Hardy Inequalities with Best Constants on Finsler Metric Measure Manifolds

Abstract

The paper is devoted to weighted $L^p$-Hardy inequalities with best constants on Finsler metric measure manifolds. There are two major ingredients. The first, which is the main part of this paper, is the Hardy inequalities concerned with distance functions in the Finsler setting. In this case, we find that besides the flag curvature, the Ricci curvature together with two non-Riemannian quantities, i.e., reversibility and S-curvature, also play an important role. And we establish the optimal Hardy inequalities not only on non-compact manifolds, but also on closed manifolds. The second ingredient is the Hardy inequalities for Finsler p-sub/superharmonic functions, in which we also investigate the existence of extremals and the Brezis–Vázquez improvement.

Appendices

Appendix A: Two Lemmas

Lemma A.1

Let $(M,o,F,\mathrm{d}\mathfrak {m})$, $\Omega$, $p,\beta$ be as in Definition 3.5. If u is a globally Lipschitz function on M with compact support in $\Omega$, then $u\in D^{1,p}(\Omega ,r^{p+\beta })$.

Proof

Since $\text {supp}(u)$ is compact, there exist a coordinate covering $\{(U_k,\phi _k)\}_{k=1}^{N<\infty }$ of $\text {supp}(u)$ and a constant $C\ge 1$ such that for each k, $U_k\subset \subset \Omega$, $\phi _k(U_k)=\mathbb {B}_{\mathbf {0}}(1)$ and

$$\begin{aligned} C^{-1} \mathrm{d}{{\,\mathrm{vol}\,}}\le \mathrm{d}\mathfrak {m}|_{U_k}\le C \mathrm{d}{{\,\mathrm{vol}\,}},\quad C^{-1}\le \frac{F^*(\omega )}{\Vert \phi ^{-1}_{k*}\omega \Vert }\le C, \quad \text { for any }\omega \in T^*{U_k}\backslash \{0\}, \end{aligned}$$

(A.1)

where $\mathrm{d}{{\,\mathrm{vol}\,}}$ and $\Vert \cdot \Vert$ are the Lebesgue measure and the Euclidean norm on the unit ball $\mathbb {B}_{\mathbf {0}}(1)$, respectively.

Choose a number $q>1$ such that $\beta q/({q-1})>-n$ if $\beta >-n$. By Lemma 3.2 and the construction above, one can easily verify $\int _{U_k}r^{{\beta q}/({q-1})} d \mathfrak {m}<\infty$ for each k.

On the other hand, let $\{\eta _k\}$ be a smooth partition of unity subordinate to $\{U_k\}$. Thus, $(\eta _ku)\circ \phi _k^{-1}$ is a globally Lipschitz function on $\mathbb {B}_{\mathbf {0}}(1)$ with respect to the Euclidean distance and hence, $(\eta _ku)\circ \phi _k^{-1}$ belongs to the Sobolev space $W^{1,pq}(\mathbb {B}_{\mathbf {0}}(1))$. Meyers–Serrin’s theorem then yields a sequence $v_{k_j}\in C_0^\infty (\mathbb {B}_{\mathbf {0}}(1))$ with $\lim _{j\rightarrow +\infty }\Vert v_{k_j}-(\eta _ku)\circ \phi _k^{-1}\Vert _{W^{1,pq}(\mathbb {B}_{\mathbf {0}}(1))}=0$. Therefore, we have $v_{k_j}\circ \phi _k\in C^\infty _0(\Omega )$ with $\text {supp}(v_{k_j}\circ \phi _k)\subset U_k$, which together with the Hölder inequality and (A.1) implies

$$\begin{aligned}&\int _\Omega |v_{k_j}\circ \phi _k-(\eta _ku)|^p r^{\beta } \mathrm{d}\mathfrak {m}\\&\quad \le \left( C\int _{\mathbb {B}_{\mathbf {0}}(1)} |v_{k_j}-(\eta _ku)\circ \phi ^{-1}_k |^{pq} \mathrm{d}{{\,\mathrm{vol}\,}}\right) ^{\frac{1}{q}} \left( \int _{U_k}r^{\frac{\beta q}{q-1}} \mathrm{d}\mathfrak {m}\right) ^{\frac{q-1}{q}}\\&\quad \le C^{\frac{1}{q}}\Vert v_{k_j}-(\eta _ku)\circ \phi _k^{-1}\Vert ^p_{W^{1,pq}(\mathbb {B}_{\mathbf {0}}(1))}\left( \int _{U_k}r^{\frac{\beta q}{q-1}} \mathrm{d}\mathfrak {m}\right) ^{\frac{q-1}{q}}\rightarrow 0, \text { as }j\rightarrow +\infty . \end{aligned}$$

