Proof. By scaling, $R(p,q)\leq R_0^*(\mathcal {D})r^{n-1}$ for any distinct $p,q\in F_wV_0$ with $w\in W_{1,n}$ and $n\geq 1$. For $w\in \{1,2\}^n$, write $[w]_l=w_1w_2\cdots w_l$ and $p_l=F_{[w]_l}(q_0)$, with $0\leq l\leq n$, then

\[ R(p_i,p_j)\leq \sum_{l=i}^{j-1} R(p_l,p_{l+1})\leq R_0^*(\mathcal{D})\sum_{l=i}^{j-1} r^l< R_0^*(\mathcal{D})\frac{r^i}{1-r}, \quad \forall 0\leq i< j\leq n. \]

In particular, this implies that $R(p,q)\leq \frac {2r^n}{1-r}R_0^*(\mathcal {D})$ for any $p,q\in F_wV_1$ and $w\in \{1,2\}^n$. Now, if $p,q\in V_1$ and ${\rm d}(p,q)<\rho ^{n+2}$, then by the remark before definition 3.1, there exist $w,w'\in \{1,2\}^n$ such that $p\in F_{w}V_1, q\in F_{w'}V_1$, and $F_{w}V_1\cap F_{w'}V_1\neq \emptyset$, which implies that $R(p,q)\leq \frac {4r^n}{1-r}R_0^*(\mathcal {D})$. As a consequence, we have

(3.3)\begin{equation} R(p,q)\leq \frac{4}{r^3(1-r)}R_0^*(\mathcal{D})\,{\rm d}(p,q)^{\frac{\log r}{\log \rho}}, \quad \forall p,q\in V_1. \end{equation}

On the other hand, for any $f\in Dom(\Psi _r\mathcal {D})$, by (RF4), we immediately have

(3.4)\begin{equation} |f(p)-f(q)|\leq \left(R(p,q)\Psi_r\mathcal{D}(f)\right)^{1/2}. \end{equation}

Combining (3.3) and (3.4), we then get $Dom(\Psi _r\mathcal {D})\subset C(\bar {V}_1)$ by a natural identification.

To show the second assertion, we let $(X,R)$ be the completion of $(V_1,R)$, and recall [Reference Kigami22, theorem 2.3.10] to get that $(\Psi _\lambda \mathcal {D},Dom(\Psi _r\mathcal {D}) )$ extends to be a resistance form on $X$. It suffices to show that the identity map $Id:V_1\to V_1$ extends to an homeomorphism from $(\bar {V}_1,d)$ to $(X,R)$ under the assumption $\rho < r<1$. First, by (3.3), $Id$ is continuous from $(V_1,d)$ to $(V_1,R)$. Next, let $f\in l(V_1)$ be a restriction of a linear function on $\mathbb {R}^2$. We have

\begin{align*} \Psi_r\mathcal{D}(f)& =\sum_{w\in W_1}r^{-|w|+1}\mathcal{D}(f\circ F_w)=\sum_{n=1}^\infty\sum_{w\in W_{1,n}}r^{{-}n+1}\mathcal{D}(f\circ F_w)\\ & =\sum_{n=1}^\infty \#W_{1,n}r^{{-}n+1}\rho^{2(n+1)}\mathcal{D}(f|_{V_0}), \end{align*}

where the last equality follows from the fact that $f$ is linear. Since $\#W_{1,n}\asymp \rho ^{-n}$, we have $\Psi _r\mathcal {D}(f)<\infty$ when $r>\rho$, so that $f\in Dom(\Psi _r\mathcal {D})$. Noticing that for any points $p\neq q\in \bar {V}_1$, we can find a linear function $f$ such that $f(p)\neq f(q)$, we have $Id$ is injective. Finally, due to the fact that $(\bar {V}_1,d)$ is a compact Hausdorff space and $(X,R)$ is the completion of $(V_1,R)$, we then have that $Id$ is an homeomorphism from $(\bar {V}_1,d)$ to $(X,R)$. This implies that $(\Psi _r\mathcal {D},Dom(\Psi _r\mathcal {D}))$ is a resistance form on $\bar {V}_1$, and (3.2) follows immediately from (3.3).

