Given $\Lambda,K,s_{1},s_{2}\in\mathbb{R}$ and
$D\in(0,\min\\{\mathfrak{r}_{\Lambda},\mathfrak{t}_{\Lambda,K}\\}]\cap\mathbb{R}$,
let

$\phi(t):=(\mathbf{c}_{\Lambda}(t)-K\mathbf{s}_{\Lambda}(t))^{-{n\over
p-1}}\mathbf{s}_{\Lambda}(t)^{-{m-n-1\over
p-1}},\quad\psi(t):=\psi(t,s_{1},s_{2}):=\int^{t}_{0}\left[\log\left({\mathbf{s}_{\Lambda}(D)\over\mathbf{s}_{\Lambda}(\tau)}\right)\right]^{s_{1}}\mathbf{s}_{\Lambda}(\tau)^{s_{2}}\mathbf{c}_{\Lambda}(\tau){\operatorname{d}}\tau.$
(5.6)

Basic properties of the function $\psi$ are collected in the following lemma.

######  Lemma 5.1.

The function $\psi$ in (5.6) is well-defined and strictly increasing in
$[0,D)$ if $s_{1}<-1$ and $s_{2}\geq-1$. In addition, if $s>0$ is such that
$s_{1}+1+s\leq 0$, then $\psi$ defined by (5.6) satisfies

$\psi(t)\leq
s^{-1}\left[\log\left({\mathbf{s}_{\Lambda}(D)\over\mathbf{s}_{\Lambda}(t)}\right)\right]^{s_{1}+1}\mathbf{s}_{\Lambda}(t)^{s_{2}+1}=s^{-1}\left[\log\left({\mathbf{s}_{\Lambda}(D)\over\mathbf{s}_{\Lambda}(t)}\right)\right]{\mathbf{s}_{\Lambda}(t)\over\mathbf{c}_{\Lambda}(t)}\psi^{\prime}(t),\
\ t\in[0,D),$ (5.7)

with equality if $s=-(s_{1}+1)$ and $s_{2}=-1$.

###### Proof.

Set

$F(t):=\psi(t)-s^{-1}\left[\log\left({\mathbf{s}_{\Lambda}(D)\over\mathbf{s}_{\Lambda}(t)}\right)\right]^{s_{1}+1}\mathbf{s}_{\Lambda}(t)^{s_{2}+1},\quad\forall\,t\in(0,D).$

Clearly, $F(0):=\lim_{t\rightarrow 0^{+}}F(t)=0$. Since
$\mathbf{s}_{\Lambda}(t)\leq\mathbf{s}_{\Lambda}(D)$ for $t\in[0,D)$, we have

$F^{\prime}(t)=s^{-1}\left[\log\left({\mathbf{s}_{\Lambda}(D)\over\mathbf{s}_{\Lambda}(t)}\right)\right]^{s_{1}}\mathbf{s}_{\Lambda}(t)^{s_{2}}\mathbf{c}_{\Lambda}(t)\left[s_{1}+1+s-(s_{2}+1)\log\left({\mathbf{s}_{\Lambda}(D)\over\mathbf{s}_{\Lambda}(t)}\right)\right]\leq
0,$

which implies $F(t)\leq F(0)=0$ for any $t\in[0,D)$, i.e., (5.7) follows. In
particular, if $s=-(s_{1}+1)$ and $s_{2}=-1$, then $F^{\prime}(t)=0$ and
hence, $F(t)=F(0)=0$. ∎

######  Theorem 5.2.

Let $(M,g)$ be an $m$-dimensional complete Riemannian manifold,
$i:\Sigma\hookrightarrow M$ be an $n$-dimensional closed totally umbilical
submanifold and $\Omega$ be a noncompact domain satisfying (3.1). Suppose that

  * •

$(M,\Sigma,\Omega)$ satisfies the upper curvature bound condition
$(\lambda,\kappa);$

  * •

there is a finite constant
$D\in(0,\min\\{\mathfrak{t}_{\lambda,\kappa},\mathfrak{r}_{\lambda}\\}]\cap\mathbb{R}$
such that $\Omega\subset\operatorname{\mathsf{T}}_{D}$.

Given $\alpha,\beta\in\mathbb{R}$ and $s>0$ satisfying $\alpha\geq p-(m-n)$
and $\beta+1+s\leq 0$, then for any $(\Lambda,K)\geq(\lambda,\kappa)$ with
$\min\\{\mathfrak{r}_{\Lambda},\mathfrak{t}_{\Lambda,K}\\}\geq D$, we have

$\displaystyle\int_{\Omega}|{\nabla}u|^{p}\left[\log\left({\mathbf{s}_{\Lambda}(D)\over\mathbf{s}_{\Lambda}(r)}\right)\right]^{p+\beta}\frac{\mathbf{c}_{\Lambda}(r)^{1-p}\mathbf{s}_{\Lambda}(r)^{\alpha}}{(\mathbf{c}_{\Lambda}(r)-K\mathbf{s}_{\Lambda}(r))^{n}}\operatorname{dvol}_{g}$
$\displaystyle\geq$ $\displaystyle\left({s\over
p}\right)^{p}\int_{\Omega}|u|^{p}\left[\log\left({\mathbf{s}_{\Lambda}(D)\over\mathbf{s}_{\Lambda}(r)}\right)\right]^{\beta}\frac{\mathbf{c}_{\Lambda}(r)\mathbf{s}_{\Lambda}(r)^{\alpha-p}}{(\mathbf{c}_{\Lambda}(r)-K\mathbf{s}_{\Lambda}(r))^{n}}\operatorname{dvol}_{g},\quad\forall\,u\in
C^{\infty}_{0}(\Omega).$ (5.8)

Moreover, if either $\Sigma\subset\Omega$ or $\Sigma=\partial\Omega$ with
$D<\min\\{\mathfrak{r}_{\Lambda},\mathfrak{t}_{\Lambda,K}\\}$, and

$\text{$\Sigma$ is compact},\quad\alpha=p-(m-n),\quad s=|\beta+1|,$ (5.9)

then (5.8) is sharp.

###### Proof.

Let $\phi,\psi$ be the functions defined on $(0,D)$ by (5.6) with

$s_{1}:=\beta<-1,\quad s_{2}:=\alpha-p+(m-n)-1\geq-1;$ (5.10)

due to Lemma 5.1, both of them are well-defined. Now we claim that Assumption
2 is satisfied. Obviously, Assumption 2/(a)(b)(f) trivially hold. Moreover,
(2.43) together with Proposition 2.1 yields

$\Delta r\geq G_{\lambda,\kappa}(r)\geq
G_{\Lambda,K}(r)=-(p-1)\frac{\phi^{\prime}(r)}{\phi(r)},\quad\forall\,x\in\mathcal{V}\mathscr{D}\Sigma\cap\Omega_{\Sigma},$

which combined with (3.2) implies Assumption 2/(d).

Given a compact set $\mathscr{K}\subset\Omega$ with
$\sup_{x\in\mathscr{K}}r(x)=:R<D$, we have $\sup_{t\in[0,R]}\psi(t)<+\infty$
and $\phi(r)^{p-1}\in L^{1}(\mathscr{K})$, which implies
$\phi(r)^{p-1}\psi(r)^{s}\in L^{1}(\mathscr{K})$ for any $s\geq 0$. Thus,
Assumption 2/(c) follows. It remains to check Assumption 2/(e). The assumption
together with (5.10) implies $s_{1}+1+s=\beta+1+s\leq 0$, which combined with
Lemma 5.1 furnishes (5.7). Therefore, we obtain

$\displaystyle\frac{\phi(r)^{p-1}|\psi^{\prime}(r)|}{|\nabla(\log\psi(r))|^{p}}=\,$
$\displaystyle(\mathbf{c}_{\Lambda}(r)-K\mathbf{s}_{\Lambda}(r))^{-n}\mathbf{s}_{\Lambda}(r)^{-(m-n-1)}\frac{\psi(r)^{p}}{|\psi^{\prime}(r)|^{p-1}}$
$\displaystyle\leq
s^{-p}\left[\log\left({\mathbf{s}_{\Lambda}(D)\over\mathbf{s}_{\Lambda}(r)}\right)\right]^{p+\beta}\frac{\mathbf{c}_{\Lambda}(r)^{1-p}\mathbf{s}_{\Lambda}(r)^{\alpha}}{(\mathbf{c}_{\Lambda}(r)-K\mathbf{s}_{\Lambda}(r))^{n}},$
(5.11) $\displaystyle\phi(r)^{p-1}|\psi^{\prime}(r)|=$
$\displaystyle\left[\log\left({\mathbf{s}_{\Lambda}(D)\over\mathbf{s}_{\Lambda}(r)}\right)\right]^{\beta}\frac{\mathbf{c}_{\Lambda}(r)\mathbf{s}_{\Lambda}(r)^{\alpha-p}}{(\mathbf{c}_{\Lambda}(r)-K\mathbf{s}_{\Lambda}(r))^{n}}.$
(5.12)

Thus, Assumption 2/(e) is verified whenever

$\left[\log\left({\mathbf{s}_{\Lambda}(D)\over\mathbf{s}_{\Lambda}(r)}\right)\right]^{p+\beta}\frac{\mathbf{c}_{\Lambda}(r)^{1-p}\mathbf{s}_{\Lambda}(r)^{\alpha}}{(\mathbf{c}_{\Lambda}(r)-K\mathbf{s}_{\Lambda}(r))^{n}},\
\left[\log\left({\mathbf{s}_{\Lambda}(D)\over\mathbf{s}_{\Lambda}(r)}\right)\right]^{\beta}\frac{\mathbf{c}_{\Lambda}(r)\mathbf{s}_{\Lambda}(r)^{\alpha-p}}{(\mathbf{c}_{\Lambda}(r)-K\mathbf{s}_{\Lambda}(r))^{n}}\in
L^{1}(\mathscr{K}).$ (5.13)

In order to check this, we notice that the assumption on $D$ yields a finite
constant $C>0$ such that on $\mathscr{K}$ we have

${\max\left\\{\frac{1}{(\mathbf{c}_{\Lambda}(r)-K\mathbf{s}_{\Lambda}(r))^{n}},\frac{\mathbf{c}_{\Lambda}(r)^{-p}}{(\mathbf{c}_{\Lambda}(r)-K\mathbf{s}_{\Lambda}(r))^{n}}\right\\}\leq\frac{C}{\operatorname{vol}(\mathbb{S}^{m-n-1})}.}$
(5.14)

