Orlicz-Besov extension and Ahlfors $n$-regular domains

TL;DR

This paper characterizes Ahlfors n-regular domains as Besov-Orlicz extension domains under certain conditions on the Young's function, linking geometric regularity with functional extension properties.

Contribution

It establishes a characterization of Ahlfors n-regular domains as Besov-Orlicz extension domains, providing necessary and sufficient conditions based on growth properties of the Young's function.

Findings

Ahlfors n-regular domains are Besov-Orlicz extension domains under certain conditions.

The necessity of Ahlfors n-regularity for Besov-Orlicz extension domains when the Young's function grows sub-exponentially.

The paper links geometric regularity with functional extension properties in the context of Besov-Orlicz spaces.

Abstract

Let $n\ge2$ and $\phi : [0,\fz) \to [0,\infty)$ be a Young's function satisfying $\sup_{x>0} \int_0^1\frac{\phi( t x)}{ \phi(x)}\frac{dt}{t^{n+1} }<\infty. $ We show that Ahlfors $n$-regular domains are Besov-Orlicz ${\dot {\bf B}}^{\phi}$ extension domains, which is necessary to guarantee the nontrivially of ${\dot {\bf B}}^{\phi}$. On the other hand, assume that $\phi$ grows sub-exponentially at $\fz$ additionally. If $\Omega$ is a Besov-Orlicz ${\dot {\bf B}}^{\phi}$ extension domain, then it must be Ahlfors $n$-regular.

Equations

$$\Vert u{\Vert }_{{\dot{\mathbf{B}}}^{\varphi }(\Omega )}:=\inf \left\{\alpha >0:{\int }_{\Omega }{\int }_{\Omega }\varphi \left(\frac{|u(x)-u(y)|}{\alpha }\right)\frac{dxdy}{|x-y{|}^{2n}}\le 1\right\}$$

$$\Vert u{\Vert }_{{\dot{\mathbf{B}}}^{\varphi }(\Omega )}={\left({\int }_{\Omega }{\int }_{\Omega }\frac{|u(x)-u(y){|}^p}{|x-y{|}^{2n}}dxdy\right)}^{1/p}.$$

$$C_{\varphi }:=\underset{x>0}{\sup }\frac{t^n}{\varphi (t)}{\int }_0^t\frac{\varphi (s)}{s^n}\frac{ds}{s}<\infty .$$

$$|B(x,r)\cap \Omega |\ge C_A(\Omega )r^n\forall x\in \Omega ,0<r<2\mathrm{missing}diam\Omega .$$

$$\underset{x\to \infty }{\mathrm{limsup}}\varphi (x)e^{-cx}=0\forall c>0.$$

$$\Vert u{\Vert }_{{\dot{\mathbf{W}}}^{s,p}(\Omega )}:={\left({\int }_{\Omega }{\int }_{\Omega }\frac{|u(x)-u(y){|}^p}{|x-y{|}^{n+sp}}dxdy\right)}^{1/p}<\infty .$$

$$|u(x)-u(y)|\leq|x-y|[g(x)+g(y)]\quad\mbox{for almost all $(x,y)\in\Omega\times\Omega$}.$$

$$\Vert u{\Vert }_{{\dot{\mathbf{M}}}^{1,p}(\Omega )}:=\underset{g\in \mathbb{D}(u,\Omega )}{\inf }\Vert g{\Vert }_{L^p(\Omega )}<\infty$$

$$\displaystyle\inf\limits_{c\in R}\int_{B\cap\Omega}\exp\left(\frac{|u-c|}{\alpha}\right)\,dx\leq C(n)|B|\quad\mbox{for any ball $B\subset{{{\mathbb{R}}}^{n}}$.}$$

$${\text{fint}}_B\exp {\left(\frac{|u(x)-u_B|}{\alpha }\right)}^{\gamma }dx\le C(n)$$

$${\int }_{\Omega }{\int }_{\Omega }\varphi \left(\frac{|u(x)-u(y)|}{\alpha }\right)\frac{dydx}{|x-y{|}^{2n}}\le 1.$$

$${\int }_{\Omega }\varphi \left(\frac{|u(x)-u(y)|}{\alpha }\right)\frac{dy}{|x-y{|}^{2n}}<\infty .$$

$${\text{fint}}_{B\cap \Omega }\varphi \left(\frac{|u(x)-u(y)|}{\alpha }\right)dy<\infty .$$

$$\varphi \left({\text{fint}}_{B\cap \Omega }\frac{|u(x)-u(y)|}{\alpha }dy\right)<\infty ,$$

$${\text{fint}}_{B\cap \Omega }|u(y)|dy\le |u(x)|+{\text{fint}}_{B\cap \Omega }|u(x)-u(y)|dy<\infty ,$$

$$H:={\int }_{\Omega }{\int }_{\Omega }\varphi \left(\frac{|u(z)-u(w)|}{\alpha }\right)\frac{dzdw}{|z-w{|}^{2n}}$$

$${\int }_W{\int }_W\varphi \left(\frac{|z-w|}{\alpha /L}\right)\frac{dzdw}{|z-w{|}^{2n}}$$

$$=n{\omega }_n{\int }_W{\int }_0^{2|\mathrm{missing}diamW|}\varphi \left(\frac{t}{\alpha /L}\right)\frac{dt}{t^{n+1}}dw$$

$$=n{\omega }_n|W|(\frac{L}{\alpha }{)}^n{\int }_0^{2L|\mathrm{missing}diamW|/\alpha }\varphi \left(s\right)\frac{ds}{s^{n+1}}$$

$$=n{\omega }_n|W|2^{-n}|\mathrm{missing}diamW{|}^{-n}\varphi \left(\frac{2L|\mathrm{missing}diamW|}{\alpha }\right).$$

$$2{\int }_{\Omega \setminus W}{\int }_V\varphi \left(\frac{1}{\alpha /L}\right)\frac{dzdw}{|z-w{|}^{2n}}$$

$${\text{fint}}_B\varphi \left(\frac{|u(x)-u_B|}{\alpha }\right)dx\le {\omega }_n^2$$

$${\text{fint}}_B|u(x)-u_B|dx\le {\varphi }^{-1}({\omega }_n^2)\Vert u{\Vert }_{{\dot{\mathbf{B}}}^{\varphi }(B)}.$$

$$\varphi \left(\frac{{\text{fint}}_B|u(x)-u_B|dx}{\alpha }\right)$$

$${\text{fint}}_B|u(x)-u_B|dx\le \alpha {\varphi }^{-1}({\omega }_n^2).$$

$${\text{fint}}_B|u(x)-u_B|dx\le {\varphi }^{-1}({\omega }_n^2)\Vert u{\Vert }_{{\dot{\mathbf{B}}}^{\varphi }(B)}$$

$$\Vert u{\Vert }_{BMO(\Omega )}=\underset{B\subset \Omega }{\sup }{\text{fint}}_B|u(x)-u_B|dx<\infty .$$

