Proof. Let \((X,d,m)\), \((\mathcal{E},\mathcal{F})\), \(\Gamma\) be given by Theorem 3, on which V(\(d_h\)), PI(\(\beta_p\)) and \(\text{cap}(\beta_p)_{\le}\) hold. By [34], the conjunction of 8 , PI(\(\beta_p\)), \(\text{cap}(\beta_p)_{\le}\) implies \(\beta_p(X,d,m)=\beta_p\). ◻

Proof of the upper bound. For notational convenience, we may assume that \(\mathrm{diam}(X)=+\infty\). For any \(R,r>0\) with \(r\le R\). If \(R/r\lesssim1\), then since \(\Psi\) is doubling and \(\Phi\) is increasing, we have \[\frac{\Psi(R)}{\Psi(r)}\lesssim1\le\left(\frac{R}{r}\right)^{p-1}\frac{\Phi(R)}{\Phi(r)}.\] Assume that \(R/r\gtrsim1\). Let \(N\) be the integer satisfying \(Nr\le R<(N+1)r\), then \(N\gtrsim1\). Take \(x,y\in X\) satisfying \(Nr<d(x,y)<(N+1)r\). By 15 , there exists \(u\in\mathcal{F}\) with \(u=1\) in \(B(x,R/(2A_{cap}))\) and \(u=0\) on \(X\backslash B(x,R/2)\) such that \[\mathcal{E}(u)\le 2C_{cap}\frac{V(x,\frac{R}{2A_{cap}})}{\Psi(\frac{R}{2A_{cap}})}\asymp\frac{\Phi(R)}{\Psi(R)}.\] By 7 , there exists a sequence \(\{y_k:0\le k\le 10N\}\) with \(y_0=x\) and \(y_{10N}=y\) such that \[d(y_k,y_{k-1})\le C_{cc}\frac{d(x,y)}{10N}\le C_{cc}\frac{(N+1)r}{10N}<C_{cc}\frac{2Nr}{10N}=\frac{1}{5}C_{cc}r\text{ for any }k=1,\ldots,10N.\] Since \(d(y_0,y_{10N})=d(x,y)>Nr\), by Lemma 1, there exists a subsequence of \(\{y_k:0\le k\le 10N\}\), denoted as \(\{x_k:0\le k\le M\}\) such that \(x_0=x\), \(x_M=y\), \(d(x_k,x_{k-1})<C_{cc}r\) for any \(k=1,\ldots,M\) and \(d(x_k,x_l)\ge \frac{2}{5}C_{cc}r\) for any distinct \(k,l=0,1,\ldots,M\). Obviously, \(M\le 10N\). On the other hand \[Nr< d(x,y)=d(x_0,x_M)\le\sum_{k=1}^Md(x_k,x_{k-1})\le M C_{cc}r,\] hence \(M\ge N/C_{cc}\), which gives \(M\asymp N\asymp R/r\).

For any \(k=0,\ldots,M\), let \(B_k=B(x_k,r)\), then \(u\equiv1\) in \(B_0\) and \(u\equiv0\) in \(B_M\), which gives \(u_{B_0}=1\) and \(u_{B_M}=0\). For any \(k=1,\ldots,M\), by 9 , 13 , and Hölder’s inequality, we have \[\begin{align} &|u_{B_k}-u_{B_{k-1}}|\le\left(\mathchoice {{\setbox 0=\displaystyle{\textstyle-}{\int}\vcenter{\textstyle- }\kern-.6\wd 0}}{{\setbox 0=\textstyle{\scriptstyle-}{\int}\vcenter{\scriptstyle- }\kern-.6\wd 0}}{{\setbox 0=\scriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}}{{\setbox 0=\scriptscriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}}\!\int_{B_k}\mathchoice {{\setbox 0=\displaystyle{\textstyle-}{\int}\vcenter{\textstyle- }\kern-.6\wd 0}}{{\setbox 0=\textstyle{\scriptstyle-}{\int}\vcenter{\scriptstyle- }\kern-.6\wd 0}}{{\setbox 0=\scriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}}{{\setbox 0=\scriptscriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}}\!\int_{B_{k-1}}|u(x)-u(y)|^pm(\mathrm{d} y)m(\mathrm{d} x)\right)^{1/p}\\ &\le \frac{1}{(m(B_k)m(B_{k-1}))^{1/p}}\left(\int_{B(x_k,2C_{cc}r)}\int_{{B(x_k,2C_{cc}r)}}|u(x)-u(y)|^pm(\mathrm{d} y)m(\mathrm{d} x)\right)^{1/p}\\ &\lesssim \frac{1}{\Phi(r)^{1/p}}\left(\int_{B(x_k,2C_{cc}r)}|u-u_{B(x_k,2C_{cc}r)}|^p\mathrm{d} m\right)^{1/p}\\ &\lesssim\left(\frac{\Psi(r)}{\Phi(r)}\right)^{1/p}\Gamma(u)(B(x_k,2A_{PI}C_{cc}r))^{1/p}, \end{align}\] hence \[\begin{align} &1=|u_{B_0}-u_{B_M}|\le\sum_{k=1}^M|u_{B_k}-u_{B_{k-1}}|\\ &\lesssim\left(\frac{\Psi(r)}{\Phi(r)}\right)^{1/p}\sum_{k=1}^M\Gamma(u)(B(x_k,2A_{PI}C_{cc}r))^{1/p}\\ &\le\left(\frac{\Psi(r)}{\Phi(r)}\right)^{1/p}M^{1-{1}/{p}}\left(\sum_{k=1}^M\Gamma(u)(B(x_k,2A_{PI}C_{cc}r))\right)^{1/p}\\ &\asymp\left(\frac{\Psi(r)}{\Phi(r)}\right)^{1/p}\left(\frac{R}{r}\right)^{1-{1}/{p}}\left(\int_X\sum_{k=1}^M1_{B(x_k,2A_{PI}C_{cc}r)}\mathrm{d}\Gamma(u)\right)^{1/p}. \end{align}\] Since \(d(x_k,x_l)\ge \frac{2}{5}C_{cc}r\) for any distinct \(k,l=0,1,\ldots,M\), by 9 , there exists some positive integer \(K\) depending only on \(C_\Phi, A_{PI}, C_{cc}\) such that \[\sum_{k=1}^M1_{B(x_k,2A_{PI}C_{cc}r)}\le K1_{\cup_{k=1}^MB(x_k,2A_{PI}C_{cc}r)}.\] Hence \[1\lesssim\left(\frac{\Psi(r)}{\Phi(r)}\right)^{1/p}\left(\frac{R}{r}\right)^{1-{1}/{p}}\mathcal{E}(u)^{1/p}\lesssim\left(\frac{\Psi(r)}{\Phi(r)}\right)^{1/p}\left(\frac{R}{r}\right)^{1-{1}/{p}}\left(\frac{\Phi(R)}{\Psi(R)}\right)^{1/p},\] which gives \[\frac{\Psi(R)}{\Psi(r)}\lesssim \left(\frac{R}{r}\right)^{p-1}\frac{\Phi(R)}{\Phi(r)}.\] ◻

Proof. For any \(z\in V\), let \[R_z=\{x\in X:d(x,z)=d(x,V)\},\] then \(B(z,\varepsilon/2)\subseteq R_z\subseteq B(z,\varepsilon)\) and \(\{R_z:z\in V\}\) is a cover of \(X\). Let \(N_z\) be an \((\varepsilon/(4A_{cap}))\)-net of \(R_z\). By 8 , there exists some positive integer \(M_1\) depending only on \(C_{VD}, A_{cap}\) such that \(\# N_z\le M_1\). For any \(w\in N_z\), by 8 and 15 , there exists \(\rho_w\in\mathcal{F}\) with \(0\le\rho_w\le1\) in \(X\), \(\rho_w=1\) in \(B(w,\varepsilon/(4A_{cap}))\), \(\rho_w=0\) on \(X\backslash B(w,\varepsilon/4)\) such that \[\mathcal{E}(\rho_w)\le 2C_{cap}\frac{V(w,\varepsilon/(4A_{cap}))}{\Psi(\varepsilon/(4A_{cap}))}\le C_1 \frac{V(z,\varepsilon)}{\Psi(\varepsilon)},\] where \(C_1>0\) depends only on \(C_{\Psi}, C_{VD}, C_{cap}, A_{cap}\).

For any \(z\in V\), let \(\varphi_z=\max_{w\in N_z}\rho_w\), then \(0\le\varphi_z\le1\) in \(X\), \(\varphi_z=1\) on \(R_z\), \(\varphi_z=0\) on \(\{x\in X:d(x,R_z)\ge\varepsilon/4\}\). By 11 , we have \(\varphi_z\in\mathcal{F}\) and \[\mathcal{E}(\varphi_z)\le C\sum_{w\in N_z}\mathcal{E}(\rho_w)\le CM_1C_1 \frac{V(z,\varepsilon)}{\Psi(\varepsilon)}=C_2\frac{V(z,\varepsilon)}{\Psi(\varepsilon)},\] where \(C>0\) depends only on \(p, M_1\), and \(C_2=CM_1C_1\). Since \(\{R_z:z\in V\}\) is a cover of \(X\), we have \(\sum_{z\in V}\varphi_z\ge1\) in \(X\). By 8 , there exists some positive integer \(M_2\) depending only on \(C_{VD}\) such that \(\#\{w\in V:\varphi_w\ne0\text{ in }B(z,\frac{5}{4}\varepsilon)\}\le M_2\) for any \(z\in V\), which implies that \(\sum_{z\in V}\varphi_z\le M_2\) in \(X\).

For any \(z\in V\), let \(\psi_z=\frac{\varphi_z}{\sum_{z\in V}\varphi_z}\). Since \(\varphi_z=0\) on \(X\backslash B(z,5\varepsilon/4)\), we have \[\psi_z=\frac{\varphi_z}{\sum_{w\in V:\varphi_w\ne0\text{ in }B(z,\frac{5}{4}\varepsilon)}\varphi_w}.\] By [34], we have \(\psi_z\in\mathcal{F}\), and there exists \(C_3>0\) depending only on \(p, M_2\) such that \[\mathcal{E}(\psi_z)\le C_3\sum_{w\in V:\varphi_w\ne0\text{ in }B(z,\frac{5}{4}\varepsilon)}\mathcal{E}(\varphi_w)\le C_3M_2C_2 \frac{V(z,\varepsilon)}{\Psi(\varepsilon)}=C_4\frac{V(z,\varepsilon)}{\Psi(\varepsilon)},\] where \(C_4=C_3M_2C_2\), hence we have [item95COenergy].

For any distinct \(z,w\in V\), we claim that \[B(w,\frac{1}{4}\varepsilon)\subseteq\{x\in X:d(x,R_z)\ge \frac{1}{4}\varepsilon\}.\] Otherwise there exists \(x\in X\) such that \(d(x,w)<\varepsilon/4\), \(d(x,R_z)<\varepsilon/4\), there exists \(y\in R_z\) such that \(d(x,y)<\varepsilon/4\), then \(d(y,w)\le d(x,y)+d(x,w)<\varepsilon/2\). But \(y\in R_z\) implies \(d(y,z)=d(y,V)\le d(y,w)<\varepsilon/2\), hence \(d(z,w)\le d(y,z)+d(y,w)<\varepsilon\), contradicting to that \(V\) is an \(\varepsilon\)-net. Hence for any \(z\in V\), \(\sum_{z\in V}\varphi_z=\varphi_z\) in \(B(z,\varepsilon/4)\), \(\psi_z=\varphi_z/\varphi_z=1\) in \(B(z,\varepsilon/4)\), \(\psi_z=0\) on \(\{x\in X:d(x,R_z)\ge\varepsilon/4\}\supseteq X\backslash B(z,5\varepsilon/4)\), hence we have [item95COspt].

Finally, [item95COunit] is obvious. ◻

Proof. The proof is standard using the telescopic technique. Since \(u\in\mathcal{F}_{loc}\), we have \(u\in L^p_{loc}(X;m)\), \(m\)-almost all points are Lebesgue points of \(u\). Let \(x,y\in(4A_{PI})^{-1}B\) be two distinct Lebesgue points of \(u\). Denote \(D=d(x,y)\in(0,R/(2A_{PI}))\). For any \(n\ge0\), let \(B_n=B(x,2^{-n}D)\), then by 8 , 13 , and Hölder’s inequality, we have \[\begin{align} &|u_{B_n}-u_{B_{n+1}}|\lesssim\left(\mathchoice {{\setbox 0=\displaystyle{\textstyle-}{\int}\vcenter{\textstyle- }\kern-.6\wd 0}}{{\setbox 0=\textstyle{\scriptstyle-}{\int}\vcenter{\scriptstyle- }\kern-.6\wd 0}}{{\setbox 0=\scriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}}{{\setbox 0=\scriptscriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}}\!\int_{B_{n}}|u-u_{B_n}|^p\mathrm{d} m\right)^{1/p}\lesssim\left(\frac{\Psi(2^{-n}D)}{m(B_n)}\int_{A_{PI}B_n}\mathrm{d}\Gamma(u)\right)^{1/p}\\ &\lesssim \left(\Psi(2^{-n}D)\frac{\Gamma(u)(A_{PI}B_n)}{m(A_{PI}B_n)}\right)^{1/p}\le\left(\Psi(2^{-n}D)M_R\Gamma(u)(x)\right)^{1/p}, \end{align}\] hence \[\begin{align} &|u(x)-u_{B(x,D)}|=\lim_{n\to+\infty}|u_{B_n}-u_{B_0}|\le\sum_{n=0}^\infty|u_{B_n}-u_{B_{n+1}}|\\ &\lesssim\sum_{n=0}^\infty\left(\Psi(2^{-n}D)M_R\Gamma(u)(x)\right)^{1/p}\lesssim\left(\Psi(D)M_R\Gamma(u)(x)\right)^{1/p}, \end{align}\] where the last inequality follows from Equation (6 ). Similarly, we have \[|u(y)-u_{B(y,D)}|\lesssim\left(\Psi(D)M_R\Gamma(u)(y)\right)^{1/p}.\] Moreover \[\begin{align} &|u_{B(x,D)}-u_{B(y,D)}|\le\left(\mathchoice {{\setbox 0=\displaystyle{\textstyle-}{\int}\vcenter{\textstyle- }\kern-.6\wd 0}}{{\setbox 0=\textstyle{\scriptstyle-}{\int}\vcenter{\scriptstyle- }\kern-.6\wd 0}}{{\setbox 0=\scriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}}{{\setbox 0=\scriptscriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}}\!\int_{B(x,D)}\mathchoice {{\setbox 0=\displaystyle{\textstyle-}{\int}\vcenter{\textstyle- }\kern-.6\wd 0}}{{\setbox 0=\textstyle{\scriptstyle-}{\int}\vcenter{\scriptstyle- }\kern-.6\wd 0}}{{\setbox 0=\scriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}}{{\setbox 0=\scriptscriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}}\!\int_{B(y,D)}|u(z_1)-u(z_2)|^pm(\mathrm{d} z_1)m(\mathrm{d} z_2)\right)^{1/p}\\ &\lesssim\left(\mathchoice {{\setbox 0=\displaystyle{\textstyle-}{\int}\vcenter{\textstyle- }\kern-.6\wd 0}}{{\setbox 0=\textstyle{\scriptstyle-}{\int}\vcenter{\scriptstyle- }\kern-.6\wd 0}}{{\setbox 0=\scriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}}{{\setbox 0=\scriptscriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}}\!\int_{B(x,2D)}\mathchoice {{\setbox 0=\displaystyle{\textstyle-}{\int}\vcenter{\textstyle- }\kern-.6\wd 0}}{{\setbox 0=\textstyle{\scriptstyle-}{\int}\vcenter{\scriptstyle- }\kern-.6\wd 0}}{{\setbox 0=\scriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}}{{\setbox 0=\scriptscriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}}\!\int_{B(x,2D)}|u(z_1)-u(z_2)|^pm(\mathrm{d} z_1)m(\mathrm{d} z_2)\right)^{1/p}\\ &\lesssim\left(\frac{1}{V(x,2D)}\int_{B(x,2D)}|u-u_{B(x,2D)}|^p\mathrm{d} m\right)^{1/p}\lesssim\left(\frac{\Psi(2D)}{V(x,2D)}\int_{B(x,2A_{PI}D)}\mathrm{d}\Gamma(u)\right)^{1/p}\\ &\lesssim\left(\Psi(D)\frac{\Gamma(u)(B(x,2A_{PI}D))}{V(x,2A_{PI}D)}\right)^{1/p}\le\left(\Psi(D)M_{R}\Gamma(u)(x)\right)^{1/p}. \end{align}\] In summary, we have \[\begin{align} &|u(x)-u(y)|\le|u(x)-u_{B(x,D)}|+|u(y)-u_{B(y,D)}|+|u_{B(x,D)}-u_{B(y,D)}|\\ &\lesssim\Psi(D)^{1/p}\left(M_{R}\Gamma(u)(x)^{1/p}+M_{R}\Gamma(u)(y)^{1/p}\right), \end{align}\] which implies \[|u(x)-u(y)|^p\lesssim\Psi(D)\left(M_{R}\Gamma(u)(x)+M_{R}\Gamma(u)(y)\right).\] ◻

Proof of the lower bound through the implication (17 ). For notational convenience, we may assume that \(\mathrm{diam}(X)=+\infty\). For any \(R,\varepsilon>0\) with \(\varepsilon\le R\). If \(R/\varepsilon\lesssim1\), then since \(\Psi\) is increasing, we have \[\left(\frac{R}{\varepsilon}\right)^p\lesssim1\le \frac{\Psi(R)}{\Psi(\varepsilon)}.\] We show that if \(R/\varepsilon\gtrsim1\), then \[\frac{\Psi(R)}{\Psi(\varepsilon)}\gtrsim \left(\frac{R}{\varepsilon}\right)^p.\]

Take \(\overline{x},\overline{y}\in X\) with \(R<d(\overline{x},\overline{y})< R+\varepsilon\), take an \(\varepsilon\)-net \(V\) with \(\{\overline{x},\overline{y}\}\in V\), let \(\{\psi_z:z\in V\}\) be the family of functions given by Lemma 2, let \[u=\sum_{z\in V}N_\varepsilon(\overline{x},z)\psi_z.\] By [item95COspt], we have \(u\equiv u(\overline{x})=N_\varepsilon(\overline{x},\overline{x})=0\) in \(B(\overline{x},\varepsilon/4)\) and \(u\equiv u(\overline{y})=N_\varepsilon(\overline{x},\overline{y})\asymp \frac{d(\overline{x},\overline{y})}{\varepsilon}\asymp \frac{R}{\varepsilon}\) in \(B(\overline{y},\varepsilon/4)\). By 8 , the above summation is locally a finite summation of functions in \(\mathcal{F}\), hence \(u\in\mathcal{F}_{loc}\).

Firstly, we show that for any \(\overline{z}\in V\) \[\Gamma(u)(B(\overline{z},\frac{5}{4}\varepsilon))\lesssim \frac{V(\overline{z},\varepsilon)}{\Psi(\varepsilon)}.\] Let \(N_{\overline{z}}=\{z\in V:B(z,\frac{5}{4}\varepsilon)\cap B(\overline{z},\frac{5}{4}\varepsilon)\ne\emptyset\}\). By 8 , there exists some positive integer \(N\) depending only on \(C_{VD}\) such that \(\# N_{\overline{z}}\le N\). By [item95COunit], we have \[u=\sum_{z\in V}N_\varepsilon(\overline{x},z)\psi_z=\sum_{z\in V}\left( N_\varepsilon(\overline{x},z)-N_\varepsilon(\overline{x},\overline{z})\right)\psi_z+N_\varepsilon(\overline{x},\overline{z}).\] In particular, in \(B(\overline{z},\frac{5}{4}\varepsilon)\), by [item95COspt], we have \[u=\sum_{z\in N_{\overline{z}}}\left( N_\varepsilon(\overline{x},z)-N_\varepsilon(\overline{x},\overline{z})\right)\psi_z+N_\varepsilon(\overline{x},\overline{z}),\] hence \[\begin{align} &\Gamma(u)(B(\overline{z},\frac{5}{4}\varepsilon))=\Gamma\left(\sum_{z\in N_{\overline{z}}}\left( N_\varepsilon(\overline{x},z)-N_\varepsilon(\overline{x},\overline{z})\right)\psi_z+N_\varepsilon(\overline{x},\overline{z})\right)(B(\overline{z},\frac{5}{4}\varepsilon))\\ &=\Gamma\left(\sum_{z\in N_{\overline{z}}}\left( N_\varepsilon(\overline{x},z)-N_\varepsilon(\overline{x},\overline{z})\right)\psi_z\right)(B(\overline{z},\frac{5}{4}\varepsilon))\\ &\le C_1\sum_{z\in N_{\overline{z}}}\Gamma\left(\left( N_\varepsilon(\overline{x},z)-N_\varepsilon(\overline{x},\overline{z})\right)\psi_z\right)(B(\overline{z},\frac{5}{4}\varepsilon))\\ &\le C_1\sum_{z\in N_{\overline{z}}}|N_\varepsilon(\overline{x},z)-N_\varepsilon(\overline{x},\overline{z})|^p\mathcal{E}(\psi_z), \end{align}\] where \(C_1>0\) depends only on \(p,N\). For any \(z\in N_{\overline{z}}\), we have \(d(z,\overline{z})<\frac{5}{2}\varepsilon\), then by 7 , we have \(|N_\varepsilon(\overline{x},z)-N_\varepsilon(\overline{x},\overline{z})|\le \lceil {\frac{5}{2}C_{cc}}\rceil\). Combining this with [item95COenergy] and 8 , we have \[\begin{align} &\Gamma(u)(B(\overline{z},\frac{5}{4}\varepsilon))\le C_1\sum_{z\in N_{\overline{z}}}\lceil{\frac{5}{2}C_{cc}} \rceil^pC_{cut}\frac{V(z,\varepsilon)}{\Psi(\varepsilon)}\\ &\le C_1\sum_{z\in N_{\overline{z}}}\lceil {\frac{5}{2}C_{cc}}\rceil^pC_{cut}C_2\frac{V(\overline{z},\varepsilon)}{\Psi(\varepsilon)}\le C_3\frac{V(\overline{z},\varepsilon)}{\Psi(\varepsilon)}, \end{align}\] where \(C_2>0\) depends only on \(C_{VD}\), and \(C_3=\lceil {\frac{5}{2}C_{cc}}\rceil^pNC_1C_2C_{cut}\).

Secondly, we show that for any \(\overline{z}\in V\) and any \(r>0\) \[\Gamma(u)(B(\overline{z},r))\lesssim \frac{V(\overline{z},r)}{\Psi(\varepsilon)}.\] If \(r\le\varepsilon/4\), then in \(B(\overline{z},r)\), we have \(u\equiv N_\varepsilon(\overline{x},\overline{z})\), hence \(\Gamma(u)(B(\overline{z},r))=0\), the result is trivial. We may assume that \(r>\varepsilon/4\), then \[\begin{align} &\Gamma(u)(B(\overline{z},r))\le\sum_{z\in V:B(z,\frac{5}{4}\varepsilon)\cap B(\overline{z},r)\ne\emptyset}\Gamma(u)(B(z,\frac{5}{4}\varepsilon))\\ &\le \sum_{z\in V:B(z,\frac{5}{4}\varepsilon)\cap B(\overline{z},r)\ne\emptyset}C_3 \frac{V(z,\varepsilon)}{\Psi(\varepsilon)}=\frac{C_3}{\Psi(\varepsilon)}\int_X\sum_{z\in V:B(z,\frac{5}{4}\varepsilon)\cap B(\overline{z},r)\ne\emptyset}1_{B(z,\varepsilon)}\mathrm{d} m. \end{align}\] By 8 , there exists some positive integer \(M\) depending only on \(C_{VD}\) such that \[\sum_{z\in V:B(z,\frac{5}{4}\varepsilon)\cap B(\overline{z},r)\ne\emptyset}1_{B(z,\varepsilon)}\le M1_{\cup_{z\in V:B(z,\frac{5}{4}\varepsilon)\cap B(\overline{z},r)\ne\emptyset}B(z,\varepsilon)},\] hence \[\begin{align} &\Gamma(u)(B(\overline{z},r))\le\frac{C_3}{\Psi(\varepsilon)}Mm\left({\bigcup_{z\in V:B(z,\frac{5}{4}\varepsilon)\cap B(\overline{z},r)\ne\emptyset}B(z,\varepsilon)}\right)\\ &\le \frac{C_3M}{\Psi(\varepsilon)}m(B(\overline{z},r+\frac{5}{2}\varepsilon))\le \frac{C_3M}{\Psi(\varepsilon)}m(B(\overline{z},11r))\le\frac{C_4}{\Psi(\varepsilon)}V(\overline{z},r), \end{align}\] where \(C_4=C_3C_{VD}^4M\).

