2·1. BBM type characterisation of heat kernel-based p-energy semi-norms

In our proof, we need the following embedding relation.

Proposition 2·1. For any $\delta \in (0,\sigma )$ , we have

(2·1) \begin{equation}\mathcal{D}(E_{p,\infty }^{\sigma })\subseteq \mathcal{D}(E_{p,p}^{\sigma-\delta }). \end{equation}

Proof. By the elementary inequality $|a-b|^{p}\leq 2^{p-1}(|a|^{p}+|b|^{p})$ , we have

(2·2) \begin{align}& \int_{M}\int_{M}|u(x)-u(y)|^{p}p_{t}(x,y)d\mu (x)d\mu (y) \nonumber \\[5pt] & \quad \leq 2^{p-1}\int_{M}\int_{M}(|u(x)|^{p}+|u(y)|^{p})p_{t}(x,y)d\mu (x)d\mu(y) \nonumber \\[5pt] & \quad = 2^{p}\int_{M}|u(x)|^{p}\left( \int_{M}p_{t}(x,y)d\mu (y)\right) d\mu(x)\leq 2^{p}||u||_{p}^p. \end{align}

Fix $\epsilon \in (0,R_{0}^{\beta ^{\ast }})$ . By (2·2),

\begin{eqnarray*}E_{p,p}^{\sigma -\delta }(u) &=&\int_{0}^{R_{0}^{\beta ^{\ast }}}\frac{1}{t^{{p(\sigma -\delta )}/{\beta ^{\ast }}}}\left(\int_{M}\int_{M}|u(x)-u(y)|^{p}p_{t}(x,y)d\mu (x)d\mu (y)\right) \frac{dt}{t}\\[5pt] &\leq &\int_{0}^{\epsilon }\frac{1}{t^{{p(\sigma -\delta )}/{\beta ^{\ast }}}}\left( \int_{M}\int_{M}|u(x)-u(y)|^{p}p_{t}(x,y)d\mu (x)d\mu (y)\right)\frac{dt}{t} \\[5pt] &&+\int_{\epsilon }^{\infty }\frac{1}{t^{{p(\sigma -\delta )}/{\beta ^{\ast }}}}\left( \int_{M}\int_{M}|u(x)-u(y)|^{p}p_{t}(x,y)d\mu (x)d\mu (y)\right)\frac{dt}{t} \\[5pt] &\leq &\int_{0}^{\epsilon }t^{-1+p\delta /{\beta ^{\ast }}}dt\sup_{t\in(0,R_{0}^{\beta ^{\ast }})}\Psi _{u}^{\sigma }(t)+2^{p}||u||_{p}^{p}\int_{\epsilon }^{\infty }\frac{dt}{t^{1+{p(\sigma -\delta )}/{\beta^{\ast }}}} \\[5pt] &=&\frac{{\beta ^{\ast }}\epsilon ^{p\delta /{\beta ^{\ast }}}}{p\delta }\sup_{t\in (0,R_{0}^{\beta ^{\ast }})}\Psi _{u}^{\sigma }(t)+2^{p}\Vert u\Vert _{p}^{p}\frac{{\beta ^{\ast }}\epsilon ^{-p(\sigma -\delta )/{\beta^{\ast }}}}{p(\sigma -\delta )} \\[5pt] &=&\frac{{\beta ^{\ast }}\epsilon ^{p\delta /{\beta ^{\ast }}}}{p\delta }E_{p,\infty }^{\sigma }(u)+\frac{2^{p}{\beta ^{\ast }}||u||_{p}^p}{p(\sigma -\delta )}\epsilon ^{-p(\sigma -\delta )/{\beta ^{\ast }}}\lt\infty ,\end{eqnarray*}

thus showing (2·1).

Theorem 2·2. Suppose that $(M,d,\mu )$ admits heat kernel $\{p_{t}\}_{t\gt0}$ satisfies property $(KE)_{\tilde{\sigma}_{p}}$ with some $\tilde{\sigma}_{p}\gt0$ . Then there exists a constant $C\geq 1$ such that for all $u\in \mathcal{D}(E_{p,\infty }^{\tilde{\sigma}_{p}})$ ,

(2·3) \begin{equation}C^{-1}{E}_{p,\infty }^{\tilde{\sigma}_{p}}(u)\leq \liminf_{\sigma \uparrow\tilde{\sigma}_{p}}(\tilde{\sigma}_{p}-\sigma ){E}_{p,p}^{\sigma }(u)\leq\limsup_{\sigma \uparrow \tilde{\sigma}_{p}}(\tilde{\sigma}_{p}-\sigma ){E}_{p,p}^{\sigma }(u)\leq C{E}_{p,\infty }^{\tilde{\sigma}_{p}}(u).\end{equation}

Proof. Let $u\in \mathcal{D}(E_{p,\infty }^{\tilde{\sigma _{p}}})$ and $\sigma \in(0,\tilde{\sigma}_{p})$ . By Proposition 2·1, $u\in \mathcal{D}(E_{p,p}^{\sigma })$ . Note that

\begin{equation*}\Psi _{u}^{\sigma }(t)=t^{p(\tilde{\sigma}_{p}-\sigma )/{\beta ^{\ast }}}\Psi _{u}^{\tilde{\sigma}_{p}}(t).\end{equation*}

For the left-hand side, by (1·15), we have

\begin{align}\liminf_{\sigma \uparrow \tilde{\sigma}_{p}}(\tilde{\sigma}_{p}-\sigma ){E}_{p,p}^{\sigma }(u)=& \liminf_{\sigma \uparrow \tilde{\sigma}_{p}}(\tilde{\sigma}_{p}-\sigma )\int_{0}^{R_{0}^{{\beta ^{\ast }}}}t^{p(\tilde{\sigma}_{p}-\sigma )/{\beta ^{\ast }}}\Psi _{u}^{\tilde{\sigma}_{p}}(t)\frac{dt}{t}\nonumber \\[3pt] \geq & \liminf_{\sigma \uparrow \tilde{\sigma}_{p}}(\tilde{\sigma}_{p}-\sigma )\int_{0}^{\tilde{\sigma}_{p}-\sigma }t^{p(\tilde{\sigma}_{p}-\sigma )/{\beta ^{\ast }}}\Psi _{u}^{\tilde{\sigma}_{p}}(t)\frac{dt}{t}\nonumber \\[3pt] \geq & \liminf_{\sigma \uparrow \tilde{\sigma}_{p}}(\tilde{\sigma}_{p}-\sigma )\int_{0}^{\tilde{\sigma}_{p}-\sigma }t^{p(\tilde{\sigma}_{p}-\sigma )/{\beta ^{\ast }}}\frac{dt}{t}\inf_{t\in (0,\tilde{\sigma}_{p}-\sigma )}\Psi _{u}^{\tilde{\sigma}_{p}}(t) \nonumber \\[3pt] =& \frac{{\beta ^{\ast }}}{p}\liminf_{\sigma \uparrow \tilde{\sigma}_{p}}(\tilde{\sigma}_{p}-\sigma )^{p(\tilde{\sigma}_{p}-\sigma )/{\beta ^{\ast }}}\inf_{t\in (0,\tilde{\sigma}_{p}-\sigma )}\Psi _{u}^{\tilde{\sigma}_{p}}(t)\nonumber \\[3pt] =& \frac{{\beta ^{\ast }}}{p}\liminf_{\sigma \uparrow \tilde{\sigma}_{p}}\inf_{t\in (0,\tilde{\sigma}_{p}-\sigma )}\Psi _{u}^{\tilde{\sigma}_{p}}(t)=\frac{{\beta ^{\ast }}}{p}\liminf_{t\rightarrow 0}\Psi _{u}^{\tilde{\sigma}_{p}}(t). \nonumber\end{align}

It follows from property $(KE)_{\tilde{\sigma}_{p}}$ that

\begin{equation*}\liminf_{\sigma \uparrow \tilde{\sigma}_{p}}(\tilde{\sigma}_{p}-\sigma ){E}_{p,p}^{\sigma }(u)\geq \frac{C{\beta ^{\ast }}}{p}E_{p,\infty }^{\tilde{\sigma}_{p}}(u).\end{equation*}

For the right-hand side, let $A\in (0,R_{0})$ be a finite positive number. By (2·2), for any $\sigma \in (0,\tilde{\sigma _{p}})$ ,

\begin{eqnarray*}{E}_{p,p}^{\sigma }(u) &=&\int_{0}^{R_{0}^{\beta ^{\ast }}}\frac{1}{t^{{p\sigma }/{\beta ^{\ast }}}}\left(\int_{M}\int_{M}|u(x)-u(y)|^{p}p_{t}(x,y)d\mu (x)d\mu (y)\right) \frac{dt}{t}\\[3pt] &\leq &\int_{0}^{A^{\beta ^{\ast }}}\frac{1}{t^{{p\sigma }/{\beta ^{\ast }}}}\left( \int_{M}\int_{M}|u(x)-u(y)|^{p}p_{t}(x,y)d\mu (x)d\mu (y)\right)\frac{dt}{t}+\int_{A^{\beta ^{\ast }}}^{\infty }\frac{2^{p}||u||_{p}^{p}}{t^{{p\sigma }/{\beta ^{\ast }}}}\frac{dt}{t} \\[3pt] &=&\int_{0}^{A^{\beta ^{\ast }}}t^{p(\tilde{\sigma}_{p}-\sigma )/{\beta^{\ast }}}\Psi _{u}^{\tilde{\sigma}_{p}}(t)\frac{dt}{t}+\frac{2^{p}{\beta^{\ast }}||u||_{p}^p}{p\sigma A^{{p\sigma }}}.\end{eqnarray*}

It follows that

\begin{align*}\limsup_{\sigma \uparrow \tilde{\sigma}_{p}}(\tilde{\sigma}_{p}-\sigma ){E}_{p,p}^{\sigma }(u)& \leq \limsup_{\sigma \uparrow \tilde{\sigma}_{p}}(\tilde{\sigma}_{p}-\sigma )\int_{0}^{A^{\beta ^{\ast }}}t^{p(\tilde{\sigma}_{p}-\sigma )/{\beta ^{\ast }}}\Psi _{u}^{\tilde{\sigma}_{p}}(t)\frac{dt}{t}\nonumber\\[3pt] &\quad +\limsup_{\sigma \uparrow \tilde{\sigma}_{p}}(\tilde{\sigma}_{p}-\sigma )\frac{2^{p}{\beta ^{\ast }}||u||_{p}^p}{p\sigma A^{{p\sigma }}}\nonumber \\[3pt] & \leq \limsup_{\sigma \uparrow \tilde{\sigma}_{p}}(\tilde{\sigma}_{p}-\sigma )\int_{0}^{A^{\beta ^{\ast }}}t^{p(\tilde{\sigma}_{p}-\sigma )/{\beta ^{\ast }}}\frac{dt}{t}\sup_{t\in (0,A^{{\beta ^{\ast }}})}\Psi _{u}^{\tilde{\sigma}_{p}}(t)+0 \end{align*}

(2·4) \begin{align} & =\frac{{\beta ^{\ast }}}{p}\limsup_{\sigma \uparrow \tilde{\sigma}_{p}}A^{p(\tilde{\sigma}_{p}-\sigma )}\sup_{t\in (0,A^{{\beta ^{\ast }}})}\Psi _{u}^{\tilde{\sigma}_{p}}(t) \nonumber \\[5pt] & =\frac{{\beta ^{\ast }}}{p}\sup_{t\in (0,A^{{\beta ^{\ast }}})}\Psi _{u}^{\tilde{\sigma}_{p}}(t)\leq \frac{{\beta ^{\ast }}}{p}E_{p,\infty }^{\tilde{\sigma}_{p}}(u). \end{align}

The proof is complete.

2·2. BBM type characterisation of Besov-Korevaar-Schoen p-energy semi-norms

The proof of this version is similar to the previous one.

Proposition 2·3. For any $\delta \in (0,\sigma )$ , we have

(2·5) \begin{equation}B_{p,\infty }^{\sigma }\subseteq B_{p,p}^{\sigma -\delta }. \end{equation}

Proof. By the elementary inequality $|a-b|^{p}\leq 2^{p-1}(|a|^{p}+|b|^{p})$ , we have

(2·6) \begin{align}& \int_{M}\frac{1}{V(x,r)}\int_{B(x,r)}|u(x)-u(y)|^{p}d\mu (y)d\mu (x)\nonumber \\[5pt] &\quad \leq 2^{p-1}\int_{M}\frac{1}{V(x,r)}\int_{B(x,r)}(|u(x)|^{p}+|u(y)|^{p})d\mu (y)d\mu (x) \nonumber \\[5pt] &\quad = 2^{p-1}\left( ||u||_{p}^p+\int_{M}\int_{M}\frac{\mathbf{1}_{\{d(y,x)\lt r\}}}{V(x,r)}|u(y)|^{p}d\mu (y)d\mu (x)\right) \nonumber \\[5pt] &\quad = 2^{p-1}\left( ||u||_{p}^p+\int_{M}\left( \int_{B(y,r)}\frac{1}{V(x,r)}d\mu (x)\right) |u(y)|^{p}d\mu (y)\right) \nonumber \\[5pt] & \quad \leq 2^{p-1}\left( ||u||_{p}^p+\sup_{x\in M}\frac{V(x,2r)}{V(x,r)}\int_{M}|u(y)|^{p}d\mu (y)\right) \leq C||u||_{p}^p. \end{align}

Fix $\epsilon \in (0,R_{0})$ . By (2·6),

\begin{eqnarray*}\lbrack u]_{B_{p,p}^{\sigma -\delta }}^{p} &=&\int_{0}^{R_{0}}r^{-(p\sigma-p\delta )}\int_{M}\frac{1}{V(x,r)}\int_{B(x,r)}|u(x)-u(y)|^{p}d\mu (y)d\mu(x)\frac{dr}{r} \\[5pt] &\leq &\int_{0}^{\epsilon }r^{-(p\sigma -p\delta )}\int_{M}\frac{1}{V(x,r)}\int_{B(x,r)}|u(x)-u(y)|^{p}d\mu (y)d\mu (x)\frac{dr}{r} \\[5pt] &&+\int_{\epsilon }^{\infty }r^{-(p\sigma -p\delta )}\int_{M}\frac{1}{V(x,r)}\int_{B(x,r)}|u(x)-u(y)|^{p}d\mu (y)d\mu (x)\frac{dr}{r} \\[5pt] &\leq &\int_{0}^{\epsilon }r^{-1+p\delta }dr\sup_{r\in (0,R_{0})}\Phi_{u}^{\sigma }(r)+C||u||_{p}^p\int_{\epsilon }^{\infty}r^{-(p\sigma -p\delta )}\frac{dr}{r} \\[5pt] &=&\frac{\epsilon ^{p\delta }}{p\delta }\sup_{r\in (0,R_{0})}\Phi_{u}^{\sigma }(r)+C||u||_{p}^p\frac{\epsilon ^{-p(\sigma -\delta )}}{p(\sigma -\delta )}=\frac{\epsilon ^{p\delta }}{p\delta }[u]_{B_{p,\infty}^{\sigma }}^{p}+C^{\prime }||u||_{p}^p\lt\infty ,\end{eqnarray*}

thus showing (2·5).

Theorem 2·4. Suppose that $(M,d,\mu )$ satisfies property $(NE)_{\sigma _{p}^{\#}}$ , then there exists a positive constant C such that for all $u\in B_{p,\infty }^{\sigma_{p}^{\#}}$ ,

(2·7) \begin{equation}C^{-1}[u]_{{B}_{p,\infty }^{\sigma _{p}^{\#}}}^{p}\leq \liminf_{\sigma\uparrow \sigma _{p}^{\#}}(\sigma _{p}^{\#}-\sigma )[u]_{{B}_{p,p}^{\sigma}}^{p}\leq \limsup_{\sigma \uparrow \sigma _{p}^{\#}}(\sigma_{p}^{\#}-\sigma )[u]_{{B}_{p,p}^{\sigma }}^{p}\leq C[u]_{{B}_{p,\infty}^{\sigma _{p}^{\#}}}^{p}. \end{equation}

Suppose that $(M,d,\mu )$ satisfies property $(\widetilde{NE})_{\sigma_p}$ with some $\sigma_{p}\gt0$ , then there exists a positive constant C such that for all $u\in B_{p,\infty }^{\sigma _{p}}$ ,

(2·8) \begin{equation}C^{-1}[u] _{{KS}_{p,\infty }^{\sigma _{p}}}^{p}\leq \liminf_{\sigma\uparrow \sigma _{p}}(\sigma _{p}-\sigma )[u]_{{B}_{p,p}^{\sigma }}^{p}\leq\limsup_{\sigma \uparrow \sigma _{p}}(\sigma _{p}-\sigma )[u]_{{B}_{p,p}^{\sigma }}^{p}\leq C[u] _{{KS}_{p,\infty }^{\sigma_{p}}}^{p}. \end{equation}

Proof. We show (2·7) first. For the left-hand side, similarly, we have that for any $\sigma _{p}\gt0,$

(2·9) \begin{align}\liminf_{\sigma \uparrow \sigma _{p}}(\sigma _{p}-\sigma )[u]_{{B}_{p,p}^{\sigma }}^{p}=& \liminf_{\sigma \uparrow \sigma _{p}}(\sigma_{p}-\sigma )\int_{0}^{R_{0}}r^{p(\sigma _{p}-\sigma )}\Phi _{u}^{\sigma_{p}}(r)\frac{dr}{r} \nonumber \\\geq & \liminf_{\sigma \uparrow \sigma _{p}}(\sigma _{p}-\sigma)\int_{0}^{\sigma _{p}-\sigma }r^{p(\sigma _{p}-\sigma )}\Phi _{u}^{\sigma_{p}}(r)\frac{dr}{r} \nonumber \\\geq & \liminf_{\sigma \uparrow \sigma _{p}}(\sigma _{p}-\sigma)\int_{0}^{\sigma _{p}-\sigma }r^{p(\sigma _{p}-\sigma )}\frac{dr}{r}\inf_{r\in (0,\sigma _{p}-\sigma )}\Phi _{u}^{\sigma _{p}}(r) \nonumber \\=& \frac{1}{p}\liminf_{\sigma \uparrow \sigma _{p}}\inf_{r\in (0,\sigma_{p}-\sigma )}\Phi _{u}^{\sigma _{p}}(r)=\frac{1}{p}\liminf_{r\rightarrow0}\Phi _{u}^{\sigma _{p}}(r). \end{align}

Therefore, by property $(NE)_{\sigma _{p}^{\#}}$ , we have for $\sigma _{p}=\sigma _{p}^{\#}$ in (2·9) that

(2·10) \begin{equation}\liminf_{\sigma \uparrow \sigma _{p}^{\#}}(\sigma _{p}^{\#}-\sigma )[u]_{{B}_{p,p}^{\sigma }}^{p}\geq \frac{C^{-1}}{p}[u]_{{B}_{p,\infty }^{\sigma_{p}^{\#}}}^{p}. \end{equation}

For the other side, let $A\in (0,R_{0})$ be a finite positive number. We have from (2·6) that

(2·11) \begin{eqnarray}\lbrack u]_{{B}_{p,p}^{\sigma }}^{p} &=&\int_{0}^{R_{0}}r^{-p\sigma }\left(\int_{M}\frac{1}{V(x,r)}\int_{B(x,r)}|u(x)-u(y)|^{p}d\mu (y)d\mu (x)\right)\frac{dr}{r} \nonumber \\&\leq &\int_{0}^{A}r^{-p\sigma }\left( \int_{M}\frac{1}{V(x,r)}\int_{B(x,r)}|u(x)-u(y)|^{p}d\mu (y)d\mu (x)\right) \frac{dr}{r}+C\int_{A}^{\infty }r^{-p\sigma }||u||_{p}^p\frac{dr}{r} \nonumber \\&=&\int_{0}^{A}r^{p(\sigma _{p}^{\#}-\sigma )}\Phi _{u}^{\sigma _{p}^{\#}}(r)\frac{dr}{r}+\frac{C||u||_{p}^p}{p\sigma A^{p\sigma }}.\end{eqnarray}

It follows that

(2·12) \begin{align}\limsup_{\sigma \uparrow \sigma _{p}^{\#}}\!(\sigma _{p}^{\#}-\sigma )[u]_{{B}_{p,p}^{\sigma }}^{p}& \leq \limsup_{\sigma \uparrow \sigma_{p}^{\#}}\!(\sigma _{p}^{\#}-\sigma )\int_{0}^{A}r^{p(\sigma _{p}^{\#}-\sigma)}\Phi _{u}^{\sigma _{p}^{\#}}(r)\frac{dr}{r}+\limsup_{\sigma \uparrow\sigma _{p}^{\#}}\!(\sigma _{p}^{\#}-\sigma )\frac{C||u||_{p}^p}{p\sigma A^{p\sigma }} \nonumber \\& \leq \limsup_{\sigma \uparrow \sigma _{p}^{\#}}(\sigma _{p}^{\#}-\sigma)\int_{0}^{A}r^{p(\sigma _{p}^{\#}-\sigma )}\frac{dr}{r}\sup_{r\in(0,R_{0})}\Phi _{u}^{\sigma _{p}^{\#}}(r)+0 \nonumber \\& =\frac{1}{p}\limsup_{\sigma \uparrow \sigma _{p}^{\#}}A^{p(\sigma_{p}^{\#}-\sigma )}\sup_{r\in (0,R_{0})}\Phi _{u}^{\sigma _{p}^{\#}}(r)=\frac{1}{p}[u]_{B_{p,\infty }^{\sigma _{p}^{\#}}}^{p}, \end{align}

thus showing (2·7).