$${\varphi }_{\gamma }(t)=e^{t^{\gamma }}-\sum _{j=0}^{[n/\gamma ]}\frac{t^{\gamma j}}{j!}$$

$${\text{fint}}_B\exp {\left(\frac{|u(x)-u_B|}{\alpha }\right)}^{\gamma }dx\le C(n)$$

$${\text{fint}}_B|u(x)-u_B{|}^{[n/\gamma ]}dx\le C(\gamma ,n)\Vert u{\Vert }_{BMO({\mathbb{R}}^n)}^{[n/\gamma ]}\le C(\gamma ,n)\Vert u{\Vert }_{{\dot{\mathbf{B}}}^{\varphi }({\mathbb{R}}^n)}^{[n/\gamma ]}.$$

$${\text{fint}}_B{\left(\frac{|u(x)-u_B|}{\alpha }\right)}^{\gamma j}dx\le 1/n$$

$${\text{fint}}_B{\varphi }_{\gamma }\left(\frac{|u(x)-u_B|}{\alpha }\right)dx\le 1/n$$

Full text

Orlicz-Besov extension and Ahlfors $n$-regular domains

Tian Liang and Yuan Zhou

Department of Mathematics, Beihang University, Beijing 100191, P.R. China

614895606@qq.com, yuanzhou@buaa.edu.cn

Abstract Let $n\geq 2$ and $\phi:[0,\infty)\to[0,\infty)$ be a Young’s function satisfying $\sup_{x>0}\int_{0}^{1}\frac{\phi(tx)}{\phi(x)}\frac{dt}{t^{n+1}}<\infty.$ We show that Ahlfors $n$-regular domains are Besov-Orlicz ${\dot{\bf B}}^{\phi}$ extension domains, which is necessary to guarantee the nontrivially of ${\dot{\bf B}}^{\phi}$. On the other hand, assume that $\phi$ grows sub-exponentially at $\infty$ additionally. If $\Omega$ is a Besov-Orlicz ${\dot{\bf B}}^{\phi}$ extension domain, then it must be Ahlfors $n$-regular.

1. Introduction

Let $\phi:[0,\infty)\to[0,\infty)$ be a Young function, that is, $\phi$ is a convex, $\phi(0)=0$, $\phi(t)>0$ for $t>0$ and $\lim_{t\to\infty}\phi(t)=+\infty$. Given any domain $\Omega\subset{{{\mathbb{R}}}^{n}}$, the Orlicz-Besov space ${\dot{\bf B}}^{\phi}(\Omega)$ consists of all measurable functions $u$ in $\Omega$ whose (semi-)norms

[TABLE]

is finite. Modulo constant functions, ${\dot{\bf B}}^{\phi}(\Omega)$ is a Banach space. We refer to [14] for the applications of Orlicz-Besov spaces in qausi-conformal geometry. Note that, in the case of $\phi(t)=t^{p}$ with $p\geq 1$, the ${\dot{\bf B}}^{\phi}(\Omega)$-norms are written as

[TABLE]

By this, when $\phi(t)=t^{p}$ with $p>n$, $\dot{\bf B}^{\phi}(\Omega)$ is exactly the Besov spaces ${\dot{\bf B}}^{n/p}_{p,p}({\Omega})$ (or fractional Sobolev spaces $\dot{W}^{n/p,p}(\Omega)$). However, when $\phi(t)=t^{p}$ with $p\leq n$, thanks to (1.1) and [4], the space ${\dot{\bf B}}^{\phi}(\Omega)$ is trivial, that is, only contains constant functions.

In general, to guarantee the nontrivially of ${\dot{\bf B}}^{\phi}(\Omega)$, we always assume

[TABLE]

Indeed, (1.2) does imply that ${\dot{\bf B}}^{\phi}(\Omega)$ contains smooth functions with compact supports, and hence nontrivial; see Lemma 2.2 below. If $\phi(t)=t^{p}$, observe that $\phi$ satisfies (1.2) if and only if $p>n$, and ${\dot{\bf B}}^{\phi}(\Omega)$ is nontrivial if and only if $p>n$. In this sense, we see that (1.2) is optimal to guarantee the nontrivially of ${\dot{\bf B}}^{\phi}(\Omega)$. There are some other interesting Young functions satisfying (1.2), for example, $t^{n}[\ln(1+t)]^{\alpha}$ with $\alpha>1$, $t^{p}[\ln(1+t)]^{\alpha}$ with $p>n$ and $\alpha\geq 1$, $t^{p}e^{ct^{\alpha}}$ with $p>n$, $c>0$ and $\alpha>0$, and $e^{ct^{\alpha}}-\sum_{j=0}^{[n/\alpha]}(ct^{\alpha})^{j}/j!$ where $c>0$ with $\alpha>0$, where $[n/\alpha]$ is the maximum of integers no bigger than $n/\alpha$.

In this paper, we obtain the following results for the Orlicz-Besov extension in Ahlfors $n$-regular domains. Recall that a domain $\Omega$ is Ahlfors $n$-regular if there exists a constant $C_{A}(\Omega)>0$ such that

[TABLE]

A domain $\Omega$ is called ${\dot{\bf B}}^{\phi}$-extension domain if any function $u\in{\dot{\bf B}}^{\phi}(\Omega)$ can be extended to as a function $\widetilde{u}\in{\dot{\bf B}}^{\phi}(\mathbb{R}^{n})$ continuously and linearly; in other words, there exists a bounded linear operator $E:{\dot{\bf B}}^{\phi}(\Omega)\to{\dot{\bf B}}^{\phi}({{{\mathbb{R}}}^{n}})$ with $Eu|_{\Omega}=u$ for any $u\in{\dot{\bf B}}^{\phi}(\Omega)$.

Theorem 1.1**.**

Let $\phi$ be a Young function satisfying (1.2).

1. (i)

If $\Omega\subset{{{\mathbb{R}}}^{n}}$ is Ahlfors $n$-regular domain, then $\Omega$ is a ${\dot{\bf B}}^{\phi}$-extension domain. 2. (ii)

Assume that $\phi$ additionally satisfies

[TABLE]

If $\Omega\subset{{{\mathbb{R}}}^{n}}$ is a ${\dot{\bf B}}^{\phi}$-extension domain, then $\Omega$ is Ahlfors $n$-regular.

