$\displaystyle\lesssim_{p,d,\beta}\int_{\Omega}\bigg{(}|\rho^{2-\beta}f_{0}|^{p}+\sum_{i=1}^{d}|\rho^{1-\beta}f_{i}|^{p}\bigg{)}\rho^{-d}\,\mathrm{d}x$
$\displaystyle\leq\big{(}N_{F}\big{)}^{p}\int_{\Omega}\rho(x)^{M}\,\mathrm{d}x<\infty\,,$

where the last inequality follows from that $p(\alpha-\beta)=d+M$. Theorem
4.13 provide a solution $u\in H_{p,\theta}^{1}(\Omega)$ of equation (4.15)
with

$\|u\|_{H_{p,\theta}^{1}(\Omega)}\lesssim_{N}\|F\|_{H_{p,\theta+2p}^{-1}(\Omega)}^{p}\lesssim_{N}\Big{(}\int_{\Omega}\rho^{M}\,\mathrm{d}x\Big{)}^{1/p}N_{F}\,,$

where $N=N(d,\alpha,M_{\alpha},\beta)$. Proposition 3.13 implies
$|u|_{0,\beta}^{(-\beta)}\lesssim_{d,\alpha,\beta}\|u\|_{H_{p,\theta}^{1}(\Omega)}$,
and therefore we obtain (4.16). By the comment at the first in this proof, we
have $u\in C^{0,\beta}(\overline{\Omega})$. ∎

The following corollary is an unweighted $L_{p}$-solvability result when $p$
is close to $2$. It is worth noting that similar results for various equations
are introduced in the literature, such as [26], utilizing the reverse Hölder
inequality. The reason to provide Corollary 4.16 is that its proof is
independent of the reverse Hölder inequality; instead, this proof relies on
the weighted solvability result (Theorem 4.13). This theorem also provides the
$L_{p}$ estimate unaffected by dilation (see (4.18)).

We denote $\mathring{W}_{p}^{1}(\Omega)$ the closure of
$C_{c}^{\infty}(\Omega)$ in $W_{p}^{1}(\Omega)$.

######  Corollary 4.16.

Let $\Omega$ satisfy (4.10) and

$\displaystyle\lambda\geq 0\quad\text{if}\quad
D_{\Omega}<\infty\quad\text{and}\quad\lambda>0\quad\text{if}\quad
D_{\Omega}=\infty\,,$ (4.17)

where $d_{\Omega}:=\sup_{x\in\Omega}d(x,\partial\Omega)$. Then there exists
$\epsilon\in(0,1)$ depending only on $d$, $\epsilon_{0}$ (in (4.10)) such that
for any $p\in(2-\epsilon,2+\epsilon)$, the following holds: For any
$f^{0},\,\ldots,\,f^{d}\in L_{p}(\Omega)$, equation (4.15) has a unique
solution $u$ in $\mathring{W}^{1}_{p}(\Omega)$. Moreover, we have

$\displaystyle\begin{split}\|\nabla
u\|_{p}+\big{(}\lambda^{1/2}+D_{\Omega}^{-1}\big{)}\|u\|_{p}\lesssim_{d,p,\epsilon_{0}}\min\big{(}\lambda^{-1/2},D_{\Omega}\big{)}\|f^{0}\|_{p}+\sum_{i=1}^{d}\|f^{i}\|_{p}\,.\end{split}$
(4.18)

###### Proof.

We first note the following two results for the capacity density condition
(4.10):

  * (a)

By Lemma 4.10.(1), there exists $\alpha\in(0,1)$ such that $\Omega$ satisfies
$\mathbf{LHMD}(\alpha)$. Due to Theorem 4.13, Statement 4.3 $(\Omega,p,d-p)$
holds for $p\in(2-\alpha_{1},2+\alpha_{1})$, and $N_{1}$ (in (4.3)) depends
only on $d,\,p,\,\gamma,\,\epsilon_{1}$.

  * (b)

It is implied by [39, Theorem 1, Theorem 2] (or see [32, Theorem 3.7,
Corollary 3.11]) that there exists $\alpha_{2}\in(0,1)$ depending only on $d$
and $\epsilon_{0}$ such that for any $p>2-\alpha_{2}$,

$\displaystyle\int_{\Omega}\Big{|}\frac{u(x)}{\rho(x)}\Big{|}^{p}\,\mathrm{d}x\leq
N(d,p,\epsilon_{0})\int_{\Omega}|\nabla
u|^{p}\,\mathrm{d}x\quad\forall\,\,u\in C_{c}^{\infty}(\Omega)\,.$ (4.19)

Put $0<\epsilon<\min(\alpha_{1},\alpha_{2})$ and consider
$p\in(2-\epsilon,2+\epsilon)$.

Step 1. Uniqueness of solutions. Since Statement 4.3 $(\Omega,p,d-p)$ holds,
it suffices to show that $\mathring{W}_{p}^{1}(\Omega)\subset
H_{p,d-p}^{1}(\Omega)$. For any $u\in\mathring{W}_{p}^{1}(\Omega)$, there
exists $\\{u_{n}\\}_{n\in\mathbb{N}}\subset C_{c}^{\infty}$ such that
$u_{n}\rightarrow u$ in $W_{p}^{1}(\Omega)$. One can choose this $\\{u_{n}\\}$
to converge to $u$ almost everywhere on $\Omega$. Consider (4.19) for
$u_{n}\in C_{c}^{\infty}(\Omega)$, and apply Fatou’s lemma, to obtain that
(4.19) holds for $u\in\mathring{W}_{p}^{1}(\Omega)$. This implies that $u\in
H_{p,d-p}^{1}(\Omega)$.

Step 2. Existence of solutions and estimate (4.18). In this Step, we use Lemma
3.12.(1), $D_{\Omega}^{\,-1}\|u\|_{p}\leq\|\rho^{-1}u\|_{p}$, and $\|\rho
f\|_{p}\leq D_{\Omega}\|f\|_{p}$, without mentioning. Additionally, we also
use the fact that

$\displaystyle\|u\|_{H_{p,d}^{\gamma}(\Omega)}\lesssim_{p,d}\|u\|_{H_{p,d-p}^{\gamma+1}(\Omega)}^{1/2}\|u\|_{H_{p,d+p}^{\gamma-1}(\Omega)}^{1/2}\,,$
(4.20)

which follows from that

$\displaystyle\qquad\sum_{n\in\mathbb{Z}}\mathrm{e}^{nd}\|\big{(}\zeta_{0,(n)}u\big{)}(\mathrm{e}^{n}\cdot)\|_{H_{p}^{\gamma}}^{p}$
$\displaystyle\lesssim_{p,d}\sum_{n\in\mathbb{Z}}\mathrm{e}^{nd}\|\big{(}\zeta_{0,(n)}u\big{)}(\mathrm{e}^{n}\cdot)\|_{H_{p}^{\gamma+1}}^{p/2}\|\big{(}\zeta_{0,(n)}u\big{)}(\mathrm{e}^{n}\cdot)\|_{H_{p}^{\gamma-1}}^{p/2}$
$\displaystyle\leq\left(\sum_{n\in\mathbb{Z}}\mathrm{e}^{n(d-p)}\|\big{(}\zeta_{0,(n)}u\big{)}(\mathrm{e}^{n}\cdot)\|_{H_{p}^{\gamma+1}}^{p}\right)^{1/2}\left(\sum_{n\in\mathbb{Z}}\mathrm{e}^{n(d+p)}\|\big{(}\zeta_{0,(n)}u\big{)}(\mathrm{e}^{n}\cdot)\|_{H_{p}^{\gamma-1}}^{p}\right)^{1/2}\,.$

To prove the existence of solutions, it is enough to find a solution in
$L_{p,d}(\Omega)\cap H_{p,d-p}^{1}(\Omega)$. Indeed, $L_{p,d}(\Omega)\cap
H_{p,d-p}^{1}(\Omega)$ is continuouly embedded in $W_{p}^{1}$, and
$C_{c}^{\infty}(\Omega)$ is dense in $L_{p,d}(\Omega)\cap
H_{p,d-p}^{1}(\Omega)$ (see Lemma 3.10.(5)).

Without loss of generality, we assume that $\lambda=0$ or $\lambda=1$ by
dilation. Note that $\epsilon_{0}$ in (4.10) is invariant even if $\Omega$ is
replaced by $r\Omega=\\{rx\,:\,x\in\Omega\\}$, for any $r>0$.

Step 2.1) Consider the case $\lambda=1$. Since Statement 4.3 $(\Omega,p,d-p)$
holds, there exists $v\in H_{p,d-p}^{2}(\Omega)$ such that $\Delta
v-v=\widetilde{\rho}^{\,-1}f^{0}$ and

$\displaystyle\|v\|_{H_{p,d-p}^{2}(\Omega)}+\|v\|_{L_{p,d+p}(\Omega)}\lesssim_{d,p,\epsilon_{0}}\left\|\widetilde{\rho}^{-1}f^{0}\right\|_{L_{p,d+p}(\Omega)}\simeq_{p,d}\|f^{0}\|_{p}\,.$
(4.21)

Due to (4.20) and (4.21), we have

$\displaystyle\|v\|_{L_{p,d+p}(\Omega)}+\|v\|_{H_{p,d}^{1}(\Omega)}\lesssim_{d,p}\|v\|_{H_{p,d-p}^{2}(\Omega)}+\|v\|_{L_{p,d+p}(\Omega)}\lesssim_{d,p,\epsilon_{0}}\|f^{0}\|_{p}\,.$
(4.22)

Put

$\widetilde{f}:=f^{0}-\Delta(\widetilde{\rho}v)+\widetilde{\rho}v=-2\Big{[}\sum_{i=1}^{d}D_{i}\big{(}vD_{i}\widetilde{\rho})\Big{]}+v\Delta\widetilde{\rho}\,,$

and observe that

$\displaystyle\begin{split}\big{\|}\widetilde{f}\big{\|}_{H_{p,d+p}^{-1}(\Omega)}&\lesssim_{d,p}\|v\|_{L_{p,d}(\Omega)}\lesssim_{d,p}\|v\|_{H_{p,d}^{1}(\Omega)}\lesssim_{d,p,\epsilon_{0}}\left\|f^{0}\right\|_{p}\,,\end{split}$

where the first and third inequalities follow from Lemma 3.12.(2) and (4.22),
respectively. Since Statement 4.3 $(\Omega,p,d-p)$ holds, there exists $w\in
H_{p,d-p}^{1}(\Omega)$ such that

$\Delta w-w=\sum_{i=1}^{d}D_{i}f^{i}+\widetilde{f}$

and

$\displaystyle\|w\|_{H_{p,d-p}^{1}(\Omega)}+\|w\|_{H^{-1}_{p,d+p}(\Omega)}\lesssim_{d,p,\epsilon_{0}}\sum_{i=1}^{d}\|f^{i}\|_{L_{p,d}(\Omega)}+\big{\|}\widetilde{f}\big{\|}_{H_{p,d+p}^{-1}(\Omega)}\lesssim\sum_{i=0}^{d}\|f^{i}\|_{p}\,.$
(4.23)

Therefore, by (4.20) and (4.23), we have

$\displaystyle\|w\|_{L_{p,d}(\Omega)}+\|w\|_{H_{p,d-p}^{1}(\Omega)}$
$\displaystyle\lesssim_{d,p}$
$\displaystyle\|w\|_{H_{p,d-p}^{1}(\Omega)}+\|w\|_{H^{-1}_{p,d+p}(\Omega)}\lesssim_{d,p,\epsilon_{0}}\sum_{i\geq
0}\left\|f^{i}\right\|_{p}\,.$ (4.24)

Put $u=v\widetilde{\rho}+w$. Then $u$ is a solution of equation (4.15) and
satisfies

$\displaystyle\|u_{x}\|_{p}+(1+D_{\Omega}^{-1})\|u\|_{p}\lesssim_{d,p}\|u\|_{L_{p,d}(\Omega)}+\|u\|_{H_{p,d-p}^{1}(\Omega)}$
(4.25) $\displaystyle\lesssim_{d,p}$
$\displaystyle\|w\|_{L_{p,d}(\Omega)}+\|w\|_{H_{p,d-p}^{1}(\Omega)}+\|v\|_{L_{p,d+p}(\Omega)}+\|v\|_{H_{p,d}^{1}(\Omega)}\lesssim_{d,p,\epsilon_{0}}\sum_{i\geq
0}\|f^{i}\|_{p}\,,$

where the last inequality follows from (4.22) and (4.24); note that (4.25)
also implies that $u\in L_{p,d}(\Omega)\cap H_{p,d-p}^{1}(\Omega)$.