Next, we show (2·8). By property $(\widetilde{NE})_{\sigma_p}$ and (2·9), we have

(2·13) \begin{equation}\liminf_{\sigma \uparrow \sigma _{p}}(\sigma _{p}-\sigma )[u]_{{B}_{p,p}^{\sigma }}^{p}\geq C^{-1}[u] _{KS_{p,\infty}^{\sigma _{p}}}^{p}.\end{equation}

Then for the other side of (2·8), by the definition of $\limsup $ , there exists $\epsilon \in (0,R_{0})$ such that

\begin{equation*}\sup_{r\in (0,\epsilon )}\Phi _{u}^{\sigma _{p}}(r)\leq2\limsup_{r\rightarrow 0}\Phi _{u}^{\sigma _{p}}(r).\end{equation*}

It follows from replacing $\sigma _{p}^{\#}$ by $\sigma _{p}$ and taking $A=\epsilon $ in (2·11) that,

(2·14) \begin{align}\limsup_{\sigma \uparrow \sigma _{p}}(\sigma _{p}-\sigma )[u]_{{B}_{p,p}^{\sigma }}^{p}& \leq \limsup_{\sigma \uparrow \sigma _{p}}(\sigma_{p}-\sigma )\int_{0}^{\epsilon }r^{p(\sigma _{p}-\sigma )}\Phi _{u}^{\sigma_{p}}(r)\frac{dr}{r}+\limsup_{\sigma \uparrow \sigma _{p}}(\sigma_{p}-\sigma )\frac{C|| u||_{p}^p}{p\sigma \epsilon ^{p\sigma }}\nonumber \\& \leq \limsup_{\sigma \uparrow \sigma _{p}}(\sigma _{p}-\sigma)\int_{0}^{\epsilon }r^{p(\sigma _{p}-\sigma )}\frac{dr}{r}\sup_{r\in(0,\epsilon )}\Phi _{u}^{\sigma _{p}}(r)+0 \nonumber \\& =\frac{1}{p}\sup_{r\in (0,\epsilon )}\Phi _{u}^{\sigma _{p}}(r)\leq \frac{2}{p}\limsup_{r\rightarrow 0}\Phi _{u}^{\sigma _{p}}(r)=\frac{2}{p}[u] _{{KS}_{p,\infty }^{\sigma _{p}}}^{p}. \end{align}

The proof is complete.

3·1. Equivalence of integral-type semi-norms

In this subsection, fix $\sigma\gt0$ . The main result of this subsection is the following.

Lemma 3·1. If $(M,d,\mu )$ admits a heat kernel $\{p_{t}\}_{t\gt0}$ satisfying (UHE) and (LHE), then $\mathcal{D}(E_{p,\infty}^{\sigma })=B_{p,\infty }^{\sigma }$ and for all $u\in B_{p,\infty}^{\sigma }$ ,

(3·1) \begin{equation}E_{p,\infty }^{\sigma }(u)\asymp \lbrack u]_{B_{p,\infty }^{\sigma }}^{p}\text{ and }[u] _{KS_{p,\infty }^{\sigma }}^{p}\asymp\limsup_{t\rightarrow 0}\Psi _{u}^{\sigma }(t). \end{equation}

In addition, $\mathcal{D}(E_{p,p}^{\sigma })=B_{p,p}^{\sigma }$ and for all $u\in B_{p,p}^{\sigma }$ ,

(3·2) \begin{equation}E_{p,p}^{\sigma }(u)\asymp \lbrack u]_{B_{p,p}^{\sigma }}^{p}. \end{equation}

We prove Lemma 3·1 by pure elementary analysis (without probability arguments) under slightly milder assumptions, and split the proof into Propositions 3·2-3·6. Part of Lemma 3·1 was also studied by K. Pietruska-Pałuba in [ Reference Pietruska-Pałuba45 , theorems 3·1 and 3·2] using probability arguments, and by Alonso Ruiz, Baudoin et al. in [ Reference Alonso-Ruiz, Baudoin, Chen, Rogers, Shanmugalingam and Teplyaev2 , proposition 4·2] under Gaussian heat kernel estimate. Recently, Cao and Qiu in [ Reference Cao and Qiu15 ] proved the equivalence of the Korevaar-Schoen norm and another type of Besov norm.

Proposition 3·2. If $(M,d,\mu )$ admits a heat kernel $\{p_{t}\}_{t\gt0}$ satisfying (LHE), then there exists a positive constant C such that for all $u\in \mathcal{D}(E_{p,p}^{\sigma })$ ,

(3·3) \begin{equation}\lbrack u]_{B_{p,p}^{\sigma }}^{p}\leq CE_{p,p}^{\sigma }(u). \end{equation}

Proof. By (LHE), we have

(3·4) \begin{align}& \frac{1}{t^{{p\sigma }/{\beta ^{\ast }}}}\int_{M}\int_{B(x, t^{{1}/{\beta^{\ast }}})}|u(x)-u(y)|^{p}p_{t}(x,y)d\mu (y)d\mu (x) \nonumber \\&\quad \geq \frac{c_{1}}{t^{p\sigma /\beta ^{\ast }}}\int_{M}\frac{1}{V(x,t^{1/\beta ^{\ast }})}\int_{B(x, t^{{1}/{\beta ^{\ast }}})}|u(x)-u(y)|^{p} \exp \left(-c_2\left(\frac{d(x,y)}{t^{1/\beta^*}}\right)^{\frac{\beta^*}{\beta^*-1}}\right)d\mu (y)d\mu (x) \nonumber \\&\quad \geq c_{1}e^{-c_2} \Phi_u^{\sigma}( t^{1/\beta^*}) . \end{align}

Therefore,

\begin{align*}E_{p,p}^{\sigma }(u)&=\int_{0}^{R_{0}^{\beta ^{\ast }}}\frac{1}{t^{{p\sigma }/{\beta ^{\ast }}}}\left( \int_{M}\int_{M}|u(x)-u(y)|^{p}p_{t}(x,y)d\mu(x)d\mu (y)\right) \frac{dt}{t} \\& \geq \beta ^{\ast }c_{1}e^{-c_2}\int_{0}^{R_{0}} \Phi_u^{\sigma}(r)\frac{dr}{r} =\beta ^{\ast}c_{1}e^{-c_2}\lbrack u]_{B_{p,p}^{\sigma }}^{p},\end{align*}

which completes the proof.

Proposition 3·3. If $(M,d,\mu )$ admits a heat kernel $\{p_{t}\}_{t\gt0}$ satisfying (UHE), then there exists a positive constant C such that for all $u\in B_{p,p}^{\sigma }$ ,

\begin{equation*}E_{p,p}^{\sigma }(u)\leq C[u]_{B_{p,p}^{\sigma }}^{p}.\end{equation*}

Proof. We decompose $B(x,t^{1/\beta ^{\ast }})^{c}$ as the union of annuli $B(x,2^{n}t^{1/\beta ^{\ast }})\setminus B(x,2^{n-1}t^{1/\beta ^{\ast }})$ for $x\in M$ , where the integers $1\leq n\leq \log _{2}(R_{0}t^{-1/\beta^{\ast }})$ , then by (1·6),

(3·5) \begin{align}& \frac{1}{t^{{p\sigma }/{\beta ^{\ast }}}}\int_{M}\int_{B(x,t^{1/\beta^{\ast }})^{c}}|u(x)-u(y)|^{p}p_{t}(x,y)d\mu (y)d\mu (x) \nonumber \\[2pt] & =\frac{1}{t^{{p\sigma }/{\beta ^{\ast }}}}\int_{M}\sum_{n=1}^{[\log_{2}(R_{0}t^{-1/\beta ^{\ast }})]}\int_{B(x,2^{n}t^{1/\beta ^{\ast}})\setminus B(x,2^{n-1}t^{1/\beta ^{\ast }})}|u(x)-u(y)|^{p}p_{t}(x,y)d\mu(y)d\mu (x) \nonumber \\[2pt] & \leq \frac{1}{t^{{p\sigma }/{\beta ^{\ast }}}}\int_{M}\frac{c_{3}}{V(x,t^{1/\beta ^{\ast }})}\sum_{n=1}^{[\log _{2}(R_{0}t^{-1/\beta ^{\ast}})]}\int_{B(x,2^{n}t^{1/\beta ^{\ast }})\setminus B(x,2^{n-1}t^{1/\beta^{\ast }})}|u(x)-u(y)|^{p}\nonumber \\[2pt] &\times \exp \left( -c_{4}2^{\frac{(n-1)\beta ^{\ast }}{\beta ^{\ast }-1}}\right) d\mu (y)d\mu (x) \nonumber \\[2pt] & \leq \sum_{n=1}^{[\log _{2}(R_{0}t^{-1/\beta ^{\ast }})]}\frac{1}{t^{{p\sigma }/{\beta ^{\ast }}}}\int_{M}\frac{c_{3}}{V(x,t^{1/\beta ^{\ast }})}\int_{B(x,2^{n}t^{1/\beta ^{\ast }})}|u(x)-u(y)|^{p}\nonumber \\[2pt] &\times \exp \left( -c_{4}2^{\frac{(n-1)\beta ^{\ast }}{\beta ^{\ast }-1}}\right) d\mu (y)d\mu (x) \nonumber\\[2pt] & \leq \sum_{n=1}^{[\log _{2}(R_{0}t^{-1/\beta ^{\ast }})]}\exp \left(-c_{4}2^{\frac{(n-1)\beta ^{\ast }}{\beta ^{\ast }-1}}\right) \frac{2^{np\sigma }}{(2^{n}t^{1/\beta ^{\ast }})^{p\sigma }}\int_{M}\frac{c_{3}C_{d}2^{n\alpha _{1}}}{V(x,2^{n}t^{1/\beta ^{\ast }})} \end{align}

\begin{align*}&\times \int_{B(x,2^{n}t^{1/\beta ^{\ast }})}|u(x)-u(y)|^{p}d\mu (y)d\mu (x) \nonumber\\& =c_{3}C_{d}\sum_{n=1}^{\infty }1_{\{n\leq \log _{2}(R_{0}t^{-1/\beta^{\ast }})\}}\exp \left( -c_{4}2^{\frac{(n-1)\beta ^{\ast }}{\beta ^{\ast }-1}}\right) 2^{(p\sigma +\alpha _{1})n}\Phi _{u}^{\sigma }(2^{n}t^{1/\beta^{\ast }}), \end{align*}

where we use the volume doubling property (1·3) in the last inequality above

(3·6) \begin{equation}\frac{V(x,2^{n}t^{1/\beta ^{\ast }})}{V(x,t^{1/\beta ^{\ast }})}\leq C_{d}2^{n\alpha _{1}}. \end{equation}

Using (1·6) again, we have

(3·7) \begin{eqnarray}&&\frac{1}{t^{{p\sigma }/{\beta ^{\ast }}}}\int_{M}\int_{B(x,t^{1/\beta^{\ast }})}|u(x)-u(y)|^{p}p_{t}(x,y)d\mu (y)d\mu (x) \nonumber \\&\leq &\frac{1}{t^{{p\sigma }/{\beta ^{\ast }}}}\int_{M}\int_{B(x,t^{1/\beta^{\ast }})}|u(x)-u(y)|^{p}\frac{c_{3}}{V(x,t^{1/\beta ^{\ast }})}\exp \left(-c_{4}\left( \frac{d(x,y)}{t^{1/\beta ^{\ast }}}\right) ^{\frac{\beta ^{\ast}}{\beta ^{\ast }-1}}\right) d\mu (y)d\mu (x) \nonumber \\&\leq &c_{3}\Phi _{u}^{\sigma }(t^{1/\beta ^{\ast }}). \end{eqnarray}

Therefore, by (3·5) and (3·7),

(3·8) \begin{align}\Psi _{u}^{\sigma }(t) &\leq c_{3}C_{d}\sum_{n=1}^{\infty }1_{\{n\leq \log_{2}(R_{0}t^{-1/\beta ^{\ast }})\}}\exp \left( -c_{4}2^{\frac{(n-1)\beta^{\ast }}{\beta ^{\ast }-1}}\right) 2^{(p\sigma +\alpha _{1})n}\Phi_{u}^{\sigma }(2^{n}t^{1/\beta ^{\ast }})+c_{3}\Phi _{u}^{\sigma}(t^{1/\beta ^{\ast }}) \nonumber \\&\leq c_{3}C_{d}\sum_{n=0}^{\infty }1_{\{n\leq \log _{2}(R_{0}t^{-1/\beta^{\ast }})\}}\exp \left( -c_{4}2^{\frac{(n-1)\beta ^{\ast }}{\beta ^{\ast }-1}}\right) 2^{(p\sigma +\alpha _{1})n}\Phi _{u}^{\sigma }(2^{n}t^{1/\beta^{\ast }}), \end{align}

it follows that

\begin{align*}& E_{p,p}^{\sigma }(u)=\int_{0}^{R_{0}^{\beta ^{\ast }}}\Psi _{u}^{\sigma}(t)\frac{dt}{t}\leq c_{3}C_{d}\sum_{n=0}^{\infty }\exp \left( -c_{4}2^{\frac{(n-1)\beta ^{\ast }}{\beta ^{\ast }-1}}\right) \int_{0}^{2^{-n\beta^{\ast }}R_{0}^{\beta ^{\ast }}}2^{(p\sigma +\alpha _{1})n}\Phi _{u}^{\sigma}(2^{n}t^{1/\beta ^{\ast }})\frac{dt}{t} \\& \quad =c_{3}\beta ^{\ast }C_{d}\sum_{n=0}^{\infty }\exp \left( -c_{4}2^{\frac{(n-1)\beta ^{\ast }}{\beta ^{\ast }-1}}\right) 2^{(p\sigma +\alpha_{1})n}\int_{0}^{R_{0}}\Phi _{u}^{\sigma }(r)\frac{dr}{r}=c_{3}\beta ^{\ast}C_{d}C_{0}[u]_{B_{p,p}^{\sigma }}^{p},\end{align*}

where

(3·9) \begin{equation}C_{0}=\sum_{n=0}^{\infty }\exp \left( -c_{4}2^{\frac{(n-1)\beta ^{\ast }}{\beta ^{\ast }-1}}\right) 2^{(p\sigma +\alpha _{1})n}\lt\infty. \end{equation}

The proof is complete.

The following two propositions deal with the ‘local’ semi-norms.

Proposition 3·4. If $(M,d,\mu )$ admits a heat kernel $\{p_{t}\}_{t\gt0}$ satisfying (LHE), then there exists a positive constant C such that for all $u\in \mathcal{D}(E_{p,\infty }^{\sigma })$ ,

\begin{equation*}\lbrack u]_{B_{p,\infty }^{\sigma }}^{p}\leq CE_{p,\infty }^{\sigma }(u).\end{equation*}

Proof. By (3·4), we have

\begin{align*}& E_{p,\infty }^{\sigma }(u)=\sup_{t\in (0,R_{0}^{\beta ^{\ast }})}\frac{1}{t^{{p\sigma }/{\beta ^{\ast }}}} \int_{M}\int_{M}|u(x)-u(y)|^{p}p_{t}(x,y)d\mu (x)d\mu (y) \\[5pt] &\quad \geq c_{1}e^{-c_2}\sup_{t\in (0,R_{0}^{\beta ^{\ast }})}\Phi_u^{\sigma}(t^{1/\beta^*}) =c_{1}e^{c_2}\sup_{r\in (0,R_{0})}\Phi_u^{\sigma}(r)=c_{1}e^{c_2}[u]_{B_{p,\infty }^{\sigma }}^{p},\end{align*}

which ends the proof.

Proposition 3·5. If $(M,d,\mu )$ admits a heat kernel $\{p_{t}\}_{t\gt0}$ satisfying (UHE), then there exists a positive constant C such that for all $u\in B_{p,\infty }^{\sigma }$ ,

\begin{equation*}E_{p,\infty }^{\sigma }(u)\leq C[u]_{B_{p,\infty }^{\sigma }}^{p}.\end{equation*}

Proof. By (3·8) and (3·6), we have

(3·10) \begin{align}& \Psi _{u}^{\sigma }(t)\leq c_{3}C_{d}\sum_{n=0}^{\infty }1_{\{n\leq \log_{2}(R_{0}t^{-1/\beta ^{\ast }})\}}\exp \left( -c_{4}2^{\frac{(n-1)\beta^{\ast }}{\beta ^{\ast }-1}}\right) 2^{(p\sigma +\alpha _{1})n}\Phi_{u}^{\sigma }(2^{n}t^{1/\beta ^{\ast }}) \nonumber \\[5pt] &\quad = c_{3}C_{d}\sum_{n=0}^{\infty }\exp \left( -c_{4}2^{\frac{(n-1)\beta^{\ast }}{\beta ^{\ast }-1}}\right) 2^{(p\sigma +\alpha_{1})n}[u]_{B_{p,\infty }^{\sigma }}^{p}=c_{3}C_{d}C_{0}[u]_{B_{p,\infty}^{\sigma }}^{p}, \end{align}

where $C_{0}$ is finite by (3·9). The proof is complete by taking the supremum in $(0,R_{0}^{\beta ^{\ast }})$ on both sides of (3·10).