Moreover, due to (2.34) and (2.31), there exists a small $\epsilon>0$ such
that

$\displaystyle\det\mathcal{A}(t,\mathbf{n})\leq
2\,\mathbf{s}_{\Lambda}(t)^{m-n-1},\text{ for
}(t,\mathbf{n})\in([0,\epsilon]\times\mathcal{V}S\Sigma)\cap\operatorname{\mathsf{F}}^{-1}(\mathscr{K}),$

which together with (2.35), (5.14) and Proposition 5.1 yield

$\displaystyle\int_{\operatorname{\mathsf{T}}_{\epsilon}\cap\mathscr{K}}\left[\log\left({\mathbf{s}_{\Lambda}(D)\over\mathbf{s}_{\Lambda}(r)}\right)\right]^{p+\beta}\frac{\mathbf{c}_{\Lambda}(r)^{1-p}\mathbf{s}_{\Lambda}(r)^{\alpha}}{(\mathbf{c}_{\Lambda}(r)-K\mathbf{s}_{\Lambda}(r))^{n}}\operatorname{dvol}_{g}\leq
2C\int^{\epsilon}_{0}\left[\log\left({\mathbf{s}_{\Lambda}(D)\over\mathbf{s}_{\Lambda}(r)}\right)\right]^{p+\beta}\mathbf{s}_{\Lambda}(r)^{s_{2}+p}\mathbf{c}_{\Lambda}(r){\operatorname{d}}r<+\infty,$
$\displaystyle\int_{\operatorname{\mathsf{T}}_{\epsilon}\cap\mathscr{K}}\left[\log\left({\mathbf{s}_{\Lambda}(D)\over\mathbf{s}_{\Lambda}(r)}\right)\right]^{\beta}\frac{\mathbf{c}_{\Lambda}(r)\mathbf{s}_{\Lambda}(r)^{\alpha-p}}{(\mathbf{c}_{\Lambda}(r)-K\mathbf{s}_{\Lambda}(r))^{n}}\operatorname{dvol}_{g}\leq
2C\int^{\epsilon}_{0}\left[\log\left({\mathbf{s}_{\Lambda}(D)\over\mathbf{s}_{\Lambda}(r)}\right)\right]^{\beta}\mathbf{s}_{\Lambda}(r)^{s_{2}}\mathbf{c}_{\Lambda}(r){\operatorname{d}}r<+\infty.$

On the other hand, by choosing an enough large $C>0$, the continuity of
$\det\mathcal{A}(t,\mathbf{n})$ implies

$\det\mathcal{A}(t,\mathbf{n})\leq C<+\infty,\quad\text{ for
}(t,\mathbf{n})\in([\epsilon,R]\times\mathcal{V}S\Sigma)\cap\operatorname{\mathsf{F}}^{-1}(\mathscr{K}).$

Thus, Proposition 5.1 again furnishes

$\displaystyle\int_{\mathscr{K}\backslash\operatorname{\mathsf{T}}_{\epsilon}}\left[\log\left({\mathbf{s}_{\Lambda}(D)\over\mathbf{s}_{\Lambda}(r)}\right)\right]^{p+\beta}\frac{\mathbf{c}_{\Lambda}(r)^{1-p}\mathbf{s}_{\Lambda}(r)^{\alpha}}{(\mathbf{c}_{\Lambda}(r)-K\mathbf{s}_{\Lambda}(r))^{n}}\operatorname{dvol}_{g}\leq
C^{2}\int^{R}_{\epsilon}\left[\log\left({\mathbf{s}_{\Lambda}(D)\over\mathbf{s}_{\Lambda}(r)}\right)\right]^{p+\beta}\mathbf{s}_{\Lambda}(r)^{\alpha}\mathbf{c}_{\Lambda}(r){\operatorname{d}}r<+\infty,$
$\displaystyle\int_{\mathscr{K}\backslash\operatorname{\mathsf{T}}_{\epsilon}}\left[\log\left({\mathbf{s}_{\Lambda}(D)\over\mathbf{s}_{\Lambda}(r)}\right)\right]^{\beta}\frac{\mathbf{c}_{\Lambda}(r)\mathbf{s}_{\Lambda}(r)^{\alpha-p}}{(\mathbf{c}_{\Lambda}(r)-K\mathbf{s}_{\Lambda}(r))^{n}}\operatorname{dvol}_{g}\leq
C^{2}\int_{\epsilon}^{R}\left[\log\left({\mathbf{s}_{\Lambda}(D)\over\mathbf{s}_{\Lambda}(r)}\right)\right]^{\beta}\mathbf{s}_{\Lambda}(r)^{\alpha-p}\mathbf{c}_{\Lambda}(r){\operatorname{d}}r<+\infty.$

From these estimates, (5.13) follows and hence, Assumption 2 holds. Therefore,
(5.8) directly follows by (5.11), (5.12) and Theorem 4.1.

We now focus on the sharpness of (5.8). If $\Sigma\subset\Omega$, then (5.10)
combined with (5.9) implies $s=-(s_{1}+1)$ and $s_{2}=-1$. Thus, it follows
that both (5.7) and (5.11) are equalities. In this way, due to Theorem 4.2 and
Remark 4.3, in order to show the sharpness of (5.8), it suffices to prove
(4.13). In fact, the compactness of $\Sigma$ implies
$c_{\mathcal{V}}(\Sigma)>0$. Let
$\delta_{\Lambda}\in(0,\min\\{D,c_{\mathcal{V}}(\Sigma)\\})$ be small enough
such that

$\min_{t\in[0,\delta_{\lambda}]}\left(\frac{\mathbf{c}_{\Lambda}(t)}{\mathbf{c}_{\Lambda}(t)-K\mathbf{s}_{\Lambda}(t)}\right)^{n}{\geq\frac{1}{2}}.$

Thus, for any $\delta\in(0,\delta_{\Lambda})$, a direct calculation together
with the equality form of (5.7) yields

$\displaystyle\lim_{{\varepsilon}\rightarrow
0^{+}}\int_{\varepsilon}^{\delta}\phi(t)^{p-1}|(\log\psi(t))^{\prime}|\mathbf{c}_{\Lambda}(t)^{n}\mathbf{s}_{\Lambda}(t)^{m-n-1}{\operatorname{d}}t=\lim_{{\varepsilon}\rightarrow
0^{+}}\int_{\varepsilon}^{\delta}\left(\frac{\mathbf{c}_{\Lambda}(t)}{\mathbf{c}_{\Lambda}(t)-K\mathbf{s}_{\Lambda}(t)}\right)^{n}\frac{\psi^{\prime}(t)}{\psi(t)}{\operatorname{d}}t$
$\displaystyle{\geq\frac{1}{2}}s\lim_{{\varepsilon}\rightarrow
0^{+}}\int_{\varepsilon}^{\delta}\left[\log\Big{(}{\mathbf{s}_{\Lambda}(D)\over\mathbf{s}_{\Lambda}(t)}\Big{)}\right]^{-1}\mathbf{s}_{\Lambda}(t)^{-1}\mathbf{c}_{\Lambda}(t){\operatorname{d}}t=+\infty,$

which shows the validity of (4.13), and hence the sharpness of (5.8).

In the other case, i.e., $\Sigma=\partial\Omega$, again both (5.7) and (5.11)
become equalities and $c_{\mathcal{V}}(\Sigma)>0$. Let
$s_{0}\in(0,\min\\{D,c_{\mathcal{V}}(\Sigma)\\})$ with

$\displaystyle\det\mathcal{A}(t,\mathbf{n})\leq
2\,\mathbf{s}_{\Lambda}(t)^{m-n-1}=2,\text{ for
}(t,\mathbf{n})\in[0,s_{0}]\times\mathcal{V}S\Sigma.$

Since $D<\mathfrak{c}_{\Lambda,K}$ and $n=m-1$, both $\psi(r)^{1+\varepsilon}$
and $\phi(r)^{p-1}$ are bounded on
$\operatorname{\mathsf{T}}_{s_{0}}\cap\Omega$. Thus, (2.35) yields

$\displaystyle\int_{{\operatorname{\mathsf{T}}_{s_{0}}}\cap\,\Omega}\psi(r)^{1+\varepsilon}\phi(r)^{p-1}|\psi^{\prime}(r)|\operatorname{dvol}_{g}\leq
C\int_{{\operatorname{\mathsf{T}}_{s_{0}}}\cap\,\Omega}\psi^{\prime}(r)\operatorname{dvol}_{g}\leq
C\int^{s_{0}}_{0}\psi^{\prime}(t){\operatorname{d}}t=C\psi(s_{0})<+\infty,$

i.e., $\psi(r)^{1+\varepsilon}\phi(r)^{p-1}|\psi^{\prime}(r)|\in
L^{1}({\operatorname{\mathsf{T}}_{s_{0}}}\cap\,\Omega)$. On the other hand, by
(5.7) and (5.12) we have

$\displaystyle\psi(r)^{-(1+\varepsilon)}\phi(r)^{p-1}|\psi^{\prime}(r)|=s^{(1+\varepsilon)}\left[\log\Big{(}{\mathbf{s}_{\Lambda}(D)\over\mathbf{s}_{\Lambda}(r)}\Big{)}\right]^{-(1+\varepsilon)(1+s_{1})+\beta}\frac{\mathbf{s}_{\Lambda}(r)^{-(1+\varepsilon)(s_{2}+1)+\alpha-p}\mathbf{c}_{\Lambda}(r)}{(\mathbf{c}_{\Lambda}(r)-K\mathbf{s}_{\Lambda}(r))^{n}}.$

Since $-(1+\varepsilon)(1+s_{1})+\beta=-1-\varepsilon(s_{1}+1)>-1$ for any
$\varepsilon>0$, Proposition 5.1 yields
$\psi^{-(1+\varepsilon)}(r)\phi(r)^{p-1}|\psi^{\prime}(r)|\in
L^{1}(\Omega\backslash{\operatorname{\mathsf{T}}_{s_{0}}})$. Thus, (3.9)
holds, which together with Corollary 3.3 implies the sharpness of (5.8).

The non-existence of extremal functions in (5.8) is a direct consequence of
Theorem 4.3. ∎

Note that $\Omega\subset\operatorname{\mathsf{T}}_{D}$ implies
$\sup_{x\in\Omega}r(x)\leq D$; however, the latter inequality in general is
not enough to follow the argument from the proof of Theorem 5.2 as the next
example shows.

######  Example 5.3.

Let $M:=\mathbb{R}\times\mathbb{S}$ be a cylinder equipped by the canonical
metric and let $(t,s)\in(-\infty,+\infty)\times[0,2\pi)$ be a natural
coordinate system of $M$. Let $\Sigma:=\mathbb{R}\times\\{0\\}$,
$A:=[-1,1]\times\\{\pi\\}$ and $\Omega:=M\backslash A$.