Note that the condition (1.3) in Theorem 1.1 (ii) allows a large class of Young functions, including $t^{n}[\ln(1+t)]^{\alpha}$ with $\alpha>1$, $t^{p}[\ln(1+t)]^{\alpha}$ with $p>n$ and $\alpha\geq 1$, $t^{p}e^{ct^{\alpha}}$ with $p>n$ and $\alpha\in(0,1)$, and $e^{ct^{\alpha}}-\sum_{j=0}^{[n/\alpha]}(ct^{\alpha})^{j}/j!$ where $c>0$ with $\alpha\in(0,1)$. But (1.3) rules out $t^{p}e^{ct^{\alpha}}$ with $p>n$ and $\alpha\geq 1$, and $e^{ct^{\alpha}}-\sum_{j=0}^{[n/\alpha]}(ct^{\alpha})^{j}/j!$ where $c>0$ with $\alpha\geq 1$,

Theorem 1.1 extends the known results for fractional Sobolev spaces $\dot{\bf W}^{n/p,p}(\Omega)$ or Besov space $\dot{\bf B}^{n/p}_{p,p}(\Omega)$. Recall that the extension problem for function spaces (including Sobolev, fracntional Sobolev, Hajlasz-Sobolev, Besov and Triebel-Lizorkin spaces) have been widely studied in the literature, see [7, 8, 20, 9, 10, 2, 11, 16, 21, 22, 5, 17, 19] and the references therein. Given function spaces $X(U)$ defined in any domain $U\subset{{{\mathbb{R}}}^{n}}$ in the same manner, define $X$-extension domains similarly to $\dot{\bf B}^{\phi}$-extension domains. It turns out that the extendability of functions in $X(\Omega)$ not only relies on the geometry of the domain but also on the analytic properties of $X$. In particular, it was essentially known that Ahlfors $n$-regular domains are fractional Sobolev $\dot{\bf W}^{s,p}$-extension domains for any $s\in(0,1)$ and $p\geq 1$; see Jonsson-Wallin [9] (also Shvartsman [17]). Here $\dot{\bf W}^{s,p}(\Omega)$ is the set of all functions with

[TABLE]

Moreover, by Shvartsman [18] and Hajłasz et al [5, 6], Ahlfors $n$-regular domains also are Hajlasz-Sobolev $\dot{\bf M}^{1,p}$-extension domain with $p\geq 1$. Recall that for a given function $u$ in $\Omega$, we say $g$ is a Hajłasz gradient of $u$ (for short $g\in\mathbb{D}(u,\Omega)$) if

[TABLE]

The Hajłasz Sobolev space $\dot{\bf M}^{1,p}(\Omega)$ is the set of all functions $u$ in $\Omega$ with

[TABLE]

Conversely, Hajłasz [5, 6] essentially proved that Hajlasz-Sobolev $\dot{\bf M}^{1,p}$-extension domains must be Ahlfors $n$-regular; and by [24], similar results hold true for fractional Sobolev $\dot{\bf W}^{s,p}$-extension for any $s\in(0,1)$ and $p\geq 1$. Note that $\dot{\bf W}^{n/p,p}(\Omega)=\dot{\bf B}^{n/p}_{p,p}(\Omega)=\bf{\dot{B}}^{\phi}(\Omega)$ for any $p>n$ and $\phi(t)=t^{p}$.

To prove Theorem 1.1 (i), it suffices to define a suitable linear extension operator and prove its boundedness. Following Jones [8], to define the extension operator we have to find suitable reflecting cubes of Whitney cubes for ${{{\mathbb{R}}}^{n}}\setminus\overline{\Omega}$. If we use the reflecting cubes the same as in [11, 5, 23, 24] which may have unbounded overlaps, we cannot prove the boundedness of the extension operator in general since the Young function may grows exponentially at $\infty$. See Remark 4.1 for details. Instead, we use the reflecting cubes of Shvartsman [17, 18], which have bounded overlaps (see Lemma 2.2), to define extension operator. The bounded overlaps of reflecting cubes allow us to use the convexity of $\phi$, and also avoid using maximal functions. With some careful analysis, we finally obtain the boundedness of extension operator.

Theorem 1.1 (ii) is proved in section 5 by borrowing some ideas from [5, 24]. Precisely, we first prove $\Omega$ supports the following imbedding: there exists positive constants $C_{I}(\Omega)$ and $C(n)$ such that

[TABLE]

whenever $u\in{\dot{\bf B}}^{\phi}(\Omega)$ and $\alpha>C_{I}(\Omega)\|u\|_{{\dot{\bf B}}^{\phi}(\Omega)}>0$. Then we calculate the precise $\|u\|_{{\dot{\bf B}}^{\phi}(\Omega)}$-norm of some cut-off functions. Using this and the sub-exponential growth of $\phi$ following the idea from [5] (see also [6, 24]), we are able to prove $\Omega$ is Ahlfors $n$-regular.

As a byproduct, we have the following result.

Corollary 1.2**.**

Suppose that $\phi$ is a Young function satisfying (1.2) and (1.3). Let $\Omega\subset{{{\mathbb{R}}}^{n}}$ be any domain. The following are equivalent:

1. (i)

$\Omega$* is Ahlfors $n$-regular;* 2. (ii)

$\Omega$* is a ${\dot{\bf B}}^{\phi}$-extension domain;* 3. (iii)

$\Omega$* supports the imbedding (1.5).*

Remark 1.3**.**

We conjecture that Theorem 1.1 (ii) holds without the additional assumption (1.3). The difficult to remove (1.3) is to find a suitable imbedding properties of ${\dot{\bf B}}^{\phi}({{{\mathbb{R}}}^{n}})$ better than (1.5) when $\phi$ does not satisfies (1.3).

Note that (1.5) is always true when $\Omega$ is a $\dot{\bf B}^{\phi}$-extension domain, but it is not enough to prove that $\Omega$ is Ahlfors $n$-regular in general. If $\phi(t)=e^{t^{\alpha}}-\sum_{j=0}^{[n/\alpha]}t^{{\gamma}j}/j!$ for $t\geq 0$, by Lemma 2.5, $\Omega$ supports the imbedding

[TABLE]

whenever $\alpha>C({\gamma},n)\|u\|_{\dot{\bf B}^{\phi_{\gamma}}({{{\mathbb{R}}}^{n}})}$. However, when $\dot{\bf B}^{\phi_{\gamma}}$-extension domain, such a imbedding is also not enough to prove $\Omega$ is Ahlfors $n$-regular.

Notation used in the following is standard. The constant $C(n,\alpha,\phi)$ would vary from line to line and is independent of parameters depending only on $n,\alpha,\phi$. Constants with subscripts would not change in different occurrences , like $C_{\phi}$. Given a domain, set $B_{\Omega}(x,r)=B(x,r)\cap\Omega$ for convenience. We denote by $u_{X}$ the average of $u$ on $X$, namely, $u_{X}=\fint_{X}u\equiv\frac{1}{|X|}\int_{X}u\,dx$. For a domain $\Omega$ and $x\in{{{\mathbb{R}}}^{n}}$, we use $d(x,\Omega)$ to describe the distance from $x$ to $\Omega$.

2. Some basic properties

We list several basic properties of Orlicz-Besov spaces.