The following proposition concerns the Korevaar-Schoen semi-norm.

Proposition 3·6. If $(M,d,\mu )$ admits a heat kernel $\{p_{t}\}_{t\gt0}$ satisfying (UHE) and (LHE), then for all $u\in B_{p,\infty }^{\sigma }$ ,

(3·11) \begin{equation}\limsup_{t\rightarrow 0}\Psi _{u}^{\sigma }(t)\asymp \limsup_{r\rightarrow0}\Phi _{u}^{\sigma }(r). \end{equation}

Proof. By Proposition 3·4, we have

(3·12) \begin{equation}\sup_{r\in (0,R_{0})}\Phi _{u}^{\sigma }(r)\leq C\sup_{t\in (0,R_{0}^{\beta^{\ast }})}\Psi _{u}^{\sigma }(t), \end{equation}

for any $R_{0}\in (0,$ diam(M)]. Therefore, if we choose $R_{0}=\epsilon $ in (3·12), then

\begin{align*}\limsup_{t\rightarrow 0}\Psi _{u}^{\sigma }(t)& =\lim_{\epsilon \rightarrow0}\sup_{t\in (0,\epsilon )}\Psi _{u}^{\sigma }(t) \\[5pt] & \geq C_{1}\lim_{\epsilon \rightarrow 0}\sup_{r\in (0,\epsilon ^{1/\beta^{\ast }})}\Phi _{u}^{\sigma }(r)=C_{1}\limsup_{r\rightarrow 0}\Phi_{u}^{\sigma }(r).\end{align*}

Next, we consider the left-hand side of (3·11). By (3·8) and the dominated convergence theorem, we have that

\begin{align*}& \limsup_{t\rightarrow 0}\frac{1}{t^{p\sigma /\beta ^{\ast }}}\int_{M}\int_{M}|u(x)-u(y)|^{p}p_{t}(x,y)d\mu (y)d\mu (x) \\&\quad \leq C_{1}\sum_{n=0 }^{\infty }\exp \left( -c_{4}2^{\frac{(n-1)\beta^{\ast }}{\beta ^{\ast }-1}}\right) \limsup_{t\rightarrow 0}\frac{1_{\{n\leq\log _{2}(R_{0}t^{-1/\beta ^{\ast }})\}}}{(2^{n}t^{1/\beta ^{\ast}})^{p\sigma }}\int_{M}\frac{2^{(p\sigma +\alpha _{1})n}}{V(x,2^{n}t^{1/\beta ^{\ast }})}\\&\qquad \times \int_{B(x,2^{n}t^{1/\beta ^{\ast}})}|u(x)-u(y)|^{p}d\mu (y)d\mu (x) \\&\quad \leq C_{1}\sum_{n=0 }^{\infty }\exp \left( -c_{4}2^{\frac{(n-1)\beta^{\ast }}{\beta ^{\ast }-1}}\right) 2^{(p\sigma +\alpha_{1})n}\limsup_{r\rightarrow 0}\Phi _{u}^{\sigma }(r)\leq C_{1}C_{0}\limsup_{r\rightarrow 0}\Phi _{u}^{\sigma }(r),\end{align*}

where $C_{1}=c_{3}C_{d}$ and the constant $C_{0}$ is defined in (3·9). The proof is complete.

3·2. Equivalence between $(KE)_{\sigma}$ and $(NE)_{\sigma}$

We start with an easy side $(NE)_{\sigma}\Rightarrow (KE)_{\sigma}$ .

Proposition 3·7. Suppose that $(M,d,\mu )$ admits a heat kernel $\{p_{t}\}_{t\gt0} $ satisfying (LHE). Then there exists $C\gt0$ such that for all $u\in \mathcal{D}(E_{p,\infty }^{\sigma })$ ,

\begin{equation*}\Psi _{u}^{\sigma }(t)\geq C\Phi _{u}^{\sigma }(t^{1/\beta ^{\ast }}).\end{equation*}

Consequently,

(3·13) \begin{equation}\liminf_{t\rightarrow 0}\Psi _{u}^{\sigma }(t)\geq C\liminf_{r\rightarrow0}\Phi _{u}^{\sigma }(r), \end{equation}

where the constant C depends only on $c_{1}$ in (LHE).

Proof. By (3·4), we have

\begin{align*}\Psi _{u}^{\sigma }(t)& \geq \frac{1}{t^{p\sigma /\beta ^{\ast }}}\int_{M}\int_{B(x,\delta t^{1/\beta ^{\ast }})}|u(x)-u(y)|^{p}p_{t}(x,y)d\mu(y)d\mu (x) \\[5pt] & \geq \frac{c_{1}C_{d}^{-1}\delta ^{\alpha _{1}}}{t^{p\sigma /\beta ^{\ast}}}\int_{M}\frac{1}{V(x,\delta t^{1/\beta ^{\ast }})}\int_{B(x,\delta t^{{1}/{\beta ^{\ast }}})}|u(x)-u(y)|^{p}d\mu (y)d\mu (x) \\[5pt] & =c_{1}C_{d}^{-1}\delta ^{p\sigma +\alpha _{1}}\Phi _{u}(\delta t^{1/\beta^{\ast }}).\end{align*}

Taking the lower limit on both sides of the above inequality, we have

\begin{equation*}\liminf_{t\rightarrow 0}\Psi _{u}^{\sigma }(t)\geq c_{1}C_{d}^{-1}\delta^{p\sigma +\alpha _{1}}\liminf_{t\rightarrow 0}\Phi _{u}^{\sigma}(t^{1/\beta ^{\ast }})=c_{1}C_{d}^{-1}\delta ^{p\sigma +\alpha_{1}}\liminf_{r\rightarrow 0}\Phi _{u}^{\sigma }(r),\end{equation*}

which implies (3·13).

We give a proof for the following claim which appeared in [ Reference Alonso-Ruiz, Baudoin, Chen, Rogers, Shanmugalingam and Teplyaev3 , The proof of Lemma 4·13].

Proposition 3·8. Suppose that $(M,d,\mu )$ admits a heat kernel $\{p_{t}\}_{t\gt0} $ satisfying (UHE) and (LHE). Then there exist $C,c\gt1$ and $c^{\prime }\gt0 $ such that for any $\delta \gt0$ and for $\mu$ -almost all $x,y\in M$ with $d(x,y)\gt\delta t^{1/\beta ^{\ast }}$ ,

\begin{equation*}p_{t}(x,y)\leq C\exp \left( -c^{\prime }\delta ^{\frac{\beta ^{\ast }}{\beta^{\ast }-1}}\right) p_{ct}(x,y).\end{equation*}

Proof. By (UHE) and the volume doubling property (1·3), we have

\begin{align*}p_{t}(x,y)& \leq \frac{c_{3}}{V(x,t^{1/\beta ^{\ast }})}\exp \left(-c_{4}\left( \frac{d(x,y)}{t^{1/\beta ^{\ast }}}\right) ^{\frac{\beta ^{\ast}}{\beta ^{\ast }-1}}\right) \\& \leq \frac{c_{3}C_{d}c^{\alpha _{1}/\beta ^{\ast }}}{V(x,(ct)^{1/\beta^{\ast }})}\exp \left( -c_{4}\left( \frac{d(x,y)}{(ct)^{1/\beta ^{\ast }}}\right) ^{\frac{\beta ^{\ast }}{\beta ^{\ast }-1}}\cdot c^{\frac{1}{\beta^{\ast }-1}}\right) \\& =\frac{c_{3}C_{d}c^{\alpha _{1}/\beta ^{\ast }}}{c_{1}}\frac{c_{1}}{V(x,(ct)^{1/\beta ^{\ast }})}\exp \left( -c_{4}\left( \frac{d(x,y)}{(ct)^{1/\beta ^{\ast }}}\right) ^{\frac{\beta ^{\ast }}{\beta ^{\ast }-1}}\cdot c^{\frac{1}{\beta ^{\ast }-1}}\right) ,\end{align*}

for $\mu $ -almost all $x,y\in M$ . Taking $\tilde{c}=c_{4}c^{\frac{1}{\beta^{\ast }-1}}-c_{2}$ , we use (LHE) to obtain

\begin{equation*}p_{t}(x,y)\leq \frac{c_{3}C_{d}c^{\alpha _{1}/\beta ^{\ast }}}{c_{1}}\exp\left( -\tilde{c}\left( \frac{d(x,y)}{(ct)^{1/\beta ^{\ast }}}\right) ^{\frac{\beta ^{\ast }}{\beta ^{\ast }-1}}\right) p_{ct}(x,y).\end{equation*}

We choose $c\gt1$ to ensure $c^{\frac{1}{\beta ^{\ast }-1}}c_{4}\gt c_{2}$ so that $\tilde{c}\gt0$ . Let $c^{\prime }={\tilde{c}}/{c^{\frac{1}{\beta ^{\ast}-1}}}$ , then for $d(x,y)\gt\delta t^{1/\beta ^{\ast }}$ ,

\begin{equation*}p_{t}(x,y)\leq \frac{c_{3}C_{d}c^{\alpha _{1}/\beta ^{\ast }}}{c_{1}}\exp\left( -c^{\prime }\left( \frac{d(x,y)}{t^{1/\beta ^{\ast }}}\right) ^{\frac{\beta ^{\ast }}{\beta ^{\ast }-1}}\right) p_{ct}(x,y)\leq C\exp \left(-c^{\prime }\delta ^{\frac{\beta ^{\ast }}{\beta ^{\ast }-1}}\right)p_{ct}(x,y),\end{equation*}

where $C={c_{3}C_{d}c^{\alpha _{1}/\beta ^{\ast }}}/{c_{1}}$ . The proof is complete.

The proof of the implication $(KE)_{\sigma}\Rightarrow (NE)_{\sigma}$ is inspired by [ Reference Alonso-Ruiz, Baudoin, Chen, Rogers, Shanmugalingam and Teplyaev3 , lemma 4·13] where $p=1$ there.

Lemma 3·9. Suppose that $(M,d,\mu )$ admits a heat kernel $\{p_{t}\}_{t\gt0} $ satisfying (UHE) and (LHE). Assuming the property $(\widetilde{KE})_{\sigma} $ with $\sigma \gt0$ , we have

(3·14) \begin{equation}\liminf_{t\rightarrow 0}\Psi _{u}^{\sigma }(t)\leq C\liminf_{r\rightarrow0}\Phi _{u}^{\sigma }(r), \end{equation}

for all $u\in \mathcal{D}(E_{p,\infty }^{\sigma })$ .

Proof. Temporally fix $u\in \mathcal{D}(E_{p,\infty }^{\sigma })$ . Without loss of generality, assume that $\liminf_{t\rightarrow 0}\Psi _{u}^{\sigma }(t)\gt0$ . Fix $\delta \gt1$ which will be determined later in (3·17). For $t\in (0,R_{0}^{\beta^{\ast }})$ , decompose

\begin{equation*}\Psi _{u}^{\sigma }(t)=\Psi _{1}(t)+\Psi _{2}(t),\end{equation*}

where

\begin{align*}\Psi _{1}(t)& =\frac{1}{t^{{p\sigma }/{\beta ^{\ast }}}}\iint_{\{d(x,y)\leq\delta t^{1/\beta ^{\ast }}\}}|u(x)-u(y)|^{p}p_{t}(x,y)d\mu (y)d\mu (x), \\\Psi _{2}(t)& =\frac{1}{t^{{p\sigma }/{\beta ^{\ast }}}}\iint_{\{d(x,y)\gt\delta t^{1/\beta ^{\ast }}\}}|u(x)-u(y)|^{p}p_{t}(x,y)d\mu (y)d\mu (x).\end{align*}

For $\Psi _{1}(t)$ , when $d(x,y)\leq \delta t^{1/\beta ^{\ast }}$ , we have

\begin{align*}\frac{c_{1}}{V(x,t^{1/\beta ^{\ast }})}\exp \left( -c_{2}\delta ^{\frac{\beta ^{\ast }}{\beta ^{\ast }-1}}\right) &\leq \frac{c_{1}}{V(x,t^{1/\beta^{\ast }})}\exp \left( -c_{2}\left( \frac{d(x,y)}{t^{1/\beta ^{\ast }}}\right) ^{\frac{\beta ^{\ast }}{\beta ^{\ast }-1}}\right) \\& \leq p_{t}(x,y)\leq \frac{c_{3}}{V(x,t^{1/\beta ^{\ast }})}\exp \left(-c_{4}\left( \frac{d(x,y)}{t^{1/\beta ^{\ast }}}\right) ^{\frac{\beta ^{\ast}}{\beta ^{\ast }-1}}\right)\\ & \leq \frac{c_{5}}{V(x,t^{1/\beta ^{\ast }})}.\end{align*}

Therefore, noting that $\delta\gt1$ , we have from (1·3) that

(3·15) \begin{align}\Psi _{1}(t)& \leq \frac{c_{3}}{t^{p\sigma /\beta ^{\ast }}}\iint_{\{d(x,y)\leq \delta t^{1/\beta ^{\ast }}\}}\frac{|u(x)-u(y)|^{p}}{V(x,t^{1/\beta ^{\ast }})}d\mu (y)d\mu (x) \nonumber \\& \leq \frac{c_{3}C_{d}\delta ^{\alpha _{1}}}{t^{p\sigma /\beta ^{\ast }}}\iint_{\{d(x,y)\leq \delta t^{1/\beta ^{\ast }}\}}\frac{|u(x)-u(y)|^{p}}{V(x,\delta t^{1/\beta ^{\ast }})}d\mu (y)d\mu (x)=C_{1}\Phi _{u}^{\sigma}(\delta t^{1/\beta ^{\ast }}), \end{align}

where $C_{1}=c_{3}C_{d}\delta ^{p\sigma +\alpha _{1}}$ .

For $\Psi _{2}(t)$ , by Proposition 3·8, we have that

(3·16) \begin{equation}\Psi _{2}(t)\leq A\Psi _{u}^{\sigma }(ct), \end{equation}

where $A=Cc^{p\sigma /\beta ^{\ast }}\exp (\!-c^{\prime }\delta ^{\frac{\beta ^{\ast }}{\beta ^{\ast }-1}}) $ can be sufficiently small by choosing sufficiently large $\delta $ (since $C,c,c^{\prime }$ are independent constants). So, from now on, we choose sufficiently large $\delta $ such that

(3·17) \begin{equation}A=Cc^{p\sigma /\beta ^{\ast }}\exp \left( -c^{\prime }\delta ^{\frac{\beta^{\ast }}{\beta ^{\ast }-1}}\right) \lt\frac{1}{2}. \end{equation}

Combining (3·15) and (3·16), we have

(3·18) \begin{equation}\Psi _{u}^{\sigma }(t)\leq C_{1}\Phi _{u}^{\sigma }(\delta t^{1/\beta ^{\ast}})+A\Psi _{u}^{\sigma }(ct). \end{equation}

By property $(\widetilde{KE})_{\sigma}$ , there exists $t_{0}\gt0$ such that for all $t\in (0,t_{0})$ ,

\begin{equation*}\Psi _{u}^{\sigma }(t)\leq \sup_{0\lt t\lt t_{0}}\Psi _{u}^{\sigma }(t)\leq2\limsup_{t\rightarrow 0}\Psi _{u}^{\sigma }(t)\leq C_{2}\liminf_{t\rightarrow 0}\Psi _{u}^{\sigma }(t),\end{equation*}

where $C_{2}$ comes from property $(\widetilde{KE})_{\sigma}$ . Therefore, we have from (3·18) that, for all $t\in (0,\frac{t_{0}}{c})$ ,

\begin{equation*}\Psi _{u}^{\sigma }(t)\leq C_{1}\Phi _{u}^{\sigma }(\delta t^{1/\beta ^{\ast}})+AC_{2}\liminf_{t\rightarrow 0}\Psi _{u}^{\sigma }(t).\end{equation*}

Fix $\delta\gt0 $ such that $AC_{2}\lt 1$ with the aforementioned requirement $A\lt{1}/{2}$ . It follows that

\begin{equation*}\liminf_{t\rightarrow 0}\Psi _{u}^{\sigma }(t)\leq C_{1}\liminf_{t\rightarrow 0}\Phi _{u}^{\sigma}(t)+AC_{2}\liminf_{t\rightarrow 0}\Psi _{u}^{\sigma }(t),\end{equation*}

which implies

\begin{equation*}\liminf_{t\rightarrow 0}\Psi _{u}^{\sigma }(t)\leq C_{3}\liminf_{t\rightarrow 0}\Phi _{u}^{\sigma }(t),\end{equation*}

where $C_{3}=\frac{C_{1}}{1-AC_{2}}$ . The proof is complete.

Proof of Theorem 1·5. Since the two-sided estimates (UHE) and (LHE) hold, we have that $\mathcal{D}(E_{p,\infty }^{\sigma })=B_{p,\infty }^{\sigma }$ for any $\sigma \gt0$ by Lemma 3·1.

By (3·11) and (3·14), $(\widetilde{KE})_{\sigma}$ implies $(\widetilde{NE})_{\sigma}$ since for all $u\in B_{p,\infty }^{\sigma }=\mathcal{D}(E_{p,\infty }^{\sigma })$ ,

(3·19) \begin{equation}\limsup_{r\rightarrow 0}\Phi _{u}^{\sigma }(r)\leq C\limsup_{t\rightarrow0}\Psi _{u}^{\sigma }(t)\leq C^{\prime }\liminf_{t\rightarrow 0}\Psi_{u}^{\sigma }(t)\leq C^{\prime \prime }\liminf_{r\rightarrow 0}\Phi_{u}^{\sigma }(r). \end{equation}

By (3·11) and (3·13), $(\widetilde{NE})_{\sigma}$ implies $(\widetilde{KE})_{\sigma}$ since

(3·20) \begin{equation}\limsup_{t\rightarrow 0}\Psi _{u}^{\sigma }(t)\leq C_{1}\limsup_{r\rightarrow 0}\Phi _{u}^{\sigma }(r)\leq C_{1}^{\prime}\liminf_{r\rightarrow 0}\Phi _{u}^{\sigma }(r)\leq C_{1}^{\prime \prime}\liminf_{t\rightarrow 0}\Psi _{u}^{\sigma }(t). \end{equation}

Therefore, by (3·19) and (3·20), we show Theorem 1·5 (i).

By (3·14) and Proposition 3·4, $(KE)_{\sigma}$ implies $(NE)_{\sigma}$ since

(3·21) \begin{equation}\sup_{r\in (0,R_{0})}\Phi _{u}^{\sigma }(r)\leq C_{2}\sup_{t\in(0,R_{0}^{\beta ^{\ast }})}\Psi _{u}^{\sigma }(t)\leq C_{2}^{\prime}\liminf_{t\rightarrow 0}\Psi _{u}^{\sigma }(t)\leq C_{2}^{\prime \prime}\liminf_{r\rightarrow 0}\Phi _{u}^{\sigma }(r), \end{equation}

where we use the simple fact $(KE)_{\sigma}$ $\Rightarrow (\widetilde{KE})_{\sigma}$ . Similarly, by (3·13) and Proposition 3·5 adjoint with the fact $(NE)_{\sigma}$ $\Rightarrow (\widetilde{NE})_{\sigma}$ , we immediately derive $(KE)_{\sigma}$ since

(3·22) \begin{equation}\sup_{t\in (0,R_{0}^{\beta ^{\ast }})}\Psi _{u}^{\sigma }(t)\leq C_{3}\sup_{r\in (0,R_{0})}\Phi _{u}^{\sigma }(r)\leq C_{3}^{\prime}\liminf_{r\rightarrow 0}\Phi _{u}^{\sigma }(r)\leq C_{3}^{\prime \prime}\liminf_{t\rightarrow 0}\Psi _{u}^{\sigma }(t). \end{equation}

Thus, Theorem 1·5 (ii) holds by (3·21) and (3·22).