Similarly, one can prove $\lim _{j\rightarrow +\infty }\int _\Omega F^{*p}(d(v_{k_j}\circ \phi _k)-d(\eta _ku)) r^{\beta +p} \mathrm{d}\mathfrak {m}=0$. Therefore, $\Vert v_{k_j}\circ \phi _k-(\eta _ku)\Vert _D\rightarrow 0$ and $(\eta _ku)\in D^{1,p}(\Omega ,r^{p+\beta })$. We conclude the proof by $u=\sum _{k=1}^N(\eta _ku)$. $\square$

Lemma A.2

Let $(M,o,F,\mathrm{d}\mathfrak {m})$ be an n-dimensional forward complete PFMMM with $\mathbf {Ric}_\infty \ge (n-1)K$ and $\mathbf {S}_o^+\ge -a$, where $a\ge 0$. Set $\frac{\pi }{\sqrt{K}}:=+\infty$ if $K\le 0$. Let (t, y) denote the polar coordinate system around o. Then we have

$$\begin{aligned} \Delta t\le (n-1)\frac{\mathfrak {s}'_K(t)}{\mathfrak {s}_K(t)}+a,\quad \mathrm{for}\, \mathrm{any }\, y\in S_oM,\quad 0<t<\min \left\{ i_y, \frac{\pi }{2\sqrt{K}} \right\} , \end{aligned}$$

which implies

$$\begin{aligned} \hat{\sigma }_o(t,y)\le e^{-\tau (y)+at}\mathfrak {s}^{n-1}_{K}(t),\quad \mathrm{for}\,\mathrm{any}\, y\in S_oM,\quad 0<t<\min \left\{ i_y, \frac{\pi }{2\sqrt{K}} \right\} , \end{aligned}$$

where $\hat{\sigma }_o(t,y)$ is defined as in (2.7) and $\mathfrak {s}_K(t)$ is the solution of $f''+K f=0$ with $f(0)=0$ and $f'(0)=1$. Hence,

$$\begin{aligned} \mathfrak {m}(B^+_o(r))\le \int _{S_oM}e^{-\tau (y)}\mathrm{d}\nu _o(y)\int _0^{r} e^{at}\mathfrak {s}^{n-1}_{K}(t)\mathrm{d}t, \text { for } 0<r<\min \left\{ i_y, \frac{\pi }{2\sqrt{K}} \right\} . \end{aligned}$$

Proof

The proof is similar to that of Wei and Wylie [33, Theorem 1.1]. First, fix $y\in S_oM$ and set

$$\begin{aligned} H(t):=\frac{\partial }{\partial t}\log \sqrt{\det g_{\nabla t}}, \ \tau (t):=\tau (\nabla t),\quad \mathbf {S}(t):=\mathbf {S}(\nabla t)=\frac{\mathrm{d}}{\mathrm{d}t}\tau (t). \end{aligned}$$

A standard argument (cf. Wu [34, (4.5)]) yields $\Delta t=H(t)-\mathbf {S}(t)$ and

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}H\le -\mathbf {Ric}(\nabla t)-\frac{H^2}{n-1}. \end{aligned}$$

(A.2)

Also set $H_K(t):=(n-1)\frac{\mathfrak {s}'_K(t)}{\mathfrak {s}_K(t)}$. By (A.2), one has

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}\left( H(t)\mathfrak {s}^2_K(t)\right) \le 2 \mathfrak {s}'_K(t)\mathfrak {s}_K(t)H(t)-\mathfrak {s}^2_K(t)\left( \mathbf {Ric}(\nabla t)+\frac{H^2}{n-1} \right) \\&\quad =-\left( \frac{\mathfrak {s}_K(t) H(t)}{\sqrt{n-1}}-\sqrt{n-1}\,\mathfrak {s}'_K(t) \right) ^2+(n-1)(\mathfrak {s}'_K(t))^2-\mathfrak {s}^2_K(t)\mathbf {Ric}(\nabla t)\\&\quad \le (n-1)(\mathfrak {s}'_K(t))^2-\mathfrak {s}^2_K(t)\left( \mathbf {Ric}_\infty (\nabla t)-\frac{\mathrm{d}}{\mathrm{d}t}\mathbf {S}(t) \right) \\&\quad \le (n-1)(\mathfrak {s}'_K(t))^2-(n-1)K\mathfrak {s}^2_K(t)+\mathfrak {s}^2_K(t)\frac{\mathrm{d}}{\mathrm{d}t}\mathbf {S}(t)\\&\quad =\frac{\mathrm{d}}{\mathrm{d}t}\left( H_K(t)\mathfrak {s}^2_K(t)\right) +\mathfrak {s}^2_K(t)\frac{\mathrm{d}}{\mathrm{d}t}\mathbf {S}(t). \end{aligned}$$