Lemma 2.1**.**

Suppose that $\phi$ is a Young function. Let $\Omega\subset{{{\mathbb{R}}}^{n}}$ be any domain. Then ${\dot{\bf B}}^{\phi}(\Omega)\subset L^{1}(B\cap\Omega)$ for any ball $B\subset{{{\mathbb{R}}}^{n}}$, in particular, ${\dot{\bf B}}^{\phi}(\Omega)\subset L^{1}(\Omega)$ when $\Omega$ is bounded.

Proof.

For any $\alpha>\|u\|_{{\dot{\bf B}}^{\phi}(\Omega)}$, we have

[TABLE]

By Fubini’s theorem, for almost all $x\in\Omega$ we have

[TABLE]

Fix such a point $x$. For any $B=B(z,r)\subset{{{\mathbb{R}}}^{n}}$ with $B\cap\Omega\neq\emptyset$, we have $|x-y|\leq|x|+|z|+r$ for all $y\in B\cap\Omega$, and hence

[TABLE]

By Jessen’s inequality, we have

[TABLE]

which implies that

[TABLE]

that is, $u\in L^{1}({B\cap\Omega})$ as desired. ∎

Lemma 2.2**.**

Suppose that $\phi$ is a Young function satisfying (1.2). Let $\Omega\subset{{{\mathbb{R}}}^{n}}$ be any domain. Then $C_{c}^{1}(\Omega)\subset\dot{\bf B}^{\phi}(\Omega)$.

Proof.

Assume that $L=\|u\|_{L^{\infty}(\Omega)}+\|Du\|_{L^{\infty}(\Omega)}>0$. Let $V={\rm\,supp\,}u\Subset W\Subset\Omega$. Then

[TABLE]

By (1.2), we have

[TABLE]

Moreover,

[TABLE]

Obviously, letting $\alpha$ sufficiently enough and using the convexity of $\phi$, we have $H\leq 1$. That is, $u\in\dot{\bf B}^{\phi}(\Omega)$ as desired. ∎

The following Poincaré type inequality is needed in Section 4. Below denote by $\omega_{n}$ the area of the unit sphere $S^{n-1}$.

Lemma 2.3**.**

Suppose that $\phi$ is a Young function. For any ball $B\subset{{{\mathbb{R}}}^{n}}$ and $u\in{\dot{\bf B}}^{\phi}(B)$, we have

[TABLE]

when $\alpha>\|u\|_{{\dot{\bf B}}^{\phi}(B)}$, and

[TABLE]

Proof.

Let $u\in{\dot{\bf B}}^{\phi}(B)$. For any $\alpha>\|u\|_{{\dot{\bf B}}^{\phi}(B)}$, by Jensen’s inequality, we have

[TABLE]

that is,

[TABLE]

Letting $\alpha\to\|u\|_{{\dot{\bf B}}^{\phi}(B)}$, we obtain

[TABLE]

as desired. ∎

As a consequence of Lemma 2.3, we have the following imbedding. Denote by $BMO(\Omega)$ the space of functions with bounded mean oscillations, that is, the collection of $u\in L^{1}_{\mathop{\mathrm{missing}}{\,loc\,}}(\Omega)$ such that

[TABLE]

Corollary 2.4**.**

Suppose that $\phi$ is a Young function. Let $\Omega\subset{{{\mathbb{R}}}^{n}}$ be any domain. We have ${\dot{\bf B}}^{\phi}(\Omega)\subset BMO(\Omega)$ and $\|u\|_{BMO(\Omega)}\leq\phi^{-1}(\omega_{n}^{2})\|u\|_{{\dot{\bf B}}^{\phi}(\Omega)}$ for all $u\in{\dot{\bf B}}^{\phi}(\Omega)$.

Note that if

[TABLE]

with ${\gamma}\geq 1$, then $\phi_{\gamma}$ a Young’s function satisfying (1.2). Denote by ${\dot{\bf B}}^{\phi_{\gamma}}({{{\mathbb{R}}}^{n}})$ the associated Orlicz-Besov space.

Lemma 2.5**.**

For ${\gamma}\geq 1$, there exists constant $C({\gamma},n)\geq 1$ such that for any $u\in{\dot{\bf B}}^{\phi_{\gamma}}({{{\mathbb{R}}}^{n}})$ and ball $B\subset{{{\mathbb{R}}}^{n}}$, we have

[TABLE]

whenever $\alpha>C({\gamma},n)\|u\|_{\dot{\bf B}^{\phi_{\gamma}}({{{\mathbb{R}}}^{n}})}$.

Proof.

By Corollary 2.4, we have $u\in BMO({{{\mathbb{R}}}^{n}})$ and $\|u\|_{BMO({{{\mathbb{R}}}^{n}})}\leq\phi^{-1}(\omega_{n}^{2})\|u\|_{{\dot{\bf B}}^{\phi}({{{\mathbb{R}}}^{n}})}$. Thus by the John-Nirenberg inequality, we have

[TABLE]

Thus for all $1\leq j\leq[n/{\gamma}]$, we have

[TABLE]

when $\alpha\geq C({\gamma},n)\|u\|^{q}_{{\dot{\bf B}}^{\phi}({{{\mathbb{R}}}^{n}})}$ for some constant $C({\gamma},n)$. Note that by Lemma 2.3, one has

[TABLE]

when $\alpha>n\omega_{n}^{2}\|u\|_{{\dot{\bf B}}^{\phi}({{{\mathbb{R}}}^{n}})}$. Since

[TABLE]

we obtain

[TABLE]

when $\alpha>[n\omega_{n}^{2}+C({\gamma},n)]\|u\|_{{\dot{\bf B}}^{\phi}({{{\mathbb{R}}}^{n}})}$. ∎

3. Whitney’s decomposition and the reflected quasi-cubes

In this section, we always let $\Omega$ be an Ahlfors $n$-regular domain. Observe that $|\partial{\Omega}|=0$; see [17, Lemma 2.1] and also [24, 5]. Moreover, ${\mathop{\mathrm{missing}}{\,diam\,}}\Omega=\infty$ if and only if $|\Omega|=\infty.$ Write $U:={{{\mathbb{R}}}^{n}}\setminus\overline{\Omega}$. Without loss of generality, we assume $U\neq\emptyset$. It’s well know that $U$ admits a Whitney decomposition.

Lemma 3.1**.**

There exists a collection ${\mathscr{W}}=\{Q_{i}\}_{i\in{\mathbb{N}}}$ of (closed) cubes satisfying

1. (i)

$U=\cup_{i\in{\mathbb{N}}}Q_{i}$, and $Q^{\circ}_{k}\cap Q^{\circ}_{i}=\emptyset$ for all $i,k\in{\mathbb{N}}$ with $i\neq k$; 2. (ii)

$l(Q_{k})\leq{\mathop{\mathrm{missing}}{\,dist\,}}(Q_{k},\partial\Omega)\leq 4\sqrt{n}l(Q_{k})$; 3. (iii)

$\frac{1}{4}l(Q_{k})\leq l(Q_{i})\leq 4l(Q_{k})$* whenever $Q_{k}\cap Q_{i}\neq\emptyset$.*

The following basic properties of Whitney’s decomposition are used quite often in Section 4. For any $Q\in\mathscr{W}$, denote by $N(Q)$ the neighbor cubes of $Q$ in $\mathscr{W}$, that is,

[TABLE]

Then, by (iii) there exists an integer $\gamma_{0}$ depending only on $n$ such that

[TABLE]

By (iii) again, for any $P,Q\in\mathscr{W}$ we know that

[TABLE]

It then follows that

[TABLE]

Indeed, by (3.2) we write

[TABLE]

By $l_{Q}\leq 4l_{P}$ given in (iii), and (3.1), we arrive at

[TABLE]

as desired.