Step 2.2) Consider the case $D_{\Omega}<\infty$, and observe that

$\displaystyle\Big{\|}f^{0}+\sum_{i\geq
1}D_{i}f^{i}\Big{\|}_{H_{p,d+p}^{-1}(\Omega)}$ $\displaystyle\lesssim_{d,p}$
$\displaystyle\|f^{0}\|_{L_{p,d+p}(\Omega)}+\sum_{i\geq
1}\|f^{i}\|_{L_{p,d}(\Omega)}$ (4.26) $\displaystyle\leq$ $\displaystyle
D_{\Omega}\|f^{0}\|_{p}+\sum_{i\geq 1}\|f^{i}\|_{p}<\infty\,.$

Since Statement 4.3 $(\Omega,p,d-p)$ holds, there exists $\widetilde{u}\in
H_{p,d-p}^{1}(\Omega)$ such that

$\Delta\widetilde{u}-\lambda\widetilde{u}=f^{0}+\sum_{i\geq 1}D_{i}f^{i}\,,$

and

$\displaystyle\|\widetilde{u}\|_{H_{p,d-p}^{1}(\Omega)}+\lambda\|\widetilde{u}\|_{H_{p,d+p}^{-1}(\Omega)}\lesssim\|f^{0}+\sum_{i\geq
1}D_{i}f^{i}\|_{H_{p,d+p}^{-1}(\Omega)}\,.$ (4.27)

By (4.20), (4.26), and (4.27), we obtain that

$\displaystyle\|\nabla\widetilde{u}\|_{L_{p}(\Omega)}+D_{\Omega}^{-1}\|\widetilde{u}\|_{L_{p}(\Omega)}+\lambda^{1/2}\|\widetilde{u}\|_{L_{p}(\Omega)}$
(4.28) $\displaystyle\lesssim_{d,p}\,$
$\displaystyle\|\widetilde{u}\|_{H_{p,d-p}^{1}(\Omega)}+\lambda\|\widetilde{u}\|_{H_{p,d+p}^{-1}(\Omega)}\lesssim_{d,p,\epsilon_{0}}D_{\Omega}\|f^{0}\|_{p}+\sum_{i\geq
1}\|f^{i}\|_{p}\,.$

Due to (4.28), we have $\widetilde{u}\in L_{p,d}(\Omega)\cap
H_{p,d-p}^{1}(\Omega)$.

Step 2.3) The existence of solutions is proved in Steps 2.1 and 2.2, for all
$\lambda$ and $D_{\Omega}$ satisfying (4.17). For the cases where
$D_{\Omega}=\infty$ and $\lambda=1$, and $D_{\Omega}<\infty$ and $\lambda=0$,
estimate (4.18) is proved in (4.25) and (4.28), respectively. Therefore, we
only need prove estimate (4.18) in the remaining case where
$D_{\Omega}<\infty$ and $\lambda=1$. Since $u$ in Step 2.1 and $\widetilde{u}$
in Step 2.2 are the same (due to the result in Step 1), (4.18) follows from
(4.25) and (4.28). ∎

###  4.2. Domain with thin exterior : Aikawa dimension

The notion of the Aikawa dimension was first introduced by Aikawa [2]. We
recall the definition of the Aikawa dimension. For a set
$E\subset\mathbb{R}^{d}$, the Aikawa dimension of $E$, denoted by
$\dim_{\mathcal{A}}(E)$, is defined by

$\dim_{\mathcal{A}}(E)=\inf\Big{\\{}\beta\geq 0\,:\,\sup_{p\in
E,\,r>0}\frac{1}{r^{\beta}}\int_{B_{r}(p)}\frac{1}{d(y,E)^{d-\beta}}\,\mathrm{d}y<\infty\Big{\\}}$

with considering $\frac{1}{0}=\infty$.

In this subsection, we assume that $d\geq 3$, and $\Omega$ satisfies

$\beta_{0}:=\dim_{\mathcal{A}}\Omega^{c}<d-2\,.$

######  Theorem 4.17.

For a constant $\beta<d-2$, if there exists a constant $A_{\beta}$ such that

$\displaystyle\sup_{p\in\Omega^{c},\,r>0}\frac{1}{r^{\beta}}\int_{B_{r}(p)}\frac{1}{d(y,\Omega^{c})^{d-\beta}}\,\mathrm{d}y\leq
A_{\beta}<\infty\,,$ (4.29)

then the function

$\phi(x):=\int_{\mathbb{R}^{d}}|x-y|^{-d+2}\rho(y)^{-d+\beta}\,\mathrm{d}y$

is a superharmonic function on $\mathbb{R}^{d}$ with
$-\Delta\phi=N(d)\rho^{-d+\beta}$. Moreover, we have

$\displaystyle N^{-1}\rho(x)^{-d+2+\beta}\leq\phi(x)\leq
N\rho(x)^{-d+2+\beta}$ (4.30)

for all $x\in\Omega$, where $N=N(d,\beta,A_{\beta})$.

Before proving Theorem 4.17, we first look at the corollaries of this theorem.

######  Corollary 4.18.

The Hardy inequality (1.2) holds on $\Omega$, where $\mathrm{C}_{0}(\Omega)$
depends only on $d$, $\beta_{0}$, $\\{A_{\beta}\\}_{\beta>\beta_{0}}$.

Actually, this corollary follows from the more general result [2, Theorem 3],
and its proof is based on Muckenhoupt’s $A_{p}$ weight theory. Considering
only Corollary 4.18, this result can be proved differently, as the following:

###### Proof of Corollary 4.18.

We first note the following inequality provided in [8, Lemma 3.5.1]: If $f\in
C_{c}^{\infty}(\mathbb{R}^{d})$ and $s>0$ is a smooth superharmonic function
on a neighborhood of $\text{supp}(f)$, then

$\displaystyle\int_{\mathbb{R}^{d}}\frac{-\Delta
s}{s}|f|^{2}\,\mathrm{d}x\leq\int_{\mathbb{R}^{d}}|\nabla
f|^{2}\,\mathrm{d}x\quad\text{for all}\quad f\in
C_{c}^{\infty}(\mathbb{R}^{d})$ (4.31)

(the proof of this inequality is based on integrating $\big{|}\nabla
f-(f/s)\nabla s\big{|}^{2}$ and performing integration by parts). Take any
$\beta\in(\beta_{0},d-2)$, and let $\phi$ be the function in Theorem 4.17, so
that

$\displaystyle-\Delta\phi\geq N_{1}\rho^{-2}\phi>0$ (4.32)

where $N_{1}=N(d,\beta,A_{\beta})>0$. Fix $f\in C_{c}^{\infty}(\Omega)$. For
$0<\epsilon<d\big{(}\text{supp}(f),\partial\Omega\big{)}$, let
$\phi^{(\epsilon)}$ be the mollification of $\phi$ in (2.1). Observe that

$-\Delta\big{(}\phi^{(\epsilon)}\big{)}\geq
N_{1}^{-1}\big{(}\rho^{-2}\phi\big{)}^{(\epsilon)}\geq
N_{1}^{-1}(\rho+\epsilon)^{-2}\phi^{(\epsilon)}\quad\text{on}\quad\mathbb{R}^{d}\,,$

where $N_{1}$ is in (4.32). By appling the monotone convergence theorem to
(4.31) with $s=\phi^{(\epsilon)}$(see Lemma 2.5.(2)), we obtain (1.2) with
$\mathrm{C}_{0}(\Omega)=N_{1}$. ∎

######  Theorem 4.19.

For any $p\in(1,\infty)$ and $\theta\in\mathbb{R}$ satisfying

$\beta_{0}<\theta<(d-2-\beta_{0})p+\beta_{0}\,,$

Statement 4.3 $(\Omega,p,\theta)$ holds. In addition, $N_{1}$ in (4.3) depends
only on $d$, $p$, $\gamma$, $\theta$, $\beta_{0}$,
$\\{\mathcal{A}_{\beta}\\}_{\beta>\beta_{0}}$.

######  Remark 4.20.

Theorem 4.19 deals with the Poisson equation in $\Omega\subset\mathbb{R}^{d}$,
$d\geq 3$. Moreover, this theorem can also be interpreted as establishing the
solvability of the Poisson equation $\Delta u-\lambda u=f$ in
$\mathbb{R}^{d}$, particularly when $f$ blows up near a set $E$ with
$\dim_{\mathcal{A}}(E)<d-2$. In other words, if $u\in
H_{p,\theta}^{2}(\Omega)$ and $f\in L_{p,\theta+2p}(\Omega)$ satisfy equation
(4.2), then

$\int_{\mathbb{R}^{d}}u(\Delta\phi-\lambda\phi)\,\mathrm{d}x=\int_{\mathbb{R}^{d}}f\phi\,\mathrm{d}x\quad\text{for
all}\quad\phi\in C_{c}^{\infty}(\mathbb{R}^{d})\,;$

we leave the proof to the reader, with a comment to utilize the test functions
$\phi_{k}:=\phi\sum_{|n|\leq k}\zeta_{0,(n)}\in C_{c}^{\infty}(\Omega)$, where
$\zeta_{0,(n)}$ is the function in (3.9).

###### Proof of Theorem 4.19.

Takd $\beta\in(\beta_{0},d-2)$ satisfying

$\beta<\theta<(d-2-\beta)p+\beta\,.$

By Corollary 4.18 and Theorem 4.17, $\Omega$ admits the Hardy inequality
(1.2), and there exists a superharmonic function $\phi$ satisfying
$\phi\simeq\rho^{-d+2+\beta}$. Therefore by Lemma 4.4, the proof is completed.
∎

###### Proof of Theorem 4.17.

We first prove (4.30). For a fixed $x\in\mathbb{R}^{d}$, put

$I_{j}=\int_{E_{j}}|x-y|^{-d+2}\rho(y)^{-d+\beta}dy\quad\text{for}\,\,\,\,j=0,\,1,\,\ldots\,,$

where $E_{0}:=B\big{(}x,2^{-1}\rho(x)\big{)}$ and
$E_{j}:=B\big{(}x,2^{j-1}\rho(x)\big{)}\setminus
B\big{(}x,2^{j-2}\rho(x)\big{)}$ for $j=1,\,2,\,\ldots$. Then
$\phi(x)=\sum_{j\in\mathbb{N}_{0}}I_{j}$. If $y\in E_{0}$ then
$\frac{1}{2}\rho(x)\leq\rho(y)\leq 2\rho(x)$, which implies

$\displaystyle
I_{0}\simeq_{d,\beta}\rho(x)^{-d+\beta}\int_{B(x,\rho(x)/2)}|x-y|^{-d+2}\,\mathrm{d}y\simeq_{d}\rho(x)^{-d+2+\beta}\,.$
(4.33)

For $I_{j}$, $j\geq 1$, take $p_{x}\in\partial\Omega$ such that
$|x-p_{x}|=\rho(x)$, and observe that

$\displaystyle I_{j}\,$ $\displaystyle\lesssim_{d}$
$\displaystyle\,\big{(}2^{j}\rho(x)\big{)}^{-d+2}\int_{B(p_{x},2^{j}\rho(x))}\rho(y)^{-d+\beta}\,\mathrm{d}y\leq
N\big{(}2^{j}\rho(x)\big{)}^{-d+2+\beta}\,,$ (4.34)

where $N=N(d,\beta,A_{\beta})$. (4.33) and (4.34) imply (4.30).