Since $\mathbf {S}_O\ge -a$, integrating by parts on the above inequality, we get

$$\begin{aligned}&\mathfrak {s}^2_K(t)\Delta t=\mathfrak {s}^2_K(t)(H(t)-\mathbf {S}(t))\le \mathfrak {s}^2_K(t)H_K(t)-\int ^t_0\frac{\mathrm{d}}{\mathrm{d}s}(\mathfrak {s}^2_K(s)) \,\mathbf {S}(s) \mathrm{d}s\nonumber \\&\quad \le \mathfrak {s}^2_K(t)H_K(t)+a\int ^t_0\frac{\mathrm{d}}{\mathrm{d}s}(\mathfrak {s}^2_K(s))\mathrm{d}s=\mathfrak {s}^2_K(t)H_K(t)+a\mathfrak {s}^2_K(t). \end{aligned}$$

(A.3)

Hence, $\Delta t\le H_K(t)+a$, which implies

$$\begin{aligned} \frac{\partial }{\partial t}\log \hat{\sigma }_o(t,y)=\Delta t\le (n-1)\frac{\mathfrak {s}'_K(t)}{\mathfrak {s}_K(t)}+a=\frac{\partial }{\partial t}\log \left[ e^{at}\mathfrak {s}^{n-1}_K(t) \right] . \end{aligned}$$

Then the estimates of $\hat{\sigma }_o(t,y)$ and $\mathfrak {m}(B^+_o(r))$ follow from a standard argument (cf. Zhao et al. [39]). $\square$

Remark 6

By a different method, Yin [37] obtained the theorem above in the case when the PFMMM is equipped by the Busemann–Hausdroff measure and satisfies $\mathbf {Ric}_\infty \ge (n-1)K$ and $\mathbf {S}_o^+\ge -a$ ($a>0$).

Appendix B: Weighted Sobolev Space

Let $(M,o,F,\mathrm{d}\mathfrak {m})$ be an n-dimensional closed reversible PFMMM and set $r(x):=d_F(o,x)$. Given $p\in (1,+\infty )$ and $\beta <-n$ with $p+\beta >-n$, by Lemma 3.2, we define a norm on $C^\infty _0(M)=C^\infty (M)$ as

$$\begin{aligned} \Vert u\Vert _{p,\beta }:=\left( \int _{{M}}|u|^p{r}^{p+\beta } \mathrm{d}\mathfrak {m}+\int _{{M}} F^{*p}(d u) {r}^{p+\beta } \mathrm{d}\mathfrak {m}\right) ^{\frac{1}{p}}. \end{aligned}$$

The weighted Sobolev space $W^{1,p}({M}, {r}^{p+\beta })$ is defined as

$$\begin{aligned} W^{1,p}({M}, {r}^{p+\beta }):=\overline{{C}^\infty _0({M})}^{\Vert \cdot \Vert _{p,\beta }}. \end{aligned}$$

In particular, $W^{1,p}(M,r^0)=:W^{1,p}(M)$, i.e., the standard Sobolev space in the sense of Hebey [13, Definition 2.1].

We also define the weighted $L^p$-space $L^p(M,r^{p+\beta })$ (resp., $L^p(TM,r^{p+\beta })$) as the completion of $C^\infty (M)$ (resp., $\Gamma ^\infty (T^*M)$, i.e., the space of the smooth sections of the cotangent bundle) under the norm

$$\begin{aligned} {[}u]_{p,\beta }:=\left( \int _{{M}}|u|^p{r}^{p+\beta } \mathrm{d}\mathfrak {m}\right) ^{\frac{1}{p}} \ \ \left( \text {resp., } [\omega ]_{p,\beta }:=\left( \int _{{M}}F^{*p}(\omega ){r}^{p+\beta } \mathrm{d}\mathfrak {m}\right) ^{\frac{1}{p}} \right) . \end{aligned}$$

And set $L^p(M):=L^p(M,r^0)$ and $L^p(TM):=L^p(TM,r^0)$.