Below we recall the reflected quasi-cubes of Whitney’s cubes as given by Shvartsman [17, Theorem 2.4]. For any $\epsilon>0$, set

[TABLE]

Obviously, $\mathscr{W}=\mathscr{W}_{\epsilon}$ for all $\epsilon>0$ if ${\mathop{\mathrm{missing}}{\,diam\,}}\Omega=\infty$, and $\mathscr{W}_{\epsilon}\subsetneq\mathscr{W}$ for any $\epsilon>0$ if ${\mathop{\mathrm{missing}}{\,diam\,}}\Omega<\infty$.

For any $Q=Q(x_{Q},l_{Q})\in\mathscr{W}_{\epsilon}$, fix any $x_{Q}^{\ast}\in\Omega$ so that ${\mathop{\mathrm{missing}}{\,dist\,}}(Q,\Omega)={\mathop{\mathrm{missing}}{\,dist\,}}(x,Q)$. By Lemma 3.1 (ii), one has

[TABLE]

Set

[TABLE]

where

[TABLE]

Below, when $\epsilon$ is small enough, we define $\widetilde{Q}^{\ast\epsilon}$ as reflected quasi-cubes of $Q\in\mathscr{W}_{\epsilon}$ so that they enjoy some nice properties; see [17, Theorem 2.4] for the proof, here we omit the details.

Lemma 3.2**.**

Let $\epsilon_{0}=[C_{A}(\Omega)/2{\gamma}_{0}]^{1/n}/(30\sqrt{n})$. Denote by $Q^{\ast}=\widetilde{Q}^{\ast\epsilon_{0}}$ as quasi-cubes of any cube $Q\in\mathscr{W}_{\epsilon_{0}}$. Then the following hold:

1. (i)

$Q^{\ast}\subset(10\sqrt{n}Q)\cap\Omega$* for any $Q\in\mathscr{W}_{\epsilon_{0}}$;* 2. (ii)

$|Q|\leq\gamma_{1}|Q^{\ast}|$* whenever $Q\in\mathscr{W}_{\epsilon_{0}}$;* 3. (iii)

$\sum\limits_{Q\in\mathscr{W}_{\epsilon_{0}}}\chi_{Q^{\ast}}\leq\gamma_{2}$.

Above $\gamma_{1}$ and $\gamma_{2}$ are positive constants depending only on $n$ and $C_{A}(\Omega)$.

If $\Omega$ is bounded, we let $Q^{\ast}=\Omega$ as the reflected quasi-cube of any cube $Q\in\mathscr{W}\setminus\mathscr{W}_{\epsilon_{0}}\neq\emptyset$. Write

[TABLE]

where $\mathscr{W}_{\epsilon_{0}}^{{}^{(0)}}=\mathscr{W}_{\epsilon_{0}}$. That is, $\mathscr{W}_{\epsilon_{0}}^{(k)}$ is the $k^{\rm th}$-neighbors of $\mathscr{W}_{\epsilon_{0}}$.

[TABLE]

Since $Q^{\ast}=\Omega$ for $Q\notin\mathscr{W}_{\epsilon_{0}}$, by Lemma 3.3 (iii) we have

[TABLE]

For $Q\in\mathscr{W}^{{}^{(k)}}_{\epsilon_{0}}\setminus\mathscr{W}_{\epsilon_{0}}$, observe that $l_{Q}\geq\frac{1}{\epsilon_{0}}{\mathop{\mathrm{missing}}{\,diam\,}}\Omega$ and $l_{Q}\leq 4^{k}l_{P}\leq\frac{4^{k}}{\epsilon_{0}}{\mathop{\mathrm{missing}}{\,diam\,}}\Omega$ for some $P\in\mathscr{W}_{\epsilon_{0}}$. Thus, by Lemma 3.1 (ii), we have

[TABLE]

for any fixed $\bar{x}\in\Omega$, and hence

[TABLE]

This yields that

[TABLE]

Finally, associated to $\mathscr{W}$, one has the following partition of unit of $U$.

Lemma 3.3**.**

There exists a family $\{\varphi_{Q}:Q\in\mathscr{W}\}$ of functions such that

1. (i)

for each $Q\in\mathscr{W}$, $0\leq\varphi_{Q}\in C^{\infty}_{0}(\frac{17}{16}Q)$; 2. (ii)

for each $Q\in\mathscr{W}$, $|\nabla\varphi_{Q}|\leq L/l_{Q}$; 3. (iii)

$\sum_{Q\in\mathscr{W}}\varphi=\chi_{U}$* .*

4. Proof of Theorem 1.1(i)

It suffices to prove the existence of a bounded linear operator $E:{\dot{\bf B}}^{\phi}(\Omega)\to{\dot{\bf B}}^{\phi}({{{\mathbb{R}}}^{n}})$ such that $Eu|_{\Omega}=u$ for all $u\in{\dot{\bf B}}^{\phi}(\Omega)$. Define the operator $E$ by

[TABLE]

for any $u\in{\dot{\bf B}}^{\phi}(\Omega)$. Recall that $\mathscr{W}$ is the Whitney cubes of $U$ as in Lemma 3.1 and $\{\varphi_{Q}\}_{Q\in W}$ as in Lemma 3.3; that ${Q^{\ast}}$ is the reflected quasi-cube of $Q\in\mathscr{W}_{\epsilon_{0}}$ as given in Lemma 3.2, and $Q^{\ast}=\Omega$ if $Q\in\mathscr{W}\setminus\mathscr{W}_{\epsilon_{0}}$ (when $\Omega$ is bounded). By Lemma 2.1, $u_{Q^{\ast}}=\frac{1}{|Q^{\ast}|}\int_{Q^{\ast}}u\,dx$ is always finite.