To prove that $-\Delta\phi=N(d)\,\phi$ in the sense of distribution, recall
that

$-\Delta_{x}\Big{(}|x-y|^{-d+2}\Big{)}=N(d)\,\delta_{0}(x-y)$

in the sense of distribution, where $\delta_{0}(\cdot)$ is the Dirac delta
distribution. Due to (4.29) and $\phi\simeq\rho^{-d+2+\beta}$, $\phi$ is
locally integrable in $\mathbb{R}^{d}$. Therefore we obtain that for any
$\zeta\in C_{c}^{\infty}(\mathbb{R}^{d})$, by the Fubini theorem,

$\displaystyle\int_{\mathbb{R}^{d}}\phi(x)\big{(}-\Delta\zeta)(x)\,\mathrm{d}x\,$
$\displaystyle=\int_{\mathbb{R}^{d}}\Big{(}\int_{\mathbb{R}^{d}}|x-y|^{-d+2}(-\Delta\zeta)(x)\,\mathrm{d}x\Big{)}\rho(y)^{-d+\beta}\,\mathrm{d}y$
$\displaystyle=N(d)\int_{\mathbb{R}^{d}}\zeta(y)\rho(y)^{-d+\beta}\,\mathrm{d}y\,.$

∎

##  5\. Application II - Various domains with fat exterior

This section presents results for the exterior cone condition, convex domains,
the exterior Reifenberg condition, and Lipschitz cones. These domains and
conditions imply the fat exterior condition.

Throughout this section, we consider a domain
$\Omega\subsetneq\mathbb{R}^{d}$, $d\geq 2$.

###  5.1. Exterior cone condition and exterior line segment condition

######  Definition 5.1 (Exterior cone condition).

For $\delta\in[0,\frac{\pi}{2})$ and $R\in(0,\infty]$, a domain
$\Omega\subset\mathbb{R}^{d}$ is said to satisfy the exterior
$(\delta,R)$-cone condition if for every $p\in\partial\Omega$, there exists a
unit vector $e_{p}\in\mathbb{R}^{d}$ such that

$\displaystyle\\{x\in B_{R}(p)\,:\,(x-p)\cdot
e_{p}\geq|x-p|\cos\delta\\}\subset\Omega^{c}\,.$ (5.1)

Note that the left hand side of (5.1) is the result of translating and
rotating the set

$\\{x=(x_{1},\ldots,x_{d})\in B_{R}(0)\,:\,x_{1}\geq|x|\cos\delta\\}\,.$

The exterior $(0,R)$-cone condition can be called the exterior $R$-line
segment condition, since if $\delta=0$, then LHS in (5.1) equals
$\\{p+re_{p}\,:\,r\in[0,R)\\}$. For examples of the exterior cone condition
and exterior line segment condition, see Figure 5.1 below.

A. Lipschitz boundaryconditionB. Exterior$(\frac{\pi}{3},\infty)$-cone
condition(doesn’t satisfy Lipschitzboundary condition)C. Exterior
$\infty$-linesegment condition(doesn’t satisfy $(\delta,R)$-cone condition,
$\forall\,\delta,R>0$) Figure 5.1. Examples for exterior cone condition

######  Example 5.2.

Suppose that there exists $K,\,R\in(0,\infty]$ such that for any
$p\in\partial\Omega$, there exists a function $f_{p}\in C(\mathbb{R}^{d-1})$
such that

$\displaystyle|f_{p}(y^{\prime})-f_{p}(z^{\prime})|\leq
K|y^{\prime}-z^{\prime}|\quad\text{for all}\quad
y^{\prime},\,z^{\prime}\in\mathbb{R}^{d-1}\,\,\,,\,\,\,\,\,\text{and}\qquad\,\,\,$
(5.2) $\displaystyle\Omega\cap
B_{R}(p)=\big{\\{}y=(y^{\prime},y_{d})\in\mathbb{R}^{d-1}\times\mathbb{R}\,:\,y_{d}>f_{p}(y^{\prime})\quad\text{and}\quad|y|<R\big{\\}}\,,$
(5.3)

where $(y^{\prime},y_{d})=(y_{1},\cdots,y_{d})$ in (5.3) is an orthonormal
coordinate system centered at $p$. Then $\Omega$ satisfies the exterior
$(\delta,R)$-cone condition, where $\delta=\textmd{arctan}(1/K)\in[0,\pi/2)$.

In addition, if $f\in C(\mathbb{R}^{d-1})$ satisfies (5.2) for $f$ instead of
$f_{p}$, then the domain

$\big{\\{}(x^{\prime},x_{n})\in\mathbb{R}^{d-1}\times\mathbb{R}\,:\,x_{n}>f(x^{\prime})\big{\\}}$

satisfies the exterior $(\delta,\infty)$-cone condition, where
$\delta=\arctan(1/K)$.

For $\delta\in(0,\pi)$, let $E_{\delta}:=\\{\sigma\in\partial
B_{1}(0)\,:\,\sigma_{1}>-\cos\delta\\}$ (see Figure 5.2 below). By
$\Lambda_{\delta}$, we denote the first Dirichlet eigenvalue of the spherical
Laplacian on $E_{\delta}$. Alternatively, $\Lambda_{\delta}$ is expressed by

$\displaystyle\Lambda_{\delta}=\inf_{f\in
F_{\pi-\delta}}\frac{\int_{0}^{\pi-\delta}|f^{\prime}(t)|^{2}(\sin
t)^{d-2}\,\mathrm{d}t}{\int_{0}^{\pi-\delta}|f(t)|^{2}(\sin
t)^{d-2}\,\mathrm{d}t}\,,$ (5.4)

where $F_{\pi-\delta}$ is the set of all non-zero Lipschitz continuous
function $f:[0,\pi-\delta]\rightarrow\mathbb{R}$ such that $f(\pi-\delta)=0$
(see [18]). We also define

$\lambda_{\delta}:=-\frac{d-2}{2}+\sqrt{\Big{(}\frac{d-2}{2}\Big{)}^{2}+\Lambda_{\delta}}\,,$

and when $d=2$, we define $\lambda_{0}=\frac{1}{2}$.

$\delta$ Figure 5.2. $E_{\delta}$

The following quantitative information of $\Lambda_{\delta}$ and
$\lambda_{\delta}$ is provided in [9]:

######  Proposition 5.3.

Let $\delta\in(0,\pi)$.

  1. (1)

If $d=2$ then
$\lambda_{\delta}=\sqrt{\Lambda_{\delta}}=\frac{\pi}{2(\pi-\delta)}>\frac{1}{2}$.

  2. (2)

If $d=4$ then
$\lambda_{\delta}=-1+\sqrt{1+\Lambda_{\delta}}=\frac{\delta}{\pi-\delta}$.

  3. (3)

For $d\geq 3$,

$\Lambda_{\delta}\geq\left(\int_{0}^{\pi-\delta}(\sin
t)^{-d+2}\Big{(}\int_{0}^{t}(\sin
r)^{d-2}\,\mathrm{d}r\Big{)}\,\,\mathrm{d}t\right)^{-1}.$

Moreover, $\Lambda_{\pi/2}=d-1$, $\lim\limits_{\delta\searrow
0}\Lambda_{\delta}=0$, and
$\lim\limits_{\delta\nearrow\pi}\Lambda_{\delta}=+\infty$.

Note that when $d=3$,
$\Lambda_{\delta}\geq\frac{1}{2}|\log\,\sin\frac{\delta}{2}|^{-1}$.

######  Remark 5.4.

For each $\delta>0$, there is a function $F\in
C\big{(}\overline{E_{\delta}}\big{)}\cap C^{\infty}(E_{\delta})$ such that

$F>0\quad\text{and}\quad\Delta_{\mathbb{S}}F+\Lambda_{\delta}F=0\quad\text{in}\,\,\,\,E_{\delta}\quad;\quad
F=0\quad\text{on}\quad\overline{E_{\delta}}\setminus E_{\delta}$

(see, e.g., [18, Section 5]), where $\Delta_{\mathbb{S}}$ is the spherical
Laplacian. Due to

$\Delta=D_{rr}+\frac{d-1}{r}D_{r}+\frac{1}{r^{2}}\Delta_{\mathbb{S}^{d-1}}$

(the representation of the Laplacian operator on $\mathbb{R}^{d}$ by the
spherical coordinate), the function
$v_{\delta}(x):=|x|^{\lambda_{\delta}}F(x/|x|)$ is harmonic in

$U_{\delta}:=\big{\\{}y\in B_{1}(0)\,:\,y_{1}>-|y|\cos\delta\big{\\}}\,,$

and vanishes on $\partial U_{\delta}\cap B_{1}(0)$.

With the help of $\lambda_{\delta}$, we state the main results of this
subsection.

######  Theorem 5.5.

Let

$\displaystyle\delta\in[0,\pi/2)\quad\text{if}\,\,\,\,d=2\quad;\quad\delta\in(0,\pi/2)\quad\text{if}\,\,\,\,d\geq
3\,,$

and let $\Omega\subset\mathbb{R}^{d}$ satisfy the exterior $(\delta,R)$-cone
condition, where

$R\in(0,\infty]\quad\text{if}\,\,\,\,\text{$\Omega$\, is\,
bounded}\,\,,\,\,\,\,\text{and}\,\,\quad
R=\infty\quad\text{if}\,\,\,\,\text{$\Omega$\, is\, unbounded.}$

Then $\Omega$ satisfies $\mathbf{LHMD}(\lambda_{\delta})$, where
$M_{\lambda_{\delta}}$ in (4.7) depends only on $d$, $\delta$, and
additionally $\mathrm{diam}(\Omega)/R$ if $\Omega$ is bounded.

Before proving Theorem 5.5, we present a corollary that directly follows from
Theorems 5.5 and 4.13.

######  Theorem 5.6.

Let $p\in(1,\infty)$. Under the same assumption of Theorem 5.5, if
$\theta\in\mathbb{R}$ satisfies

$-2-(p-1)\lambda_{\delta}<\theta-d<-2+\lambda_{\delta}\,,$

then Statement 4.3 $(\Omega,p,\theta)$ holds. In addition, $N_{1}$ in (4.3)
depends only on $d$, $p$, $\theta$, $\gamma$, $\delta$, and additionally
$\mathrm{diam}(\Omega)/R$ if $\Omega$ is bounded.

To prove Theorem 5.5, we use the boundary Harnack principle on Lipschitz
domains.