Finally, in Lemma 3, let \(B=B(\overline{x},4A_{max}R)\), then \(\overline{x},\overline{y}\in A_{max}^{-1}B=B(\overline{x},4R)\), since \(B(\overline{x},\frac{\varepsilon}{4})\cup B(\overline{y},\frac{\varepsilon}{4})\subseteq B(\overline{x},4R)\), and \(u\equiv u(\overline{x})=N_\varepsilon(\overline{x},\overline{x})=0\) in \(B(\overline{x},\frac{\varepsilon}{4})\), \(u\equiv u(\overline{y})=N_\varepsilon(\overline{x},\overline{y})\asymp \frac{R}{\varepsilon}\) in \(B(\overline{y},\frac{\varepsilon}{4})\), we may assume that \(\overline{x}, \overline{y}\) are Lebesgue points of \(u\), then \[\begin{align} |u(\overline{x})-u(\overline{y})|^p\le C_{max}\Psi(d(\overline{x},\overline{y}))\left(M_{4A_{max}R}\Gamma(u)(\overline{x})+M_{4A_{max}R}\Gamma(u)(\overline{y})\right), \end{align}\] where \[\begin{align} &M_{4A_{max}R}\Gamma(u)(\overline{x})=\sup_{r\in(0,4A_{max}R)}\frac{\Gamma(u)(B(\overline{x},r))}{V(\overline{x},r)}\le \frac{C_4}{\Psi(\varepsilon)},\\ &M_{4A_{max}R}\Gamma(u)(\overline{y})=\sup_{r\in(0,4A_{max}R)}\frac{\Gamma(u)(B(\overline{y},r))}{V(\overline{y},r)}\le \frac{C_4}{\Psi(\varepsilon)}. \end{align}\] Hence \[\left(\frac{R}{\varepsilon}\right)^p\asymp N_\varepsilon(\overline{x},\overline{y})^p=|u(\overline{x})-u(\overline{y})|^p\le C_{max}\Psi(d(\overline{x},\overline{y}))\frac{2C_4}{\Psi(\varepsilon)}\asymp\frac{\Psi(R)}{\Psi(\varepsilon)},\] which gives \[\left(\frac{R}{\varepsilon}\right)^p\lesssim \frac{\Psi(R)}{\Psi(\varepsilon)}.\] ◻

Proof. Let \(C_1=C_{cap}\), \(A=A_{cap}\) be the constants in 15 . Let \(L=4A+4\) and let \(V\) be an \(\frac{r}{L}\)-net. For any \(v\in V\), by 15 , there exists a cutoff function \(\phi_v\in \mathcal{F}\) for \(B(v,\frac{r}{L})\subseteq B(v,\frac{Ar}{L})\) such that \[\mathcal{E}(\phi_v)=\int_{B(v,\frac{Ar}{L})}\mathrm{d}\Gamma(\phi_v)\le 2C_1\frac{V(v,\frac{r}{L})}{\Psi(\frac{r}{L})}.\] Let \({V}_1=V\cap B(x,R+\frac{r}{2})\), then by 8 , we have \(\#V_1<+\infty\). Let \[\phi=\max_{v\in{V}_1}\phi_v,\] then by 11 , we have \(\phi\in \mathcal{F}\). Moreover, \(0\le\phi\le1\) in \(X\), \(\phi=1\) in \(B(x,R+\frac{r}{2}-\frac{r}{L})\supseteq\overline{B(x,R)}\), \(\mathrm{supp}(\phi)\subseteq B(x,R+\frac{r}{2}+\frac{Ar}{L})\subseteq B(x,R+r)\), hence \(\phi\in \mathcal{F}\) is a cutoff function for \(B(x,R)\subseteq B(x,R+r)\) and \[\mathrm{supp}(\Gamma(\phi))\subseteq B(x,R+\frac{r}{2}+\frac{Ar}{L})\backslash{B(x,R+\frac{r}{2}-\frac{r}{L})}.\] Let \(W={V}_1\backslash B(x,R+\frac{r}{2}-\frac{r}{L}-\frac{Ar}{L})\), then \[\phi=\max_{v\in W}\phi_v\text{ in }B(x,R+\frac{r}{2}+\frac{Ar}{L})\backslash{B(x,R+\frac{r}{2}-\frac{r}{L})},\] hence \[\begin{align} &\mathcal{E}(\phi)=\int_{B(x,R+\frac{r}{2}+\frac{Ar}{L})\backslash{B(x,R+\frac{r}{2}-\frac{r}{L})}}\mathrm{d}\Gamma(\phi)\overset{(\star)}{\scalebox{2}[1]{\le}}\sum_{v\in W}\int_{{B(x,R+\frac{r}{2}+\frac{Ar}{L})\backslash{B(x,R+\frac{r}{2}-\frac{r}{L})}}}\mathrm{d}\Gamma(\phi_v)\\ &\le\sum_{v\in W}\int_{{B(v,\frac{Ar}{L})}}\mathrm{d}\Gamma(\phi_v)\le 2C_1\sum_{v\in W}\frac{V(v,\frac{r}{L})}{\Psi(\frac{r}{L})}=\frac{2C_1}{\Psi(\frac{r}{L})}\int_X\left(\sum_{v\in W}1_{B(v,\frac{r}{L})}\right)\mathrm{d} m, \end{align}\] where \((\star)\) follows from 11 . By 8 , there exists some positive integer \(N\) depending only on \(C_{VD}\) such that \[\sum_{v\in W}1_{B(v,\frac{r}{L})}\le N1_{\cup_{v\in W}B(v,\frac{r}{L})},\] where \[{\bigcup_{v\in W}B(v,\frac{r}{L})}\subseteq B(x,R+\frac{r}{2}+\frac{r}{L})\backslash \overline{B(x,R+\frac{r}{2}-\frac{r}{L}-\frac{Ar}{L}-\frac{r}{L})}\subseteq B(x,R+r)\backslash\overline{B(x,R)}.\] Therefore, we have \[\begin{align} &\mathrm{cap}\left(B(x,R),X\backslash B(x,R+r)\right)\le \mathcal{E}(\phi)\le \frac{2C_1N}{\Psi(\frac{r}{L})}m\left(B(x,R+r)\backslash\overline{B(x,R)}\right)\\ &\le \frac{2C_1NC_2}{\Psi({r})}m \left(B(x,R+r)\backslash\overline{B(x,R)}\right)=C \frac{m \left(B(x,R+r)\backslash\overline{B(x,R)}\right)}{\Psi(r)}, \end{align}\] where \(C_2>0\) depends only on \(C_\Psi,L\), and \(C=2C_1NC_2\). ◻

Proof of the lower bound through the implication (18 ). For notational convenience, we may assume that \(\mathrm{diam}(X)=+\infty\). Let \(A=A_{cap}\) be the constant in \(\text{cap}(\Psi)\). For any \(R,r>0\) with \(r\le R\), without loss of generality, we may assume that \(R/r\gtrsim1\), let \(N\) be the integer satisfying \(Nr\le (A-1)R<(N+1)r\), then \(N\asymp R/r\gtrsim1\). By 8 and 15 , applying Lemma 4, there exists \(C_1>0\) such that for any \(n=0,\ldots,N-1\), there exists a cutoff function \(\phi_n\in \mathcal{F}\) for \(B(x,R+nr)\subseteq B(x,R+(n+1)r)\) such that \[\mathcal{E}(\phi_n)\le C_1 \frac{m \left(B(x,R+(n+1)r)\backslash \overline{B(x,R+nr)}\right)}{\Psi(r)}.\] Let \(\phi=\frac{1}{N}\sum_{n=0}^{N-1}\phi_n\), then \(\phi\in \mathcal{F}\) is a cutoff function for \(B(x,R)\subseteq B(x,R+Nr)\), hence also a cutoff function for \(B(x,R)\subseteq B(x,AR)\). By 16 , we have \[\begin{align} &\frac{1}{C_{cap}}\frac{V(x,R)}{\Psi(R)}\le \mathrm{cap}\left(B(x,R),X\backslash B(x,AR)\right)\le \mathcal{E}(\phi)\overset{(\star)}{\scalebox{2}[1]{\le}}\frac{1}{N^p}\sum_{n=0}^{N-1}\mathcal{E}\left(\phi_n\right)\\ &\le \frac{C_1}{N^p} \sum_{n=0}^{N-1}\frac{m \left(B(x,R+(n+1)r)\backslash \overline{B(x,R+nr)}\right)}{\Psi(r)}\le \frac{C_1}{N^p} \frac{V (x,AR)}{\Psi(r)}\overset{(\diamond)}{\scalebox{2}[1]{\le}} \frac{C_1C_2}{N^p} \frac{V(x,R)}{\Psi(r)}, \end{align}\] where \((\star)\) follows from 11 , \((\diamond)\) follows from 8 , and \(C_2>0\) depends only on \(C_{VD}, A\). Therefore, we have \[\frac{\Psi(R)}{\Psi(r)}\gtrsim N^p\asymp \left(\frac{R}{r}\right)^p.\] ◻

Proof. By [item95tree2], any two distinct \(n\)-cells intersect at most one point and any \(n\)-cell contains at most two points shared with other \(n\)-cells, hence there exist non-negative integers \(M,N\le\sup_{\mathbb{Z}}\mathbf{b}-1\), \(\mathbf{t},\mathbf{s}\in K\), and distinct \(n\)-cells \(\widetilde{K}_1\), …\(\widetilde{K}_M\) and \(\widehat{K}_1\), …\(\widehat{K}_N\) such that \(\widetilde{K}_k\cap K=\{\mathbf{t}\}\) for any \(k=1,\ldots,M\), and \(\widehat{K}_l\cap K=\{\mathbf{s}\}\) for any \(l=1,\ldots,N\), here it is possible that \(M\) or \(N\) is 0, which means that \(K\) has only one point shared with other \(n\)-cells. Let \(\mathbf{t}^{(k)}\in\widetilde{K}_k\) (resp. \(\mathbf{s}^{(l)}\in\widehat{K}_l\)) denote the other point in \(\widetilde{K}_k\) (resp. \(\widehat{K}_l\)) which is shared with other \(n\)-cells, if such a point exists; see Figure 6.

It is obvious that \(\mathcal{N}(K)=K\cup\cup_{k=1}^M\widetilde{K}_k\cup\cup_{l=1}^N\widehat{K}_l\). Let \(\phi_K\) be given by \(\phi_K=1\) on \(K\), \(\phi_K=0\) on \(\mathcal{T}(\mathbf{b})\backslash\mathcal{N}(K)\), on each \(\widetilde{K}_k\), \[\phi_K=\frac{\mathbf{d}_{\mathcal{T}(\mathbf{b})}(c(\cdot,\mathbf{t},\mathbf{t}^{(k)}),\mathbf{t}^{(k)})}{\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{t},\mathbf{t}^{(k)})}=\frac{1}{2^n}{\mathbf{d}_{\mathcal{T}(\mathbf{b})}(c(\cdot,\mathbf{t},\mathbf{t}^{(k)}),\mathbf{t}^{(k)})},\] and on each \(\widehat{K}_l\), \[\phi_K=\frac{\mathbf{d}_{\mathcal{T}(\mathbf{b})}(c(\cdot,\mathbf{s},\mathbf{s}^{(l)}),\mathbf{s}^{(l)})}{\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{s},\mathbf{s}^{(l)})}=\frac{1}{2^n}{\mathbf{d}_{\mathcal{T}(\mathbf{b})}(c(\cdot,\mathbf{s},\mathbf{s}^{(l)}),\mathbf{s}^{(l)})}.\] It is obvious that \(\phi_K\in C_c(\mathcal{T}(\mathbf{b}))\) is well-defined. For \(\widetilde{K}_k\), on \([\mathbf{t},\mathbf{t}^{(k)}]\), we have \(\phi_K=\frac{\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\cdot,\mathbf{t}^{(k)})}{\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{t},\mathbf{t}^{(k)})}=\frac{1}{2^n}{\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\cdot,\mathbf{t}^{(k)})}\), and on each component of \(\widetilde{K}_k\backslash[\mathbf{t},\mathbf{t}^{(k)}]\), since \(c(\cdot,\mathbf{t},\mathbf{t}^{(k)})\) is constant, we have \(\phi_K\) is constant, which gives \[\int_{\widetilde{K}_k}|\nabla_\mathcal{T}\phi_K|^p\mathrm{d}\lambda_{\mathcal{T}(\mathbf{b})}=\int_{[\mathbf{t},\mathbf{t}^{(k)}]}\frac{1}{2^{pn}}\mathrm{d}\lambda_{\mathcal{T}(\mathbf{b})}=\frac{1}{2^{(p-1)n}}.\] Similarly, we have \[\int_{\widehat{K}_l}|\nabla_\mathcal{T}\phi_K|^p\mathrm{d}\lambda_{\mathcal{T}(\mathbf{b})}=\frac{1}{2^{(p-1)n}}.\] Hence \[\begin{align} &\int_{\mathcal{T}(\mathbf{b})}|\nabla_\mathcal{T}\phi_K|^p\mathrm{d}\lambda_{\mathcal{T}(\mathbf{b})}\\ &=\sum_{k=1}^M\int_{\widetilde{K}_k}|\nabla_\mathcal{T}\phi_K|^p\mathrm{d}\lambda_{\mathcal{T}(\mathbf{b})}+\sum_{l=1}^N\int_{\widehat{K}_l}|\nabla_\mathcal{T}\phi_K|^p\mathrm{d}\lambda_{\mathcal{T}(\mathbf{b})}\\ &=(M+N)\frac{1}{2^{(p-1)n}}\le \frac{2\left(\sup_{\mathbb{Z}}\mathbf{b}-1\right)}{2^{(p-1)n}}, \end{align}\] and \(\phi_K\in\mathcal{F}^\mathcal{T}\). Moreover, for any \(f\in\mathcal{F}^\mathcal{T}\subseteq C(\mathcal{T}(\mathbf{b}))\), there exists \(x_0\in\mathcal{N}(K)\) such that \(|f(x_0)|^p=\mathchoice {{\setbox 0=\displaystyle{\textstyle-}{\int} \vcenter{\textstyle- }\kern-.6\wd 0}}{{\setbox 0=\textstyle{\scriptstyle-}{\int} \vcenter{\scriptstyle- }\kern-.6\wd 0}}{{\setbox 0=\scriptstyle{\scriptscriptstyle-}{\int} \vcenter{\scriptscriptstyle- }\kern-.6\wd 0}}{{\setbox 0=\scriptscriptstyle{\scriptscriptstyle-}{\int} \vcenter{\scriptscriptstyle- }\kern-.6\wd 0}}\!\int_{\mathcal{N}(K)}|f|^p\mathrm{d} \mathbf{m}_{\mathcal{T}(\mathbf{b})}\), and \[\int_{\mathcal{N}(K)}|f|^p|\nabla_\mathcal{T}\phi_K|^p\mathrm{d}\lambda_{\mathcal{T}(\mathbf{b})}=\frac{1}{2^{pn}}\left(\sum_{k=1}^M\int_{[\mathbf{t},\mathbf{t}^{(k)}]}|f|^p\mathrm{d}\lambda_{\mathcal{T}(\mathbf{b})}+\sum_{l=1}^N\int_{[\mathbf{s},\mathbf{s}^{(l)}]}|f|^p\mathrm{d}\lambda_{\mathcal{T}(\mathbf{b})}\right).\] For any \(x\in\mathcal{N}(K)\), we have \([x,x_0]\subseteq\mathcal{N}(K)\) and \(\mathbf{d}_{\mathcal{T}(\mathbf{b})}(x,x_0)\le3\cdot2^n\), hence by Hölder’s inequality, we have \[\begin{align} &|f(x)-f(x_0)|\le\int_{[x,x_0]}|\nabla_\mathcal{T} f|\mathrm{d}\lambda_{\mathcal{T}(\mathbf{b})}\\ &\le \mathbf{d}_{\mathcal{T}(\mathbf{b})}(x,x_0)^{1-\frac{1}{p}}\left(\int_{[x,x_0]}|\nabla_\mathcal{T} f|^p\mathrm{d}\lambda_{\mathcal{T}(\mathbf{b})}\right)^{\frac{1}{p}}\le\left(3\cdot2^n\right)^{1-\frac{1}{p}}\left(\int_{\mathcal{N}(K)}|\nabla_\mathcal{T} f|^p\mathrm{d}\lambda_{\mathcal{T}(\mathbf{b})}\right)^{\frac{1}{p}}, \end{align}\] which gives \[\begin{align} &|f(x)|^p\le2^{p-1}\left(|f(x)-f(x_0)|^p+|f(x_0)|^p\right)\\ &\le2^{p-1}\left(\left(3\cdot2^n\right)^{p-1}\int_{\mathcal{N}(K)}|\nabla_\mathcal{T} f|^p\mathrm{d}\lambda_{\mathcal{T}(\mathbf{b})}+\mathchoice {{\setbox 0=\displaystyle{\textstyle-}{\int}\vcenter{\textstyle- }\kern-.6\wd 0}}{{\setbox 0=\textstyle{\scriptstyle-}{\int}\vcenter{\scriptstyle- }\kern-.6\wd 0}}{{\setbox 0=\scriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}}{{\setbox 0=\scriptscriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}}\!\int_{\mathcal{N}(K)}|f|^p\mathrm{d} \mathbf{m}_{\mathcal{T}(\mathbf{b})}\right). \end{align}\] Hence \[\begin{align} &\int_{\mathcal{N}(K)}|f|^p|\nabla_\mathcal{T}\phi_K|^p\mathrm{d}\lambda_{\mathcal{T}(\mathbf{b})}\\ &\le \frac{1}{2^{pn}}\cdot2^{p-1}\left(\left(3\cdot2^n\right)^{p-1}\int_{\mathcal{N}(K)}|\nabla_\mathcal{T} f|^p\mathrm{d}\lambda_{\mathcal{T}(\mathbf{b})}+\mathchoice {{\setbox 0=\displaystyle{\textstyle-}{\int}\vcenter{\textstyle- }\kern-.6\wd 0}}{{\setbox 0=\textstyle{\scriptstyle-}{\int}\vcenter{\scriptstyle- }\kern-.6\wd 0}}{{\setbox 0=\scriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}}{{\setbox 0=\scriptscriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}}\!\int_{\mathcal{N}(K)}|f|^p\mathrm{d} \mathbf{m}_{\mathcal{T}(\mathbf{b})}\right)\cdot(M+N)\cdot2^n\\ &\le2^p3^{p-1}(\sup_{\mathbb{Z}}\mathbf{b}-1)\int_{\mathcal{N}(K)}|\nabla_\mathcal{T} f|^p\mathrm{d}\lambda_{\mathcal{T}(\mathbf{b})}+\frac{2^p(\sup_{\mathbb{Z}}\mathbf{b}-1)}{2^{(p-1)n}}\mathchoice {{\setbox 0=\displaystyle{\textstyle-}{\int}\vcenter{\textstyle- }\kern-.6\wd 0}}{{\setbox 0=\textstyle{\scriptstyle-}{\int}\vcenter{\scriptstyle- }\kern-.6\wd 0}}{{\setbox 0=\scriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}}{{\setbox 0=\scriptscriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}}\!\int_{\mathcal{N}(K)}|f|^p\mathrm{d} \mathbf{m}_{\mathcal{T}(\mathbf{b})}. \end{align}\] ◻

Proof. We only give the proof of the uniform denseness of \(\mathcal{F}^\mathcal{T}\cap C_c(\mathcal{T}(\mathbf{b}))\) in \(C_c(\mathcal{T}(\mathbf{b}))\). The other parts are easy and similar to the proof of \((\int_{\mathbb{R}}|f'(x)|^p\mathrm{d} x,W^{1,p}(\mathbb{R}))\) is a \(p\)-energy on \(\mathbb{R}\) with a \(p\)-energy measure given by \((f,A)\mapsto\int_A|f'(x)|^p\mathrm{d} x\).

Indeed, since \(\mathcal{F}^\mathcal{T}\cap C_c(\mathcal{T}(\mathbf{b}))\) is a sub-algebra of \(C_c(\mathcal{T}(\mathbf{b}))\), by the Stone-Weierstraß theorem, we only need to show that \(\mathcal{F}^\mathcal{T}\cap C_c(\mathcal{T}(\mathbf{b}))\) separates points and vanishes nowhere. For any distinct \(\mathbf{t},\mathbf{s}\in\mathcal{T}(\mathbf{b})\), there exist \(n\in\mathbb{Z}\) sufficiently small and an \(n\)-cell \(K\) such that \(\mathbf{t}\in K\) and \(\mathbf{s}\not\in\mathcal{N}(K)\). Let \(\phi_K\in\mathcal{F}^\mathcal{T}\cap C_c(\mathcal{T}(\mathbf{b}))\) be given by Lemma 5, then \(\phi_K(\mathbf{t})=1\ne0=\phi_K(\mathbf{s})\). ◻

Proof. For any \(x,y\in {B_{\mathcal{T}(\mathbf{b})}}\), by the tree property, we have \([x,y]\subseteq{B_{\mathcal{T}(\mathbf{b})}}\), by Hölder’s inequality, we have \[\begin{align} &|f(x)-f(y)|\le\int_{[x,y]}|\nabla_\mathcal{T} f|\mathrm{d}\lambda_{\mathcal{T}(\mathbf{b})}\\ &\le \mathbf{d}_{\mathcal{T}(\mathbf{b})}(x,y)^{1-\frac{1}{p}}\left(\int_{[x,y]}|\nabla_\mathcal{T} f|^p\mathrm{d}\lambda_{\mathcal{T}(\mathbf{b})}\right)^{\frac{1}{p}}\le(2r)^{1-\frac{1}{p}}\left(\int_{{B_{\mathcal{T}(\mathbf{b})}}}|\nabla_\mathcal{T} f|^p\mathrm{d}\lambda_{\mathcal{T}(\mathbf{b})}\right)^{\frac{1}{p}}. \end{align}\] Hence \[\begin{align} &\int_{B_{\mathcal{T}(\mathbf{b})}}\left| f-\mathchoice {{\setbox 0=\displaystyle{\textstyle-}{\int}\vcenter{\textstyle- }\kern-.6\wd 0}}{{\setbox 0=\textstyle{\scriptstyle-}{\int}\vcenter{\scriptstyle- }\kern-.6\wd 0}}{{\setbox 0=\scriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}}{{\setbox 0=\scriptscriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}}\!\int_{B_{\mathcal{T}(\mathbf{b})}}f\mathrm{d}\mathbf{m}_{\mathcal{T}(\mathbf{b})}\right|^p\mathrm{d}\mathbf{m}_{\mathcal{T}(\mathbf{b})}\\ &\le \frac{1}{\mathbf{m}_{\mathcal{T}(\mathbf{b})}({B_{\mathcal{T}(\mathbf{b})}})}\int_{B_{\mathcal{T}(\mathbf{b})}}\int_{B_{\mathcal{T}(\mathbf{b})}}|f(x)-f(y)|^p\mathbf{m}_{\mathcal{T}(\mathbf{b})}(\mathrm{d} x)\mathbf{m}_{\mathcal{T}(\mathbf{b})}(\mathrm{d} y)\\ &\le {\mathbf{m}_{\mathcal{T}(\mathbf{b})}({B_{\mathcal{T}(\mathbf{b})}})}(2r)^{p-1}\int_{{B_{\mathcal{T}(\mathbf{b})}}}|\nabla_\mathcal{T} f|^p\mathrm{d}\lambda_{\mathcal{T}(\mathbf{b})}\asymp r^{p-1}V_{\mathbf{b}}(r)\int_{{B_{\mathcal{T}(\mathbf{b})}}}|\nabla_\mathcal{T} f|^p\mathrm{d}\lambda_{\mathcal{T}(\mathbf{b})}. \end{align}\] ◻

Proof. Write \(B_{\mathcal{T}(\mathbf{b})}=B_{\mathcal{T}(\mathbf{b})}(x,r)\). Let \(n\) be the integer satisfying \(2^{n-1}\le r<2^{n}\), there exists an \(n\)-cell \(K\) such that \(x\in K\). Let \(\mathcal{N}\left({{K}}\right)=\bigcup_{\widetilde{K}:n\text{-cell,}\widetilde{K}\cap {K}\ne\emptyset}\widetilde{K}\) be the \(n\)-cell neighborhood of \(K\) in \(\mathcal{T}(\mathbf{b})\), then \(B_{\mathcal{T}(\mathbf{b})}(x,r)\subseteq\mathcal{N}(K)\), see Figure 6. For any \(n\)-cell \(\widetilde{K}\) with \(\widetilde{K}\cap K\ne\emptyset\), by Lemma 5, there exists \(\phi_{\widetilde{K}}\in\mathcal{F}^\mathcal{T}\cap C_c(\mathcal{T}(\mathbf{b}))\) with \(0\le\phi_{\widetilde{K}}\le1\) in \(\mathcal{T}(\mathbf{b})\), \(\phi_{{\widetilde{K}}}=1\) on \({\widetilde{K}}\), \(\phi_{{\widetilde{K}}}=0\) on \(\mathcal{T}(\mathbf{b})\backslash\mathcal{N}({\widetilde{K}})\) such that \(\mathcal{E}^\mathcal{T}(\phi_{{\widetilde{K}}})\le \frac{2(\sup_\mathbb{Z}\mathbf{b}-1)}{2^{(p-1)n}}\).