Obviously, $E$ is linear, $Eu|_{\Omega}=u$ in $\Omega$, and moreover, if $\|u\|_{{\dot{\bf B}}^{\phi}(\Omega)}=0$, then $u$ and hence $Eu$ must be a constant function essentially. Thus, to prove the boundedness of $E:{\dot{\bf B}}^{\phi}(\Omega)\to{\dot{\bf B}}^{\phi}({{{\mathbb{R}}}^{n}})$, by the definition of the norm $\|\cdot\|_{{\dot{\bf B}}^{\phi}({{{\mathbb{R}}}^{n}})}$, we only need to find a constant $M>0$ depending only on $n$, $C_{A}(\Omega)$ and $\phi$ such that

[TABLE]

whenever $\|u\|_{{\dot{\bf B}}^{\phi}(\Omega)}=1$ and $\alpha>M$. Below we assume that $\|u\|_{{\dot{\bf B}}^{\phi}(\Omega)}=1$ Since $|\partial\Omega|=0$, one writes

[TABLE]

To get (4.1), it suffices to find constants $M_{i}\geq 1$ depending only on $n$, $C_{A}(\Omega)$ and $\phi$ such that $H_{i}(\alpha)\leq 1/4$ whenever $\alpha\geq M_{i}$ for $i=1,2,3$. Indeed, by taking $M=M_{1}+M_{2}+M_{3}$, we have $H(\alpha)\leq 1$ whenever $\alpha\geq M$.

Firstly, we may let $M_{1}=4$. Indeed, if $\alpha>4$ that is, $\alpha/4>1$, by the convexity of $\phi$ and $\|u\|_{{\dot{\bf B}}^{\phi}(\Omega)}=1$, we have

[TABLE]

To find $M_{2}$ and $M_{3}$, we consider two cases: ${\mathop{\mathrm{missing}}{\,diam\,}}\Omega=\infty$ and ${\mathop{\mathrm{missing}}{\,diam\,}}\Omega<\infty$.

Case ${\mathop{\mathrm{missing}}{\,diam\,}}\Omega=\infty$. To find $M_{2}$, for any $x\in U$ and $y\in{\Omega}$, since $\sum\limits_{Q\in\mathscr{W}}\varphi_{Q}(x)=1$ by Lemma 3.3, one has

[TABLE]

and hence, by the convexity of $\phi$ and Jensen’s inequality,

[TABLE]

If $\varphi_{Q}(x)\neq 0$, then $x\in\frac{17}{16}Q$. For $z\in Q^{\ast}$, by $Q^{\ast}\subset 10\sqrt{n}Q$, we have $|x-z|\leq 20nl(Q)$. Since $|x-y|\geq d(x,\,\Omega)\geq l(Q)$, we have $|x-z|\leq 20n|x-y|$, that is ,

[TABLE]

So we have

[TABLE]

Thus, by Lemma 3.2 (ii) we write

[TABLE]

Since $\varphi_{Q}\leq\chi_{\frac{9}{8}Q}$ as given in Lemma 3.3, by (3.3) we have

[TABLE]

which implies that

[TABLE]

By $\sum_{Q\in\mathscr{W}}\chi_{Q^{\ast}}\leq\gamma_{2}$ as in Lemma 3.2 (iii), we obtain

[TABLE]

Take $M_{2}=8\gamma_{1}4^{n}\gamma_{0}\gamma_{2}(21n)^{2n}$. By the convexity of $\phi$ again, if $\alpha>M_{2}$, we have $H_{2}(\alpha)\leq 1/4$.

To find $M_{3}$, for each $x\in U$ set

[TABLE]

Write

[TABLE]

Below, we show that there exists $M_{3i}\geq 1$ such that if $\alpha>M_{3i}$, then $H_{3i}\leq 1/8$ for $i=1,2$. If this is true, then letting $M_{3}:=\max\{M_{31},M_{32}\}$, for $\alpha>M_{3}$ we have $H_{3}\leq\frac{1}{4}$ as desired.

To find $M_{31}$, for $x\in U$ and $y\in X_{1}(x)$, since

[TABLE]

we have

[TABLE]

Applying the convexity of $\phi$ and Jensen’s inequality, one obtains

[TABLE]

For $x\in Q$ and $z\in Q^{\ast}$, by $Q^{\ast}\subset 10\sqrt{n}Q$, we have $|x-z|\leq 10nl_{Q}\leq 10nd(x,\Omega)$. Similarly, for $y\in P$, and $w\in P^{\ast}$, we have $|y-w|\leq 10nd(y,\Omega)$. If $y\in X(x)$, that is, $132n|x-y|\geq\max\{d(x,\Omega),d(y,\Omega)\}$, we further have

[TABLE]

Thus

[TABLE]

By $|Q|\leq\gamma_{1}|Q^{\ast}|$ and $|P|\leq\gamma_{1}|P^{\ast}|$ as given in Lemma 2.2 (ii), we have

[TABLE]

By Lemma 3.3 and (3.3) we have

[TABLE]

Thus

[TABLE]

Observing $\sum\limits_{Q\in\mathscr{W}}\chi_{Q^{\ast}}\leq{\gamma}_{2}$ as in Lemma 3.2 (iii), we arrive at

[TABLE]

Letting $M_{31}=8(2641n)^{2n}{\gamma}_{1}^{2}{\gamma}_{2}^{2}(4^{n}\gamma_{0})^{2}$. If $\alpha>M_{31}$, by the convexity of $\phi$ again we have $H_{31}(\alpha)\leq 1/8$.

To find $M_{32}$, write

[TABLE]

Let $x\in U$ and $y\in X_{2}(x)\cap P$ for some $P\in\mathscr{W}$. Since

[TABLE]

we write

[TABLE]

Note that by Lemma 3.3,

[TABLE]

One gets

[TABLE]

Moreover, we have

[TABLE]

Indeed, since $y\in X_{2}(x)$, that is, $|x-y|\leq\frac{1}{132n}\max\{d(x,\Omega),d(y,\Omega)\}$, taking $\bar{y}\in\bar{\Omega}$ with $|y-\bar{y}|=d(y,\Omega)$ we have

[TABLE]

which implies

[TABLE]

Similarly, we have

[TABLE]

Thus,

[TABLE]

If $y\in\frac{17}{16}Q$, by $y\in P$ we have $Q\in N(P)$, and hence

[TABLE]

Thus

[TABLE]

which implies that $x\in\frac{9}{8}Q$. Moreover, if $x\in\frac{17}{16}Q$, similarly we have $y\in\frac{9}{8}Q$, and hence $Q\in N(P)$. We conclude that

[TABLE]

when $Q\notin N(P)$, and

[TABLE]

when $Q\in N(P)$, This gives (4.3).