######  Lemma 5.7 (see Theorem 1 of [54]).

Let $D$ be a bounded Lipschitz domain, $A$ be a relatively open subset of
$\partial D$, and $U$ be a subdomain of $D$ with $\partial U\cap\partial
D\subset A$.Then there exists $N=N(D,A,U)>0$ such that if $u,v$ are positive
harmonic funtion on $D$, and vanish on $E$, then

$\frac{u(x)}{v(x)}\leq N\frac{u(x_{0})}{v(x_{0})}\quad\text{for any}\quad
x_{0},\,x\in U\,.$

###### Proof of Theorem 5.5.

By Lemma 4.9, it is sufficient to prove that there exists a constant $M>0$
such that

$\displaystyle w(x,p,r)\leq
M\left(\frac{|x-p|}{r}\right)^{\lambda_{\delta}}\quad\text{for
all}\,\,\,x\in\Omega\cap B(p,r)$

whenever $p\in\partial\Omega$ and $r\in(0,R)$. For any $p\in\partial\Omega$,
there exists a unit vector $e_{p}\in\mathbb{R}^{d}$ such that

$C_{p}:=\\{y\in B_{R}(p)\,:\,(y-p)\cdot
e_{p}\geq|y-p|\cos\delta\\}\subset\Omega^{c}\,.$

Since

$\Omega\cap B_{r}(p)\subset B_{r}(p)\setminus
C_{p}\quad\text{and}\quad\Omega\cap\partial B_{r}(p)\subset\partial
B_{r}(p)\setminus C_{p}\,,$

we have

$\displaystyle w(x,p,r)\leq w\big{(}x,\,B_{r}(p)\setminus C_{p}\,,\,\partial
B_{r}(p)\setminus C_{p}\,\big{)},$ (5.5)

by directly applying the definition of $w(\cdot,p,r)$ (see (4.6)). Consider a
rotation map $T$ such that $T(e_{p})=(-1,0,\ldots,0)$, and put
$T_{0}(x)=r^{-1}T(x-p)$. Then

$\displaystyle w\big{(}x,\,B_{r}(p)\setminus C_{p}\,,\,\partial
B_{r}(p)\setminus
C_{p}\,\big{)}=w\big{(}T_{0}(x),U_{\delta},E_{\delta}\big{)},$ (5.6)

where

$U_{\delta}=\\{y\in
B_{1}(0)\,:\,y_{1}>-|y|\cos\delta\\}\,\,\,\,\text{and}\,\,\,\,E_{\delta}=\\{y\in\partial
B_{1}(0)\,:\,y_{1}>-|y|\cos\delta\\}\,.$

Due to (5.5) and (5.6), it is sufficient to show that there exists a constant
$M>0$ depending only on $d$ and $\delta$ such that

$\displaystyle w(x,U_{\delta},E_{\delta})\leq
M|x|^{\lambda_{\delta}}\quad\text{for all}\quad x\in U_{\delta}\,,$ (5.7)

Case 1: $\delta>0$. Put $v(x)=|x|^{\lambda_{\delta}}F_{0}(x/|x|)$ where
$F_{0}$ is the first Dirichlet eigenfunction of spherical laplacian on
$E_{\delta}\subset\partial B_{1}(0)$, with $\sup_{E_{\delta}}F_{0}=1$ (see
Remark 5.4). Note that $U_{\delta}$ is a bounded Lipschitz domain, and
$w(\,\cdot\,,U_{\delta},E_{\delta})$ and $v$ are positive harmonic functions
on $U_{\delta}$, and vanish on $\partial U_{\delta}\cap B_{1}$. By applying
Lemma 5.7 for $D=U_{\delta}$, $A=(\partial U_{\delta})\cap B_{1}(0)$, and
$U=U_{\delta}\cap B_{1/2}(0)$, we obtain that there exists a constant
$N_{0}=N_{0}(d,\delta)>0$ such that

$w(x,U_{\delta},E_{\delta})\leq N_{0}v(x)\leq
N_{0}|x|^{\lambda_{\delta}}\quad\text{for}\quad x\in U_{\delta}\cap
B_{1/2}(0).$

Therefore (5.7) is obtained, where
$M_{0}=\max\big{(}N_{0},2^{\lambda_{0}}\big{)}$.

Case 2: $\delta=0$ and $d=2$. We consider $\mathbb{R}^{2}$ as $\mathbb{C}$.
Note

$U_{0}=\\{r\mathrm{e}^{i\theta}\,:\,r\in(0,1),\,\theta\in(-\pi,\pi)\\}\,\,,\,\,\,\,E_{0}=\\{\mathrm{e}^{i\theta}\,:\,\theta\in(-\pi,\pi)\\}\,.$

Observe that a function $s$ is a classical superharmonic function on $U_{0}$
if and only if $s(z^{2})$ is a classical superharmonic function on
$B_{1}(0)\cap\mathbb{R}_{+}^{2}$ (use Lemma 2.4). It is implied by the
definition of PWB solutions (see (4.6)) that

$w(z^{2},U_{0},E_{0})=w\big{(}z,B_{1}(0)\cap\mathbb{R}_{+}^{2},\partial
B_{1}(0)\cap\mathbb{R}_{+}^{2}\big{)}\,.$

Since the map $z=(z_{1},z_{2})\mapsto z_{1}$ is harmonic on
$B_{1}(0)\cap\mathbb{R}^{2}_{+}$, by Lemma 5.7 with
$D=B_{1}(0)\cap\mathbb{R}^{2}_{+}$, we obtain that

$\displaystyle w\big{(}z,B_{1}(0)\cap\mathbb{R}_{+}^{2},\big{(}\partial
B_{1}(0)\big{)}\cap\mathbb{R}_{+}^{2}\big{)}\leq N|z|\quad\text{for}\,\,\,z\in
B_{1/2}(0)\cap\mathbb{R}_{+}^{2}\,,$ (5.8)

where $N$ depends on nothing. Therefore the proof is completed. ∎

###  5.2. Convex domains

Recall that a set $E\subset\mathbb{R}^{d}$ is said to be convex if
$(1-t)x+ty\in E$ for any $x,\,y\in E$ and $t\in[0,1]$.

######  Lemma 5.8.

For an open set $\Omega\subset\mathbb{R}^{d}$, $\Omega$ is convex if and only
if for any $p\in\partial\Omega$, there exists a unit vector
$e_{p}\in\mathbb{R}^{d}$ such that

$\displaystyle\Omega\subset\\{x\,:\,(x-p)\cdot e_{p}<0\\}=:U_{p}\,.$ (5.9)

###### Proof.

Let $\Omega$ be a convex domain, and fix $p\in\partial\Omega$. Since the set
$\\{p\\}$ is convex and disjoint from $\Omega$, the hyperplane separation
theorem (see, e.g., [48, Theorem 3.4.(a)]) implies that there exists a unit
vector $e_{p}\in\mathbb{R}^{d}$ such that (5.9) holds. Conversely, suppose
that for any $p\in\partial\Omega$, there exists a unit vector $e_{p}$
satisfying (5.9). Then $E:=\bigcap_{p\in\partial\Omega}U_{p}$ is convex,
$\Omega\subset E$, and $E\cap\partial\Omega=\emptyset$. These imply
$E=\Omega$; if not, $E\cap\partial\Omega\neq\emptyset$ which is a
contradiction. Therefore our claim is proven. ∎

######  Theorem 5.9.

Let $\Omega\subset\mathbb{R}^{d}$ be a convex domain. Then $\Omega$ satisfies
$\mathbf{LHMD}(1)$ where $M_{1}$ in (4.7) depends only on $d$.

###### Proof.

The argument to obtain (5.8) also implies that for any $d\in\mathbb{N}$,

$w\big{(}x,B_{1}(0)\cap\mathbb{R}_{+}^{d},(\partial
B_{1}(0))\cap\mathbb{R}_{+}^{d}\big{)}\leq N(d)|x|\quad\text{for
all}\,\,\,x\in B_{1}(0)\cap\mathbb{R}^{d}_{+}\,.$

By translation, dilation, and rotation, we obtain that for a convex domain
$\Omega$ and $p\in\partial\Omega$,

$w(x,p,r)\leq w(x,B_{r}(p)\cap U_{p},\big{(}\partial B_{r}(p)\big{)}\cap
U_{p})\leq N(d)\frac{|x-p|}{r}$

for all $x\in B_{r}(p)\cap\Omega$, where $U_{p}$ is the set on the right-hand
side of (5.9). Therefore, the proof is completed. ∎

This result also implies that the Hardy inequality (1.2) holds on $\Omega$,
where $\mathrm{C}_{0}(\Omega)$ depends only on $d$ (see Lemma 4.10); it is
worth noting that Marcus, Mizel, and Pinchover [41, Theorem 11] provided that
for a convex domain $\Omega$, (1.2) holds where $\mathrm{C}_{0}(\Omega)=4$,
and $\mathrm{C}_{0}(\Omega)$ cannot be chosen less than 4.

By combining Theorems 4.13 and 5.9, we obtain the following result:

######  Theorem 5.10.

Let $\Omega\subset\mathbb{R}^{d}$ be a convex domain. For any $p\in(1,\infty)$
and $\theta\in\mathbb{R}$ with

$-p-1<\theta-d<-1\,,$

Statement 4.3 $(\Omega,p,\theta)$ holds. In addition, $N_{1}$ in (4.3) depends
only on $d$, $p$, $\gamma$, $\theta$. In particular, $\Omega$ is not
necessarily bounded, and $N_{1}$ is independent of $\Omega$.

###  5.3. Exterior Reifenberg condition

The notion of the vanishing Reifenberg condition was introduced by Reifenberg
[47] and has been extensively studied in the literature (see, e.g., [11, 12,
24, 50] and Subsubsection 1.2.3 of this paper). The following definition can
be found in [11, 24]: For $\delta\in(0,1)$ and $R>0$, a domain
$\Omega\subset\mathbb{R}^{d}$ is said to satisfy the $(\delta,R)$-Reifenberg
condition, if for every $p\in\partial\Omega$ and $r\in(0,R]$, there exists a
unit vector $e_{p,r}\in\mathbb{R}^{d}$ such that

$\displaystyle\begin{split}&\Omega\cap B_{r}(p)\subset\\{x\in
B_{r}(p)\,:\,(x-p)\cdot e_{p,r}<\delta r\\}\quad\text{and}\\\ &\Omega\cap
B_{r}(p)\supset\\{x\in B_{r}(p)\,:\,(x-p)\cdot e_{p,r}>-\delta
r\\}\,.\end{split}$ (5.10)

In addition, $\Omega$ is said to satisfy the vanishing Reifenberg condition if
for any $\delta\in(0,1)$, there exists $R_{\delta}>0$ such that $\Omega$
satisfies the $(\delta,R_{\delta})$-Reifenberg condition. Note that the
vanishing Reifenberg condition is weaker than the $C^{1}$-boundary condition
(see Example 5.13.(2) and (3)).

In this subsection, we present the totally vanishing exterior Reifenberg
condition, which is a generalization of the vanishing Reifenberg condition. We
also obtain a result for the Poisson equation on domains satisfying the
totally vanishing exterior Reifenberg condition (see Theorem 5.18).

######  Definition 5.11 (Exterior Reifenberg condition).

  1. (1)

By $\mathbf{ER}_{\Omega}$ we denote the set of all
$(\delta,R)\in[0,1]\times\mathbb{R}_{+}$ satisfying the following: For each
$p\in\partial\Omega$, and each connected component $\Omega_{p,R}^{(i)}$ of
$\Omega\cap B(p,R)$, there exists a unit vector
$e_{p,R}^{(i)}\in\mathbb{R}^{d}$ such that

$\displaystyle\Omega_{p,R}^{(i)}\subset\\{x\in B_{R}(p)\,:\,(x-p)\cdot
e_{p,R}^{(i)}<\delta R\\}\,.$ (5.11)

By $\delta(R):=\delta_{\Omega}(R)$ we denote the infimum of $\delta$ such that
$(\delta,R)\in\mathbf{ER}_{\Omega}$.

  2. (2)

For $\delta\in[0,1]$, we say that $\Omega$ satisfies the totally
$\delta$-exterior Reifenberg condition (abbreviate to
‘$\langle\mathrm{TER}\rangle_{\delta}$’), if there exist constants
$0<R_{0}\leq R_{\infty}<\infty$ such that

$\displaystyle\delta_{\Omega}(R)\leq\delta\quad\text{whenever}\quad R\leq
R_{0}\,\,\,\text{or}\,\,\,R\geq R_{\infty}\,.$ (5.12)

  3. (3)

We say that $\Omega$ satisfies the totally vanishing exterior Reifenberg
condition (abbreviate to ‘$\langle\mathrm{TVER}\rangle$’), if $\Omega$
satisfies $\langle\mathrm{TER}\rangle_{\delta}$ for all $\delta\in(0,1]$. In
other word,

$\lim_{R\rightarrow
0}\delta_{\Omega}(R)=\lim_{R\rightarrow\infty}\delta_{\Omega}(R)=0\,.$

The main theorem in this subsection concerns domains satisfying
$\langle\mathrm{TER}\rangle_{\delta}$ for sufficiently small $\delta>0$.
However, our main interest is the condition $\langle\mathrm{TVER}\rangle$. For
a comparison between the Refenberg condition and
$\langle\mathrm{TVER}\rangle$, see Figure 5.3 and Example 5.13 below.

