(2^{-k})\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,2^{-k+1})}g^{p}d\mu\right)^{\frac{1}{p}}$
$\displaystyle\leq c\left(\sum_{k\geq
m-1}h_{\omega}(2^{-k+1})^{-\frac{1}{p}}\phi(2^{-k})\right)\sup_{k\geq
m-1}h_{\omega}(2^{-k+1})^{\frac{1}{p}}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,2^{-k+1})}g^{p}d\mu\right)^{\frac{1}{p}}$
$\displaystyle\leq c\left(\sum_{k\geq
m-1}\omega(2^{-k})^{-1}\right)\sup_{k\geq
m-1}h_{\omega}(2^{-k+1})^{\frac{1}{p}}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,2^{-k+1})}g^{p}d\mu\right)^{\frac{1}{p}}$
(3.23) $\displaystyle\leq c\sup_{k\geq
m-1}h_{\omega}(2^{-k+1})^{\frac{1}{p}}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{B(x,2^{-k+1})}g^{p}d\mu\right)^{\frac{1}{p}}.$

Now for every $x\in E$, there is a ball $B(x,2^{-k_{x}+1})$ with $k_{x}\geq
m-1$ such that

$\frac{\mu(B(x,2^{-k_{x}+1})}{h_{\omega}(2^{-k_{x}+1})}\leq
c\int_{B(x,2^{-k_{x}+1})}g^{p}d\mu.$

By $5B$-covering lemma, pick a collection of disjoint balls
$B_{i}=B(x_{i},2^{-k_{x_{i}}+1})$ of radii $r_{i}=2^{-k_{x_{i}}+1}\leq
2^{-m+2}$ such that $E\subset\cup_{i}5B_{i}.$ Now, summing over $i$ and using
the disjointness of the balls, doubling condition as well as the increasing
property of $\phi,$ we obtain

$\displaystyle\|g\|^{p}_{L^{p}(X)}$ $\displaystyle\geq
c\sum_{i}\frac{\mu(B_{i})}{h_{\omega}(r_{i})}$ $\displaystyle\geq
c\sum_{i}\frac{\mu(5B_{i})}{h_{\omega}(5r_{i})}$

and hence $H^{h_{\omega}}_{5\cdot 2^{-m+2}}\leq c\|g\|^{p}_{L^{p}(X)}.$ Taking
$\delta\rightarrow 0$ in (3.17) and applying (3.19) we obtain desired result.
∎