Thus, $\Omega$ is a noncompact domain with $\sup_{x\in\Omega}r(x)=\pi$.
Moreover, it is obvious that

$r(x)=\pi,\quad\forall\,x\in(\mathbb{R}\backslash[-1,1])\times\\{\pi\\}\subset\Omega.$

Choose $\lambda=\Lambda=0=K=\kappa$ and $\alpha=p-(m-n)=p-1$ and let
$(\phi,\psi)$ be as in the proof of Theorem 5.2. For the compact set
$\mathscr{K}:=[2,3]\times\mathbb{S}\subset\Omega$, since $\beta<-1$, by (5.12)
we have

$\int_{\mathscr{K}}\phi(r)^{p-1}|\psi^{\prime}(r)|\operatorname{dvol}_{g}=2\int_{0}^{\pi}\left[\log\left({\pi\over
r}\right)\right]^{\beta}r^{-1}{\operatorname{d}}r=+\infty,$

i.e., $\phi(r)^{p-1}|\psi^{\prime}(r)|\notin
L_{\operatorname{loc}}^{1}(\Omega)$.

##  6\. Application II: Sharp Hardy inequalities on manifolds with lower
curvature bounds

In this section we study Hardy inequalities on manifolds with curvature
bounded from below. The generic setting of the present section is described in
the following definition.

######  Definition 6.1.

Let $(M,g)$ be an $m$-dimensional Riemannian manifold,
$i:\Sigma\hookrightarrow M$ be an $n$-dimensional closed submanifold and
$\Omega$ be a non-empty domain in $M$ satisfying (3.1). Given
$\lambda,\kappa\in\mathbb{R}$, the triple $(M,\Sigma,\Omega)$ satisfies the
lower curvature bound condition $(\lambda,\kappa)$ if one of the following
conditions holds:

  1. (a)

${\operatorname{Ric}}_{M}\geq\lambda$ and $n=0$;

  2. (b)

${\operatorname{Ric}}_{M}\geq\lambda$, $n=m-1$ and
$\operatorname{{tr}}\mathfrak{A}^{\mathbf{n}}\geq n\kappa$ for any inward
normal vector $\mathbf{n}$ w.r.t. $\Omega$;

  3. (c)

$\sec_{M}\geq\lambda$, $1\leq n\leq m-2$ and
$\operatorname{{tr}}\mathfrak{A}^{\mathbf{n}}\geq n\kappa$ for any
$\mathbf{n}\in\mathcal{V}S\Sigma$.

######  Remark 6.1.

We use $n\kappa$ to denote the lower bound of
$\operatorname{{tr}}\mathfrak{A}^{\mathbf{n}}$. More precisely,

  * •

(a) means that $\Sigma$ is a point and hence, the Weingarten map always
vanishes, when $\kappa$ can be chosen as any real number;

  * •

if $\kappa=0$ in (b), then $\Sigma$ is called weakly mean convex (cf. [26,
10]);

  * •

(c) implies $\kappa\leq 0$, because
$n\kappa\leq\operatorname{{tr}}\mathfrak{A}^{-\mathbf{n}}=-\operatorname{{tr}}\mathfrak{A}^{\mathbf{n}}\leq-n\kappa$
for any $\mathbf{n}\in\mathcal{V}S\Sigma$. Moreover, if $\kappa=0$, the above
inequality yields

$0=\operatorname{{tr}}\mathfrak{A}^{\mathbf{n}}=n\langle
H,\mathbf{n}\rangle,\quad\forall\,\mathbf{n}\in\mathcal{V}S\Sigma,$

which implies that $H=0$, i.e., $\Sigma$ is a minimal submanifold.

As in the previous section, although we do not emphasize the sign of $\kappa$,
we assume that $\kappa\leq 0$ whenever $1\leq n\leq m-2$.

###  6.1. Hardy inequalities involving double-curvature weights

######  Theorem 6.1.

Let $(M,g)$ be an $m$-dimensional complete Riemannian manifold,
$i:\Sigma\hookrightarrow M$ be an $n$-dimensional closed submanifold and
$\Omega$ be a non-empty domain in $M$ satisfying (3.1). Denote
$r(x):=d(\Sigma,x)$ the distance function from $\Sigma$. Suppose that
$(M,\Sigma,\Omega)$ satisfies the lower curvature bound condition
$(\lambda,\kappa)$. Given any pair
$(\Lambda,K)\leq(\lambda,\min\\{\kappa,0\\})$ with $\Lambda<\lambda/4$ if
$\lambda>0$, the following statements hold$:$

  1. (i)

for $p>1$, $\beta<-(m-n)$, there holds

$\int_{\Omega_{\Sigma}}|{\nabla}u|^{p}\frac{\mathbf{c}_{\Lambda}(r)^{1-p}\mathbf{s}_{\Lambda}(r)^{p+\beta}}{(\mathbf{c}_{\Lambda}(r)-K\mathbf{s}_{\Lambda}(r))^{n}}\operatorname{dvol}_{g}\geq\Big{|}{\beta+m-n\over
p}\Big{|}^{p}\int_{\Omega_{\Sigma}}|u|^{p}\frac{\mathbf{c}_{\Lambda}(r)\mathbf{s}_{\Lambda}(r)^{\beta}}{(\mathbf{c}_{\Lambda}(r)-K\mathbf{s}_{\Lambda}(r))^{n}}\operatorname{dvol}_{g},$
(6.1)

for any $u\in C^{\infty}_{0}(\Omega_{\Sigma})$. In particular, if $\Sigma$ is
compact and either $\Omega=M$ or $\partial\Omega=\Sigma$, then (6.1) is sharp
and there are no extremal functions in $C^{\infty}_{0}(\Omega_{\Sigma});$

  2. (ii)

for $p>1$, $\beta<-(m-n)$ with $p+\beta>-(m-n)$, inequality (6.1) holds for
every $u\in C^{\infty}_{0}(\Omega,\Sigma)$. Furthermore, if $\Sigma$ is
compact and either $\Omega=M$ or $\partial\Omega=\Sigma$, then (6.1) is sharp
over $C^{\infty}_{0}(\Omega,\Sigma)$ and there are no extremal functions in
$C^{\infty}_{0}(\Omega,\Sigma)$.

###### Proof.

(i) Let us define

$\phi(t):=(\mathbf{c}_{\Lambda}(t)-K\mathbf{s}_{\Lambda}(t))^{-\frac{n}{p-1}}\mathbf{s}_{\Lambda}(t)^{-\frac{m-n-1}{p-1}},\quad\psi(t):=\mathbf{s}_{\Lambda}(t)^{\beta+m-n},\quad
t\in(0,\min\\{\mathfrak{r}_{\Lambda},\mathfrak{t}_{\Lambda,K}\\}).$ (6.2)

In addition, for any $\lambda,\kappa\in\mathbb{R}$ and
$\mathbf{n}\in\mathcal{V}S\Sigma$, it follows from (2.38), (2.25), (2.23) and
(2.24) that

$c_{\mathcal{V}}(\mathbf{n})\leq\mathfrak{t}_{\lambda,\kappa}\leq
2\mathfrak{r}_{\lambda}\begin{cases}<\mathfrak{r}_{\Lambda}=\min\\{\mathfrak{r}_{\Lambda},\mathfrak{t}_{\Lambda,K}\\},&\text{
if }\lambda>0;\\\ \\\
=\mathfrak{r}_{\Lambda}=\min\\{\mathfrak{r}_{\Lambda},\mathfrak{t}_{\Lambda,K}\\}=+\infty,&\text{
if }\lambda\leq 0.\end{cases}$ (6.3)

Therefore, Assumption 3/(a)(b)(e) directly follow. Moreover, Theorem 2.1 and
Proposition 2.1 provide

$\displaystyle\Delta r\leq G_{\lambda,\kappa}(r)\leq
G_{\Lambda,K}(r)=-(p-1)\frac{\phi^{\prime}(r)}{\phi(r)},\quad\forall\,x\in\mathcal{V}\mathscr{D}\Sigma\backslash\Sigma,$

which implies together with (3.2) Assumption 3/(c). Since $\phi(r),\psi(r)$
are locally bounded and $\psi^{\prime}(r)$ is locally Lipschitz in
$M\backslash\Sigma$, it is a direct consequence that

$\phi(r)^{p-1},\phi(r)^{p-1}\psi(r),\phi(r)^{p-1}|\psi^{\prime}(r)|\in
L^{1}_{\operatorname{loc}}(\Omega_{\Sigma}).$

In particular, $|\psi^{\prime}(r)|$ is always nonzero and locally bounded in
$M\backslash\Sigma$. Thus, a direct calculation yields

$\displaystyle\frac{\phi(r)^{p-1}|\psi^{\prime}(r)|}{|\nabla\log(\psi(r))|^{p}}=\frac{\phi(r)^{p-1}\psi(r)^{p}}{|\psi^{\prime}(r)|^{p-1}}\in
L^{1}_{\operatorname{loc}}(M\backslash\Sigma),$ (6.4)

which shows the validity of Assumption 1/(d). Thus, Assumption 1 holds as it
is weaker than Assumption 3. Accordingly, (6.1) follows directly by Theorem
3.1.

According to Corollary 3.3, in order to obtain the sharpness, it suffices to
prove (3.12). First, Proposition 2.1 implies for any
$t\in(0,\mathfrak{t}_{\lambda,\kappa})$ and small $\varepsilon>0$ that

$\displaystyle\left.\log\left(\left[\mathbf{c}_{\lambda}(s)-\kappa\mathbf{s}_{\lambda}(s)\right]^{n}\mathbf{s}^{m-n-1}_{\lambda}(s)\right)\right|_{s=\varepsilon}^{s=t}=\int^{t}_{{\varepsilon}}G_{\lambda,\kappa}(s){\operatorname{d}}s$
$\displaystyle\leq$
$\displaystyle\int^{t}_{{\varepsilon}}G_{\Lambda,K}(s){\operatorname{d}}s=\left.\log\left(\left[\mathbf{c}_{\Lambda}(s)-K\mathbf{s}_{\Lambda}(s)\right]^{n}\mathbf{s}^{m-n-1}_{\Lambda}(s)\right)\right|_{s=\varepsilon}^{s=t},$

which furnishes

$\left[\mathbf{c}_{\lambda}(t)-\kappa\mathbf{s}_{\lambda}(t)\right]^{n}\mathbf{s}^{m-n-1}_{\lambda}(t)\leq\left[\mathbf{c}_{\Lambda}(t)-K\mathbf{s}_{\Lambda}(t)\right]^{n}\mathbf{s}^{m-n-1}_{\Lambda}(t).$
(6.5)

Due to (6.3) and the compactness of $\Sigma$, we have

$0<c_{\mathcal{V}}(\Sigma)\leq\sup_{\mathbf{n}\in\mathcal{V}S\Sigma}c_{\mathcal{V}}(\mathbf{n})\leq\mathfrak{t}_{\lambda,\kappa}\leq
2\mathfrak{r}_{\lambda}\leq\mathfrak{r}_{\Lambda},$

where we used the assumption that $\Lambda<\lambda/4$ whenever $\lambda>0$.
Therefore, for any $s_{0}\in(0,c_{\mathcal{V}}(\Sigma))$, by (2.38) and (6.5)
we obtain