Lemma B.1

If $u\in W^{1,p}({M},{r}^{p+\beta })$, then $u\in W^{1,1}(M)$. Moreover, the differential $\varpi$ of u in $W^{1,p}({M},{r}^{p+\beta })$ is the distributional derivative of u, i.e., $\varpi \in L^1(TM)$ and

$$\begin{aligned} \int _{{M}} \langle X,\varpi \rangle \mathrm{d}\mathfrak {m}=-\int _{{M}}u {{\,\mathrm{div}\,}}X \mathrm{d}\mathfrak {m},\quad \text { for any smooth vector field }X. \end{aligned}$$

Proof

Since $({\beta +p})/({1-p})>-n$, Lemma 3.2 implies that ${r}^{\frac{\beta +p}{1-p}}$ is integrable. Given $f\in L^p({M},{r}^{\beta +p})$, the Hölder inequality yields

$$\begin{aligned} \int _M|f| \mathrm{d}\mathfrak {m}&=\int _M |f|{r}^{\frac{\beta +p}{p}}{r}^{-\frac{\beta +p}{p}}\mathrm{d}\mathfrak {m}\le \left( \int _M |f|^p {r}^{p+\beta }\mathrm{d}\mathfrak {m} \right) ^{\frac{1}{p}}\left( \int _M {r}^{\frac{p+\beta }{1-p}} \mathrm{d}\mathfrak {m} \right) ^{\frac{p-1}{p}}. \end{aligned}$$

(B.1)

Consequently, if $u\in W^{1,p}({M},{r}^{p+\beta })$, (B.1) implies $u\in L^{1}({M})$ and its differential $\varpi \in L^{1}(T{M})$. On the other hand, there exists a sequence $u_j\in C^\infty _0({M})$ such that $[u_j-u]_{p,\beta }\rightarrow 0$ and $[\mathrm{d}u_j-\varpi ]_{p,\beta }\rightarrow 0$. Thus, for any smooth vector field X, (B.1) together with the compactness of M yields

$$\begin{aligned}&\left| \int _{{M}} \langle X,\varpi \rangle -(-u{{\,\mathrm{div}\,}}X) \mathrm{d}\mathfrak {m} \right| \\&\quad =\left| \int _{{M}} \langle X,\varpi \rangle - \langle X,\mathrm{d}u_j\rangle + \langle X,\mathrm{d}u_j\rangle -(-u{{\,\mathrm{div}\,}}X) \mathrm{d}\mathfrak {m} \right| \\&\quad \le \int _M \left| \langle X, \varpi -\mathrm{d}u_j\rangle \right| \mathrm{d}\mathfrak {m}+ \int _M \left| (u_j-u){{\,\mathrm{div}\,}}X \right| \mathrm{d}\mathfrak {m}\\&\quad \le \max _M F(X) \int _{M}F^*(\varpi -\mathrm{d}u_j)\mathrm{d}\mathfrak {m} +\max _M|{{\,\mathrm{div}\,}}X| \int _{M}|u_j- u|\mathrm{d}\mathfrak {m} \\&\quad \le \left( \max _M F(X)+\max _M|{{\,\mathrm{div}\,}}X|\right) \left( \int _M {r}^{\frac{p+\beta }{1-p}} \mathrm{d}\mathfrak {m} \right) ^{\frac{p-1}{p}}\left( [\varpi -d u_j]_{p,\beta }^p+[u_j -u]_{p,\beta }^p \right) \rightarrow 0. \end{aligned}$$

Furthermore, (B.1) also implies that $u_j\rightarrow u$ in $W^{1,1}({M})$ and hence, the lemma follows. $\square$

Lemma B.2

If $u\in W^{1,p}({M},{r}^{\beta +p})$, then $u_+:=\max \{u,0\}$, $u_-:=-\min \{u,0\}$ and $|u|=u_+-u_-$ are all in $W^{1,p}({M},{r}^{\beta +p})$.