VanishingReifenberg conditionTotally vanishing exteriorReifenberg
condition(Definition 1.6)Totally vanishing exteriorReifenberg
condition(Definition 5.11) Figure 5.3. Totally vanishing exterior Reifenberg
condition

######  Lemma 5.12.

For any $R>0$, $\big{(}\delta(R),R\big{)}\in\mathbf{ER}_{\Omega}$.

###### Proof.

Take a sequence $\\{\delta_{n}\\}_{n\in\mathbb{N}}$ such that
$(\delta_{n},R)\in\mathbf{ER}_{\Omega}$ and $\delta_{n}\rightarrow\delta(R)$
as $n\rightarrow\infty$. Since $(\delta_{n},R)\in\mathbf{ER}_{\Omega}$, for
any $p\in\partial\Omega$ and any connected component of $\Omega\cap B(p,R)$,
denoted by $\Omega_{p,R}$, there exists a unit vector $e_{n}$ such that

$\displaystyle\Omega_{p,R}\subset\\{x\in B_{R}(p)\,:\,(x-p)\cdot
e_{n}<\delta_{n}R\\}\,.$ (5.13)

Since $\\{e_{n}\\}_{n\in\mathbb{N}}\subset\partial B(0,1)$, there exists a
subsequence $\\{e_{n_{k}}\\}_{k\in\mathbb{N}}$ such that
$e_{p}:=\lim_{k\rightarrow\infty}e_{n_{k}}$ exists in $\partial B(0,1)$. It is
impliled by (5.13) that

$\displaystyle\Omega_{p,R}\subset\\{x\in B_{R}(p)\,:\,(x-p)\cdot
e_{p}<\delta(R)R\\}\,.$

Therefore $\big{(}\delta(R),R\big{)}\in\mathbf{ER}_{\Omega}$. ∎

######  Example 5.13.

  1. (1)

If $\Omega$ satisfies the $(\delta,R_{1})$-Reifenberg condition, then
$\delta(R)\leq\delta$ for all $R\leq R_{1}$, indeed the first line of (5.10)
implies (5.11) with $e_{p,r}^{(i)}=e_{p,r}$. Moreover, if $\Omega$ is bounded,
then Proposition 5.14 implies $\delta(R)\leq\mathrm{diam}(\Omega)/R$.
Therefore, if $\Omega$ is a bounded domain satisfying the vanishing Reifenberg
condition, then $\Omega$ also satisfies $\langle\mathrm{TVER}\rangle$.

  2. (2)

By $\lambda_{*}(\mathbb{R}^{d-1})$, we denote the little Zygmund class, which
is the set of all $f\in C(\mathbb{R}^{d-1})$ such that

$\lim_{h\rightarrow
0}\sup_{x\in\mathbb{R}^{d-1}}\frac{|f(x+h)-2f(x)+f(x-h)|}{|h|}=0\,.$

For $f\in\lambda_{*}(\mathbb{R}^{d-1})$, put

$\Omega=\\{(x^{\prime},x_{d})\in\mathbb{R}^{d-1}\times\mathbb{R}\,:\,x_{d}>f(x^{\prime})\\}\,.$

Then, as mentioned in [12, Example 1.4.3] (see also [16, Theorem 6.3]),
$\Omega$ satisfies the vanishing Reifenberg condition, which implies
$\lim_{R\rightarrow 0}\delta_{\Omega}(R)=0$. Moreover, since
$A:=\|f\|_{C(\mathbb{R}^{d-1})}<\infty$, Proposition 5.14 implies that
$\delta(R)\leq\frac{2\|f\|_{C(\mathbb{R}^{d-1})}}{R}$. Therefore $\Omega$
satisfies $\langle\mathrm{TVER}\rangle$.

  3. (3)

Suppose that $\Omega$ is bounded, and for any $p\in\partial\Omega$ there
exists $R>0$ and $f\in\lambda_{\ast}(\mathbb{R}^{d-1})$ such that

$\Omega\cap
B(p,R)=\big{\\{}y=(y^{\prime},y_{n})\in\mathbb{R}^{d-1}\times\mathbb{R}\,:\,|y|<R\,\,\,\text{and}\,\,\,y_{n}>f(y^{\prime})\big{\\}}\,,$

where $(y^{\prime},y_{n})=(y_{1},\ldots,y_{n})$ is an orthonormal coordinate
system centered at $p$. Then $\Omega$ satisfies the vanishing Reifenberg
condition, and therefore $\Omega$ satisfies $\langle\mathrm{TVER}\rangle$.

  4. (4)

Let $\Omega$ satisfy the exterior $R_{0}$-ball condition, i.e., there exists
$R_{0}>0$ such that for any $p\in\partial\Omega$, there exists
$q\in\mathbb{R}^{d}$ satisfying $|p-q|=R_{0}$ and
$B(q,R_{0})\subset\Omega^{c}$. Then $\delta(R)\leq\frac{R}{2R_{0}}$, and
therefore $\lim_{R\rightarrow 0}\delta(R)=0$.

  5. (5)

If a domain $\Omega$ is an intersection of domains satisfying the totally
vanishing Reifenberg condition, then $\Omega$ satisfies
$\langle\mathrm{TVER}\rangle$.

All of the following examples are valid even if $\langle\mathrm{TVER}\rangle$
is defined by (1.9) instead of (5.11).

A sufficient condition for $\lim_{R\rightarrow\infty}\delta_{\Omega}(R)=0$ is
that $\delta_{\Omega}(R)\lesssim 1/R$. We provide an equivalent condition for
$\Omega$ to satisfy $\delta_{\Omega}(R)\lesssim 1/R$.

######  Proposition 5.14.

$\displaystyle\sup_{R>0}R\,\delta_{\Omega}(R)=\sup_{p\in\partial\Omega}d\big{(}p,\partial(\Omega_{\mathrm{c.h.}})\big{)}\,,$

where $\Omega_{\mathrm{c.h.}}$ is the convex hull of $\Omega$, i.e.,

$\Omega_{\mathrm{c.h.}}:=\big{\\{}(1-t)x+ty\,:\,x,\,y\in\Omega\,\,,\,\,t\in[0,1]\big{\\}}\,.$

######  Remark 5.15.

It follows from the definition of $\delta_{\Omega}(R)$ that
$R\delta_{\Omega}(R)$ increases as $R\rightarrow\infty$. Therefore if
$\delta_{\Omega}(r_{0})>0$ for some $r_{0}>0$, then $\delta_{\Omega}(R)\gtrsim
1/R$ as $R\rightarrow\infty$. As a result, due to Proposition (5.14), an
equivalent condition for $\delta(R)$ to have minimal nontrivial decay (i.e.,
$\delta_{\Omega}(R)\simeq 1/R$) is that
$\sup_{p\in\partial\Omega}d\big{(}p,\partial(\Omega_{\mathrm{c.h.}})\big{)}<\infty$.

###### Proof of Proposition 5.14.

We only need to prove that for any $N_{0}>0$,

$\displaystyle\sup_{R>0}R\,\delta_{\Omega}(R)\leq
N_{0}\quad\Longleftrightarrow\quad\sup_{p\in\partial\Omega}d\big{(}p,\partial(\Omega_{\mathrm{c.h.}})\big{)}\leq
N_{0}\,.$ (5.14)

Step 1. We first claim that LHS of (5.14) holds if and only if for any
$p\in\partial\Omega$, there exists a unit vector $e_{p}$ such that

$\displaystyle\Omega\subset\\{x\in\mathbb{R}^{d}\,:\,(x-p)\cdot
e_{p}<N_{0}\\}\,.$ (5.15)

The ‘if’ part is obvious. Therefore, we only need to prove the ‘only if’ part.
Assume that LHS of (5.14) holds. Fix $p\in\partial\Omega$, and take
$\\{\widetilde{\Omega}_{n}\\}_{n\in\mathbb{N}}$ such that
$\widetilde{\Omega}_{n}$ is a connceted component of $\Omega\cap B_{n}(p)$,
and
$\widetilde{\Omega}_{1}\subset\widetilde{\Omega}_{2}\subset\widetilde{\Omega}_{3}\subset\cdots$.
Since $\Omega$ is a domain, $\Omega$ is path connected, which implies

$\displaystyle\bigcup_{n\in\mathbb{N}}\widetilde{\Omega}_{n}=\Omega\,.$ (5.16)

Since $R\delta(R)\leq N_{0}$, for each $n\in\mathbb{N}$, there exists
$e_{n}\in\partial B_{1}(0)$ such that

$\displaystyle\widetilde{\Omega}_{n}\subset\\{x\in\mathbb{R}^{d}\,:\,(x-p)\cdot
e_{n}<N_{0}\\}$ (5.17)

(see Lemma 5.12). Since $\partial B_{1}(0)$ is compact, there exists a
subsequence $\\{e_{n_{k}}\\}$ which converges to a certain point,
$e_{p}\in\partial B_{1}(0)$. Due to (5.16) and (5.17), we obtain that (5.15)
holds for this $e_{p}$.

Step 2. Due to (5.14), we only need to prove the following: For
$p\in\partial\Omega$, (5.15) holds for some $e_{p}\in\partial B_{1}(0)$ if and
only if $d\big{(}p,\partial(\Omega_{\mathrm{c.h.}})\big{)}\leq N_{0}$.

To prove the ‘only if’part, suppose (5.15) and observe that

$p\in\partial\Omega\subset\overline{\Omega_{\mathrm{c.h.}}}\subset\\{x\in\mathbb{R}^{d}\,:\,(x-p)\cdot
e_{p}\leq N_{0}\\}\,.$

Put $\alpha_{0}:=\sup\\{\alpha\geq 0\,:\,p+\alpha
e_{p}\in\overline{\Omega_{\mathrm{c.h.}}}\\}$. Then
$p+\alpha_{0}e_{p}\in\partial(\Omega_{\mathrm{c.h.}})$, and therefore
$d\big{(}p,\partial(\Omega_{\mathrm{c.h.}})\big{)}\leq\alpha_{0}\leq N_{0}$.

To prove the ‘if’ part, suppose that there exists
$q\in\partial(\Omega_{\mathrm{c.h.}})$ such that

$|p-q|=d\big{(}p,\partial(\Omega_{\mathrm{c.h.}})\big{)}\leq N_{0}\,.$

Due to Lemma 5.8 and that $\Omega_{\mathrm{c.h.}}$ is a convex domain, there
is a unit vector $\widetilde{e}_{q}$ such that

$\Omega_{\mathrm{c.h.}}\subset\\{x\in\mathbb{R}^{d}\,:\,(x-q)\cdot\widetilde{e}_{q}<0\\}\,.$

This implies that for any $x\in\Omega\subset\Omega_{\mathrm{c.h.}}$,

$(x-p)\cdot\widetilde{e}_{q}<(q-p)\cdot\widetilde{e}_{q}\leq|p-q|\leq
N_{0}\,.$

Therefore (5.15) holds for $e_{p}:=\widetilde{e}_{q}$. ∎

######  Remark 5.16.

From Step 1 in the proof of Proposition 5.14, one can observe that this
proposition remains valid even if the definition of $\delta_{\Omega}(R)$ is
replaced by the infimum of $\delta>0$ such that, for any $p\in\partial\Omega$,
there exists a unit vector $e_{p,R}$ satisfying (1.9) for $r=R$.

Now, we state the main result of this subsection. We temporarily assume
Theorem 5.17 (they are proved at the end of this subsection) and prove Theorem
5.18.

######  Theorem 5.17.

For any $\epsilon\in(0,1)$, there exists $\delta>0$ depending only on
$d,\,\epsilon$ such that if $\Omega$ satisfies
$\langle\mathrm{TER}\rangle_{\delta}$, then $\Omega$ satisfies
$\mathbf{LHMD}(1-\epsilon)$ where $M_{1-\epsilon}$ in (4.7) depends only on
$d$, $\epsilon$, $\delta$, and $R_{0}/R_{\infty}$, where $R_{0}$ and
$R_{\infty}$ are constants in (5.12).