Proof. (a) is obvious since all $\mathcal {D}\in \mathcal {M}$ are comparable up to multiplicative constants, we only need to prove (b). Let $f\in Dom(\Psi _{r_2}\mathcal {D})$, and choose $n$ large enough so that

(4.1)\begin{equation} \sum_{l=n}^\infty \sum_{w\in W_{1,l}}r_2^{{-}l+1}\mathcal{D}(f\circ F_w)<\varepsilon. \end{equation}

For convenience, we rename the vertices $\{F_wq_0\}_{w\in W_{1,n}}$ to be $\{p_i\}_{i=1}^{N}$ with $N=\# W_{1,n}$, so that for each $i$, $p_i$ is on the left of $p_{i+1}$. Then, noticing that the effective resistance between $q_1$ and $p_1$ (symmetrically, $q_2$ and $p_N$) is bounded above by a multiple of $r^{-n}$, by (RF4), it is not hard to see

\begin{align*} & r_2^{{-}n}\left(\sum_{i=1}^{N-1}\left(f(p_i)-f(p_{i+1})\right)^2+\left(f(q_1)-f(p_{1})\right)^2+\left(f(q_2)-f(p_N)\right)^2\right)\\ & \leq c_1\left(\sum_{l=n}^\infty \sum_{w\in W_{1,l}}r_2^{{-}l+1}\mathcal{D}(f\circ F_w)\right)< c_1\varepsilon, \end{align*}

where $c_1$ is a constant depending on $\mathcal {D}$ and $r_2$, but not on $n$.

Write $x_i$ for the $x$-coordinate of $p_i$, so we have $0< x_1< x_2<\cdots < x_N<1$. We introduce a piecewise linear function $u$ on $\mathbb {R}^2$ such that

1. 1. $u(x,y)$ depends only on $x$;

2. 2. $u(q_1)=f(q_1)$, $u(q_2)=f(q_2)$, and $u(p_i)=f(p_i)$, $1\leq i\leq N$;

3. 3. $u(x,0)$ is linear on each interval $(0,x_1)$, $(x_N,1)$ and $(x_i,x_{i+1})$, $1\leq i\leq N-1$.

We define $g\in l(V_1)$ as

\[ g(p)=\begin{cases} f(p),\text{ if }p\in \bigcup_{l=1}^{n-1}\bigcup_{w\in W_{1,l}}\{F_wq_0\},\\ u(p),\text{ if }p\in \bigcup_{l=n}^{\infty}\bigcup_{w\in W_{1,l}}\{F_wq_0\}. \end{cases} \]

By a similar estimate to that applied in the proof of proposition 3.3, one can check that $g\in Dom(\Psi _{r_1}\mathcal {D})$, and

\begin{align*} & \sum_{l=n}^\infty \sum_{w\in W_{1,l}}r_2^{{-}l+1}\mathcal{D}(g\circ F_w)\\ & \quad \leq c_2r_2^{{-}n}\left(\sum_{i=1}^{N-1}\left(f(p_i)-f(p_{i+1})\right)^2+\left(f(q_1)-f(p_{1})\right)^2+\left(f(q_2)-f(p_N)\right)^2\right), \end{align*}

where $c_2$ depends only on $\mathcal {D}$ and $r_2$. So we have $\Psi _{r_2}\mathcal {D}(f-g)\leq c_3\varepsilon$ for some constant $c_3$. Since $\varepsilon$ can be arbitrarily small, we have that $Dom(\Psi _{r_1}\mathcal {D})$ is dense in $Dom(\Psi _{r_2}\mathcal {D})$. Finally, the claim that $Dom(\Psi _{r_1}\mathcal {D})$ is dense in $C(\bar {V}_1)$ follows from the same argument.