Proof

Since $u_-=(-u)_+$, it suffices to prove $u_+\in W^{1,p}({M},{r}^{\beta +p})$. Choose a sufficiently large constant $q>1$ such that $\frac{q(\beta +p)}{q-1}>-n$. For any $f\in L^{pq}(M)$, the Hölder inequality together with Lemma 3.2 yields

$$\begin{aligned} \int _M |f|^p{r}^{\beta +p}\mathrm{d}\mathfrak {m}\le \left( \int _M |f|^{pq} \mathrm{d}\mathfrak {m}\right) ^{\frac{1}{q}} \left( \int _M {r}^{\frac{q(\beta +p)}{q-1}}\mathrm{d}\mathfrak {m}\right) ^{\frac{q-1}{q}}. \end{aligned}$$

(B.2)

First we consider the case when $u\in C^\infty _0({M})$. The standard theory yields a subsequence $u_j\in C^\infty _0({M})$ such that $u_j\rightarrow u_+$ in $W^{1,pq}({M})$ (cf. Hebey [13, Lemma 2.5]), which together with (B.2) implies $u_j\rightarrow u_+$ in $W^{1,p}({M}, {r}^{p+\beta })$. Hence, $u_+\in W^{1,p}({M}, {r}^{p+\beta })$.

For the general case (i.e., $u\in W^{1,p}({M},{r}^{\beta +p})$), choose a sequence $u_j\in C^\infty _0({M})$ such that $\Vert u_j- u\Vert _{p,\beta }\rightarrow 0$. From above, we have $u_{j+}=\max \{u_j,0\}\in W^{1,p}({M}, {r}^{p+\beta })$. Since $\max \{s,t\}=\frac{1}{2}(s+t-|s-t|)$, the triangle inequality yields $\Vert u_{j+}-u_+\Vert _{p,\beta }\le \Vert u_j-u\Vert _{p,\beta }\rightarrow 0$. Hence, $u_+\in W^{1,p}({M},{r}^{\beta +p})$. $\square$

Since M is closed, the following result follows from Lemma B.2 directly.

Corollary B.3

Given $u\in W^{1,p}({M},{r}^{\beta +p})$, then $u_\epsilon :{=}\max \{u-\epsilon ,0\}\in W^{1,p}({M},{r}^{\beta +p})$, for any $\epsilon >0$.

Now set ${M_o}:=M\backslash \{o\}$. Define the weighted Sobolev space $W^{1,p}({M_o},r^{\beta +p})$ as the completion of $C^\infty _0({M_o})$ with respect to the norm

$$\begin{aligned} \Vert u\Vert _{{M_o},p,\beta }:=\left( \int _{{M_o}}|u|^p{r}^{p+\beta } \mathrm{d}\mathfrak {m}+\int _{{M_o}} F^{*p}(\mathrm{d} u)^p {r}^{p+\beta } \mathrm{d}\mathfrak {m}\right) ^{\frac{1}{p}}. \end{aligned}$$

Lemma B.4

If $u\in W^{1,p}({M},{r}^{\beta +p})$ with compact support in ${M_o}$, then $u|_{{M_o}}\in W^{1,p}({M_o},{r}^{\beta +p})$.

Proof

Since $\text {supp}(u)\subset {M_o}$ is compact, one can choose a cut-off function $\eta \in C^\infty _0({M})$ such that $\text {supp}(u)\subsetneqq \text {supp}(\eta )\subset {M_o}$ and $\eta |_{\text {supp}(u)}=1$.

On the other hand, since $u\in W^{1,p}({M},{r}^{\beta +p})$, there exists a sequence $u_i\in C^\infty _0({M})$ with $\Vert u_i- u\Vert _{p,\beta }\rightarrow 0$. Note that if $\Vert \eta u_i- \eta u\Vert _{{M_o},p,\beta }\rightarrow 0$, then $u|_{{M_o}}=\eta u\in W^{1,p}({M_o},{r}^{\beta +p})$ and the lemma follows. Hence, it suffices to show $\Vert \eta u_i- \eta u\Vert _{{M_o},p,\beta }\rightarrow 0$.

A direct calculation together with the triangle inequality (i.e., $F^*(\omega _1+\omega _2)\le F^*(\omega _1)+F^*(\omega _2)$) furnishes