######  Theorem 5.18.

For any $p\in(1,\infty)$ and $\theta\in\mathbb{R}$ with $-p-1<\theta-d<-1$,
there exists $\delta>0$ depending only on $d$, $p$, $\epsilon$ such that if
$\Omega$ satisfies $\langle\mathrm{TER}\rangle_{\delta}$, then Statement 4.3
$(\Omega,p,\theta)$ holds. In addition, $N_{1}$ in (4.3) depends only on $d$,
$p$, $\gamma$, $\theta$, and $R_{0}/R_{\infty}$, where $R_{0}$ and
$R_{\infty}$ are constants in (5.12). In particular, if $\Omega$ satisfying
$\langle\mathrm{TVER}\rangle$, then Statement 4.3 $(\Omega,p,\theta)$ holds
for all $p\in(1,\infty)$ and $\theta\in\mathbb{R}$ with $-p-1<\theta-d<-1$.

###### Proof.

Take $\epsilon\in(0,1)$ such that

$\displaystyle-p-1+(p-1)\epsilon<\theta-d<-1-\epsilon\,,$ (5.18)

and put $\delta$ as the constant in Theorem 5.17 for this $\epsilon$. Consider
a domain $\Omega$ satisfying $\langle\mathrm{TER}\rangle_{\delta}$. By Theorem
5.17, this $\Omega$ satisfies $\mathbf{LHMD}(1-\epsilon)$. Therefore Theorem
4.13 and (5.18) imply that Statement 4.3 $(\Omega,p,\theta)$ holds with
$N_{1}=N(d,p,\gamma,\theta,R_{0}/R_{\infty})$. ∎

######  Remark 5.19.

Kenig and Toro [25, Lemma 2.1] established that if a bounded domain satisfies
the vanishing Reifenberg condition, then this domain also satisfies
$\mathbf{LHMD}(1-\epsilon)$ for all $\epsilon\in(0,1)$.

To prove Theorem 5.17, we need the following lemma:

######  Lemma 5.20.

If $(\delta,R)\in\mathbf{ER}_{\Omega}$, then there exists a continuous
function $w_{p,R}:\Omega\rightarrow(0,1]$ satisfying the following:

  1. (1)

$w_{p,R}$ is a classical superharmonic function on $\Omega$ .

  2. (2)

$w_{p,R}=1\text{ on }\\{x\in\Omega\,:\,|x-p|>(1-\delta)R\\}$ .

  3. (3)

$w_{p,R}\leq M\delta$ on $\Omega\cap B(p,\delta R)$ .

Here, $M$ is a constant depending only on $d$. In particular, $M$ is
independent of $\delta$.

###### Proof of Lemma 5.20.

If $\delta>1/8$, then by putting $w_{p,R}\equiv 1$ and $M=8$, this lemma is
proved. Therefore we only need to consider the case $\delta\leq 1/8$. For a
fixed $p\in\partial\Omega$, let $\big{\\{}\Omega^{(i)}_{p,R}\big{\\}}$ be the
set of all connected components of $\Omega\cap B(p,R)$. For each $i$, take a
unit vector $e_{p,R}^{(i)}$ satisfying (5.11). Put

$\displaystyle q=p+R(\delta+1/4)e^{(i)}_{p,R}\,,$ (5.19)

so that

$\displaystyle|p-q|=R(\delta+1/4)\quad\text{and}\quad\Omega_{p,R}^{(i)}\cap
B(q,R/4)\neq\emptyset$ (5.20)

(see Figure 5.4 below).

$\Omega_{p,R}^{(i)}$$p$$q$$e_{p,R}^{(i)}$ Figure 5.4. $q$ and $B(q,R/4)$ in
(5.19), (5.20)

Put $W^{(i)}(x)=F_{0}\big{(}4R^{-1}|x-q|\big{)}/F_{0}(2)$, where

$\displaystyle\begin{split}F_{0}(t)=\log(t)\quad\text{if}\quad d=2\quad;\quad
F_{0}(t)=1-t^{2-d}\quad\text{if}\quad d\geq 3\,,\end{split}$ (5.21)

so that $\Delta W^{(i)}=0$ on $\mathbb{R}^{d}\setminus\\{q\\}$. Observe that

$\displaystyle 0\leq W^{(i)}(x)\,$ $\displaystyle\leq
M_{0}\big{(}4R^{-1}|x-q|-1\big{)}\quad$ $\displaystyle\text{if}\,\,\,|x-q|\geq
R/4\,;$ $\displaystyle W^{(i)}(x)\,$ $\displaystyle\geq 1\quad$
$\displaystyle\text{if}\,\,\,|x-q|\geq R/2\,,$

where $M_{0}$ is a constant depends only on $d$. Due to (5.20) and that
$\delta<\frac{1}{8}$, for $x\in\Omega_{p,R}^{(i)}$,

$\displaystyle\text{if\quad$|x-p|\leq\delta R$}\,\,,\,\,\,$
$\displaystyle\text{then\quad$\frac{R}{4}\leq|x-q|\leq\frac{R}{4}+2\delta
R$}\,;$ $\displaystyle\text{if\quad$|x-p|\geq(1-\delta)R$}\,\,,\,\,\,$
$\displaystyle\text{then\quad$|x-q|\geq\frac{(3-8\delta)R}{4}\geq\frac{R}{2}$}\,.$

Therefore we obtain that

$\displaystyle 0\leq\,$ $\displaystyle W^{(i)}(x)\leq 8M_{0}\delta\quad$
$\displaystyle\text{if}\,\,\,\,|x-p|\leq\delta R\,;$ $\displaystyle
W^{(i)}(x)\geq 1\quad$ $\displaystyle\text{if}\,\,\,\,|x-p|\geq(1-\delta)R\,.$

Put

$\displaystyle w_{p,R}(x)=\begin{cases}W^{(i)}(x)\wedge
1&\text{if}\,\,\,x\in\Omega^{(i)}_{p,R}\\\
1&\text{if}\,\,\,x\in\Omega\setminus B(p,R)\,.\end{cases}$

Then $w_{p,R}$ is continuous on $\Omega$, and satisfies (2) and (3) of this
lemma. (1) of this lemma follows from (5.21) and Lemma 4.5. ∎

###### Proof of Theorem 5.17.

Let $M>0$ be the constant in Lemma 5.20. For given $\epsilon\in(0,1)$, take
small enough $\delta\in(0,1)$ such that $M\delta<\delta^{1-\epsilon}$. We
assume that $\Omega$ satisfies (5.12) for this $\delta$. By using dilation and
Lemma 5.12, without loss of generality, we assume that
$(\delta,R)\in\mathbf{ER}_{\Omega}$ whenever
$R\leq\widetilde{R}_{0}:=R_{0}/R_{\infty}\,\,(\leq 1)$ or $R\geq 1$.

Note that for $(\delta,R)\in\mathbf{ER}_{\Omega}$, due to Lemma 5.20 and the
definition of PWB solutions (4.6), $w(\cdot,p,R)\leq
M\delta\leq\delta^{1-\epsilon}$ on $\Omega\cap\partial B_{\delta R}(p)$.
Therefore, by Lemma 4.6.(3),

$\displaystyle w(\cdot,p,R)\leq\delta^{1-\epsilon}w(\cdot,p,\delta
R)\quad\text{on}\quad\Omega\cap B_{\delta R}(p)\,.$ (5.22)

The proof is completed by establishing (4.7) for $\alpha:=1-\epsilon$ and
$M_{1-\epsilon}$ depending only on $\delta$ and $\widetilde{R}_{0}$. We prove
(4.7) by dividing $r$ and $|x-p|$ into the following five cases:

Case 1: $r\leq\widetilde{R}_{0}$. Take $n_{0}\in\mathbb{N}_{0}$ such that
$\delta^{n_{0}+1}r\leq|x-p|<\delta^{n_{0}}r$. Since
$(\delta,\delta^{k}r)\in\mathbf{ER}_{\Omega}$ for all $k\geq 0$, it follows
from (5.22) that

$w(x,p,r)\leq\delta^{n_{0}(1-\epsilon)}w(x,p,\delta^{n_{0}}r)\leq\delta^{n_{0}(1-\epsilon)}\leq\bigg{(}\frac{|x-p|}{\delta\,r}\bigg{)}^{1-\epsilon}\,.$

Case 2: $|x-p|<\widetilde{R}_{0}<r\leq 1$. By Lemmas 4.6.(1) and (3) and the
result in Case 1, we have

$w(x,p,r)\leq
w(x,p,\widetilde{R}_{0})\lesssim_{\delta,\widetilde{R}_{0}}|x-p|^{1-\epsilon}\leq\Big{(}\frac{|x-p|}{r}\Big{)}^{1-\epsilon}$

Case 3: $\widetilde{R}_{0}\leq|x-p|<r\leq 1$. It directly follows that

$w(x,p,r)\leq
1\leq\bigg{(}\frac{|x-p|}{\widetilde{R}_{0}\,r}\bigg{)}^{1-\epsilon}\,.$

Case 4: $|x-p|<1<r$. Take $n_{0}\in\mathbb{N}_{0}$ such that
$\delta^{n_{0}+1}r\leq 1<\delta^{n_{0}}r$. Then
$(\delta,\delta^{k}r)\in\mathbf{ER}_{\Omega}$ for all
$k=0,\,1,\,\ldots,\,n_{0}-1$. Therefore we have

$\displaystyle
w(x,p,r)\leq\delta^{n_{0}(1-\epsilon)}w(x,p,\delta^{n_{0}}r)\leq\delta^{n_{0}(1-\epsilon)}w(x,p,1)\lesssim_{\delta,\widetilde{R}_{0}}\delta^{n_{0}(1-\epsilon)}|x-p|^{1-\epsilon}\,,$

where the first inequality follows from (5.22), the second follows from Lemmas
4.6.(1) and (3), and the last follows from the result in Cases 2 and 3. Since
$\delta^{n_{0}}\leq 1/(\delta r)$, we have
$w(x,p,r)\lesssim_{\delta,\widetilde{R}_{0}}\big{(}|x-p|/r\big{)}^{1-\epsilon}$.

Case 5: $1\leq|x-p|<r$. Take $n_{0}\in\mathbb{N}_{0}$ such that
$\delta^{n_{0}+1}r\leq|x-p|<\delta^{n_{0}}r$. Since $1<|x-p|$, we have
$(\delta,\delta^{k}r)\in\mathbf{ER}_{\Omega}$ for all
$k=0,\,1,\,\ldots,\,n_{0}-1$. This implies that

$w(x,p,r)\leq\delta^{n_{0}(1-\epsilon)}w(x,p,\delta^{n_{0}}r)\leq\delta^{n_{0}(1-\epsilon)}\leq\bigg{(}\frac{|x-p|}{\delta\,r}\bigg{)}^{1-\epsilon}\,.$

∎

##  Appendix A Auxiliary results

######  Lemma A.1.

Let $p\in(1,\infty)$ and $u\in C(\mathbb{R}^{d})$ satisfy (2.3).

  1. (1)

$|u|^{p/2-1}u\in W_{2}^{1}(\mathbb{R}^{d})$ and
$D_{i}(|u|^{p/2-1}u)=\frac{p}{2}|u|^{p/2-1}(D_{i}u)1_{\\{u\neq 0\\}}$.