Let \[\label{eq95tree95cutoff} \phi_\mathcal{T}=\max_{\widetilde{K}:n\text{-cell, }\widetilde{K}\cap {K}\ne\emptyset}\phi_{\widetilde{K}}.\tag{19}\] Since \[\label{eq95tree95num} \#\{\widetilde{K}:n\text{-cell, }\widetilde{K}\cap {K}\ne\emptyset\}\le 2\sup_{\mathbb{Z}}\mathbf{b}-1,\tag{20}\] we have \(\phi_\mathcal{T}\in\mathcal{F}^\mathcal{T}\cap C_c(\mathcal{T}(\mathbf{b}))\) is well-defined, \(0\le\phi_\mathcal{T}\le1\) in \(\mathcal{T}(\mathbf{b})\), \(\phi_\mathcal{T}=1\) on \(\mathcal{N}(K)\supseteq B_{\mathcal{T}(\mathbf{b})}(x,r)\), \(\phi_\mathcal{T}=0\) on \(\mathcal{T}(\mathbf{b})\backslash\cup_{\widetilde{K}:n\text{-cell, }\widetilde{K}\cap {K}\ne\emptyset}\mathcal{N}(\widetilde{K})\). Since \(\mathcal{N}(\widetilde{K})\subseteq B_{\mathcal{T}(\mathbf{b})}(x,2^{n+2})\) for any \(n\)-cell \(\widetilde{K}\) with \(\widetilde{K}\cap K\ne\emptyset\), we have \(\phi_\mathcal{T}=0\) on \(\mathcal{T}(\mathbf{b})\backslash B_{\mathcal{T}(\mathbf{b})}(x,2^{n+2})\supseteq \mathcal{T}(\mathbf{b})\backslash B_{\mathcal{T}(\mathbf{b})}(x,8r)\). Moreover, there exists \(C\ge1\) depending only on \(p, \sup_{\mathbb{Z}}\mathbf{b}\) such that \[\begin{align} &\mathcal{E}^\mathcal{T}(\phi_\mathcal{T})\le C\sum_{\widetilde{K}:n\text{-cell, }\widetilde{K}\cap {K}\ne\emptyset}\mathcal{E}^\mathcal{T}(\phi_{\widetilde{K}})\\ &\le C\left(2\sup_{\mathbb{Z}}\mathbf{b}-1\right)\frac{2(\sup_\mathbb{Z}\mathbf{b}-1)}{2^{(p-1)n}}\le \frac{4C\left(\sup_{\mathbb{Z}}\mathbf{b}\right)^2}{r^{p-1}}. \end{align}\] Hence \[\begin{align} &\mathrm{cap}_{\mathcal{T}(\mathbf{b})}(B_{\mathcal{T}(\mathbf{b})}(x,r),\mathcal{T}(\mathbf{b})\backslash(B_{\mathcal{T}(\mathbf{b})}(x,8r)))\le\frac{4C\left(\sup_{\mathbb{Z}}\mathbf{b}\right)^2}{r^{p-1}}\\ &={4C\left(\sup_{\mathbb{Z}}\mathbf{b}\right)^2}\frac{V_\mathbf{b}(r)}{r^{p-1}V_\mathbf{b}(r)}\asymp\frac{\mathbf{m}_{\mathcal{T}(\mathbf{b})}(B_{\mathcal{T}(\mathbf{b})}(x,r))}{r^{p-1}V_\mathbf{b}(r)}. \end{align}\] Moreover, for any \(f\in\mathcal{F}^\mathcal{T}\), by Lemma 5, we have \[\int_{\mathcal{N}(\widetilde{K})}|f|^p|\nabla_\mathcal{T}\phi_{\widetilde{K}}|^p\mathrm{d}\lambda_{\mathcal{T}(\mathbf{b})}\lesssim\int_{\mathcal{N}(\widetilde{K})}|\nabla_\mathcal{T} f|^p\mathrm{d}\lambda_{\mathcal{T}(\mathbf{b})}+\frac{1}{2^{(p-1)n}}\mathchoice {{\setbox 0=\displaystyle{\textstyle-}{\int}\vcenter{\textstyle- }\kern-.6\wd 0}}{{\setbox 0=\textstyle{\scriptstyle-}{\int}\vcenter{\scriptstyle- }\kern-.6\wd 0}}{{\setbox 0=\scriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}}{{\setbox 0=\scriptscriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}}\!\int_{\mathcal{N}(\widetilde{K})}|f|^p\mathrm{d}\mathbf{m}_{\mathcal{T}(\mathbf{b})}.\] Therefore, we have \[\begin{align} &\int_{B_{\mathcal{T}(\mathbf{b})}(x,8r)}|f|^p|\nabla_\mathcal{T}\phi_\mathcal{T}|^p\mathrm{d}\lambda_{\mathcal{T}(\mathbf{b})}\\ &\overset{(\star)}{\scalebox{2}[1]{\lesssim}}\sum_{\widetilde{K}:n\text{-cell, }\widetilde{K}\cap {K}\ne\emptyset}\int_{B_{\mathcal{T}(\mathbf{b})}(x,8r)}|f|^p|\nabla_\mathcal{T}\phi_{\widetilde{K}}|^p\mathrm{d}\lambda_{\mathcal{T}(\mathbf{b})}\\ &\overset{(\dagger)}{\scalebox{2}[1]{=}}\sum_{\widetilde{K}:n\text{-cell, }\widetilde{K}\cap {K}\ne\emptyset}\int_{\mathcal{N}(\widetilde{K})}|f|^p|\nabla_\mathcal{T}\phi_{\widetilde{K}}|^p\mathrm{d}\lambda_{\mathcal{T}(\mathbf{b})}\\ &\lesssim\sum_{\widetilde{K}:n\text{-cell, }\widetilde{K}\cap {K}\ne\emptyset}\left(\int_{\mathcal{N}(\widetilde{K})}|\nabla_\mathcal{T} f|^p\mathrm{d}\lambda_{\mathcal{T}(\mathbf{b})}+\frac{1}{2^{(p-1)n}}\mathchoice {{\setbox 0=\displaystyle{\textstyle-}{\int}\vcenter{\textstyle- }\kern-.6\wd 0}}{{\setbox 0=\textstyle{\scriptstyle-}{\int}\vcenter{\scriptstyle- }\kern-.6\wd 0}}{{\setbox 0=\scriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}}{{\setbox 0=\scriptscriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}}\!\int_{\mathcal{N}(\widetilde{K})}|f|^p\mathrm{d}\mathbf{m}_{\mathcal{T}(\mathbf{b})}\right)\\ &\overset{(\diamond)}{\scalebox{2}[1]{\lesssim}}\int_{B_{\mathcal{T}(\mathbf{b})}(x,8r)}|\nabla_\mathcal{T} f|^p\mathrm{d}\lambda_{\mathcal{T}(\mathbf{b})}+\frac{1}{r^{p-1}V_\mathbf{b}(r)}\int_{B_{\mathcal{T}(\mathbf{b})}(x,8r)}|f|^p\mathrm{d}\mathbf{m}_{\mathcal{T}(\mathbf{b})}, \end{align}\] where \((\star)\) follows from Equations (19 ) and (20 ), \((\dagger)\) follows from the fact that \(\mathcal{N}(\widetilde{K})\subseteq B_{\mathcal{T}(\mathbf{b})}(x,8r)\), and \((\diamond)\) follows from Equation (20 ) and the fact that \(\mathbf{m}_{\mathcal{T}(\mathbf{b})}(\mathcal{N}(\widetilde{K}))\asymp V_\mathbf{b}(r)\). ◻

Proof. We fix two points \((\mathbf{u}^{(1)},\mathbf{t}^{(1)})\) and \((\mathbf{u}^{(2)},\mathbf{t}^{(2)})\) from the equivalence classes \([(\mathbf{u}^{(1)},\mathbf{t}^{(1)})]\) and \([(\mathbf{u}^{(2)},\mathbf{t}^{(2)})]\). If \(\mathbf{u}^{(1)}=\mathbf{u}^{(2)}\), let \(\Gamma=\{\mathbf{u}^{(1)}\}\times[\mathbf{t}^{(1)},\mathbf{t}^{(2)}]\), then \(\mathcal{Q}(\Gamma)\) is a path connecting \(x,y\), and \(\mathcal{H}^1(\Gamma)=\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{t}^{(1)},\mathbf{t}^{(2)})<+\infty\). Hence we may assume that \(\mathbf{u}^{(1)}\ne\mathbf{u}^{(2)}\).

Let \(I=\left\{k\in\mathbb{Z}:\mathbf{u}^{(1)}(k)\ne\mathbf{u}^{(2)}(k)\right\}\). Since \(\mathbf{u}^{(1)}(k)\to0\), \(\mathbf{u}^{(2)}(k)\to0\) as \(k\to+\infty\), we have \(I\subseteq\mathbb{Z}\) is bounded from above. Write \(I=\{n_1,n_2,\ldots:n_1>n_2>\ldots\}\). Roughly speaking, \(I\) is the set of all the levels of wormholes needed to jump to go from \(x\) to \(y\), our proof is to jump through wormholes sorted by level: \(n_1,n_2,\ldots\).

Let \(\mathbf{s}^{(0)}=\mathbf{t}^{(1)}\), \(\mathbf{v}^{(0)}=\mathbf{u}^{(1)}\). Let \(\mathbf{s}^{(1)}\in\mathcal{T}(\mathbf{b})\) be a nearest level-\(n_1\) wormhole to \(\mathbf{s}^{(0)}\)⁷, then \(\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{s}^{(0)},\mathbf{s}^{(1)})\le2^{n_1-1}\). Let \(\mathbf{v}^{(1)}\in\mathcal{U}(\mathbf{g})\) be given by \(\mathbf{v}^{(1)}|_{\mathbb{Z}\backslash\{n_1\}}=\mathbf{v}^{(0)}|_{\mathbb{Z}\backslash\{n_1\}}\) and \(\mathbf{v}^{(1)}(n_1)=\mathbf{u}^{(2)}(n_1)\), then \([(\mathbf{v}^{(0)},\mathbf{s}^{(1)})]=[(\mathbf{v}^{(1)},\mathbf{s}^{(1)})]\).

Assume we have constructed \(\mathbf{s}^{(1)},\ldots,\mathbf{s}^{(l)}\), \(\mathbf{v}^{(1)},\ldots,\mathbf{v}^{(l)}\). Let \(\mathbf{s}^{(l+1)}\in\mathcal{T}(\mathbf{b})\) be a nearest level-\(n_{l+1}\) wormhole to \(\mathbf{s}^{(l)}\), then \(\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{s}^{(l)},\mathbf{s}^{(l+1)})\le2^{n_{l+1}-1}\). Let \(\mathbf{v}^{(l+1)}\in\mathcal{U}(\mathbf{g})\) be given by \(\mathbf{v}^{(l+1)}|_{\mathbb{Z}\backslash\{n_{l+1}\}}=\mathbf{v}^{(l)}|_{\mathbb{Z}\backslash\{n_{l+1}\}}\) and \(\mathbf{v}^{(l+1)}(n_{l+1})=\mathbf{u}^{(2)}(n_{l+1})\), then \([(\mathbf{v}^{(l)},\mathbf{s}^{(l+1)})]=[(\mathbf{v}^{(l+1)},\mathbf{s}^{(l+1)})]\). Since \(\mathbf{v}^{(0)}=\mathbf{u}^{(1)}\) and \(\mathbf{v}^{(l+1)}\) is obtained by replacing the \(n_{l+1}\)-th term of \(\mathbf{v}^{(l)}\) by \(\mathbf{u}^{(2)}(n_{l+1})\), we have \(\mathbf{d}_{\mathcal{U}(\mathbf{g})}(\mathbf{v}^{(l)},\mathbf{u}^{(2)})\le2^{n_{l+1}}\).

If \(I\) is a finite set, then \(\mathbf{v}^{(\# I)}=\mathbf{u}^{(2)}\). Let \[\Gamma=\left(\bigcup_{k=1}^{\# I}\{\mathbf{v}^{(k-1)}\}\times[\mathbf{s}^{(k-1)},\mathbf{s}^{(k)}]\right)\cup\left(\{\mathbf{u}^{(2)}\}\times[\mathbf{s}^{(\# I)},\mathbf{t}^{(2)}]\right),\] then \(\mathcal{Q}(\Gamma)\) is a path connecting \(x,y\), and \[\mathcal{H}^1(\Gamma)\le\sum_{k=1}^{\# I}2^{n_k-1}+\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{s}^{(\#I)},\mathbf{t}^{(2)})<+\infty.\]

If \(I\) is an infinite set, then we obtain infinite sequences \(\{\mathbf{s}^{(l)}\}\), \(\{\mathbf{v}^{(l)}\}\). Since
\(\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{s}^{(l)},\mathbf{s}^{(l+1)})\le2^{n_{l+1}-1}\), we have \(\{\mathbf{s}^{(l)}\}\) converges to some \(\mathbf{s}^{(\infty)}\) in \((\mathcal{T}(\mathbf{b}),\mathbf{d}_{\mathcal{T}(\mathbf{b})})\). Since \(\mathbf{d}_{\mathcal{U}(\mathbf{g})}(\mathbf{v}^{(l)},\mathbf{u}^{(2)})\le2^{n_{l+1}}\), we have \(\{\mathbf{v}^{(l)}\}\) converges to \(\mathbf{u}^{(2)}\) in \((\mathcal{U}(\mathbf{g}),\mathbf{d}_{\mathcal{U}(\mathbf{g})})\). Let \[\Gamma=\left(\bigcup_{k=1}^{+\infty}\{\mathbf{v}^{(k-1)}\}\times[\mathbf{s}^{(k-1)},\mathbf{s}^{(k)}]\right)\cup\left(\{\mathbf{u}^{(2)}\}\times[\mathbf{s}^{(\infty)},\mathbf{t}^{(2)}]\right),\] then \(\mathcal{Q}(\Gamma)\) is a path connecting \(x,y\), and \[\mathcal{H}^1(\Gamma)\le\sum_{k=1}^{+\infty}2^{n_k-1}+\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{s}^{(\infty)},\mathbf{t}^{(2)})<+\infty.\] ◻

Proof. We fix two points \((\mathbf{u},\mathbf{t})\), \((\mathbf{v},\mathbf{s})\) from the equivalence classes \([(\mathbf{u},\mathbf{t})]\), \([(\mathbf{v},\mathbf{s})]\). If \(\mathbf{u}=\mathbf{v}\), let \(\Gamma=\{\mathbf{u}\}\times[\mathbf{t},\mathbf{s}]\), then \(\mathcal{Q}(\Gamma)\) is obviously a geodesic connecting \([(\mathbf{u},\mathbf{t})]\), \([(\mathbf{v},\mathbf{s})]\), and \(\mathcal{H}^1(\Gamma)=\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{t},\mathbf{s})=\mathbf{d}_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{u},\mathbf{t})],[(\mathbf{v},\mathbf{s})])\), we say such a geodesic is monotone. Hence we may assume that \(\mathbf{u}\ne\mathbf{v}\).

Let \(I=\left\{k\in\mathbb{Z}:\mathbf{u}(k)\ne\mathbf{v}(k)\right\}\). Since \(\mathbf{u}(k)\to0\), \(\mathbf{v}(k)\to0\) as \(k\to+\infty\), we have \(I\subseteq\mathbb{Z}\) is bounded from above. Write \(I=\{n_1,n_2,\ldots:n_1>n_2>\ldots\}\). Roughly speaking, any path connecting \([(\mathbf{u},\mathbf{t})]\), \([(\mathbf{v},\mathbf{s})]\) needs to jump through wormholes with all the levels in \(I\). We only have the following two cases.

1. \([\mathbf{t},\mathbf{s}]\cap\mathcal{W}_{n}\ne\emptyset\) for any \(n\in I\), or \([\mathbf{t},\mathbf{s}]\) does contain all the “necessary" wormholes.

2. \([\mathbf{t},\mathbf{s}]\cap\mathcal{W}_{n}=\emptyset\) for some \(n\in I\), or \([\mathbf{t},\mathbf{s}]\) does not contain all the “necessary" wormholes.

For (1), we construct a path connecting \([(\mathbf{u},\mathbf{t})]\), \([(\mathbf{v},\mathbf{s})]\) with length \(\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{t},\mathbf{s})\), which would imply this path is indeed a geodesic connecting \([(\mathbf{u},\mathbf{t})]\), \([(\mathbf{v},\mathbf{s})]\) and
\(\mathbf{d}_{\mathcal{L}(\mathbf{g},\mathbf{b})}([\mathbf{u},\mathbf{t}],[\mathbf{v},\mathbf{s}])=\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{t},\mathbf{s})\). If \(\# I=1\), let \(\mathbf{t}^{(1)}\in[\mathbf{t},\mathbf{s}]\cap\mathcal{W}_{n_1}\) and \(\Gamma=(\{\mathbf{u}\}\times[\mathbf{t},\mathbf{t}^{(1)}])\cup(\{\mathbf{v}\}\times[\mathbf{t}^{(1)},\mathbf{s}])\), then \(\mathcal{Q}(\Gamma)\) is a path connecting \([(\mathbf{u},\mathbf{t})]\), \([(\mathbf{v},\mathbf{s})]\) with length \(\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{t},\mathbf{s})\). Assume that \(\# I\ge2\). Take \(\mathbf{t}^{(1)}\in[\mathbf{t},\mathbf{s}]\cap\mathcal{W}_{n_1}\) and \(\mathbf{t}^{(2)}\in[\mathbf{t},\mathbf{s}]\cap\mathcal{W}_{n_2}\). By interchanging the roles of \(\mathbf{t}\) and \(\mathbf{s}\), we may assume that \([\mathbf{t},\mathbf{t}^{(1)}]\cap[\mathbf{t}^{(2)},\mathbf{s}]=\emptyset\). Since \(n_1>n_2\), we have \([\mathbf{t}^{(1)},\mathbf{t}^{(2)}]\cap\mathcal{W}_n\ne\emptyset\) for any \(n>n_2\).

If \(\#I=+\infty\), for any \(l\ge3\), there exists \(\mathbf{t}^{(l)}\in[\mathbf{t}^{(1)},\mathbf{t}^{(2)}]\cap\mathcal{W}_{n_l}\) satisfying that \(\mathbf{t}^{(l+1)}\in[\mathbf{t}^{(l)},\mathbf{t}^{(2)}]\), \(\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{t}^{(l)},\mathbf{t}^{(l+1)})=2^{n_{l+1}-1}\), and \(\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{t}^{(1)},\mathbf{t}^{(3)})=2^{n_3-1}\).⁸ Hence \(\{\mathbf{t}^{(l)}\}\) converges to some \(\mathbf{t}^{(\infty)}\in[\mathbf{t}^{(1)},\mathbf{t}^{(2)}]\) in \((\mathcal{T}(\mathbf{b}),\mathbf{d}_{\mathcal{T}(\mathbf{b})})\), see Figure 7.

Let \(\mathbf{u}^{(1)}\in\mathcal{U}(\mathbf{g})\) be given by \(\mathbf{u}^{(1)}|_{\mathbb{Z}\backslash\{n_1\}}=\mathbf{u}|_{\mathbb{Z}\backslash\{n_1\}}\) and \(\mathbf{u}^{(1)}(n_1)=\mathbf{v}(n_1)\). Let \(\mathbf{u}^{(2)}\in\mathcal{U}(\mathbf{g})\) be given by \(\mathbf{u}^{(2)}|_{\mathbb{Z}\backslash\{n_2\}}=\mathbf{v}|_{\mathbb{Z}\backslash\{n_2\}}\) and \(\mathbf{u}^{(2)}(n_2)=\mathbf{u}(n_2)\). Let \(\mathbf{u}^{(3)}\in\mathcal{U}(\mathbf{g})\) be given by \(\mathbf{u}^{(3)}|_{\mathbb{Z}\backslash\{n_3\}}=\mathbf{u}^{(1)}|_{\mathbb{Z}\backslash\{n_3\}}\) and \(\mathbf{u}^{(3)}(n_3)=\mathbf{v}(n_3)\). For any \(l\ge3\), let \(\mathbf{u}^{(l+1)}\in\mathcal{U}(\mathbf{g})\) be given by \(\mathbf{u}^{(l+1)}|_{\mathbb{Z}\backslash\{n_{l+1}\}}=\mathbf{u}^{(l)}|_{\mathbb{Z}\backslash\{n_{l+1}\}}\) and \(\mathbf{u}^{(l+1)}(n_{l+1})=\mathbf{v}(n_{l+1})\). Then \(\{\mathbf{u}^{(l)}\}\) converges to \(\mathbf{u}^{(2)}\) in \((\mathcal{U}(\mathbf{g}),\mathbf{d}_{\mathcal{U}(\mathbf{g})})\). Let \[\begin{align} &\Gamma=\left(\{\mathbf{u}\}\times[\mathbf{t},\mathbf{t}^{(1)}]\right)\cup\left(\{\mathbf{u}^{(1)}\}\times[\mathbf{t}^{(1)},\mathbf{t}^{(3)}]\right)\\ &\cup\left(\bigcup_{l=3}^{+\infty}\{\mathbf{u}^{(l)}\}\times[\mathbf{t}^{(l)},\mathbf{t}^{(l+1)}]\right)\cup\left(\{\mathbf{u}^{(2)}\}\times[\mathbf{t}^{(\infty)},\mathbf{t}^{(2)}]\right)\cup\left(\{\mathbf{v}\}\times[\mathbf{t}^{(2)},\mathbf{s}]\right), \end{align}\] then \(\mathcal{Q}(\Gamma)\) is a path connecting \([(\mathbf{u},\mathbf{t})]\), \([(\mathbf{v},\mathbf{s})]\) with length \(\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{t},\mathbf{s})\), we say \(\mathcal{Q}(\Gamma)\) is monotone. If \(\#I<+\infty\), we construct \(\Gamma\) using a similar finite union, which also gives the desired result, we also say \(\mathcal{Q}(\Gamma)\) is monotone.

For (2), \([\mathbf{t},\mathbf{s}]\cap\mathcal{W}_{n}=\emptyset\) for some \(n\in I\), let \(n^*=n_l\) be the maximum of such \(n\), take \(\mathbf{t}^{(1)}\in\mathcal{W}_{n^*}\), which is closest to \([\mathbf{t},\mathbf{s}]\), while such \(\mathbf{t}^{(1)}\) may not be unique, only finitely many exist, and we choose one arbitrarily. Let \(\mathbf{c}=c(\mathbf{t}^{(1)},\mathbf{t},\mathbf{s})\), then \(\mathbf{t}^{(1)}\ne\mathbf{c}\), see Figure 8.

Firstly, if \(([\mathbf{t}^{(1)},\mathbf{t}]\cup[\mathbf{t}^{(1)},\mathbf{s}])\cap\mathcal{W}_n\ne\emptyset\) for any \(n\in I\), let \[\begin{align} I_1&=\left\{n\in I:[\mathbf{t},\mathbf{t}^{(1)}]\cap\mathcal{W}_n\ne\emptyset\right\},\\ I_2&=\left\{n\in I\backslash I_1:[\mathbf{t}^{(1)},\mathbf{s}]\cap\mathcal{W}_n\ne\emptyset\right\}, \end{align}\] then \(I=I_1\sqcup I_2\). Let \(\mathbf{u}^{(1)}\in\mathcal{U}(\mathbf{g})\) be given by \(\mathbf{u}^{(1)}|_{\mathbb{Z}\backslash I_1}=\mathbf{u}|_{\mathbb{Z}\backslash I_1}\) and \(\mathbf{u}^{(1)}|_{I_1}=\mathbf{v}|_{I_1}\), then by (1), there exist

• a monotone geodesic \(\gamma^{(1)}\) connecting \([(\mathbf{u},\mathbf{t})]\), \([(\mathbf{u}^{(1)},\mathbf{t}^{(1)})]\) with length \(\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{t},\mathbf{t}^{(1)})\),

• a monotone geodesic \(\gamma^{(2)}\) connecting \([(\mathbf{u}^{(1)},\mathbf{t}^{(1)})]\), \([(\mathbf{v},\mathbf{s})]\) with length \(\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{t}^{(1)},\mathbf{s})\).

Hence \(\gamma^{(1)}\cup\gamma^{(2)}\) is a path connecting \([(\mathbf{u},\mathbf{t})]\), \([(\mathbf{v},\mathbf{s})]\) with length \[\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{t},\mathbf{t}^{(1)})+\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{t}^{(1)},\mathbf{s})=\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{t},\mathbf{s})+2\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{t}^{(1)},[\mathbf{t},\mathbf{s}]).\] Since any path connecting \([(\mathbf{u},\mathbf{t})]\), \([(\mathbf{v},\mathbf{s})]\) needs to jump through at least one level-\(n^*\) wormhole, we have \[\begin{align} \mathbf{d}_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{u},\mathbf{t})], [(\mathbf{v},\mathbf{s})])&\ge\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{t},\mathbf{s})+2\min_{\mathbf{r}\in\mathcal{W}_{n^*}}\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{r},[\mathbf{t},\mathbf{s}])\\ &=\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{t},\mathbf{s})+2\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{t}^{(1)},[\mathbf{t},\mathbf{s}]). \end{align}\] Hence \(\gamma^{(1)}\cup\gamma^{(2)}\) is a geodesic connecting \([(\mathbf{u},\mathbf{t})]\), \([(\mathbf{v},\mathbf{s})]\) and \[\mathbf{d}_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{u},\mathbf{t})], [(\mathbf{v},\mathbf{s})])=\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{t},\mathbf{s})+2\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{t}^{(1)},[\mathbf{t},\mathbf{s}]),\] we say such a geodesic has one inversion.

Secondly, if \(([\mathbf{t}^{(1)},\mathbf{t}]\cup[\mathbf{t}^{(1)},\mathbf{s}])\cap\mathcal{W}_n=\emptyset\) for some \(n\in I\), let \(n^{**}\) be the maximum of such \(n\), then \(n^{**}<n^*=n_l\). We claim that \(n^{**}=n_{l+1}\). Otherwise, we have \(([\mathbf{t}^{(1)},\mathbf{t}]\cup[\mathbf{t}^{(1)},\mathbf{s}])\cap\mathcal{W}_{n_{l+1}}\ne\emptyset\), take \(\mathbf{t}^{(2)}\in([\mathbf{t}^{(1)},\mathbf{t}]\cup[\mathbf{t}^{(1)},\mathbf{s}])\cap\mathcal{W}_{n_{l+1}}\). Since \(n_l>n_{l+1}\), we have \([\mathbf{t}^{(1)},\mathbf{t}^{(2)}]\cap\mathcal{W}_n\ne\emptyset\) for any \(n>n_{l+1}\), in particular, \([\mathbf{t}^{(1)},\mathbf{t}]\cup[\mathbf{t}^{(1)},\mathbf{s}]\supseteq[\mathbf{t}^{(1)},\mathbf{t}^{(2)}]\) intersects \(\mathcal{W}_n\) for any \(n>n_{l+1}\), \(([\mathbf{t}^{(1)},\mathbf{t}]\cup[\mathbf{t}^{(1)},\mathbf{s}])\cap\mathcal{W}_n\ne\emptyset\) for any \(n\in I\), contradicting to our assumption.

Take \(\mathbf{t}^{(2)}\in\mathcal{W}_{n^{**}}\), which is closest to \([\mathbf{t},\mathbf{s}]\). Let \(\mathbf{c}^{(1)}=c(\mathbf{t},\mathbf{t}^{(1)},\mathbf{t}^{(2)})\), \(\mathbf{c}^{(2)}=c(\mathbf{s},\mathbf{t}^{(1)},\mathbf{t}^{(2)})\). By assumption, we only have the case \(\mathbf{c}^{(1)},\mathbf{c}^{(2)}\in[\mathbf{t}^{(1)},\mathbf{t}^{(2)}]\backslash\{\mathbf{t}^{(1)},\mathbf{t}^{(2)}\}\), see Figure 9, however, we do not exclude the cases \(\mathbf{c}^{(1)}=\mathbf{c}^{(2)}\), \(\mathbf{t}=\mathbf{c}^{(1)}\) or \(\mathbf{s}=\mathbf{c}^{(2)}\). By interchanging the roles of \(\mathbf{t}\) and \(\mathbf{s}\), we may assume that \(\mathbf{c}^{(1)}\subseteq[\mathbf{t}^{(1)},\mathbf{c}^{(2)}]\).

Since \(\mathbf{t}^{(1)}\in\mathcal{W}_{n^*}=\mathcal{W}_{n_l}\), \(\mathbf{t}^{(2)}\in\mathcal{W}_{n^{**}}=\mathcal{W}_{n_{l+1}}\) and \(n_l>n_{l+1}\), we have \([\mathbf{t}^{(1)},\mathbf{t}^{(2)}]\cap\mathcal{W}_n\ne\emptyset\) for any \(n>n_{l+1}\), which implies \(([\mathbf{t},\mathbf{t}^{(1)}]\cup[\mathbf{t}^{(1)},\mathbf{t}^{(2)}]\cup[\mathbf{t}^{(2)},\mathbf{s}])\cap\mathcal{W}_n\ne\emptyset\) for any \(n\in I\). Let \[\begin{align} I_1&=\left\{n\in I:[\mathbf{t},\mathbf{t}^{(1)}]\cap\mathcal{W}_n\ne\emptyset\right\},\\ I_2&=\left\{n\in I\backslash I_1:[\mathbf{t}^{(1)},\mathbf{t}^{(2)}]\cap\mathcal{W}_n\ne\emptyset\right\},\\ I_3&=\left\{n\in I\backslash (I_1\sqcup I_2):[\mathbf{t}^{(2)},\mathbf{s}]\cap\mathcal{W}_n\ne\emptyset\right\}, \end{align}\] then \(I=I_1\sqcup I_2\sqcup I_3\). Let \(\mathbf{u}^{(1)}\in\mathcal{U}(\mathbf{g})\) be given by \(\mathbf{u}^{(1)}|_{\mathbb{Z}\backslash I_1}=\mathbf{u}|_{\mathbb{Z}\backslash I_1}\) and \(\mathbf{u}^{(1)}|_{I_1}=\mathbf{v}|_{I_1}\). Let \(\mathbf{u}^{(2)}\in\mathcal{U}(\mathbf{g})\) be given by \(\mathbf{u}^{(2)}|_{\mathbb{Z}\backslash I_2}=\mathbf{u}^{(1)}|_{\mathbb{Z}\backslash I_2}\) and \(\mathbf{u}^{(2)}|_{I_2}=\mathbf{v}|_{I_2}\), then by (1), there exist

• a monotone geodesic \(\gamma^{(1)}\) connecting \([(\mathbf{u},\mathbf{t})]\), \([(\mathbf{u}^{(1)},\mathbf{t}^{(1)})]\) with length \(\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{t},\mathbf{t}^{(1)})\),

• a monotone geodesic \(\gamma^{(2)}\) connecting \([(\mathbf{u}^{(1)},\mathbf{t}^{(1)})]\), \([(\mathbf{u}^{(2)},\mathbf{t}^{(2)})]\)
  with length \(\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{t}^{(1)},\mathbf{t}^{(2)})\),

• a monotone geodesic \(\gamma^{(3)}\) connecting \([(\mathbf{u}^{(2)},\mathbf{t}^{(2)})]\), \([(\mathbf{v},\mathbf{s})]\) with length \(\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{t}^{(2)},\mathbf{s})\).