Proof of theorem 4.2. Let $r_n\to r\in (\rho,1)$ and $\mathcal {D}_n\to \mathcal {D}\in \mathcal {M}$. Also, let $u\in l(V_0)$. First, we show that

(4.2)\begin{equation} \limsup_{n\to\infty}\mathcal{R}(r_n,\mathcal{D}_n)(u)\leq \mathcal{R}(r,\mathcal{D})(u). \end{equation}

We define $f$ to be the unique function in $Dom(\Psi _r\mathcal {D})$ such that $f|_{V_0}=u$ and

\[ \mathcal{R}(r,\mathcal{D})(u)=\Psi_r\mathcal{D}(f). \]

By proposition 4.3, for any $\varepsilon >0$, there is $f_\varepsilon$ such that $f_\varepsilon |_{V_0}=u$, $f_\varepsilon \in Dom(\Psi _{r_n}\mathcal {D}_n)$ for any $n\geq 1$, and

\[ \Psi_r\mathcal{D}(f_\varepsilon)\leq\Psi_r\mathcal{D}(f)+\varepsilon. \]

As a consequence, we have

\[ \limsup_{n\to\infty}\mathcal{R}(r_n,\mathcal{D}_n)(u)\leq \lim_{n\to\infty}\Psi_{r_n}\mathcal{D}_n(f_\varepsilon)=\Psi_r\mathcal{D}(f_\varepsilon)\leq \mathcal{R}(r,\mathcal{D})(u)+\varepsilon, \]

where the equality is due to the dominated convergence theorem. Since $\varepsilon$ can be arbitrarily chosen, we get (4.2).

Next, for each $n$, let $f_n$ be the unique function in $Dom(\Psi _{r_n}\mathcal {D}_n)$ such that $f_n|_{V_0}=u$ and

\[ \mathcal{R}(r_n,\mathcal{D}_n)(u)=\Psi_{r_n}\mathcal{D}_n(f_n). \]

Then $\{f_n\}_{n\geq 1}$ is uniformly bounded by the Markov property (RF5). In addition, $\Psi _{r_n}\mathcal {D}_n(f_n)\leq \mathcal {R}(r_*,\mathcal {D}_n)(u)$ with $r_*=\inf _{n\geq 1} r_n$, so $\{\Psi _{r_n}\mathcal {D}_n(f_n)\}_{n\geq 1}$ is a bounded sequence. By estimates (3.2) and (3.4), we have

\[ |f_n(p)-f_n(q)|\leq c\left({\rm d}(p,q)^{\frac{\log r^*}{\log \rho}}\sup_{n\geq 1}\Psi_{r_n}\mathcal{D}_n(f_n)\right)^{1/2},\quad\forall n\geq 1,\forall p,q\in \bar{V}_1, \]

where $r^*=\sup _{n\geq 1}r_n$ and $c^2=\sup _{n\geq 1}\{\frac {4}{{r_n^{3}}(1-r_n)}R_0^*(\mathcal {D}_n)\}$, and so $\{f_n\}_{n\geq 1}$ is also equicontinuous. Thus, there is a subsequence $\{f_{n_k}\}_{k\geq 1}$ such that $f_{n_k}$ converges uniformly to a function $f\in C(\bar {V}_1)$. Clearly, $f$ is an extension of $u$. By Fatou's lemma,

\[ \mathcal{R}(r,\mathcal{D})(u)\leq\Psi_r\mathcal{D}(f)\leq \liminf_{k\to\infty}\Psi_{r_{n_k}}\mathcal{D}_{n_k}(f_{n_k})=\liminf_{k\to\infty}\mathcal{R}(r_{n_k},\mathcal{D}_{n_k})(u). \]

Combining this with (4.2), we see that

\[ \mathcal{R}(r,\mathcal{D})(u)=\lim_{k\to\infty}\mathcal{R}(r_{n_k},\mathcal{D}_{n_k})(u). \]

Since the argument works for any sequence $(r'_n,\mathcal {D}'_n)\to (r,\mathcal {D})$, we actually have

\[ \mathcal{R}(r,\mathcal{D})(u)=\lim_{n\to\infty}\mathcal{R}(r_n,\mathcal{D}_n)(u). \]

The theorem follows immediately since $u$ can be any function in $l(V_0)$.