$\displaystyle\int_{{\operatorname{\mathsf{T}}_{s_{0}}}\cap\Omega}\psi(r)^{-{\varepsilon}}\phi(r)^{p-1}|\nabla(\log\psi(r))|\operatorname{dvol}_{g}$
$\displaystyle=$
$\displaystyle\int_{\Sigma}{\operatorname{dvol}}_{i^{*}g}(x)\int_{\mathcal{V}S_{x}\Sigma}d\nu_{x}(\mathbf{n})\int_{0}^{s_{0}}|\alpha|\frac{\mathbf{c}_{\Lambda}(t)\mathbf{s}_{\Lambda}(t)^{-(m-n)-\alpha\varepsilon}}{(\mathbf{c}_{\Lambda}(t)-K\mathbf{s}_{\Lambda}(t))^{n}}\det\mathcal{A}(t,\mathbf{n}){\operatorname{d}}t$
$\displaystyle\leq$
$\displaystyle|\alpha|\int_{\Sigma}{\operatorname{dvol}}_{i^{*}g}(x)\int_{\mathcal{V}S_{x}\Sigma}d\nu_{x}(\mathbf{n})\int^{s_{0}}_{0}\mathbf{s}_{\Lambda}(t)^{-\alpha\varepsilon-1}\mathbf{c}_{\Lambda}(t){\operatorname{d}}t$
$\displaystyle=$
$\displaystyle\frac{1}{\varepsilon}\operatorname{vol}(\mathbb{S}^{m-n-1})\operatorname{vol}_{i^{*}g}(\Sigma)\mathbf{s}_{\Lambda}(s_{0})^{-\alpha\varepsilon}<+\infty,$

where $\alpha:=\beta+m-n<0$. A similar computation yields

$\displaystyle\int_{{\operatorname{\mathsf{T}}_{s_{0}}}\backslash\Omega}\psi(r)^{2+{\varepsilon}}\phi(r)^{p-1}|\nabla(\log\psi(r))|\operatorname{dvol}_{g}\leq|\alpha|\int_{\Sigma}{\operatorname{dvol}}_{i^{*}g}(x)\int_{\mathcal{V}S_{x}\Sigma}d\nu_{x}(\mathbf{n})\int_{s_{0}}^{c_{\mathcal{V}}(\mathbf{n})}\mathbf{s}_{\Lambda}(t)^{\alpha(2+\varepsilon)-1}\mathbf{c}_{\Lambda}(t){\operatorname{d}}t<+\infty.$

Therefore, (3.12) holds, which implies the sharpness of (6.1). In addition,
Theorem 3.4 implies the non-existence of extremal functions for (6.1) in
$C^{\infty}_{0}(\Omega_{\Sigma})$.

(ii) We first show (6.1) for $C^{\infty}_{0}(\Omega,\Sigma)$ and its
sharpness. According to Theorem 4.4 and Remark 4.5, it suffices to show that
Assumption 3 is satisfied. Moreover, comparing Assumption 1 and Assumption 3,
we only need to prove that

$\frac{\phi(r)^{p-1}|\psi^{\prime}(r)|}{|\nabla(\log\psi(r))|^{p}}\in
L^{1}_{\operatorname{loc}}(M),\quad\text{ $\frac{\partial}{\partial
r}\log\left(\frac{\phi(r)^{p-1}|\psi^{\prime}(r)|}{|\nabla(\log\psi(r))|^{p}}\right)$
is locally bounded in $M\backslash\Sigma$}.$ (6.6)

The latter property from (6.6) easily follows, since
$\mathbf{c}_{\Lambda}(r)$, $\mathbf{s}_{\Lambda}(r)$ and
$(\mathbf{c}_{\Lambda}(r)-K\mathbf{s}_{\Lambda}(r))$ are nonzero and locally
bounded in $M\backslash\Sigma$. For the former property of (6.6), due to
(6.4), we only need to check its local integrability around $\Sigma$.
Therefore, without loss of generality, we may assume that $\Sigma$ is compact;
otherwise, we choose a compact subset of $\Sigma$. Now, the first limit of
(2.34) together with the compactness of $\mathcal{V}S\Sigma$ provides a small
$\epsilon\in(0,c_{\mathcal{V}}(\Sigma))$ such that

$\det\mathcal{A}(t,\mathbf{n})\leq
2t^{m-n-1},\quad\forall\,(t,\mathbf{n})\in(0,\epsilon)\times\mathcal{V}S\Sigma.$
(6.7)

Since $\mathbf{s}_{\Lambda}(t)\sim t$ as $t\to 0^{+}$, by choosing eventually
a smaller $\epsilon$, we may assume

$\frac{\phi(t)^{p-1}|\psi^{\prime}(t)|}{|(\log\psi(t))^{\prime}|^{p}}=\frac{\mathbf{c}_{\Lambda}(t)^{1-p}\mathbf{s}_{\Lambda}(t)^{p+\beta}}{(\mathbf{c}_{\Lambda}(t)-K\mathbf{s}_{\Lambda}(t))^{n}}\leq
2t^{p+\beta},\quad\forall\,t\in(0,\epsilon).$ (6.8)

Due to $p+\beta>-(m-n)$, the volume expression (2.35) combined with (6.7) and
(6.8) yields

$\displaystyle
0\leq\int_{\operatorname{\mathsf{T}}_{\epsilon}}\frac{\phi(r)^{p-1}|\psi^{\prime}(r)|}{|\nabla(\log\psi(r))|^{p}}\operatorname{dvol}_{g}\leq
4\operatorname{vol}(\mathbb{S}^{m-n-1})\operatorname{vol}_{i^{*}g}(\Sigma)\int^{\epsilon}_{0}t^{p+\beta+m-n-1}{\operatorname{d}}t<+\infty,$

which together with (6.4) implies
$\frac{\phi(r)^{p-1}|\psi^{\prime}(r)|}{|\nabla(\log\psi(r))|^{p}}\in
L^{1}_{\operatorname{loc}}(M)$.

It remains to show the nonexistence of extremal functions. Owing to the
compactness of $\Sigma$ and (2.34), choose a small
$\delta\in(0,c_{\mathcal{V}}(\Sigma))$ such that

$\frac{\det\mathcal{A}(t,\mathbf{n})}{(\mathbf{c}_{\Lambda}(t)-K\mathbf{s}_{\Lambda}(t))^{n}}\geq\frac{1}{2}\mathbf{s}_{\Lambda}(t)^{m-n-1},\quad\text{
for }(t,\mathbf{n})\in[0,\delta]\times\mathcal{V}S\Sigma.$

Thus, for any $\varepsilon>0$ we have

$\displaystyle\int^{\delta}_{\varepsilon}\phi(t)^{p-1}|(\log\psi(t))^{\prime}|\det\mathcal{A}(t,\mathbf{n}){\operatorname{d}}t\geq\frac{|\alpha|}{2}\int^{\delta}_{\varepsilon}\mathbf{s}_{\Lambda}(t)^{-1}\mathbf{c}_{\Lambda}(t){\operatorname{d}}t\rightarrow+\infty,\quad\text{
as }\varepsilon\rightarrow 0^{+},$

which implies Condition (iii) in Remark 4.5 and the nonexistence of extremal
functions. ∎

######  Remark 6.2.

It should be noticed that (6.3) implies
$M\subset\overline{\operatorname{\mathsf{T}}_{\mathfrak{t}_{\lambda,\kappa}}}$.
In fact, for every $x\in\mathcal{V}\mathscr{C}\Sigma\backslash\Sigma$, there
exists a unique minimal normal geodesic from $\Sigma$ to $x$ with
$\dot{\gamma}(0)=\mathbf{n}\in\mathcal{V}S\Sigma$, see Lemma 2.1. Clearly, the
length of $\gamma$ is not larger than $c_{\mathcal{V}}(\mathbf{n})$, which
combined with (6.3) implies
$M\subset\overline{\operatorname{\mathsf{T}}_{\mathfrak{t}_{\lambda,\kappa}}}$.
Obviously, this result can be considered as a submanifold version of the
Bonnet-Myers theorem.

######  Remark 6.3.

In the assumption of Theorem 6.1 we require that $\Lambda<\lambda/4$ if
$\lambda>0$; however, this condition can be relaxed to “$\Lambda\leq\lambda/4$
if $\lambda>0$” whenever $n>0$. In fact, such an assumption is needed to
establish (6.3), i.e.,

$c_{\mathcal{V}}(\mathbf{n})<\min\\{\mathfrak{r}_{\Lambda},\mathfrak{t}_{\Lambda,K}\\},$
(6.9)

which guarantees the fact that the functions $\phi,\psi$ defined in (6.2)
satisfy Assumption 3. When $n>0$ and $\lambda>0$, in view of (2.22) and
(2.25), we have

$c_{\mathcal{V}}(\mathbf{n})\leq\mathfrak{t}_{\lambda,\kappa}=\mathfrak{c}_{\lambda,\kappa}<2\mathfrak{r}_{\lambda}\leq\mathfrak{r}_{\Lambda}=\min\\{\mathfrak{r}_{\Lambda},\mathfrak{t}_{\Lambda,K}\\},$

i.e., (6.9) remains valid, and the same argument as above can be followed.
Furthermore, if $n>0$, $\kappa=0$ and $\lambda>0$, a similar argument applies
whenever $\Lambda<\lambda$.

######  Remark 6.4.

Although one could expect that (6.1) can be extended to
$C^{\infty}_{0}(\Omega)$ (i.e., Theorem 4.1), it is easy to see that
Assumption 2/(c) fails. Moreover, if $\Omega=M$ is a compact manifold,
constant functions belong to $C^{\infty}_{0}(\Omega)$, for which (6.1) clearly
fails.

An interesting application of Theorem 6.1 to spheres is as follows, providing
three different kinds of Hardy inequalities.

######  Example 6.1.

Let $\operatorname{\mathbb{M}}_{\lambda}=\mathbb{S}^{m}$ and let $\Sigma$ be
the great sphere $\mathbb{S}^{m-1}\subset\mathbb{S}^{m}$ (i.e., $n=m-1$). Let
$\Omega$ be an open hemisphere with $\partial\Omega=\Sigma$. Obviously,
$(M,\Sigma,\Omega)$ satisfies the (non-optimal) lower curvature bound
condition $(\lambda,\kappa)=(1,0)$.