  2. (2)

$|u|^{p}\in W_{1}^{2}(\mathbb{R}^{d})$ and

$\displaystyle\begin{split}D_{i}\big{(}|u|^{p}\big{)}\,&=p|u|^{p-2}uD_{i}u1_{\\{u\neq
0\\}}\,\,;\\\
D_{ij}\big{(}|u|^{p}\big{)}\,&=\big{(}p|u|^{p-2}uD_{ij}u+p(p-1)|u|^{p-2}D_{i}uD_{j}u\big{)}\,1_{\\{u\neq
0\\}}\,.\end{split}$ (A.1)

###### Proof.

This proof is a variant of [34, Lemma 2.17]. Take a sequence of nonnegative
functions $\\{g_{n}\\}_{n\in\mathbb{N}}\subset C(\mathbb{R})$ such that
$g_{n}=0$ on a neighborhood of $0$ for each $n\in\mathbb{N}$, and
$g_{n}(s)\nearrow|s|^{p/2-1}1_{s\neq 0}$ for all $s\in\mathbb{R}$. Put

$\displaystyle F_{n}(t):=\,\,\int^{t}_{0}g_{n}(s)\,\mathrm{d}s\quad,\quad
G_{n}(t):=\int_{0}^{t}\big{(}g_{n}(s)\big{)}^{2}\,\mathrm{d}s\,.$

Recall the assumption (2.3), and denote $A=\sup|u|$. Since $0\leq
g_{n}(s)\leq|s|^{p/2-1}$, the Lebesgue dominated convergence theorem implies
that $F_{n}(t)\rightarrow\,\frac{2}{p}\,|t|^{p/2-1}t$ and
$G_{n}(t)\rightarrow\,\frac{1}{p-1}|t|^{p-2}t$ uniformly for $t\in[-A,A]$.
Furthermore, there absolute values increase as $n\rightarrow\infty$. Since
$F_{n}\big{(}u(\,\cdot\,)\big{)}$ and $G_{n}\big{(}u(\,\cdot\,)\big{)}$ vanish
on a neighborhood of $\\{u=0\\}$, they are supported on a compact subset of
$\\{u\neq 0\\}$, and continuously differentiable with

$D_{i}\big{(}F_{n}(u)\big{)}=g_{n}(u)D_{i}u\,1_{\\{u\neq
0\\}}\quad\text{and}\quad
D_{i}\big{(}G_{n}(u)\big{)}=\big{(}g_{n}(u)\big{)}^{2}D_{i}u\,1_{\\{u\neq
0\\}}\,.$

(1) Integrate by parts to obtain

$\displaystyle\int_{\mathbb{R}^{d}}|g_{n}(u)\nabla u\,1_{\\{u\neq
0\\}}|^{2}\,\mathrm{d}x\,$ $\displaystyle=-\int_{\mathbb{R}^{d}}G_{n}(u)\Delta
u\,1_{\\{u\neq 0\\}}\,\mathrm{d}x$
$\displaystyle\leq\frac{1}{p-1}\int_{\\{u\neq 0\\}}|u|^{p-1}|\Delta
u|\,\mathrm{d}x\,.$

By the monotone convergence theorem, we have $|u|^{p/2-1}|\nabla u|\in
L_{2}(\mathbb{R}^{d})$. We denote $v=\frac{2}{p}|u|^{p/2-1}u$. For any
$\zeta\in C_{c}^{\infty}(\mathbb{R}^{d})$, we have

$\displaystyle-\int_{\mathbb{R}^{d}}v\cdot D_{i}\zeta\,\mathrm{d}x$
$\displaystyle=-\lim_{n\rightarrow\infty}\int_{\mathbb{R}^{d}}F_{n}(u)\cdot
D_{i}\zeta\,\mathrm{d}x$
$\displaystyle=\lim_{n\rightarrow\infty}\int_{\\{u\neq
0\\}}g_{n}(u)D_{i}u\cdot\zeta\,\mathrm{d}x=\int_{\\{u\neq
0\\}}|u|^{p/2-1}D_{i}u\cdot\zeta\,\mathrm{d}x\,.$

Here, the first and the last equalities follow from the Lebesgue dominated
convergence theorem, because $|F_{n}(u)|\leq|v|$ and
$|g_{n}(u)D_{i}u|\leq|u|^{p/2-1}|\nabla u|\in L_{2}(\mathbb{R}^{d})$.
Therefore $v\in W_{2}^{1}(\mathbb{R}^{d})$ and
$D_{i}v=|u|^{p/2-1}D_{i}u\,1_{\\{u\neq 0\\}}$.

(2) It follows from (1) of this lemma that $|u|^{p}\in
W_{1}^{1}(\mathbb{R}^{d})$ with
$D_{i}\big{(}|u|^{p}\big{)}=p|u|^{p-2}uD_{i}u1_{u\neq 0}$. For any $\zeta\in
C_{c}^{\infty}$, we have

$\displaystyle\frac{1}{p-1}\int_{\\{u\neq 0\\}}|u|^{p-2}uD_{i}u\cdot
D_{j}\zeta\,\mathrm{d}x$ $\displaystyle=$
$\displaystyle\,\lim_{n\rightarrow\infty}\int_{\mathbb{R}^{d}}G_{n}(u)D_{i}u\cdot
D_{j}\zeta\,\mathrm{d}x$ $\displaystyle=$
$\displaystyle\,-\lim_{n\rightarrow\infty}\int_{\mathbb{R}^{d}}\Big{(}|g_{n}(u)|^{2}D_{i}uD_{j}u+G_{n}(u)D_{ij}u\Big{)}\zeta\,\mathrm{d}x$
$\displaystyle=$ $\displaystyle\,-\int_{\\{u\neq
0\\}}\Big{(}|u|^{p-2}D_{i}uD_{j}u+\frac{1}{p-1}|u|^{p-2}uD_{ij}u1_{\\{u\neq
0\\}}\Big{)}\zeta\,\mathrm{d}x\,.$

Here, the first and last inequalities follow from the Lebesgue dominated
convergence theorem, because $|G_{n}(u)|\leq\frac{1}{p-1}|u|^{p-1}$ and
$|g_{n}(u)|\leq|u|^{p/2-1}$ (recall the assumption for $u$, and (1) of this
lemma). Therefore $|u|^{p}\in W_{1}^{2}(\mathbb{R}^{d})$ with (A.1). ∎

######  Lemma A.2.

There exist linear maps

$\Lambda_{0}\,:\,\Psi H_{p,\theta}^{\gamma}\rightarrow\Psi
H_{p,\theta}^{\gamma+1}(\Omega)\quad\text{and}\quad\,\Lambda_{1},\,\ldots,\,\Lambda_{d}:\Psi
H_{p,\theta}^{\gamma}\rightarrow\Psi H_{p,\theta-p}^{\gamma+1}(\Omega)$

such that for any $f\in\Psi H_{p,\theta}^{\gamma}(\Omega)$,
$f=\Lambda_{0}f+\sum_{i=1}^{d}D_{i}(\Lambda_{i}f)$ and

$\displaystyle\begin{gathered}\|\Lambda_{0}f\|_{\Psi
H_{p,\theta}^{\gamma+1}(\Omega)}+\sum_{i=1}^{d}\|\Lambda_{i}f\|_{\Psi
H_{p,\theta-p}^{\gamma+1}(\Omega)}\leq N\|f\|_{\Psi
H_{p,\theta}^{\gamma}(\Omega)}\,,\end{gathered}$ (A.3)

where $N=N(d,p,\gamma,\theta,\mathrm{C}_{2}(\Psi))$.

###### Proof.

Step 1. We first prove the case $\Psi\equiv 1$. Consider linear operators from
$H_{p}^{\gamma}$ to $H_{p}^{\gamma+1}$ defined by $L_{0}:=(1-\Delta)^{-1}$ and
$L_{i}:=-D_{i}(1-\Delta)^{-1}$ for $i=1,\,\ldots,\,d$. They satisfy that for
any $g\in H_{p}^{\gamma}$,

$\displaystyle
L_{0}g+\sum_{i=1}^{d}D_{i}L_{i}g=g\quad\text{and}\quad\sum_{i=0}^{d}\|L_{i}g\|_{H_{p}^{\gamma+1}}\lesssim_{d,p,\gamma}\|g\|_{H_{p}^{\gamma}}\,.$
(A.4)

We denote
$\zeta_{1}(t)=\zeta_{0}(\mathrm{e}^{-1}t)+\zeta_{0}(t)+\zeta_{0}(\mathrm{e}t)$
and
$\zeta_{1,(n)}(x):=\zeta_{1}\big{(}\mathrm{e}^{-n}\widetilde{\rho}(x)\big{)}$.
Put

$\displaystyle\widetilde{\Lambda}_{0}f(x):=\,$
$\displaystyle\sum_{n\in\mathbb{Z}}\zeta_{1,(n)}(x)\,L_{0}\Big{[}\big{(}\zeta_{0,(n)}f\big{)}(\mathrm{e}^{n}\cdot)\Big{]}(\mathrm{e}^{-n}x)$
$\displaystyle-\sum_{k=1}^{d}\sum_{n\in\mathbb{Z}}\mathrm{e}^{n}\big{(}D_{k}\zeta_{1,(n)}\big{)}(x)\,L_{k}\Big{[}\big{(}\zeta_{0,(n)}f\big{)}(\mathrm{e}^{n}\cdot)\Big{]}(\mathrm{e}^{-n}x)\,,$
$\displaystyle\widetilde{\Lambda}_{i}f(x):=\,$
$\displaystyle\sum_{n\in\mathbb{Z}}\mathrm{e}^{n}\zeta_{1,(n)}(x)\,L_{i}\Big{[}\big{(}\zeta_{0,(n)}f\big{)}(\mathrm{e}^{n}\cdot)\Big{]}(\mathrm{e}^{-n}x)\,,$

for $i=1,\,\ldots,\,d$. Due to (A.4), we have

$\displaystyle\widetilde{\Lambda}_{0}f+\sum_{i=1}^{d}D_{i}\widetilde{\Lambda}_{i}f=\,$
$\displaystyle\sum_{n\in\mathbb{Z}}\left(\zeta_{1,(n)}(\,\cdot\,)\times\Big{[}\big{(}L_{0}+\sum_{i=1}^{d}D_{i}L_{i}\big{)}\big{[}(\zeta_{0,(n)}f)(\mathrm{e}^{n}\cdot)\big{]}\Big{]}(\mathrm{e}^{-n}\,\cdot\,)\right)$
$\displaystyle=\,$
$\displaystyle\sum_{n\in\mathbb{Z}}\Big{[}\zeta_{1,(n)}\zeta_{0,(n)}f\Big{]}=\sum_{n\in\mathbb{Z}}\zeta_{0,(n)}f=f\,.$

In addition, we also obtain

$\displaystyle\big{\|}\big{(}\zeta_{0,(n)}\widetilde{\Lambda}_{0}f\big{)}(\mathrm{e}^{n}\cdot)\big{\|}_{H_{p}^{\gamma+1}}^{p}+\sum_{i=1}^{d}\mathrm{e}^{-np}\big{\|}\big{(}\zeta_{0,(n)}\widetilde{\Lambda}_{i}f\big{)}(\mathrm{e}^{n}\cdot)\big{\|}_{H_{p}^{\gamma+1}}^{p}$
(A.5) $\displaystyle\lesssim_{N}$ $\displaystyle\sum_{i=0}^{d}\sum_{|k|\leq
2}\Big{\|}\big{(}\zeta_{0,(n)}\zeta_{1,(n+k)}\big{)}(\mathrm{e}^{n}\cdot)\times
L_{i}\Big{[}\big{(}\zeta_{0,(n+k)}f\big{)}(\mathrm{e}^{n+k}\cdot)\Big{]}(\mathrm{e}^{-k}\cdot)\Big{\|}_{H_{p}^{\gamma+1}}^{p}$
$\displaystyle+\mathrm{e}^{np}\sum_{i=0}^{d}\sum_{|k|\leq
2}\Big{\|}\big{(}\zeta_{0,(n)}D\zeta_{1,(n+k)}\big{)}(\mathrm{e}^{n}\cdot)\times
L_{i}\Big{[}\big{(}\zeta_{0,(n+k)}f\big{)}(\mathrm{e}^{n+k}\cdot)\Big{]}(\mathrm{e}^{-k}\cdot)\Big{\|}_{H_{p}^{\gamma+1}}^{p}$
$\displaystyle\lesssim_{N}$ $\displaystyle\sum_{|k|\leq
2}\sum_{i=0}^{d}\Big{\|}L_{i}\Big{[}\big{(}\zeta_{0,(n+k)}f\big{)}(\mathrm{e}^{n+k}\cdot)\Big{]}\Big{\|}_{H_{p}^{\gamma+1}}^{p}\lesssim_{d,p,\gamma}\sum_{|k|\leq
2}\big{\|}\big{(}\zeta_{0,(n+k)}f\big{)}(\mathrm{e}^{n+k}\cdot)\big{\|}_{H_{p}^{\gamma}}^{p}\,,$

Here, the first and second inequalities follow from that

$\displaystyle\big{\|}\big{(}\zeta_{0,(n)}\zeta_{1,(n+k)}\big{)}(\mathrm{e}^{n}\cdot)\big{\|}_{C^{m}(\mathbb{R}^{d})}+\mathrm{e}^{n}\big{\|}\big{(}\zeta_{0,(n)}\cdot
D\zeta_{1,(n+k)}\big{)}(\mathrm{e}^{n}\cdot)\big{\|}_{C^{m}(\mathbb{R}^{d})}\leq
N(d,k,m)\,,$

where $N(d,k,l)=0$ if $|k|\geq 3$ (note (3.10) and that
$\text{supp}(\zeta_{1})\subset[\mathrm{e}^{-2},\mathrm{e}^{2}]$). (A.5)
implies (A.3) for $\Psi\equiv 1$ and $\widetilde{\Lambda}_{i}$ instead of
$\Lambda_{i}$.