Hence \(\gamma^{(1)}\cup\gamma^{(2)}\cup\gamma^{(3)}\) is a path connecting \([(\mathbf{u},\mathbf{t})]\), \([(\mathbf{v},\mathbf{s})]\) with length \[\begin{align} &\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{t},\mathbf{t}^{(1)})+\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{t}^{(1)},\mathbf{t}^{(2)})+\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{t}^{(2)},\mathbf{s})\\ &=\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{t},\mathbf{s})+2\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{t}^{(1)},[\mathbf{t}, \mathbf{s}])+2\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{t}^{(2)},[\mathbf{t},\mathbf{s}]). \end{align}\] Since any path connecting \([(\mathbf{u},\mathbf{t})]\), \([(\mathbf{v},\mathbf{s})]\) needs to jump through at least one level-\(n^*\) wormhole and at least one level-\(n^{**}\) wormhole, we have \[\begin{align} \mathbf{d}_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{u},\mathbf{t})], [(\mathbf{v},\mathbf{s})])&\ge\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{t},\mathbf{s})+2\min_{\mathbf{r}\in\mathcal{W}_{n^*}}\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{r},[\mathbf{t},\mathbf{s}])+2\min_{\mathbf{r}\in\mathcal{W}_{n^{**}}}\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{r},[\mathbf{t},\mathbf{s}])\\ &=\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{t},\mathbf{s})+2\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{t}^{(1)},[\mathbf{t},\mathbf{s}])+2\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{t}^{(2)},[\mathbf{t},\mathbf{s}]). \end{align}\] Hence \(\gamma^{(1)}\cup\gamma^{(2)}\cup\gamma^{(3)}\) is a geodesic connecting \([(\mathbf{u},\mathbf{t})]\), \([(\mathbf{v},\mathbf{s})]\) and \[\mathbf{d}_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{u},\mathbf{t})], [(\mathbf{v},\mathbf{s})])=\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{t},\mathbf{s})+2\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{t}^{(1)},[\mathbf{t},\mathbf{s}])+2\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{t}^{(2)},[\mathbf{t},\mathbf{s}]),\] we say such a geodesic has two inversions. ◻

Proof. [item95maxonept] Suppose there exist distinct \(\mathbf{s}^{(1)},\mathbf{s}^{(2)}\) in this set, since \(\mathbf{s}^{(1)},\mathbf{s}^{(2)}\in B_{\mathcal{T}(\mathbf{b})}(\mathbf{t},r)\), we have \(\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{s}^{(1)},\mathbf{s}^{(2)})<2r<2^{n_0+1}\). However, since \(\mathbf{s}^{(1)},\mathbf{s}^{(2)}\in\cup_{n\ge n_0+2}\mathcal{W}_n\) are distinct, we have \(\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{s}^{(1)},\mathbf{s}^{(2)})\ge2^{n_0+1}\), which gives \(2^{n_0+1}\le\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{s}^{(1)},\mathbf{s}^{(2)})<2^{n_0+1}\), contradiction.

[item95minonept] If \(\gamma\) is monotone as in [item95geo95mono], assume that \(\gamma\) connects \([(\mathbf{u},\mathbf{t})]\), \([(\mathbf{v},\mathbf{s})]\in\mathcal{L}(\mathbf{g},\mathbf{b})\), then \(\pi^\mathcal{T}(\gamma)=[\mathbf{t},\mathbf{s}]\) and \(l(\gamma)=\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{t},\mathbf{s})\). Since for any \(n\in\mathbb{Z}\), each \(n\)-cell in \((\mathcal{T}(\mathbf{b}),\mathbf{d}_{\mathcal{T}(\mathbf{b})})\) has diameter \(2^n\) and any point in \((\mathcal{T}(\mathbf{b}),\mathbf{d}_{\mathcal{T}(\mathbf{b})})\) has distance at most \(2^{n-1}\) from \(\mathcal{W}_n\), we have for any \(n\in\mathbb{Z}\) with \(2^n\le \mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{t},\mathbf{s})=l(\gamma)\), \(\pi^\mathcal{T}(\gamma)\cap\mathcal{W}_n=[\mathbf{t},\mathbf{s}]\cap\mathcal{W}_n\ne\emptyset\). If \(\gamma\) has one inversion as in [item95geo95oneinv], then \(\gamma=\gamma^{(1)}\cup\gamma^{(2)}\), where \(\gamma^{(1)}, \gamma^{(2)}\) are two monotone geodesics and \(l(\gamma)=l(\gamma^{(1)})+l(\gamma^{(2)})\), then either \(l(\gamma^{(1)})\ge l(\gamma)/2\) or \(l(\gamma^{(2)})\ge l(\gamma)/2\), we may consider the first case, the second follows analogously. By the result for monotone geodesics, for any \(n\in\mathbb{Z}\) with \(2^n\le l(\gamma)/2\le l(\gamma^{(1)})\), we have \(\pi^{\mathcal{T}}(\gamma)\cap\mathcal{W}_n\supseteq\pi^{\mathcal{T}}(\gamma^{(1)})\cap\mathcal{W}_n\ne\emptyset\). If \(\gamma\) has two inversions as in [item95geo95twoinv], then \(\gamma=\gamma^{(1)}\cup\gamma^{(2)}\cup\gamma^{(3)}\), where \(\gamma^{(1)}, \gamma^{(2)}, \gamma^{(3)}\) are three monotone geodesics and \(l(\gamma)=l(\gamma^{(1)})+l(\gamma^{(2)})+l(\gamma^{(3)})\), then there exists \(i\in\{1,2,3\}\) such that \(l(\gamma^{(i)})\ge l(\gamma)/3\). By the result for monotone geodesics, for any \(n\in\mathbb{Z}\) with \(2^n\le l(\gamma)/3\le l(\gamma^{(i)})\), we have \(\pi^{\mathcal{T}}(\gamma)\cap\mathcal{W}_n\supseteq\pi^{\mathcal{T}}(\gamma^{(i)})\cap\mathcal{W}_n\ne\emptyset\). In summary, since \(2^{n_0}\le l(\gamma)/4\), for any \(n\le n_0\), we have \(\pi^\mathcal{T}(\gamma)\cap\mathcal{W}_n\ne\emptyset\). ◻

Proof. Firstly, we show that \(\mathcal{Q}\) is \(3\)-Lipschitz. For any \((\mathbf{u},\mathbf{t})\), \((\mathbf{v},\mathbf{s})\in\mathcal{P}(\mathbf{g},\mathbf{b})\), there exists a geodesic connecting \([(\mathbf{u},\mathbf{t})]\), \([(\mathbf{v},\mathbf{s})]\in\mathcal{L}(\mathbf{g},\mathbf{b})\) given by Proposition 9, along with three associated cases.

For [item95geo95mono], we have \[\mathbf{d}_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{u},\mathbf{t})], [(\mathbf{v},\mathbf{s})])=\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{t},\mathbf{s})\le\mathbf{d}_{\mathcal{P}(\mathbf{g},\mathbf{b})}((\mathbf{u},\mathbf{t}), (\mathbf{v},\mathbf{s})).\]

For [item95geo95oneinv], we have \[\mathbf{d}_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{u},\mathbf{t})], [(\mathbf{v},\mathbf{s})])=\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{t},\mathbf{s})+2\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{t}^{(1)},[\mathbf{t},\mathbf{s}]),\] where \(\mathbf{t}^{(1)}\in\mathcal{W}_{n^*}\) is given by the remark following Proposition 9. Since \[\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{t}^{(1)},[\mathbf{t},\mathbf{s}])\le2^{n^*-1}\le 2^{\max\{n:\mathbf{u}(n)\ne\mathbf{v}(n)\}-1}=\frac{1}{2}\mathbf{d}_{\mathcal{U}(\mathbf{g})}(\mathbf{u},\mathbf{v}),\] we have \[\mathbf{d}_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{u},\mathbf{t})], [(\mathbf{v},\mathbf{s})])\le\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{t},\mathbf{s})+\mathbf{d}_{\mathcal{U}(\mathbf{g})}(\mathbf{u},\mathbf{v})\le2\mathbf{d}_{\mathcal{P}(\mathbf{g},\mathbf{b})}((\mathbf{u},\mathbf{t}), (\mathbf{v},\mathbf{s})).\]

For [item95geo95twoinv], we have \[\mathbf{d}_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{u},\mathbf{t})], [(\mathbf{v},\mathbf{s})])=\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{t},\mathbf{s})+2\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{t}^{(1)},[\mathbf{t},\mathbf{s}])+2\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{t}^{(2)},[\mathbf{t},\mathbf{s}]),\] where \(\mathbf{t}^{(1)}\in\mathcal{W}_{n^*}\), \(\mathbf{t}^{(2)}\in\mathcal{W}_{n^{**}}\) are given by the remark following Proposition 9. Since \[\begin{align} \mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{t}^{(1)},[\mathbf{t},\mathbf{s}])&\le2^{n^*-1}\le 2^{\max\{n:\mathbf{u}(n)\ne\mathbf{v}(n)\}-1}=\frac{1}{2}\mathbf{d}_{\mathcal{U}(\mathbf{g})}(\mathbf{u},\mathbf{v}),\\ \mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{t}^{(2)},[\mathbf{t},\mathbf{s}])&\le2^{n^{**}-1}\le 2^{\max\{n:\mathbf{u}(n)\ne\mathbf{v}(n)\}-1}=\frac{1}{2}\mathbf{d}_{\mathcal{U}(\mathbf{g})}(\mathbf{u},\mathbf{v}), \end{align}\] we have \[\mathbf{d}_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{u},\mathbf{t})], [(\mathbf{v},\mathbf{s})])\le\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{t},\mathbf{s})+\mathbf{d}_{\mathcal{U}(\mathbf{g})}(\mathbf{u},\mathbf{v})+\mathbf{d}_{\mathcal{U}(\mathbf{g})}(\mathbf{u},\mathbf{v})\le3\mathbf{d}_{\mathcal{P}(\mathbf{g},\mathbf{b})}((\mathbf{u},\mathbf{t}), (\mathbf{v},\mathbf{s})).\]

Secondly, for any \([(\mathbf{u},\mathbf{t})]\in\mathcal{L}(\mathbf{g},\mathbf{b})\), \(r>0\), let \(n\) be the integer satisfying \(2^{n-1}\le r<2^n\). By Lemma 8 [item95maxonept], \(B_{\mathcal{T}(\mathbf{b})}(\mathbf{t},r)\cap\cup_{k\ge n+2}\mathcal{W}_k\) consists of at most one point.

If \(B_{\mathcal{T}(\mathbf{b})}(\mathbf{t},r)\cap\cup_{k\ge n+2}\mathcal{W}_k=\emptyset\). Take \((\mathbf{u},\mathbf{t})\) in \([(\mathbf{u},\mathbf{t})]\). We claim that \[\mathcal{Q}^{-1}(B_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{u},\mathbf{t})],r))\subseteq\left\{\mathbf{v}\in\mathcal{U}(\mathbf{g}):\mathbf{v}|_{\mathbb{Z}\cap[n+2,+\infty)}=\mathbf{u}|_{\mathbb{Z}\cap[n+2,+\infty)}\right\}\times B_{\mathcal{T}(\mathbf{b})}(\mathbf{t},r).\] Indeed, we prove that if \((\mathbf{v},\mathbf{s})\in\mathcal{P}(\mathbf{g},\mathbf{b})\) satisfies \(\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{s},\mathbf{t})\ge r\) or \(\mathbf{v}|_{\mathbb{Z}\cap[n+2,+\infty)}\ne\mathbf{u}|_{\mathbb{Z}\cap[n+2,+\infty)}\), then \(\mathbf{d}_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{v},\mathbf{s})],[(\mathbf{u},\mathbf{t})])\ge r\). For the first case, it is obvious that \(\mathbf{d}_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{v},\mathbf{s})],[(\mathbf{u},\mathbf{t})])\ge\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{s},\mathbf{t})\ge r\). For the second case, since \(\mathbf{v}|_{\mathbb{Z}\cap[n+2,+\infty)}\ne\mathbf{u}|_{\mathbb{Z}\cap[n+2,+\infty)}\), any path connecting \([(\mathbf{v},\mathbf{s})],[(\mathbf{u},\mathbf{t})]\) needs to jump through at least one level-\(k\) wormhole with \(k\ge n+2\), however, \(B_{\mathcal{T}(\mathbf{b})}(\mathbf{t},r)\cap\cup_{k\ge n+2}\mathcal{W}_k=\emptyset\), we have such path has length at least \(r\), which implies that \(\mathbf{d}_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{v},\mathbf{s})],[(\mathbf{u},\mathbf{t})])\ge r\). Hence \[\begin{align} &\mathcal{Q}^{-1}(B_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{u},\mathbf{t})],r))\subseteq \overline{B}_{\mathcal{U}(\mathbf{g})}(\mathbf{u},2^{n+1})\times B_{\mathcal{T}(\mathbf{b})}(\mathbf{t},r)\\ &\subseteq B_{\mathcal{U}(\mathbf{g})}(\mathbf{u},8r)\times B_{\mathcal{T}(\mathbf{b})}(\mathbf{t},r)\subseteq B_{\mathcal{P}(\mathbf{g},\mathbf{b})}((\mathbf{u},\mathbf{t}),8r). \end{align}\] Moreover, for any \(\mathbf{v}\in B_{\mathcal{U}(\mathbf{g})}(\mathbf{u},8r)\), \(\mathbf{s}\in B_{\mathcal{T}(\mathbf{b})}(\mathbf{t},r)\), we have \[\begin{align} &\mathbf{d}_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{v},\mathbf{s})],[(\mathbf{u},\mathbf{t})])\le\mathbf{d}_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{v},\mathbf{s})],[(\mathbf{u},\mathbf{s})])+\mathbf{d}_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{u},\mathbf{s})],[(\mathbf{u},\mathbf{t})])\\ &\le3\mathbf{d}_{\mathcal{P}(\mathbf{g},\mathbf{b})}((\mathbf{v},\mathbf{s}),(\mathbf{u},\mathbf{s}))+\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{s},\mathbf{t})=3\mathbf{d}_{\mathcal{U}(\mathbf{g})}(\mathbf{v},\mathbf{u})+\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{s},\mathbf{t})<3\cdot 8r+r=25r, \end{align}\] which gives \[B_{\mathcal{U}(\mathbf{g})}(\mathbf{u},8r)\times B_{\mathcal{T}(\mathbf{b})}(\mathbf{t},r)\subseteq \mathcal{Q}^{-1}(B_{{\mathcal{L}(\mathbf{g},\mathbf{b})}}([(\mathbf{u},\mathbf{t})],25r)).\]

If \(B_{\mathcal{T}(\mathbf{b})}(\mathbf{t},r)\cap\cup_{k\ge n+2}\mathcal{W}_k=\{\mathbf{t}^{(1)}\}\) is a one-point set, where \(\mathbf{t}^{(1)}\in\mathcal{W}_m\) for some \(m\ge n+2\). Take \((\mathbf{u},\mathbf{t})\) in \([(\mathbf{u},\mathbf{t})]\). For any \(k=1,\ldots,\mathbf{g}(m)\), let \(\mathbf{u}^{(k)}\in\mathcal{U}(\mathbf{g})\) be given by \(\mathbf{u}^{(k)}|_{\mathbb{Z}\backslash\{m\}}=\mathbf{u}|_{\mathbb{Z}\backslash\{m\}}\) and \(\mathbf{u}^{(k)}(m)=k-1\), then \([(\mathbf{u}^{(k)},\mathbf{t}^{(1)})]=[(\mathbf{u},\mathbf{t}^{(1)})]\). We claim that \[\begin{align} &\mathcal{Q}^{-1}(B_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{u},\mathbf{t})],r))\\ &\subseteq\left(\bigcup_{k=1}^{\mathbf{g}(m)}\left\{\mathbf{v}\in\mathcal{U}(\mathbf{g}):\mathbf{v}|_{\mathbb{Z}\cap[n+2,+\infty)}=\mathbf{u}^{(k)}|_{\mathbb{Z}\cap[n+2,+\infty)}\right\}\right)\times B_{\mathcal{T}(\mathbf{b})}(\mathbf{t},r). \end{align}\] Indeed, we prove that if \((\mathbf{v},\mathbf{s})\in\mathcal{P}(\mathbf{g},\mathbf{b})\) satisfies \(\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{s},\mathbf{t})\ge r\) or \(\mathbf{v}|_{\mathbb{Z}\cap[n+2,+\infty)}\ne\mathbf{u}^{(k)}|_{\mathbb{Z}\cap[n+2,+\infty)}\) for any \(k=1,\ldots,\mathbf{g}(m)\), then \(\mathbf{d}_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{v},\mathbf{s})],[(\mathbf{u},\mathbf{t})])\ge r\). For the first case, it is obvious that \(\mathbf{d}_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{v},\mathbf{s})],[(\mathbf{u},\mathbf{t})])\ge\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{s},\mathbf{t})\ge r\). For the second case, we have \(\mathbf{v}|_{(\mathbb{Z}\cap[n+2,+\infty))\backslash\{m\}}\ne\mathbf{u}|_{(\mathbb{Z}\cap[n+2,+\infty))\backslash\{m\}}\), otherwise, \(\mathbf{v}|_{\mathbb{Z}\cap[n+2,+\infty)}=\mathbf{u}^{(k)}|_{\mathbb{Z}\cap[n+2,+\infty)}\) with \(k=\mathbf{v}(m)+1\). Then there exists \(p\ge n+2\) with \(p\ne m\) such that \(\mathbf{v}(p)\ne\mathbf{u}(p)\), any path connecting \([(\mathbf{v},\mathbf{s})],[(\mathbf{u},\mathbf{t})]\) needs to jump through at least one level-\(p\) wormhole, however, since \(B_{\mathcal{T}(\mathbf{b})}(\mathbf{t},r)\cap\cup_{k\ge n+2}\mathcal{W}_k=\{\mathbf{t}^{(1)}\}\subseteq\mathcal{W}_m\), we have \(B_{\mathcal{T}(\mathbf{b})}(\mathbf{t},r)\cap\mathcal{W}_p=\emptyset\), such path has length at least \(r\), which implies that \(\mathbf{d}_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{v},\mathbf{s})],[(\mathbf{u},\mathbf{t})])\ge r\). Hence \[\begin{align} &\mathcal{Q}^{-1}(B_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{u},\mathbf{t})],r))\subseteq \left(\bigcup_{k=1}^{\mathbf{g}(m)} \overline{B}_{\mathcal{U}(\mathbf{g})}(\mathbf{u}^{(k)},2^{n+1})\right)\times B_{\mathcal{T}(\mathbf{b})}(\mathbf{t},r)\\ &\subseteq\left(\bigcup_{k=1}^{\mathbf{g}(m)}B_{\mathcal{U}(\mathbf{g})}(\mathbf{u}^{(k)},8r)\right)\times B_{\mathcal{T}(\mathbf{b})}(\mathbf{t},r)\subseteq\bigcup_{k=1}^{\mathbf{g}(m)}B_{\mathcal{P}(\mathbf{g},\mathbf{b})}((\mathbf{u}^{(k)},\mathbf{t}),8r). \end{align}\] Moreover, for any \(\mathbf{v}\in B_{\mathcal{U}(\mathbf{g})}(\mathbf{u}^{(k)},8r)\), \(\mathbf{s}\in B_{\mathcal{T}(\mathbf{b})}(\mathbf{t},r)\), since \([(\mathbf{u}^{(k)},\mathbf{t}^{(1)})]=[(\mathbf{u},\mathbf{t}^{(1)})]\) and \(\mathbf{t}^{(1)}\in B_{\mathcal{T}(\mathbf{b})}(\mathbf{t},r)\), we have \[\begin{align} &\mathbf{d}_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{v},\mathbf{s})],[(\mathbf{u},\mathbf{t})])\\ &\le\mathbf{d}_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{v},\mathbf{s})],[(\mathbf{u}^{(k)},\mathbf{s})])\\ &+\mathbf{d}_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{u}^{(k)},\mathbf{s})],[(\mathbf{u}^{(k)},\mathbf{t}^{(1)})])+\mathbf{d}_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{u}^{(k)},\mathbf{t}^{(1)})],[(\mathbf{u},\mathbf{t})])\\ &\le3\mathbf{d}_{\mathcal{P}(\mathbf{g},\mathbf{b})}((\mathbf{v},\mathbf{s}),(\mathbf{u}^{(k)},\mathbf{s}))+\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{s},\mathbf{t}^{(1)})+\mathbf{d}_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{u},\mathbf{t}^{(1)})],[(\mathbf{u},\mathbf{t})])\\ &=3\mathbf{d}_{\mathcal{U}(\mathbf{g})}(\mathbf{v},\mathbf{u}^{(k)})+\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{s},\mathbf{t}^{(1)})+\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{t}^{(1)},\mathbf{t})<3\cdot8r+2r+r=27r, \end{align}\] which gives \[\left(\bigcup_{k=1}^{\mathbf{g}(m)}B_{\mathcal{U}(\mathbf{g})}(\mathbf{u}^{(k)},8r)\right)\times B_{\mathcal{T}(\mathbf{b})}(\mathbf{t},r)\subseteq\mathcal{Q}^{-1}\left(B_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{u},\mathbf{t})],27r)\right).\]

Therefore, in both of the aforementioned cases, \(\mathcal{Q}^{-1}(B_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{u},\mathbf{t})],r))\) can always be contained in the union of \(\sup_\mathbb{Z}\mathbf{g}\) balls with radius \(8r\).

In summary, \(\mathcal{Q}:(\mathcal{P}(\mathbf{g},\mathbf{b}),\mathbf{d}_{\mathcal{P}(\mathbf{g},\mathbf{b})})\to(\mathcal{L}(\mathbf{g},\mathbf{b}),\mathbf{d}_{\mathcal{L}(\mathbf{g},\mathbf{b})})\) is David-Semmes regular.

Finally, for any Cauchy sequence \(\{[\mathbf{q}_n]\}_{n\ge1}\) in \(\mathcal{L}(\mathbf{g},\mathbf{b})\), we have \(\{[\mathbf{q}_n]\}_{n\ge1}\) is bounded in \(\mathcal{L}(\mathbf{g},\mathbf{b})\), hence \(\mathcal{Q}^{-1}(\{[\mathbf{q}_n]\}_{n\ge1})\) is bounded in \(\mathcal{P}(\mathbf{g},\mathbf{b})\). Since \(\mathcal{P}(\mathbf{g},\mathbf{b})\) is complete and locally compact, \(\mathcal{Q}^{-1}(\{[\mathbf{q}_n]\}_{n\ge1})\) has a limit point \(\mathbf{q}\in\mathcal{P}(\mathbf{g},\mathbf{b})\). By the Lipschitz continuity of \(\mathcal{Q}\), \([\mathbf{q}]=\mathcal{Q}(\mathbf{q})\in\mathcal{L}(\mathbf{g},\mathbf{b})\) is the limit of \(\{[\mathbf{q}_n]\}_{n\ge1}\) in \(\mathcal{L}(\mathbf{g},\mathbf{b})\), hence \(\mathcal{L}(\mathbf{g},\mathbf{b})\) is complete.

For any bounded closed set \(K\) in \(\mathcal{L}(\mathbf{g},\mathbf{b})\), we have \(\mathcal{Q}^{-1}(K)\) is bounded closed in \(\mathcal{P}(\mathbf{g},\mathbf{b})\). Since \(\mathcal{P}(\mathbf{g},\mathbf{b})\) is locally compact, we have \(\mathcal{Q}^{-1}(K)\) is compact in \(\mathcal{P}(\mathbf{g},\mathbf{b})\). By the continuity of \(\mathcal{Q}\), we have \(K=\mathcal{Q}(\mathcal{Q}^{-1}(K))\) is compact in \(\mathcal{L}(\mathbf{g},\mathbf{b})\), hence \(\mathcal{L}(\mathbf{g},\mathbf{b})\) is locally compact.

Since \(\mathcal{P}(\mathbf{g},\mathbf{b})\) is separable, let \(C\) be a countable dense subset of \(\mathcal{P}(\mathbf{g},\mathbf{b})\), then \(\mathcal{Q}(C)\) is a countable dense subset of \(\mathcal{L}(\mathbf{g},\mathbf{b})\). Hence \(\mathcal{L}(\mathbf{g},\mathbf{b})\) is separable. ◻

Proof. We only give the proof of (1). The proof of (2) is easy and also similar to the proof of \((\int_{\mathbb{R}}|f'(x)|^p\mathrm{d} x,W^{1,p}(\mathbb{R}))\) is a \(p\)-energy on \(\mathbb{R}\) with a \(p\)-energy measure given by \((f,A)\mapsto\int_A|f'(x)|^p\mathrm{d} x\), see also [46] or [47] for the construction of a Dirichlet form on the original Laakso space in [23].

Indeed, since \(\mathcal{C}^\mathcal{L}\) is a sub-algebra of \(C_c(\mathcal{L}(\mathbf{g},\mathbf{b}))\), by the Stone-Weierstraß theorem, we only need to show that \(\mathcal{C}^\mathcal{L}\) separates points and vanishes nowhere. For any distinct \(x,y\in\mathcal{L}(\mathbf{g},\mathbf{b})\), let \(D=\mathbf{d}_{\mathcal{L}(\mathbf{g},\mathbf{b})}(x,y)>0\), by the following Proposition 14, there exists \(\phi_{\mathcal{L}}\in\mathcal{C}^{\mathcal{L}}\) such that \(\phi_\mathcal{L}=1\) in \(B_{\mathcal{L}(\mathbf{g},\mathbf{b})}(x,D/256)\) and \(\phi_\mathcal{L}=0\) on \(\mathcal{L}(\mathbf{g},\mathbf{b})\backslash B_{\mathcal{L}(\mathbf{g},\mathbf{b})}(x,D)\), in particular, we have \(\phi_\mathcal{L}(x)=1\ne0=\phi_\mathcal{L}(y)\). ◻

Proof. Let \(\gamma_0\) be a geodesic connecting \(x,y\) given by Proposition 9. Let \(D=\mathbf{d}_{\mathcal{L}(\mathbf{g},\mathbf{b})}(x,y)\) and \(k_0\) the integer satisfying \(2^{k_0+3}\le D<2^{k_0+4}\). Let \(z=\gamma_0(D/2)\), then \(\gamma_0|_{[0,D/2]}\) is a geodesic connecting \(x, z\) given by Proposition 9, and \(\gamma_0|_{[D/2,D]}\) is a geodesic connecting \(z, y\) given by Proposition 9. Moreover, \(l(\gamma_0|_{[0,D/2]})=l(\gamma_0|_{[D/2,D]})=D/2\). Since \(2^{k_0+2}\le D/2\), by Lemma 8 [item95minonept], for any \(k\le k_0\), \(\pi^\mathcal{T}(\gamma_0|_{[0,D/2]})\cap\mathcal{W}_k\ne\emptyset\), \(\pi^\mathcal{T}(\gamma_0|_{[D/2,D]})\cap\mathcal{W}_k\ne\emptyset\).