Proof. (RF1) and (RF5) are trivial. We only need to verify (RF2)–(RF4). For convenience, we focus on $(\mathcal {D}^{(2)},\mathcal {F}^{(2)})$ only, while for larger $m$, the same proof works inductively. (RF2). Let $\{f_k\}_{k\geq 1}$ be a Cauchy sequence in $\mathcal {F}^{(2)}$. Then, $f_k|_{\bar {V}_1}$ converges in $\mathcal {F}^{(1)}$ to some $\tilde {f}$ in $\mathcal {F}^{(1)}$, since $(\mathcal {D}^{(1)},\mathcal {F}^{(1)})$ is a resistance form. Also, for each $w\in W_1$, $f_k\circ F_w$ converges in $\mathcal {F}^{(1)}$ to a function $\tilde {f}_w$. Now, define $f\in l(\bar V_2)$ such that $f\circ F_w=\tilde {f}_w$ and $f|_{\bar {V_1}\setminus V_1}=\tilde {f}$. We show that $f\in C(\bar V_2)$. It suffices to prove that $f$ is continuous at any point $p\in \bar {V}_1\setminus V_1$. In fact, for any $\varepsilon$, there exists $\delta$ and $N$ such that 1. for $q\in B_{\delta }(p)\cap \bar {V}_1$, we have $|f(p)-f(q)|<\varepsilon$; 2. for $w\in \bigcup _{n=N}^\infty W_{1,n}$ and $q,q'\in F_w\bar {V}_1$, we have $|f(q)-f(q')|<\varepsilon$. This follows from the fact that $\mathcal {D}^{(1)}(f\circ F_w)\leq r_w\sup _{k\geq 1}\mathcal {D}^{(2)}(f_k)$. The continuity of $f$ follows immediately. Lastly, by using Fatou's lemma, we can directly check that $f_k$ converges to $f$ in $\mathcal {F}^{(2)}$. (RF3). First, we observe that the minimal energy extension of $f\in \mathcal {F}^{(1)}$ to $l(V_2)$ is continuous by a same argument as in (RF2). Let $V$ be a finite set and $u\in l(V)$. First, we always have $f_1\in \mathcal {F}^{(1)}$ such that $f_1|_{V\cap \bar {V}_1}=u|_{V\cap \bar {V}_1}$. Then we can extend $f_1$ to be a desired function in $\mathcal {F}^{(2)}$. (RF4). Let $p,q\in \bar {V}_2$ and $f\in \mathcal {F}^{(2)}$. If $p\in \bar {V}_1$, we let $p'=p$; otherwise we choose $p'\in V_1$ so that $p,p'\in F_w\bar {V_1}$ for some $w\in W_1$, and thus

\[ \mathcal{D}^{(2)}(f)\geq r_w^{{-}1}\mathcal{D}^{(1)}(f\circ F_w)\geq c_1\left(f(p)-f(p')\right)^2, \]

for some $c_1>0$. Also, we define $q'$ in the same manner. Note that $\mathcal {D}^{(2)}(f)\geq \mathcal {D}^{(1)}(f)\geq c_2(f(p')-f(q'))^2$ for some $c_2>0$, it then follows that

\begin{align*} & \mathcal{D}^{(2)}(f)\geq \min\{c_1,c_2\}\left(\left(f(p)-f(p')\right)^2+\left(f(p')-f(q')\right)^2+\left(f(q')-f(q)\right)^2\right)\\ & \quad \geq c_3\left(f(p)-f(q)\right)^2, \end{align*}

for some $c_3>0$. (RF4) follows immediately. Thus, we have proved that $(\mathcal {D}^{(2)},\mathcal {F}^{(2)})$ is a resistance form on $\bar V_2$. The claim that $R_2(p,q)=R_1(p,q)$ for $p,q\in \bar V_1$ is obvious. The same arguments can be used inductively for $m\geq 3$.