## References

  * [1] F. Cobos, D. L. Fernandez, Hardy-Sobolev spaces and Besov spaces with a function parameter, In Function Spaces and Applications., (Lund, 1986); Lecture Notes in Math., 1302; Springer. Berlin, Germany (1988), 158–170.
  * [2] A. M. Caetano, W. Farkas, Local growth envelopes of Besov spaces of generalized smoothness, Z. Anal. Anwend. (2006), 25: 265–298.
  * [3] F. Cobos, T. Kühn, Approximation and entropy numbers in Besov spaces of generalized smoothness., J. Approx. Theory. (2009), 160: 56–70.
  * [4] A. M. Caetano, H. G. Leopold, Local growth envelopes of Triebel–Lizorkin spaces of generalized smoothness., J. Fourier Anal. Appl. (2006), 12: 427–445.
  * [5] A. M. Caetano, S. D. Moura, Local growth envelopes of spaces of generalized smoothness., The subcritical case. Math. Nachr. (2004), 273: 43–57.
  * [6] A.M. Caetano, S.D.Moura, Local growth envelopes of spaces of generalized smoothness., The critical case., Math. Inequal. Appl. (2004), 7: 573–606.
  * [7] W. Farkas, H. G. Leopold, characterisations of function spaces of generalized smoothness., Ann. Mat. Pura Appl. (2006), 185: 1–62.
  * [8] P. Hajłasz, Sobolev spaces on an arbitrary metric space, Potential Anal. (1996), 5: 403-415.
  * [9] T. Heikkinen, L. Ihnatsyeva, H. Tuominen, Measure density and extension of Besov and Triebel-Lizorkin functions, J. Fourier Anal. Appl. (2016), 22(2): 334–382.
  * [10] T. Heikkinen, P. Koskela, H. Tuominen, Approximation and quasicontinuity of Besov and Triebel-Lizorkin functions, Trans. Amer. Math. Soc. (2017), 369(5): 3547–3573.
  * [11] J. Hu, A note on Hajłasz–Sobolev spaces on fractals., J. Math. Anal. Appl. (2003), 280: 91–101.
  * [12] G. A. Kaljabin and P. I. Lizorkin, Spaces of functions of generalized smoothness, Math. Nachr. (1987), 133: 7-32.
  * [13] N. Karak, Triebel-Lizorkin capacity and Hausdorff measure in metric spaces., Math. Slovaca. (2020), 70(3): 617–624.
  * [14] N. Karak, D. Mondal, Besov and Triebel-Lizorkin Capacity in Metric Spaces., Math. Slovaca. (2023), 73(4): 937-948.
  * [15] J. Kinnunen, V.Latvala, Lebesgue points for Sobolev functions on metric spaces., Rev. Mat. Iberoamericana. (2002), 18: 685-700.
  * [16] J. Kinnunen, O. Martio, The Sobolev capacity on metric spaces., Ann. Acad. Sci. Fenn. Math. (1996), 21: 367-382.
  * [17] P. Koskela, D. Yang, Y. Zhou, Pointwise characterizations of Besov and Triebel-Lizorkin spaces and quasiconformal mappings., Adv. Math. (2011), 226(4): 3579–3621.
  * [18] Z. Li, D. Yang, W. Yuan, Lebesgue points of Besov and Triebel-Lizorkin spaces with generalized smoothness, Mathematics. (2021), 9: Art. No. 2724.
  * [19] Z. Li, D. Yang, W. Yuan, Pointwise characterization of Besov and Triebel–Lizorkin spaces with generalized smoothness and their applications., Acta Math. Sin. (Engl. Ser.). (2022), 38(4): 623-661.
  * [20] M. Mocanu, Lebesgue points for Orlicz–Sobolev functions on metric measure spaces, An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.). (2011), 57: 175-186.
  * [21] Y. Netrusov, Metric estimates for the capacities of sets in Besov spaces, Trudy Mat. Inst. Steklov (1989), 190: 159–185 (Russian); English transl.: Proc. Steklov Inst. Math. (1992), 1: 167-192.
  * [22] Y. Netrusov, Estimates of capacities associated with Besov spaces, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) (1992), 201: 124-156 (in Russian); English transl.: J. Math. Sci. (1996), 78(2): 199-217.
  * [23] J. Nuutinen, The Besov capacity in metric spaces, Ann. Polon. Math. (2016), 117(1): 59-78.
  * [24] M. A. Prokhorovich, Capacities and Lebesgue points for Hajłasz–Sobolev fractional classes on metric measure spaces, Vestsī Nats. Akad. Navuk Belarusī Ser. Fīz. Mat. Navuk. (2006), 124: 19-23.
  * [25] J. Pohelhuis and A. Torchinsky, Medians, continuity, and vanishing oscillation, Studia Math. (2012), 213(3): 227-242.
  * [26] C. A. Rogers, Hausdroff measures, Cambridge Mathematical Library, Cambridhe University Press, Cambridge, Reprint of the 1970 original, With a foreword by Falconer, K. J. (1998).
  * [27] D. Yang, New characterizations of Hajłasz–Sobolev spaces on metric spaces., Sci. China Ser. A. (2003), 46: 675–689.

Nijjwal Karak
Department of Mathematics,
Birla Institute of Technology and Science-Pilani, Hyderabad Campus,
Hyderabad-500078, India
<EMAIL_ADDRESS><EMAIL_ADDRESS>

Debarati Mondal
Department of Mathematics,
Birla Institute of Technology and Science-Pilani, Hyderabad Campus,
Hyderabad-500078, India
<EMAIL_ADDRESS>