$$\begin{aligned}&\Vert \eta u_i-\eta u\Vert _{{M_o},p,\beta }^p = \int _{\text {supp}\eta }|\eta u_i-\eta u|^p {r}^{\beta +p} \mathrm{d}\mathfrak {m}+\int _{\text {supp}\eta }F^{*p}\left( \mathrm{d}(\eta u_i-\eta u)\right) {r}^{\beta +p}\mathrm{d}\mathfrak {m}\\&\quad \le \int _{{M}}| u_i- u|^p {r}^{\beta +p} \mathrm{d}\mathfrak {m}+2^p\left[ \int _{\text {supp}\eta } F^{*p}\left( (u_i- u)\mathrm{d}\eta \right) {r}^{\beta +p}\mathrm{d}\mathfrak {m}\right. \\&\qquad \left. +\int _{\text {supp}\eta }F^{*p}\left( \eta \mathrm{d}(u_i-u)\right) {r}^{\beta +p}\mathrm{d}\mathfrak {m} \right] \\&\quad \le \int _{{M}}| u_i- u|^p {r}^{\beta +p} \mathrm{d}\mathfrak {m}+2^p\left[ \Vert F^{*p}(\mathrm{d}\eta )\Vert _{\infty }\int _{M} |u_i- u|^p {r}^{\beta +p}\mathrm{d}\mathfrak {m}\right. \\&\qquad \left. +\int _{M}F^{*p}(\mathrm{d}u_i-\mathrm{d}u){r}^{\beta +p}\mathrm{d}\mathfrak {m} \right] \\&\quad \le \left[ 2^p (\Vert F^{*p}(\mathrm{d}\eta )\Vert _{\infty }+1)+1 \right] \Vert u_i-u\Vert _{p,\beta }^p\rightarrow 0. \end{aligned}$$

$\square$

Proof of Proposition 3.15

Without loss of generality, we may prove the proposition in the case when $u\ge 0$. Thus, Lemma B.2 implies $u=u_+\in W^{1,p}({M}, {r}^{\beta +p})\cap C({M})$.

For each $\epsilon \in (0,1)$, set $u_\epsilon (x):=\max \{u-\epsilon ,0\}$. Since u is continuous with $u(o)=0$, there exists a small $\delta >0$ such that $u_\epsilon =0$ in $B_o(\delta )$, which implies that $\text {supp}(u_\epsilon )$ is a compact subset of ${M_o}$. Corollary B.3 then yields $u_\epsilon |_{{M_o}} \in W^{1,p}({M_o},{r}^{\beta +p})$. By a direct calculation, we have

$$\begin{aligned}&\left\| u|_{{M_o}}-u_\epsilon |_{{M_o}}\right\| _{{M_o},p,\beta }^p\\&=\left\| u- u_\epsilon \right\| ^p_{p,\beta }=\int _{{M}}|u-u_\epsilon |^p{r}^{p+\beta } \mathrm{d}\mathfrak {m}\\&\qquad + \int _{{M}} F^{*p}(d (u-u_\epsilon )) {r}^{p+\beta } \mathrm{d}\mathfrak {m}\le \epsilon ^p \int _M {r}^{\beta +p} \mathrm{d}\mathfrak {m}\\&\qquad +\int _{{M}}\chi _{\{0\le u\le \epsilon \}}|u|^p{r}^{\beta +p} \mathrm{d}\mathfrak {m} + \int _{{M}} \chi _{\{0\le u\le \epsilon \}} F^{*p}(\mathrm{d}u) {r}^{p+\beta } \mathrm{d}\mathfrak {m}. \end{aligned}$$

Now the assumption together with the dominated convergence theorem yields

$$\begin{aligned}&\lim _{\epsilon \rightarrow 0^+}\int _{{M}}\chi _{\{0\le u\le \epsilon \}}|u|^p{r}^{\beta +p} \mathrm{d}\mathfrak {m}=\int _{{M}}\lim _{\epsilon \rightarrow 0^+}\chi _{\{0\le u\le \epsilon \}}|u|^p{r}^{\beta +p} \mathrm{d}\mathfrak {m}=0,\\&\lim _{\epsilon \rightarrow 0^+}\int _{{M}} \chi _{\{0\le u\le \epsilon \}} F^{*p}(\mathrm{d}u) {r}^{p+\beta } \mathrm{d}\mathfrak {m}=\int _{{M}} \lim _{\epsilon \rightarrow 0^+}\chi _{\{0\le u\le \epsilon \}} F^{*p}(\mathrm{d}u) {r}^{p+\beta } \mathrm{d}\mathfrak {m}=0, \end{aligned}$$

which imply $\left\| u|_{{M_o}}-u_\epsilon |_{{M_o}}\right\| _{{M_o},p,\beta }^p\rightarrow 0$ as $\epsilon \rightarrow 0^+$ and hence, $u|_{{M_o}}\in W^{1,p}({M_o},{r}^{\beta +p})$. $\square$