Remark 3·10. We say that a heat kernel $\{p_{t}\}_{t\gt0}$ satisfies the near diagonal lower estimate (NLE), if there exists $\delta \in (0,\infty)$ such that, for all $t\in (0,R_{0}^{\beta^{\ast}}) $ and $\mu $ -almost all $x,y\in M$ :

(3·23) \begin{equation}p_{t}(x,y)\geq \frac{c_{1}}{V(x,t^{1/\beta ^{\ast }})}\ \ \ \text{whenever }\ d(x,y)\leq \delta t^{1/\beta ^{\ast }}. \end{equation}

We conjecture that Lemma 3·1 and Lemma 3·9 cannot be derived from the weaker heat kernel estimates (UHE) and (NLE) without further assumptions (like the chain condition).

By Theorem 1·5 and Remark 1·2, we have the following interesting corollary for $p=2$ , which is important for the construction of local Dirichlet forms based on Besov norms.

4·1. Equivalent discrete semi-norms of fractal glue-ups

We first introduce connected homogeneous p.c.f. self-similar sets in $\mathbb{R}^{d}$ ( $d\geq 2$ ). Let $\{\phi _{i}\}_{i=1}^{N}$ $(N\geq 2)$ be an IFS where each $\phi _{i}\;:\;\mathbb{R}^{d}\rightarrow \mathbb{R}^{d}$ is of the form

(4·1) \begin{equation}\phi _{i}(x)=\rho (x-b_{i})+b_{i}, \end{equation}

with $\rho \in (0,1)$ , $b_{i}\in \mathbb{R}^{d}$ for $i=1,...,N$ . Let K be the attractor of the iterated function system (IFS) $\{\phi _{i}\}_{i=1}^{N}$ , namely, K is the unique non-empty compact set in $\mathbb{R}^{d}$ satisfying

\begin{equation*}K=\bigcup_{i=1}^{N}\phi _{i}(K).\end{equation*}

To explain what ‘p.c.f’ is, we introduce the natural symbolic space associated with the IFS $\{\phi _{i}\}_{i=1}^{N}$ . Let $W=\{1,2,...,N\}$ , $W_{n}$ be the set of words with length n over W, and $W^{\mathbb{N}}$ be the set of all infinite words over W. For $w=\mathrm{w_{1}w_{2}}\cdots\in W^{\mathbb{N}}$ , the canonical projection $\pi :$ $W^{\mathbb{N}}\rightarrow K$ is defined by $\pi (w)\;:\!=\;\bigcap\limits_{n\in \mathbb{N}^{\ast }}F_{\mathrm{w_{1}...w_{n}}}(K),$ where $F_{\mathrm{w_{1}...w_{n}}}\;:\!=\;F_{\mathrm{w_{1}}}\circ \cdots\circ F_{\mathrm{w_{n}}}$ . Then the critical set $\Gamma $ and the post-critical set $\mathcal{P}$ is defined by

\begin{equation*}\Gamma =\pi ^{-1}\left( \bigcup_{1\leq i\lt j\leq N}\left( \phi _{i}(K)\cap\phi _{j}(K)\right) \right) ,\quad \mathcal{P}=\bigcup_{m\geq 1}\tau^{m}(\Gamma ),\end{equation*}

where $\tau $ is the left shift by one index on $W^{\mathbb{N}}$ (see for example [ Reference Kigami36 , definition 1·3·13]). We say that the IFS $\{\phi _{i}\}_{i=1}^{N}$ is post-critically finite (p.c.f.) if $\mathcal{P}$ is finite. Define

(4·2) \begin{equation}V_{0}=\pi (\mathcal{P}),\quad V_{w}\;:\!=\;F_{w}\left( V_{0}\right), \quad V_{n}=\bigcup_{w\in W_{n}}V_{w} ,\quad V_{\ast }=\bigcup_{n\geq 1}V_{n}, \end{equation}

then K is the closure of $V_{\ast }$ with respect to the Euclidean metric. For $w\in W_n$ ,

\begin{equation*}K_{w}\;:\!=\;F_{w}\left( K\right) \end{equation*}

is called an n-cell of K. Without loss of generality, we always assume that

(4·3) \begin{equation}\textrm{diam}(K)=1\end{equation}

by an affine transformation on $\{b_{i}\}_{i=1}^{N}$ . From now on, denote by K a homogeneous p.c.f. self-similar set that is connected. It is known that a homogeneous p.c.f. IFS in the form of (4·1) satisfies the open set condition (OSC) (see for example [ Reference Deng and Lau19 , theorem 1·1]). Hence the Hausdorff dimension of K is

\begin{equation*}\alpha =\mathrm{dim}_{H}(K)=-\log N/\log \rho,\end{equation*}

and the $\alpha$ -dimensional Hausdorff measure on K, denote by $\mu $ , is $\alpha $ -regular.

It is known from [ Reference Gu and Lau27 , proposition 2·5] that Condition (H) holds for connected homogeneous p.c.f. self-similar sets. That is, there exists $c\gt0$ depending only on K such that, for any two words w and $w^{\prime }$ with the same length $m\geq 1$ , $K_{w}\cap K_{w^{\prime}}=\emptyset $ implies that $\mathrm{dist}\left( K_{w},K_{w^{\prime }}\right)\geq c\rho ^{m}$ .

In order to unify the notation for bounded and unbounded fractals, we introduce the concept of fractal glue-ups. The definition is adapted from the idea of fractafold in [ Reference Strichartz and Teplyaev48 ] although it is not the same. Given a compact set $K\subset \mathbb{R}^d$ and a set F of similitudes on $\mathbb{R}^d$ , define

\begin{equation*}K^F\;:\!=\;\bigcup_{f\in F}f(K)\subset \mathbb{R}^d.\end{equation*}

In this paper, we need the following requirements:

1. (1) isometries: all similitudes in F are isometries;

2. (2) just-touching property: for any $f\neq g\in F$ ,

   \begin{equation*}f(K)\bigcap g(K)=f(V_0)\bigcap g(V_0);\end{equation*}

3. (3) condition (H): There exists a constant $C_{H}\in (0,1)$ such that, if $|x-y|\lt C_{H}\rho ^{m}$ and $x\in f_{1}(K_{w})$ and $y\in f_{2}(K_{\tilde{w}}) $ for two words w and $\tilde{w}$ with the same length $m\geq0$ and $f_1,f_2\in F$ , then $f_{1}(K_{w})$ intersects $f_{2}(K_{\tilde{w}})$ ;

4. (4) connectedness: $K^{F}$ is connected.

From now on, whenever we mention a fractal glue-up $K^{F}$ , we automatically assume that these four conditions are satisfied. We call each $f(K)$ a tile ( $f\in F$ ). The above conditions (2) and (3) imply the following uniform finitely-joint property.

Proof. Denote all the tiles that intersect B(x,R) by $f_{n}(K),\ 1\leq n\leq N_{0} $ (here $N_{0}$ might be $\infty $ ). Denote by t the smallest integer satisfying

\begin{equation*}N^t\gt|V_0|.\end{equation*}

Choose a level-t cell $\phi_{w}(K)$ ( $|w|=t$ ) that does not intersect $V_{0}$ . Such a cell exists by the pigeonhole principle. Condition (H) implies that all $f_{n}(\phi _{w}(K))$ are separated by distance $C_{H}\rho ^{t}$ for $1\leq n\leq N_{0}$ , but they all belong to the ball $B(x,R+1)$ by (4·3). Let $C_K(R)$ be the maximal cardinality of disjoint balls of radii $C_{H}\rho ^{t}/2$ inside a ball with diameter $R+1$ in $\mathbb{R}^d$ . Since there are $N_{0}$ disjoint balls $B(f_{n}(\phi _{w}(v)),C_{H}\rho ^{t}/2)$ inside $B(x,R+1)$ , we obtain $N_{0}\leq C_K(R)$ . The second inclusion follows from choosing $R=1$ and $x\in f(K)$ .

Example 4.3. (Fractal blow-up). Let $\{\phi _{i}\}_{i=1}^{N}$ be defined as in (4·1). Define

\begin{equation*}K_{l}\;:\!=\;\underbrace{\phi _{1}^{-1}\circ \phi _{1}^{-1}\circ \cdots \circ \phi_{1}^{-1}}_{l}K=\rho^{-l}K=\bigcup _{f\in F_{l}}f(K),\end{equation*}

where $F_{l}=\left\{x+\sum_{j=1}^{l}\rho ^{-j}c_{j}\;:\; c_{j}\in \{(1-\rho)b_{i}\}_{i=1}^{N}\right\}$ .

Clearly, $F_{l}$ and $K_{l}$ are increasing sequences, so that $K=K_{0}\subseteq K_{1}\subseteq K_{2}\subseteq \cdots $ .

The final fractal blow-up of K given by

\begin{equation*}K_{\infty }=\bigcup _{l=1}^{\infty }K_{l}= \lim_{l\rightarrow\infty} K_l,\end{equation*}

is in the form $K^{F}$ with $F=\bigcup_{l=1}^{\infty }F_{l}= \lim_{l\rightarrow\infty} F_l$ .

From now on, we assume that K is a connected homogeneous p.c.f. self-similar set when we say that $K^{F}$ is a fractal glue-up. In what follows, we use the superscript F to distinguish the underlying space K and $K^{F}$ for the energies and norms etc.

Let

(4·4) \begin{equation}\mu ^{F}\;:\!=\;\sum_{f\in F}\mu \circ f^{-1}, \end{equation}

be the glue-up measure on $K^{F}$ . Then $\mu ^{F}$ is $\alpha $ -regular by our assumptions on F. Let

(4·5) \begin{equation}\lbrack u]_{B_{p,\infty }^{\sigma ,F}}^{p}\;:\!=\;\sup_{n\geq 0}\Phi _{u}^{\sigma,F}(C_H\rho ^{n}), \end{equation}

where $\Phi _{u}^{\sigma ,F}(r)\;:\!=\;r^{-(p\sigma +\alpha)}\int_{K^{F}}\int_{B(x,r)}|u(x)-u(y)|^{p}d\mu ^{F}(y)d\mu ^{F}(x)$ , and

\begin{equation*}B_{p,\infty }^{\sigma ,F}\;:\!=\;\{u\in L^{p}(K^{F},\mu ^{F})\;:\;[u]_{B_{p,\infty}^{\sigma ,F}}\lt\infty \}.\end{equation*}

It is easy to see that

(4·6) \begin{equation}\sup_{n\geq 0}\Phi _{u}^{\sigma ,F}(C_H\rho ^{n})\asymp \sup_{r\in(0,C_H)}\Phi _{u}^{\sigma ,F}(r), \end{equation}

which implies that

(4·7) \begin{equation}B_{p,\infty }^{\sigma ,F}=B_{p,\infty }^{\sigma }(K^{F})\text{ and }[u]_{B_{p,\infty }^{\sigma ,F}}\asymp \lbrack u]_{B_{p,\infty }^{\sigma}(K^{F})}(\text{with}\ R_{0}=C_H\ \text{in}\ (1{\cdot}10)). \end{equation}

Also, we have

(4·8) \begin{equation}\liminf_{n\rightarrow \infty }\Phi _{u}^{\sigma ,F}(\rho ^{n})\asymp\liminf_{r\rightarrow 0}\Phi _{u}^{\sigma ,F}(r)\text{ and }\limsup_{n\rightarrow \infty }\Phi _{u}^{\sigma ,F}(\rho ^{n})\asymp\limsup_{r\rightarrow 0}\Phi _{u}^{\sigma ,F}(r). \end{equation}

We need the following Morrey-Sobolev inequality to show that $B_{p,\infty}^{\sigma }(K^F)$ essentially embeds into $C(K^F)$ when $\sigma \gt\alpha /p$ .

Lemma 4·4 ([ Reference Baudoin11 , theorem 3·2]). Let $(M,d,\mu )$ be a metric measure space with $\mu $ satisfying (1·4). When $\sigma\gt\alpha /p$ , for any $u\in B_{p,\infty }^{\sigma }(M)$ , there exists a continuous version $\tilde{u}\in C^{(p\sigma -\alpha )/p}(M)$ satisfying $\tilde{u}=u$ $\mu $ -almost everywhere in M and

(4·9) \begin{equation}|\tilde{u}(x)-\tilde{u}(y)|\leq C|x-y|^{(p\sigma -\alpha )/p}\sup_{r\in(0,3d(x,y)]}\Phi _{u}^{\sigma }(r)\leq C|x-y|^{(p\sigma -\alpha)/p}[u]_{B_{p,\infty }^{\sigma }(M)}, \end{equation}

for all x, $y\in M$ with $d(x,y)\lt R_{0}/3$ , where C is a positive constant. Here $C^{\beta }(M)$ denotes the class of Hölder continuous functions of order $\beta $ on M.

Next, we estimate

\begin{equation*}I_{\infty ,n}^{F}(u)\;:\!=\;\int_{K^{F}}\int_{B(x,C_{H}\rho^{n})}|u(x)-u(y)|^{p}d\mu ^{F}(y)d\mu ^{F}(x).\end{equation*}

By (4·5), for all $u\in B_{p,\infty }^{\sigma ,F}$ ,

(4·10) \begin{equation}\rho ^{-n(p\sigma +\alpha )}I_{\infty ,n}^{F}(u)=C_{H}^{p\sigma +\alpha}\Phi _{u}^{\sigma ,F}(C_{H}\rho ^{n}). \end{equation}

Define

\begin{equation*}\mu _{m}\;:\!=\;\frac{1}{|V_{m}|}\sum_{a\in V_{m}}\delta _{a}\end{equation*}

and

(4·11) \begin{equation}\mu _{m}^{F}\;:\!=\;\sum_{f\in F}\mu_m\circ f^{-1}=\sum_{f\in F}\frac{1}{|V_{m}|}\sum_{a\in V_{m}}\delta _{f(a)}, \end{equation}

where $\delta _{a}$ is the Dirac measure at point a and $|A|$ denotes the cardinality of a finite set A. Let

(4·12) \begin{equation}I_{m,n}^{F}(u)\;:\!=\;\int_{K^{F}}\int_{B(x,C_{H}\rho ^{n})}|u(x)-u(y)|^{p}d\mu_{m}^{F}(y)d\mu _{m}^{F}(x). \end{equation}

We will use these discrete sums to estimate the integral $I_{\infty ,n}^{F}(u)$ .

Lemma 4·5. Let $K^{F}$ be a fractal glue-up. Then for all $u\in C(K^F)$ ,

(4·13) \begin{equation}\liminf_{m\rightarrow \infty }I_{m,n}^{F}(u)\geq I_{\infty ,n}^{F}(u).\end{equation}

The proof of this lemma is based on the proof of [ Reference Gu and Lau27 , lemma 3·2]. The key is to establish the following Proposition 4·6, which is not obvious since K is uncountable. We may assume that K is not contained in any hyperplane $H\subset \mathbb{R}^{d}$ . Otherwise, we could reduce the IFS to H by restricting each similitude $\phi_{i}$ to H and applying a reversible affine transformation from H to $\mathbb{R}^{d^{\prime }}$ , where $d^{\prime }$ is the Hausdorff dimension of H. The attractor of this reduced IFS is a scaled copy of K. Thus we can apply [ Reference Gu and Lau27 , lemma 3·2].

Proof. Let

\begin{equation*}R_{x}(a,b)=\overline{B(x,a)}\setminus B(x,b).\end{equation*}

Fix $x_{w}\in K_{w}$ for each $w\in W_n$ . Then

\begin{equation*}\bigcup_{x\in K_{w}}\partial B(x,r)\subset R_{x_{w}}(r-\rho ^{n},r+\rho^{n}),\end{equation*}

since for any $y\in \partial B(x,r)$ , $d(x,y)=r$ and $d(x,x_{w})\leq \rho^{n}$ , thus $d(y,x_{w})\in \lbrack r-\rho ^{n},r+\rho ^{n}].$ Note that

\begin{equation*}E(r)\;:\!=\;\bigcup_{x\in K}\{x\}\times \partial B(x,r)\subset \bigcup_{w\in W_{n}}K_{w}\times R_{x_{w}}(r-\rho ^{n},r+\rho ^{n}),\end{equation*}

thus

(4·14) \begin{align}\mu \times \mu (E(r))& \leq \sum_{w\in W_{n}}\mu (K_{w})\mu(R_{x_{w}}(r-\rho ^{n},r+\rho ^{n}))\nonumber \\&\leq \mu (K)\sup_{w\in W_{n}}\mu (R_{x_{w}}(r-\rho ^{n},r+\rho ^{n}))\nonumber \\&\leq \mu (K)\sup_{x\in K}\mu (R_{x}(r-\rho ^{n},r+\rho ^{n})).\end{align}

We claim that

\begin{equation*}\sup_{x\in K}\mu (R_{x}(r-\rho ^{n},r+\rho ^{n}))\rightarrow 0\text{ as }n\rightarrow \infty .\end{equation*}

Otherwise, if $\mu (R_{x_{n}}(r-\rho ^{n},r+\rho ^{n}))\geq \delta \gt0$ for a subsequence $\{x_{n}\}_n\subset K$ , we may choose a subsequence $\{x_{n_{i}}\}_i$ that converge to $x_{\infty }\in K$ due to the compactness of K, and further require that $d(x_{n_{i}},x_{\infty })$ decrease in i. Then

\begin{equation*}A_{i}\;:\!=\;R_{x_{\infty }}(r-\rho ^{n_{i}}-d(x_{n_{i}},x_{\infty }),r+\rho^{n_{i}}+d(x_{n_{i}},x_{\infty }))\supset R_{x_{n_{i}}}(r-\rho^{n_{i}},r+\rho ^{n_{i}})\end{equation*}

and $A_{i}\downarrow \partial B(x_{\infty },r)$ , thus

\begin{equation*}\mu (\partial B(x_{\infty },r))=\lim_{i\rightarrow \infty }\mu (A_{i})\geq\delta ,\end{equation*}

which contradicts [ Reference Gu and Lau27 , proposition 2·4] that $\mu(\partial B(x,r))=0$ for any $x\in K$ .

The proof is complete by letting $n\rightarrow\infty$ in (4·14).