Given $p>1$ and $\beta<-1$, by choosing different pairs of
$(\Lambda,0)\leq(1,0)$ (i.e., $\Lambda=1/4,0,-1$), Theorem 6.1 together with
Remark 6.3 implies for every $u\in C^{\infty}_{0}(\Omega)$ that

$\displaystyle\int_{\Omega}|\nabla u|^{p}r^{\beta+p}\operatorname{dvol}_{g}$
$\displaystyle\geq\left|\frac{\beta+1}{p}\right|^{p}\int_{\Omega}|u|^{p}r^{\beta}\operatorname{dvol}_{g},$
$\displaystyle\int_{\Omega}|{\nabla}u|^{p}\cos\left(\frac{r}{2}\right)^{2-m-p}\sin\left(\frac{r}{2}\right)^{p+\beta}\operatorname{dvol}_{g}$
$\displaystyle\geq\Big{|}{\beta+1\over{2p}}\Big{|}^{p}\int_{\Omega}|u|^{p}\cos\left(\frac{r}{2}\right)^{2-m}\sin\left(\frac{r}{2}\right)^{\beta}\operatorname{dvol}_{g},$
$\displaystyle\int_{\Omega}|{\nabla}u|^{p}\cosh(r)^{2-m-p}\sinh(r)^{p+\beta}\operatorname{dvol}_{g}$
$\displaystyle\geq\Big{|}{\beta+1\over
p}\Big{|}^{p}\int_{\Omega}|u|^{p}\cosh(r)^{2-m}\sinh(r)^{\beta}\operatorname{dvol}_{g}.$

In particular, all of them are sharp and no extremal functions exist in
$C^{\infty}_{0}(\Omega)$.

######  Remark 6.5.

A similar observation can be made as in Remark 5.3. Indeed, the domain
$\Omega$ in Theorem 6.1 is arbitrary but the sharpness depends on
$c_{\mathcal{V}}(\Sigma)>0$. In addition, if

$\sup_{x\in\Omega}r(x)<+\infty,$

holds instead of $c_{\mathcal{V}}(\Sigma)>0$, one can prove formally the same
improved sharp Hardy inequality as (5.5) on the spaces
$C^{\infty}_{0}(\Omega_{\Sigma})$ and $C^{\infty}_{0}(\Omega,\Sigma)$ instead
of $C^{\infty}_{0}(\Omega)$, which is another natural extension of the result
of Barbatis et al. [1] (see also Chen et al. [10]).

###  6.2. Hardy inequalities involving logarithmic double-curvature weights

The pair $(\phi,\psi)$ is defined as follows. Given
$\Lambda,K,s_{1},s_{2}\in\mathbb{R}$ and
$D\in(0,\min\\{\mathfrak{r}_{\Lambda},\mathfrak{t}_{\Lambda,K}\\})$,
$L\in(0,D]$, let

$\phi(t):=(\mathbf{c}_{\Lambda}(t)-K\mathbf{s}_{\Lambda}(t))^{-{n\over
p-1}}\mathbf{s}_{\Lambda}(t)^{-{m-n-1\over
p-1}},\quad\psi(t):=\psi(t,s_{1},s_{2}):=\int^{L}_{t}\left[\log\left({\mathbf{s}_{\Lambda}(D)\over\mathbf{s}_{\Lambda}(\tau)}\right)\right]^{s_{1}}\mathbf{s}_{\Lambda}(\tau)^{s_{2}}\mathbf{c}_{\Lambda}(\tau){\operatorname{d}}\tau.$
(6.10)

######  Lemma 6.1.

The function $\psi$ in (6.10) is well-defined and strictly decreasing in
$(0,L]$ either when $L<D$ and $s_{1},s_{2}\in\mathbb{R}$, or when $L=D$ and
$s_{1}>-1$ and $s_{2}\in\mathbb{R}$. Moreover, given $L\in(0,D]$, if $s>0$,
$s_{1}>-1$ and $s_{2}\leq-1$ satisfy

$s_{1}+1\geq
s+(s_{2}+1)\log{\mathbf{s}_{\Lambda}(D)\over\mathbf{s}_{\Lambda}(L)},$

then

$\psi(t)\leq
s^{-1}\left[\log\left({\mathbf{s}_{\Lambda}(D)\over\mathbf{s}_{\Lambda}(t)}\right)\right]^{s_{1}+1}\mathbf{s}_{\Lambda}(t)^{s_{2}+1}=s^{-1}\left[\log\left({\mathbf{s}_{\Lambda}(D)\over\mathbf{s}_{\Lambda}(t)}\right)\right]{\mathbf{s}_{\Lambda}(t)\over\mathbf{c}_{\Lambda}(t)}|\psi^{\prime}(t)|,\
\ t\in(0,L],$ (6.11)

with equality if $s_{2}=-1$, $s=s_{1}+1$ and $L=D$.

###### Proof.

The well-definiteness of $\psi$ is based on Proposition 5.1; moreover, it is a
decreasing positive function, as

$\psi^{\prime}(t)=-\left[\log\left({\mathbf{s}_{\Lambda}(D)\over\mathbf{s}_{\Lambda}(t)}\right)\right]^{s_{1}}\mathbf{s}_{\Lambda}(t)^{s_{2}}\mathbf{c}_{\Lambda}(t)<0,\quad
t\in(0,L).$ (6.12)

Let

$F(t):=\psi(t)-s^{-1}\left[\log\left({\mathbf{s}_{\Lambda}(D)\over\mathbf{s}_{\Lambda}(t)}\right)\right]^{s_{1}+1}\mathbf{s}_{\Lambda}(t)^{s_{2}+1}.$

Since $\mathbf{s}_{\Lambda}$ is increasing in $[0,D]$, a direct calculation
together with (6.12) yields

$F^{\prime}(t)=s^{-1}\left[\log\left({\mathbf{s}_{\Lambda}(D)\over\mathbf{s}_{\Lambda}(t)}\right)\right]^{s_{1}}\mathbf{s}_{\Lambda}(t)^{s_{2}}\mathbf{c}_{\Lambda}(t)\left[s_{1}+1-s-(s_{2}+1)\log\left({\mathbf{s}_{\Lambda}(D)\over\mathbf{s}_{\Lambda}(t)}\right)\right]\geq
0.$

Clearly, $\psi(L)=0$ and $F(L)\leq 0$. Thus, $F(t)\leq F(L)\leq 0$ for any
$t\in(0,L]$, which is exactly (6.11). In particular, if $s_{2}=-1$,
$s=s_{1}+1$ and $L=D$, then $F^{\prime}(t)=0$ and $F(t)=F(L)=F(D)=0$. ∎

In addition, elementary computations imply the fact that
$\Lambda\mapsto\frac{\mathbf{s}_{\Lambda}(D)}{\mathbf{s}_{\Lambda}(L)}$ is
decreasing in $\Lambda\in(-\infty,\lambda)$ whenever $L\in(0,D]$; therefore,
we have the following monotonicity property which plays a crucial role in the
sequel.

######  Lemma 6.2.

Given $\lambda\in\mathbb{R}$, $D\in(0,\mathfrak{r}_{\lambda})$ and
$L\in(0,D]$, we have

$\frac{\mathbf{s}_{\lambda}(D)}{\mathbf{s}_{\lambda}(L)}\leq\frac{\mathbf{s}_{\Lambda}(D)}{\mathbf{s}_{\Lambda}(L)},\quad\forall\,\Lambda\leq\lambda.$

We now derive the following Hardy inequality with logarithmic weights.

######  Theorem 6.2.

Let $(M,g)$ be an $m$-dimensional complete Riemannian manifold,
$i:\Sigma\hookrightarrow M$ be an $n$-dimensional closed submanifold and
$\Omega$ be a non-empty domain in $M$ satisfying (3.1) and
$\sup_{x\in\Omega}r(x)<+\infty$. Suppose $(M,\Sigma,\Omega)$ satisfies the
lower curvature bound condition $(\lambda,\kappa)$. If
$D\in(0,\min\\{\mathfrak{r}_{\lambda},\mathfrak{t}_{\lambda,\kappa}\\})$ and
$s,\alpha,\beta\in\mathbb{R}$ are such that

$\sup_{x\in\Omega}r(x)\leq D,\quad s>0,\quad\beta>-1,\quad\alpha-p+m-n\leq
0,\quad\beta+1\geq
s+(\alpha-p+m-n)\log\left({\mathbf{s}_{\lambda}(D)\over\sup_{x\in\Omega}\mathbf{s}_{\lambda}(r(x))}\right),$

then for any $(\Lambda,K)\leq(\lambda,\min\\{\kappa,0\\})$ with
$\Lambda<\lambda/4$ if $\lambda>0$, we have

$\displaystyle\int_{\Omega_{\Sigma}}|{\nabla}u|^{p}\left[\log\left({\mathbf{s}_{\Lambda}(D)\over\mathbf{s}_{\Lambda}(r)}\right)\right]^{p+\beta}\frac{\mathbf{c}_{\Lambda}(r)^{1-p}\mathbf{s}_{\Lambda}(r)^{\alpha}}{(\mathbf{c}_{\Lambda}(r)-K\mathbf{s}_{\Lambda}(r))^{n}}\operatorname{dvol}_{g}$
$\displaystyle\geq$ $\displaystyle\left({s\over
p}\right)^{p}\int_{\Omega_{\Sigma}}|u|^{p}\left[\log\left({\mathbf{s}_{\Lambda}(D)\over\mathbf{s}_{\Lambda}(r)}\right)\right]^{\beta}\frac{\mathbf{c}_{\Lambda}(r)\mathbf{s}_{\Lambda}(r)^{\alpha-p}}{(\mathbf{c}_{\Lambda}(r)-K\mathbf{s}_{\Lambda}(r))^{n}}\operatorname{dvol}_{g},\quad\forall\,u\in
C^{\infty}_{0}(\Omega_{\Sigma}).$ (6.13)

Moreover, if either $\Omega=M$ or $\Sigma=\partial\Omega$, and

$\text{$\Sigma$ is compact},\quad\alpha=p-(m-n),\quad
s=\beta+1,\quad\sup_{x\in\Omega}r(x)=D,$ (6.14)

then (6.13) is sharp.