Step 2. For $f\in\Psi H_{p,\theta}^{\gamma}$ ($\Leftrightarrow\Psi^{-1}f\in
H_{p,\theta}^{\gamma}(\Omega)$), put

$\displaystyle\Lambda_{0}f=\Psi\widetilde{\Lambda}_{0}(\Psi^{-1}f)-\sum_{k=1}^{d}\big{(}D_{k}\Psi\big{)}\cdot\widetilde{\Lambda}_{k}(\Psi^{-1}f)\quad;\quad\Lambda_{i}f=\Psi\widetilde{\Lambda}_{i}(\Psi^{-1}f)\,.$

for $i=1,\,\cdots,\,d$. Then we have

$\Big{(}\Lambda_{0}+\sum_{i=1}^{d}D_{i}\Lambda_{i}\Big{)}f=\Psi\Big{(}\widetilde{\Lambda}_{0}+\sum_{i=1}^{d}D_{i}\widetilde{\Lambda}_{i}\Big{)}\big{(}\Psi^{-1}f\big{)}=f\,.$

Moreover, Lemma 3.10.(3) and (A.5) imply that

$\displaystyle\|\Psi^{-1}\Lambda_{0}f\|_{H_{p,\theta}^{\gamma+1}(\Omega)}+\sum_{i=1}^{d}\|\Psi^{-1}\Lambda_{i}f\|_{H_{p,\theta-p}^{\gamma+1}(\Omega)}$
$\displaystyle\lesssim\,\,$
$\displaystyle\|\widetilde{\Lambda}_{0}(\Psi^{-1}f)\|_{H_{p,\theta}^{\gamma+1}(\Omega)}+\sum_{i=1}^{d}\|\widetilde{\Lambda}_{i}(\Psi^{-1}f)\|_{H_{p,\theta-p}^{\gamma+1}(\Omega)}\lesssim\|\Psi^{-1}f\|_{H_{p,\theta}^{\gamma}(\Omega)}\,.$

Therefore, the proof is completed. ∎

######  Lemma A.3.

Let $\eta\in C_{c}^{\infty}(\mathbb{R}^{d})$ satisfy $\eta=1$ on $B_{1}(0)$
and $\mathrm{supp}(\eta)\subset B_{2}(0)$. For each $i\in\mathbb{N}$, let
$N(i)\in\mathbb{N}$ be a constant satisfying

$\mathrm{supp}\Big{(}\sum_{|n|\leq
i}\zeta_{0,(n)}\Big{)}\subset\big{\\{}x\in\Omega\,:\,\big{(}N(i)/2\big{)}^{-1}\leq\rho(x)\leq
N(i)/2\big{\\}}\,.$

Let $\Lambda_{i}$, $\Lambda_{i,j}$, $\Lambda_{i,j,k}$ are linear functionals
on $\mathcal{D}^{\prime}(\Omega)$ defined as

$\displaystyle\Lambda_{i}f:=\Big{(}\sum_{|n|\leq
i}\zeta_{0,(n)}\Big{)}f\,\,\,,\,\,\,\,\Lambda_{i,j}f=\eta(j^{-1}\cdot\,)\Lambda_{i}f\,\,\,,\,\,\,\,\Lambda_{i,j,k}f=\big{(}\Lambda_{i,j}f\big{)}^{(N(i)^{-1}k^{-1})}\,,$

where $\big{(}\Lambda_{i,j}f\big{)}^{(\epsilon)}$ is defined in the same way
as in (2.1). Then for any $p\in(1,\infty)$, $\gamma$, $\theta\in\mathbb{R}$,
and regular Harnack function $\Psi$, the following hold:

  1. (1)

For any $f\in\mathcal{D}^{\prime}(\Omega)$, $\Lambda_{i,j,k}f\in
C_{c}^{\infty}(\Omega)$ .

  2. (2)

For any $f\in\Psi H_{p,\theta}^{\gamma}(\Omega)$,

$\displaystyle\sup_{i}\|\Lambda_{i}f\|_{\Psi
H_{p,\theta}^{\gamma}(\Omega)}\leq N_{1}\|f\|_{\Psi
H_{p,\theta}^{\gamma}(\Omega)}$
$\displaystyle\,\sup_{j}\|\Lambda_{i,j}f\|_{\Psi
H_{p,\theta}^{\gamma}(\Omega)}\leq N_{2}\|f\|_{\Psi
H_{p,\theta}^{\gamma}(\Omega)}$ (A.6)
$\displaystyle\,\sup_{k}\|\Lambda_{i,j,k}f\|_{\Psi
H_{p,\theta}^{\gamma}(\Omega)}\leq N_{3}\|f\|_{\Psi
H_{p,\theta}^{\gamma}(\Omega)}\,,$

where $N_{1}$, $N_{2}$, $N_{3}$ are constants independent of $f$.

  3. (3)

For any $f\in\Psi H_{p,\theta}^{\gamma}(\Omega)$,

$\displaystyle\lim_{k\rightarrow\infty}\Lambda_{i,j,k}f=\Lambda_{i,j}f\,\,\,,\,\,\,\lim_{j\rightarrow\infty}\Lambda_{i,j}f=\Lambda_{i}f\,\,\,,\,\,\,\lim_{i\rightarrow\infty}\Lambda_{i}f=f\quad\text{in}\quad\Psi
H_{p,\theta}^{\gamma}(\Omega)\,.$ (A.7)

###### Proof.

(1) It follows directly from the properties of distributions.

(2), (3) Step 1: $\Lambda_{i}$. Let $f\in H_{p,\theta}^{\gamma}(\Omega)$. From
(3.10), one can observe that

$\displaystyle\|f-\Lambda_{i}f\|_{\Psi
H_{p,\theta}^{\gamma}(\Omega)}^{p}\lesssim_{N}\sum_{|n|\geq
i-1}\mathrm{e}^{n\theta}\big{\|}\big{(}\Psi^{-1}f\zeta_{0,(n)}\big{)}(\mathrm{e}^{n}\cdot)\big{\|}_{H_{p}^{\gamma}}^{p}\leq\|f\|_{\Psi
H_{p,\theta}^{\gamma}(\Omega)}^{p}\,,$

where $N=N(d,p,\gamma,\theta)$. Therefore we have

$\sup_{i}\|\Lambda_{i}f\|_{\Psi H_{p,\theta}^{\gamma}(\Omega)}\leq
N\|f\|_{\Psi
H_{p,\theta}^{\gamma}(\Omega)}\quad\text{and}\quad\lim_{i\rightarrow\infty}\|f-\Lambda_{i}f\|_{\Psi
H_{p,\theta}^{\gamma}(\Omega)}=0\,.$

Step 2: $\Lambda_{i,j}$. The definition of $H_{p,\theta}^{\gamma}(\Omega)$
implies that for any $A>1$, if $F\in\mathcal{D}^{\prime}(\Omega)$ or
$F\in\mathcal{D}^{\prime}(\mathbb{R}^{d})$, and $F$ is supported on
$\\{x\in\Omega\,:\,A^{-1}\leq\rho(x)\leq A\\}$, then

$\displaystyle\|F\|_{H_{p,\theta}^{\gamma}(\Omega)}\simeq_{N}\|F\|_{H_{p}^{\gamma}}\,,$
(A.8)

where $N=N(d,p,\theta,\gamma,A)$. For each $i\in\mathbb{N}$,
$\Psi^{-1}\Lambda_{i}f$ and $\Psi^{-1}\Lambda_{i,j}f$ are supported on

$\big{\\{}x\in\Omega\,:\,N(i)^{-1}\leq\rho(x)\leq N(i)\big{\\}}\,.$

Therefore $\Psi^{-1}\Lambda_{i}f\in H_{p}^{\gamma}$. Since
$\Psi^{-1}\Lambda_{i,j}f=\eta(j^{-1}\,\cdot)\Psi^{-1}\Lambda_{i}f$, we obtain
that

$\displaystyle\begin{gathered}\lim_{j\rightarrow\infty}\big{\|}\Psi^{-1}\Lambda_{i}f-\Psi^{-1}\Lambda_{i,j}f\big{\|}_{H_{p}^{\gamma}}=0\quad\text{and}\quad\big{\|}\Psi^{-1}\Lambda_{i,j}f\big{\|}_{H_{p}^{\gamma}}\lesssim_{N_{2}}\big{\|}\Psi^{-1}\Lambda_{i}f\big{\|}_{H_{p}^{\gamma}}\,,\end{gathered}$

where $N_{2}=N(d,p,\gamma,\theta,i,\eta)$. Due to (A.8), (A.6) and (A.7) for
$\Lambda_{i,j}$ are proved.

Step 3: $\Lambda_{i,j,k}$. Put

$K_{i,j}=\\{x\in\Omega\,:\,N(i)^{-1}\leq\rho(x)\leq N(i)\,,\,\,|x|\leq
2j\\}\,,$

which is a compact subset of $\Omega$, and $\Lambda_{i,j}f$ and
$\Lambda_{i,j,k}f$ are supported on there. Since $\Psi$ and $\Psi^{-1}$ belong
to $C^{\infty}(\Omega)$, we obtain that

$\displaystyle\|\Lambda_{i,j}f\|_{\Psi
H_{p,\theta}^{\gamma}(\Omega)}:=\|\Psi^{-1}\Lambda_{i,j}f\|_{H_{p,\theta}^{\gamma}(\Omega)}\simeq_{N}\|\Psi^{-1}\Lambda_{i,j}f\|_{H_{p}^{\gamma}}\simeq_{N}\|\Lambda_{i,j}f\|_{H_{p}^{\gamma}},$
(A.9)

where $N=N(d,p,\gamma,\theta,i,j,\Psi)$; it also holds for $\Lambda_{i,j,k}f$
and $\Lambda_{i,j}f-\Lambda_{i,j,k}f$, instead of $\Lambda_{i,j}f$.

Since $\Lambda_{i,j,k}f$ is a mollification of $\Lambda_{i,j}f$, we have

$\displaystyle\|\Lambda_{i,j,k}f\|_{H_{p}^{\gamma}}\lesssim_{N_{3}}\|\Lambda_{i,j}f\|_{H_{p}^{\gamma}}\quad\text{and}\quad\lim_{k\rightarrow\infty}\big{\|}\Lambda_{i,j}f-\Lambda_{i,j,k}f\big{\|}_{H_{p}^{\gamma}}=0\,,$

where $N_{3}=N(d,p,\theta,\gamma,\Psi,i,j,\eta)$. ∎

## Acknowledgment

The author wishes to express deep appreciation to Prof. Kyeong-Hun Kim for
valuable comments. The author is also sincerely grateful to Dr. Jin Bong Lee
for careful assistance in proofreading and editing this paper.

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