Note that by Lemma 3.1(iii), $\sum\limits_{Q\in\mathscr{W}}\chi_{\frac{17}{16}Q}(x)\leq\gamma_{0}$. From the convexity of $\phi$ and (4.3) it follows that

[TABLE]

Therefore, we obtain

[TABLE]

Observe that for $x\in\frac{9}{8}Q$ and $y\in P\cap X_{2}(x)$, by $d(x,\Omega)\leq 4\sqrt{n}l_{Q}$ we have

[TABLE]

By the assumption (1.2) for $\phi$, we have

[TABLE]

Using the above inequality and (3.3), one has

[TABLE]

For each $P\in\mathscr{W}$ and $Q\in N(P)$, by Jessen’s inequality

[TABLE]

Note that by Lemma 3.2 (i), $P^{\ast}\subset 10\sqrt{n}P$ and $Q^{\ast}\subset 10\sqrt{n}Q$. Thus for any $z\in P^{\ast}$ and $w\in Q^{\ast}$, by $Q\in N(P)$, we have

[TABLE]

Since $|Q|\leq\gamma_{1}|Q^{\ast}|$ and $|P|\leq\gamma_{1}|P^{\ast}|$ as given in Lemma 3.2 (ii), one gets

[TABLE]

Therefore,

[TABLE]

and hence

[TABLE]

With $\sum_{Q\in\mathscr{W}}\chi_{Q^{\ast}}\leq\gamma_{2}$ as given in Lemma 2.2 (iii), we obtain

[TABLE]

Letting $M_{32}=8L\gamma_{0}C_{\phi}n\omega_{n}4^{2n}(50n)^{2n}(\gamma_{1})^{2}(\gamma_{2})^{2}$. If $\alpha>M_{32}$, we have $H_{32}(\alpha)\leq 1/8$ as desired.

Case ${\mathop{\mathrm{missing}}{\,diam\,}}\Omega<\infty$.

To find $M_{2}$, write

[TABLE]

Recall that $V^{(2)}$ is defined by (3.4) in Section 3. It suffices to find $M_{2i}$ such that $H_{2i}\leq 1/8$ for $i=1,2$.

Regards of $H_{22}(\alpha)$, observe that for any $Q\in\mathscr{W}\setminus\mathscr{W}_{\epsilon_{0}}^{(2)}$, we have $N(Q)\cap\mathscr{W}_{\epsilon_{0}}=\emptyset$, and hence $P^{\ast}=\Omega$ for all $P\in N(Q)$. Thus, for any $x\in U\setminus V^{(2)}$, by Lemma 3.3 and Lemma 3.1 we have

[TABLE]

Thus

[TABLE]

By Jensen’s inequality, one gets

[TABLE]

For any $x\in U\setminus V^{(2)}$ and $y\in\Omega$, since there exists $Q\in\mathscr{W}\setminus\mathscr{W}_{\epsilon_{0}}$ so that $x\in Q$, one always has

[TABLE]

Moreover, by the Ahlfors $n$-regular assumption, it holds that $|\Omega|\geq C_{A}(\Omega)|{\mathop{\mathrm{missing}}{\,diam\,}}\Omega|^{2}$. Thus,

[TABLE]

from which, we conclude that

[TABLE]

Letting $M_{22}=8\omega_{n}\frac{1}{C}_{A}(\Omega)\epsilon_{0}^{n}$, by the convexity of $\phi$ again, for $\alpha>M_{22}$ we have $H_{22}(\alpha)\leq 1/8$.

Regards of $H_{21}(\alpha)$, observe that

[TABLE]

whenever $x\in V^{(2)}$. With aid of this and following, line by line, the argument to get (4.2) for $H_{2}(\alpha)$ in the case ${\mathop{\mathrm{missing}}{\,diam\,}}\Omega=\infty$, one has

[TABLE]

Here we omit the details. Since

[TABLE]

we have

[TABLE]

Set $M_{21}=16\gamma_{1}4^{n}\gamma_{0}[\gamma_{2}+(\epsilon_{0}+64\sqrt{n})^{n}](21n)^{2n}$. By the convexity of $\phi$ again, if $\alpha>M_{21}$, we have $H_{21}(\alpha)\leq 1/8$ as dsired.

To find $M_{3}$, notice that

[TABLE]

We write

[TABLE]

Since $Eu(x)=Eu(y)=u_{\Omega}$ for $x,y\in U\setminus V^{(2)}$, we have $H_{33}(\alpha)=0$. It suffices to find $M_{3i}$ such that $H_{3i}(\alpha)$ for all $\alpha>M_{3i}$ and $i=1,2$.

Regard of $H_{31}(\alpha)$, similarly to $H_{3}(\alpha)$ in the case ${\mathop{\mathrm{missing}}{\,diam\,}}\Omega=\infty$ and taking $M_{31}$ as $M_{3}$ there with $\gamma_{2}$ replaced by $\gamma_{2}+(\epsilon_{0}+4^{5}\sqrt{n})^{n}$, we can show that if $\alpha\geq M_{31}$, then $H_{31}(\alpha)\leq 1/8$. Here we omit the details.

For $H_{32}(\alpha)$, note that for $y\in U\backslash V^{(2)}$, we have $Eu(y)=u_{\Omega}$. Thus

[TABLE]

By Jessen’s inequality, one has

[TABLE]

For any $x\in V^{(2)}$ and $y\in U\backslash V^{(3)}$ note that $|x-y|\geq l(Q)\geq\frac{1}{\epsilon_{0}}{\mathop{\mathrm{missing}}{\,diam\,}}\Omega$, where $Q\in\mathscr{W}_{\epsilon_{0}}^{(3)}\setminus\mathscr{W}_{\epsilon_{0}}^{(2)}$ and $y\in Q$. Thus

[TABLE]

Since $|\Omega|\geq C_{A}(\Omega){\mathop{\mathrm{missing}}{\,diam\,}}\Omega$, one has

[TABLE]

Note that for any $x\in V^{(2)}$ there exists a $P_{i}\in\mathscr{W}^{(i)}_{\epsilon_{0}}$ such that $x\in P_{2}$ and $P_{i}\in N(P_{i-1})$ for $i=1,2$. Since $l(P_{0})\leq\frac{1}{\epsilon_{0}}{\mathop{\mathrm{missing}}{\,diam\,}}\Omega$, by Lemma 3.1 we know that $l(P_{2})\leq 4^{2}\frac{1}{\epsilon_{0}}{\mathop{\mathrm{missing}}{\,diam\,}}\Omega$. Thus for $y\in\Omega$, one has

[TABLE]

Therefore,

[TABLE]

If $\alpha>M_{32}=\frac{8}{C_{A}(\Omega)}4^{8n}\epsilon_{0}^{-n}n^{n}M_{21}$, we have $H_{32}(\alpha)\leq 1/8$. This completes the proof of Theorem 1.1 (i).

Remark 4.1**.**

We emphasis that the bounded overlaps of reflecting cubes $Q^{\ast}$ in Lemma 3.2 (iii) play central roles in the proof of the boundedness of extension operator $E:\dot{\bf B}^{\phi}(\Omega)$ to $\dot{\bf B}^{\phi}({{{\mathbb{R}}}^{n}})$.

Similarly to [24, 5, 11] and the reference therein, one may define the extension operator $\widetilde{E}u$ similarly to $Eu$ but replacing $Q^{\ast}$ in $Eu$ with $Q(x_{Q}^{\ast},l_{Q})\cap\Omega$, where $x^{\ast}_{Q}$ is the nearest point in $\Omega$ of $Q\in\mathscr{W}$. Note that $\{Q(x_{Q}^{\ast},l_{Q})\cap\Omega,Q\in\mathscr{W}\}$ does not have bounded overlap property as in Lemma 3.2(iii) in general.