For any \(k\le k_0\), there exist only finitely many \(t\in[0,D]\) such that \(\pi^\mathcal{T}(\gamma_0(t))\in\mathcal{W}_k\), let \[\begin{align} t_{k,1}&=\min\{t\in[0,D]:\pi^\mathcal{T}(\gamma_0(t))\in\mathcal{W}_k\},\\ t_{k,2}&=\max\{t\in[0,D]:\pi^\mathcal{T}(\gamma_0(t))\in\mathcal{W}_k\}, \end{align}\] then \(t_{k,1}\in[0,D/2]\), \(t_{k,2}\in[D/2,D]\). Moreover, \(t_{k,1}\in[0,2^{k+2}]\), \(t_{k,2}\in[D-2^{k+2},D]\). Indeed, for any \(t\in[0,D]\), \(\gamma_0|_{[0,t]}\) is a geodesic given by Proposition 9 with length \(t\), by Lemma 8 [item95minonept], for any \(t\ge2^{k+2}\), \(\pi^{\mathcal{T}}(\gamma_0|_{[0,t]})\cap\mathcal{W}_k\ne\emptyset\), hence \(t_{k,1}\in[0,2^{k+2}]\). Similarly, \(t_{k,2}\in[D-2^{k+2},D]\). Since \(\{\mathcal{W}_k\}_{k\in\mathbb{Z}}\) are disjoint, we have \(\{\{t_{k,1},t_{k,2}\}\}_{k\le k_0}\) are also disjoint.

For any \(t\in(0,D)\), let \[k(t)=\min\{\lfloor \log_2t\rfloor,\lfloor{\log_2(D-t)} \rfloor\}-3,\] then for any \(k\le k(t)\), \(t_{k,1}\le 2^{k+2}\le2^{k(t)+2}\le t/2<t\), \(t_{k,2}\ge D-2^{k+2}\ge D-2^{k(t)+2}\ge D-(D-t)/2=(D+t)/2>t\), hence \(t\in(t_{k,1},t_{k,2})\).

Let \((\Omega,\mathcal{F},\mathbb{P})\) be a probability space on which there exist independent random variables \(\{\xi_{k}:k\le k_0\}\) satisfying that \(\xi_{k}\) has the uniform distribution \(\mathrm{Unif}(\{0,1,\ldots,\mathbf{g}(k)-1\})\). Write \(\gamma_0(t)=[(\mathbf{v}_0(t),\mathbf{s}_0(t))]\) for \(t\in[0,D]\). For any \(\omega\in\Omega\), let \(\gamma(\omega)=[(\mathbf{v}(\omega),\mathbf{s}(\omega))]:[0,D]\to\mathcal{L}(\mathbf{g},\mathbf{b})\), where \(\mathbf{v}(\omega):[0,D]\to\mathcal{U}(\mathbf{g})\), \(\mathbf{s}(\omega):[0,D]\to\mathcal{T}(\mathbf{b})\), be given as follows. Let \(\mathbf{s}(\omega)=\mathbf{s}_0\). Recall that \[\mathcal{U}(\mathbf{g})=\left\{\mathbf{u}:\mathbb{Z}\to\mathbb{Z}|\mathbf{u}(k)\in\{0,1,\ldots,\mathbf{g}(k)-1\}\text{ for any }k\in\mathbb{Z}\text{ and }\lim_{k\to+\infty}\mathbf{u}(k)=0\right\}.\] We define the value of \(\mathbf{v}(\omega)(t)(n)\) for \(t\in[0,D]\) and \(n\in\mathbb{Z}\) as follows. For any \(t\in[0,D]\), let \(\mathbf{v}(\omega)(t)|_{\mathbb{Z}\cap[k_0+1,+\infty)}=\mathbf{v}_0(t)|_{\mathbb{Z}\cap[k_0+1,+\infty)}\). For any \(k\le k_0\), let \[\mathbf{v}(\omega)(t)(k)= \begin{cases} \mathbf{v}_0(t)(k)&\text{if }t\in[0,t_{k,1}]\cup(t_{k,2},D],\\ \xi_{k}(\omega)&\text{if }t\in(t_{k,1},t_{k,2}]. \end{cases}\]

We claim that \(\gamma(\omega)\) is a geodesic connecting \(x,y\). It is obvious that \(\gamma(\omega)(0)=x\), \(\gamma(\omega)(D)=y\). We show that \(\gamma(\omega):[0,D]\to(\mathcal{L}(\mathbf{g},\mathbf{b}),\mathbf{d}_{\mathcal{L}(\mathbf{g},\mathbf{b})})\) is continuous. For any \(t\in(0,D)\), there exists an open interval \(U\subset(0,D)\) containing \(t\) such that \(k(\cdot)\ge k(t)-1\) in \(U\), then for any \(k\le k(t)-1\), we have \(U\subseteq(t_{k,1},t_{k,2})\), which implies \(\mathbf{v}(\omega)(\cdot)|_{\mathbb{Z}\cap(-\infty,k(t)-1]}\) is constant in \(U\). By definition, \(U\) can be written as a finite union of intervals such that \(\mathbf{v}(\omega)(\cdot)|_{\mathbb{Z}\cap[k(t),+\infty)}\) is constant on each interval,⁹ hence \(\mathbf{v}(\omega)(\cdot)\) is constant on each interval, where the discontinuity between adjacent intervals comes from the jumps through wormholes, however, by the definition of \(R_\mathcal{L}\), the function of the equivalence classes \(\gamma(\omega)(\cdot)=[(\mathbf{v}(\omega)(\cdot),\mathbf{s}(\omega)(\cdot))]=[(\mathbf{v}(\omega)(\cdot),\mathbf{s}_0(\cdot))]\) is indeed continuous in \(U\), hence \(\gamma(\omega)\) is continuous at \(t\) for any \(t\in(0,D)\). At \(0\), we have \[\begin{align} &\mathbf{d}_{\mathcal{L}(\mathbf{g},\mathbf{b})}(\gamma(\omega)(t),\gamma(\omega)(0))=\mathbf{d}_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{v}(\omega)(t),\mathbf{s}_0(t))],[(\mathbf{v}(\omega)(0),\mathbf{s}_0(0))])\\ &\le 3\mathbf{d}_{\mathcal{P}(\mathbf{g},\mathbf{b})}((\mathbf{v}(\omega)(t),\mathbf{s}_0(t)),(\mathbf{v}_0(0),\mathbf{s}_0(0)))\\ &=3\max\left\{\mathbf{d}_{\mathcal{U}(\mathbf{g})}(\mathbf{v}(\omega)(t),\mathbf{v}_0(0)),\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{s}_0(t),\mathbf{s}_0(0))\right\}, \end{align}\] where the above inequality follows from Lemma 9. Since \(\lim_{t\downarrow0}\mathbf{d}_{\mathcal{T}(\mathbf{b})}(\mathbf{s}_0(t),\mathbf{s}_0(0))=0\), we only need to consider the term \(\mathbf{v}(\omega)\). For any \(\varepsilon>0\), let \(k_1\in\mathbb{Z}\) satisfy \(2^{k_1}<\varepsilon\). By definition, \([0,D]\) can be written as a finite union of intervals such that \(\mathbf{v}(\omega)(\cdot)|_{\mathbb{Z}\cap[k_1+1,+\infty)}\) is constant on each interval, take \((0,\delta)\) on which \(\mathbf{v}(\omega)(\cdot)|_{\mathbb{Z}\cap[k_1+1,+\infty)}\) is constant, by choosing another element from the equivalence class \([(\mathbf{v}_0(0),\mathbf{s}_0(0))]\), we may assume that \(\mathbf{v}_0(0)|_{\mathbb{Z}\cap[k_1+1,+\infty)}\) is this constant, then \(\mathbf{d}_{\mathcal{U}(\mathbf{g})}(\mathbf{v}(\omega)(t),\mathbf{v}_0(0))\le2^{k_1}<\varepsilon\) for any \(t\in(0,\delta)\), hence \(\gamma(\omega)\) is continuous at \(0\). Similarly, \(\gamma(\omega)\) is continuous at \(D\). Therefore, \(\gamma(\omega):[0,D]\to(\mathcal{L}(\mathbf{g},\mathbf{b}),\mathbf{d}_{\mathcal{L}(\mathbf{g},\mathbf{b})})\) is continuous, \(\gamma(\omega)\) is a path connecting \(x, y\). By construction, \(l(\gamma(\omega))=l(\gamma_0)=D\), hence \(\gamma(\omega)\) is a geodesic connecting \(x, y\).

Let \(\Gamma_{x,y}=\{\gamma(\omega):\omega\in\Omega\}\) and \(\mathbb{P}^{x,y}=\mathbb{P}\). We show \(\Gamma_{x,y}\) and \(\mathbb{P}^{x,y}\) give the desired result. Except for the end point \(y\), all geodesics in \(\Gamma_{x,y}\) are contained within the ball \(B=B_{\mathcal{L}(\mathbf{g},\mathbf{b})}(x,D)\). For any non-negative measurable function \(h\) on \(\mathcal{L}(\mathbf{g},\mathbf{b})\), we have \[\begin{align} &\int_{\Gamma_{x,y}}\left(\int_0^Dh(\gamma(t))\mathrm{d} t\right)\mathrm{d}\mathbb{P}^{x,y}=\int_{\Omega}\left(\int_0^Dh(\gamma(\omega)(t))1_{B}(\gamma(\omega)(t))\mathrm{d} t\right)\mathbb{P}(\mathrm{d}\omega)\\ &=\int_0^D\left(\int_\Omega (h1_B)\left([(\mathbf{v}(\omega)(t),\mathbf{s}(\omega)(t))]\right)\mathbb{P}(\mathrm{d}\omega)\right)\mathrm{d} t\\ &=\int_0^D\left(\int_\Omega (h1_B)\left([(\mathbf{v}(\omega)(t),\mathbf{s}_0(t))]\right)\mathbb{P}(\mathrm{d}\omega)\right)\mathrm{d} t. \end{align}\]

For fixed \(t\in(0,D)\), let us consider the distribution \(\mathbb{P}_{\mathbf{v}(\cdot)(t)}\) of \(\mathbf{v}(\cdot)(t)\) under \(\mathbb{P}\). Recall that for any \(k\le k(t)\), we have \(t\in(t_{k,1},t_{k,2})\), then \(\mathbf{v}(\cdot)(t)(k)=\xi_k\). For any \(\mathbf{u}\in\mathcal{U}(\mathbf{g})\), \(n\in\mathbb{Z}\), we have \[\mathbb{P}\left[\omega:\mathbf{v}(\omega)(t)\in{B}_{\mathcal{U}(\mathbf{g})}(\mathbf{u},2^n)\right]=\mathbb{P}\left[\omega:\mathbf{v}(\omega)(t)|_{\mathbb{Z}\cap[n,+\infty)}=\mathbf{u}|_{\mathbb{Z}\cap[n,+\infty)}\right].\] If \(n<k(t)\), then \[\begin{align} &\mathbb{P}\left[\omega:\mathbf{v}(\omega)(t)|_{\mathbb{Z}\cap[n,+\infty)}=\mathbf{u}|_{\mathbb{Z}\cap[n,+\infty)}\right]\\ &\le\mathbb{P}\left[\omega:\xi_k(\omega)=\mathbf{u}(k)\text{ for any }k=n,\ldots,k(t)\right]=\frac{1}{\prod_{k=n}^{k(t)}\mathbf{g}(k)}. \end{align}\] If \(n\ge1\), then \[\frac{1}{\prod_{k=n}^{k(t)}\mathbf{g}(k)}=\frac{\prod_{k=1}^{n-1}\mathbf{g}(k)}{\prod_{k=1}^{k(t)}\mathbf{g}(k)}\asymp \frac{V_\mathbf{g}(2^n)}{V_\mathbf{g}(2^{k(t)})}.\] If \(n\le0\), then \[\begin{align} &\frac{1}{\prod_{k=n}^{k(t)}\mathbf{g}(k)}=\frac{1}{\prod_{k=n}^{0}\mathbf{g}(k)}\frac{\prod_{k=n}^{0}\mathbf{g}(k)}{\prod_{k=n}^{k(t)}\mathbf{g}(k)}\\ &= \begin{cases} \frac{1}{\prod_{k=n}^{0}\mathbf{g}(k)}\frac{1}{\prod_{k=1}^{k(t)}\mathbf{g}(k)}&\text{if }k(t)\ge1,\\ \frac{1}{\prod_{k=n}^{0}\mathbf{g}(k)}\prod_{k=k(t)+1}^{0}\mathbf{g}(k)&\text{if }k(t)\le0, \end{cases}\\ &\asymp \begin{cases} V_\mathbf{g}(2^n)\frac{1}{V_{\mathbf{g}}(2^{k(t)})}&\text{if }k(t)\ge1,\\ V_\mathbf{g}(2^n)\frac{1}{V_{\mathbf{g}}(2^{k(t)})}&\text{if }k(t)\le0, \end{cases}=\frac{V_\mathbf{g}(2^n)}{V_{\mathbf{g}}(2^{k(t)})}. \end{align}\] Hence \[\mathbb{P}\left[\omega:\mathbf{v}(\omega)(t)|_{\mathbb{Z}\cap[n,+\infty)}=\mathbf{u}|_{\mathbb{Z}\cap[n,+\infty)}\right]\lesssim \frac{V_\mathbf{g}(2^n)}{V_{\mathbf{g}}(2^{k(t)})}.\] If \(n\ge k(t)\), then \[\mathbb{P}\left[\omega:\mathbf{v}(\omega)(t)|_{\mathbb{Z}\cap[n,+\infty)}=\mathbf{u}|_{\mathbb{Z}\cap[n,+\infty)}\right]\le1=\frac{V_\mathbf{g}(2^n)}{V_\mathbf{g}(2^n)}\le \frac{V_\mathbf{g}(2^n)}{V_{\mathbf{g}}(2^{k(t)})}.\] In summary, we have \[\mathbb{P}\left[\omega:\mathbf{v}(\omega)(t)\in{B}_{\mathcal{U}(\mathbf{g})}(\mathbf{u},2^n)\right]\lesssim \frac{V_\mathbf{g}(2^n)}{V_\mathbf{g}(2^{k(t)})}\asymp\frac{1}{V_{\mathbf{g}}(2^{k(t)})}\mathbf{m}_{\mathcal{U}(\mathbf{g})}(B_{\mathcal{U}(\mathbf{g})}(\mathbf{u},2^n)),\] which gives \(\mathbb{P}_{\mathbf{v}(\cdot)(t)}\) is absolutely continuous with respect to \(\mathbf{m}_{\mathcal{U}(\mathbf{g})}\) with Radon derivative \(\lesssim \frac{1}{V_\mathbf{g}(2^{k(t)})}\). Since for any \(\omega\), \[2^{k(t)}\asymp t\wedge (D-t)=\mathbf{d}_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{v}(\omega)(t),\mathbf{s}_0(t))],x)\wedge \mathbf{d}_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{v}(\omega)(t),\mathbf{s}_0(t))],y),\] we have \(V_\mathbf{g}(\mathbf{d}_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\cdot,\mathbf{s}_0(t))],x)\wedge \mathbf{d}_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\cdot,\mathbf{s}_0(t))],y))\mathbb{P}_{\mathbf{v}(\cdot)(t)}\) is absolutely continuous with respect to \(\mathbf{m}_{\mathcal{U}(\mathbf{g})}\) with Radon derivative \(\lesssim1\). Therefore \[\begin{align} &\int_0^D\left(\int_\Omega (h1_B)\left([(\mathbf{v}(\omega)(t),\mathbf{s}_0(t))]\right)\mathbb{P}(\mathrm{d}\omega)\right)\mathrm{d} t\\ &=\int_0^D\left(\int_{\mathcal{U}(\mathbf{g})} (h1_B)\left([(\mathbf{u},\mathbf{s}_0(t))]\right)\mathbb{P}_{\mathbf{v}(\cdot)(t)}(\mathrm{d}\mathbf{u})\right)\mathrm{d} t\\ &=\int_0^D\left(\int_{\mathcal{U}(\mathbf{g})} (h1_B)\left([(\mathbf{u},\mathbf{s}_0(t))]\right)\right.\\ &\left.\frac{V_\mathbf{g}(\mathbf{d}_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{u},\mathbf{s}_0(t))],x)\wedge \mathbf{d}_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{u},\mathbf{s}_0(t))],y))}{V_\mathbf{g}(\mathbf{d}_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{u},\mathbf{s}_0(t))],x)\wedge \mathbf{d}_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{u},\mathbf{s}_0(t))],y))} \mathbb{P}_{\mathbf{v}(\cdot)(t)}(\mathrm{d}\mathbf{u})\right)\mathrm{d} t\\ &\lesssim \int_0^D\left(\int_{\mathcal{U}(\mathbf{g})}\frac{(h1_B)\left([(\mathbf{u},\mathbf{s}_0(t))]\right)}{V_\mathbf{g}(\mathbf{d}_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{u},\mathbf{s}_0(t))],x)\wedge \mathbf{d}_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{u},\mathbf{s}_0(t))],y))}\mathbf{m}_{\mathcal{U}(\mathbf{g})}(\mathrm{d}\mathbf{u})\right)\mathrm{d} t\\ &=\int_{\mathcal{U}(\mathbf{g})}\left(\int_0^D\frac{(h1_B)\left([(\mathbf{u},\mathbf{s}_0(t))]\right)}{V_\mathbf{g}(\mathbf{d}_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{u},\mathbf{s}_0(t))],x)\wedge \mathbf{d}_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{u},\mathbf{s}_0(t))],y))}\mathrm{d} t\right)\mathbf{m}_{\mathcal{U}(\mathbf{g})}(\mathrm{d}\mathbf{u})\\ &\le\int_{\mathcal{U}(\mathbf{g})}\left(\int_{\mathcal{T}(\mathbf{b})}\frac{(h1_B)\left([(\mathbf{u},\mathbf{t})]\right)}{V_\mathbf{g}(\mathbf{d}_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{u},\mathbf{t})],x)\wedge \mathbf{d}_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{u},\mathbf{t})],y))}\lambda_{\mathcal{T}(\mathbf{b})}(\mathrm{d}\mathbf{t})\right)\mathbf{m}_{\mathcal{U}(\mathbf{g})}(\mathrm{d}\mathbf{u})\\ &=\int_{B}\frac{h(z)}{V_\mathbf{g}(\mathbf{d}_{\mathcal{L}(\mathbf{g},\mathbf{b})}(x,z))\wedge V_\mathbf{g}(\mathbf{d}_{\mathcal{L}(\mathbf{g},\mathbf{b})}(y,z))}\lambda_{\mathcal{L}(\mathbf{g},\mathbf{b})}(\mathrm{d} z). \end{align}\] ◻

Proof. For notational convenience, let us denote \(B(\cdot,\cdot)=B_{\mathcal{L}(\mathbf{g},\mathbf{b})}(\cdot,\cdot)\), \(d=\mathbf{d}_{\mathcal{L}(\mathbf{g},\mathbf{b})}\), \(m=\mathbf{m}_{\mathcal{L}(\mathbf{g},\mathbf{b})}\), \(\lambda=\lambda_{\mathcal{L}(\mathbf{g},\mathbf{b})}\), \(\nabla=\nabla_\mathcal{L}\). Since \(m(B(x_0,r))\asymp V_\mathbf{g}(r)V_\mathbf{b}(r)\), we have \[\begin{align} &\int_{B(x_0,r)}\lvert f-\mathchoice {{\setbox 0=\displaystyle{\textstyle-}{\int}\vcenter{\textstyle- }\kern-.6\wd 0}}{{\setbox 0=\textstyle{\scriptstyle-}{\int}\vcenter{\scriptstyle- }\kern-.6\wd 0}}{{\setbox 0=\scriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}}{{\setbox 0=\scriptscriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}}\!\int_{B(x_0,r)}f\mathrm{d} m\rvert^p\mathrm{d} m\le\int_{B(x_0,r)}\mathchoice {{\setbox 0=\displaystyle{\textstyle-}{\int}\vcenter{\textstyle- }\kern-.6\wd 0}}{{\setbox 0=\textstyle{\scriptstyle-}{\int}\vcenter{\scriptstyle- }\kern-.6\wd 0}}{{\setbox 0=\scriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}}{{\setbox 0=\scriptscriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}}\!\int_{B(x_0,r)}|f(x)-f(y)|^pm(\mathrm{d} y)m(\mathrm{d} x)\\ &\asymp\frac{1}{V_\mathbf{g}(r)V_\mathbf{b}(r)}\int_{B(x_0,r)}\int_{B(x_0,r)}|f(x)-f(y)|^pm(\mathrm{d} y)m(\mathrm{d} x). \end{align}\] For any distinct \(x,y\in B(x_0,r)\), let \(\Gamma_{x,y}\), \(\mathbb{P}^{x,y}\) be given by Proposition 11, then for any \(\gamma\in\Gamma_{x,y}\), we have \[\begin{align} &|f(x)-f(y)|^p\le\left(\int_0^{d(x,y)}|\nabla f(\gamma(t))|\mathrm{d} t\right)^p\\ &\le d(x,y)^{p-1}\int_0^{d(x,y)}|\nabla f(\gamma(t))|^p\mathrm{d} t\le (2r)^{p-1}\int_0^{d(x,y)}|\nabla f(\gamma(t))|^p\mathrm{d} t. \end{align}\] Taking expectation over \(\gamma\in\Gamma_{x,y}\) under \(\mathbb{P}^{x,y}\), by Proposition 11, we have \[\begin{align} &|f(x)-f(y)|^p\le(2r)^{p-1}\int_{\Gamma_{x,y}}\left(\int_0^{d(x,y)}|\nabla f(\gamma(t))|^p\mathrm{d} t\right)\mathrm{d} \mathbb{P}^{x,y}\\ &\lesssim r^{p-1}\int_{B(x,d(x,y))}\frac{|\nabla f(z)|^p}{V_\mathbf{g}(d(x,z))\wedge V_\mathbf{g}(d(y,z))}\lambda(\mathrm{d} z). \end{align}\] Hence \[\begin{align} &\int_{B(x_0,r)}\lvert f-\mathchoice {{\setbox 0=\displaystyle{\textstyle-}{\int}\vcenter{\textstyle- }\kern-.6\wd 0}}{{\setbox 0=\textstyle{\scriptstyle-}{\int}\vcenter{\scriptstyle- }\kern-.6\wd 0}}{{\setbox 0=\scriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}}{{\setbox 0=\scriptscriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}}\!\int_{B(x_0,r)}f\mathrm{d} m\rvert^p\mathrm{d} m\\ &\lesssim\frac{r^{p-1}}{V_\mathbf{g}(r)V_\mathbf{b}(r)}\int_{B(x_0,r)}\int_{B(x_0,r)}\left(\int_{B(x,d(x,y))}\frac{|\nabla f(z)|^p}{V_\mathbf{g}(d(x,z))\wedge V_\mathbf{g}(d(y,z))}\lambda(\mathrm{d} z)\right)m(\mathrm{d} x)m(\mathrm{d} y)\\ &\le\frac{r^{p-1}}{V_\mathbf{g}(r)V_\mathbf{b}(r)}\int_{B(x_0,4r)}\int_{B(x_0,4r)}\left(\int_{B(x_0,4r)}\frac{|\nabla f(z)|^p}{V_\mathbf{g}(d(x,z))\wedge V_\mathbf{g}(d(y,z))}\lambda(\mathrm{d} z)\right)m(\mathrm{d} x)m(\mathrm{d} y)\\ &=\frac{r^{p-1}}{V_\mathbf{g}(r)V_\mathbf{b}(r)}\\ &\cdot\int_{B(x_0,4r)}|\nabla f(z)|^p\left(\int_{B(x_0,4r)}\int_{B(x_0,4r)}\frac{1}{V_\mathbf{g}(d(x,z))\wedge V_\mathbf{g}(d(y,z))}m(\mathrm{d} x)m(\mathrm{d} y)\right)\lambda(\mathrm{d} z), \end{align}\] where for any \(z\in B(x_0,4r)\), we have \[\begin{align} &\int_{B(x_0,4r)}\int_{B(x_0,4r)}\frac{1}{V_\mathbf{g}(d(x,z))\wedge V_\mathbf{g}(d(y,z))}m(\mathrm{d} x)m(\mathrm{d} y)\\ &\le\int_{B(x_0,4r)}\int_{B(x_0,4r)}\frac{1_{d(x,z)\le d(y,z)}+1_{{d(x,z)\ge d(y,z)}}}{V_\mathbf{g}(d(x,z))\wedge V_\mathbf{g}(d(y,z))}m(\mathrm{d} y)m(\mathrm{d} x)\\ &=2\int_{B(x_0,4r)}\frac{m(B(x_0,4r)\backslash B(z,d(x,z)))}{V_\mathbf{g}(d(x,z))}m(\mathrm{d} x)\le 2m(B(x_0,4r))\int_{B(z,8r)}\frac{m(\mathrm{d} x)}{V_\mathbf{g}(d(x,z))}\\ &=2m(B(x_0,4r))\sum_{n\le3}\int_{B(z,2^nr)\backslash B(z,2^{n-1}r)}\frac{m(\mathrm{d} x)}{V_\mathbf{g}(d(x,z))}\le2m(B(x_0,4r))\sum_{n\le3}\frac{m(B(z,2^nr))}{V_\mathbf{g}(2^{n-1}r)}\\ &\asymp V_\mathbf{g}(r)V_\mathbf{b}(r)\sum_{n\le3}\frac{V_\mathbf{g}(2^nr)V_\mathbf{b}(2^nr)}{V_\mathbf{g}(2^{n-1}r)}\asymp V_\mathbf{g}(r)V_\mathbf{b}(r)\sum_{n\le3}V_\mathbf{b}(2^nr)\asymp V_\mathbf{g}(r)V_\mathbf{b}(r)^2, \end{align}\] which gives \[\begin{align} &\int_{B(x_0,r)}\lvert f-\mathchoice {{\setbox 0=\displaystyle{\textstyle-}{\int}\vcenter{\textstyle- }\kern-.6\wd 0}}{{\setbox 0=\textstyle{\scriptstyle-}{\int}\vcenter{\scriptstyle- }\kern-.6\wd 0}}{{\setbox 0=\scriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}}{{\setbox 0=\scriptscriptstyle{\scriptscriptstyle-}{\int}\vcenter{\scriptscriptstyle- }\kern-.6\wd 0}}\!\int_{B(x_0,r)}f\mathrm{d} m\rvert^p\mathrm{d} m\\ &\lesssim \frac{r^{p-1}}{V_\mathbf{g}(r)V_\mathbf{b}(r)}\int_{B(x_0,4r)}|\nabla f(z)|^pV_\mathbf{g}(r)V_\mathbf{b}(r)^2\lambda(\mathrm{d} z)\\ &=r^{p-1}V_\mathbf{b}(r)\int_{B(x_0,4r)}|\nabla f(z)|^p\lambda(\mathrm{d} z). \end{align}\] ◻

Proof. Write \(B_{\mathcal{L}(\mathbf{g},\mathbf{b})}=B_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{u},\mathbf{t})],r)\), where \((\mathbf{u},\mathbf{t})\in\mathcal{P}(\mathbf{g},\mathbf{b})\). Let \(n\) be the integer satisfying \(2^{n-1}\le r< 2^n\). By Lemma 9, we have the following dichotomies. For \(B_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{u},\mathbf{t})],r)\), either \(B_{\mathcal{T}(\mathbf{b})}(\mathbf{t},r)\cap\cup_{k\ge n+2}\mathcal{W}_k=\emptyset\), then \[\label{eq95dic1r1} \mathcal{Q}^{-1}(B_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{u},\mathbf{t})],r))\subseteq B_{\mathcal{U}(\mathbf{g})}(\mathbf{u},8r)\times B_{\mathcal{T}(\mathbf{b})}(\mathbf{t},r),\tag{22}\] or \(B_{\mathcal{T}(\mathbf{b})}(\mathbf{t},r)\cap\cup_{k\ge n+2}\mathcal{W}_k=\{\mathbf{t}^{(1)}\}\), where \(\mathbf{t}^{(1)}\in\mathcal{W}_m\) for some \(m\ge n+2\), then \[\label{eq95dic1r2} \mathcal{Q}^{-1}(B_{{\mathcal{L}(\mathbf{g},\mathbf{b})}}([(\mathbf{u},\mathbf{t})],r))\subseteq\bigcup_{k=1}^{\mathbf{g}(m)}B_{\mathcal{U}(\mathbf{g})}(\overline{\mathbf{u}}^{(k)},8r)\times B_{\mathcal{T}(\mathbf{b})}(\mathbf{t},r),\tag{23}\] where \(\overline{\mathbf{u}}^{(1)}\), …, \(\overline{\mathbf{u}}^{(\mathbf{g}(m))}\in\mathcal{U}(\mathbf{g})\) are all the points satisfying that \([(\overline{\mathbf{u}}^{(1)},\mathbf{t}^{(1)})]=\ldots=[(\overline{\mathbf{u}}^{(\mathbf{g}(m))},\mathbf{t}^{(1)})]=[(\mathbf{u},\mathbf{t}^{(1)})]\).