Proof. Noticing that $\{w\tau :\tau \in W_1\}\subset {W}_{m+1}$, the first statement follows. The second statement follows from the first statement and proposition 3.3: for any $p,q\in F_w\bar {V}_1$,

\[ R_{m+1}(p,q)\leq r_w R_1(F_w^{{-}1}p, F_w^{{-}1}q)\leq cr_wd(F_w^{{-}1}p, F_w^{{-}1}q)^\theta=c\,{\rm d}(p,q)^\theta, \]

holds for some constant $c>0$, where the first inequality follows from the first statement, and the second inequality follows from proposition 3.3.

Proof. First, we claim that there is a constant $c_1>0$ such that

\[ \tilde{R}(p,q)\leq c_1\rho_w^\theta,\quad \forall w\in \tilde{W}_*, \forall p,q\in F_w \mathcal{G}\cap (\bigcup_{m\geq 0}\bar{V}_m). \]

We first consider the case $q\in F_w\bar {V}_1$. Assume that $p\in F_w\bar {V}_n$ for some $n\geq 1$, then we can find $\tau \in W_{n-1}$ such that $p\in F_wF_\tau \bar {V}_1$. We can then find a sequence

\[ q=p_0,p_1,\cdots,p_{|\tau|+1}=p, \]

such that $p_i\in F_wF_{[\tau ]_{i-1}}\bar {V}_1\cap F_wF_{[\tau ]_{i}}\bar {V}_1$ for $1\leq i\leq |\tau |$. As a consequence, by using lemma 6.3, we see that

\[ \tilde{R}(p,q)\leq \sum_{i=0}^{|\tau|} c_2\,{\rm d}(p_i,p_{i+1})^\theta\leq \sum_{i=0}^{|\tau|} c_2(\rho_w\rho^i)^\theta\leq \frac {c_2}{1-\rho^\theta}\rho_w^\theta, \]

where $c_2$ is the same constant in lemma 6.3. For general $q$, we only need to set $c_1=\frac {2c_2}{1-\rho ^\theta }\rho _w^\theta$. Now, let $p,q\in \bigcup _{m=0}^\infty \bar {V}_m$. We choose $w,w'\in \tilde W_*$ such that $p\in F_w\mathcal {G}$, $q\in F_{w'}\mathcal {G}$ and

\[ \rho {\rm d}(p,q)\leq\rho_w,\rho_{w'}<\rho^{{-}1}\,{\rm d}(p,q). \]

In addition, we can find a chain

\[ w=w^{(0)},w^{(1)},\cdots,w^{(k)}=w' \]

such that $\min \{\rho _w,\rho _{w'}\}\leq \rho _{w^{(i)}}<\rho ^{-2}\min \{\rho _w,\rho _{w'}\}$ of length at most $c_3$, where $c_3$ is a constant independent of $p,q$. By choosing a sequence $p=p_0,p_1,\cdots,p_{k+1}=q$ such that $p_i\in F_{w^{(i-1)}}\bar {V}_1\cap F_{w^{(i)}}\bar {V}_1$, $1\leq i\leq k$, we get the desired estimate as above.

Proof. First, we claim that $(\mathcal {E},\mathcal {F})$ is a resistance form on $\bigcup _{m\geq 0}\bar {V}_m$. (RF1) and (RF5) are obvious given lemma 6.2. Observing that by iterating the minimal energy extension, we can extend any $f\in \mathcal {F}^{(m)}$ to $f\in \mathcal {F}$ thanks to the upper-bound estimate of the resistance metric in lemma 6.4. (RF2), (RF3) and (RF4) are then easy to show with lemma 6.2. In addition, we see that

\[ \tilde{R}(p,q)=R(p,q):=\sup_{f\in \mathcal{F}}\frac{|f(p)-f(q)|^2}{\mathcal{E}(f)},\quad \forall p,q\in \bigcup_{m\geq 0}\bar{V}_m. \]