In the case $\phi(t)=t^{p}$ with $p>n$, similarly to [24], one may prove that $\widetilde{E}$ is bounded from $\dot{\bf B}^{n/p}_{pp}(\Omega)$ to $\dot{\bf B}^{n/p}_{pp}({{{\mathbb{R}}}^{n}})$. The point is prove that

[TABLE]

where $\mathcal{M}$ is certain Hardy-Littlewood maixmal operator. See page 968 in the proof of [24, Theorem 1.1].

For general $\phi$ in Theorem 1.1, some appropriate estimates of of $\phi(\frac{\widetilde{E}u(x)-\widetilde{E}u(y)}{\alpha})\frac{1}{|x-y|^{2n}}$ via certain maximal functions are not available for us. We do not know if it is possible to obtain the boundedness of $\widetilde{E}$ from ${\bf B}^{\phi}(\Omega)$ to $\dot{\bf B}^{\phi}({{{\mathbb{R}}}^{n}})$. Note the our proof of the boundedness of $E$ does not work for $\widetilde{E}$ since $\{Q(x_{Q}^{\ast},l_{Q})\cap\Omega,Q\in\mathscr{W}\}$ does not have the bounded overlap property.

5. Proof of Theorem 1.1 (ii)

We divide the proof into 3 steps.

Step 1. Since $\Omega$ is a ${\dot{\bf B}}^{\phi}$-extension domain, there exists a bounded linear extension operator $E:{\dot{\bf B}}^{\phi}(\Omega)\to{\dot{\bf B}}^{\phi}({{{\mathbb{R}}}^{n}})$. For any $u\in{\dot{\bf B}}^{\phi}(\Omega)$, we have $Eu\in{\dot{\bf B}}^{\phi}({{{\mathbb{R}}}^{n}})$ with $Eu=u$ in $\Omega$, $\|Eu\|_{{\dot{\bf B}}^{\phi}({{{\mathbb{R}}}^{n}})}\leq\|E\|\|u\|_{{\dot{\bf B}}^{\phi}(\Omega)}$. By Lemma 2.3, we have $\widetilde{u}\in BMO({{{\mathbb{R}}}^{n}})$ and $\|Eu\|_{BMO({{{\mathbb{R}}}^{n}})}\leq\phi^{-1}({|S^{1}|^{2}})\|Eu\|_{{\dot{\bf B}}^{\phi}({{{\mathbb{R}}}^{n}})}$. From the John-Nirenberg inequality, it follows that

[TABLE]

Thus,

[TABLE]

Step 2. For $x\in\Omega$ and $0<r<t<{\mathop{\mathrm{missing}}{\,diam\,}}\Omega$, set the function

[TABLE]

We have the following.

Lemma 5.1**.**

Suppose that $\phi$ is a Young function satisfying (1.2). For $x\in\Omega$ and $0<r<t<{\mathop{\mathrm{missing}}{\,diam\,}}\Omega$, we have $u_{x,r,t}\in{\dot{\bf B}}^{\phi}(\Omega)$ with

[TABLE]

Proof of Lemma 5.1.

Write

[TABLE]

If suffices to find a constant $M$ depending only on $n$ such that for $\alpha=M[\phi^{-1}\left(\frac{(t-r)^{n}}{|B(x,t)\cap\Omega|}\right)]^{-1}$, we have $H_{1}\leq\frac{1}{2}$ and $H_{2}(\alpha)\leq\frac{1}{2}$.

Write

[TABLE]

Observe that

[TABLE]

Applying (1.2), we have

[TABLE]

On the other hand,

[TABLE]

Thus

[TABLE]

If $\alpha=M[\phi^{-1}\left(\frac{(t-r)^{n}}{|B(x,t)\cap\Omega|}\right)]^{-1}$ and $M\geq 2(nC_{\phi}+1)\omega_{n}$, we have

[TABLE]

Write

[TABLE]

Note that $\Omega\setminus B(x,t)\subset\Omega\setminus B(z,t-|z-x|)$, we have

[TABLE]

Hence,

[TABLE]

Notice that

[TABLE]

an hence

[TABLE]

Therefore,

[TABLE]

If $\alpha=M[\phi^{-1}\left(\frac{(t-r)^{n}}{|B(x,t)\cap\Omega|}\right)]^{-1}$ and $M\geq 8(C_{\phi}4^{n}+1)\omega_{n}$, we have

[TABLE]

as desired. ∎

Step 3. Let $x\in\Omega$ and $0<r<2{\mathop{\mathrm{missing}}{\,diam\,}}\Omega$. Let $b_{0}=1$ and $b_{j}\in(0,1)$ for $j\in N$ such that

[TABLE]

Let $u_{j}=u_{x,b_{j+1}r,b_{j}r}$ for $j\geq 1$ be as in Lemma 5.1. By (5.1), we have

[TABLE]

For any $c\in R$, we know that $|u_{j}-c|\geq 1/2$ either on $B(x,\,b_{j+1}r\,)\cap{\Omega}$ or on $[B(x,b_{j-1}r)\diagdown B(x,\,b_{j}r\,)]\cap\Omega$, and note that, by (5.2),

[TABLE]

Thus, for any $j\geq 1$, we have

[TABLE]

that is,

[TABLE]

Since

[TABLE]

we have

[TABLE]

and hence

[TABLE]

By (1.3), for any $\delta>0$, we have $\phi(t)\leq C(\delta)e^{\delta t}$ for all $t\geq 0$. Taking $\delta_{0}=1/2C(\phi,n,\Omega)$, that is, $C(\phi,n,\Omega)\delta_{0}=1/2$, we obtain

[TABLE]

Thus

[TABLE]

If $b_{1}\geq 1/10$, we get

[TABLE]

as desired.

If $b_{1}<1/10$, we can know that exists a point $x^{\prime}\in B(x,r)\cap\Omega$ satisfying $|x-x^{\prime}|=b_{1}r+r/5$. Let $R=2r/5$, then $B(x,b_{1}r)\subset B(x^{\prime},R)\subset B(x,r)$ and $B(x,b_{1}r)\cap B(x^{\prime},R/2)=\emptyset$. Thus

[TABLE]

By this, if $\left|B(x^{\prime},b^{\prime}_{1}R)\cap\Omega\right|=\frac{1}{2}\left|B(x^{\prime},R)\cap\Omega\right|$, then $b_{1}^{\prime}\geq 1/2$. Applying the result when $b_{1}\geq 1/10$ to the $B(x^{\prime},R/2)$ and $b_{1}^{\prime}\geq 1/2$, we get

[TABLE]

as desired. This completes the proof of Theorem 1.1 (ii).