For \(B_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{u},\mathbf{t})],8r)\), either \(B_{\mathcal{T}(\mathbf{b})}(\mathbf{t},8r)\cap\cup_{k\ge n+5}\mathcal{W}_k=\emptyset\), then \[\mathcal{Q}^{-1}(B_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{u},\mathbf{t})],8r))\subseteq B_{\mathcal{U}(\mathbf{g})}(\mathbf{u},64r)\times B_{\mathcal{T}(\mathbf{b})}(\mathbf{t},8r)\subseteq\mathcal{Q}^{-1}(B_{{\mathcal{L}(\mathbf{g},\mathbf{b})}}([(\mathbf{u},\mathbf{t})],256r)),\] or \(B_{\mathcal{T}(\mathbf{b})}(\mathbf{t},8r)\cap\cup_{k\ge n+5}\mathcal{W}_k=\{\mathbf{s}^{(1)}\}\), where \(\mathbf{s}^{(1)}\in\mathcal{W}_l\) for some \(l\ge n+5\), then \[\begin{align} &\mathcal{Q}^{-1}(B_{{\mathcal{L}(\mathbf{g},\mathbf{b})}}([(\mathbf{u},\mathbf{t})],8r))\nonumber\\ &\subseteq\bigcup_{k=1}^{\mathbf{g}(l)}B_{\mathcal{U}(\mathbf{g})}\left(\widetilde{\mathbf{u}}^{(k)},64r\right)\times B_{\mathcal{T}(\mathbf{b})}(\mathbf{t},8r)\nonumber\\ &\subseteq\mathcal{Q}^{-1}(B_{{\mathcal{L}(\mathbf{g},\mathbf{b})}}([(\mathbf{u},\mathbf{t})],256r))\label{eq95dic2r2}, \end{align}\tag{24}\] where \(\widetilde{\mathbf{u}}^{(1)}\), …, \(\widetilde{\mathbf{u}}^{(\mathbf{g}(l))}\in\mathcal{U}(\mathbf{g})\) are all the points satisfying that \([(\widetilde{\mathbf{u}}^{(1)},\mathbf{s}^{(1)})]=\ldots=[(\widetilde{\mathbf{u}}^{(\mathbf{g}(l))},\mathbf{s}^{(1)})]=[(\mathbf{u},\mathbf{s}^{(1)})]\). To write in a unified way, denote \(\mathbf{u}^{(1)}\), …, \(\mathbf{u}^{(\sup_\mathbb{Z}\mathbf{g})}\in\mathcal{U}(\mathbf{g})\), such that \[\begin{align} &\mathcal{Q}^{-1}(B_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{u},\mathbf{t})],8r))\nonumber\\ &\subseteq\bigcup_{k=1}^{\sup_\mathbb{Z}\mathbf{g}}B_{\mathcal{U}(\mathbf{g})}\left(\mathbf{u}^{(k)},64r\right)\times B_{\mathcal{T}(\mathbf{b})}\left(\mathbf{t},8r\right)\nonumber\\ &\subseteq\mathcal{Q}^{-1}(B_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{u},\mathbf{t})],256r))\label{eq95dic2r}. \end{align}\tag{25}\] Then \(\{\mathbf{u}^{(1)},\ldots,\mathbf{u}^{(\sup_\mathbb{Z}\mathbf{g})}\}\) is either the one-point set \(\{\mathbf{u}\}\) or the set \(\{\widetilde{\mathbf{u}}^{(1)},\ldots,\widetilde{\mathbf{u}}^{(\mathbf{g}(l))}\}\). It is possible that \(\{\overline{\mathbf{u}}^{(\cdot)}\}\), \(\{\widetilde{\mathbf{u}}^{(\cdot)}\}\) are different, but \(\{\overline{\mathbf{u}}^{(\cdot)}\}\), \(\{\widetilde{\mathbf{u}}^{(\cdot)}\}\) both contain \(\mathbf{u}\).

We claim that \[\label{eq95claim951r} \mathcal{Q}^{-1}(B_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{u},\mathbf{t})],r))\subseteq\bigcup_{k=1}^{\sup_\mathbb{Z}\mathbf{g}}B_{\mathcal{U}(\mathbf{g})}\left(\mathbf{u}^{(k)},64r\right)\times B_{\mathcal{T}(\mathbf{b})}(\mathbf{t},r).\tag{26}\] Indeed, we consider the dichotomy for \(B_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{u},\mathbf{t})],r)\). For the first case, by Equation (22 ), we have \[\mathcal{Q}^{-1}(B_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{u},\mathbf{t})],r))\subseteq B_{\mathcal{U}(\mathbf{g})}(\mathbf{u},8r)\times B_{\mathcal{T}(\mathbf{b})}(\mathbf{t},r)\subseteq\bigcup_{k=1}^{\sup_\mathbb{Z}\mathbf{g}}B_{\mathcal{U}(\mathbf{g})}\left(\mathbf{u}^{(k)},64r\right)\times B_{\mathcal{T}(\mathbf{b})}(\mathbf{t},r),\] because \(\{\mathbf{u}^{(\cdot)}\}\) always contains \(\mathbf{u}\). For the second case, if \(m\ge n+5\), then \[B_{\mathcal{T}(\mathbf{b})}(\mathbf{t},8r)\cap\cup_{k\ge n+5}\mathcal{W}_k\supseteq B_{\mathcal{T}(\mathbf{b})}(\mathbf{t},r)\cap\cup_{k\ge n+5}\mathcal{W}_k=\{\mathbf{t}^{(1)}\},\] we are in the second case of the dichotomy for \(B_{{\mathcal{L}(\mathbf{g},\mathbf{b})}}([(\mathbf{u},\mathbf{t})],8r))\), \(\mathbf{t}^{(1)}=\mathbf{s}^{(1)}\), \(m=l\), and \(\{\overline{\mathbf{u}}^{(\cdot)}\}\), \(\{\widetilde{\mathbf{u}}^{(\cdot)}\}\) coincide. Then by Equation (23 ), we have \[\begin{align} &\mathcal{Q}^{-1}(B_{{\mathcal{L}(\mathbf{g},\mathbf{b})}}([(\mathbf{u},\mathbf{t})],r))\subseteq\bigcup_{k=1}^{\mathbf{g}(m)}B_{\mathcal{U}(\mathbf{g})}(\overline{\mathbf{u}}^{(k)},8r)\times B_{\mathcal{T}(\mathbf{b})}(\mathbf{t},r)\\ &=\bigcup_{k=1}^{\mathbf{g}(l)}B_{\mathcal{U}(\mathbf{g})}(\widetilde{\mathbf{u}}^{(k)},8r)\times B_{\mathcal{T}(\mathbf{b})}(\mathbf{t},r)\subseteq\bigcup_{k=1}^{\mathbf{g}(l)}B_{\mathcal{U}(\mathbf{g})}(\widetilde{\mathbf{u}}^{(k)},64r)\times B_{\mathcal{T}(\mathbf{b})}(\mathbf{t},r)\\ &=\bigcup_{k=1}^{\sup_\mathbb{Z}\mathbf{g}}B_{\mathcal{U}(\mathbf{g})}\left(\mathbf{u}^{(k)},64r\right)\times B_{\mathcal{T}(\mathbf{b})}(\mathbf{t},r). \end{align}\] If \(m\in\{n+2,n+3,n+4\}\), then for any \(\overline{\mathbf{u}}^{(k)}\), we have \(\mathbf{d}_{\mathcal{U}(\mathbf{g})}(\overline{\mathbf{u}}^{(k)},\mathbf{u})\le2^m\le2^{n+4}\), hence \[\begin{align} &\bigcup_{k=1}^{\mathbf{g}(m)}B_{\mathcal{U}(\mathbf{g})}(\overline{\mathbf{u}}^{(k)},8r)\subseteq\bigcup_{k=1}^{\mathbf{g}(m)} B_{\mathcal{U}(\mathbf{g})}({\mathbf{u}},\mathbf{d}_{\mathcal{U}(\mathbf{g})}(\overline{\mathbf{u}}^{(k)},\mathbf{u})+8r)\\ &\subseteq B_{\mathcal{U}(\mathbf{g})}({\mathbf{u}},2^{n+4}+8r)\subseteq B_{\mathcal{U}(\mathbf{g})}({\mathbf{u}},40r)\subseteq\bigcup_{k=1}^{\sup_\mathbb{Z}\mathbf{g}}B_{\mathcal{U}(\mathbf{g})}\left(\mathbf{u}^{(k)},64r\right), \end{align}\] also because \(\{\mathbf{u}^{(\cdot)}\}\) always contains \(\mathbf{u}\). Then by Equation (23 ), we have \[\mathcal{Q}^{-1}(B_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{u},\mathbf{t})],r))\subseteq\bigcup_{k=1}^{\sup_\mathbb{Z}\mathbf{g}}B_{\mathcal{U}(\mathbf{g})}\left(\mathbf{u}^{(k)},64r\right)\times B_{\mathcal{T}(\mathbf{b})}(\mathbf{t},r).\] Therefore, we finish the proof of Equation (26 ).

By the capacity upper bound on \(\mathcal{T}(\mathbf{b})\), Proposition 8, there exists \(\phi_\mathcal{T}\in\mathcal{F}^\mathcal{T}\cap C_c(\mathcal{T}(\mathbf{b}))\) with \(0\le\phi_\mathcal{T}\le1\) in \(\mathcal{T}(\mathbf{b})\), \(\phi_\mathcal{T}=1\) in \(B_{\mathcal{T}(\mathbf{b})}(\mathbf{t},r)\), \(\phi_\mathcal{T}=0\) on \(\mathcal{T}(\mathbf{b})\backslash B_{\mathcal{T}(\mathbf{b})}(\mathbf{t},8r)\) such that \(\mathcal{E}^\mathcal{T}(\phi_\mathcal{T})\lesssim \frac{1}{r^{p-1}}\). Let \(\phi_\mathcal{L}:\mathcal{L}(\mathbf{g},\mathbf{b})\to\mathbb{R}\) be given by \[\phi_\mathcal{L}([(\mathbf{v},\cdot)])= \begin{cases} \phi_\mathcal{T}&\text{if }\mathbf{v}\in \bigcup_{k=1}^{\sup_\mathbb{Z}\mathbf{g}}B_{\mathcal{U}(\mathbf{g})}\left(\mathbf{u}^{(k)},64r\right),\\ 0&\text{if }\mathbf{v}\not\in \bigcup_{k=1}^{\sup_\mathbb{Z}\mathbf{g}}B_{\mathcal{U}(\mathbf{g})}\left(\mathbf{u}^{(k)},64r\right). \end{cases}\]

Firstly, we show that \({\phi_\mathcal{L}}\) is well-defined. We only need to show that if \[\mathbf{v}^{(1)}\in\bigcup_{k=1}^{\sup_\mathbb{Z}\mathbf{g}}B_{\mathcal{U}(\mathbf{g})}\left(\mathbf{u}^{(k)},64r\right),\mathbf{v}^{(2)}\not\in\bigcup_{k=1}^{\sup_\mathbb{Z}\mathbf{g}}B_{\mathcal{U}(\mathbf{g})}\left(\mathbf{u}^{(k)},64r\right),\] satisfy \([(\mathbf{v}^{(1)},\mathbf{s})]=[(\mathbf{v}^{(2)},\mathbf{s})]\), then \(\phi_\mathcal{T}(\mathbf{s})=0\). By assumption, we have \(\mathbf{d}_{\mathcal{U}(\mathbf{d})}(\mathbf{v}^{(1)},\mathbf{v}^{(2)})\ge64r\ge2^{n+5}\), hence \(\mathbf{s}\in\cup_{k\ge n+5}\mathcal{W}_k\). Suppose \(\mathbf{s}\in B_{\mathcal{T}(\mathbf{b})}(\mathbf{t},8r)\), then we are in the second case of the dichotomy for \(B_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{u},\mathbf{t})],8r)\), hence \(\mathbf{s}=\mathbf{s}^{(1)}\), \(\{\mathbf{v}^{(1)},\mathbf{v}^{(2)}\}\subseteq\{\widetilde{\mathbf{u}}^{(1)},\ldots,\widetilde{\mathbf{u}}^{(\mathbf{g}(l))}\}\), in particular, we have \[\mathbf{v}^{(2)}\in\bigcup_{k=1}^{\mathbf{g}(l)}B_{\mathcal{U}(\mathbf{g})}\left(\widetilde{{\mathbf{u}}}^{(k)},64r\right)=\bigcup_{k=1}^{\sup_\mathbb{Z}\mathbf{g}}B_{\mathcal{U}(\mathbf{g})}\left(\mathbf{u}^{(k)},64r\right),\] contradiction. Hence \(\mathbf{s}\not\in B_{\mathcal{T}(\mathbf{b})}(\mathbf{t},8r)\), \(\phi_\mathcal{T}(\mathbf{s})=0\). Therefore, \(\phi_\mathcal{L}\) is well-defined.

Secondly, it is obvious that \(\phi_\mathcal{L}\in C_c(\mathcal{L}(\mathbf{g},\mathbf{b}))\). We claim that \(\phi_\mathcal{L}\in\mathcal{C}^\mathcal{L}_{n+4}\subseteq\mathcal{C}^\mathcal{L}\). Indeed, let \(\widetilde{\phi}_\mathcal{L}=\phi_\mathcal{L}\circ\mathcal{Q}\in C_c(\mathcal{P}(\mathbf{g},\mathbf{b}))\). For any \(\mathbf{v}\in\mathcal{U}(\mathbf{g})\), \(\widetilde{\phi}_\mathcal{L}(\mathbf{v},\cdot)\) is either \(\phi_\mathcal{T}\) or \(0\), hence in \(\mathcal{F}^\mathcal{T}\). For any \(\mathbf{s}\in\mathcal{T}(\mathbf{b})\), for any \(\mathbf{v}^{(1)}\), \(\mathbf{v}^{(2)}\in\mathcal{U}(\mathbf{g})\) with \(\mathbf{d}_{\mathcal{U}(\mathbf{g})}(\mathbf{v}^{(1)},\mathbf{v}^{(2)})\le2^{n+4}\), suppose \[\mathbf{v}^{(1)}\in\bigcup_{k=1}^{\sup_\mathbb{Z}\mathbf{g}}B_{\mathcal{U}(\mathbf{g})}\left(\mathbf{u}^{(k)},64r\right),\mathbf{v}^{(2)}\not\in\bigcup_{k=1}^{\sup_\mathbb{Z}\mathbf{g}}B_{\mathcal{U}(\mathbf{g})}\left(\mathbf{u}^{(k)},64r\right),\] then \(\mathbf{d}_{\mathcal{U}(\mathbf{g})}(\mathbf{v}^{(1)},\mathbf{v}^{(2)})\ge64r\ge2^{n+5}>2^{n+4}\ge \mathbf{d}_{\mathcal{U}(\mathbf{g})}(\mathbf{v}^{(1)},\mathbf{v}^{(2)})\), contradiction. Hence \(\widetilde{\phi}_\mathcal{L}(\mathbf{v}^{(1)},\mathbf{s})=\widetilde{\phi}_\mathcal{L}(\mathbf{v}^{(2)},\mathbf{s})\) is either \(\phi_\mathcal{T}(\mathbf{s})\) or \(0\), that is, \(\widetilde{\phi}_\mathcal{L}(\cdot,\mathbf{s})\) is constant on any closed ball with radius \(2^{n+4}\). Therefore \(\phi_\mathcal{L}\in\mathcal{C}^\mathcal{L}_{n+4}\subseteq\mathcal{C}^\mathcal{L}\).

Finally, it is obvious that \(0\le\phi_\mathcal{L}\le1\) in \(\mathcal{L}(\mathbf{g},\mathbf{b})\). By Equation (25 ), we have \(\phi_\mathcal{L}=0\) on \(\mathcal{L}(\mathbf{g},\mathbf{b})\backslash B_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{u},\mathbf{t})],256r)\). By Equation (26 ), we have \(\phi_\mathcal{L}=1\) in \(B_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{u},\mathbf{t})],r)\). Moreover \[\mathcal{E}^\mathcal{L}(\phi_\mathcal{L})\asymp V_\mathbf{g}(r)\mathcal{E}^\mathcal{T}(\phi_\mathcal{T})\lesssim V_\mathbf{g}(r)\frac{1}{r^{p-1}}=\frac{V_\mathbf{g}(r)V_\mathbf{b}(r)}{r^{p-1}V_\mathbf{b}(r)}\asymp \frac{\mathbf{m}_{\mathcal{L}(\mathbf{g},\mathbf{b})}(B_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{u},\mathbf{t})],r))}{r^{p-1}V_{\mathbf{b}}(r)},\] hence \[\mathrm{cap}_{\mathcal{L}(\mathbf{g},\mathbf{b})}(B_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{u},\mathbf{t})],r),\mathcal{L}(\mathbf{g},\mathbf{b})\backslash B_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{u},\mathbf{t})],256r))\lesssim\frac{\mathbf{m}_{\mathcal{L}(\mathbf{g},\mathbf{b})}(B_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{u},\mathbf{t})],r))}{r^{p-1}V_{\mathbf{b}}(r)}.\] Moreover, for any \(f\in\mathcal{C}^\mathcal{L}\), we have \(\widehat{f}=f\circ\mathcal{Q}\in C_c(\mathcal{P}(\mathbf{g},\mathbf{b}))\) satisfies \(\widehat{f}(\mathbf{v},\cdot)\in\mathcal{F}^\mathcal{T}\) for any \(\mathbf{v}\in\mathcal{U}(\mathbf{g})\). Let \(U=\bigcup_{k=1}^{\sup_\mathbb{Z}\mathbf{g}}B_{\mathcal{U}(\mathbf{g})}\left(\mathbf{u}^{(k)},64r\right)\). By Proposition 8, we have \[\begin{align} &\int_{B_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{u},\mathbf{t})],256r)}|f|^p|\nabla_\mathcal{L}\phi_\mathcal{L}|^p\mathrm{d}\lambda_{\mathcal{L}(\mathbf{g},\mathbf{b})}\nonumber\\ &=\int_{U}\left(\int_{B_{\mathcal{T}(\mathbf{b})}(\mathbf{t},8r)}|\widehat{f}(\mathbf{v},\cdot)|^p|\nabla_\mathcal{T}\phi_\mathcal{T}|^p\mathrm{d}\lambda_{\mathcal{T}(\mathbf{b})}\right)\mathbf{m}_{\mathcal{U}(\mathbf{g})}(\mathrm{d}\mathbf{v})\nonumber\\ &\lesssim\int_U \left(\int_{B_{\mathcal{T}(\mathbf{b})}(\mathbf{t},8r)}|\nabla_\mathcal{T}\widehat{f}(\mathbf{v},\cdot)|^p\mathrm{d}\lambda_{\mathcal{T}(\mathbf{b})}\right.\nonumber\\ &\left.+\frac{1}{r^{p-1}V_\mathbf{b}(r)}\int_{B_{\mathcal{T}(\mathbf{b})}(\mathbf{t},8r)}|\widehat{f}(\mathbf{v},\cdot)|^p\mathrm{d}\mathbf{m}_{\mathcal{T}(\mathbf{b})}\right)\mathbf{m}_{\mathcal{U}(\mathbf{g})}(\mathrm{d}\mathbf{v})\nonumber\\ &\le\int_{B_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{u},\mathbf{t})],256r)}|\nabla_\mathcal{L} f|^p\mathrm{d}\lambda_{\mathcal{L}(\mathbf{g},\mathbf{b})}\nonumber\\ &+\frac{1}{r^{p-1}V_\mathbf{b}(r)}\int_{B_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{u},\mathbf{t})],256r)}|f|^p\mathrm{d}\mathbf{m}_{\mathcal{L}(\mathbf{g},\mathbf{b})}\label{eq95Laa95CScalC}, \end{align}\tag{27}\] where the last inequality follows from Equation (25 ). For any \(f\in\mathcal{F}^\mathcal{L}\), since \(\mathcal{C}^\mathcal{L}\) is \((\mathcal{E}^\mathcal{L}_1(\cdot)=\mathcal{E}^\mathcal{L}(\cdot)+\lVert \cdot\rVert^p_{L^p(\mathcal{L}(\mathbf{g},\mathbf{b});\mathbf{m}_{\mathcal{L}(\mathbf{g},\mathbf{b})})})\)-dense in \(\mathcal{F}^\mathcal{L}\), by Proposition 15, there exist a quasi-continuous modification \(\widetilde{f}\) of \(f\), \(\{f_n\}\subseteq\mathcal{C}^\mathcal{L}\) which is \(\mathcal{E}^\mathcal{L}_1\)-convergent to \(f\), and an increasing sequence of closed sets \(\{F_k\}\) with \(\mathrm{cap}_1\left(\mathcal{L}(\mathbf{g},\mathbf{b})\backslash F_k\right)\downarrow0\), such that \(\{f_n\}\) is uniformly convergent to \(\widetilde{f}\) on each \(F_k\). Hence \(\mathrm{cap}_1\left(\mathcal{L}(\mathbf{g},\mathbf{b})\backslash\cup_kF_k\right)=0\) and \(\lim_{n\to+\infty}f_n(x)=\widetilde{f}(x)\) for any \(x\in\cup_kF_k\). By Proposition 17, \(\Gamma^\mathcal{L}(\phi_\mathcal{L})\left(\mathcal{L}(\mathbf{g},\mathbf{b})\backslash\cup_kF_k\right)=0\). Recall that \(\mathrm{d}\Gamma^\mathcal{L}(\phi_\mathcal{L})=|\nabla_\mathcal{L}\phi_\mathcal{L}|^p\mathrm{d}\lambda_{\mathcal{L}(\mathbf{g},\mathbf{b})}\). By Fatou’s lemma, we have \[\begin{align} &\int_{B_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{u},\mathbf{t})],256r)}|\widetilde{f}|^p|\nabla_\mathcal{L}\phi_\mathcal{L}|^p\mathrm{d}\lambda_{\mathcal{L}(\mathbf{g},\mathbf{b})}=\int_{B_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{u},\mathbf{t})],256r)}\lim_{n\to+\infty}|f_n|^p|\nabla_\mathcal{L}\phi_\mathcal{L}|^p\mathrm{d}\lambda_{\mathcal{L}(\mathbf{g},\mathbf{b})}\\ &\le\varliminf_{n\to+\infty}\int_{B_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{u},\mathbf{t})],256r)}|{f}_n|^p|\nabla_\mathcal{L}\phi_\mathcal{L}|^p\mathrm{d}\lambda_{\mathcal{L}(\mathbf{g},\mathbf{b})}\\ &\overset{(\star)}{\scalebox{2}[1]{\lesssim}}\varliminf_{n\to+\infty}\left(\int_{B_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{u},\mathbf{t})],256r)}|\nabla_\mathcal{L} f_n|^p\mathrm{d}\lambda_{\mathcal{L}(\mathbf{g},\mathbf{b})}\right.\\ &\left.+\frac{1}{r^{p-1}V_\mathbf{b}(r)}\int_{B_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{u},\mathbf{t})],256r)}|f_n|^p\mathrm{d}\mathbf{m}_{\mathcal{L}(\mathbf{g},\mathbf{b})}\right)\\ &\overset{(\diamond)}{\scalebox{2}[1]{=}}\int_{B_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{u},\mathbf{t})],256r)}|\nabla_\mathcal{L} f|^p\mathrm{d}\lambda_{\mathcal{L}(\mathbf{g},\mathbf{b})}+\frac{1}{r^{p-1}V_\mathbf{b}(r)}\int_{B_{\mathcal{L}(\mathbf{g},\mathbf{b})}([(\mathbf{u},\mathbf{t})],256r)}|f|^p\mathrm{d}\mathbf{m}_{\mathcal{L}(\mathbf{g},\mathbf{b})}, \end{align}\] where \((\star)\) follows from Equation (27 ) for \(\{f_n\}\), and \((\diamond)\) follows from the fact that \(\{f_n\}\) is \(\mathcal{E}^\mathcal{L}_1\)-convergent to \(f\). ◻

Proof. By replacing \(\theta_1, \theta_2\) by \(\lfloor{\theta_1} \rfloor, \lceil {\theta_2}\rceil\), respectively, we may assume that \(\theta_1\le\theta_2\) are non-negative integers. We construct \(\mathbf{a}(n)\) for \(n\ge0\), a similar construction applies for \(n\le0\).