###### Proof.

In view of the assumption, choose

$L:=\sup_{x\in\Omega}r(x)\leq
D<\min\\{\mathfrak{r}_{\lambda},\mathfrak{t}_{\lambda,\kappa}\\},\quad
s_{1}:=\beta>-1,\quad s_{2}:=\alpha-p+(m-n)-1\leq-1.$ (6.15)

Due to Lemma 6.1, the pair of functions $(\phi,\psi)$ constructed by (6.10) is
well-defined on $(0,L]$. Note that (6.3) remains valid and hence, Assumption
1/(a)(b)(e) hold. Moreover, according to Theorem 2.1 and Proposition 2.1, we
have

$\displaystyle\Delta r\leq G_{\lambda,\kappa}(r)\leq
G_{\Lambda,K}(r)=-(p-1)\frac{\phi^{\prime}(r)}{\phi(r)},\quad\forall\,x\in\mathcal{V}\mathscr{D}\Sigma,$

which combined with (3.2) and (6.12) yields Assumption 1/(c). Now we are going
to show that Assumption 1/(d) holds. Note that $\phi(r),\psi(r)$ are locally
bounded and the latter is locally Lipschitz in $\Omega_{\Sigma}$. Thus, one
has $\phi(r)^{p-1},\phi(r)^{p-1}\psi(r),\phi(r)^{p-1}|\psi^{\prime}(r)|\in
L^{1}_{\operatorname{loc}}(\Omega_{\Sigma}).$ Moreover, by the assumption and
Lemma 6.2 we obtain

$\displaystyle s_{1}+1$ $\displaystyle=\beta+1\geq
s+(\alpha-p+m-n)\log\left({\mathbf{s}_{\lambda}(D)\over\sup_{x\in\Omega}\mathbf{s}_{\lambda}(r(x))}\right)$
$\displaystyle=s+(s_{2}+1)\log\left({\mathbf{s}_{\lambda}(D)\over\mathbf{s}_{\lambda}(L)}\right)\geq
s+(s_{2}+1)\log\left({\mathbf{s}_{\Lambda}(D)\over\mathbf{s}_{\Lambda}(L)}\right),$

which combined with Lemma 6.1 yields (6.11). Thus, we have

$\displaystyle\frac{\phi(r)^{p-1}|\psi^{\prime}(r)|}{|\nabla(\log\psi(r))|^{p}}=\,$
$\displaystyle(\mathbf{c}_{\Lambda}(r)-K\mathbf{s}_{\Lambda}(r))^{-n}\mathbf{s}_{\Lambda}(r)^{-(m-n-1)}\frac{\psi(r)^{p}}{|\psi^{\prime}(r)|^{p-1}}$
$\displaystyle\leq$ $\displaystyle
s^{-p}\left[\log\left({\mathbf{s}_{\Lambda}(D)\over\mathbf{s}_{\Lambda}(r)}\right)\right]^{p+\beta}\frac{\mathbf{c}_{\Lambda}(r)^{1-p}\mathbf{s}_{\Lambda}(r)^{\alpha}}{(\mathbf{c}_{\Lambda}(r)-K\mathbf{s}_{\Lambda}(r))^{n}},$
(6.16)

where the latter is obviously locally bounded in $\Omega_{\Sigma}$. Thus,
Assumption 1/(d) holds. From above, Assumption 1 is satisfied and hence,
(6.13) follows by (3.5) and (6.16).

Assume either $\Omega=M$ or $\Sigma=\partial\Omega$; we show the sharpness of
(6.13) under (6.14). Due to (6.15), we have

$s_{1}=\beta>-1,\quad s_{2}=-1,\quad s=\beta+1>0,\quad
L=\sup_{x\in\Omega}r(x)=D,$

which combined with Lemma 6.1 yields

$\psi(t)=(\beta+1)^{-1}\left[\log\left({\mathbf{s}_{\Lambda}(D)\over\mathbf{s}_{\Lambda}(t)}\right)\right]^{\beta+1},\quad\forall\,t\in(0,D],$
(6.17)

which furnishes $\psi(0^{+})=+\infty$ and $\psi^{\prime}(t)<0$. Since now
$\Omega$ is a bounded domain, in view of Corollary 3.3, it remains to show
(3.12). Indeed, the compactness of $\Sigma$ implies
$c_{\mathcal{V}}(\Sigma)>0$. Choose
$s_{0}\in(0,\min\\{c_{\mathcal{V}}(\Sigma),D\\})$. Thus, (6.17) together with
(2.38) and (6.5) (i.e., the integral version of Proposition 2.1) yields

$\displaystyle\int_{\operatorname{\mathsf{T}}_{s_{0}}\cap\,\Omega}\psi(r)^{-(1+{\varepsilon})}\phi(r)^{p-1}|\psi^{\prime}(r)|\operatorname{dvol}_{g}$
$\displaystyle\leq$
$\displaystyle(\beta+1)^{1+{\varepsilon}}\int_{r<s_{0}}\left[\log\left({\mathbf{s}_{\Lambda}(D)\over\mathbf{s}_{\Lambda}(r)}\right)\right]^{-(1+{\varepsilon})(\beta+1)+\beta}\frac{\mathbf{c}_{\Lambda}(r)\mathbf{s}_{\Lambda}(r)^{-(m-n)}}{(\mathbf{c}_{\Lambda}(r)-K\mathbf{s}_{\Lambda}(r))^{n}}\operatorname{dvol}_{g}$
$\displaystyle\leq$
$\displaystyle(\beta+1)^{1+{\varepsilon}}\operatorname{vol}(\mathbb{S}^{m-n-1})\operatorname{vol}_{i^{*}g}(\Sigma)\int_{0}^{s_{0}}\left[\log\left({\mathbf{s}_{\Lambda}(D)\over\mathbf{s}_{\Lambda}(r)}\right)\right]^{-(1+{\varepsilon})(\beta+1)+\beta}\mathbf{s}_{\Lambda}(r)^{-1}\mathbf{c}_{\Lambda}(r){\operatorname{d}}r<+\infty,$

where the last inequality follows by $-(1+{\varepsilon})(\beta+1)+\beta<-1$
and Proposition 5.1. In the same way, by $(1+{\varepsilon})(\beta+1)+\beta>-1$
and Lemma 6.1 we have

$\displaystyle\int_{\Omega\backslash{\operatorname{\mathsf{T}}_{s_{0}}}}\psi(r)^{1+{\varepsilon}}\phi(r)^{p-1}|\psi^{\prime}(r)|\operatorname{dvol}_{g}\leq(\beta+1)^{-(1+{\varepsilon})}\int_{r>s_{0}}\left[\log\left({\mathbf{s}_{\Lambda}(D)\over\mathbf{s}_{\Lambda}(r)}\right)\right]^{(1+{\varepsilon})(\beta+1)+\beta}\frac{\mathbf{c}_{\Lambda}(r)\mathbf{s}_{\Lambda}(r)^{-(m-n)}}{(\mathbf{c}_{\Lambda}(r)-K\mathbf{s}_{\Lambda}(r))^{n}}\operatorname{dvol}_{g}$
$\displaystyle\leq(\beta+1)^{-(1+{\varepsilon})}\operatorname{vol}(\mathbb{S}^{m-n-1})\operatorname{vol}_{i^{*}g}(\Sigma)\int_{s_{0}}^{D}\left[\log\left({\mathbf{s}_{\Lambda}(D)\over\mathbf{s}_{\Lambda}(r)}\right)\right]^{(1+{\varepsilon})(\beta+1)+\beta}\mathbf{s}_{\Lambda}(r)^{-1}\mathbf{c}_{\Lambda}(r){\operatorname{d}}r<+\infty.$

Thus, (3.12) and the sharpness follow. ∎

######  Remark 6.6.

On account of Lemma 3.2 and Corollary 3.2, by the same calculation one can
show that (6.13) is also sharp when $\Sigma$ is compact with
$\Sigma\subset\Omega$, $\partial\Omega=\\{r=D\\}$, $\alpha=p-(m-n)$ and
$s=\beta+1$.

##  Appendix A Monotonicity of $G_{\lambda,\kappa}$

In this section, we provide the proof of Proposition 2.1 that states some
important monotonicity properties of the function $G_{\lambda,\kappa}$ defined
by (2.26), i.e.,

$\displaystyle
G_{\lambda,\kappa}(t)=\left\\{\begin{array}[]{lll}\sqrt{\lambda}\left[-n{\sqrt{\lambda}+\kappa\cot(\sqrt{\lambda}t)\over\sqrt{\lambda}\cot(\sqrt{\lambda}t)-\kappa}+(m-n-1)\cot(\sqrt{\lambda}t)\right],&&\text{
if }\lambda>0,\\\ \\\ -{n\kappa\over 1-\kappa t}+{m-n-1\over t},&&\text{ if
}\lambda=0,\\\ \\\
\sqrt{-\lambda}\left[-n{-\sqrt{-\lambda}+\kappa\coth(\sqrt{-\lambda}t)\over\sqrt{-\lambda}\coth(\sqrt{-\lambda}t)-\kappa}+(m-n-1)\coth(\sqrt{-\lambda}t)\right],&&\text{
if }\lambda<0.\end{array}\right.$ (A.6)

Although the proof is elementary, for the sake of completeness, we provide its
proof.

###### Proof of Proposition 2.1.

The monotonicity in $\kappa$ follows by a similar calculation as in the proof
of Theorem 2.1 (see Step 1). Hence, we focus on the monotonicity in $\lambda$.

Case 1: $\lambda>0$. For convenience, set $z:=\sqrt{\lambda}$. Thus, in view
of (A.6), we study the monotonicity of

$z\mapsto z\left[-n{z+\kappa\cot(zt)\over
z\cot(zt)-\kappa}+(m-n-1)\cot(zt)\right]=:\mathcal{G}_{\kappa}(z,t).$

A direct calculation yields

$\displaystyle\frac{\partial\mathcal{G}_{\kappa}}{\partial z}(z,t)=$
$\displaystyle-n{z^{2}[\cot(zt)+zt\text{csc}^{2}(zt)]+\kappa^{2}[zt\text{csc}^{2}(zt)-\cot(zt)]-2\kappa
z\over(z\cot(zt)-\kappa)^{2}}$
$\displaystyle+(m-n-1)[\cot(zt)-zt\csc^{2}(zt)].$ (A.7)

On the other hand, we have the identities

$\displaystyle
A(x):=x\csc^{2}(x)+\cot(x)=\frac{x+\sin(x)\cos(x)}{\sin^{2}(x)}=\frac{x+\frac{1}{2}\sin(2x)}{\sin^{2}(x)}>0,\quad\forall\,x>0.$
$\displaystyle
B(x):=x\csc^{2}(x)-\cot(x)=\frac{x-\sin(x)\cos(x)}{\sin^{2}(x)}=\frac{x-\frac{1}{2}\sin(2x)}{\sin^{2}(x)}>0,\quad\forall\,x>0.$
(A.8)

Since $\sin^{2}x\leq x^{2}$, we have $x^{2}\csc^{2}x-1\geq 0$ and hence,

$\displaystyle z^{2}A(x)+\kappa^{2}B(x)$ $\displaystyle\geq
2z|\kappa|A(x)B(x)=2z|\kappa|\left(x^{2}\csc^{4}x-\cot^{2}x\right)$
$\displaystyle=2z|\kappa|\left(\csc^{2}x(x^{2}\csc^{2}x-1)+1\right)\geq
2z|\kappa|\geq 2\kappa z,$

which together with (A.7) and (A.8) implies
$\frac{\partial\mathcal{G}_{\kappa}}{\partial z}(z,t)<0$, i.e.,
$G_{\lambda,\kappa}(t)$ is decreasing in $\lambda$. Furthermore, by using
$\lim_{\lambda\rightarrow 0^{+}}\sqrt{\lambda}\cot(\sqrt{\lambda}t)=1/t$, we
obtain