Proof of Lemma 4·5. Let $F=\{f_{i}\}_{i\geq 1}$ , and let $F_{k}=\{f_{i}\}_{1\leq i\leq k}$ . As $\mu _{m}$ weak $\ast $ -converges to $\mu $ , we know by definition that the finite glue-up measure $\mu _{m}^{F_{k}}$ weak $\ast $ -converges to $\mu^{F_{k}}$ on $K^{F_{k}}$ (the pieces $f_{i}(K)$ are mutually measure-disjoint), and that $\mu ^{F_{k}}$ is exactly the restriction of $\mu ^{F}$ to $K^{F_{k}}$ . Using the same argument as in that of [ Reference Gu and Lau27 , lemma 3·2], for $u\in C(K^F)$ , we have

\begin{align*}\lim_{m\rightarrow \infty }I_{m,n}^{F_{k}}(u)&=\lim_{m\rightarrow \infty}\int_{K^{F_{k}}}\int_{B(x,C_{H}\rho ^{n})}|u(a)-u(b)|^{p}d\mu_{m}^{F_{k}}(b)d\mu _{m}^{F_{k}}(a) \\&=\int_{K^{F_{k}}}\int_{B(x,C_{H}\rho ^{n})}|u(a)-u(b)|^{p}d\mu^{F_{k}}(b)d\mu ^{F_{k}}(a)\;:\!=\;I_{\infty ,n}^{F_{k}}(u).\end{align*}

Note that the nonnegative sequence $I_{m,n}^{F_{k}}(u)\uparrow I_{m,n}^{F}(u)$ when $k\rightarrow \infty $ . Denote $I_{m ,n}^{F_{0}}(u)=0$ , then

(4·15) \begin{equation}I_{m,n}^{F}(u)=\sum_{k=0}^{\infty }(I_{m,n}^{F_{k+1}}(u)-I_{m,n}^{F_{k}}(u)) \end{equation}

and, by noting that

\begin{equation*}\liminf_{m\rightarrow \infty }(I_{m,n}^{F_{k+1}}(u)-I_{m,n}^{F_{k}}(u))=I_{\infty,n}^{F_{k+1}}(u)-I_{\infty ,n}^{F_{k}}(u),\end{equation*}

our conclusion directly follows from Fatou’s Lemma.

We are now in the position to prove the first estimate lemma.

Lemma 4·7. Let $K^{F}$ be a fractal glue-up. If $\sigma \gt\alpha /p$ , then for all $u\in B_{p,\infty }^{\sigma ,F}$ ,

(4·16) \begin{equation}I_{\infty ,n}^{F}(u)\leq C\rho ^{2n\alpha }\sum_{k=n}^{\infty}E_{k}^{(p),F}(u)\leq C^{\prime }\rho ^{n(p\sigma +\alpha )}\sup_{k\geq n}\mathcal{E}_{k}^{\sigma ,F}(u). \end{equation}

Proof. Let $I_{m,n}^{F}(u)$ be defined as in (4·12) ( $m\gt n)$ . Since $K^{F}$ is connected and satisfies the condition $(\mathrm{H})$ , $|x-y|\leq C_{H}\rho^{n}$ implies that x,y lie in the same or the neighbouring n-cells. For every $w\in W_{n}$ and $x\in f(K_{w})$ , if $y\in B(x,C_{H}\rho ^{n})$ , then there exists $(g,\tilde{w})\in F\times W_{n}$ such that $y\in g(K_{\tilde{w}})$ and

\begin{equation*}g(K_{\tilde{w}})\cap f(K_{w})\neq \emptyset .\end{equation*}

Therefore,

\begin{align*}I_{m,n}^{F}(u)\leq & \sum_{|w|=n}\sum_{f\in F}\int_{f(K_{w})}\left( \sum_{|\tilde{w}|=n}\sum_{g\in F}\int_{g(K_{\tilde{w}})}|u(x)-u(y)|^{p}1_{\{g(K_{\tilde{w}})\cap f(K_{w})\neq \emptyset \}}d\mu _{m}^{F}(y)\right) d\mu_{m}^{F}(x) \\=& \sum_{|w|=n}\sum_{f\in F}\sum_{|\tilde{w}|=n}\sum_{g\in F}\sum_{x\in f(K_{w}\cap V_{m})}\sum_{y\in g(K_{\tilde{w}}\cap V_{m})}\frac{1_{\{g(K_{\tilde{w}})\cap f(K_{w})\neq \emptyset \}}}{|V_{m}|^{2}}|u(x)-u(y)|^{p}.\end{align*}

When $g(K_{\tilde{w}})\cap f(K_{w})\neq \emptyset $ , we can find a common vertex of these two cells and denote it by $z\in f(V_{w})\cap g(V_{\tilde{w}})$ , due to the just-touching property when $f\neq g$ and the p.c.f. structure when $f=g$ . Furthermore, by the inequality $|u(x)-u(y)|^{p}\leq 2^{p-1}\left( |u(x)-u(z)|^{p}+|u(z)-u(y)|^{p}\right) $ ,

\begin{equation*}1_{\{g(K_{\tilde{w}})\cap f(K_{w})\neq \emptyset \}}|u(x)-u(y)|^{p}\leq2^{p-1}\sum_{z\in f(V_{w})\cap g(V_{\tilde{w}})}\left(|u(x)-u(z)|^{p}+|u(z)-u(y)|^{p}\right) .\end{equation*}

It follows that

(4·17) \begin{align*}I_{m,n}^{F}(u)&\leq 2^{p-1}\sum_{|w|=n}\sum_{f\in F}\sum_{|\tilde{w}|=n}\sum_{g\in F}\sum_{x\in f(K_{w}\cap V_{m})}\sum_{y\in g(K_{\tilde{w}}\cap V_{m})}\sum_{z\in f(V_{w})\cap g(V_{\tilde{w}})}\frac{1}{|V_{m}|^{2}}\nonumber \\& \times \left( |u(x)-u(z)|^{p}+|u(z)-u(y)|^{p}\right) \nonumber \\[5pt] & \leq C_{1}\sum_{|w|=n}\sum_{f\in F}\sum_{|\tilde{w}|=n}\sum_{g\in F}\sum_{z\in f(V_{w})\cap g(V_{\tilde{w}})}\sum_{x\in f(K_{w}\cap V_{m})}\sum_{y\in g(K_{\tilde{w}}\cap V_{m})}N^{-2m}\nonumber \\& \times \left(|u(x)-u(z)|^{p}+|u(z)-u(y)|^{p}\right) \nonumber \\[5pt] & \leq C_{1}N^{-2m}\sum_{|w|=n}\sum_{f\in F}\sum_{z\in f(V_{w})}\sum_{x\in f(K_{w}\cap V_{m})}|u(x)-u(z)|^{p}\nonumber \\& \times \left( \sum_{|\tilde{w}|=n}\sum_{g\in F}1_{\{z\in f(V_{w})\cap g(V_{\tilde{w}})\}}|g(K_{\tilde{w}}\cap V_{m})|\right) \\& +C_{1}N^{-2m}\sum_{|\tilde{w}|=n}\sum_{g\in F}\sum_{z\in g(V_{\tilde{w}})}\sum_{y\in g(K_{\tilde{w}}\cap V_{m})}|u(z)-u(y)|^{p}\nonumber \\& \times \left(\sum_{|w|=n}\sum_{f\in F}1_{\{z\in f(V_{w})\cap g(V_{\tilde{w}})\}}|f(K_{w}\cap V_{m})|\right) \nonumber \\[5pt] & \leq C_{2}N^{-(m+n)} \Bigg(\sum_{|w|=n}\sum_{f\in F}\sum_{z\in f(V_{w})}\sum_{x\in f(K_{w}\cap V_{m})}|u(x)-u(z)|^{p}\nonumber \\& \quad \qquad +\sum_{|\tilde{w}|=n}\sum_{g\in F}\sum_{z\in g(V_{\tilde{w}})}\sum_{y\in g(K_{\tilde{w}}\cap V_{m})}|u(z)-u(y)|^{p} \Bigg) \nonumber \\[5pt] & = 2C_{2}N^{-(m+n)}\sum_{|w|=n}\sum_{f\in F}\sum_{x\in f(K_{w}\cap V_{m})}\sum_{z\in f(V_{w})}|u(x)-u(z)|^{p}, \end{align*}

where we use Proposition 4·1 for each common vertex $z\in f(V_{w})\cap g(V_{\tilde{w}})$ in the last inequality, so that

\begin{equation*}\sum_{|\tilde{w}|=n}\sum_{g\in F}1_{\{z\in f(V_{w})\cap g(V_{\tilde{w}})\}}\end{equation*}

is uniformly bounded for any $(f,w)\in F\times W_{n}$ , and the estimates

(4·18) \begin{equation}|V_{m}| \asymp N^{m}=\rho ^{-\alpha m},\end{equation}

(4·19) \begin{equation}\text{ }|K_{w}\cap V_{m}| \asymp N^{m-n}\text{ for every }w\in W_{n}.\end{equation}

For $x\in f(V_{m}),z\in f(V_{n})$ , one can naturally fix a decreasing sequence of cells $K_{w_{k}}$ with $|w_{k}|=k$ for $k=n,\ldots ,m$ and vertices $x_{k}(x,z)\in V_{w_{k}}$ such that, $z=x_{n}(x,z),\ x=x_{m}(x,z)$ , so by Hölder’s inequality we have

(4·20) \begin{align}|u(z)-u(x)|^{p}\leq & \left( \sum_{k=n}^{m-1}N^{(n-k)q/p}\right)^{p/q}\left(\sum_{k=n}^{m-1}N^{k-n}|u(x_{k}(x,z))-u(x_{k+1}(x,z))|^{p}\right) \nonumber \\\leq & C_{3}\sum_{k=n}^{m-1}N^{k-n}|u(x_{k}(x,z))-u(x_{k+1}(x,z))|^{p},\end{align}

where $q=p/(p-1)$ . Note that for every pair $(a,b)\in f(K_{w^{\prime }}\cap V_{k})\times f(K_{w^{\prime }}\cap V_{k+1})$ , the cardinality of $(x,z)\in{\kern.8pt} f(K_{w}\cap V_{m})\times f(V_{w})$ with $(x_{k}(x,z),x_{k+1}(x,z))=(a,b)$ is no greater than $C^{\prime }N^{m-k}$ for some $C^{\prime }\gt0$ , due to the p.c.f. structure. Plugging (4·20) into (4·17), we obtain

(4·21) \begin{align*}I_{m,n}^{F}(u)& \leq C_{4}N^{-(m+n)}\sum_{|w|=n}\sum_{f\in F}\sum_{(x,z)\in f(K_{w}\cap V_{m})\times f(V_{w})}\sum_{k=n}^{m-1}N^{k-n}|u(x_{k}(x,z))-u(x_{k+1}(x,z))|^{p} \nonumber\\& \leq C_{4}N^{-(m+n)}\sum_{|w|=n}\sum_{k=n}^{m-1}\sum_{f\in F}\sum\limits_{\substack{|w^{\prime }|=k\\ f(K_{w^{\prime }})\subset f(K_{w})}}\sum\limits_{\substack{(x,z)\in f(K_{w}\cap V_{m})\times f(V_{w}),\\ (a,b)\in f(K_{w^{\prime }}\cap V_{k})\times f(K_{w^{\prime }}\cap V_{k+1})}}N^{k-n}|u(a)-u(b)|^{p} \\& \leq C_{5}N^{-(m+n)}\sum_{|w|=n}\sum_{k=n}^{m-1}\sum_{f\in F}\sum\limits_{\substack{|w^{\prime }|=k\\ f(K_{w^{\prime }})\subset f(K_{w})}}\sum_{a,b\in f(K_{w^{\prime }}\cap V_{k+1})}N^{k-n}\cdot N^{m-k}|u(a)-u(b)|^{p} \nonumber \\ & =C_{5}\rho ^{2n\alpha }\sum_{k=n}^{m-1}\sum_{|w|=n}\sum_{f\in F}\sum\limits_{\substack{|w^{\prime }|=k\\ f(K_{w^{\prime }})\subset f(K_{w})}}\sum_{a,b\in f(K_{w^{\prime }}\cap V_{k+1})}|u(a)-u(b)|^{p} \nonumber \\ & \leq C_{6}\rho ^{2n\alpha }\sum_{k=n}^{m-1}\sum_{f\in F}E_{k+1}^{(p)}(u\circ f)\leq C_{6}\rho ^{2n\alpha}\sum_{k=n}^{m}E_{k}^{(p),F}(u), \end{align*}

where we use (1·18) in the last line. It follows that

\begin{equation*}I_{m,n}^{F}(u)\leq C_{6}\rho ^{2n\alpha }\rho ^{n(p\sigma -\alpha)}\sum_{k=n}^{m}\rho ^{-n(p\sigma -\alpha )}E_{k}^{(p),F}(u)\leq C_{7}\rho^{n(p\sigma +\alpha )}\sup_{k\geq n}\mathcal{E}_{k}^{\sigma ,F}(u),\end{equation*}

which completes the proof by letting $m\rightarrow \infty $ and using Lemma 4·5.

The following simple proposition is required.

Proposition 4·8. For all $u\in B_{p,\infty }^{\sigma ,F}$ , we have

(4·22) \begin{equation}\liminf_{n\rightarrow \infty }\rho ^{-n(p\sigma +\alpha )}I_{\infty,n}^{F}(u) \geq \rho ^{p\sigma +\alpha }C_{H}^{p\sigma +\alpha}\liminf_{n\rightarrow \infty }\Phi _{u}^{\sigma ,F}(\rho ^{n}),\end{equation}

(4·23) \begin{equation}\limsup_{n\rightarrow \infty }\rho ^{-n(p\sigma +\alpha )}I_{\infty,n}^{F}(u) \geq \rho ^{p\sigma +\alpha }C_{H}^{p\sigma +\alpha}\limsup_{n\rightarrow \infty }\Phi _{u}^{\sigma ,F}(\rho ^{n}).\end{equation}

Proof. Since $C_{H}\in (0,1)$ , assume that $\rho ^{s+1}\leq C_{H}\lt\rho ^{s}$ for some nonnegative integer s. Then, for any integer n,

(4·24) \begin{align}& \rho ^{-n(p\sigma +\alpha )}I_{\infty ,n}^{F}(u)\nonumber \\[5pt] &\quad =\rho ^{-n(p\sigma+\alpha )}\int_{K^{F}}\int_{B(x,C_{H}\rho ^{n})}|u(x)-u(y)|^{p}d\mu^{F}(y)d\mu ^{F}(x) \nonumber \\[5pt] &\quad \geq \rho ^{-n(p\sigma +\alpha )}\int_{K^{F}}\int_{B(x,\rho^{n+s+1})}|u(x)-u(y)|^{p}d\mu ^{F}(y)d\mu ^{F}(x) \\[5pt] &\quad =\rho ^{(s+1)(p\sigma +\alpha )}\rho ^{-(n+s+1)(p\sigma +\alpha)}\int_{K^{F}}\int_{B(x,\rho ^{n+s+1})}|u(x)-u(y)|^{p}d\mu ^{F}(y)d\mu^{F}(x) \nonumber \\[5pt] &\quad \gt\rho ^{p\sigma +\alpha }C_{H}^{p\sigma +\alpha }\rho ^{-(n+s+1)(p\sigma+\alpha )}\int_{K^{F}}\int_{B(x,\rho ^{n+s+1})}|u(x)-u(y)|^{p}d\mu^{F}(y)d\mu ^{F}(x). \nonumber\end{align}

Taking $\liminf_{n\rightarrow \infty }$ and $\limsup_{n\rightarrow \infty }$ on both sides, we obtain (4·22) and (4·23).

Denote

\begin{equation*}I_{n}^{F}(u)\;:\!=\;\int_{K^{F}}\int_{\{\rho ^{n+1}\leq d(x,y)\lt\rho^{n}\}}|u(x)-u(y)|^{p}d\mu ^{F}(y)d\mu ^{F}(x).\end{equation*}

Lemma 4·9. Let $K^{F}$ be a fractal glue-up. If $\sigma \gt\alpha /p$ , then for any $\delta \in (0,p\sigma -\alpha )$ and $u\in B_{p,\infty}^{\sigma ,F}$ ,

(4·25) \begin{equation}\mathcal{E}_{n}^{\sigma ,F}(u)\leq C\sum_{k=0}^{\infty }\rho ^{(-2\alpha-\delta )k}\rho ^{-(p\sigma +\alpha )n}I_{k+n}^{F}\leq C^{\prime}\sup_{k\geq 0}\Phi _{u}^{\sigma ,F}(\rho ^{n+k}). \end{equation}

Proof. For $a,b\in f(K_{w})$ , we have $|u(a)-u(b)|^{p}\leq2^{p-1}(|u(a)-u(x)|^{p}+|u(x)-u(b)|^{p})$ , where $x\in f(K_{w})$ and $f\in F$ . Approximate F with increasing finite sets $F_{j}$ so that $F=\lim_{j\rightarrow \infty }F_{j}$ . Integrating with respect to x and dividing by $\mu ^{F}(f(K_{w}))$ , we have

(4·26) \begin{align}E_{n}^{(p),F_{j}}(u)& =\sum_{f\in F_{j}}\sum_{a,b\in f(V_{w}),|w|=n}|u(a)-u(b)|^{p} \nonumber \\& \leq 2^{p-1}\sum_{f\in F_{j}}\sum_{a,b\in f(V_{w}),a\neq b,|w|=n}\nonumber \\&\quad \times \left(\frac{1}{\mu ^{F}(f(K_{w}))}\int_{f(K_{w})}|u(a)-u(x)|^{p}+|u(x)-u(b)|^{p}d\mu ^{F}(x)\right) \nonumber \\& \leq 2^{p-1}|V_{0}|\sum_{f\in F_{j}}\sum_{a\in f(V_{w}),|w|=n}\frac{1}{\mu^{F}(f(K_{w}))}\int_{f(K_{w})}|u(a)-u(x)|^{p}d\mu ^{F}(x). \end{align}

For every $a\in f(V_{w})$ with $|w|=n$ , we can fix a decreasing sequence of cells $\{f(K_{w_{k}})\}_{k=n}^{m}$ where $|w_{k}|=k$ for all large enough $m\gt n$ , such that $a\in \cap _{k=n}^{m}f(K_{w_{k}})$ with $w_{n}=w$ . We choose $x_{k}\in f(K_{w_{k}})$ for $k=n,n+1,\ldots,m$ . Let $\delta \in (0,p\sigma-\alpha )$ . By Hölder’s inequality,

(4·27) \begin{align}& |u(a)-u(x_{n})|^{p} \nonumber \\&\quad \leq 2^{p-1}|u(a)-u(x_{m})|^{p}+2^{p-1}\left( \sum_{k=n}^{m-1}\rho^{\delta (k-n)q/p}\right) ^{p/q}\left( \sum_{k=n}^{m-1}\rho ^{\delta(n-k)}|u(x_{k})-u(x_{k+1})|^{p}\right) , \end{align}

where $q=p/(p-1)$ . Integrating (4·27) with respect to $x_{k}\in K_{w_{k}}$ and dividing by $\mu ^{F}(f(K_{w_{k}}))$ ,

\begin{align*}E_{n}^{(p),F_{j}}(u)&\leq C_{1}\sum_{f\in F_{j}}\sum_{a\in f(V_{w}),|w|=n}\frac{2^{p-1}}{\mu ^{F}(f(K_{w_{m}}))}\int_{f(K_{w_{m}})}|u(a)-u(x_{m})|^{p}d\mu ^{F}(x_{m}) \\& \quad +C_{2}\sum_{f\in F_{j}}\sum_{k=n}^{m-1}\frac{\rho ^{\delta (n-k)}}{\mu^{F}(f(K_{w_{k}}))\mu ^{F}(f(K_{w_{k+1}}))}\\& \quad \times \int_{f(K_{w_{k}})}\int_{f(K_{w_{k+1}})}|u(x_{k})-u(x_{k+1})|^{p}d\mu ^{F}(x_{k+1})d\mu^{F}(x_{k}).\end{align*}