Let \(\mathbf{a}(0)=2^{\theta_1}\). Denote \(\mathbf{A}(n)=\Pi_{k=0}^n\mathbf{a}(k)\) if \(\mathbf{a}(0),\ldots,\mathbf{a}(n)\) have already been defined.

By assumption, for any \(n\ge0\), we have \(\frac{\Phi(2^n)}{\Phi(1)}\ge \frac{1}{C}2^{\theta_1 n}\). If \(\frac{\Phi(2^n)}{\Phi(1)}\le C2^{\theta_1n}\) for any \(n\ge0\), then let \(\mathbf{a}(n)=2^{\theta_1}\) for any \(n\ge1\), the result is obvious. Otherwise, let \(n_1=\min\{n\ge0:\frac{\Phi(2^n)}{\Phi(1)}>C2^{\theta_1n}\}\), then \(n_1>0\), let \(\mathbf{a}(n)=2^{\theta_1}\) for any \(n=1,\ldots,n_1\). Since \(\frac{\Phi(2^{n_1-1})}{\Phi(1)}\le C2^{\theta_1(n_1-1)}\), we have \[\frac{\Phi(2^{n_1})}{\Phi(1)}=\frac{\Phi(2^{n_1})}{\Phi(2^{n_1-1})}\frac{\Phi(2^{n_1-1})}{\Phi(1)}\le (C2^{\theta_2})(C2^{\theta_1(n_1-1)})=C^22^{\theta_2-\theta_1}\mathbf{A}(n_1).\] By assumption, for any \(k\ge0\), we have \[\frac{\Phi(2^{n_1+k})}{\Phi(1)}=\frac{\Phi(2^{n_1+k})}{\Phi(2^{n_1})}\frac{\Phi(2^{n_1})}{\Phi(1)}\le(C2^{\theta_2k})\left(C^22^{\theta_2-\theta_1}\mathbf{A}(n_1)\right)=C^32^{\theta_2-\theta_1}\mathbf{A}(n_1)2^{\theta_2k}.\] By definition, \(\frac{\Phi(2^{n_1})}{\Phi(1)}>C\mathbf{A}(n_1)\ge \frac{1}{C}\mathbf{A}(n_1)\). If \(\frac{\Phi(2^{n_1+k})}{\Phi(1)}\ge \frac{1}{C}\mathbf{A}(n_1)2^{\theta_2k}\) for any \(k\ge0\), then let \(\mathbf{a}(n)=2^{\theta_2}\) for any \(n>n_1\), the result is obvious. Otherwise, let \(m_1=\min\{n_1+k:k\ge0,\frac{\Phi(2^{n_1+k})}{\Phi(1)}<\frac{1}{C}\mathbf{A}(n_1)2^{\theta_2k}\}\), then \(m_1>n_1\), let \(\mathbf{a}(n)=2^{\theta_2}\) for any \(n=n_1+1,\ldots,m_1\). Since \(\frac{\Phi(2^{m_1-1})}{\Phi(1)}\ge\frac{1}{C}\mathbf{A}(n_1)2^{\theta_2(m_1-1-n_1)}\), we have \[\frac{\Phi(2^{m_1})}{\Phi(1)}=\frac{\Phi(2^{m_1})}{\Phi(2^{m_1-1})}\frac{\Phi(2^{m_1-1})}{\Phi(1)}\ge\left(\frac{1}{C}2^{\theta_1}\right)\left(\frac{1}{C}\mathbf{A}(n_1)2^{\theta_2(m_1-1-n_1)}\right)=\frac{1}{C^22^{\theta_2-\theta_1}}\mathbf{A}(m_1).\] By assumption, for any \(k\ge0\), we have \[\frac{\Phi(2^{m_1+k})}{\Phi(1)}=\frac{\Phi(2^{m_1+k})}{\Phi(2^{m_1})}\frac{\Phi(2^{m_1})}{\Phi(1)}\ge\left(\frac{1}{C}2^{\theta_1k}\right)\left(\frac{1}{C^22^{\theta_2-\theta_1}}\mathbf{A}(m_1)\right)=\frac{1}{C^32^{\theta_2-\theta_1}}\mathbf{A}(m_1)2^{\theta_1k}.\] By definition, \(\frac{\Phi(2^{m_1})}{\Phi(1)}<\frac{1}{C}\mathbf{A}(n_1)2^{\theta_2(m_1-n_1)}=\frac{1}{C}\mathbf{A}(m_1)\le C\mathbf{A}(m_1)\).

Assume that we have constructed \(n_l, m_l\). If \(\frac{\Phi(2^{m_l+k})}{\Phi(1)}\le C\mathbf{A}(m_l)2^{\theta_1k}\) for any \(k\ge0\), then let \(\mathbf{a}(n)=2^{\theta_1}\) for any \(n>m_l\), the result is obvious. Otherwise, let \(n_{l+1}=\min\{m_l+k:k\ge0,\frac{\Phi(2^{m_l+k})}{\Phi(1)}>C\mathbf{A}(m_l)2^{\theta_1k}\}\), then \(n_{l+1}>m_l\), let \(\mathbf{a}(n)=2^{\theta_1}\) for any \(n=m_l+1,\ldots,n_{l+1}\). Since \(\frac{\Phi(2^{n_{l+1}-1})}{\Phi(1)}\le C\mathbf{A}(m_l)2^{\theta_1(n_{l+1}-1-m_l)}\), we have \[\frac{\Phi(2^{n_{l+1}})}{\Phi(1)}=\frac{\Phi(2^{n_{l+1}})}{\Phi(2^{n_{l+1}-1})}\frac{\Phi(2^{n_{l+1}-1})}{\Phi(1)}\le (C2^{\theta_2})\left(C\mathbf{A}(m_l)2^{\theta_1(n_{l+1}-1-m_l)}\right)=C^22^{\theta_2-\theta_1}\mathbf{A}(n_{l+1}).\] By assumption, for any \(k\ge0\), we have \[\begin{align} &\frac{\Phi(2^{n_{l+1}+k})}{\Phi(1)}=\frac{\Phi(2^{n_{l+1}+k})}{\Phi(2^{n_{l+1}})}\frac{\Phi(2^{n_{l+1}})}{\Phi(1)}\\ &\le\left(C2^{\theta_2k}\right)\left(C^22^{\theta_2-\theta_1}\mathbf{A}(n_{l+1})\right)=C^32^{\theta_2-\theta_1}\mathbf{A}(n_{l+1})2^{\theta_2k}. \end{align}\] By definition, \(\frac{\Phi(2^{n_{l+1}})}{\Phi(1)}>C\mathbf{A}(m_l)2^{\theta_1(n_{l+1}-m_l)}=C\mathbf{A}(n_{l+1})\ge \frac{1}{C}\mathbf{A}(n_{l+1})\). If \(\frac{\Phi(2^{n_{l+1}+k})}{\Phi(1)}\ge \frac{1}{C}\mathbf{A}(n_{l+1})2^{\theta_2k}\) for any \(k\ge0\), then let \(\mathbf{a}(n)=2^{\theta_2}\) for any \(n>n_{l+1}\), the result is obvious. Otherwise, let \(m_{l+1}=\min\{n_{l+1}+k:k\ge0,\frac{\Phi(2^{n_{l+1}+k})}{\Phi(1)}<\frac{1}{C}\mathbf{A}(n_{l+1})2^{\theta_2k}\}\), then \(m_{l+1}>n_{l+1}\), let \(\mathbf{a}(n)=2^{\theta_2}\) for any \(n=n_{l+1}+1,\ldots,m_{l+1}\). Since \(\frac{\Phi(2^{m_{l+1}-1})}{\Phi(1)}\ge\frac{1}{C}\mathbf{A}(n_{l+1})2^{\theta_2(m_{l+1}-1-n_{l+1})}\), we have \[\begin{align} &\frac{\Phi(2^{m_{l+1}})}{\Phi(1)}=\frac{\Phi(2^{m_{l+1}})}{\Phi(2^{m_{l+1}-1})}\frac{\Phi(2^{m_{l+1}-1})}{\Phi(1)}\\ &\ge\left(\frac{1}{C}2^{\theta_1}\right)\left(\frac{1}{C}\mathbf{A}(n_{l+1})2^{\theta_2(m_{l+1}-1-n_{l+1})}\right)=\frac{1}{C^22^{\theta_2-\theta_1}}\mathbf{A}(m_{l+1}). \end{align}\] By assumption, for any \(k\ge0\), we have \[\begin{align} &\frac{\Phi(2^{m_{l+1}+k})}{\Phi(1)}=\frac{\Phi(2^{m_{l+1}+k})}{\Phi(2^{m_{l+1}})}\frac{\Phi(2^{m_{l+1}})}{\Phi(1)}\\ &\ge\left(\frac{1}{C}2^{\theta_1k}\right)\left(\frac{1}{C^22^{\theta_2-\theta_1}}\mathbf{A}(m_{l+1})\right)=\frac{1}{C^32^{\theta_2-\theta_1}}\mathbf{A}(m_{l+1})2^{\theta_1k}. \end{align}\] By definition, \(\frac{\Phi(2^{m_{l+1}})}{\Phi(1)}<\frac{1}{C}\mathbf{A}(n_{l+1})2^{\theta_2(m_{l+1}-n_{l+1})}=\frac{1}{C}\mathbf{A}(m_{l+1})\le C\mathbf{A}(m_{l+1})\).

By the principle of induction, we obtain \(\mathbf{a}(n)\) for \(n\ge0\) satisfying that for any \(n\ge0\) \[\frac{1}{C^32^{\theta_2-\theta_1}}\prod_{k=0}^n\mathbf{a}(k)\le \frac{\Phi(2^n)}{\Phi(1)}\le{C^32^{\theta_2-\theta_1}}\prod_{k=0}^n\mathbf{a}(k).\] ◻

Proof of Theorem 3. By assumption, for any \(R,r>0\) with \(r\le R\), we have \[\frac{1}{C}\frac{R}{r}\le \frac{\frac{\Psi(R)}{R^{p-1}}}{\frac{\Psi(r)}{r^{p-1}}}\le C \frac{\Phi(R)}{\Phi(r)}\le CC_\Phi\left(\frac{R}{r}\right)^{\log_2C_\Phi}.\] By Lemma 11, there exists \(\mathbf{b}:\mathbb{Z}\to\{2^1,2^{\lceil{\log_2C_\Phi} \rceil}\}\) such that \[\frac{\frac{\Psi(2^n)}{(2^n)^{p-1}}}{\frac{\Psi(1)}{1^{p-1}}}\asymp\prod_{k=0}^n\mathbf{b}(k)\text{ for any }n\ge0,\] \[\frac{\frac{\Psi(1)}{1^{p-1}}}{\frac{\Psi(2^n)}{(2^n)^{p-1}}}\asymp \prod_{k=n}^0\mathbf{b}(k)\text{ for any }n\le0,\] hence \(\frac{\Psi(r)}{r^{p-1}}\asymp V_\mathbf{b}(r)\), or equivalently, \(\Psi(r)\asymp r^{p-1}V_\mathbf{b}(r)\).

Moreover, by assumption, for any \(R,r>0\) with \(r\le R\), we have \[\frac{1}{C}\le \frac{\frac{R^{p-1}\Phi(R)}{\Psi(R)}}{\frac{r^{p-1}\Phi(r)}{\Psi(r)}}=\left(\frac{R}{r}\right)^{p-1}\frac{\Phi(R)}{\Phi(r)}\frac{\Psi(r)}{\Psi(R)}\le C_\Phi \left(\frac{R}{r}\right)^{\log_2C_\Phi+p-1}.\] By Lemma 11 again, there exists \(\mathbf{g}:\mathbb{Z}\to\{2^0,2^{\lceil {\log_2C_\Phi+p}\rceil-1}\}\) such that \(\frac{r^{p-1}\Phi(r)}{\Psi(r)}\asymp V_\mathbf{g}(r)\), hence \(\Phi(r)\asymp r^{1-p}\Psi(r)V_\mathbf{g}(r)\asymp V_\mathbf{g}(r)V_\mathbf{b}(r)\).

Let \((\mathcal{L}(\mathbf{g},\mathbf{b}),\mathbf{d}_{\mathcal{L}(\mathbf{g},\mathbf{b})},\mathbf{m}_{\mathcal{L}(\mathbf{g},\mathbf{b})})\) be the corresponding Laakso-type space, and \((\mathcal{E}^\mathcal{L},\mathcal{F}^\mathcal{L})\) the corresponding \(p\)-energy with a \(p\)-energy measure \(\Gamma^\mathcal{L}\). Then by Proposition 9, Equation (21 ), Proposition 13, and Proposition 14, we have \((\mathcal{L}(\mathbf{g},\mathbf{b}),\mathbf{d}_{\mathcal{L}(\mathbf{g},\mathbf{b})})\) is a geodesic space, and 9 , 13 , 14 hold. In particular, for any \(d_h,\beta_p>0\) satisfying \(p\le\beta_p\le d_h+(p-1)\), let \(\Phi:r\mapsto r^{d_h}\), \(\Psi:r\mapsto r^{\beta_p}\), then for any \(R,r>0\) with \(r\le R\), we have \[\left(\frac{R}{r}\right)^p\le\left(\frac{R}{r}\right)^{\beta_p}=\frac{\Psi(R)}{\Psi(r)}=\left(\frac{R}{r}\right)^{\beta_p-d_h}\frac{\Phi(R)}{\Phi(r)}\le\left(\frac{R}{r}\right)^{p-1}\frac{\Phi(R)}{\Phi(r)},\] hence V(\(d_h\)), PI(\(\beta_p\)), \(\text{CS}(\beta_p)\) hold. ◻

Proof. It is obvious that \(\mathcal{L}_A\) is a non-empty closed convex subset of \(\mathcal{F}\), by [53], there exists a unique element \(e_A\in\mathcal{L}_A\) such that \(\mathcal{E}_1(e_A)=\inf_{u\in\mathcal{L}_A}\mathcal{E}_1(u)=\mathrm{cap}_1(A)\). Since \((e_A\vee 0)\wedge1\in\mathcal{L}_A\) and \(\mathcal{E}_1((e_A\vee 0)\wedge1)\le\mathcal{E}_1(e_A)\), which follows from the Markovian property, by the uniqueness of \(e_A\) in \(\mathcal{L}_A\), we have \(e_A=(e_A\vee 0)\wedge1\), which gives \(0\le e_A\le1\) \(m\)-a.e. in \(X\) and \(e_A=1\) \(m\)-a.e. on \(A\).

For any \(v\in\mathcal{F}\) with \(v\ge0\) \(m\)-a.e. on \(A\), for any \(t>0\), we have \(e_A+tv\in\mathcal{L}_A\), hence \(\mathcal{E}_1(e_A+tv)\ge\mathcal{E}_1(e_A)\), which gives \(\mathcal{E}_1(e_A;v)=\frac{1}{p}\lim_{t\downarrow0}\frac{1}{t}\left(\mathcal{E}_1(e_A+tv)-\mathcal{E}_1(e_A)\right)\ge0\).

Assume that \(u\in\mathcal{F}\) satisfies \(u=1\) \(m\)-a.e. on \(A\) and \(\mathcal{E}_1(u;v)\ge0\) for any \(v\in\mathcal{F}\) with \(v\ge0\) \(m\)-a.e. on \(A\). Since \(u-e_A=e_A-u=0\) \(m\)-a.e. on \(A\), by assumption we have \(\mathcal{E}_1(u;u-e_A)\ge0\), \(\mathcal{E}_1(u;e_A-u)\ge0\), which gives \(\mathcal{E}_1(u;u-e_A)=0\), similarly, we have \(\mathcal{E}_1(e_A;u-e_A)=0\). Suppose that \(u\ne e_A\), consider the function \(\varphi(t)=\mathcal{E}_1 \left((1-t)e_A+tu\right)=\mathcal{E}_1(e_A+t(u-e_A))\), \(t\in\mathbb{R}\), then \(\varphi'(t)=p\mathcal{E}_1(e_A+t(u-e_A);u-e_A)\), hence \(\varphi'(0)=p\mathcal{E}_1(e_A;u-e_A)=0\), \(\varphi'(1)=p\mathcal{E}_1(u;u-e_A)=0\). However, since \(\mathcal{E}_1(u-e_A)>0\), by (28 ) for \(\mathcal{E}_1\), we have \(\varphi'\) is strictly increasing, contradicting to \(\varphi'(0)=\varphi'(1)=0\). Therefore, \(u=e_A\). ◻

Proof. For (1), it is obvious by definition. For (2), we may assume that \(\mathrm{cap}_1(A)\), \(\mathrm{cap}_1(B)\) are finite. By Lemma 13, there exist unique \(e_A\in\mathcal{L}_A\), \(e_B\in\mathcal{L}_B\) such that \(\mathrm{cap}_1(A)=\mathcal{E}_1(e_A)\), \(\mathrm{cap}_1(B)=\mathcal{E}_1(e_B)\). Then \(e_A\vee e_B\in\mathcal{L}_{A\cup B}\), \(e_A\wedge e_B\in\mathcal{L}_{A\cap B}\), by 11 , we have \[\mathrm{cap}_1(A\cup B)+\mathrm{cap}_1(A\cap B)\le\mathcal{E}_1(e_A\vee e_B)+\mathcal{E}_1(e_A\wedge e_B)\le\mathcal{E}_1(e_A)+\mathcal{E}_1( e_B)=\mathrm{cap}_1(A)+\mathrm{cap}_1(B).\] For (3), by (1), we have “\(\ge\)". To show”\(\le\)", we may assume that the RHS is finite. By Lemma 13, for any \(n\), there exists \(e_n\in\mathcal{L}_{A_n}\) such that \(\mathrm{cap}_1(A_n)=\mathcal{E}_1(e_n)\). Since \(\{e_n\}\) is \(\mathcal{E}_1\)-bounded and \((\mathcal{F},\mathcal{E}_1^{1/p})\) is a reflexive Banach space, by the Banach–Alaoglu theorem (see [53]), there exists a subsequence, still denoted by \(\{e_n\}\), which is \(\mathcal{E}_1\)-weakly-convergent to some element \(e\in\mathcal{F}\). By the Mazur’s lemma, here we refer to the version in [54], for any \(n\ge1\), since \(\{e_m\}_{m\ge n}\) is \(\mathcal{E}_1\)-weakly-convergent to \(e\), there exist \(I_n\ge n\), \(\alpha^{(n)}_i\ge0\) for \(i=n,\ldots, I_n\) with \(\sum_{i=n}^{I_n}\alpha^{(n)}_i=1\) such that \(\mathcal{E}_1(\sum_{i=n}^{I_n}\alpha^{(n)}_ie_i-e)<\frac{1}{n}\), let \(f_n=\sum_{i=n}^{I_n}\alpha^{(n)}_ie_i\), then \(\{f_n\}\) is \(\mathcal{E}_1\)-convergent to \(e\), which implies the \(L^p\)-convergence. Since \(f_n=1\) \(m\)-a.e. on \(A_n\), by passing to a subsequence to have the a.e. convergence, we have \(e=1\) \(m\)-a.e. on \(A_n\) for any \(n\), which implies \(e=1\) \(m\)-a.e. on \(\cup_nA_n\), \(e\in\mathcal{L}_{\cup_nA_n}\). Hence \[\mathrm{cap}_1(\cup_nA_n)\le\mathcal{E}_1(e)\overset{(\star)}{\scalebox{2}[1]{\le}}\varliminf_{n\to+\infty}\mathcal{E}_1(e_n)=\sup_n\mathrm{cap}_1(A_n),\] where \((\star)\) follows from the fact that \(\{e_n\}\) is \(\mathcal{E}_1\)-weakly-convergent to \(e\).

By [17], a \([0,+\infty]\)-valued set function defined on \(\mathcal{O}\) satisfying the above three conditions gives a Choquet capacity by Equation (30 ). Since \((X,d)\) is a locally compact separable metric space, by [17], we have Equation (31 ). ◻

Proof. For simplicity, we may assume that \(G=X\). For any \(k\ge1\), there exists a closed set \(F_k\) with \(\mathrm{cap}_1(X\backslash F_k)<\frac{1}{k}\) such that \(u|_{F_k}\) is continuous. Since \(\mathcal{L}_{X\backslash F_k}=\mathcal{L}_{X\backslash \mathrm{supp}(1_{F_k}m)}\), we have \(\mathrm{cap}_1(X\backslash F_k)=\mathrm{cap}_1(X\backslash \mathrm{supp}(1_{F_k}m))\). By replacing \(F_k\) by \(\mathrm{supp}(1_{F_k}m)\), we may assume that \(F_k=\mathrm{supp}(1_{F_k}m)\). By replacing \(F_k\) by \(\cup_{i=1}^kF_i\), we may assume that \(\{F_k\}\) is increasing, then \(\mathrm{cap}_1(X\backslash\cup_kF_k)=0\). We claim that \(u(x)\ge0\) for any \(x\in\cup_kF_k\). Indeed, suppose that \(u(x)<0\) for some \(x\in F_k\), since \(u|_{F_k}\) is continuous, there exists an open neighborhood \(U\) of \(x\) such that \(u<0\) on \(U\cap F_k\). Since \(\mathrm{supp}(1_{F_k}m)=F_k\), we have \(m(U\cap F_k)>0\), contradicting to \(u\ge0\) \(m\)-a.e. in \(X\). ◻

Proof. We follow the same argument as in the proof of [17]. Firstly, for any \(u\in\mathcal{F}\cap C(X)\) and any \(\lambda>0\), we have \(\{x\in X:|u(x)|>\lambda\}\) is an open set, on which \(\frac{|u|}{\lambda}\ge1\), hence \[\label{eq95quasi95capbd} \mathrm{cap}_1 \left(\left\{x\in X:|u(x)|>\lambda\right\}\right)\le\mathcal{E}_1 \left(\frac{|u|}{\lambda}\right)\le \frac{1}{\lambda^p}\mathcal{E}_1(u).\tag{32}\] Secondly, for any \(u\in\mathcal{F}\), since \(\mathcal{C}\subseteq\mathcal{F}\cap C_c(X)\) is \(\mathcal{E}_1\)-dense in \(\mathcal{F}\), there exists \(\{u_n\}\subseteq\mathcal{C}\) which is \(\mathcal{E}_1\)-convergent to \(u\) and satisfies that \(\mathcal{E}_1(u_n-u_{n+1})<\frac{1}{2^{(p+1)n}}\) for any \(n\). Since \(u_n-u_{n+1}\in\mathcal{F}\cap C(X)\), let \[U_n=\left\{x\in X:|u_n(x)-u_{n+1}(x)|>\frac{1}{2^n}\right\},\] then \(U_n\) is an open set, and by above, we have \[\mathrm{cap}_1(U_n)\le \frac{1}{\left(\frac{1}{2^n}\right)^p}\mathcal{E}_1(u_n-u_{n+1})\le \frac{1}{2^n}.\] Let \(F_k=X\backslash\cup_{n=k}^{+\infty}U_n\), then \(\{F_k\}\) is an increasing sequence of closed sets and \[\mathrm{cap}_1(X\backslash F_k)=\mathrm{cap}_1(\cup_{n=k}^{+\infty}U_n)\le\sum_{n=k}^{+\infty}\mathrm{cap}_1(U_n)\le\sum_{n=k}^{+\infty}\frac{1}{2^n}=2^{1-k}\to0.\] For any \(k\), we have \[F_k=\bigcap_{n=k}^{+\infty}\left\{x\in X:|u_n(x)-u_{n+1}(x)|\le \frac{1}{2^n}\right\},\] hence there exists a real-valued function \(v\) defined on \(\cup_kF_k\) such that \(\{u_n\}\) is uniformly convergent to \(v\) on each \(F_k\), which implies \(\lim_{n\to+\infty}u_n(x)=v(x)\) for any \(x\in\cup_kF_k\). Since \(u_n|_{F_k}\) is continuous, by the standard \(3\varepsilon\)-argument, we have \(v|_{F_k}\) is continuous for any \(k\), which gives \(v\) is quasi-continuous. Since \(\{u_n\}\) is \(\mathcal{E}_1\)-convergent to \(u\), in particular, also \(L^p\)-convergent to \(u\), by passing to a subsequence to have the a.e. convergence, we have \(u=v\) \(m\)-a.e. in \(X\), that is, \(v\) is a modification of \(u\). ◻

Proof. By Proposition 15, there exists \(\{u_n\}\subseteq\mathcal{F}\cap C_c(X)\) satisfying that \(\{u_n\}\) is \(\mathcal{E}_1\)-convergent to \(u\), and for any \(\varepsilon>0\), there exists an open set \(G\) with \(\mathrm{cap}_1(G)<\varepsilon\) such that \(\left\{u_n\right\}\) converges uniformly to \(u\) on \(X\backslash G\). For any \(\lambda_1\in(0,\lambda)\), there exists \(N\ge1\) such that for any \(n>N\), we have \[\left\{x\in X:|u(x)|>\lambda\right\}\subseteq\left\{x\in X\backslash G:|u_n(x)|>\lambda_1\right\}\cup G\subseteq\left\{x\in X:|u_n(x)|>\lambda_1\right\}\cup G.\] For any \(n\), by Equation (32 ), we have \[\mathrm{cap}_1\left(\left\{x\in X:|u_n(x)|>\lambda_1\right\}\right)\le\frac{1}{\lambda_1^p}\mathcal{E}_1(u_n),\] hence \[\mathrm{cap}_1\left(\left\{x\in X:|u(x)|>\lambda\right\}\right)\le\mathrm{cap}_1\left(\left\{x\in X:|u_n(x)|>\lambda_1\right\}\right)+\mathrm{cap}_1(G)\le\frac{1}{\lambda_1^p}\mathcal{E}_1(u_n)+\varepsilon.\] Firstly, letting \(n\to+\infty\), secondly, letting \(\lambda_1\uparrow\lambda\), finally, letting \(\varepsilon\downarrow0\), we have \[\mathrm{cap}_1\left(\left\{x\in X:|u(x)|>\lambda\right\}\right)\le\frac{1}{\lambda^p}\mathcal{E}_1(u).\] ◻

Proof. Take a subsequence \(\{u_{n_k}\}\) satisfying \(\mathcal{E}_1(u_{n_k}-u_{n_{k+1}})<\frac{1}{2^{(p+1)k}}\), then by Lemma 16, we have \[\mathrm{cap}_1\left(\left\{x\in X:|u_{n_k}(x)-u_{n_{k+1}}(x)|>\frac{1}{2^k}\right\}\right)\le\frac{1}{\left(\frac{1}{2^k}\right)^p}\mathcal{E}_1(u_{n_k}-u_{n_{k+1}})\le\frac{1}{2^k}.\] Let \[G_k=\left\{x\in X:|u_{n_k}(x)-u_{n_{k+1}}(x)|>\frac{1}{2^k}\right\},\] then for any \(n\), we have \[\mathrm{cap}_1\left(\bigcup_{k\ge n}G_k\right)\le\sum_{k\ge n}\mathrm{cap}_1\left(G_k\right)\le\sum_{k\ge n}\frac{1}{2^k}=2^{1-n}\to0,\] and \[\bigcap_{k\ge n}G_k^c=\left\{x\in X:|u_{n_k}(x)-u_{n_{k+1}}(x)|\le\frac{1}{2^k}\right\},\] hence there exists a real-valued function \(u\) defined on \(\bigcup_{n\ge1}\bigcap_{k\ge n}G_k^c\) such that \(\{u_{n_k}\}\) converges uniformly to \(u\) on \(\bigcap_{k\ge n}G_k^c\) for any \(n\), which implies \(\lim_{k\to+\infty}u_{n_k}(x)=u(x)\) for any \(x\in\bigcup_{n\ge1}\bigcap_{k\ge n}G_k^c\), \(\left\{u_{n_k}\right\}\) converges to \(u\) q.e. on \(X\), \(\left\{u_{n_k}\right\}\) converges to \(u\) \(m\)-a.e. in \(X\). Since \(\left\{u_n\right\}\) is an \(\mathcal{E}_1\)-Cauchy sequence, there exists \(v\in\mathcal{F}\) such that \(\left\{u_n\right\}\) is \(\mathcal{E}_1\)-convergent to \(v\), there exists a subsequence of \(\left\{u_{n_k}\right\}\) that converges to \(v\) \(m\)-a.e. in \(X\). Hence \(u=v\) \(m\)-a.e. in \(X\), \(u\in\mathcal{F}\) and \(\{u_n\}\) is \(\mathcal{E}_1\)-convergent to \(u\).