Next, to prove that $(\mathcal {E},\mathcal {F})$ is a resistance form on $\mathcal {G}$, we need to show that $\mathcal {F}$ separates points in $\mathcal {G}$, just like in proposition 3.3. It suffices to prove that $\mathcal {F}$ is dense in $C(\mathcal {G})$. Let $u\in C(\mathcal {G})$, we fix $N$ large enough so that $|u(x)-u(y)|<\varepsilon$ if $x,y \in F_wK$ and $|w|\geq N$. We can apply proposition 4.3 to create $f\in \mathcal {F}$ such that $\|f-u\|_{L^\infty (\mathcal {G})}< 2\varepsilon$. First, we find $f_1\in \mathcal {F}^{(1)}$ such that 1. $\|f_1-u|_{\bar {V_1}}\|_{L^\infty }<\varepsilon$; 2. $f_1(p)=u(p)$ for any $p\in \bigcup _{n=1}^N\bigcup _{w\in W_{1,n}} F_wV_0$. Then we apply harmonic extension to $f_1$ on $\bar {V}_2\setminus \bigcup _{n=1}^N\bigcup _{w\in W_{1,n}} F_w\bar {V}_1$. On the cells $F_w\bar {V}_1$ with $|w|< N$, we apply the same construction to get $f_2$, but with $N-2$ replacing $N$ this time. After $k=[N/2]+1$ times, we get $f_{k}\in \mathcal {F}^{(k)}$ such that $\|f_k-u|_{\bar {V}_k}\|_{L^\infty }<2\varepsilon$. Since all cells have size smaller than $\rho ^N$, by harmonically extending, we get $f\in \mathcal {F}$ such that $\|f-u\|_{L^\infty (\mathcal {G})}<2\varepsilon$. Thus, $(\mathcal {E},\mathcal {F})$ is regular resistance form on $\mathcal {G}$. It remains to show that the form is strongly local. Let $f,g\in \mathcal {F}$ with $supp(f)\cap supp(g)=\emptyset$, then there exists $\varepsilon >0$ such that $d(supp(f),supp(g))>\varepsilon$. Thus, we have $\mathcal {D}^{(n)}(f,g)=0$ for large $n$, because the supports of $f$ and $g$ are suitably separated by $n$-cells for large $n$. By taking the limit, we see that $\mathcal {E}(f,g)=0$. Clearly $1\in \mathcal {F}$ with $\mathcal {E}(1)=0$, and it follows that the form is strongly local.

Proof. The claimed properties of $(\mathcal {E},\mathcal {F})$ are immediate consequences of the construction.

The uniqueness follows by a well-known argument, but in the infinite graph version. Let $(\mathcal {E}',\mathcal {F}')$ be another form satisfying the above properties with $\theta '$ replacing $\theta$. Define $\mathcal {D}'$ to be the trace of $\mathcal {E}'$ onto $V_0$, and write $r'_w=\rho _w^{\theta '}$, $r'=\rho ^{\theta '}$. For any $u\in l(V_0)$, let $h_u$ be the harmonic extension of $u$ to $\mathcal {F}'$, then we can see that

\begin{align*} \mathcal{D}'(u)& =\mathcal{E}'(h_u)=\sum_{w\in W_1} {r'}_w^{{-}1}\mathcal{E}'(h_u\circ F_w)\geq \sum_{w\in W_1}{r'}_w^{{-}1}\mathcal{D}'\left((h_u\circ F_w)|_{V_0}\right)\\ & \quad \geq{r'}^{{-}2}\mathcal{R}_{r'}\mathcal{D}'(u), \end{align*}

where $\mathcal {R}_{r'}$ is the renormalization map introduced in definition 4.1, and we use properties (a) and (b) in the inequalities.