Using Lemma 4·4, (4·7) and (4·18), the first term of the right-hand side above vanishes as $m\rightarrow \infty $ since

\begin{align*}& \sum_{a\in f(V_{w}),|w|=n}\frac{2^{p-1}}{\mu ^{F}(f(K_{w_{m}}))}\int_{f(K_{w_{m}})}|u(a)-u(x_{m})|^{p}d\mu ^{F}(x_{m}) \\&\quad \leq C|f(V_{n})|2^{p-1}\rho ^{(p\sigma -\alpha )m}[u]_{B_{p,\infty}^{\sigma }(K^{F})}^{p}\leq C^{\prime }|f(V_{n})|2^{p-1}\rho ^{(p\sigma-\alpha )m}[u]_{B_{p,\infty }^{\sigma ,F}}^{p} \\&\quad \leq C_{3}\rho ^{(p\sigma -\alpha )m-\alpha n}[u]_{B_{p,\infty }^{\sigma,F}}^{p}\rightarrow 0.\end{align*}

Letting $m\rightarrow \infty $ , we have

(4·28) \begin{align}E_{n}^{(p),F_{j}}(u)& \leq C_{4}\sum_{k=n}^{\infty }\rho ^{\delta (n-k)}\rho^{-2\alpha k}\int_{K^{F}}\int_{B(x,\rho ^{k})}|u(x)-u(y)|^{p}d\mu^{F}(y)d\mu ^{F}(x) \nonumber \\& =C_{4}\sum_{k=n}^{\infty }\rho ^{\delta (n-k)}\rho ^{-2\alpha k}\sum_{l=k}^{\infty }I_{l}^{F}(u)=C_{4}\sum_{k=0}^{\infty }\rho ^{-\delta k}\rho ^{-2\alpha (n+k)}\sum_{l=k}^{\infty }I_{l+n}^{F}(u) \nonumber \\& =C_{4}\sum_{l=0}^{\infty }\left( \sum_{k=0}^{l}\rho ^{-\delta k}\rho^{-2\alpha (n+k)}\right) I_{l+n}^{F}(u)\nonumber \\& =C_{4}\sum_{l=0}^{\infty }\left(\sum_{k=0}^{l}\rho ^{\delta (l-k)}\rho ^{2\alpha (l-k)}\right) \rho^{-\delta l}\rho ^{-2\alpha (l+n)}I_{l+n}^{F}(u) \nonumber \\& \leq C_{5}\sum_{l=0}^{\infty }\rho ^{-\delta l}\rho ^{-2\alpha(l+n)}I_{l+n}^{F}(u)=C_{5}\sum_{k=0}^{\infty }\rho ^{-\delta k}\rho^{-2\alpha (k+n)}I_{k+n}^{F}(u). \end{align}

Letting $j\rightarrow \infty $ , we have

\begin{eqnarray*}\mathcal{E}_{n}^{\sigma ,F}(u) &=&\rho ^{-(p\sigma -\alpha)n}E_{n}^{(p),F}(u)\leq C_{5}\sum_{k=0}^{\infty }\rho ^{(-2\alpha -\delta)k}\rho ^{-(p\sigma +\alpha )n}I_{k+n}^{F}(u) \\&\leq &C_{5}\sum_{k=0}^{\infty }\rho ^{(-2\alpha -\delta )k}\rho ^{-(p\sigma+\alpha )n}\int_{K^{F}}\int_{B(x,\rho ^{n+k})}|u(x)-u(y)|^{p}d\mu^{F}(y)d\mu ^{F}(x) \\&\leq &C_{5}\sum_{k=0}^{\infty }\rho ^{(p\sigma +\alpha -2\alpha -\delta)k}\sup_{k\geq 0}\Phi _{u}^{\sigma ,F}(\rho ^{n+k}) \\&=&C_{5}\left( \sum_{k=0}^{\infty }\rho ^{(p\sigma -\alpha -\delta)k}\right) \sup_{k\geq 0}\Phi _{u}^{\sigma ,F}(\rho ^{n+k})=\frac{C_{5}}{1-\rho ^{p\sigma -\alpha -\delta }}\sup_{k\geq 0}\Phi _{u}^{\sigma ,F}(\rho^{n+k}),\end{eqnarray*}

where the second inequality follows from enlarging the integral region.

We need the following proposition to manage the transitions between different levels of vertex energies.

Proposition 4·10. Let K be a connected homogeneous p.c.f. self-similar set, then there exists $C\gt0$ such that for all $u\in L^{p}(K,\mu )$ and nonnegative integer $n$ ,

\begin{equation*}E_{n}^{(p)}(u)\leq CE_{n+1}^{(p)}(u).\end{equation*}

Moreover, there exists $C\gt0$ such that for all $u\in L^{p}(K^{F},\mu ^{F})$ ,

(4·29) \begin{equation}E_{n}^{(p),F}(u)\leq CE_{n+1}^{(p),F}(u). \end{equation}

Proof. We start with the basic case $n=0$ . For any $x,y\in V_{0}$ , we can fix a vertex-disjoint path $x=z_{1}(x,y),z_{2}(x,y),\ldots ,z_{k}(x,y)=y$ such that $z_{j}(x,y)$ and $z_{j+1}(x,y)$ ( $1\leq j\leq k$ ) belong to the same $V_{w}$ for some $w\in W_{1}$ , and $2\leq k\leq |V_{1}|$ . Using Jensen’s inequality,

\begin{align*}|u(x)-u(y)|^{p}& \leq(k-1)^{p-1}\sum_{j=1}^{k-1}|u(z_{j}(x,y))-u(z_{j+1}(x,y))|^{p} \\& \leq |V_{1}|^{p-1}\sum_{j=1}^{k-1}|u(z_{j}(x,y))-u(z_{j+1}(x,y))|^{p}.\end{align*}

It follows that

\begin{align*}E_{0}^{(p)}(u)& \leq \sum_{x,y\in V_{0}}|V_{1}|^{p-1}\sum_{j=1}^{k-1}|u(z_{j}(x,y))-u(z_{j+1}(x,y))|^{p} \\& \leq |V_{0}|^{2}|V_{1}|^{p-1}\sum_{a,b\in V_{w},|w|=1}|u(a)-u(b)|^{p} \\& =|V_{0}|^{2}|V_{1}|^{p-1}E_{1}^{(p)}(u),\end{align*}

since for each pair $(a,b)\in V_{w}$ with $|w|=1$ , there exists at most one $j\in \{1,2,...k\}$ (depending on (x,y)) with $(z_{j}(x,y),z_{j+1}(x,y))=(a,b)$ for any $x,y\in V_{0}$ . Thus, we show the desired for $n=0$ with $C\;:\!=\;|V_{0}|^{2}|V_{1}|^{p-1}$ for any u. The rest simply follows from that

\begin{align*}E_{n}^{(p)}(u)& =\sum_{x,y\in V_{w},|w|=n}|u(x)-u(y)|^{p}=\sum_{|w|=n}E_{0}^{(p)}(u\circ \phi _{w}) \\& \leq C\sum_{|w|=n}E_{1}^{(p)}(u\circ \phi _{w})=C\sum_{a,b\in V_{wi},|w|=n,|i|=1}|u(a)-u(b)|^{p}=CE_{n+1}^{(p)}(u).\end{align*}

The second inclusion follows from the definition of $K^{F}$ .

In [ Reference Gao, Yu and Zhang23 , theorem 1·4], for a homogeneous p.c.f. self-similar set K, we established the equivalence

(4·30) \begin{equation}\sup\limits_{n\geq 0}\rho ^{-n(p\sigma -\alpha )}E_{n}^{(p)}(u)\asymp\lbrack u]_{B_{p,\infty }^{\sigma }(K)}^{p}\text{ (}p\sigma \gt\alpha \text{)}, \end{equation}

where $E_{n}^{(p)}(u)$ is from (1·17).

Using Lemma 4·7 and Lemma 4·9, we immediately derive such equivalence for $K^{F}$ . Define the discrete local p-energy for $K^{F}$ by

\begin{equation*}\mathcal{E}_{p,\infty }^{\sigma ,F}(u)\;:\!=\;\sup\limits_{n\geq 0}\rho^{-n(p\sigma -\alpha )}E_{n}^{(p),F}(u)=\sup_{n\geq 0}\mathcal{E}_{n}^{\sigma ,F}(u).\end{equation*}

Corollary 4·11. Let $K^{F}$ be a fractal glue-up. If $\sigma\gt\alpha /p$ , then for all $u\in B_{p,\infty }^{\sigma ,F}$ ,

(4·31) \begin{equation}\limsup_{n\rightarrow \infty }\mathcal{E}_{n}^{\sigma ,F}(u)\asymp\limsup_{n\rightarrow \infty }\Phi _{u}^{\sigma ,F}(\rho ^{n}),\ \mathcal{E}_{p,\infty }^{\sigma ,F}(u)\asymp \lbrack u]_{B_{p,\infty}^{\sigma ,F}}^{p}.\end{equation}

Proof. Taking limsup and sup (of n) on both sides of (4·16) and (4·25), we have

(4·32) \begin{equation}\limsup_{n\rightarrow \infty }\rho ^{-n(p\sigma +\alpha )}I_{\infty,n}^{F}(u) \leq C\limsup_{n\rightarrow \infty }\mathcal{E}_{n}^{\sigma,F}(u)\leq C^{\prime }\limsup_{n\rightarrow \infty }\Phi _{u}^{\sigma,F}(\rho ^{n}), \end{equation}

(4·33) \begin{equation}\sup\limits_{n\geq 0}\rho ^{-n(p\sigma +\alpha )}I_{\infty ,n}^{F}(u) \leq C\sup\limits_{n\geq 0}\mathcal{E}_{n}^{\sigma ,F}(u). \end{equation}

By (4·23), we have

\begin{equation*}\limsup_{n\rightarrow \infty }\rho ^{-n(p\sigma +\alpha )}I_{\infty,n}^{F}(u)\geq \rho ^{p\sigma +\alpha }C_{H}^{p\sigma +\alpha}\limsup_{n\rightarrow \infty }\Phi _{u}^{\sigma ,F}(\rho ^{n}),\end{equation*}

thus showing

\begin{equation*}\limsup_{n\rightarrow \infty }\mathcal{E}_{n}^{\sigma ,F}(u)\asymp\limsup_{n\rightarrow \infty }\Phi _{u}^{\sigma ,F}(\rho ^{n})\end{equation*}

by (4·32). On the other hand, we have from (4·10) and (4·5) that

(4·34) \begin{equation}\sup\limits_{n\geq 0}\rho ^{-n(p\sigma +\alpha )}I_{\infty,n}^{F}(u)=C_{H}^{p\sigma +\alpha }\sup\limits_{n\geq 0}\Phi _{u}^{\sigma,F}(C_{H}\rho ^{n})=C_{H}^{p\sigma +\alpha }[u]_{B_{p,\infty }^{\sigma,F}}^{p}. \end{equation}

Since $C_{H}\in (0,1)$ , assume that $\rho ^{s+1}\leq C_{H}\lt\rho ^{s}$ for some nonnegative integer s. Replacing n by $n+s+1$ in (4·25), we obtain

(4·35) \begin{equation}\mathcal{E}_{n+s+1}^{\sigma ,F}(u)\leq C\sup_{k\geq 0}\Phi _{u}^{\sigma,F}(\rho ^{n+s+1+k})\leq C^{\prime }\sup_{k\geq 0}\Phi _{u}^{\sigma,F}(C_{H}\rho ^{n+k}). \end{equation}

By (4·29),

\begin{align*}\mathcal{E}_{n+s+1}^{\sigma ,F}(u)&=\rho ^{-(n+s+1)(p\sigma -\alpha)}E_{n+s+1}^{(p),F}(u)\geq C^{-s-1}\rho ^{-(n+s+1)(p\sigma -\alpha)}E_{n}^{(p),F}(u)\\&=C^{-s-1}\rho ^{-(s+1)(p\sigma -\alpha )}\mathcal{E}_{n}^{\sigma ,F}(u),\end{align*}

thus by (4·35),

(4·36) \begin{equation}\mathcal{E}_{n}^{\sigma ,F}(u)\leq C^{\prime \prime }C_{H}^{p\sigma -\alpha}\sup_{k\geq 0}\Phi _{u}^{\sigma ,F}(C_{H}\rho ^{n+k}). \end{equation}

We conclude from (4·33), (4·34) and (4·36) that

\begin{equation*}\mathcal{E}_{p,\infty }^{\sigma ,F}(u)\asymp \lbrack u]_{B_{p,\infty}^{\sigma ,F}}^{p}.\end{equation*}

The proof is complete.

4·2. Property (E) holds for nested fractals

For a set V, let $l (V)=\{u\;:$ u maps V into $\mathbb{R}\}$ .

We know from [ Reference Gao, Yu and Zhang23 , remark 3·2] that, if property (E) holds with $\sigma$ , then $\sigma=\sigma _{p}^{*}(K)$ , and for all $n\geq m$ , $\mathcal{E}_{m}^{\sigma_{p}^{*}}(u)\leq C\mathcal{E}_{n}^{\sigma _{p}^{*}}(u)$ . Thus we omit the index $\sigma$ . The following proposition shows the importance of property (E).

Proposition 4·13. If a connected homogeneous p.c.f. self-similar set K satisfies property (E), then

(4·37) \begin{equation}\sigma _{p}^{\ast }(K)=\sigma _{p}^{\#}(K)=\sigma _{p}^{\#}(K^{F}).\end{equation}

Moreover, $K^{F}$ satisfies property $(VE)_{\sigma _{p}^{\#}(K)}$ , and $B_{p,\infty }^{\sigma _{p}^{\#}(K),F}$ contains non-constant functions.

For the reverse inclusion $\sigma _{p}^{\#}(K)\leq \sigma _{p}^{\#}(K^{F})$ to hold, we use property (E)(ii) to find a non-constant function in $B_{p,\infty }^{\sigma _{p}^{\#}(K),F}$ . To see this, we pick a contraction $f\in F$ , and determine the values of u on $f(V_{1})$ by $u(f(V_{0}))=0$ while $u(f(V_{1}\setminus V_{0}))=1$ . We can then extend u from $f(V_{1})$ to f(K) and make sure that $u\circ f\in B_{p,\infty}^{\sigma _{p}^{\#}(K)}$ by property (E)(ii), and simply define $u=0$ outside of f(K) on $K^{F}$ . Then u is an extended non-trivial function in $B_{p,\infty }^{\sigma _{p}^{\#}(K),F}$ since by (4·31) and (4·30),

\begin{equation*}\lbrack u]_{B_{p,\infty }^{\sigma _{p}^{\#}(K),F}}^{p}\asymp \mathcal{E}_{p,\infty }^{\sigma _{p}^{\#}(K),F}(u)=\mathcal{E}_{p,\infty }^{\sigma_{p}^{\#}(K)}(u\circ f)\asymp \lbrack u\circ f]_{B_{p,\infty }^{\sigma_{p}^{\#}(K)}(K)}^{p}\lt\infty.\end{equation*}

Thus we show (4·37) and that $B_{p,\infty }^{\sigma _{p}^{\#}(K),F}$ contains non-constant functions.

Finally, since K satisfies property (E), that is for all n,

\begin{equation*}\mathcal{E}_{n}^{\sigma _{p}^{\#}}(u)\leq C\liminf_{k\rightarrow \infty }\mathcal{E}_{k}^{\sigma _{p}^{\#}}(u),\end{equation*}

summing over F gives that for all n,

\begin{equation*}\mathcal{E}_{n}^{\sigma _{p}^{\#},F}(u)\leq C\sum_{f\in F}\liminf_{k\rightarrow \infty }\mathcal{E}_{k}^{\sigma _{p}^{\#}}(u\circ f)\leq C\liminf_{k\rightarrow \infty }\sum_{f\in F}\mathcal{E}_{k}^{\sigma_{p}^{\#}}(u\circ f)=C\liminf_{k\rightarrow \infty }\mathcal{E}_{k}^{\sigma_{p}^{\#},F}(u),\end{equation*}

where we use Fatou’s lemma to obtain the desired.

When property (E) fails, $\sigma _{p}^{\#}=\sigma _{p}^{\ast }$ does not always hold for connected homogeneous p.c.f. self-similar sets (see [ Reference Gu and Lau28 ]). Moreover, we mention in passing that, the critical domain $B_{p,\infty }^{\sigma _{p}^{\ast }}$ can be trivial for some metric measure spaces (bounded or unbounded).

We present the following key lemma for nested fractals, a class of connected homogeneous p.c.f. self-similar sets defined in [ Reference Kumagai38 ] satisfying conditions (A-0 $)\sim (A $ -3) and $|V_{0}|\geq 2$ therein.

Proof. In this proof, we use the same terminology and notions as in [ Reference Cao, Gu and Qiu16 ]. Let K be a nested fractal. We fix the components $r_{i}$ of $\mathbf{r}$ in [ Reference Cao, Gu and Qiu16 , theorem 6·3] to be the same number r, so it is ‘ $\mathscr{G}$ -symmetric’ (see [ Reference Cao, Gu and Qiu16 , section 3] for definition). Then [ Reference Cao, Gu and Qiu16 , theorem 6·3] states that, ‘condition $\left( \mathbf{A}\right) $ ’ (see [ Reference Cao, Gu and Qiu16 , the beginning of Section 4] for definition) holds for affine nested fractals, including K. By multiplying $\mathbf{r}$ with a constant (still denoted by $\mathbf{r}$ ), [ Reference Cao, Gu and Qiu16 , theorem 4·2] guarantees that

(4·38) \begin{equation}\mathcal{T}E=E \end{equation}

(see [ Reference Cao, Gu and Qiu16 , definition 2·8, definition 3·1] for related definitions), thus showing ‘condition $\left( \mathbf{A}^{\prime }\right) $ ’ (see [ Reference Cao, Gu and Qiu16 , the beginning of Section 5]). So by [ Reference Cao, Gu and Qiu16 , lemma 5·4], we can fix

(4·39) \begin{equation}r\lt 1 \end{equation}

to further satisfy condition $\left( \mathbf{A}^{\prime }\right) $ on K.

The equivalence of E and $E_{0}^{(p)}$ (for functions $u\in l(V_{0})$ , see the statement of [ Reference Cao, Gu and Qiu16 , theorem 5·1]) is stated in the proof of [ Reference Cao, Gu and Qiu16 , proposition 5·3 (b)], which implies that

(4·40) \begin{equation}\Lambda ^{n}E(u)\asymp \Lambda ^{n}E_{0}^{(p)}(u)=r^{-n}E_{n}^{(p)}(u)\end{equation}

for all $n\geq 0$ and all functions $u\in l(V_{n})$ by the definition of $\Lambda $ in [ Reference Cao, Gu and Qiu16 , definition 3·1] and $E_{n}^{(p)}$ given in (1·17).

Now fix

(4·41) \begin{equation}\sigma =\frac{\log _{\rho }r+\alpha }{p}, \end{equation}

so that $r=\rho ^{p\sigma -\alpha }$ .