$\displaystyle\lim_{\lambda\rightarrow
0^{+}}G_{\lambda,\kappa}(t)=-{n\kappa\over 1-\kappa t}+{m-n-1\over
t}=G_{0,\kappa}(t).$

Case 2: $\lambda<0$. By setting $w:=\sqrt{-\lambda}$ in (A.6), we need to
study the monotonicity of

$w\mapsto w\left[-n{-w+\kappa\coth(wt)\over
w\coth(wt)-\kappa}+(m-n-1)\coth(wt)\right]=:\mathcal{G}_{\kappa}(w,t).$

Similarly, we have

$\displaystyle\frac{\partial\mathcal{G}_{\kappa}}{\partial w}(w,t)=$
$\displaystyle
n{w^{2}[wt\text{csch}^{2}(wt)+\coth(wt)]+\kappa^{2}[\coth(wt)-wt\text{csch}^{2}(wt)]-2\kappa
w\over(w\coth(wt)-\kappa)^{2}}$
$\displaystyle+(m-n-1)[\coth(wt)-wt\text{csch}^{2}(wt)].$ (A.9)

Let

$\displaystyle A(x):=\coth(x)+x\text{csch}^{2}(x)>0,\quad
B(x):=\coth(x)-x\text{csch}^{2}(x)>0,\quad\forall\,x>0.$ (A.10)

Since $1+\text{csch}^{2}x=\coth^{2}x$ and $\sinh^{2}x\geq x^{2}$ for any
$x\geq 0$, we have

$\displaystyle w^{2}A(x)+\kappa^{2}B(x)$ $\displaystyle\geq
2w|\kappa|A(x)B(x)=2w|\kappa|(\coth^{2}x-x^{2}\text{csch}^{4}x)$
$\displaystyle=2w|\kappa|(\text{csch}^{2}x(1-x^{2}\text{csch}^{2}x)+1)\geq
2w|\kappa|\geq 2w\kappa.$

which together with (A.9) and (A.10) yields
$\frac{\partial\mathcal{G}_{\kappa}}{\partial w}(w,t)>0$, which concludes the
proof. ∎

##  Appendix B Weighted Sobolev spaces

The results in this section – which are of interest of their own right – are
natural extensions of the ones from Chen et al. [10, Appendix A], where we
establish some properties of the weighted Sobolev spaces defined by Definition
4.1 under Assumption 3. Throughout the section, we assume the validity of
Assumption 3.

A standard argument together with Remark 4.4 yields the following result.

######  Lemma B.1.

Let
${\operatorname{d\mu}}:=\frac{\phi(r)^{p-1}|\psi^{\prime}(r)|}{|\nabla(\log\psi(r))|^{p}}{\operatorname{dvol}}_{g}$.
Then ${\operatorname{d\mu}}$ is a $\sigma$-finite measure on $M$. Moreover,
$\mu(E)=0$ if and only if $\operatorname{vol}_{g}(E)=0$ for any Borel set
$E\subset\Sigma\cup\Omega$. In particular, $\mu(\Sigma)=0$.

In the sequel, we prefer to use ${\operatorname{d\mu}}$ rather than
$\operatorname{dvol}_{g}$ to investigate weighted Sobolev spaces. Let
$U\subset M$ be an open set. For any $s\in(1,\infty)$, we define
$L^{s}({U},\phi,\psi)$ (resp., $L^{s}(T{U},\phi,\psi)$) as the completion of
$C_{0}({U})$ (resp., $\Gamma_{0}(T{U})$, i.e., the space of continuous tangent
vector fields with compact support in ${U}$) under the norm

$\displaystyle[u]_{s,\mu}:=\left(\int_{{{U}}}|u|^{s}{\operatorname{d\mu}}\right)^{\frac{1}{s}},\
\ \ \left(\text{ resp.,
}[X]_{s,\mu}:=\left(\int_{{{U}}}|X|^{s}{\operatorname{d\mu}}\right)^{\frac{1}{s}}\right).$

The standard theory of Lebesgue spaces yields the following result.

######  Theorem B.1.

Both $L^{s}({U},\phi,\psi)$ and $L^{s}(T{U},\phi,\psi)$ are reflexive. Hence,
if a sequence $\\{u_{i}\\}$ $($resp., $\\{X_{i}\\}$$)$ is bounded in
$L^{s}({U},\phi,\psi)$ $($resp., $L^{s}(T{U},\phi,\psi)$$),$ then there exists
a subsequence $\\{u_{i_{k}}\\}$ $($resp., $\\{X_{i_{k}}\\}$$)$ and $u\in
L^{s}({U},\phi,\psi)$ $($resp., $X\in L^{s}(T{U},\phi,\psi)$$)$ such that
$u_{i_{k}}\rightarrow u$ weakly in $L^{s}({U},\phi,\psi)$ $($resp.,
$X_{i_{k}}\rightarrow X$ weakly in $L^{s}(T{U},\phi,\psi)$$)$ as
$k\rightarrow\infty$.

Let $W^{1,p}(U,\phi,\psi)$ be as in Definition 4.1; thus, for any $u\in
C^{\infty}({U})$, we have $\|u\|^{p}_{p}=[u]^{p}_{p,\mu}+[\nabla
u]^{p}_{p,\mu}$.

######  Remark B.1.

It is easy to check that the weak gradient in the sense of
$W^{1,p}({U},\phi,\psi)$ satisfies the following properties:

  1. (a)

$\nabla(\lambda u+\zeta v)=\lambda\nabla u+\zeta\nabla v$ for any
$\lambda,\zeta\in\mathbb{R}$ and $u,v\in W^{1,p}({U},\phi,\psi)$;

  2. (b)

For any smooth vector field $X$ with compact support in ${U}_{\Sigma}$, one
has

$\int_{{{U}}}g(\nabla
u,X){\operatorname{d\mu}}=-\int_{{{U}}}u\operatorname{div}_{\mu}X{\operatorname{d\mu}},\quad\forall\,u\in
W^{1,p}({U},\phi,\psi),$ (B.1)

where
$\operatorname{div}_{\mu}(X){\operatorname{d}}\mu:={\operatorname{d}}(X\rfloor{\operatorname{d}}\mu)$,
i.e.,

$\operatorname{div}_{\mu}(X)=\operatorname{div}(X)+\langle X,\nabla
r\rangle\frac{\partial}{\partial
r}\log\left(\frac{\phi(r)^{p-1}|\psi^{\prime}(r)|}{|\nabla(\log\psi(r))|^{p}}\right).$
(B.2)

In particular, $|\operatorname{div}_{\mu}(X)|<+\infty$ follows by Assumption
3/(f) and the compactness of $\operatorname{supp}X$.

In the sequel, $U$ is either $M$ or $\Omega_{\Sigma}$. The same argument as in
Corollary 3.2 yields the following result.

######  Lemma B.2.

For any globally Lipschitz function $u$ with compact support in $U$, we have
$u\in W_{0}^{1,p}({U},\phi,\psi)$.

Due to Lemma B.2, the following result can be proved in the same way as in
Hebey [19, Theorem 2.7].

######  Theorem B.2.

$W^{1,p}(M,\phi,\psi)=W_{0}^{1,p}(M,\phi,\psi)$.

Theorem B.2 combined with the same proof of Zhao [34, Lemma B.4] produces the
following result.

######  Corollary B.3.

If $u\in W^{1,p}(M,\phi,\psi)$ with compact support in ${\Omega_{\Sigma}}$,
then $u|_{\Omega_{\Sigma}}\in W_{0}^{1,p}(\Omega_{\Sigma},\phi,\psi)$.

In order to show the reflexivity of $W^{1,p}(M,\phi,\psi)$, we recall Mazur’s
lemma (cf. Renardy and Rogers [31, Lemma 10.19]).

######  Lemma B.3 (Mazur’s lemma).

Assume that $X$ is a Banach space and that $x_{i}\rightarrow x$ weakly in $X$
as $i\rightarrow\infty$. Then there exists a sequence of convex combinations

$\tilde{x}_{i}=\sum_{j=i}^{m_{i}}a_{i,j}x_{j},\ a_{i,j}\geq 0,\
\sum_{j=i}^{m_{i}}a_{i,j}=1,$

such that $\tilde{x}_{i}\rightarrow x$ strongly in $X$ as
$i\rightarrow\infty$.

######  Theorem B.4.

$W^{1,p}(M,\phi,\psi)$ is reflexive, i.e., given a bounded sequence
$\\{u_{i}\\}$ in $W^{1,p}(M,\phi,\psi)$, there exists a subsequence
$\\{u_{i_{k}}\\}$ and $u\in W^{1,p}(M,\phi,\psi)$ such that
$u_{i_{k}}\rightarrow u$ weakly in $L^{p}(M,\phi,\psi)$ and $\nabla
u_{i_{k}}\rightarrow\nabla u$ weakly in $L^{p}(TM,\phi,\psi)$ as
$k\rightarrow\infty$.

###### Proof.

Since $\\{u_{i}\\}$ is a bounded sequence in $W^{1,p}(M,\phi,\psi)$, Theorem
B.1 yields a subsequence $\\{u_{i_{k}}\\}$ and $u\in L^{p}(M,\phi,\psi)$ and a
vector field $X\in L^{p}(TM,\phi,\psi)$ such that $u_{i_{k}}\rightarrow u$
weakly in $L^{p}(M,\phi,\psi)$ and $\nabla u_{i_{k}}\rightarrow X$ weakly in
$L^{p}(TM,\phi,\psi)$ as $k\rightarrow\infty$. Consider the Banach space
$L^{p}(M,\phi,\psi)\times L^{p}(TM,\phi,\psi)$ endowed with the product
metric. Obviously,

$(u_{i_{k}},\nabla u_{i_{k}})\rightarrow(u,X)\text{ weakly in
}L^{p}(M,\phi,\psi)\times L^{p}(TM,\phi,\psi).$

Thus, by Lemma B.3 we have a sequence of convex combinations such that

$\sum_{k=l}^{m_{l}}a_{l,k}(u_{i_{k}},\nabla u_{i_{k}})\rightarrow(u,X)\text{
strongly in }L^{p}(M,\phi,\psi)\times L^{p}(TM,\phi,\psi),\text{ as
}l\rightarrow\infty.$

Set $v_{l}:=\sum_{k=l}^{m_{l}}a_{l,k}u_{i_{k}}\in W^{1,p}(M,\phi,\psi)$. Then
both $v_{l}\rightarrow u$ strongly and $\nabla v_{l}\rightarrow X$ strongly in
the corresponding $[\cdot]_{p,\mu}$-norms. Hence, $(v_{l})$ is a Cauchy
sequence in $W^{1,p}(M,\phi,\psi)$, which implies $\nabla u=X$ and $u\in
W^{1,p}(M,\phi,\psi)$. ∎

Theorem B.4 together with the dominated convergence theorem implies the
following two consequences; we omit their proofs because they are standard.