On the other hand, we can perform the harmonic extension of $u$ in two steps: first, we extend $u$ to $f_1\in C(\bar {V}_1)$ so that $f_1$ minimizes $\Psi _{r'}\mathcal {D}'$, then we take harmonic extension of $f_1$ on each cell $F_w\mathcal {G},w\in W_1$, to $f\in C(\mathcal {G})$, by using property (a) and the Markov property (RF5). In addition, $f\in \mathcal {F}'$ by the property (c). Then, by property (b),

\[ {r'}^{{-}2}\mathcal{R}_{r'}\mathcal{D}'(u)={r'}^{{-}2}\Psi_{r'}\mathcal{D}'(f_1)=\sum_{w\in W_1}{r'}_w^{{-}1}\mathcal{E}'(f\circ F_w)=\mathcal{E}'(f)\geq \mathcal{D}'(u). \]

Thus, we get $\mathcal {R}_{r'}\mathcal {D}'={r'}^2\mathcal {D}'$, which implies that $\mathcal {D}'=\mathcal {D}$ and $\theta '=\theta$ by theorem 5.1. Finally, by a similar argument, one can easily find that the restriction of $\mathcal {E}'$ to $\bar {V}_m$ is $\mathcal {D}^{(m)}$, and the claim that $\mathcal {E}'=\mathcal {E}$ follows immediately by taking the limit.

Proof. The proof is essentially the same as that for PCF self-similar sets [Reference Hambly and Kumagai18]. We reproduce it here for convenience of readers. First, we claim that for any $f\in \mathcal {F}$, $\|f\|^2_{L^2(\mathcal {G})}\leq c_1(\mathcal {E}(f)+\|f\|^2_{L^1(\mathcal {G})})$ for some constant $c_1>0$ independent of $f$. In fact, let $\bar f=\int _{\mathcal {G}}f\,{\rm d}\mu _H$ and $g=f-\bar f$, it suffices to check that $\|g\|_{L^2(\mathcal {G})}^2\leq c_2\mathcal {E}(f)$ for some $c_2>0$, which follows from

\begin{align*} \|g\|^2_{L^2(\mathcal{G})}& =\frac 12 \int_{\mathcal{G}}\int_{\mathcal{G}}\left(g(x)-g(y)\right)^2\,{\rm d}\mu_H(x)\,{\rm d}\mu_H(y)\\ & =\frac 12 \int_{\mathcal{G}}\int_{\mathcal{G}}\left(f(x)-f(y)\right)^2\,{\rm d}\mu_H(x)\,{\rm d}\mu_H(y)\leq c_2 \mathcal{E}(f), \end{align*}

where the last inequality is due to (RF4) and proposition 3.3. Next, write $f_w=f\circ F_w$ for $w\in \tilde W_*$ for short. Then for $0< s<1$,

\begin{align*} \|f\|^2_{L^2(\mathcal{G})}& \leq \sum_{w\in \hat{W}_s}\rho_w^{d_H}\|f_w\|^2_{L^2(\mathcal{G})}\leq c_1\sum_{w\in \hat{W}_s}\rho_w^{d_H}\left(\mathcal{E}(f_w)+\|f_w\|^2_{L^1(\mathcal{G})}\right)\\ & \leq c_3s^{d_H+\theta}\sum_{w\in \hat{W}_s}\rho_w^{-\theta}\mathcal{E}(f_w)+c_4s^{{-}d_H}\sum_{w\in \hat{W}_s}(\rho_w^{d_H}\|f_w\|_{L^1(\mathcal{G})})^2\\ & \leq c_5\left(s^{d_H+\theta}\mathcal{E}(f)+s^{{-}d_H}\|f\|_{L^1(\mathcal{G})}^2\right), \end{align*}

for some $c_3-c_5>0$, where in the last inequality, we use the observation that $\sum _{w\in \hat {W}_s}\rho _w^{-\theta }\mathcal {E}(f_w)\leq c'\mathcal {E}(f)$ for some $c'\geq 1$. In the case that $\mathcal {E}(f)>\|f\|^2_{L^1(\mathcal {G})}$, we choose $s$ such that $s^{2d_H+\theta }\mathcal {E}(f)=\|f\|^2_{L^1(\mathcal {G})}$, then $\|f\|^2_{L^2(\mathcal {G})}\leq 2c_5\mathcal {E}(f)^{\frac {d_H}{2d_H+\theta }}\|f\|_{L^1(\mathcal {G})}^{\frac {2d_H+2\theta }{2d_H+\theta }}$, and so the desired result follows immediately. In the case that $\mathcal {E}(f)\leq \|f\|^2_{L^1(\mathcal {G})}$, we have $\|f\|^2_{L^2(\mathcal {G})}\leq c_1(\mathcal {E}(f)+\|f\|_{L^1(\mathcal {G})}^2)\leq 2c_1\|f\|^2_{L^1(\mathcal {G})}$, and the result still follows.