Property (E)(i) holds with this $\sigma $ , by using the monotonicity property in [ Reference Cao, Gu and Qiu16 , proposition 5·3 (a)] and (4·40) that

(4·42) \begin{equation}E_{0}^{(p)}(u)\asymp E(u)\leq \Lambda E(u)\leq \cdots \leq \Lambda^{n}E(u)\asymp r^{-n}E_{n}^{(p)}(u)=\rho ^{-n(p\sigma -\alpha)}E_{n}^{(p)}(u)=\mathcal{E}_{n}^{\sigma}(u). \end{equation}

To see Property (E)(ii), we show the existence of piecewise harmonic functions (see definition in [ Reference Cao, Gu and Qiu16 , section 5·1]) using a standard iterative construction as in [ Reference Herman, Peirone and Strichartz31 ]. The iteration part is that, for any boundary value of a cell $u_{w}\in l(f_{w}(V_{0}))$ , by (4·38), we can take its next-level harmonic extension $\tilde{u}_{w}\in l(f_{w}(V_{1}))$ such that $\Lambda E(\tilde{u}_{w})=E(u_{w})$ and $\tilde{u}_{w}|_{f_{w}(V_{0})}=u_{w}$ . The iteration starts from $V_{0}$ to $V_{1}$ , and from each level-1 cell to level-2 cells in the above way, then finally to $V_{\ast }$ . Such an extension satisfies the definition of piecewise harmonic functions in [ Reference Cao, Gu and Qiu16 , section 5·1], so we know by [ Reference Cao, Gu and Qiu16 , section 5·2] that it embeds into C(K) and this gives an extension to K instead of $V_{\ast }$ .

Therefore, we conclude from above that K satisfies property (E) with $\sigma $ defined as in (4·41). By (4·37), we have $\sigma =\sigma_{p}^{\#}$ , thus

(4·43) \begin{equation}r=\rho ^{p\sigma -\alpha }=\rho ^{p\sigma _{p}^{\#}-\alpha }. \end{equation}

The inclusion $\sigma _{p}^{\#}\gt\alpha /p$ follows from (4·39).

Remark 4·15. This lemma also indicates that, $E_{p,\infty }^{\sigma _{p}^{\#}}(u)$ , $[u]_{B_{p,\infty}^{\sigma _{p}^{\#}}}^p$ and the homogeneous discrete p-energy in [ Reference Cao, Gu and Qiu16 , theorem 5·1] are equivalent for a nested fractal K. It is known from [ Reference Barlow4 , theorem 8·18] that K admits a heat kernel satisfying (1·9), so (3·1) holds by Lemma 3·1. The critical exponents $\sigma_{p}^{\#}(K)=\sigma _{p}^{\ast }(K)$ due to property (E) by Proposition 4·13. Using $\Lambda ^{n}E_{0}^{(p)}(u)\asymp \Lambda^{n}E(u) $ in (4·40), the p-energy in [ Reference Cao, Gu and Qiu16 , theorem 5·1] with $r_{i}=r$ is equivalent to $\lim_{n\rightarrow \infty}\Lambda ^{n}E(u)$ (which exists by [ Reference Cao, Gu and Qiu16 , proposition 5·3]), where $r=\rho ^{p\sigma _{p}^{\#}(K)-\alpha }$ is given by (4·43). By (4·30) and (4·42),

\begin{equation*}[u]_{B_{p,\infty }^{\sigma _{p}^{\#}}}^{p}\asymp \lim_{n\rightarrow \infty}\Lambda ^{n}E(u).\end{equation*}

4·3. Equivalence between $(VE)_{\sigma}$ and $(NE)_{\sigma}$

Our goal is to show the equivalence between $(VE)_{\sigma}$ and $(NE)_{\sigma}$ for $K^{F}$ . It is quite different and more delicate to obtain the lower limit equivalence, compared with the equivalence in Corollary 4·11 using the upper limit.

Lemma 4·16. If a fractal glue-up $K^{F}$ satisfies $(\widetilde{VE})_\sigma$ with $\sigma \gt\alpha /p$ , then there exists $C\gt0$ such that for all $u\in B_{p,\infty }^{\sigma ,F}$ ,

(4·44) \begin{equation}\liminf_{n\rightarrow \infty }\mathcal{E}_{n}^{\sigma ,F}(u)\leq C\liminf_{n\rightarrow \infty }\Phi _{u}^{\sigma ,F}(\rho ^{n}).\end{equation}

Proof. Since $C_{H}\in (0,1)$ , we can assume that $\rho ^{s+1}\leq C_{H}\lt\rho ^{s}$ for some $s\in \mathbb{N}$ . For all $n\gt0$ , we have by (4·24) that

(4·45) \begin{equation}I_{\infty ,n}^{F}(u)\geq \int_{K^{F}}\int_{B(x,\rho^{n+s+1})}|u(x)-u(y)|^{p}d\mu ^{F}(y)d\mu ^{F}(x)\geq I_{n+s+1}^{F}(u).\end{equation}

Note that there exists a positive integer $n_{0}$ such that

\begin{equation*}\sup_{k\geq n_{0}}\mathcal{E}_{k}^{\sigma ,F}(u)\leq 2\limsup_{n\rightarrow\infty }\mathcal{E}_{n}^{\sigma ,F}(u).\end{equation*}

By (4·45), we have for all $n\geq n_{0}+s+1$ that

(4·46) \begin{eqnarray}I_{n}^{F}(u) &\leq &I_{\infty ,n-s-1}^{F}(u) \nonumber \\&\leq &C\rho ^{(n-s-1)(p\sigma +\alpha )}\sup_{k\geq n-s-1}\mathcal{E}_{k}^{\sigma ,F}(u)\text{ (using \eqref{eq6.3})} \nonumber \\&\leq &CC_{H}^{-(p\sigma +\alpha )}\rho ^{n(p\sigma +\alpha )}\sup_{k\geq n_{0}}\mathcal{E}_{k}^{\sigma ,F}(u)\leq 2CC_{H}^{-(p\sigma +\alpha )}\rho^{n(p\sigma +\alpha )}\limsup_{n\rightarrow \infty }\mathcal{E}_{n}^{\sigma,F}(u) \nonumber \\&\leq &C_{1}\rho ^{n(p\sigma +\alpha )}\liminf_{n\rightarrow \infty }\mathcal{E}_{n}^{\sigma ,F}(u)\text{ (by property }(\widetilde{VE})_\sigma\text{)}. \end{eqnarray}

By Lemma 4·9, for any positive integer $L\geq n_{0}+s+1 $ and $0\lt\delta \lt p\sigma -\alpha $ ,

\begin{align*}\mathcal{E}_{n}^{\sigma ,F}(u)& \leq C_{2}\sum_{k=0}^{\infty }\rho^{(-2\alpha -\delta )k}\rho ^{-(p\sigma +\alpha )n}I_{k+n}^{F}(u) \\& = C_{2}\sum_{k=0}^{L}\rho ^{(-2\alpha -\delta )k}\rho ^{-(p\sigma +\alpha)n}I_{k+n}^{F}(u)+C_{2}\sum_{k=L}^{\infty }\rho ^{(-2\alpha -\delta )k}\rho^{-(p\sigma +\alpha )n}I_{k+n}^{F}(u) \\& \leq C_{2}\sum_{k=0}^{L}\rho ^{(-2\alpha -\delta )k}\rho ^{-(p\sigma+\alpha )n}I_{k+n}^{F}(u)+C_{1}C_{2}\sum_{k=L}^{\infty }\rho ^{(p\sigma-\alpha -\delta )k}\liminf_{n\rightarrow \infty }\mathcal{E}_{n}^{\sigma,F}(u); \\& =C_{2}\left( \sum_{k=0}^{L}\rho ^{(-2\alpha -\delta )k}\rho ^{-(p\sigma+\alpha )n}I_{k+n}^{F}(u)+\frac{C_{1}\rho ^{(p\sigma -\alpha -\delta )L}}{1-\rho ^{(p\sigma -\alpha -\delta )}}\liminf_{n\rightarrow \infty }\mathcal{E}_{n}^{\sigma ,F}(u)\right),\end{align*}

where we use (4·46) in the third line. Taking $\liminf_{n\rightarrow\infty }$ in the right-hand side above, we have

\begin{equation*}C_{3}\liminf_{n\rightarrow \infty }\mathcal{E}_{n}^{\sigma ,F}(u)\leq\liminf_{n\rightarrow \infty }\sum_{k=0}^{L}\rho ^{-(2\alpha +\delta )k}\rho^{-(p\sigma +\alpha )n}I_{k+n}^{F}(u),\end{equation*}

where $C_{3}\;:\!=\;{1}/{C_{2}}-{C_{1}\rho ^{(p\sigma -\alpha -\delta )L}}/({1-\rho ^{(p\sigma -\alpha -\delta )}})$ .

Fix a large integer L such that $C_{3}\gt0$ , then

\begin{align*}C_{3}\rho ^{(2\alpha +\delta )L}\liminf_{n\rightarrow \infty }\mathcal{E}_{n}^{\sigma ,F}(u)& \leq \rho ^{(2\alpha +\delta )L}\liminf_{n\rightarrow\infty }\left( \sum_{k=0}^{L}\rho ^{-n(p\sigma +\alpha )}\rho ^{-(2\alpha+\delta )k}I_{n+k}^{F}(u)\right) \\& \leq \liminf_{n\rightarrow \infty }\rho ^{-n(p\sigma +\alpha)}\sum_{k=0}^{L}I_{n+k}^{F}(u)\leq \liminf_{n\rightarrow \infty }\Phi_{u}^{\sigma ,F}(\rho ^{n}),\end{align*}

thus showing (4·44).

Next, we show the reverse inequality of (4·44) by using (4·29) under property $(\widetilde{NE})_{\sigma}$ .

Lemma 4·17. If property $(\widetilde{NE})_{\sigma}$ holds for a fractal glue-up $K^{F}$ with $\sigma \gt\alpha /p$ , then there exists $C\gt0$ such that for all $u\in B_{p,\infty }^{\sigma ,F}$ ,

(4·47) \begin{equation}\liminf_{n\rightarrow \infty }\Phi _{u}^{\sigma ,F}(\rho ^{n})\leq C\liminf_{n\rightarrow \infty }\mathcal{E}_{n}^{\sigma ,F}(u).\end{equation}

Proof. By (4·21), we have

(4·48) \begin{align}\rho ^{-n(p\sigma +\alpha )}I_{m,n}^{F}(u)&\leq C\rho ^{-n(p\sigma -\alpha)}\sum_{k=n}^{m}\rho ^{k(p\sigma -\alpha )}\rho ^{-k(p\sigma -\alpha)}E_{k}^{(p),F}(u) \nonumber \\&\leq C\sum_{k=n}^{m}\rho ^{(k-n)(p\sigma -\alpha )}\mathcal{E}_{k}^{\sigma,F}(u)\leq C\sum_{k=0}^{\infty }\rho ^{k(p\sigma -\alpha )}\mathcal{E}_{k+n}^{\sigma ,F}(u). \end{align}

Note that there exists $n_{0}\gt0$ such that for all positive integer $L\gt n_{0}$ ,

\begin{eqnarray*}\sup_{n\geq L}\mathcal{E}_{n}^{\sigma ,F}(u) &\leq &2\limsup_{n\rightarrow\infty }\mathcal{E}_{n}^{\sigma ,F}(u) \\&\asymp &\limsup_{n\rightarrow \infty }\Phi _{u}^{\sigma ,F}(\rho ^{n})\text{ (by Corollary 4.11)} \\&\leq &C\liminf_{n\rightarrow \infty }\Phi _{u}^{\sigma ,F}(\rho ^{n})\text{ (by property }(\widetilde{NE})_{\sigma}\text{)}.\end{eqnarray*}

It follows from (4·48) that

\begin{eqnarray*}\rho ^{-n(p\sigma +\alpha )}I_{m,n}^{F}(u) &\leq &C\sum_{k=0}^{\infty }\rho^{k(p\sigma -\alpha )}\mathcal{E}_{k+n}^{\sigma ,F}(u) \\&=&C\sum_{k=0}^{L}\rho ^{k(p\sigma -\alpha )}\mathcal{E}_{k+n}^{\sigma,F}(u)+C\sum_{k=L+1}^{\infty }\rho ^{k(p\sigma -\alpha )}\mathcal{E}_{k+n}^{\sigma ,F}(u) \\&\leq &C\sum_{k=0}^{L}\rho ^{k(p\sigma -\alpha )}\mathcal{E}_{k+n}^{\sigma,F}(u)+C\sum_{k=L+1}^{\infty }\rho ^{k(p\sigma -\alpha )}\sup_{n\geq L}\mathcal{E}_{n}^{\sigma ,F}(u) \end{eqnarray*}

\begin{eqnarray*}&\leq &C\sum_{k=0}^{L}\rho ^{k(p\sigma -\alpha )}\mathcal{E}_{k+n}^{\sigma,F}(u)+C^{\prime }\sum_{k=L+1}^{\infty }\rho ^{k(p\sigma -\alpha)}\liminf_{n\rightarrow \infty }\Phi _{u}^{\sigma ,F}(\rho ^{n}) \\&=&C\sum_{k=0}^{L}\rho ^{k(p\sigma -\alpha )}\mathcal{E}_{k+n}^{\sigma,F}(u)+C^{\prime }\frac{\rho ^{(p\sigma -\alpha )(L+1)}}{1-\rho ^{p\sigma-\alpha }}\liminf_{n\rightarrow \infty }\Phi _{u}^{\sigma ,F}(\rho ^{n}).\end{eqnarray*}

Using Lemma 4·5, we have

\begin{align*}\rho ^{-n(p\sigma +\alpha )}I_{\infty ,n}^{F}(u)\leq & \rho ^{-n(p\sigma+\alpha )}\liminf_{m\rightarrow \infty }I_{m,n}^{F}(u) \\\leq & C\sum_{k=0}^{L}\rho ^{k(p\sigma -\alpha )}\mathcal{E}_{k+n}^{\sigma,F}(u)+\frac{C^{\prime }\rho ^{(p\sigma -\alpha )(L+1)}}{1-\rho ^{p\sigma-\alpha }}\liminf_{n\rightarrow \infty }\Phi _{u}^{\sigma ,F}(\rho ^{n}).\end{align*}

Combining this with (4·22), we have

\begin{align*}\rho ^{p\sigma +\alpha }C_{H}^{p\sigma +\alpha }\liminf_{n\rightarrow \infty}\Phi _{u}^{\sigma ,F}(\rho ^{n})&\leq C\liminf_{n\rightarrow \infty}\sum_{k=0}^{L}\rho ^{k(p\sigma -\alpha )}\mathcal{E}_{k+n}^{\sigma ,F}(u)\\&+\frac{C^{\prime }\rho ^{(p\sigma -\alpha )(L+1)}}{1-\rho ^{p\sigma -\alpha }}\liminf_{n\rightarrow \infty }\Phi _{u}^{\sigma ,F}(\rho ^{n}).\end{align*}

For sufficiently large L ( $\gt n_{0}$ ), we have

\begin{equation*}C^{\prime \prime }\;:\!=\;\frac{1}{C}\left( \rho ^{p\sigma +\alpha }C_{H}^{p\sigma+\alpha }-\frac{C^{\prime }\rho ^{(p\sigma -\alpha )(L+1)}}{1-\rho ^{p\sigma-\alpha }}\right) \gt0.\end{equation*}

Then for this L, we obtain

(4·49) \begin{align}C^{\prime \prime }\liminf_{n\rightarrow \infty }\Phi _{u}^{\sigma ,F}(\rho^{n})& \leq \liminf_{n\rightarrow \infty }\sum_{k=0}^{L}\rho ^{k(p\sigma-\alpha )}\mathcal{E}_{k+n}^{\sigma ,F}(u) \nonumber \\& =\liminf_{n\rightarrow \infty }\sum_{k=0}^{L}\rho ^{k(p\sigma -\alpha)}\rho ^{-(k+n)(p\sigma -\alpha )}E_{k+n}^{(p),F}(u) \nonumber \\& =\liminf_{n\rightarrow \infty }\sum_{k=0}^{L}\rho ^{-n(p\sigma -\alpha)}E_{k+n}^{(p),F}(u). \end{align}

Using (4·29), there exists $c\gt0$ such that for all positive integer n and $u\in B_{p,\infty }^{\sigma ,F}$ ,

\begin{equation*}E_{n}^{(p),F}(u)\leq cE_{n+1}^{(p),F}(u).\end{equation*}

Applying the above inequality to (4·49), we have

\begin{align*}C^{\prime \prime }\liminf_{n\rightarrow \infty }\Phi _{u}^{\sigma ,F}(\rho^{n})& \leq \liminf_{n\rightarrow \infty }C_{L}\ \rho ^{-(n+L)(p\sigma-\alpha )}E_{n+L}^{(p),F}(u) \\& =C_{L}\liminf_{n\rightarrow \infty }\mathcal{E}_{n+L}^{(p),F}(u)=C_{L}\liminf_{n\rightarrow \infty }\mathcal{E}_{n}^{(p),F}(u),\end{align*}

where $C_{L}=(({1-c^{L}})/({1-c}))\rho ^{L(p\sigma -\alpha )}$ does not depend on n, which gives (4·47).

Proof of Theorem 1·7. Let $\sigma \gt\alpha /p$ . By (4·31), (4·44) and (4·8), $(\widetilde{VE})_{\sigma}$ implies $(\widetilde{NE})_{\sigma}$ since

(4·50) \begin{equation}\limsup_{r\rightarrow 0}\Phi _{u}^{\sigma ,F}(r) \leq C\limsup_{n\rightarrow\infty }\mathcal{E}_{n}^{\sigma ,F}(u)\leq C^{\prime }\liminf_{n\rightarrow\infty }\mathcal{E}_{n}^{\sigma ,F}(u)\leq C^{\prime \prime}\liminf_{r\rightarrow 0}\Phi _{u}^{\sigma ,F}(r). \end{equation}

By (4·31), (4·8) and (4·47), $(\widetilde{NE})_{\sigma}$ implies $(\widetilde{VE})_{\sigma}$ since

(4·51) \begin{equation}\limsup_{n\rightarrow \infty }\mathcal{E}_{n}^{\sigma ,F}(u)\leq C_{1}\limsup_{r\rightarrow 0}\Phi _{u}^{\sigma ,F}(r)\leq C_{1}^{\prime}\liminf_{r\rightarrow 0}\Phi _{u}^{\sigma ,F}(r)\leq C_{1}^{\prime \prime}\liminf_{n\rightarrow \infty }\mathcal{E}_{n}^{\sigma ,F}(u).\end{equation}

Thus, by (4·50) and (4·51), we show Theorem 1·7 (i).