It remains to show that \(u\) is quasi-continuous. Indeed, since \(\{u_n\}\) is a sequence of quasi-continuous functions, there exists an increasing sequence of closed sets \(\left\{F_k\right\}\) with \(\mathrm{cap}_1(F_k)\downarrow0\) such that \(u_n|_{F_k}\) is continuous for any \(n, k\). For any \(\varepsilon>0\), take \(n, l\ge1\) such that \(2^{1-n}<\varepsilon/2\) and \(\mathrm{cap}_1(X\backslash F_l)<\varepsilon/2\). Then \(\mathrm{cap}_1\left(\cup_{k\ge n}G_k\right)\le{2^{1-n}}<{\varepsilon}/{2}\), there exists an open set \(U\supseteq\cup_{k\ge n}G_k\) such that \(\mathrm{cap}_1(U)<\varepsilon/2\), then \(F_l\cap U^c\) is a closed set and \[\mathrm{cap}_1(X\backslash(F_l\cap U^c))=\mathrm{cap}_1((X\backslash F_l)\cup U)\le\mathrm{cap}_1(X\backslash F_l)+\mathrm{cap}_1(U)<\frac{\varepsilon}{2}+\frac{\varepsilon}{2}=\varepsilon.\] Since \(u_{n_k}|_{F_l}\) is continuous, we have \(u_{n_k}|_{F_l\cap U^c}\) is continuous. Since \(\left\{u_{n_k}\right\}\) converges uniformly to \(u\) on \(\cap_{k\ge n}G_k^c\supseteq U^c\), we have \(\left\{u_{n_k}\right\}\) converges uniformly to \(u\) on \(F_l\cap U^c\). Hence \(u|_{F_l\cap U^c}\) is continuous. Therefore, \(u\) is quasi-continuous. ◻

Proof. It is obvious that \(\mathcal{L}_A\) is a non-empty convex subset of \(\mathcal{F}\). We claim that \(\mathcal{L}_A\) is closed in \((\mathcal{F},\mathcal{E}_1^{1/p})\). Indeed, for any \(\mathcal{E}_1\)-Cauchy sequence \(\{u_n\}\subseteq\mathcal{L}_A\), by Corollary 2, there exist a subsequence \(\{u_{n_k}\}\) and a function \(u\in\widetilde{\mathcal{F}}\) such that \(\{\widetilde{u}_{n_k}\}\) converges to \(u\) q.e. on \(X\) and \(\{u_n\}\) is \(\mathcal{E}_1\)-convergent to \(u\). Since \(\widetilde{u}_{n_k}\ge1\) q.e. on \(A\), we have \(u\ge1\) q.e. on \(A\), which gives \(u\in\mathcal{L}_A\), \(\mathcal{L}_A\) is closed. By the same argument as in the proof of Lemma 13, there exists a unique element \(e_A\in\mathcal{L}_A\) such that \(\mathcal{E}_1(e_A)=\inf_{u\in\mathcal{L}_A}\mathcal{E}_1(u)\). Moreover, \(0\le e_A\le 1\) \(m\)-a.e. in \(X\), \(\widetilde{e}_A=1\) q.e. on \(A\), \(e_A\) is the unique element \(u\in\mathcal{F}\) satisfying \(\widetilde{u}=1\) q.e. on \(A\) and \(\mathcal{E}_1(u;v)\ge0\) for any \(v\in\mathcal{F}\) with \(\widetilde{v}\ge0\) q.e. on \(A\).

It remains to show that \(\mathcal{E}_1(e_A)=\mathrm{cap}_1(A)\), or equivalently, \[\mathcal{E}_1(e_A)=\inf_{u\in\mathcal{L}_A}\mathcal{E}_1(u)=\inf_{B\in\mathcal{O}:A\subseteq B}\mathrm{cap}_1(B)=\mathrm{cap}_1(A).\] “\(\le\)": For any \(B\in\mathcal{O}\) with \(A\subseteq B\), be Lemma 13, there exists \(e_B\in\mathcal{L}_B\) with \(e_B=1\) \(m\)-a.e. on \(B\) such that \(\mathrm{cap}_1(B)=\mathcal{E}_1(e_B)\). By Lemma 15, \(\widetilde{e}_B=1\) q.e. on \(B\supseteq A\), hence \(e_B\in\mathcal{L}_A\), which gives \(\inf_{u\in\mathcal{L}_A}\mathcal{E}_1(u)\le\mathcal{E}_1(e_B)=\mathrm{cap}_1(B)\). Taking the infimum over all such \(B\), we have the desired result.

“\(\ge\)": Since \(\widetilde{e}_A\) is quasi-continuous and \(\widetilde{e}_A=1\) q.e. on \(A\), for any \(\varepsilon>0\), there exists an open set \(G\) with \(\mathrm{cap}_1(G)<\varepsilon\) such that \(\widetilde{e}_A|_{X\backslash G}\) is continuous and \(\widetilde{e}_A=1\) on \(A\backslash G\). For any \(\delta\in(0,1)\), there exists an open set \(U\) with \(U\backslash G\supseteq A\backslash G\) such that \(\widetilde{e}_A>1-\delta\) on \(U\backslash G\), then \(U\cup G\) is an open set satisfying \(U\cup G\supseteq A\) and \(U\cup G\subseteq\left\{x\in X:\widetilde{e}_A(x)>1-\delta\right\}\cup G\). By Lemma 16, we have \[\mathrm{cap}_1(U\cup G)\le\mathrm{cap}_1(\left\{x\in X:\widetilde{e}_A(x)>1-\delta\right\})+\mathrm{cap}_1(G)\le\frac{1}{(1-\delta)^p}\mathcal{E}_1(e_A)+\varepsilon,\] hence \[\mathrm{cap}_1(A)\le\mathrm{cap}_1(U\cup G)\le\frac{1}{(1-\delta)^2}\mathcal{E}_1(e_A)+\varepsilon.\] Letting \(\varepsilon\downarrow0\) and \(\delta\downarrow0\), we have the desired result. ◻

Proof. “[cond95quasi95Radon]\(\Rightarrow\)[cond95quasi95posFCX]" and”[cond95quasi95posFX]\(\Rightarrow\)[cond95quasi95posFCX]": Obvious.

“[cond95quasi95posFCX]\(\Rightarrow\)[cond95quasi95posFX]": For any \(v\in\mathcal{F}\) with \(v\ge0\) \(m\)-a.e. in \(X\), there exists \(\{v_n\}\subseteq\mathcal{F}\cap C_c(X)\) which is \(\mathcal{E}_1\)-convergent to \(v\). By [52], we have \(\left\{v_n^+\right\}\) is \(\mathcal{E}_1\)-convergent to \(v^+=v\). Since \(v_n^+\in\mathcal{F}\cap C_c(X)\) satisfies \(v_n^+\ge0\) on \(X\), by assumption, we have \(\mathcal{E}_1(u;v_n^+)\ge0\), which gives \(\mathcal{E}_1(u;v)=\lim_{n\to+\infty}\mathcal{E}_1(u;v_n^+)\ge0\).

“[cond95quasi95posFCX]\(\Rightarrow\)[cond95quasi95Radon]": The idea is to construct a positive linear functional on \(C_c(X)\) to apply the Riesz representation theorem. For any \(v\in C_c(X)\), there exist \(\left\{v_n\right\}\subseteq\mathcal{F}\cap C_c(X)\) that converges uniformly to \(v\), and \(\varphi\in\mathcal{F}\cap C_c(X)\) with \(0\le\varphi\le1\) on \(X\) and \(\varphi=1\) on \(\mathrm{supp}(v)\), then \(\left\{\varphi v_n\right\}\subseteq\mathcal{F}\cap C_c(X)\) converges uniformly to \(\varphi v=v\) and \(\mathrm{supp}(\varphi v_n)\subseteq\mathrm{supp}(\varphi)\) for any \(n\). For any \(n, m\), we have \(\lVert v_n-v_m\rVert_{L^\infty(X;m)}\varphi\pm\varphi(v_n-v_m)\in\mathcal{F}\cap C_c(X)\) and \(\lVert v_n-v_m\rVert_{L^\infty(X;m)}\varphi\pm\varphi(v_n-v_m)\ge0\) on \(X\). By assumption, we have \[\mathcal{E}_1(u;\lVert v_n-v_m\rVert_{L^\infty(X;m)}\varphi\pm\varphi(v_n-v_m))\ge0,\] hence \[|\mathcal{E}_1(u;\varphi v_n)-\mathcal{E}_1(u;\varphi v_m)|\le \lVert v_n-v_m\rVert_{L^\infty(X;m)}\mathcal{E}_1(u;\varphi)\to0\] as \(n,m\to+\infty\), that is, \(\{\mathcal{E}_1(u;\varphi v_n)\}\) is a Cauchy sequence. Define \[l(v)=\lim_{n\to+\infty}\mathcal{E}_1(u;\varphi v_n).\] It is obvious that \(l(v)\) is well-defined, that is, the limit is independent of the choice of \(\left\{v_n\right\}\) and \(\varphi\), and that \(l\) is a linear functional on \(C_c(X)\). Moreover, \(l(v)=\mathcal{E}_1(u;v)\) for any \(v\in\mathcal{F}\cap C_c(X)\).

We claim that \(l(v)\ge0\) if \(v\in C_c(X)\) satisfies \(v\ge0\) on \(X\). Since \(\{v_n\}\subseteq\mathcal{F}\cap C_c(X)\) converges uniformly to \(v\) and \(v\ge0\) on \(X\), we have \(\{v_n^+\}\subseteq\mathcal{F}\cap C_c(X)\) also converges uniformly to \(v\), for any \(\varepsilon>0\), there exists \(N\) such that \(\varphi v_n^+\ge-\varepsilon\varphi\) on \(X\) for any \(n>N\), then \(\mathcal{E}_1(u;\varphi v_n^+)\ge-\varepsilon\mathcal{E}_1(u;\varphi)\). Hence \[l(v)=\lim_{n\to+\infty}\mathcal{E}_1(u;\varphi v_n^+)\ge-\varepsilon\mathcal{E}_1(u;\varphi).\] Letting \(\varepsilon\downarrow0\), we have \(l(v)\ge0\). Therefore, \(l\) is a positive linear functional on \(C_c(X)\). By the Riesz representation theorem, there exists a positive Radon measure \(\mu\) on \(X\) such that \[l(v)=\int_Xv\mathrm{d}\mu\text{ for any }v\in C_c(X).\] In particular, for any \(v\in\mathcal{F}\cap C_c(X)\), we have \[\mathcal{E}_1(u;v)=l(v)=\int_Xv\mathrm{d}\mu.\]

To prove Equation (?? ) for any Borel set \(A\), by the regular property of \(\mu\) and \(\mathrm{cap}_1\), we only need to prove for any compact set \(K\). Indeed, since \(K\) is compact, we have \(\mathcal{L}_K\ne\emptyset\), by Lemma 17, we have \(\mathrm{cap}_1(K)<+\infty\), which gives \[\mathrm{cap}_1(K)=\inf_{U\in \mathcal{O}:K\subseteq U}\mathrm{cap}_1(U)=\inf_{U\in \mathcal{O}:\mathcal{L}_U\ne\emptyset,K\subseteq U}\mathrm{cap}_1(U).\] For any \(U\in \mathcal{O}\) satisfying \(\mathcal{L}_U\ne\emptyset\) and \(K\subseteq U\), by Lemma 13, there exists \(e_U\in \mathcal{F}\) satisfying the conditions therein. Take \(\{u_n\}\subseteq\mathcal{F}\cap C_c(X)\) which is \(\mathcal{E}_1\)-convergent to \(e_U\). Since \(0\le e_U\le 1\) \(m\)-a.e. in \(X\), by replacing \(u_n\) by \((u_n\vee0)\wedge1\) and applying [52], we may assume that \(0\le u_n\le 1\) on \(X\). Moreover, there exists \(\varphi\in\mathcal{F}\cap C_c(X)\) such that \(0\le\varphi\le1\) on \(X\), \(\varphi=1\) on \(K\), and \(\mathrm{supp}(\varphi)\subseteq U\). Let \(v_n=\varphi+u_n-\varphi u_n\), then \(v_n\in\mathcal{F}\cap C_c(X)\), \(v_n\ge0\) on \(X\), and \(v_n=1+u_n-u_n=1\) on \(K\). By 12 , we have \[\begin{align} &{\mathcal{E}(\varphi u_n)}^{1/p}\le C_p \max\{\lVert \varphi\rVert_{L^\infty(X;m)},\lVert u_n\rVert_{L^\infty(X;m)}\} \left(\mathcal{E}(\varphi)^{1/p}+\mathcal{E}(u_n)^{1/p}\right)\\ &\le C_p \left(\mathcal{E}(\varphi)^{1/p}+\sup_n\mathcal{E}(u_n)^{1/p}\right)<+\infty, \end{align}\] hence \(\sup_{n}\mathcal{E}(\varphi u_n)<+\infty\). Since \(\{\varphi u_n\}\) is \(L^p(X;m)\)-convergent to \(\varphi e_U\), by [52], we have \(\{\varphi u_n\}\) is \(\mathcal{E}_1\)-weakly-convergent to \(\varphi e_U\). By the Mazur’s lemma (see [54]), for any \(n\), there exist \(I_n\ge n\), \(\lambda^{(n)}_k\ge0\) for \(k=n,\ldots,I_n\) with \(\sum_{k=n}^{I_n}\lambda_k^{(n)}=1\), such that \(\{\sum_{k=n}^{I_n}\lambda^{(n)}_k\varphi u_k\}_n\) is \(\mathcal{E}_1\)-convergent to \(\varphi e_U\). Let \(w_n=\sum_{k=n}^{I_n}\lambda^{(n)}_kv_k\), then \(w_n\in\mathcal{F}\cap C_c(X)\), \(w_n\ge0\) on \(X\), \(w_n=1\) on \(K\), and \(\{w_n\}\) is \(\mathcal{E}_1\)-convergent to \(\varphi+e_U-\varphi e_U=e_U+\varphi(1-e_U)=e_U+\varphi(1-1)=e_U\), by noting that \(e_U=1\) \(m\)-a.e. in \(U\supseteq \mathrm{supp}(\varphi)\). Hence \[\begin{align} &\mu(K)=\int_K w_n \mathrm{d}\mu\le \int_X w_n \mathrm{d}\mu\overset{(\star)}{\scalebox{2}[1]{=}}\mathcal{E}_1(u;w_n)\le \mathcal{E}_1(u)^{(p-1)/p}\mathcal{E}_1(w_n)^{1/p}\\ &\to \mathcal{E}_1(u)^{(p-1)/p}\mathcal{E}_1(e_U)^{1/p}=\mathcal{E}_1(u)^{(p-1)/p}\mathrm{cap}_1(U)^{1/p}, \end{align}\] where \((\star)\) follows from [cond95quasi95Radon] for \(w_n\). Taking the infimum over all such \(U\), we have \(\mu(K)\le \mathcal{E}_1(u)^{(p-1)/p}\mathrm{cap}_1(K)^{1/p}\).

In particular, for any subset \(A\subseteq X\), if \(\mathrm{cap}_1(A)=0\), then for any \(n\ge1\), there exists an open set \(G_n\supseteq A\) such that \(\mathrm{cap}_1(G_n)<\frac{1}{n}\), then \(A\subseteq\cap_{n\ge1}G_n\) and \[\mu(\cap_{n\ge1}G_n)\le\mu(G_n)\le\mathcal{E}_1(u)^{(p-1)/p}\mathrm{cap}_1(G_n)^{1/p}\le\mathcal{E}_1(u)^{(p-1)/p}\left(\frac{1}{n}\right)^{1/p}\to0,\] hence \(\mu(\cap_{n\ge1}G_n)=0\), which gives \(\mu(A)=0\).

Finally, for any \(v\in\mathcal{F}\), by Proposition 15, there exists \(\{v_n\}\subseteq\mathcal{F}\cap C_c(X)\) such that \(\{v_n\}\) is \(\mathcal{E}_1\)-convergent to \(v\) and \(\{v_n\}\) converges to \(\widetilde{v}\) q.e. on \(X\). By above, the q.e. convergence implies the \(\mu\)-a.e. convergence, hence by Fatou’s lemma, for any \(n\), we have \[\begin{align} &\int_X|v_n-\widetilde{v}|\mathrm{d}\mu=\int_X\lim_{m\to+\infty}|v_n-v_m|\mathrm{d}\mu\le\varliminf_{m\to+\infty}\int_X|v_n-v_m|\mathrm{d}\mu=\varliminf_{m\to+\infty}\mathcal{E}_1(u;|v_n-v_m|)\\ &\le\varliminf_{m\to+\infty}\mathcal{E}_1(u)^{(p-1)/p}\mathcal{E}_1(|v_n-v_m|)^{1/p}\le\varliminf_{m\to+\infty}\mathcal{E}_1(u)^{(p-1)/p}\mathcal{E}_1(v_n-v_m)^{1/p}, \end{align}\] which implies that \(\widetilde{v}\in L^1(X;\mu)\) and \(\{v_n\}\) is \(L^1(X;\mu)\)-convergent to \(\widetilde{v}\), hence \[\int_X\widetilde{v}\mathrm{d}\mu=\lim_{n\to+\infty}\int_Xv_n\mathrm{d}\mu=\lim_{n\to+\infty}\mathcal{E}_1(u;v_n)=\mathcal{E}_1(u;v).\] ◻

Proof. “[cond95quasi95RadonF]\(\Rightarrow\)[cond95quasi95posFCF]" and”[cond95quasi95posFF]\(\Rightarrow\)[cond95quasi95posFCF]": Obvious.

“[cond95quasi95posFCF]\(\Rightarrow\)[cond95quasi95RadonF]": It is obvious that [cond95quasi95posFCX] in Proposition 16 holds. By Proposition 16, there exists a positive Radon measure \(\mu\) on \(X\) such that \(\mathcal{E}(u;v)=\int_Xv\mathrm{d}\mu\) for any \(v\in\mathcal{F}\cap C_c(X)\). It remains to show that \(\mathrm{supp}(\mu)\subseteq F\). Suppose that \(\mathrm{supp}(\mu)\not\subseteq F\). Since \(F\) is closed, there exist a compact set \(K\) and an open set \(G\) satisfying \(K\subseteq G\subseteq X\backslash F\) and \(\mu(K)>0\). There exists \(v\in\mathcal{F}\cap C_c(X)\) with \(0\le v\le1\) on \(X\), \(v=1\) on \(K\) and \(\mathrm{supp}(v)\subseteq G\). Then \(v=0\) on \(F\), by assumption, we have \(\mathcal{E}_1(u;v)=0\), which implies \[0=\mathcal{E}_1(u;v)=\int_Xv\mathrm{d}\mu\ge\mu(K)>0,\] contradiction. Hence \(\mathrm{supp}(\mu)\subseteq F\).

“[cond95quasi95RadonF]\(\Rightarrow\)[cond95quasi95posFF]": It is obvious that [cond95quasi95Radon] in Proposition 16 holds. For any \(v\in\mathcal{F}\) with \(\widetilde{v}\ge0\) q.e. on \(F\), by Proposition 16, we have \(\widetilde{v}\in L^1(X;\mu)\) and \(\widetilde{v}\ge0\) \(\mu\)-a.e. on \(F\). By assumption, applying Equation (?? ), we have \[\mathcal{E}_1(u;v)=\int_X\widetilde{v}\mathrm{d}\mu=\int_F\widetilde{v}\mathrm{d}\mu\ge0.\] ◻

Proof. Let \(\overline{\mathcal{C}}^K\) be the \(\mathcal{E}_1\)-closure of \(\mathcal{C}^K\) in \(\mathcal{F}\), then \(\overline{\mathcal{C}}^K\) is a non-empty closed convex subset of \(\mathcal{F}\), by [53] and its proof, there exists a unique element \(u\in\overline{\mathcal{C}}^K\) such that \(\mathcal{E}_1(u)=\inf_{u\in\overline{\mathcal{C}}^K}\mathcal{E}_1(u)=\inf_{u\in\mathcal{C}^K}\mathcal{E}_1(u)\), and any minimizing sequence in \(\mathcal{C}^K\) is an \(\mathcal{E}_1\)-Cauchy sequence which is \(\mathcal{E}_1\)-convergent to \(u\).

To prove that \(\mathcal{E}_1(u)=\mathrm{cap}_1(K)\), by Lemma 17, we only need to prove that \(\widetilde{u}=1\) q.e. on \(K\) and \(\mathcal{E}_1(u;v)\ge0\) for any \(v\in\mathcal{F}\) with \(\widetilde{v}\ge0\) q.e. on \(K\).

Firstly, we show that \(\widetilde{u}=1\) q.e. on \(K\). Let \(\phi:\mathbb{R}\to\mathbb{R}\), \(t\mapsto(t\vee0)\wedge1\). For any minimizing sequence \(\{u_n\}\) in \(\mathcal{C}^K\), we have \(\{\phi(u_n)\}\subseteq\mathcal{C}^K\) and \(\mathcal{E}_1(u)\le\mathcal{E}_1(\phi(u_n))\le\mathcal{E}_1(u_n)\to\mathcal{E}_1(u)\), hence \(\{\phi(u_n)\}\) is also a minimizing sequence in \(\mathcal{C}^K\), which is an \(\mathcal{E}_1\)-Cauchy sequence that \(\mathcal{E}_1\)-converges to \(u\). Moreover, \(0\le\phi(u_n)\le1\) on \(X\) and \(\phi(u_n)=1\) on \(K\). By Corollary 2, there exists a subsequence \(\left\{\phi(u_{n_k})\right\}\) that converges to \(\widetilde{u}\) q.e. on \(X\). Since \(\phi(u_{n_k})=1\) on \(K\), we have \(\widetilde{u}=1\) q.e. on \(K\).

Secondly, we show that \(\mathcal{E}_1(u;v)\ge0\) for any \({v}\in\mathcal{F}\) with \(\widetilde{v}\ge0\) q.e. on \(K\). By Lemma 18, we only need to show that \(\mathcal{E}_1(u;v)\ge0\) for any \(v\in\mathcal{F}\cap C_c(X)\) with \(v\ge0\) on \(K\). For any minimizing sequence \(\{u_n\}\) in \(\mathcal{C}^K\), we have \(\{u_n\}\) is \(\mathcal{E}_1\)-convergent to \(u\), for any \(\varepsilon>0\), we have \(\{u_n+\varepsilon v\}\subseteq\mathcal{C}^K\), hence \(\mathcal{E}_1(u)=\inf_{u\in\mathcal{C}^K}\mathcal{E}_1(u)\le\mathcal{E}_1(u_n+\varepsilon v)\). Letting \(n\to+\infty\), we have \(\mathcal{E}_1(u)\le\mathcal{E}_1(u+\varepsilon v)\), hence \(\mathcal{E}_1(u;v)=\frac{1}{p}\lim_{\varepsilon\downarrow0}\frac{1}{\varepsilon}\left(\mathcal{E}_1(u+\varepsilon v)-\mathcal{E}_1(u)\right)\ge0\). ◻

Proof. Since \(\mathcal{F}\cap C_c(X)\) is \(\mathcal{E}_1\)-dense in \(\mathcal{F}\), we only need to show that for any \(u\in\mathcal{F}\cap C_c(X)\), for any compact subset \(K\subseteq X\) with \(\mathrm{cap}_1(K)=0\), we have \(\Gamma(u)(K)=0\). Indeed, by Lemma 19, there exists \(\{\phi_n\}\subseteq\mathcal{F}\cap C_c(X)\) with \(0\le\phi_n\le1\) on \(X\), \(\phi_n=1\) on \(K\) such that \(\lim_{n\to+\infty}\mathcal{E}_1(\phi_n)=\mathrm{cap}_1(K)=0\). By [52], we have \[0\le\Gamma(u)(K)\le\int_X\phi_n\mathrm{d}\Gamma(u)=\mathcal{E}(u;u\phi_n)-\left(\frac{p-1}{p}\right)^{p-1}\mathcal{E} \left(|u|^{\frac{p}{p-1}};\phi_n\right),\] where \[\lvert\mathcal{E}\left(|u|^{\frac{p}{p-1}};\phi_n\right)\rvert\le\mathcal{E} \left(|u|^{\frac{p}{p-1}}\right)^{{(p-1)}/{p}}\mathcal{E}(\phi_n)^{{1}/{p}}\to0.\] Since \(\{\phi_n\}\) is \(L^p(X;m)\)-convergent to \(0\) and \(u\in\mathcal{F}\cap C_c(X)\), we have \(\{u\phi_n\}\) is \(L^p(X;m)\)-convergent to \(0\). Since \[\begin{align} &\mathcal{E}(u\phi_n)^{{1}/{p}}\le C_p\max\{\lVert u\rVert_{L^\infty(X;m)},\lVert \phi_n\rVert_{L^\infty(X;m)}\}\left(\mathcal{E}(\phi_n)^{{1}/{p}}+\mathcal{E}(u)^{{1}/{p}}\right)\\ &\le C_p\max\{\lVert u\rVert_{L^\infty(X;m)},1\}\left(\sup_n\mathcal{E}(\phi_n)^{{1}/{p}}+\mathcal{E}(u)^{{1}/{p}}\right)<+\infty, \end{align}\] we have \(\sup_{n}\mathcal{E}(u\phi_n)<+\infty\). By [52], we have \(\{u\phi_n\}\) is \(\mathcal{E}_1\)-weakly-convergent to 0, which gives \(\lim_{n\to+\infty}\mathcal{E}(u;u\phi_n)=0\). Hence \(0\le\Gamma(u)(K)\le\lim_{n\to+\infty}\int_X\phi_n\mathrm{d}\Gamma(u)=0\), that is, \(\Gamma(u)(K)=0\). ◻