Proof. We already have the estimate $R(p,q)\leq c_2\,{\rm d}(p,q)^\theta$ from lemma 6.4 and theorem 6.5. Now we show $R(p,B^c_s(p))\geq c_3s^\theta$ for $p\in \mathcal {G}$ and $0< s<1$. Define

\begin{align*} & U_{p,s,0}=\bigcup_{w\in \hat{W}_{p,s,0}} F_w\mathcal{G} \text{ with }\hat{W}_{p,s,0}=\{w\in \hat{W}_{s\rho^2}:p\in F_w\mathcal{G}\},\\ & U_{p,s,1}=\bigcup_{w\in \hat{W}_{p,s,1}} F_w\mathcal{G} \text{ with }\hat{W}_{p,s,1}=\{w\in \hat{W}_{s\rho^2}:F_w\mathcal{G}\cap U_{p,s,0}\neq \emptyset\}. \end{align*}

Clearly, we have $U_{p,s,0}\subset U_{p,s,1}\subset B_{s}(p)$. Since $(\mathcal {E},\mathcal {F})$ is regular, there exists $f_{p,s}\in \mathcal {F}$ so that $f_{p,s}|_{{U}^c_{p,s,1}}=0$ and $f_{p,s}|_{U_{p,s,0}}=1$. As $\mathcal {G}$ satisfies the finite type property, there exists a finite class $\{(p_i,s_i)\}_{i=1}^N$ such that for any $p\in \mathcal {G}$ and $0< s<1$, there exists $1\leq i\leq N$ and an affine map $\psi$ such that $\psi :U_{p,s,l}\to U_{p_i,s_i,l}$ for $l=0,1$, which maps cells corresponding to $\hat W_{p,s,l}$ to those corresponding to $\hat W_{p_i,s_i,l}$. In addition, we require that $\psi$ maps the boundary of $U_{p,s,l}$ to the boundary of $U_{p_i,s_i,l}$, which only depend on how the outside cells of approximately same size intersect $U_{p_i,s_i,1}$. Thus, we can assume that

\[ f_{p,s}(q)=\begin{cases} f_{p_i,s_i}\circ\psi (q), & \text{ if }q\in U_{p,s,1},\\ 0, & \text{ if }q\in U^c_{p,s,1}.\end{cases} \]

By a similar observation as in lemma 6.3, there exists $m\in \mathbb {Z}$ such that

\[ \mathcal{D}^{(n)}(f_{p_i,s_i})\leq \rho_\psi^{\theta}\mathcal{D}^{(n+m)}(f_{p,s}), \]

where $\rho _\psi$ is the similarity ratio of $\psi$. So we have $\mathcal {E}(f_{p,s})=\rho _{\psi }^{-\theta }\mathcal {E}(f_{p_i,s_i})\leq c_3^{-1}s^{-\theta }$ for some constant $c_3$ independent of $p,s,i$. Since $f_{p,s}|_{B_s^c(p)}=0$ and $f_{p,s}(p)=1$, we get the estimate $R(p,B_s^c(p))\geq c_3s^\theta$.

Finally, the estimates $R(p,q)\geq c_1\,{\rm d}(p,q)^\theta$ follows from the fact that $R(p,q)\geq R(p,B_{{\rm d}(p,q)}^c(p))\geq c_3{{\rm d}(p,q)}^\theta$, and $R(p,B_s^c(p))\leq c_4s^\theta$ follows from the fact that $R(p,B_s^c(p))\leq R(p,q)\leq c_2s^\theta$ for some $q\in \mathcal {G}$ satisfying ${\rm d}(p,q)=s$.