By (4·31) and (4·44) adjoint with $(VE)_{\sigma}\Rightarrow (\widetilde{VE})_{\sigma}$ , we immediately have from (4·6), (4·8) that

(4·52) \begin{equation}\sup_{r\in (0,C_H)}\Phi _{u}^{\sigma ,F}(r)\leq C_{2}\sup_{n\geq 0 }\mathcal{E}_{n}^{\sigma ,F}(u)\leq C_{2}^{\prime }\liminf_{n\rightarrow \infty }\mathcal{E}_{n}^{\sigma ,F}(u)\leq C_{2}^{\prime \prime}\liminf_{r\rightarrow 0}\Phi _{u}^{\sigma ,F}(r). \end{equation}

Similarly, by (4·31) and (4·47) adjoint with the fact $(NE)_{\sigma}\Rightarrow (\widetilde{NE})_{\sigma}$ , we immediately have

(4·53) \begin{equation}\sup_{n\geq 0}\mathcal{E}_{n}^{\sigma ,F}(u)\leq C_{3}\sup_{n\geq 0}\Phi_{u}^{\sigma ,F}(C_H\rho ^{n})\leq C_{3}^{\prime }\liminf_{n\rightarrow\infty }\Phi _{u}^{\sigma ,F}(\rho ^{n})\leq C_{3}^{\prime \prime}\liminf_{n\rightarrow \infty }\mathcal{E}_{n}^{\sigma ,F}(u).\end{equation}

Thus, by (4·52) and (4·53), we show Theorem 1·7 (ii).

4·4. Consequences of $(VE)_{\sigma}$ : BBM type characterisation and Gagliardo-Nirenberg inequality

Let us state the following embedding inequality given by Baudoin, using our notions.

Lemma 4·19 ([ Reference Baudoin11 , theorem 4·3]). Suppose that property $(NE)_{\sigma_{p}^{\#}}$ holds with $R_0=\infty$ for the metric measure space $(M,d,\mu )$ , and there exists $R\gt0$ such that $\inf_{x\in M}\mu (B(x,R))\gt0$ . When $p\sigma_{p}^{\#}\neq \alpha_1 $ , where $\alpha_1$ is from (1·3), let $q={p\alpha_1 }/({\alpha_1 -p\sigma_{p}^{\#}})$ . For $r,s\in (0,\infty ]$ , $\theta\in (0,1]$ satisfying:

\begin{equation*}\frac{1}{r}=\frac{\theta }{q}+\frac{1-\theta }{s},\end{equation*}

there exists a constant $C\gt0$ such that for any $f\in B_{p,\infty }^{\sigma_{p}^{\#}}$ ,

(4·54) \begin{equation}\Vert f\Vert _{r}\leq C(\Vert f\Vert _{p}+[f]_{B_{p,\infty }^{\sigma_{p}^{\#}}})^{\theta }\Vert f\Vert _{s}^{1-\theta }. \end{equation}

By Theorem 1·5 and Theorem 1·7, we have the following corollary.

The following theorem can be regarded as a summary of our paper when the underlying space is a fractal.

Theorem 4·21. Let K be a connected homogeneous p.c.f. self-similar set with property (E), and let $K^{F}$ be a fractal glue-up. Then $K^{F}$ satisfies $(NE)_{\sigma_{p}^{\#}(K)}$ with $R_{0}=C_H$ and $B_{p,\infty }^{\sigma_{p}^{\#}(K^{F})}(K^{F})\;:\!=\;B_{p,\infty }^{\sigma _{p}^{\#}}(K^{F})$ contains non-constant functions. Also, BBM type characterisation (2·7) holds for $K^{F}$ with $R_{0}=C_H$ , that is, for $R_{0}=C_H$ , there exists a positive constant C such that for all $u\in B_{p,\infty }^{\sigma_{p}^{\#}}(K^{F})$ ,

(4·55) \begin{align}C^{-1}[u]_{{B}_{p,\infty }^{\sigma _{p}^{\#}}(K^{F})}^{p}&\leq\liminf_{\sigma \uparrow \sigma _{p}^{\#}(K^{F})}\left( \sigma_{p}^{\#}(K^{F})-\sigma \right) [u]_{{B}_{p,p}^{\sigma }(K^{F})}^{p}\leq\limsup_{\sigma \uparrow \sigma _{p}^{\#}(K^{F})}\left( \sigma_{p}^{\#}(K^{F})-\sigma \right) [u]_{{B}_{p,p}^{\sigma }(K^{F})}^{p}\nonumber \\&\leq C[u]_{{B}_{p,\infty }^{\sigma _{p}^{\#}}(K^{F})}^{p}. \end{align}

Furthermore, if K is a nested fractal and $K^{F}=K_{\infty }$ , where $K_{\infty }$ is defined in Example 4·3, then $(NE)_{\sigma_{p}^{\#}(K)}$ and $(KE)_{\sigma_{p}^{\#}(K)}$ hold for $K_{\infty }$ with $R_{0}=\infty $ , and the Gagliardo-Nirenberg inequality (4·54) holds.

Since a nested fractal K satisfies property (E) by Lemma 4·14, these inclusions hold for nested fractals. Next, we state how to upgrade $R_0=C_H$ to $R_0=\infty $ for $(NE)_{\sigma_{p}^{\#}}$ on $K_{\infty }$ .

For $n\in \mathbb{N}$ , define the scaling function $g_{n}:K^{F}\rightarrow \rho ^{-n}K^{F}$ by $g_{n}(x)=\rho ^{-n}x.$ Due to the global self-similarity of $K_{\infty }$ , we have

(4·56) \begin{equation}g_{n}(K_{\infty })=K_{\infty }. \end{equation}

For any $u\in B_{p,\infty }^{\sigma _{p}^{\#} ,F}$ , clearly $u\circ g_{n}\in B_{p,\infty }^{\sigma _{p}^{\#} ,F}$ . By definition, for all $r\gt0$ ,

(4·57) \begin{eqnarray}\Phi _{u\circ g_{n}}^{\sigma _{p}^{\#} ,F}(\rho ^{n}r) &=&(\rho ^{n}r)^{-p\sigma _{p}^{\#}-\alpha }\int_{K_{\infty }}\int_{B(x,\rho^{n}r)}|u(g_{n}(x))-u(g_{n}(y))|d\mu ^{F}(y)d\mu ^{F}(x) \nonumber \\&=&(\rho ^{n}r)^{-p\sigma _{p}^{\#} -\alpha }\int_{K_{\infty}}\int_{B(g_{n}(x),r)}|u(g_{n}(x))-u(y)|d(\mu ^{F}\circ g_{n}^{-1})(y)d\mu^{F}(x) \nonumber \\&=&(\rho ^{n}r)^{-p\sigma _{p}^{\#} -\alpha }\int_{g_{n}(K_{\infty})}\int_{B(x,r)}|u(x)-u(y)|d\mu ^{F}(\rho ^{n}y)d\mu ^{F}(\rho ^{n}x) \nonumber\\&\asymp &(\rho ^{n}r)^{-p\sigma _{p}^{\#} -\alpha }\rho ^{2n\alpha }\int_{K_{\infty}}\int_{B(x,r)}|u(x)-u(y)|d\mu ^{F}(y)d\mu ^{F}(x)\ \nonumber \\&=&\rho ^{-n(p\sigma _{p}^{\#} -\alpha )}\Phi _{u}^{\sigma _{p}^{\#} ,F}(r), \end{eqnarray}

where we use (4·56) and that $\mu ^{F}$ is $\alpha $ -regular in the fourth line. For any $u\in B_{p,\infty }^{\sigma _{p}^{\#} ,F}$ , there exists $n\lt\infty$ such that

\begin{equation*}\sup_{r\in (0,\infty )}\Phi _{u}^{\sigma _{p}^{\#} ,F}(r)\leq 2\sup_{r\in (0,C_{H}\rho^{-n})}\Phi _{u}^{\sigma _{p}^{\#} ,F}(r).\end{equation*}

Combined with (4·57),

(4·58) \begin{equation}\sup_{r\in (0,\infty )}\Phi _{u}^{\sigma _{p}^{\#} ,F}(r)\leq C\sup_{r^{\prime }\in(0,C_{H})}\Phi _{u\circ g_{n}}^{\sigma _{p}^{\#} ,F}(r^{\prime })\rho ^{n(p\sigma _{p}^{\#}-\alpha )}. \end{equation}

As $(NE)_{\sigma _{p}^{\#}}$ holds for $R_{0}=C_{H}$ ,

\begin{eqnarray*}\sup_{r^{\prime }\in (0,C_{H})}\Phi _{u\circ g_{n}}^{\sigma_{p}^{\#},F}(r^{\prime })\rho ^{n(p\sigma _{p}^{\#}-\alpha )} &\leq&C\liminf_{r^{\prime }\rightarrow 0}\Phi _{u\circ g_{n}}^{\sigma_{p}^{\#},F}(r^{\prime })\rho ^{n(p\sigma _{p}^{\#}-\alpha )} \\&=&C\liminf_{r^{\prime }\rightarrow 0}\Phi _{u}^{\sigma _{p}^{\#},F}(\rho^{-n}r^{\prime })\text{ (by (4$\cdot$57))} \\&=&C\liminf_{r\rightarrow 0}\Phi _{u}^{\sigma _{p}^{\#},F}(r),\text{ }\end{eqnarray*}

where the constant C does not depend on u and n. Thus $(NE)_{\sigma _{p}^{\#}}$ also holds with $R_{0}=\infty $ by (4·58). Since $\mu ^{F}$ is $\alpha $ -regular, we know by Lemmas 4·14 and 4·19 that (4·54) holds true. Finally, for a nested fractal K, it is known by [ Reference Kumagai38 ] (see also [ Reference Fitzsimmonsm, Hambly and Kumagai21 , theorem 1·1]) that $K_{\infty }$ admits a heat kernel satisfying (1·9), thus $(KE)_{\sigma _{p}^{\#}}$ is verified by Theorem 1·5. The proof is complete.

When $p=2$ , we use heat kernel estimates to verify weak-monotonicity properties in the following corollary, which is analytical (compared with the algebraic resistance-estimate arguments for $(VE)_{\sigma}$ ).

We use an example based on [ Reference Hambly and Kumagai30 , section 4] to show that Corollaries 4·20 and 4·22 are also meaningful for some non-nested fractal blow-ups. That is, there exists a connected homogeneous p.c.f. self-similar set which is not a nested fractal, such that, its fractal blow-up admits a heat kernel satisfying (1·9).

Example 4·23. Let S be a unit equilateral triangle in $\mathbb{R}^2$ . Its vertices are located at $q_0=(0, 0)$ , $q_1=(1, 0)$ , and $q_2=(1/2, \sqrt{3}/2)$ . Divide S into a mesh of sub-triangles of side length $1/12$ , and pick 34 sub-triangles as shown in Figure 1. Let $\{\phi_i\}_{i=1}^{34}$ be the IFS on $\mathbb{R}^2$ that maps S to these 34 sub-triangles with attractor K. Then $K=\bigcup_{i=1}^{34}\phi_i(K)$ is a connected homogeneous p.c.f. self-similar set, which is also a uniformly finitely ramified fractal (u.f.r. fractal for short, see definition in [ Reference Hambly and Kumagai30 , section 2]). However, K is not a nested fractal. Its Hausdorff dimension is $ \alpha = \dim_H(K) = {\log 34}/{\log 12} $ , and the $ \alpha $ -dimensional Hausdorff measure exists since the open set condition is satisfied. Define $V_0=\{q_{k}\}_{k=0}^2$ and $V_{n+1}=\bigcup_{i=1}^{34}\phi_i(V_n)$ as in (4·2). Let $K_\infty$ be the fractal blow-up of K (defined in Example 4·3 as $ \bigcup_{l=1}^\infty K_l $ , where $ K_l = 12^l K $ ). Let $d_\infty$ be the shortest path length between any two points in $ K_\infty $ . By the global self-similarity of blow-ups, $ d_\infty $ is bi-Lipschitz equivalent to the Euclidean metric $ d_{\mathbb{R}^2}|_{K_\infty} $ .

Fig. 1.

A uniformly finitely ramified fractal.

Define $V=\bigcup_{i=0}^\infty 12^n V_n$ and $E \;:\!=\; \{(x, y) \in V \times V : d(x, y) = 1\}$ . Consider the graph (V, E) with weights $\mu$ satisfying the $p_0$ -condition in [ Reference Barlow and Bass8 , formula (1·5)]. By substituting each edge in E with an isometric copy of the line segment [0, 1] (referred to as a cable), and connecting them in a natural way at the vertices, we obtain an unbounded connected closed set $X_0 \subset \mathbb{R}^2$ , known as the cable system associated with (V, E). The metric $d_0$ is defined on $X_0\times X_0$ by employing the Euclidean distance along each cable, and this is extended to a metric over $X_0\times X_0$ that is consistent with the graph distance on V. We define $\Phi_0(r)\;:\!=\;r \vee r^{\alpha}$ and $\Psi_0(r)\;:\!=\;r^2 \vee r^{d_w}$ . Denote by $m_0$ the Hausdorff measure on $(X_0,d_0)$ , then we have $m_0(B(x,r))\asymp\Phi_0(r)$ by [ Reference Barlow and Bass8 , lemma 2·1], where $B(x,r)\;:\!=\;\{z\in X_0\;:\; d_0(x,z)\lt r\}$ . Define a regular Dirichlet form $(\mathcal{E}_0,\mathcal{F}_0)$ corresponding to the cable process on $(X_0,d_0,m_0)$ as in [ Reference Barlow and Bass8 , section 2], which is conservative and strongly local. By [ Reference Hambly and Kumagai30 , corollary 4·14], there exist constants $c_1$ , $c_2$ , $c_3$ and $c_4\gt0$ such that for all $x,y\in V$ , $k\geq d_0(x,y)$

(4·59) \begin{equation} \begin{aligned} p_k(x, y) & \leq c_{1} k^{-\alpha/d_w} \exp\left(-c_{2} \left( \frac{d_0(x, y)^{d_w}}{k} \right)^{1/(d_w-1)}\right), \\ p_k(x, y) + p_{k+1}(x, y) & \geq c_{3} k^{-\alpha/d_w} \exp\left(-c_{4} \left( \frac{d_0(x, y)^{d_w}}{k} \right)^{1/(d_w-1)}\right).\end{aligned}\end{equation}

Therefore, (4·59) implies [ Reference Barlow and Bass8 , formulae (1·12) and (1·13)]. By [ Reference Barlow and Bass8 , theorem 1·2, Corollary 2·5 and Lemma 2·6], the elliptic Harnack inequality $(\mathrm{EHI})$ and the mean exit time estimate $(\mathrm{E}_{\Psi_0})$ (see [ Reference Grigor’yan and Telcs26 , definition 3·10]) hold for the cable process. Thus the metric measure Dirichlet space $ (X_0, d_0, m_0, \mathcal{E}_0,\mathcal{F}_0)$ admits a heat kernel $\{p_t\}_{t\gt0}$ satisfying HKE( $\Psi_0$ ) by [ Reference Grigor’yan and Telcs26 , theorem 5·15], where the condition $HKE(\Psi)$ (for the metric measure space (X,d,m)) is defined in [ Reference Chen18 , definition 1·1] as follows: there exist $c_{1},c_{2},c_{3}, \delta\gt0$ and a heat kernel $\{p_t\}_{t\gt0}$ such that for any $t\gt0$ ,

(4·60) \begin{equation} {} {}p_{t}(x,y) \leq \frac{c_{1}}{V(x,\Psi^{-1}(t))} \exp\left(-c_{2}t\Phi\left(c_{3}\frac{d(x, y)}{t}\right)\right) {} {}\qquad \mbox{for}\,{{m}-{\rm a.e.}\,x,y \in X} {}\end{equation}

(4·61) \begin{equation}\text{and }\ p_{t}(x,y) \geq \frac{c_{1}^{-1}}{V(x,\Psi^{-1}(t))} {} {}\qquad \mbox{for} \,{{m}-{\rm a.e.}\ x,y\in X with \, d(x,y) \leq \delta \Psi^{-1}(t)}, {}\end{equation}

where $V(x,r)\;:\!=\;m(\{z\in X\;:\; d(z,x)\lt r\})$ and

\begin{equation*} {} {}\Phi(s)\;:\!=\;\Phi_{\Psi}(s)\;:\!=\;\sup_{r\gt0}\left({\frac{s}{r}-\frac{1}{\Psi(r)}}\right). {}\end{equation*}

Next, we construct a sequence of metric measure spaces $ (X_n, d_n, m_n, \mathcal{E}_n,\mathcal{F}_n)$ ( $n\in \mathbb{N}$ ), by

\begin{align*}&X_n = X_0,\; d_n = 12^{-n} d_0\; \text{and}\; m_n\;:\!=\;12^{-\alpha n} m_0,\\&\mathcal{E}_n=12^{(d_w-\alpha)n} \mathcal{E}_0,\; u,v\in \mathcal{F}_n\;:\!=\;\mathcal{F}_0.\end{align*}

To apply the pointed Gromov-Hausdorff convergence results in [ Reference Chen18 ], we need to verify that the spaces $ (X_n, d_n, m_n, q_0) $ satisfy Conditions (A1)-(A6) in [ Reference Chen18 , theorem 1·2]. For the scaled sequence $ (X_n, d_n, m_n) $ , the heat kernel $ p^{(n)}_t( x, y) $ is related to the original heat kernel $\{p_t\}_{t\gt0}$ by

\[ p^{(n)}_t(x, y) = 12^{\alpha n} \cdot p_{12^{d_w n} t}( x, y). \]

This scaling preserves the sub-Gaussian form, so $ (X_n, d_n, m_n,\mathcal{E}_n,\mathcal{F}_n)$ satisfies $ HKE(\Psi_n) $ with constants independent of n and $ \Psi_n(r) \;:\!=\; 12^{-d_w n} \Psi_0(12^n r)$ . Therefore, condition (A1) in [ Reference Chen18 , theorem 1·2] holds. It is easy to verify that conditions (A2-A5) in [ Reference Chen18 , theorem 1·2] also hold.

It remains to verify condition (A6) in [ Reference Chen18 , theorem 1·2]. Define

\[\widetilde{X}_{n}\;:\!=\;12^{-n}X_{0}=\{12^{-n}x\;:\;x\in X_{0}\},\ n\geq 1\]

and

\begin{equation*}\widetilde{d}_{n}(x,y)\;:\!=\;12^{-n}d_{0}(12^{n}x,12^{n}y),\ \forall x,y\in \widetilde{X}_{n}.\end{equation*}

This scaling satisfies the consistency relation with $ d_\infty $ :

\begin{equation*} d_\infty(x, y) = \widetilde{d}_{n}(x, y) \quad \forall x, y \in \widetilde{X}_{n}.\end{equation*}

Note that $\bigcup_{n=1}^\infty X_n=K_\infty$ and the map $x\mapsto 12^{-n}x$ is a bijective isometry from $(X_n, d_n) $ to $(\widetilde{X}_{n},\widetilde{d}_{n})$ . Clearly, $(\widetilde{X}_{n},\widetilde{d}_{n})$ is a proper length space, since the intrinsic metric $ \widetilde{d}_{n}$ coincides with the shortest path length between points, so $ (\widetilde{X}_{n},\widetilde{d}_{n}, q_0) $ is an eventually proper sequence of pointed length spaces. Similar to [ Reference Chen18 , example 8·1(6) on Page 41], we know that $(\widetilde{X}_{n},\widetilde{d}_{n},q_0)$ pointed Gromov-Hausdorff converge to $(K_{\infty},d_{\infty},q_0)$ .

Finally, applying [ Reference Chen18 , theorem 1·2 and remark 1·3 (i)], we conclude that there is a measure $m_{\infty}$ on $(K_{\infty},d_{\infty})$ satisfies $m_{\infty}(B_{\infty}(x,r))\asymp r^{\alpha}$ , and there is a regular strongly local Dirichlet form on $L^{2}(K_{\infty},d_{\infty},m_{\infty})$ that satisfies (1·9).