######  Corollary B.5.

If $u\in{W^{1,p}}(M,\phi,\psi)$, then $u_{+}:=\max\\{u,0\\}$,
$u_{-}:=-\min\\{u,0\\}$ and $|u|=u_{+}-u_{-}$ are all in
${W^{1,p}}(M,\phi,\psi)$. In particular,
$\nabla|u|=\operatorname{sgn}(u)\nabla u$.

######  Corollary B.6.

Given $u\in W^{1,p}(M,\phi,\psi)$, we have $\max\\{0,\min\\{u,1\\}\\}\in
W^{1,p}(M,\phi,\psi)$. Moreover, for any $\lambda>0$, set
$u_{\lambda}:=\max\\{-\lambda,\min\\{u,\lambda\\}\\}$. Then
$u_{\lambda}\rightarrow u$ in $W^{1,p}(M,\phi,\psi)$ as
$\lambda\rightarrow+\infty$.

######  Proposition B.1.

Given $u,v\in W^{1,p}(M,\phi,\psi)$, the following statements hold$:$

  1. (i)

$\max\\{u,v\\}\in W^{1,p}(M,\phi,\psi)$ and
$|\nabla\max\\{u,v\\}|\leq\max\\{|\nabla u|,|\nabla v|\\}$;

  2. (ii)

if $\|u\|_{\infty}+\|v\|_{\infty}<+\infty$, then $uv\in W^{1,p}(M,\phi,\psi)$
and $\nabla(uv)=u\nabla v+v\nabla u$.

###### Proof.

Since $\max\\{a,b\\}=\frac{1}{2}(a+b+|a-b|)$, (i) immediately follows by
Corollary B.5 and Remark B.1/(a). In order to prove (ii), by Theorem B.2 we
have two sequences $\\{u_{i}\\},\\{v_{i}\\}$ in $C^{\infty}_{0}(M)$ such that
$u_{i}\rightarrow u$ and $v_{i}\rightarrow v$ under $\|\cdot\|_{p}$. Let
$u^{*}_{i}:=\max\\{-\lambda,\min\\{u_{i},\lambda\\}\\}$ and
$v^{*}_{i}:=\max\\{-\lambda,\min\\{v_{i},\lambda\\}\\}$, where $\lambda$ is a
constant satisfying $\|u\|_{\infty}+\|v\|_{\infty}\leq\lambda$. Thus,
$u^{*}_{i}v^{*}_{i}$ is a Lipschitz function with compact support, which
together with Lemma B.2 and Theorem B.2 implies $u^{*}_{i}v^{*}_{i}\in
W^{1,p}(M,\phi,\psi)$. Moreover, it follows from Corollary B.6 that
$u^{*}_{i},v^{*}_{i}\in W^{1,p}(M,\phi,\psi)$ and $u^{*}_{i}\rightarrow u$,
$v^{*}_{i}\rightarrow v$ under $\|\cdot\|_{p}$. Since
$\|u^{*}_{i}\|_{\infty}+\|v^{*}_{i}\|_{\infty}\leq 2\lambda$, it is easy to
check $u^{*}_{i}v^{*}_{i}\rightarrow uv$ and
$\nabla(u^{*}_{i}v^{*}_{i})\rightarrow v\nabla u+u\nabla v$ under the
corresponding $[\cdot]_{p,\mu}$-norms. Hence, $\\{u^{*}_{i}v^{*}_{i}\\}$ is a
Cauchy sequence in $W^{1,p}(M,\phi,\psi)$. We conclude the proof by
$u^{*}_{i}v^{*}_{i}\rightarrow uv$ in $\|\cdot\|_{p}$. ∎

######  Theorem B.7.

Suppose that $\\{u_{i}\\}$ is a bounded sequence in $W^{1,p}(M,\phi,\psi)$ and
$u_{i}\rightarrow u$ pointwise a.e. in ${M}$. Then $u\in
W^{1,p}(M,\phi,\psi)$, $u_{i}\rightarrow u$ weakly in $L^{p}(M,\phi,\psi)$ and
$\nabla u_{i}\rightarrow\nabla u$ weakly in $L^{p}(TM,\phi,\psi)$.

###### Proof.

Choose an arbitrary subsequence $\\{u_{i_{k}}\\}$ of $\\{u_{i}\\}$. According
to Theorem B.4, there is a subsequence of $\\{u_{i_{k}}\\}$, still denoted by
$\\{u_{i_{k}}\\}$, and a function $v\in W^{1,p}(M,\phi,\psi)$ such that
$u_{i_{k}}\rightarrow v$ weakly in $L^{p}(M,\phi,\psi)$ and $\nabla
u_{i_{k}}\rightarrow\nabla v$ weakly in $L^{p}(TM,\phi,\psi)$. We now show
that $v=u$ a.e. in ${M}$.

Lemma B.3 together with the proof of Theorem B.4 provides a sequence of convex
combinations with

$u^{*}_{k}:=\sum_{j=k}^{m_{k}}a_{k,j}u_{i_{j}}\rightarrow v,\ \ \nabla
u^{*}_{k}=\sum_{j=k}^{m_{k}}a_{k,j}\nabla u_{i_{j}}\rightarrow\nabla v,\text{
as }k\rightarrow\infty,$

in the corresponding $[\cdot]_{p,\mu}$-norms. Thus, the standard theory of
$L^{p}$-space together with Lemma B.1 yields a subsequence of
$\\{u^{*}_{k}\\}$ converging pointwise to $v$ a.e. in ${M}$. On the other
hand, since $\\{u_{j}\\}$ converges to $u$ point-wisely, we see that
$\\{u^{*}_{k}\\}$ converges to $u$ point-wisely as well. Consequently, $v=u$
a.e. and therefore $\nabla u=\nabla v$ a.e. in ${M}$.

Therefore, we proved that for every subsequence of $\\{u_{i}\\}$, there is
another subsequence $\\{u_{i_{k}}\\}$ such that $u_{i_{k}}\rightarrow u$
weakly in $L^{p}(M,\phi,\psi)$ and $\nabla u_{i_{k}}\rightarrow\nabla u$
weakly in $L^{p}(TM,\phi,\psi)$, which indicates that the original sequence
$\\{u_{i}\\}$ also satisfies such a property. ∎

######  Corollary B.8.

Given $u\in W^{1,p}(M,\phi,\psi)$, we have $\max\\{u-\epsilon,0\\}\in
W^{1,p}(M,\phi,\psi)$ for any $\epsilon>0$.

###### Proof.

Due to Theorem B.2, there exists a sequence $\\{u_{i}\\}$ in
$C^{\infty}_{0}(M)$ such that $u_{i}\rightarrow u$ under $\|\cdot\|_{p}$ and
$\|u_{i}\|_{p}\leq\|u\|_{p}+1$. By passing to a subsequence, we may assume
that $\\{u_{i}\\}$ converges to $u$ point-wisely. Let
$v_{i}:=\max\\{u_{i}-\epsilon,0\\}$. Since $v_{i}$ is a Lipschitz function
with compact support, Lemma B.2 yields $v_{i}\in W^{1,p}(M,\phi,\psi)$.
Moreover, note that $\|v_{i}\|_{p}\leq\|u_{i}\|_{p}\leq\|u\|_{p}+1$ and
$\\{v_{i}\\}$ converges to $\max\\{u-\epsilon,0\\}$ point-wisely, which
implies $\max\\{u-\epsilon,0\\}\in W^{1,p}(M,\phi,\psi)$ through Theorem B.7.
∎

Now we introduce the capacity with respect to $\|\cdot\|_{p}$.

######  Definition B.1.

The $p$-capacity of a set $E\subset M$ is defined by

$\displaystyle\operatorname{Cap}_{p}(E):=\inf_{u\in\mathscr{A}(E)}\|u\|^{p}_{p},$

where $\mathscr{A}(E)=\\{u\in{W^{1,p}}(M,\phi,\psi)\,:\,u\geq 1\text{ on a
neighborhood of }E\\}$. In particular, set $\operatorname{Cap}_{p}(E):=\infty$
if $\mathscr{A}(E)=\emptyset$.

Proceeding as in the proof of Kinnunen and Martio [22, Remark 3.1] and using
Corollary B.6, we have the following result.

######  Proposition B.2.

Let $\mathscr{A}^{\prime}(E):=\\{u\in{W^{1,p}}(M,\phi,\psi)\,:\,0\leq u\leq
1\text{ and }u=1\text{ on a neighborhood of }E\\}$. Then
$\operatorname{Cap}_{p}(E)=\inf_{u\in\mathscr{A}^{\prime}(E)}\|u\|^{p}_{p}.$

The following result collects the basic properties of
$\operatorname{Cap}_{p}$; since the proofs are standard, based on Theorem B.7,
Proposition B.1 and Lemma B.3, we omit them.

######  Theorem B.9.

The $p$-capacity is an outer measure, that is

  1. (i)

$\operatorname{Cap}_{p}(\emptyset)=0$;

  2. (ii)

$\operatorname{Cap}_{p}(E_{1})\leq\operatorname{Cap}_{p}(E_{2})$ if
$E_{1}\subset E_{2}$;

  3. (iii)

$\operatorname{Cap}_{p}\left(\bigcup_{i=1}^{\infty}E_{i}\right)\leq\sum_{i=1}^{\infty}\operatorname{Cap}_{p}(E_{i})$;

  4. (iv)

$\operatorname{Cap}_{p}(\cdot)$ is outer regular, i.e.,
$\operatorname{Cap}_{p}(E)=\inf\left\\{\operatorname{Cap}_{p}(O)\,:\,O\text{
is open with }E\subset O\right\\}$.

######  Definition B.2.

We say that a property holds $p$-quasieverywhere in a set $E$, if it holds
except for a subset of $p$-capacity zero in $E$. In addition, a function
$u:M\rightarrow[-\infty,+\infty]$ is $p$-quasicontinuous if for any
$\epsilon>0$, there is a set $E$ such that
$\operatorname{Cap}_{p}(E)<\epsilon$ and the restriction $u|_{M\backslash E}$
is continuous.

A suitable modification of the argument from Chen et al. [10, Theorem A.16],
based on the the above notions, provides the following result:

######  Theorem B.10.

If $u\in W^{1,p}(M,\phi,\psi)$ is $p$-quasicontinuous and $u=0$
$p$-quasieverywhere in $M\backslash{\Omega_{\Sigma}}$, then
${u|_{{\Omega_{\Sigma}}}}\in W^{1,p}_{0}(\Omega_{\Sigma},\phi,\psi)$.

Declarations.

Conflict of interest. The authors state that there is no conflict of interest.

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